Dynamics, Volume 3 (Handbook of Surface Science)

Handbook of Surface Science Volume 3 Handbook of Surface Science Series editors S. HOLLOWAY Surface Science Research ...

Author: Eckart Hasselbrink

276 downloads 1916 Views 14MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Handbook of Surface Science Volume 3

Handbook of Surface Science Series editors S. HOLLOWAY Surface Science Research Centre Liverpool, UK

N.V. RICHARDSON Director, Surface Science Research Centre Liverpool, UK

ELSEVIER Amsterdam • Lausanne • New York • Oxford • Shannon • Singapore • Tokyo

Volume 3

Dynamics

Volume editors E. HASSELBRINK Fachbereich Chemie and Centre for NanoIntegration (CeNIDE) Universität Duisburg-Essen D-45117 Essen, Germany

and B.I. LUNDQVIST Center for Atomic-scale Materials Design Department of Physics Technical University of Denmark DK-2800 Lyngby, Denmark Department of Applied Physics Chalmers University of Technology SE-412 96 Göteborg, Sweden

2008

ELSEVIER Amsterdam • Lausanne • New York • Oxford • Shannon • Singapore • Tokyo

North-Holland is an imprint of Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK First edition 2008 Copyright © 2008 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-444-52056-2 ISSN: 1573-4331 For information on all North-Holland publications visit our website at elsevierdirect.com Printed and bound in U.K. 08 09 10 11 12

10 9 8 7 6 5 4 3 2 1

General Preface How many times has it been said that surface science has come of age? Rather than being a fledgling area of study, it is now patently clear that the investigation of solid surfaces and related interfacial problems is a unique field with profound implications for basic (dare one say academic!) scientific study and the understanding of materials. Surface science provides major support to many technologically ambitious industries. It is widely recognised that it underpins the fabrication of electronic devices but this also extends to any industry working in nanotechnology. Surface science makes significant contributions to product development and problem solving in many materials-based industries where surface finish, cleanliness, adhesion, wear or friction are important. An understanding of surface processes is vital in chemical industries because of the importance of heterogeneous catalysis and is likely to make major contributions to the growing optoelectronics and molecules sensing industries. In social terms, an improved understanding of surfaces will facilitate the development of better catalysts and sensors for improvement of our environment. Most recently, surface science has begun to make real contributions to the understanding of biological surfaces. In this series, we have brought together some of the key players that have made seminal contributions to the study of solid surfaces and their interactions with foreign species. Because of the broad scope of surface science, even when restricted to the solid surfaces which we hope to cover in this series, we have been coerced into ‘packaging’ the subject in what is, it must be said, a rather arbitrary way. No doubt different editors would have chosen different themes around which to base individual volumes. The first volume addressed the static geometry of surfaces while the second focused on their electronic structure. Now, this third volume is dedicated to the study of the dynamics of systems at surfaces and interfaces and we have been lucky to attract the foremost practitioners in the field to contribute both editorial expertise and articles. The study of dynamical processes at surfaces has its own rich history going right back in the area of chemical dynamics to the inspirational work of Eyring and Polanyi published at the start of the previous century. Since the advent of UHV and all of its trappings, studying the actual time dependence of processes became less interesting than following the kinetics of processes and even the vast majority of scattering experiments were analysed in the stationary state language of quantum mechanics. In the late 70s, there was a change in attitude to dynamics at surface which was inspired by a number of research summer schools organised by one of the co-editors of this volume. These dynamics summer camps were for many of us an opportunity to express our ignorance and start to ask the correct questions v

vi

General Preface

such that, towards the end of the 80s, the first fruits of a new approach began to appear in many areas of the subject. By the middle of the 90s, the biannual Dynamics Gordon Research Conference had firmly given a seal of approval to this endeavour and ushered in new swathes of activity in such diverse areas as chemical reactions, photoreactions, material growth, inelastic tunnelling, etc.: we are very fortunate to have these among the many topics covered by this volume. This year of publication, 2008, sees the current Nobel Prize for Chemistry in the firm grasp of Professor Gerhard Ertl who according to the citation “. . . has succeeded in providing a detailed description of how chemical reactions take place on surfaces . . .”. It really is most fitting that one of the leading proponents of studying the dynamics of surface processes should achieve this ultimate accolade by the community. Stephen Holloway and Neville Richardson February 2008

Preface to Volume 3 The quest to develop a microscopic understanding of chemical processes and change at surfaces motivates the study of dynamics. Surfaces play important roles for everyday life, because of the chemical processes which occur at surface and since most materials processing modifies the surface of some solid. In heterogeneous catalysis, surfaces play the most prominent role in a chemical context. However, corrosion and erosion are also chemical processes in which surfaces play a decisive role. Materials processing involves removal of material from the surface, chemical and physical modification of the surface near region, and deposition of material on surfaces. Thus, a predictive description of chemical and physical processes and changes at surfaces is highly desirable. This requires a microscopic understanding of the underlying processes. The holy grail is an account for complex and interwoven chemical scenaria on an atom-by-atom basis from first principles. However, well designed fundamental studies provide the basis for developing the conceptual guidelines which allow us to interpret the phenoma occurring in complex “real” chemical environments, and – maybe more importantly – allow us to dispel misconceptions. The metal–gas interface is the common prototype system in surface science, wellspecified thanks to accurate preparation, characterization, and ultra-high-vacuum (UHV) conditions. This together with an extremely well-characterized “gas” provide the platform. A mechanistic understanding of chemical processes is developed by tracing the temporal evolution leading from some characterisable state to another. For this purpose the individual steps connecting some initial state with the final product state need to be identified. This results in detailed and well-resolved experimental data, which challenge the development of advanced theory, where a detailed comparison on the quantum level between experimental and theoretical results is possible for a steadily growing number of surface processes. Thanks to a multitude of accurate experiments and advanced microscopic theory, surfaces now provide excellent platforms to sharpen our theoretical tools. These tools can often be applied generally, to, e.g., catalysis, nanotechnology, biology, photonics, and materials. At surfaces, concepts like resonant tunneling, electron–hole separation, and charge transfer can be made precise at the quantum level, and then they can be applied in other areas. Since the early 1980s, the field of surface science has changed dramatically, after the discovery of STM in 1983 by Binnig and Rohrer. It then got possible to image almost routinely surfaces and surface bound species with atomic-scale resolution. In 1991, Eigler and Schweizer showed that matter could be manipulated on an atom by atom basis. Semiconductor-device technology, with its dedication to strive for the smaller, faster, vii

viii

Preface to Volume 3

cheaper, and better, is marching closer and closer to surfaces. Surface science has always been about nanoscale science. The previous two volumes of this handbook series are about the physical and electronic structures of surfaces, primarily static phenomena. Dynamics, the topic of the present volume, builds heavily on the insights obtained in these areas. However, in many instances, dynamics studies also enlarged our knowledge about the physical and electronic structure of surfaces. Studies on the dynamics at surface have greatly profited from a tight interaction of experimental and theoretical work. The dispute between predictions made based on theoretical calculation and the hard facts derived from experiments has been the most prolific driving force for refining theoretical concepts and advancing sophisticated experiments. Among other techniques, molecular beams, laser spectroscopy and scanning tunneling microscopy have proved to be the powerful tools for unraveling individual steps of processes at surfaces. Hence, this volume starts with chapters by Aart Kleyn, Dick Manson, Mats Persson, Greg Sitz and George Darling summarizing research from the experimental and theoretical perspective which explores the interaction and the energy flow in molecule-surface collisions which is fundamental for energy exchange and dissociative adsorption. Next, Vladimir Zhdanov discusses how lateral interactions between the adsorbates on a surface influences the reaction kinetics. Thomas Bligaard, Jens Nørskov, and Bengt Lundqvist build upon the previous chapters and apply microscopic concepts to what rules heterogeneous catalysis. Ronald Imbihl delves into the aspect of non-linear dynamics in surface reactions. Bengt Lundqvist’s, Anders Hellman’s, and Igor Zorić’s chapter deals with the aspect of charge transfer and non-adiabaticity in surface chemical reactions. Richard Berndt and Jörg Kröger examine recent research regarding the dynamics of electronic states at surfaces to which laser based, STM and theoretical efforts have contributed. Nicolás Lorente, Eckart Hasselbrink, Andrew Mayne and Gérald Dujardin report from various perspectives on the dynamics of processes in which the surface is manipulated by photons or alternatively the electrons that tunnel through an STM. These processes exploit various aspects of molecule-surface interaction and their studies shed new light on these. Harald Brune and Kurt Kolasinski present different aspects of growth on surfaces. Brune examines how different regimes of the kinetics can be utilized to control the physical structure in metal on metal growth to ultimately obtain nanostructures. Kolasinski more generally looks at semiconductor materials and also epitaxy. And finally, Herbert Urbassek gives an introduction to the interaction of particles at non-thermal energies with surfaces. Moreover, we are very grateful to have Bill Gadzuk contributing a general introduction to the field. And last, but certainly not least, the volume contains an interview with 2007 Nobel laureate in Chemistry, Gerhard Ertl, in which he speaks about the role of dynamics for the Surface Science as a whole and the future prospects of the field. The chapters in the volume have been prepared with a particular potential reader in mind: the beginning graduate student. Hence, they do not attempt to serve as comprehensive reviews, but rather focus on developing the lines of thought which connect different research and the conclusions which can be drawn from the results obtained. We hope that these readers will find that the collection of chapters in this Handbook provides a sound introduction into half a decade of research into dynamics at surfaces. We would like to express our gratitude to our colleagues and contributors, for their help getting this volume together and published. Special thanks are due to the editorial staff, especially Anita Koch,

Preface to Volume 3

ix

at Elsevier Science for their expert assistance, and their patience. We are indebt to Sabrina Tauchmann for her help in getting the artwork of the book prepared. Eckart Hasselbrink and Bengt Lundqvist

This page intentionally left blank

Contents of Volume 3 General Preface . . . . . . . . . . . . . . . . . Preface to Volume 3 . . . . . . . . . . . . . . . Contents of Volume 3 . . . . . . . . . . . . . . . Contributors to Volume 3 . . . . . . . . . . . . Interviewing Nobel Prize Winner Gerhard Ertl

. . . . .

v vii xi xiii xv

1. J.W. Gadzuk Fundamental Atomic-Scale Issues/Processes Pertinent to Dynamics at Surfaces

1

2. A.W. Kleyn Basic Mechanisms in Atom–Surface Interactions . . . . . . . . . . . . . . . . .

29

3. J.R. Manson Energy Transfer to Phonons in Atom and Molecule Collisions with Surfaces . .

53

4. M. Persson and S. Andersson Physisorption Dynamics at Metal Surfaces . . . . . . . . . . . . . . . . . . . .

95

5. G.R. Darling Intra-molecular Energy Flow in Gas–Surface Collisions . . . . . . . . . . . . .

141

6. G.O. Sitz Inelastic Scattering of Heavy Molecules from Surfaces . . . . . . . . . . . . . .

197

7. V.P. Zhdanov Reaction Dynamics and Kinetics: TST, Non-equilibrium and Non-adiabatic Effects, Lateral Interactions, etc. . . . . . . . . . . . . . . . . . . . . . . . . . .

231

8. T. Bligaard, J.K. Nørskov and B.I. Lundqvist Understanding Heterogeneous Catalysis from the Fundamentals . . . . . . . .

269

9. R. Imbihl Non-linear Dynamics in Catalytic Reactions . . . . . . . . . . . . . . . . . . .

341

10. B.I. Lundqvist, A. Hellman and I. Zorić Electron Transfer and Nonadiabaticity . . . . . . . . . . . . . . . . . . . . . .

429

11. R. Berndt and J. Kröger Dynamics of Electronic States at Metal Surfaces . . . . . . . . . . . . . . . . .

525

xi

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

xii

Contents of Volume 3

12. N. Lorente Theory of Tunneling Currents and Induced Dynamics at Surfaces . . . . . . . .

575

13. E. Hasselbrink Photon Driven Chemistry at Surfaces . . . . . . . . . . . . . . . . . . . . . . .

621

14. A.J. Mayne and G. Dujardin STM Manipulation and Dynamics . . . . . . . . . . . . . . . . . . . . . . . . .

681

15. H. Brune Creating Metal Nanostructures at Metal Surfaces Using Growth Kinetics . . .

761

16. K.W. Kolasinski Growth and Etching of Semiconductors . . . . . . . . . . . . . . . . . . . . . .

787

17. H.M. Urbassek Sputtering and Laser Ablation . . . . . . . . . . . . . . . . . . . . . . . . . . .

871

Author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

915 991

Contributors to Volume 3

Stig Andersson, Department of Physics, University of Göteborg SE-412 96, Göteborg, Sweden R. Berndt, Institut für Experimentelle und Angewandte Physik, Christian-AlbrechtsUniversität zu Kiel, Olshausenstrasse 40, D-24098 Kiel, Germany T. Bligaard, Center for Atomic-scale Materials Design, Department of Physics NanoDTU, Technical University of Denmark, DK-2800 Lyngby, Denmark Harald Brune, Institut de Physique des Nanostructures (IPN), Ecole Polytechnique Fédérale de Lausanne (EPFL), Station 3, CH-1015 Lausanne, Switzerland G.R. Darling, Surface Science Research Centre, Department of Chemistry, The University of Liverpool, Liverpool L69 3BX, UK Gérald Dujardin, Laboratoire de Photophysique Moléculaire, CNRS, Bat. 210, Univ. Paris Sud, 91405 Orsay, France J.W. Gadzuk, Electron and Optical Physics Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA Eckart Hasselbrink, Fachbereich Chemie and Centre for NanoIntegration (CeNIDE), Universität Duisburg-Essen, D-45117 Essen, Germany Anders Hellman, Department of Applied Physics, Chalmers University of Technology, SE412 96, Göteborg, Sweden and Competence Center for Catalysis, Chalmers University of Technology, SE-412 96, Göteborg, Sweden R. Imbihl, Institut für Physikalische Chemie und Elektrochemie, Leibniz-Universität Hannover, Callinstrasse 3-3a, D-30167 Hannover, Germany Aart W. Kleyn, FOM Institute for Plasma Physics Rijnhuizen, Euratom-FOM Association, P.O. Box 1207, 3430 BE Nieuwegein, The Netherlands and Leiden Institute of Chemistry, Gorlaeus Laboratories, Leiden University, P.O. Box 9502, 2300 RA Leiden, The Netherlands Kurt W. Kolasinski, Department of Chemistry, West Chester University, West Chester, PA 19383, USA J. Kröger, Institut für Experimentelle und Angewandte Physik Christian-AlbrechtsUniversität zu Kiel Olshausenstrasse 40, D-24098 Kiel, Germany xiii

xiv

Contributors to Volume 3

N. Lorente, Centro de Investigaciones en Nanociencia y Nanotecnología (CSIC - ICN), Campus de la Universitat Autònoma de Barcelona, 08193 Bellaterra, Spain Bengt I. Lundqvist, Center for Atomic-scale Materials Design, Department of Physics, Technical University of Denmark, DK-2800 Kgs. Lyngby, Denmark and Center for Atomic-scale Materials Design, Department of Applied Physics, Chalmers University of Technology, SE-412 96 Göteborg, Sweden J.R. Manson, Department of Physics and Astronomy, Clemson University, Clemson, SC 29634, USA Andrew J. Mayne, Laboratoire de Photophysique Moléculaire, CNRS, Bat. 210, Univ. Paris Sud, 91405 Orsay, France J.K. Nørskov, Center for Atomic-scale Materials Design, Department of Physics NanoDTU, Technical University of Denmark, DK-2800 Lyngby, Denmark Mats Persson, Surface Science Research Centre and Department of Chemistry, The University of Liverpool, Liverpool L69 3BX, United Kingdom Greg O. Sitz, Department of Physics, University of Texas, Austin, TX 78712, USA Herbert M. Urbassek, Fachbereich Physik, Universität Kaiserslautern, Erwin-SchrödingerStraße, D-67663 Kaiserslautern, Germany Vladimir P. Zhdanov, Department of Applied Physics, Chalmers University of Technology, S-412 96 Göteborg, Sweden and Boreskov Institute of Catalysis, Russian Academy of Sciences, Novosibirsk 630090, Russia Igor Zorić, Department of Applied Physics, Chalmers University of Technology, SE-412 96, Göteborg, Sweden

Interviewing Nobel Prize Winner Gerhard Ertl On October 10, 2007 it was announced that Germany’s Gerhard Ertl wins the Nobel Chemistry Prize of 2007. On Ertl’s 71st birthday, he got to know that it was awarded “for his studies of chemical processes on solid surfaces” (http://nobelprize.org/nobel_prizes/chemistry/ laureates/2007/). His work is related to large parts of Volume 3, and his contributions have been very important for the development of Dynamics at Surfaces. Further, most of the contributors to the volume have connections to him. Several of them have gotten inspiration from him and learned from his insights. Coauthors Harald Brune, Ronald Imbihl, Kurt Kolasinski and Eckart Hasselbrink are either his pupils or have spent longer periods of time with him, resulting in numerous joint publications. The volume editors sincerely congratulate professor Ertl for having been awarded the 2007 Nobel Prize in Chemistry. It is an honor that time is spared for this interview for Volume 3 of the Handbook of Surface Science. We see this as a great opportunity to promote surfaces science in general and dynamics at surfaces in particular. Thus the introduction to this volume relates to Ertl and his Nobel Prize. The prize announcement came at the real finish of the writing of the book. In order not to introduce unnecessary delay, a format of edited interviews was chosen. All over the world, much is written about the Nobel Prizes, but perhaps the medial activity is particularly intense in Sweden. Therefore this chapter is written as (i) an account of two broadcasts in the Swedish Radio, covering some general issues, and (ii) a brief interview with a couple of focused scientific issues, for which Ertl’s broad perspective and insight is particularly interesting for other participants in the field of dynamics at surfaces. In News from the Science Radio on Friday November 9 at 08:37, the following was broadcast: ‘And now we go to Berlin, where our reporter has met this year’s Nobel Prize winner in chemistry, Gerhard Ertl! He is a gray-headed 71-years old man, who nowadays walks with a stick through the corridors of the Fritz-Haber Institute in Berlin. Gerhard Ertl, the single chemistry-prize winner of this year, has devoted his life to the chemistry on solid surfaces. And this means quite a lot! Everything from car catalysts, to ozone break down, to production of artificial fertilizer concerns chemistry on solid surfaces. xv

xvi

Interviewing Nobel Prize Winner Gerhard Ertl

Professor Ertl far from emphasizing his own excellence, but gladly surface chemistry as such, what it has done, and what it can do. A hundred years ago surface chemists succeeded to trap the nitrogen in our air and to make fertilizer from it. This was a determining factor for the development of Mankind. Without artificial fertilizers, our Earth had not at all been able to feed so many lives. And now we face new questions of our destiny: The climate and the energy supply. We have to make it without oil. And whether the solution is called solar cells, hydrogen used in fuel cells, or fusion reactors that need extremely endurable walls, so is surface chemistry important also now. “Surface Science should play a major role in the answers to and solutions of the problems of the future”, Ertl concludes.’ Another feature broadcast from the Swedish Radio about the Nobel Prize winner was slightly longer, maybe based on the same interview, (which we shortened somewhat). Some extracts are given below. To distinguish commentator (I) and Ertl (E), it is interesting to note that, while the former uses the term Surface Chemistry (I), Ertl says Surface Science (E). I: In one of the pompous old buildings Gerhard Ertl resides, this year’s Nobel Laureate in Chemistry. Since he retired three years ago, he has moved out of his director’s office into Fritz Haber’s old office. He has worked about half days. At least before the message about the Nobel Prize arrived. E: Before I was a retired professor sitting here at my desk, writing a book and thinking about the development of my topic and my subject. And then the call from Sweden came in. Everything changed. I was overflooded with telephone calls, with emails, with nice mails with requests and invitations. I: Thinking of what he has done, devoted his life since the sixties to surface chemistry, it is almost remarkable that he can give such a simple and clear picture of what surface chemistry basically is. E: If you look at a piece of metal or just a solid, then the atoms that are inside are surrounded by nearest neighbors, that means chemical bonds that are saturated. However, if you then cut the solid into two halves, then you get two surfaces, where the outermost atoms are not surrounded by a maximal number of atomic neighbors. That means that they have free valencies that can take part in chemical reactions with atoms or molecules in the surrounding gas, for instance, the air. This is surface chemistry. It is restricted to the topmost atomic layer! I: In this absolute boundary between, for instance, metal and gas, one is also at the boundary between physics and chemistry. This was what attracted the young GE once. From the beginning he is a physicist, but he has all the time carried along the chemistry interest, which caught him already in his boy’s years, when he made experiments at home. E: But my mother was not very happy about it, because it was sometimes smelling strangely and sometimes it was exploding. And she was frightened.


So I had to refrain myself and to stop that. I went over to physics and started to build radio sets. I: Then there were physics studies at the university, but the attraction from chemistry was left. And so it became physical chemistry for Ertl, and surface chemistry... Even if he has a lot to say about what good things surface chemistry has done and can do to mankind, Ertl’s own engagement is entirely devoted to pure curiosity. E: I was always going back to the university, so we never had in mind any applications or happiness of mankind or so. But I think that answering questions is the main task of science, and we persisted for the whole time. And if something comes out of it, which is of use to other ones, that’s fine, but it was not the major motivation. I: Ertl emphasizes the role of Fritz Haber, Ertl’s predecessor at FHI, about 100 years ago. Haber was certainly scientific leader of the German program with chemical weapons, but this is not what Ertl wants to emphasize but rather that Haber, with the help of surface chemistry, succeeded to produce artificial fertilizers for plants. Today we are used to artificial fertilizers but then it was a challenge to trap nitrogen from the air and transform it into nitrogen in a form that the plants can use. This reaction was necessary to support a growing world population. E: They found this reaction in the laboratory in 1908 for the first time, and it was immediately taken over by industry, and they developed the first industrial plants, which stood in operation 1913. And in the meantime, the production of ammonia, which is the basis of fertilizer industry, increased continuously, parallel to the increase in world population. Without this Haber–Bosch reaction the world would have looked differently today. I: This was 50 years before Ertl started his research career. He entered the stage in the sixties, when the situation was quite different from that of Fritz Haber. At this time new techniques became available, not created to answer the questions of the chemists but fully useful it appeared. Ertl started to use them immediately and looked for answers to new questions. In 1974, at about the same time as ABBA won the European Song Contest, Ertl was at a symposium that just concerned the chemistry behind fertilization. A leading scientist talked about the past 50 years of research in the area. E: He came to the conclusion that we still don’t know how the nitrogen molecule is activated to get the nitrogen into ammonia. After 50 years! And I went back and told my own students that this is a problem that we should be able to solve now with our techniques. That is how I got interested in that. And a year later or so we knew that the nitrogen first has to be split apart into two nitrogen atoms, which are bonded to the iron surface – the catalyst is made by iron – and which react, in the next step, with hydrogen atoms to make ammonia. The final result is ammonia that gets off the surface into the gas phase. And when the ammonia plant producers now know how the reaction proceeds in detail, they get better control over their fabrication process. It is very important for them to know what is really going on, they say.

xvii

xviii


I: Gerhard Ertl considers himself lucky, to have come at the right time to these new techniques and to always have had good coworkers to work with. When I go out into the corridors of the FHI to look for these collaborators, it appears that many in the group have left! I do not find much of equipment either. It might be more worrying that his laboratory is getting depleted of people? Isn’t it disappointing that old colleagues leave? That is not what Ertl thinks. E: Many, many of the young scientists working with me have in the meantime become professors at other universities and started their own groups. I am very proud of that, because I think this is the best thing that you can do as a scientist, not make any discovery but educate other people. I: What research is Ertl most proud of? Then he talks about something that I have seen pictures of in the corridor: Peculiar patterns, circles, and spirals on the walls! Things that develop under certain circumstances in chemical reactions at surfaces, he explains. When the reaction does not proceed equally fast everywhere, but spreads like waves in the sea. The pictures that I see are from reactions of carbon monoxide oxidation on a platinum surface. This is, in fact, what happens in a car catalyst. Do these patterns appear in my car? E: No, he, he. It might happen in your car, but the conditions, at which your carexhaust catalyst is working are such, that you always have an optimum transformation of the carbon monoxide to carbon dioxide. Rather it happens in your heart, because the knowledge about these spiral and circle patterns is something general, because the underlying physical laws are the same everywhere. So if you have propagating concentration waves, for example, this cannot only be a chemical reaction. It can also be a biological system. Also currents in your heart propagating across the heart muscle. Underlying are the same physical phenomena. That is why the theory that is available here, and which has also been available at our department, can be applied to quite different fields. I: Thus there is not only Chemistry and Physics that occupy Ertl. Medicine and biology lie stumblingly close. It appears to apply to more than these patterns. When I ask “What do you think of surface chemistry during the next 10 years?”, he does not only talk about Chemistry, then he talks about biology as something self-evident. However, in a more direct way. Surface chemistry is not only about a solid surface against a gas but also about a solid surface against a liquid, like in our body, for example. E: There are many open questions also concerning the interaction of molecules with cell membranes and biological systems. I: So the next Nobel Prize in surface science might be in biology? E: I don’t know. It is definitely not impossible. I: In a 50 years’ perspective, what does surface chemistry do then? It solves the really big problems of our future, Ertl seems to mean. When the World has to solve the climate challenge, and replace fossil fuels, when it can concern everything from efficient and cheap solar cells to hydrogen storage and fuel cells, or endurable reactor walls in future fusion reactors that do not give longliving radioactive waste.


E: In each of these challenges, there are surfaces and surface-related problems. Surface Science will play a major role in the solution of these problems. So much for the radio interviews. Now to our questions: Q: We, the volume editors, sincerely congratulate you for having been awarded the 2007 Nobel Prize in Chemistry. We feel honored that you spare the time for this interview which will accompany the Volume 3 of the Handbook of Surface Science. The Swedish Royal Academy of Sciences summarizes the laudatio as follows: “for his studies of chemical processes on solid surfaces”. Do you think the phrase summarizes properly your life times achievements as scientist? A: More or less. But we worked also to some extend to more physical processes of surface science, such as deexcitation of metastable atoms, exoelectron emission, or electron dynamics by ultrafast laser techniques. Q: Has during the latest 50 years the research on dynamics at surface developed in a way that you like? A: Progress in this field was always promoted by the development of novel experimental tools – from state resolved molecular beam measurements to femtosecond laser techniques, to quote just two examples. Q: Is the dynamics at surface completed as a research field? A: Certainly not. I am still waiting for experiments on the time-resolved spectroscopy of the transition state of surface reaction. Q: A recent review expresses some disappointment over that “the experimental study of gas–surface dynamics has unfortunately dropped off precipitously in the last several years”. Do you agree with the assessment? Are you also worried? A: I think this is a consequence of the fact that the whole field has to some extent matured in the meantime. Novel experiments became more elaborate and hence fewer groups can compete at the forefront. Q: Once one could at Surface Science conferences count many hundreds of 3–4-letter-acronym tools. Is it today sufficient with DFT and STM? Will STM and DFT solve the remaining problems of surface dynamics? A: Certainly not. Both methods provide only limited information which has to be supplemented by the results of other studies in order to obtain a complete picture. Q: You and your group have extensively studied nonadiabatic processes in Surface Science. Yet the attitude in the important subfield of gas–surface reactions is still an adiabatic one. Do you see any problems? Any solutions? A: Chemical reactions are usually activated by thermal energy and hence mostly proceed in the electronic ground state. The same holds also for ‘normal’ catalytic reactions. The study of non-adiabatic effects is more related to the physical elementary processes of energy exchange between the various degrees of freedom. Q: Would you still recommend to young scientists to enter the field of surface science and the area of dynamics in particular?

xix

xx


A: Yes, definitely. There are still many new challenges, such as the study of more complex systems such as in biology or of effects of molecular dynamics which will initiate novel experimental techniques as well as theoretical concepts. The trend will be directed also more towards surfaces of soft matter in contact with the liquid phase, so that optical methods will become more important.

CHAPTER 1

Fundamental Atomic-Scale Issues/Processes Pertinent to Dynamics at Surfaces

J.W. GADZUK Electron and Optical Physics Division National Institute of Standards and Technology Gaithersburg, MD 20899, USA

© 2008 Elsevier B.V. All rights reserved DOI: 10.1016/S1573-4331(08)00001-2

Handbook of Surface Science Volume 3, edited by E. Hasselbrink and B.I. Lundqvist

Contents 1.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1.2. General quantum dynamics for surfaces . . . . . . . . . 1.3. Trajectorizing . . . . . . . . . . . . . . . . . . . . . . . . 1.4. Diabatic transitions . . . . . . . . . . . . . . . . . . . . . 1.5. Anderson orthogonality, Friedel, phase shifts and friction 1.6. Electron-resonance-enhanced dynamics . . . . . . . . . . 1.7. Final thoughts . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

Abstract An introduction to some basic concepts and building blocks that facilitate the understanding of atomic scale dynamics at surfaces is presented. The focus is mainly towards processes defined in terms of both nuclear and electronic involvement. Reduction of the general quantum dynamics problem into something tractable is taken up first within the Born–Oppenheimer separation of fast and slow variables. The role of non-adiabatic processes is of special importance in condensed matter systems owing to the continuum of electron–hole pair excited states and this is a continuing theme throughout this introductory chapter. The basis for the commonly invoked trajectory model is examined. A paradigm-establishing example of a curvehopping, diabatic transition model for the simple process of sticking is used to illustrate several points involving molecule–surface collisions. Surface processes involving significant dynamic charge transfer are dealt with taking advantage of both some unlikely similarities with fundamental processes triggered in core level spectroscopies and also the many modern advances in understanding the role of electron friction in condensed phase chemical dynamics. The importance of electron-resonance-enhanced dynamics is taken up and illustrated with a simple, generic, classical model for resonance-enhanced processes here applied to bondbreaking/desorption. This introduction concludes with some final historical thoughts.

2

3 4 8 10 13 19 24 25

3

1.1. Introduction Surface dynamics is here regarded as the phenomenology of atomic level “things happening” at solid surfaces, by which is meant the time evolution of an ensemble of atoms, molecules, and condensed phase entities in which there is sufficient energy available for chemical bonds to be broken, site-to-site atom transfer to be possible, and/or dissociation/desorption to occur. In addition, there should be adequate heat baths (e.g. substrate phonons, electron–hole pairs, secondary intra-molecular degrees-of-freedom, etc.) into which excess energy can be dissipated thus allowing stable particles to be formed as a result of the “happening”. Arguably an ultimate realization of surface dynamics might be thought of as a “chemical” reaction (Pearson, 1971; Nikitin and Zülicke, 1978; Salem, 1982; Hoffman, 1988; Marcus, 1993; Wyatt and Zhang, 1996; Nakamura, 2006), following in the tradition of nuclear reaction dynamics (Wigner, 1946, 1955; Blatt and Weisskopf, 1952). The reaction could be as simple as the sticking of an energetic flux of atoms or molecules to a surface (i.e. A∗ + S → (AS)∗ ) or as complex as surface-induced molecular dissociative adsorption followed by fragment rearrangement/fusion. Two abstract technical perspectives on this have been offered. d’Agliano and coworkers (1975) state that various chemical compounds correspond to minima in the electronic eigenenergy hypersurface in the space of the nuclear coordinates of the constituent atoms (ad-particles). In many cases a chemical reaction may be viewed as a Brownian motion of the system’s representative point in this space from one minimum to another. In contrast, Metiu et al. (1979) have noted that a reaction can be treated as an electronic transition between two different electronic states corresponding to two different quasi-adiabatic potential surfaces, one for reactants and one for products. The first view describes processes/reactions well treated in the adiabatic representation whereas the second characterizes those better visualized within the diabatic representation (O’Malley, 1971), both of which will be explained shortly. The purpose of this opening chapter is to introduce some basic terminology, fundamentals, and simple physics associated with a number of elementary molecular processes occurring mainly at metal surfaces that will be helpful in subsequent detailed topics within this volume (Lundqvist et al., 1979; Metiu and Gadzuk, 1981; Nørskov, 1981; Kleyn, 1992; Brivio and Grimley, 1993; Jacobs, 1995; Billing, 1999; Ertl, 2000; Tully, 2000a, 2000b; Kroes et al., 2002; Asscher and Zeiri, 2003; Groß, 2003; Wodtke et al., 2008). No attempt will be made to provide a uniform and/or comprehensive survey of either the literature or the field. Instead snapshots of a few historically interesting and important points will be presented that convey some of the spirit that guided the development of surface dynamics over the past several decades. When useful, illustrative simple analytic models will be featured as preludes to more computerintensive studies emphasized in subsequent chapters. To this end the introductory chapter is structured as follows. Section 1.2 will deal with the “big picture” reduction of the general quantum dynamics problem in which aspects of the Born–Oppenheimer separation, adiabatic vs diabatic representations, and substrate electron–hole (e–h) pair and phonon dissipation/friction will be introduced. The important and enlightening (if not always intellectually satisfying) use of trajectory-based models will be illustrated in the third section. An idealized model problem illustrating some connections between trajectories which include diabatic transitions (possibly due to charge transfer) and atomic

4

J.W. Gadzuk

sticking will be outlined in Section 1.4. The dissipative interaction between a moving particle/time-dependent perturbation and the substrate e–h pairs has been of major interest for more than a century, going back to the early days of exo-electron emission (Greber, 1997). Since the ground-breaking work of Müller-Hartmann et al. (1971a, 1971b), various theoretical studies have been carried out in which the dissipative interactions are characterized in terms of such things as charge transfer, phase shifts, friction and/or Anderson (1967) orthogonality/X-ray edges. In many cases these are just seemingly different ways of including the same physical ingredients, as will be illustrated in Section 1.5. This extra attention to such issues seems particularly appropriate since a number of striking new direct observations of hot-electron surface dynamics have been reported within the past few years after decades of data-less speculation (Gergen et al., 2001; Nienhaus, 2002). An example of resonance-enhanced hot-electron processes is taken up in Section 1.6 as it represents both an important class of problems under study by surface dynamicists and STM atom movers/nano-manipulators (Bartels et al., 1998; Hla and Rieder, 2003; Ueba and Persson, 2004; Barth et al., 2005; Mayne et al., 2006; Otero et al., 2006) and also it presents an enlightening show of the relative roles of various time scales; resonance lifetimes, interaction time durations or time delays, characteristic vibrational times in the reactant, resonance, and product states. Some final remarks are offered in Section 1.7.

1.2. General quantum dynamics for surfaces The formal theoretical challenge first requires specification of a many body Hamiltonian that describes the initial coupled electron-nuclear system of the surface and external atoms or “reactants”, including both nuclear motion (e.g. phonons, TVR => translational, vibrational, rotational) and also electronic excitations, quasi-discrete excited states or resonances associated with the external atoms/molecules as well as the continuum of substrate electron–hole pair excitations (Metiu and Gadzuk, 1981). In order to obtain manageable but meaningful eigenstates from such all-inclusive Hamiltonians, physically motivated simplifications are prudent and necessary. The nature of the simplifications are guided by one’s chosen theoretical strategy; for instance large scale numerical simulation, analytic modeling, or something in between. The complexity of this task has been captured in Fig. 1.1, Reduction of the General Quantum Dynamics Problem (Benderskii et al., 1994). Although proposed within the context of low temperature chemical reactions, it is equally relevant to most condensed phase reaction scenarios at surfaces. In general the complete system is first divided into slow and fast subsystems in which the coordinates of the slow system can be treated parametrically within the world of the fast system. The most obvious separation between electron and nuclear motion is based on the mass differences and this is known as the Born–Oppenheimer approximation (Tully, 2000b; Wodtke et al., 2004). For a given fixed nuclear configuration R = {R , R , . . .}, ∼ ∼1 ∼2 R ) consists of the entire Hamiltonian of the system the electronic Hamiltonian Hel (r∼; ∼

with the exception of the kinetic energy operator of the slow system. Adiabatic (Born– Oppenheimer) electronic wave functions Φj (r∼; ∼ R ) are defined to be eigenfunctions of


5

Fig. 1.1. Roadmap for reduction of the general (including surface) quantum dynamics problem into a theoretically tractable model (Benderskii et al., 1994).

Hel (r∼; ∼ R ) for fixed ∼ R ; that is Hel (r∼; ∼ R )Φj (r∼; ∼ R ) = εj (R )Φj (r∼; ∼ R) ∼ ), the adiabatic PES corresponding to the electronic state j , is the total energy where εj (R ∼ of the system when in nuclear configuration R and electronic state j . With the BO sepa∼ ration of fast and slow subsystems, the exact total system wave function Ψ (r∼, ∼ R ) can be expressed as Ψ (r∼, ∼ R) =

aj Φj (r∼; ∼ R )Ωj (R ) ∼

j

where Ωj (R ) is the wave function describing nuclear motion on the PES εj (R ). The ∼ ∼ ground state PES ε0 (R ) provides the force field used in most numerical trajectory stud∼

6

J.W. Gadzuk

ies whether based on classical non-dissipative Newton’s or dissipative Langevin equations (Adelman and Garrison, 1976; Adelman and Doll, 1977; Adelman, 1979; Li et al., 1993; Tully, 2000a, 2000b) or on wave packet propagation via the time-dependent Schrödinger equation (Heller, 1981; Gadzuk, 1988; Tannor, 2007). The εj =0 (R ) are the PES govern∼ ing nuclear motion when the system is in an electronically excited state (Tannor and Rice, 1988; Moiseyev et al., 2004), either quasi-localized on the external atom/molecule, extended as in a superposition of substrate e–h pairs, or some combination of both. Adiabatic dynamics apply to processes in which only a single electron state is involved, hence nuclear motion on a single PES. The consequent dynamics is limited to energy redistribution entirely within the domain of (TVR and phonon) nuclear motion (Jacobs, 1995; Rettner et al., 1996), thoroughly discussed here in the chapters by Kleyn and by Manson. While presenting a formidable computational challenge, it is well defined and properly posed and thus will not be further considered in this introduction. Fundamental problems arise when dealing with situations in which there are significant changes in the electronic charge distribution (i.e. charge transfer, polarization, distortion, etc.) along a trajectory or reaction path since this, in one way or another can only be dealt with by inclusion of electronic excited-states, hence non-adiabatic dynamics. Within the context of the adiabatic representation, this is treated through proper inclusion of the so-far-neglected slow system kinetic energy operators. Since Ωi (R )|∇R2 |Ωj (R ) = 0, available energy ∼ ∼ ∼

(εΩi − εΩj = εΩ = 0) within the slow system can then be used to create the electron excited states in a proper energy-conserving process. The problem in computational studies is that since there can be a distribution of excited electronic final states, there is not a single PES that can be legitimately incorporated within the chosen equations of motion for the slow particles. Various Ehrenfest-theorem-based averaging or mean-field approximations are frequently used (Sawada et al., 1985; Jackson, 1988; Li et al., 2005; Shenvi et al., 2006), but it remains troubling that these procedures cannot satisfy the correct asymptotic branching ratio limits. Tully and coworkers (Tully, 1990; Wodtke et al., 2004; Cheng et al., 2007) have insightfully addressed these issues. The diabatic representation provides an alternative description which includes many of the so-called non-adiabatic transitions in a natural and straightforward way, particularly for processes in which either fast electronic transitions occur within a spatially localized region of configuration space or they do not occur at all. For a system in an initial electronic state = i in which the nuclear motion is governed by εi (R ) such transitions to states ∼ ), the j = i serves as a switching procedure after which motion evolves according to εj (R ∼

= R ) new PES. The point (1 dimension) or seam (2D) where εi (R = ∼ R c ) = εj (R ∼ ∼ ∼c

defines the curve crossing region where the electronic states i and j are degenerate. This condition allows for an i → j “diabatic transition” that looks like elastic passage through the crossing point/seam if the available kinetic energy of the “reactants” is sufficient to permit accessing the curve crossing. The more spatially localized is the region in which the electronic transitions occur, the quicker is the passage through the transition region and thus the switch in PES, if it occurs, is better characterized within the sudden approximation limit (Gadzuk and Nørskov, 1984). Parenthetically note that describing a process in which no electron transition occurs may require introduction of


7

Fig. 1.2. Energy-level diagram for an atom or molecule characterized by its highest occupied (image-potential-upshifted) valence level Vi and lowest unoccupied (downshifted) affinity level A interacting with a metal surface. The lowering of the affinity level allows a Fermi-level electron from the substrate to tunnel into the atom/molecule thus creating a negative ion. This possibility is turned on at a separation Rc where the shifted affinity level coincides with the Fermi level.

non-adiabatic transitions; an awkward need for an artificial transition in the adiabatic representation to account for the fact that nothing has happened. An important realization of a diabatic transition occurs in atomic/molecular scattering from surfaces in which either the single electron valence or electron affinity levels of the incident particle are shifted (due to image potentials) and cross the substrate Fermi level at some distance z = Rc from the surface, as depicted and described in Fig. 1.2 (Gadzuk and Metiu, 1980). Long range charge transfer is the hallmark of such processes (Lang and Nørskov, 1983; van Wunnik and Los, 1983; Gadzuk and Nørskov, 1984; Brako and Newns, 1989; Los and Geerlings, 1990; Marston et al., 1993; Lundqvist, 1996; Borisov et al., 1998; Gauyacq et al., 2007). Further discussion on the topics of charge transfer and diabatic transitions will be given later. The issue of energy dissipation is always an important factor in surface dynamics since the substrate can be regarded as a large heat sink/source. For the “chemical” types of processes of interest here in which the energies are on the less-than-10 eV scale, the relevant substrate excitations are lattice vibrations (phonons 0.1 eV) and valence electron excitations (zero-gap e–h pairs with 0 εe–h εFermi and possibly plasmons) (Liebsch, 1997a; Chulkov et al., 2006). Both phonons and intra-molecular vibrations are dealt with by first letting the nuclear coordinates ∼ Rj = R +R where ∼ R j,0 is the static equilib∼ j,0 ∼j rium configuration for electronic state j and ∼ uj represents small displacements about the local minima. Typically the lattice problem is then reduced to an ensemble of harmonic oscillators in the frequency range 0 < h¯ ω < h¯ ωD ≈ 10−2 eV with a phonon density of states that varies quadratically with frequency. Although not as intuitively obvious, the valence e–h pair excitations of the solid, within the random phase approximation, can also be considered as harmonically oscillating density fluctuations which are then “bosonized” but with an e–h pair DOS 2 ρe–h (ω) = ρel (εel )ρhole (εh )δ(εel + εh − h¯ ω) dεel dεh ≈ hωρ (1.1) ¯ el (εFermi ) varying linearly with ω and quadratically with ρel (εFermi ), the Fermi level electron DOS (one factor for the electron, one for the hole) (Müller-Hartmann et al., 1971a). Thus the

8

J.W. Gadzuk

problem of energy transfer between an energetic external entity and the substrate becomes one of coupling to excitable boson systems, either phonons or e–h pairs (Gunnarsson and Schönhammer, 1982; Cini and D’Andrea, 1988). Invariably the coupling is taken to be proportional to the boson displacement which enables modeling in terms of driven displaced oscillators (Nikitin, 1974; Gazdy et al., 1987; Harris, 1987; Mahan, 1990). Furthermore it is known that the dissipation experienced by a moving object that is linearly coupled to a harmonic bath can be treated as a viscous force (Hänggi et al., 1990; Li et al., 1993), which fact has encouraged theoretical studies of both phonon and electronic friction in surface dynamics. The relationship between the friction and atomistic scattering approaches will be returned to.

1.3. Trajectorizing There are many dissipative surface scattering processes in which the energy loss to the “receiving” mode is small relative to the available kinetic energy of the incident particle while in the spatial region where the dissipative forces are strong. Under these circumstances a reasonable approximation to the classical mechanical trajectory can be obtained by just solving the appropriate equations of motion for the energetic “reactant” neglecting dissipation or recoil of the particle due to its small energy loss relative to its incident energy (since from ε = p 2 /2m the change in momentum p ∼ ε/ε 1/2 ). This is implemented in the following way. Suppose that the incident particle with mass M and coordinate ∼ Rp moves in some static potential field Vp = Vp (R ) and is coupled to the dissipative har∼p monic oscillations (labeled with displacement coordinates ∼ u) via some Vint = Vint (u , R ). ∼ ∼p

The classical dynamics of the system is then specified by the set of coupled equations of motion ¨ = − 1 ∇R p Vp (R p ) + ∇R p Vint (u, R p ) R (1.2) ∼p ∼ ∼ ∼ ∼ ∼ M and 1 u¨ + ω02 ∼ (1.3) u = − ∇u Vint (u , R ). ∼ ∼ ∼p m ∼ In Eq. (1.2), if ∼ u is set equal to ∼ u0 , its constant mean value, then ∼ R p becomes independent ) and of the solution to Eq. (1.3) and can in principle be solved for any prescribed Vp (R ∼p

; u ) to yield R =∼ R p (t). This is known as the trajectory approximation in which Vint (R ∼ p ∼0 ∼p

the time dependence of the oscillator motion does not influence the motion of the particle during the period of strong interaction (Müller-Hartmann et al., 1971b; Lucas and Šunjić, 1972; Blandin et al., 1976; Brako and Newns, 1980; Gadzuk and Metiu, 1980; Nørskov, 1981; De Pristo, 1983; Leung et al., 1984, Schönhammer and Gunnarsson, 1980, 1981, 1984; Harris, 1987; Los and Geerlings, 1990; Burke et al., 1993; Jackson, 1993; Trail et al., 2003). With regards the oscillator system, taking Vint (u , R ) to be linear in oscillator displace∼ ∼p is equivalent to a time dependent function, ments about ∼ u0 and using the fact that R ∼p


9

[R (t); u ], that of a the oscillator equation of motion, Eq. (1.3), is: ∼ u¨ + ω02 ∼ u = − m1 Vint ∼p ∼0

is evaluated at the original osforced harmonic oscillator where the force functional Vint cillator equilibrium. It is a standard exercise to show that an oscillator subjected to such a time-dependent force (here provided by the incident particle on trajectory ∼ R p (t)) will

experience an energy gain εclass (ω) = where Vint (ω) =

(ω)|2 |Vint 2m

R (t); ∼ Vint u0 eiωt dt ∼p

(1.4)

(1.5)

is the Fourier transform of the time-dependent driving force on the oscillator system due to the “trajectorized” incident energetic particle (Marion, 1965; Mahan, 1990). The correspondence principle provides an elegant and exact connection between the energy gain of the forced classical harmonic oscillator and the vibrational excitation probability distribution of the equivalent quantum mechanical harmonic oscillator subjected to the same forcing function (Nikitin, 1974; Gentry, 1979; Billing, 1999). In terms of the parameter 2 β = εclass (ω)/h¯ ω = Vint (1.6) (ω) /2mh¯ ω, it has been demonstrated many times that the probability for a 0 → n HO transition is P0→n = e−β (β n /n!),

(1.7)

a Poisson distribution. Furthermore, when β 1, the Poisson distribution becomes a Gaussian whose width = β 1/2 hω. The dynamics of a specific process enter ¯ into Eq. (1.7) solely through Eqs. (1.2) and (1.4)–(1.6) which express the functional [R (t); u ]. The forced HO has a dependence of β on the Fourier transform of Vint ∼p ∼0 rich history in surface dynamics where it has frequently been presented in exquisite formal detail while still being fundamentally no more than an extension of the basic displaced HO just discussed here (Langreth, 1970; Lucas and Šunjić, 1972; Harris, 1987; Gadzuk, 1988; Wingreen et al., 1989; Gadzuk, 1991; Jackson, 1991, 1993; Groß, 2003). Since the interaction between the driving particle and the oscillator has been taken linear in oscillator displacement, the rate of single-quantum oscillator excitation in the scattering event can be perturbatively estimated from Fermi’s Golden rule 2 2π h¯ ω R(ω) = (1.8) λint (ω) ρ(ω) coth 2kT h¯ h¯ 1/2 which also includes non-zero bath temperature and λint (ω) ≡ ( 2mω ) Vint (ω). As an illustrative example, consider an incident particle that experiences a timedependent Gaussian (on-and-off) dissipative interaction with the oscillator system 2 1

Vint ∼ (1.9) R p (t) = λ0 exp − t/τp (εin ) . 2

10

J.W. Gadzuk

The Gaussian width parameter τp serves the dual role of both the incident-energydependent switch-on and also the interaction-duration time. Since the Fourier transform of a Gaussian returns a Gaussian,

Vint (1.10) (ω) = 2πτp2 λ0 exp −ω2 τp2 /2 . The low temperature rate for e–h pair production that follows from Eq. (1.1) and (1.8)– (1.10) is 2π 2 2 ρel (εFermi )τp2 λ20 exp −ω2 τp2 ≡ R0 τp2 exp −ω2 τp2 . m The overall rate for a process requiring an activation energy ε∗ or greater is √ ∗ ∞ ε τp π exp −ω2 τp2 dω = R0 τp 1 − erf Rtot (ε ∗ , τp ) = R0 τp2 ∗ 2 h¯ ε /h¯ Re–h (ω, τp ) =

(1.11)

supporting the early results due to Brako and Newns (1980). With regards some general physical principles, note from Eq. (1.11) that large τp is equivalent to a slow, adiabatic switching process in which there is little e–h pair excitation except for low energy pairs with hω ¯ < h¯ /τp that see this switching as fast and thus non-adiabatic. More will be said about this in Section 1.5. Finally it must be cautioned that trajectory models, though useful, easy, and enlightening when appropriate, are inherently non-energy conserving. Although energy was delivered to the oscillator systems, the incident exciting particle was not allowed to slow down, as required due to energy-conserving recoil. Prudence must be exercised to assure that spurious consequences of this are not taken seriously.

1.4. Diabatic transitions Consider the following simple and transparent example of a diabatic inelastic scattering event in which a single particle on an initial PES is incident upon a substrate. Somewhere along its trajectory a hop is possible onto another PES due to a substrate-induced diabatic transition between coupled electronic states of the particle and substrate. Energy dissipation due to substrate excitation as the particle moves on the second PES, determines whether the particle remains trapped on this curve and hence determines the “reaction probability”. The model, considered as a multiple-pass Landau–Zener problem including dissipation, is a condensed phase realization of the historic gas phase “surface-hopping trajectory theory” (Tully and Preston, 1971; Sholl and Tully, 1998) that has more recently been extended to mixed classical/quantum systems (Cheng et al., 2007). Suppose an atom or molecule far removed from a surface is prepared in some internal state (say its electronic, vibrational, and rotational ground state) and then is allowed to interact with the surface subject to the constraint that it remains in this internal state. The total energy of the coupled system, as a function of the position of the molecular center-ofmass with respect to the surface might appear as the repulsive curve labeled 1 in Fig. 1.3. Now imagine that the same particle is put into some electronically excited state (excited


11

Fig. 1.3. Top: Diabatic potential energy curve as a function of z, the normal distance from the surface, for an incident atom or molecule initially in some electronic state 1 and with kinetic energy Ki . As drawn, curve 1 corresponds to an electronic state giving rise to a strictly repulsive surface interaction, curve 2 to an electronic state which strongly adsorbs at an equilibrium separation R0 . The probability for a diabatic electronic transition from curve 1 to 2 is maximum at the crossing point Z = Rc . Bottom: Some elastic trajectories for a particle moving in the potentials shown at the top.

neutral, positive or negative ion, or dissociated molecule for instance) and this object is then allowed to interact with the substrate, again subject to the frozen-internal-state constraint. A possible potential curve labeled 2 in Fig. 1.3 shows a strongly attractive potential well, lower in energy close to the surface than if the particle was in electronic state 1. For instance, if state 2 corresponds to an ion, then in spite of the fact that V2 (z = ∞) exceeds V1 (z = ∞) by Vi −ϕ(ϕ−A) for a positive (negative) ion with ϕ the substrate work function and Vi /A the particle ionization potential/electron affinity, the resulting V+/− = −e2 /4z image potential attraction more than compensates for this when z ≈ Rc . Furthermore, a continuum of roughly parallel curves are required if substrate e–h pair excitations are considered and a set of discretely spaced parallel curves if internal vibrations of the incident molecules are important, in the spirit of the iconic Bauer–Fisher–Gilmore (1969) “networks” first used to treat the multiple curve-crossings associated with vibrational excitation in atom–molecule and then molecule–surface charge-transfer collisions (Gadzuk and Holloway, 1985; Qian et al., 1995).

12

J.W. Gadzuk

An elementary “reactive event” can be thought of as follows (Gadzuk, 1982). A thermal beam of particles is placed in motion (one dimensional for simplicity) on curve 1, incident upon the surface. In the vicinity of the curve crossing point z = Rc , the electronic states 1 and 2 are degenerate so an energy conserving electronic transition between these states is possible with probability ≡ P 12 = P 21 depending upon the specifics of the substrate–particle interaction. For the case of ion formation, the crossing point corresponds to that place where the image-potential-shifted ionization or affinity level hits the Fermi-level, thus turning on the possibility for electron charge transfer between surface and particle (Nørskov and Lundqvist, 1979; Gadzuk and Metiu, 1980; Nørskov, 1981; Lang and Nørskov, 1983; van Wunnik and Los, 1983; Gadzuk and Holloway, 1985; Brako and Newns, 1989; Los and Geerlings, 1990; Marston et al., 1993; Lundqvist, 1996; Qian et al., 1995; Cheng et al., 2007). Consequently its magnitude drops roughly exponentially with distance Rc due to the exponential decay of overlapping wave-functions required for charge transfer to occur. This transition is referred to as a diabatic transition from state 1 to 2. Note that the center-of-mass trajectory branches near Rc , some being reflected without undergoing the 1 → 2 transition. Upon reflection at z = RT , the incident particle, now in state 2, hits the crossing point from the left where another branching is experienced. The time interval between passes at the crossing point is approximately Tv = 2π/ω0 where ω0 is the vibrational frequency associated with the bound state well. From Fig. 1.3, it is easy to see that the probability for scattering is given by the (geometric series) sum over all trajectories 2 + P12 (1 − P12 )P12 + P12 (1 − P12 )2 P12 + · · · Pout = (1 − P12 ) + P12

which is expressed compactly as 2 Pout = (1 − P12 ) + P12

N−1

(1 − P12 )n = 1 − P12 (1 − P12 )N

(1.12)

n=0

where N = the number of trajectories. If there are no dissipative features restricting this number, then N = ∞ and with 1 − P12 < 1, Pout = 1 which certainly makes sense. What goes in must come out if no dissipation, sources, or sinks are present. Of course the sticking probability PS = 1 − Pout = 0 in this trivial case. For the incident particle to remain trapped in the bound-state well of curve 2 (“product side”), some form of energy dissipation (e.g. substrate pairs or phonons, intra-molecular vibrations or rotations) must be possible, allowing for decay of the initial vibrational state to a lower lying one below either the dissociative continuum threshold or the activation barrier. Within the context of the present model, dissipation limits the number of possible outgoing trajectories included in the Eq. (1.12) summation to those whose residence on curve 2 is completed within a time duration less than a decay time τn = h¯ /γn for the nth vibrational state of the inner potential well. For a discrete harmonic oscillator state coupled to a dissipative continuum, the width of the nth level is approximately γn ≈ nγ0

(1.13)

where γ0 is the 0 → 1 fundamental width as can be observed in optical spectroscopy studies. The number of oscillations within the inner well and hence the number of trajectories


13

is related to the decay time by N Tvibe = τn and thus N = hω ¯ 0 /2πγn .

(1.14)

Since Ps + Pout = 1, from Eq. (1.12) the non-zero sticking probability is Ps = P12 (1 − P12 )N

(1.15)

with N given by Eq. (1.14). In order to obtain an explicit expression for Ps , one must know P12 which depends upon the specific chemistry of the system being considered. Here we are interested in system-independent general characteristics of the “surface-hopping trajectory model”. For example, the value of P12 yielding the maximum sticking probability max = 1/(1 + N ) which when inserted in Eq. (1.15) yields an upper bound on the is P12 sticking probability N 1 N max Ps = (1.16) . 1+N 1+N Equation (1.16) is to be interpreted as the maximum sticking probability for a system whose parameters fall between N and N + 1. For N = 1, Psmax = 1/4 which just says that if the incident particle is to have a large (Ps 0.25) sticking probability, the dissipation must occur before the first roundtrip is completed (N < 1). Upon consideration this is quite reasonable. First, in order for the particle to gain entry into the dissipative well, P12 must be large. However, if this is the case, then the probability for exiting the well on the first attempt from the left will also be large. Unless the particle is damped before this attempt, it will escape and then the sticking probability will be small. Lastly the relevant value of n must be specified. If curve 2 represents a state in which the substrate is in its ground state and if the potential well is harmonic, then it is apparent from Fig. 1.3 that nh¯ ω0 = εdesorp + kT .

(1.17)

Combining Eqs. (1.13), (1.14), and (1.17), the final simple criterion for large sticking probability is N = (h¯ ω0 )2 / 2πγ0 (εdesorp + kT ) < 1, (1.18) given in terms of the easily accessible, system-specific quantities ω0 the vibrational frequency characteristic of the ad-particle-surface bond, γ0 the 0 → 1 line-width associated with this bond, and εdesorp the desorption energy. Note that for the entirely realistic values of h¯ ω0 = 0.1 eV, γ0 = 0.001 eV, and εdesorp = 1.0 eV, N ∼ 1 suggesting that the stick-no stick criteria given by Eq. (1.18) should be sensitive to variations in the physical quantities within the parameter range realized in real systems (Persson and Harris, 1987; Brako and Newns, 1989; DePristo and Kara, 1990; Jackson, 1993).

1.5. Anderson orthogonality, Friedel, phase shifts and friction The role of substrate conduction band e–h pair excitations in fundamental surface processes such as sticking, molecular dissociation, adsorption, inelastic beam scattering, etc. was

14

J.W. Gadzuk

frequently questioned/raised in “modern terms” throughout the decade of the 70s and early 80s (Müller-Hartmann et al., 1971b; d’Agliano et al., 1975; Lundqvist et al., 1979; Nørskov and Lundqvist, 1979; Gadzuk and Metiu, 1980; Schönhammer and Gunnarsson, 1980, 1981; Brako and Newns, 1980, 1981; Nørskov, 1981; Metiu and Gadzuk, 1981; Schönhammer and Gunnarsson, 1983, 1984). Many of these studies focused on phenomena in which charge transfer between an incident particle and the surface was the defining characteristic of the process. The motivation was at least in part due to perceived similarities between a charge-transfer event involving the massive incident particle interacting with the e–h pair continuum of the substrate and the related ionization (a form of “charge transfer”) that occurred in X-ray absorption and/or photoelectron spectroscopy of core levels, cutting edge issues at that time (Langreth, 1970; Mahan, 1974; Gadzuk and Šunjić, 1975; Langreth, 1984; Gunnarsson and Schönhammer, 1985). In both cases, the e–h pairs see a transient switching on of a localized atomic-like potential (the difference between the potential of the initial neutral and the final charged particle). This dynamic potential, similar in both cases, excites e–h pairs. A particularly intriguing point was raised by the Anderson (1967) Orthogonality Theorem (AOT) which says that the many body ground state of an infinitely extended Fermi system/electron gas is orthogonal to the ground state of the same system within which a localized potential has been introduced. One consequence of the AOT is that permanent switching on of a potential in an electron gas always results in excitation of e–h pairs with zero-probability for perfectly elastic switching, independent of the details of the switching process. Another consequence examined in detail by Müller-Hartmann et al. (1971a) is that the strength of the no-loss line, i.e. electron gas “Debye–Waller factor” (Brako and Newns, 1980; Gadzuk, 1981; Nørskov, 1981; Schönhammer and Gunnarsson, 1984), is always less than unity when a localized potential is turned on and then off after a finite time delay, again independent of the specific dynamics of the switching. In other words, introducing and then withdrawing a scattering potential into an infinite Fermi system cannot be done without some finite probability for leaving the system in an excited state. Thus truly adiabatic processes in such systems are rigorously impossible. This is a consequence of the gapless excitation spectrum. No matter how slowly is the switching, it appears fast to even slower low energy e–h pairs. A physical realization of this effect should occur in particle scattering from metallic surfaces. An incident particle plays the role of the switched-on potential and likewise the same reflected particle that of the switched-off potential. Realizing that such an event cannot happen adiabatically, there is then the possibility that sufficient energy could be transferred to the e–h pair excitations (from the incident particle kinetic energy) that the particle would remain trapped at the surface. Consider first the core hole problem. If a potential is suddenly switched on within an extended electron gas/Fermi system, then in the long-time limit, the electron-gas is necessarily raised to an excited state according to the distribution 2 P (ω) = Vloc ; h¯ ω|0; 0 (1.19) where |Vloc ; ε and |0; 0 are many-body excited and ground states with and without the localized potential. In this form, P (ω) given by Eq. (1.19) plays the role of a many-body Franck–Condon factor. If the potential is a localized atomic-like potential such as that from a suddenly created core hole, then the excitation probability distribution displays


15

the Mahan–Nozières–de Dominicis (MND) X-ray edge singularity about which much has been written (Nozières and de Dominicis, 1969; Mahan, 1974; Ohtaka and Tanabe, 1990). Without going into great detail, the idea is that many low energy pairs can be produced upon localized hole creation in analogy with discrete atomic shake-up (Gadzuk and Šunjić, 1975), and this transforms the absorption threshold into a divergent-but-integrable form which ND solved exactly in the long-time limit (i.e. right at threshold) as P (ω) = N

1 ω1−α

(0 < ω ωc = εc /h¯ )

(1.20)

where N = α/ωcα is a normalization (per unit energy) factor, εc a cutoff energy of order the conduction bandwidth, and α=2

∞

(2l + 1)(δl /π)2

(1.21)

l=0

where δl is the l-wave Fermi-level electron phase shift associated with the localized hole potential. A similar expression can be obtained for a localized potential outside a metallic surface in which case α and ωc become functions of the hole–surface separation (Gumhalter and Newns, 1975; Gadzuk and Metiu, 1980). The main useful point to be stressed here is the inherent identity between the overlap integrals, the e–h pair excitation spectrum, and the electron-gas Franck–Condon factors and their connections with the core hole phase shifts. Although the sudden-limit provides useful guidelines and also an upper limit to the distribution of excited states produced, a softer introduction of the localized potential into the Fermi system might be expected when thermal beams of atoms or molecules are incident upon a surface, as for instance in the recent experimental (Gergen et al., 2001; Nienhaus, 2002) and theoretical (Gadzuk, 2002; Trail et al., 2003) chemicurrent studies. In the comprehensive theory of the dynamic response of the electron gas to such potentials by M-HRT (1971), the influence of the switching rate is considered. In contrast to the sudden-on case (Eq. (1.20)), they find that the distribution of e–h pair states generated by the same localized potential that is now exponentially rather than suddenly switched on (V (t) ≈ V0 exp(+ηt), −∞ < t < 0; = V0 , t 0) is P (ω) =

exp(−ω/η) (Γ (α)ηα )ω1−α

(1.22)

which redistributes weight from the high energy tail (where ω > η) back into the low energy region. Essentially this guarantees that there will be no Fermi system response until the perturbation is acting. Intuitively, one expects 1/η, the characteristic time scale, to be of the order of z/vel, where z is the width of the region outside the surface within which the potential switches (0.1 z 1 Å) and vel ≈ (2D/M)1/2 1.4 × 1014 D 1/2 Nm Å/sec is a typical velocity in this region (D = potential well depth or adsorption energy in eV and Nm = mass number). The switching region is where the particle-surface interaction changes significantly over a narrow spatial range such as at a curve crossing or seam for a charge transfer process. With realistic choices of systemdefining parameters, hη ¯ falls in the range 0.1 eV hη ¯ 1.0 eV. Hot electron–hole pair energy distributions obtained from Eq. (1.22) for a range of localized potential strengths,

16

J.W. Gadzuk

Fig. 1.4. Hot electron–hole pair energy distribution (Eq. (1.22)) per incident event as a function of ω/η, for a range of potential strengths, as conveyed by α.

as embodied in the magnitude of their phase shifts hence α values (Eq. (1.21)), are shown in Fig. 1.4. Clearly the switching rate η sets the upper limit of the phenomena. For given η, the stronger the potential, the greater is the probability of hot electron production. Both consequences are in accord with intuitively based expectations (Gadzuk, 2002). As an alternative to the abstract “switching on” of a localized potential so far discussed, when addressing actual surface processes, electronic excitation is also intimately connected with the rate at which new states appear (to the substrate states) to be created or destroyed below the Fermi level as a result of the incident particle-surface interactions (Schönhammer and Gunnarsson, 1981; Nørskov, 1981; Lundqvist, 1996). As an illustrative example consider a simple spin-less Anderson (1961)–Fano (1961) model (Mahan, 1990)

Va k z(t) ca† c k + H.C. εk nk + H = εa z(t) na + k

∼

∼ ∼

k

∼

∼

∼

in which the spatial dependence of εa , the incident particle electronic level, and Va k , its ∼ (exponentially-dependent) coupling with the substrate conduction band continuum states are taken to be time-dependent through the trajectorizing procedure. The “c” and “n” operators obey appropriate Fermi commutation rules. One of the standard consequences of the model is that the original discrete state is shifted and broadened into a resonance characterized by a local DOS (Schönhammer and Gunnarsson, 1980) 1 Γa [z(t)] ρa ε; z(t) = π (ε − εa [z(t)])2 + Γa [z(t)]2 in which Γa (z) ∼ ρel (εFermi )|VakF (z)|2 . Figure 1.5 shows what Lundqvist (1996) has called the “canonized picture of the resonance model of adsorbate electronic structure” originally used to describe alkali atom adsorption on metal surfaces (Gadzuk, 1970). Since εa (z = ∞), the ionization energy, is less than the substrate work-function, charge transfer from atom to metal occurs. In Fig. 1.6 a complimentary example is shown in which the electron affinity level of the external atom/molecule is image-potential-downshifted, crossing the substrate Fermi level at the position z = zc where electron transfer from substrate


17

Fig. 1.5. Energy-level diagram descriptive of an interacting alkali atom–metal surface system which displays the image-potential-upshifted (E) and broadened (Γ ) valence electron level.

to external particle (“harpooning”) is expected (Lang and Nørskov, 1983). Also shown in the lower panel are diabatic potential energy curves for the neutral and negative ion particle interaction with the surface which illustrate the connection between the curve-crossing point and the affinity-level crossing at the Fermi level. At zero temperature the resonance level is partially filled according to εFermi 2 Γa (z) Nel (z) = ρa,σ (ε; z) dε = tan−1 , π εa (z) −∞ σ now allowing for implicit spin degeneracy. For a virtual level of an impurity immersed in a homogeneous host, the displaced charge would be related to δl (εF ), the phase shifts associated with the localized potential as seen by Fermi level electrons, by the Friedel sum rule (Langreth, 1966; Schönhammer and Gunnarsson, 1983) Nel =

∞ 2 (2l + 1)δl (εF ). π

(1.23)

l=0

To the extent that a single s-wave scatterer provides a qualitatively useful surface picture, δ0 = π2 Nel , so with Eq. (1.21), the X-ray edge singularity index is 2 1 2 −1 tan (Γ a /εa ) αs ≈ 2(δ0 /π) = 2 π in terms of the Fermi level resonance characteristics.

18

J.W. Gadzuk

Fig. 1.6. Top: Variation of an atomic affinity level εa with distance z from a metal surface. εa crosses the Fermi level εF at zc , and approaches the free atom affinity level −A asymptotically. As the atom moves away from the surface, the width of the level decreases. Bottom: Diabatic potential energy curves for the same system. Close to the surface a state with a negative ion A− and positive metal M+ is the ground state. At zc , where εa (z) = εF , this diabatic state crosses another where A and M are neutral. Further out the neutral state is the ground state. In the crossing region, neither of the two diabatic states are solutions to the full Hamiltonian of the system and it is here that diabatic transitions between the two states are most likely to occur. The new thing is that to each diabatic state, there is a continuum of higher lying states (dashed in the figure), corresponding to electron–hole pair excitations in the metal. The M+ /A− state thus crosses infinitely many states and can make a transition to any of them. Except when the transition takes place at the lowest M/A state, the system ends up with a neutral atom ejected (M/A state) and electron–hole pair excitations in the metal (Lang and Nørskov, 1983).

Schönhammer and Gunnarsson (1980, 1984) have provided additional insightful perspective on the role of the incident atom/molecule potential acting on the substrate electronic excitations. In terms of the resonance level model, they note that time interval for the Fermi level crossing at time t = t0 when the shifted level hits the Fermi level is εa 2Γa z dz a tc 2Γa [z(t0 )]/˙εa [z(t)]|t=t0 . Since ε˙ a = dε dz × dt vz ( z ), then tc vz εa . Noting that a discrete shift of εa ∼ 2Γa would move the broadened level from above to below the Fermi level, tc z/vz which is identical in spirit with 1/η, the switching-on rate discussed in conjunction with Eq. (1.22) and Fig. 1.4. Finally, for interacting physical systems which do not experience significant charge transfer or rearrangement, the encounter may be sufficiently adiabatic to permit use of a single electronic ground state PES in the equations of motion. Dissipation of excess inci-


19

dent particle kinetic energy into substrate e–h pairs may be thought of as viscous damping (Adelman and Doll, 1977). Consider the simple example of particle motion through a homogeneous medium with a position-independent electronic viscosity ηel . The friction force Fel = ηel v results in energy loss ε = Fel x over a path length x. The resulting rate of energy loss is ˙ε = Fel x˙ = ηel v 2 where the electron gas friction has previously been shown (d’Agliano et al., 1975; Blandin et al., 1976) to be representable by ηel

∞ 8 m εFermi (2l + 1) sin2 (δl − δl+1 ) 3π M h¯ l=0

(1.24)

in terms of the phase shifts that also appear in the X-ray edge exponent and the Friedel sum rule (Eqs. (1.21) and (1.23)) (Schönhammer, 1991). Equation (1.24) has been extended to incorporate inhomogeneous surface events in which the position (hence time dependence in a trajectorized model) functionality of ηel has also been taken into account (Schaich, 1975; Plihal and Langreth, 1999). Needless to say, pedagogical simplicity is sacrificed. On the other hand, the friction approach is popular when appropriate (Wodtke et al., 2004) because it provides a well defined Langevin dynamics (Adelman and Doll, 1977; Li et al., 1993) that contains something tractable for the electronic structure specialists to calculate, namely the phase shifts and thus with Eq. (1.24), the e–h pair viscosity (Nourtier, 1977; Persson and Persson, 1980; Schönhammer and Gunnarsson, 1983; Hellsing and Persson, 1984; Bönig and Schönhammer, 1989; Head-Gordon and Tully, 1992, 1995; Persson, 1993; Liebsch, 1997b; Plihal and Langreth, 1999; Trail et al., 2002). Friction-based theories have been used for modeling many femtosecond-laser-induced surface processes in which the ultra-fast, ultra-intense laser pulse creates a highly nonequilibrium initial distribution of extreme hot electrons which can then frictionally drive chemically-interesting nuclear motion (Dai and Ho, 1995; Brandbyge et al., 1995; Newns et al., 1996; Saalfrank, 2006). In particular, laser-induced-hot-electron-desorption studies have proliferated both because of their intrinsic interest and importance in surface dynamics and also because the internal electronic and TVR quantum states of the product (desorbate) can in principle be measured (Dai and Ho, 1995; Saalfrank, 2006; Frischkorn and Wolf, 2006). Still one must heed the cautionary warning of Wodtke et al. (2004) that the [frictional] theory rests on a ‘weak coupling approximation’. As a result, it is suspect in cases for which non-adiabatic couplings are strong, such as at avoided crossings or when electron hops occur.

1.6. Electron-resonance-enhanced dynamics Electron resonance enhanced processes in atomic scale systems can occur when an energetic electron temporarily occupies a “chemical-bond-significant” orbital for a time duration that is long enough to force atomic displacement resulting in highly excited, dissociated, re-arranged, or otherwise altered final states. The study of such processes has a venerable history within the context of inelastic electron spectroscopy, first with gas phase (Birtwistle and Herzenberg, 1971; Domcke and Cederbaum, 1980) and then adsorbed molecules (Gadzuk, 1988; Palmer and Rous, 1992; Bach et al., 2003; Saalfrank, 2006).

20

J.W. Gadzuk

Going beyond spectroscopy, Tannor and Rice (1988) have proposed using an excited electronic state of a molecular system to control nuclear motion and thus the ultimate outcome of a process on the electronic ground state PES. The force upon the constituent atoms when they are placed on an appropriately chosen excited state PES results in the desired nuclear displacement. By controlling the time duration for propagation on the excited state PES, e.g. by pump-dump laser pulse sequencing in Femtochemistry (Manz and Wöste, 1995; Zewail, 2000), by “engineering” resonance lifetimes in condensed matter systems (Rous, 1999; Marinica et al., 2006; Chulkov et al., 2006), control of product branching ratios and/or internal product quantum state distributions can, in principle, be attained. Consider the simple displaced HO model shown in Fig. 1.7 illustrating that the ultimate excitation on the ground state PES depends both on δz, the magnitude of the oscillator potential displacement, on td , the time spent on the displaced excited state PES, and on dissipative relaxation ˜ d is an odd (even) due to an assumed condensed phase environment. If td is such that ωt integral multiple of π, then maximum (minimum) oscillator excitation occurs. Thus a commensurability between ω, ˜ the excited state oscillator frequency, and the delay time between pumping and dumping is a most crucial factor. In molecular systems the magnitude of δz is determined by the change in electronic character associated with the V0 → V ∗ switch. For instance, if the transition involves a bonding-to-antibonding excitation or an ionization or electron attachment process in which the charge state within the bonding region changes, then the change in equilibrium positions of the atoms hence δz will be large and significant vibrational excitation can occur as a result of the transient switch, possibly leading to bond breaking, dissociation, and/or desorption. On the other hand, if the electronic transitions involve electronic states that do not contribute to the bond represented by the oscillators, then δz will be small and the resulting excitation will be insignificant. This has nicely been discussed by Kock et al. (2002) within the context of pump-probe charge transfer events in condensed phase systems. The principle of selectivity is more dramatically illustrated in Fig. 1.7b for a system showing multiple minima in the electronic eigenenergy hypersurface in the space of the. . . constituent atoms (aka the adiabatic ground state PES) where in this illustration, multiple refers to a simple double well. Thermal “shake and bake” (statistical) chemistry amounts to climbing the barriers separating minima and establishing some equilibrium between the populations in the various localized minima (Hänggi et al., 1990; Zare, 1994). Tannor-Rice (1988) suggest creating electronic excited states with PES such as Vexc that allow relatively free nuclear motion amongst the configurations defined by the several minima in the ground state PES. By controlling t, the time delay before the Vexc → V0 deexcitation occurs, one controls kAB , the rate constant for the A → B reaction and thus the ultimate configuration hence chemical species. Since such multiple-well potentials form the basis for isomerization dynamics of adsorbed molecules such as NH3 (Paramonov and Saalfrank, 1999) and for atom transfer on surfaces (Gao et al., 1997; Gadzuk, 2006) amongst other things, it seems appropriate to at least call attention to their importance here. Returning to the simple example of inelastic resonance excitation, useful parallels between hot-electron-induced resonant desorption (Gadzuk, 1991, 1995), resonance Raman (Heller et al., 1982), dissociative attachment (O’Malley, 1971), inelastic resonant electronmolecule scattering (Domcke and Cederbaum, 1980; Gadzuk, 1988, 1991; Palmer and


21

(a)

(b) Fig. 1.7. Potential curves and time evolution characterizing transient dynamics basic to resonance excitation and to Tannor–Rice selectivity. (a): Wavepacket evolution involving double switching from initial harmonic oscillator ground state on V0 to (at t = 0) and from (at t = td ) a displaced HO excited state (Vexc ). The degree of final state excitation can be controlled by choice of td . (b): Double well potential representing possible “products” A and B that are selectively accessed using the potential Vexc associated with an appropriately chosen electronic excited state and controlling t, the time delay for time evolution on Vexc . The resulting rate constant kAB is then a function of t.

Rous, 1992), resonant tunneling (Wingreen et al., 1989; Gadzuk, 1991), molecular electronics (Joachim and Ratner, 2005; Galperin et al., 2006), and compound decay in nuclear reactions (Wigner, 1946) all point to a common generalizable inelastic electron-transfer theory. The cross section for these processes is often written as a product of a resonant capture cross section for formation of the negative molecular ion or compound nuclear state with resonance lifetime τR multiplied by a unitarity-preserving branching ratio or decay probability into a distribution of allowed final states which, in the case of resonant desorption, might include all the energetically allowed TVR states of the originally adsorbed molecule. The total cross section for resonance scattering in which an electron with incident energy εin experiences an energy loss ε (assumed to be εin ) is

22

J.W. Gadzuk

then σtot (εin , ε; τR ) σres (εin , ε; τR )P (ε; τR )

(1.25)

where σres is the resonant electron capture/scattering cross section evaluated at some appropriate fixed molecular geometry and P (ε; τR ) is the (normalized) molecular excitation probability distribution. A simple single channel resonant cross section invariably has the structure Γin (εin )Γfin (εin − ε) exp(iδR (εin )) σres (εin , ε; τR ) σ0 (1.26) Im Γ /2 εin − εa + iΓ /2 with Γin (εin )/h¯ [Γfin (εin − ε)/h¯ ] the electron transition rate at the initial [final] energy into [out of] the resonance state, εa the (“frozen nuclei”) resonance energy, δR (εin ) a possible phase factor due to geometrical/trajectory considerations that gives rise to asymmetric Fano-like (Fano, 1961) rather than symmetric Breit–Wigner lineshapes (e.g. STM lineshapes of Kondo atoms in quantum corrals (Gadzuk and Plihal, 2003)), and σ0 a slowly varying function of εin across the resonance which can be treated as a constant scale factor. Neglecting the explicit energy dependence of the level width functions [i.e. Γin (εin ) ≈ Γfin (εin − ε) ≡ Γ ] as is common practice, σres (Eq. (1.26)), when integrated over εin is independent of Γ , hence τR . This allows that the resonance-lifetime-dependence of the total yield per capture event for delivering an amount of energy D to the ground state oscillator is εmax PD (τR ) = (1.27) P (ε; τR ) d(ε). D

As a specific example appropriate to dynamics at surfaces, consider resonance enhanced desorption, in which case V0 might be a Morse or a HO potential truncated on one side with a well depth/desorption energy = D. Both classical mechanics and semi classical Gaussian wave-packet dynamics (Heller, 1981; Gadzuk, 1988) provide insightful platforms for further considerations. Initially the desorbate is bound to the surface in the vibrational ground state of V0 (z), the molecule–surface potential energy curve associated with the electronic ground state of the system, as shown in Fig. 1.8 (Gadzuk, 1996). At some time taken as t = 0, a hot electron, perhaps laser excited from the substrate (Saalfrank, 2006; Frischkorn and Wolf, 2006) or from an external STM tip (Seideman, 2003; Hla and Rieder, 2003; Ueba and Persson, 2004; Mayne et al., 2006), becomes trapped in the unoccupied shape resonance responsible for the electron affinity level. The presence of this extra electron strengthens and compresses the molecule–surface bond, resulting in the potential curve labeled Vexc in Fig. 1.8. This displaced oscillator potential provides the forces acting upon the classical point mass (or Gaussian wave packet) representing the molecule translational motion. After a time delay t > 0, the trapped electron exits and the now-moving particle (wave packet) is returned to V0 displaced from its equilibrium position, possibly high enough up on the repulsive wall of V0 to result in desorption, as accounted for by a yet-to-be-determined distribution = P des (t), a function of the time delay spent on the displaced oscillator negative-ion potential, which is related to PD (τR ) specified by Eq. (1.27). Under the near classical conditions appropriate to widely studied systems such as NO/Pt, the periodic function Pdes (t)[= Pdes (t + T ) with T ⇒ Tvibe =


23

Fig. 1.8. Resonance-state phase-space trajectory (top) in relation to motion on real-space potential curves Vexc of the intermediate state and V0 of the ground state. Return to V0 at τ1 results in bound state vibrational excitation whereas return at τ2 results in excitation to the desorption continuum, where τ1 < τon < τ2 < Tvibe /2.

2π/ω0 , the vibrational period on Vexc ] is well represented by a top-hat structure switched on from Pdes = 0 to Pdes = 1 at t = ton (z) and back off at t = T − ton (z). The time interval between ton and T − ton represents that part of the periodic phase-space trajectory, shown as the marked region at the top of Fig. 1.8, which returns the translationally moving point back onto the desorptive wall of V0 . Formally, ton is obtained from the solution of pz2 (ton )/2M + V0 (z(ton )) = D, where both the momentum pz (ton ) and the displacement z(ton ) have been acquired under the influence of Vexc , as suggested by the phase space trajectory in Fig. 1.8. For the special case of displaced (and half-truncated) HOs of the same D frequency, ton is given by ton = ω10 cos−1 (1 − 2 2 ) which depends upon the ground Mω0 z

state through the value of D and upon the excited state via z. In order to account for the continuous distribution of negative ion survival times (hence aforementioned time delays), PD (τR ) should be obtained as an average over all delay times, not just the single so-called resonance lifetime, weighted by the probability that the intermediate state is still populated at the given moment when it is returned to the ground state. Thus the classical equivalent of the desorption/bond-breaking probability per resonance event is taken as

1 PD (τR ) = τR

∞

d(t) exp(−t/τR )Pdes (t),

(1.28)

0

where an exponential survival probability, consistent with Lorentzian lineshapes, has been adopted. This averaging procedure has proven to be surprisingly useful and reliable in a wide variety of surface dynamics applications (Gadzuk, 1988; Saalfrank, 1996; Finger and Saalfrank, 1997; Guo et al., 1999; Bach et al., 2003). Invoking the periodic top-hat form for Pdes (t), the integral in Eq. (1.28) is straightforward and the resulting

24

J.W. Gadzuk

Fig. 1.9. Resonance-enhanced desorption probability per electron capture event versus resonance lifetime (in units of Tvibe ), as given by Eq. (1.29), for selected values of τon , hence z, as labeled.

expression for the desorption yield is sinh([T − 2ton (z)]/2τR ) , PD (τR ) = sinh(T /2τR )

(1.29)

a very compact analytic synopsis of resonant desorption dynamics as envisioned within the displaced HO paradigm (Gadzuk, 1995). This solution clearly relates the desorption yield to the three relevant time scales T , τR , and ton (or alternatively, two time scales T and τR and the force upon the negative ion for which ton (z) is a measure) in a physically significant way. The numerical implications of Eq. (1.29) are shown in Fig. 1.9 in the form of PD (τR ) versus 2πτR /T (or equivalently, ω0 τR ) plots for various values of ton (z), as labeled. The general form of these results unambiguously validate intuitive expectations. An initial increase in the bond-breaking/desorption yield is obtained as the resonance lifetime increases, saturating at a maximum value set by ton , with smaller ton signifying longer interaction time on Vexc hence greater excitation probability and thus desorption yield, all other things being equal. This simple mechanical picture (Gadzuk, 1996) should be useful in providing thumbnail sketches of other molecular realizations beyond the present.

1.7. Final thoughts This short introduction to some of the fundamental atomic-scale issues/processes pertinent to dynamics at surfaces has provided a window mainly into the theoretical basis of surface processes involving both electronic and nuclear transitions/rearrangements. This is a combined reflection of the author’s prejudices and/or the importance of the limited set of examples chosen to analytically illustrate in as simple and general a way as possible what the issues of surface dynamics are all about. As announced in the first section, it would be impossible to present a truly representative, uniform, and fair overview of surface dynamics within the limited confines of a single chapter in this volume. In fact, that is the purpose of this volume in its entirety. In addition, many books, proceedings, dedicated (both to people and to topics) journal issues, and reviews in serial publications provide excellent entries into the field.


25

There are a few things missing from this chapter that should be mentioned. Nothing of substance has been said about numerical simulations, molecular dynamics, large scale electronic structure calculations, or (can you believe it?) density functional theory (Kohn, 1999). This omission is not accidental. For the most part the role of DFT and electronic structure calculation is to provide the force fields/PES and “realistic” interactions that are inputs into the dynamics models. Tully (2000a, 2000b) has spoken directly to this: The Born–Oppenheimer approximation separates the theoretical study of molecules into two parts, the electronic structure part. . . and the nuclear motion part. This has produced a separation among theoretical chemists themselves: the electronic structure theorists and the statistical mechanics/dynamics theorists, i.e., those that compute εj (R ) and those that ∼ use it. While this is an overstatement and there have been many theorists with a foot in each camp, this bifurcation has been unhealthy for chemistry. Theoretical chemists of the future will need to be expert in both electronic structure and statistical mechanics/dynamics. Finally, rather than providing a too-long list of references, it is suggested that the reader consult an electronic data-base such as the Web of Science for the purpose of accessing the publication lists of principle players in the development of theoretical surface dynamics. I have benefited hugely from the wisdom, both personal and written, of many sustained contributors to surface dynamics and heartily recommend a visit to their publications list and then their publications. An imperfect but useful “Theoretical Family Tree” would include: Patriarchs: Steve Adelman, Wilhelm Brenig, Jimmy Doll, Peter Feibelman, Bill Gadzuk, Tom Grimley, Olle Gunnarsson, John Harris, Walter Kohn, Norton Lang, David Langreth, Bengt Lundqvist, Dennis Newns, Kurt Schönhammer, Marijan Šunjić, John Tully. Offspring: Pedro Echenique, Axel Groß, Steve Holloway, Bret Jackson, Peter Nordlander, Jens Nørskov, Bo Persson, Mats Persson. Computational: Don Hamann, Uzi Landman, John Pendry, Mathias Scheffler. Chemical physics: Jean-Pierre Gauyacq, Martin Head-Gordon, Eric Heller, Ronnie Kosloff, Horia Metiu, Abe Nitzan, Peter Saalfrank.

References Adelman, S.A., Garrison, B.J., 1976. J. Chem. Phys. 65, 3751. Adelman, S.A., Doll, J.D., 1977. Acct. Chem. Res. 10, 378. Adelman, S.A., 1979. J. Chem. Phys. 71, 4471. Anderson, P.W., 1961. Phys. Rev. 124, 41. Anderson, P.W., 1967. Phys. Rev. Lett. 18, 1049. Asscher, M., Zeiri, Y., 2003. J. Phys. Chem. B 107, 6903. Bach, C., Klüner, T., Groß, A., 2003. Appl. Phys. A 78, 231. Bartels, L., Meyer, G., Rieder, K.-H., Velic, D., Knoesel, E., Hotzel, A., Wolf, M., Ertl, G., 1998. Phys. Rev. Lett. 80, 2004. Barth, J.V., Costantini, G.C., Kern, K., 2005. Nature 437, 671. Bauer, E., Fisher, E.R., Gilmore, F.R., 1969. J. Chem. Phys. 51, 4173. Benderskii, V.A., Makarov, D.E., Wight, C.A., 1994. Adv. Chem. Phys. 138, 1. Billing, G.D., 1999. Dynamics of Molecule–Surface Interactions. Wiley, New York. Birtwistle, D.T., Herzenberg, A., 1971. J. Phys. B 4, 53.

26

J.W. Gadzuk

Blandin, A., Nourtier, A., Hone, D.W., 1976. J. Phys. (Paris) 37, 369. Blatt, J.M., Weisskopf, V.F., 1952. Theoretical Nuclear Physics. Wiley, New York. pp. 311–457. Bönig, L., Schönhammer, K., 1989. Phys. Rev. B 39, 7413. Borisov, A.G., Kazansky, A.K., Gauyacq, J.P., 1998. Phys. Rev. Lett. 80, 1996. Brako, R., Newns, D.M., 1980. Solid State Comm. 33, 633. Brako, R., Newns, D.M., 1981. J. Phys. C: Solid State Phys. 14, 3065. Brako, R., Newns, D.M., 1989. Rep. Prog. Phys. 52, 655. Brandbyge, M., Hedegård, P., Heinz, T.F., Misewich, J.A., Newns, D.M., 1995. Phys. Rev. B 52, 6042. Brivio, G.P., Grimley, T.B., 1993. Surf. Sci. Rep. 17, 1. Burke, K., Gumhalter, B., Langreth, D.C., 1993. Phys. Rev. B 47, 12852. Cheng, H., Shenvi, N., Tully, J.C., 2007. Phys. Rev. Lett. 99, 053201. Chulkov, E.V., Borisov, A.G., Gauyacq, J.P., Sánchez-Portal, D., Silkin, V.M., Zhukov, V.P., Echenique, P.M., 2006. Chem. Rev. 106, 4160. d’Agliano, E.G., Kumar, P., Schaich, W., Suhl, H., 1975. Phys. Rev. B 11, 2122. Cini, M., D’Andrea, A., 1988. J. Phys. C: Solid State Phys. 21, 193. Dai, H.-L., Ho, W. (Eds.), 1995. Laser Spectroscopy and Photo-Chemistry on Metal Surfaces, Part I and II. World Scientific, Singapore. De Pristo, A.E., 1983. J. Chem. Phys. 78, 1237. DePristo, A.E., Kara, A., 1990. Adv. Chem. Phys. 77, 163. Domcke, W., Cederbaum, L.S., 1980. J. Phys. B 13, 2829. Ertl, G., 2000. Adv. Catalysis 45, 1. Fano, U., 1961. Phys. Rev. 124, 1866. Finger, K., Saalfrank, P., 1997. Chem. Phys. Lett. 268, 291. Frischkorn, C., Wolf, M., 2006. Chem. Rev. 106, 4207. Gadzuk, J.W., 1970. Phys. Rev. B 1, 2110. Gadzuk, J.W., Šunjić, M., 1975. Phys. Rev. B 12, 524. Gadzuk, J.W., Metiu, H., 1980. Phys. Rev. B 22, 2603. Gadzuk, J.W., 1981. Phys. Rev. B 24, 1866. Gadzuk, J.W., 1982. Surf. Sci. 118, 180. Gadzuk, J.W., Nørskov, J.K., 1984. J. Chem. Phys. 81, 2828. Gadzuk, J.W., Holloway, S., 1985. Phys. Scripta 32, 413. Gadzuk, J.W., 1988. Ann. Rev. Phys. Chem. 39, 395. Gadzuk, J.W., 1991. Phys. Rev. B 44, 13466. Gadzuk, J.W., 1995. Surf. Sci. 342, 345. Gadzuk, J.W., 1996. Phys. Rev. Lett. 76, 4234. Gadzuk, J.W., 2002. J. Phys. Chem. B 106, 8265. Gadzuk, J.W., Plihal, M., 2003. Phys. Rev. B 68, 235413. Gadzuk, J.W., 2006. Phys. Rev. B 73, 085401. Galperin, M., Nitzan, A., Ratner, M.A., 2006. Phys. Rev. B 73, 045314. Gao, S., Persson, M., Lundqvist, B.I., 1997. Phys. Rev. B 55, 4825. Gauyacq, J.P., Borisov, A.G., Bauer, M., 2007. Prog. Surf. Sci. 82, 244. Gazdy, B., Sun, Q., Bowman, J.M., 1987. J. Chem. Phys. 87, 6618. Gentry, W.R., 1979. Vibrational excitation II: Classical and semiclassical methods. In: Bernstein, R.B. (Ed.), Atom–Molecule Collision Theory. Plenum, New York, pp. 391–425. Gergen, B., Weyers, S.J., Nienhaus, H., Weinberg, W.H., McFarland, E.W., 2001. Science 294, 2521. Greber, T., 1997. Surf. Sci. Rep. 28, 1. Groß, A., 2003. Theoretical Surface Science: A Microscopic Perspective. Springer, Berlin. Gumhalter, B., Newns, D.M., 1975. Phys. Lett. 53A, 137. Gunnarsson, O., Schönhammer, K., 1982. Phys. Rev. B 25, 2503. Gunnarsson, O., Schönhammer, K., 1985. Aspects of core level X-ray photoemission for large systems. In: Langreth, D.L., Suhl, H. (Eds.), Many-Body Phenomena at Surfaces. Academic, Orlando, pp. 221–238. Guo, H., Saalfrank, P., Seideman, T., 1999. Prog. Surf. Sci. 62, 239. Hänggi, P., Talkner, P., Borkovec, M., 1990. Rev. Mod. Phys. 62, 251. Harris, 1987. Phys. Scripta 36, 156.


27

Head-Gordon, M., Tully, J.C., 1992. J. Chem. Phys. 96, 3939. Head-Gordon, M., Tully, J.C., 1995. J. Chem. Phys. 103, 10137. Heller, E.J., 1981. Acc. Chem. Res. 14, 368. Heller, E.J., Sundberg, R.L., Tannor, D., 1982. J. Phys. Chem. 86, 1822. Hellsing, B., Persson, M., 1984. Phys. Scripta 29, 360. Hla, W., Rieder, K.-H., 2003. Ann. Rev. Phys. Chem. 54, 307. Hoffman, R., 1988. Rev. Mod. Phys. 60, 601. Jackson, B., 1988. J. Chem. Phys. 88, 1383. Jackson, B., 1991. J. Chem. Phys. 94, 5126. Jackson, B., 1993. J. Chem. Phys. 98, 9905. Jacobs, D.C., 1995. J. Phys.: Condens. Matter 7, 1023. Joachim, C., Ratner, M.A., 2005. Proc. Nat. Acad. Sci. 102, 8801. Kleyn, A.W., 1992. J. Phys.: Condens. Matter 4, 8375. Kock, C.P., Klüner, T., Kosloff, R., 2002. J. Chem. Phys. 116, 7983. Kohn, W., 1999. Rev. Mod. Phys. 71, 1253. Kroes, G.-J., Gross, A., Baerends, E.-J., Scheffler, M., McCormack, D.A., 2002. Acc. Chem. Res. 35, 193. Lang, N.D., Nørskov, J.K., 1983. Phys. Scripta T6, 15. Langreth, D.C., 1966. Phys. Rev. 150, 516. Langreth, D.C., 1970. Phys. Rev. B 1, 471. Langreth, D.C., 1984. Some elementary theoretical concepts in core-hole spectroscopy. In: Langreth, D.L., Suhl, H. (Eds.), Many-Body Phenomena at Surfaces. Academic, Orlando, pp. 191–220. Leung, K.M., Schön, G., Rudolph, P., Metiu, H., 1984. J. Chem. Phys. 81, 3307. Li, X.L., Ford, G.W., O’Connell, R.F., 1993. Am. J. Phys. 61, 924. Li, X., Tully, J.C., Schlegel, H.B., Frisch, M.J., 2005. J. Chem. Phys. 123, 084106. Liebsch, A., 1997a. Electronic Excitations at Metal Surfaces. Plenum, New York. Liebsch, A., 1997b. Phys. Rev. B 55, 13263. Los, J., Geerlings, J.J.C., 1990. Phys. Repts. 190, 133. Lucas, A.A., Šunjić, M., 1972. Prog. Surf. Sci. 2, 75. Lundqvist, B.I., Gunnarsson, O., Hjelmberg, H., Nørskov, J.K., 1979. Surf. Sci. 89, 196. Lundqvist, B.I., 1996. Electron transfer processes at surfaces. In: Ilisca, E., Makoshi, H. (Eds.), Electronic Processes at Solid Surfaces. World Scientific, Singapore, pp. 1–40. Mahan, G.D., 1974. Solid State Phys. 29, 75. Mahan, G.D., 1990. Many-Particle Physics. Plenum, New York. Manz, J., Wöste, L. (Eds.), 1995. Femtosecond Chemistry, vols. 1 and 2. VCH, Weinheim. Marcus, R.A., 1993. Rev. Mod. Phys. 65, 599. Marinica, D.C., Ramseyer, C., Teillet-Billy, D., Gauyacq, J.P., 2006. Surf. Sci. 600, 803. Marion, J.B., 1965. Classical Dynamics of Particles and Systems. Academic, New York. pp. 128–164. Marston, J.B., Anderson, D.R., Behringer, E.R., Cooper, B.H., DiRubio, C.A., Kimmel, G.A., Richardson, C., 1993. Phys. Rev. B 48, 7809. Mayne, A.J., Dujardin, G., Comtet, G., Riedel, D., 2006. Chem. Rev. 106, 4355. Metiu, H., Ross, J., Whitesides, G.M., 1979. Angew. Chem. (Int. Ed. Engl.) 18, 377. Metiu, H., Gadzuk, J.W., 1981. J. Chem. Phys. 74, 2641. Moiseyev, N., Scheit, S., Cederbaum, L.S., 2004. J. Chem. Phys. 121, 722. Müller-Hartmann, E., Ramakrishnan, T.V., Toulouse, G., 1971a. Phys. Rev. B 3, 1102. Müller-Hartmann, E., Ramakrishnan, T.V., Toulouse, G., 1971b. Solid State Comm. 9, 99. Nakamura, H., 2006. J. Phys. Chem. A 110, 10929. Newns, D.M., Heinz, T.F., Misewich, J.A., Brandbyge, M., Hedegård, P., 1996. Electron transfer processes at surfaces. In: Ilisca, E., Makoshi, H. (Eds.), Electronic Processes at Solid Surfaces. World Scientific, Singapore, pp. 41–58. Nikitin, E.E., 1974. Theory of Elementary Atomic and Molecular Processes in Gases. Clarendon, Oxford. pp. 52–57. Nikitin, E.E., Zülicke, L., 1978. Selected Topics of the Theory of Chemical Elementary Processes. SpringerVerlag, Berlin. Nienhaus, H., 2002. Surf. Sci. Rep. 45, 1.

28

J.W. Gadzuk

Nørskov, J.K., Lundqvist, B.I., 1979. Surf. Sci. 89, 251. Nørskov, J.K., 1981. J. Vac. Sci. Technol. 18 (2), 420. Nourtier, A., 1977. J. Phys. (Paris) 38, 479. Nozières, P., de Dominicis, C.T., 1969. Phys. Rev. 178, 1097. O’Malley, T.F., 1971. Adv. At. Mol. Phys. 7, 233. Ohtaka, K., Tanabe, Y., 1990. Rev. Mod. Phys. 62, 929. Otero, R., Rosei, F., Besenbacher, F., 2006. Ann. Rev. Phys. Chem. 57, 497. Palmer, R.E., Rous, P.J., 1992. Rev. Mod. Phys. 64, 383. Paramonov, G.K., Saalfrank, P., 1999. Chem. Phys. Lett. 301, 509. Pearson, R.G., 1971. Acct. Chem. Res. 4, 152. Persson, B.N.J., Persson, M., 1980. Solid State Comm. 36, 175. Persson, B.N.J., 1993. Phys. Rev. B 48, 18143. Persson, M., Harris, J., 1987. Surf. Sci. 187, 67. Plihal, M., Langreth, D.C., 1999. Phys. Rev. B 60, 5969. Qian, J., Jacobs, D.C., Tannor, D.J., 1995. J. Chem. Phys. 103, 10764. Rettner, C.T., Auerbach, D.J., Tully, J.C., Kleyn, A.W., 1996. J. Phys. Chem. 100, 13021. Rous, P.J., 1999. Phys. Rev. Lett. 83, 5086. Saalfrank, P., 1996. Chem. Phys. 211, 265. Saalfrank, P., 2006. Chem. Rev. 106, 4116. Salem, L., 1982. Electrons in Chemical Reactions: First Principles. Wiley, New York. Sawada, S.I., Nitzan, A., Metiu, H., 1985. Phys. Rev. B 32, 851. Schaich, W.L., 1975. Surf. Sci. 49, 221. Schönhammer, K., Gunnarsson, O., 1980. Phys. Rev. B 22, 1629. Schönhammer, K., Gunnarsson, O., 1981. Phys. Rev. B 24, 7084. Schönhammer, K., Gunnarsson, O., 1983. Phys. Rev. B 27, 5113. Schönhammer, K., Gunnarsson, O., 1984. Energy dissipation at metal surfaces: The electron hole-pair mechanism. In: Langreth, D.L., Suhl, H. (Eds.), Many-Body Phenomena at Surfaces. Academic, Orlando, pp. 421– 451. Schönhammer, K., 1991. Phys. Rev. B 43, 11323. Seideman, T., 2003. J. Phys.: Condens. Matter 15, R521. Shenvi, N., Cheng, H., Tully, J.C., 2006. Phys. Rev. A 74, 062902. Sholl, D.S., Tully, J.C., 1998. J. Chem. Phys. 109, 7702. Tannor, D.J., Rice, S.A., 1988. Adv. Chem. Phys. 70, 441. Tannor, D.J., 2007. Introduction to Quantum Mechanics: A Time-Dependent Perspective. University Science Books, Sausolito. Trail, J.R., Graham, M.C., Bird, D.M., Persson, M., Holloway, S., 2002. Phys. Rev. Lett. 88, 166802. Trail, J.R., Bird, D.M., Persson, M., Holloway, S., 2003. J. Chem. Phys. 119, 4539. Tully, J.C., Preston, R.K., 1971. J. Chem. Phys. 76, 2412. Tully, J.C., 1990. J. Chem. Phys. 93, 1061. Tully, J.C., 2000a. Annu. Rev. Phys. Chem. 51, 153. Tully, J.C., 2000b. Theor. Chem. Acc. 103, 173. Ueba, H., Persson, B.N.J., 2004. Surf. Sci. 566, 1. van Wunnik, J.N.M., Los, J., 1983. Phys. Scripta T6, 27. Wigner, E.P., 1946. Phys. Rev. 70, 606. Wigner, E.P., 1955. Am. J. Phys. 23, 371. Wingreen, N.S., Jacobsen, K.W., Wilkins, J.W., 1989. Phys. Rev. B 40, 11834. Wodtke, J., Tully, C., Auerbach, D.J., 2004. Int. Rev. Phys. Chem. 23, 513. Wodtke, A.M., Matsiev, D., Auerbach, D.J., 2008. Prog. Surf. Sci. 83, 167. Wyatt, R.E., Zhang, J.Z.H. (Eds.), 1996. Dynamics of Molecules and Chemical Reactions. Marcel Dekker, New York. Zare, R.N., 1994. Science 266, 215. Zewail, A.H., 2000. J. Phys. Chem. A 104, 5660.

CHAPTER 2

Basic Mechanisms in Atom–Surface Interactions

Aart W. KLEYN FOM Institute for Plasma Physics Rijnhuizen Euratom-FOM Association P.O. Box 1207, 3430 BE Nieuwegein The Netherlands E-mail: [email protected] Leiden Institute of Chemistry Gorlaeus Laboratories Leiden University P.O. Box 9502, 2300 RA Leiden The Netherlands



Contents 2.1. 2.2. 2.3. 2.4.

Introduction . . . . . . . . . . . . . . . . . . . . Need for simple models . . . . . . . . . . . . . Interaction potentials . . . . . . . . . . . . . . . Limiting cases in the scattering dynamics . . . . 2.4.1. Full molecular dynamics . . . . . . . . . 2.4.2. Hard cube model . . . . . . . . . . . . . 2.4.3. Hard sphere scattering . . . . . . . . . . 2.4.4. Rainbow scattering . . . . . . . . . . . . 2.4.5. Washboard model . . . . . . . . . . . . . 2.4.6. Different surface regions . . . . . . . . . 2.5. Limiting cases in adsorption dynamics . . . . . 2.5.1. Cube models . . . . . . . . . . . . . . . . 2.5.2. Displacement trapping and implantation 2.6. Conclusion . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

30

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

31 31 31 34 34 35 36 38 41 44 45 45 49 50 51

31

2.1. Introduction Detailed calculations and analysis do not always provide further insight. In contrast, simplified models can deepen one’s insight into the underlying physics of a problem, even though the numerical agreement between a calculation using a simple model and experiment may not be perfect. In this chapter, I will introduce a number of simple models to describe the dynamics of gas–surface scattering and adsorption. My introduction will be based upon lectures given on the topic over the years. The survey of the literature may be incomplete. The work discussed mainly concerns studies of scattering and adsorption at low index metal surfaces, and usually involving projectiles that are lighter than the individual surface atoms; typically atoms with mass 20–40 hitting surfaces consisting of atoms with mass 100–200. The energy range is between thermal to hyperthermal, with energies up to 100 eV. The models will be confined to atom scattering. In some cases scattering of molecules is considered as well, but for a review on the role of the molecular degrees of freedom, such as rotation and vibration, the reader is referred to other chapters in this volume. The focus of this chapter will be completely on simple models, experimental methods will be ignored. Most of the experiments are carried out using molecular beam methods (Scoles et al., 1988a, 1988b; Kleyn, 2003a, 2003b; Campargue, 2001; Weaver et al., 2003; Barker and Auerbach, 1984; Vattuone et al., 2003; McClure et al., 2003; Hodgson, 2003; Hayden and Mormiche, 2003; Arumainayagam and Madix, 1991; Rettner et al., 1996). 2.2. Need for simple models In spite of enormous advances in computational power ‘complete’ modelling of gas– surface interactions remains very difficult. Ideally, the forces between the interacting atoms can be computed using quantum chemical methods. Assuming that electronically inelastic effects do not occur, in other words that the Born–Opperheimer approximation is applicable, the interaction dynamics can be computed by full molecular dynamics methods, or quantum mechanical methods. In the case of classical methods most aspects of the physical interaction can be included, obviously excluding quantization. In the case of quantum calculations, if surface motion effects and electronic excitations can be discarded, a full description can be obtained. A very recent example for a ‘complete’ description of the hydrogen platinum system is given by Nieto et al. (2006). However, in many cases such a complete picture cannot be provided. In addition, it is often very hard to extract a good picture of the relevant physical mechanisms from a large and ‘complete’ calculation. To sharpen one’s physical intuition and get an idea about which mechanisms dominate the physics of the interaction it is useful to develop simple models. In fact, several of these models to be discussed below were developed at the time when high performance computation was not available, but their validity at the conceptual level remains. 2.3. Interaction potentials To derive simple models we start with the assumption that for most aspects of the interaction classical mechanics can provide an adequate description. In addition, we as-

32

A.W. Kleyn

sume the Born–Oppenheimer approximation to be valid; i.e., electronically inelastic effects are neglected, and the electrons can be considered to be infinitely faster than the heavy nuclei. In this case all effects due to the presence of the electrons can be contained in the interaction potential between the interacting nuclei (Gross, 2003; Kroes, 1999; Kroes et al., 2002). These potentials can be computed using the methods of quantum chemistry and the result of such a calculation is shown in Fig. 2.1 (Kirchner et al., 1994). Shown is the interaction potential between an Ar atom and a cluster of Ag atoms. Two cases are shown; in the left panel the interaction for an Ar approaching a Ag atom in the cluster atop. Several types of calculations are shown: clusters of different sizes, and a slab calculation. The calculations have been done for fixed geometries, described in the original paper. An actual calculation of the dynamics involves many more geometries than for which the quantum chemical calculations have been performed, and the computed potential needs to be parameterized for use in a dynamics calculation. Several approximations to these potentials can be envisaged, and the ones tested here start with a summation over pair potentials. To account for more complex three and multi-body interactions correction terms can be added. In the case of the work of Fig. 2.1, these were rather simple. More complex potentials as input to this summation might include Brenner-type potentials, which allow for bond making and bond breaking, which is outside the scope of this article (Brenner, 1990; Graves and Humbird, 2002; Gou et al., 2006). In the left panel of Fig. 2.1, the potential can be approximated very well by a summation of a simple two body potential, where in addition only one term between the Ar and the atop Ag-atom is relevant. In the right hand panel of Fig. 2.1, the situation for approach of the threefold hollow site is depicted. Now there is not a single dominating interaction and at least 3 Ag atoms contribute to the interaction. Describing this interaction by a summation over 3 pair potentials is not satisfactory and correction terms are needed. Nevertheless, pair potentials give again a starting point. To get a better feel for the interaction mechanisms such quantum chemical potentials should be plotted in a different fashion, shown in Fig. 2.2 (Lahaye et al., 1995). Here results are shown for the interaction between Ar and a Ag(111) surface, now showing contours of equal potential energy as a function of the 2 dimensional Ar position. The potential is shown along two different azimuthal directions of the Ag(111) crystal, the close packed [1, 0, −1] direction and the more open [1, 1, −2] direction. The atomic plane containing the Ag-atoms is located at height z = 0 Å. Above the surface a weak physisorption well at −0.1 eV can be discerned for both azimuths. The zero energy contour is almost flat. For increasing repulsive energy the potential turns more corrugated and at 500 eV the interaction around each Ag atom becomes spherically symmetric. Also the onset for penetration of the crystal below 100 eV can be seen. In the [1, 1, −2] direction the same features can be seen and in this cases also the influence of Ag-atoms that are outside the plane of the drawing, for instance at x = 2.5 Å. From these pictures the limiting cases to be discussed below already emerge: the flat surface or cube model, the corrugated or washboard model, and the rainbow scattering model.

Basic Mechanisms in Atom–Surface Interactions Fig. 2.1. Calculated potential between an Ar atom and a silver surface for the Ar approaching an on-top site of the silver (left panel) and approaching a three fold hollow site. The symbols and lines represent various calculations using Ag-clusters or slabs, and various approximations of the results using pair wise additive potentials. The results show that pair wise addition works very well in the repulsive region. It fails for the three fold hollow site in the attractive region and a more refined parameterization of the potential is needed. From Kirchner et al. (1994).

33

34

A.W. Kleyn

Fig. 2.2. The upper two panels show the contour plots of the Ar–Ag(111) potential energy surface along two azimuthal orientations of the surface. The X-axis displays the lateral displacement along the direction shown above the figure. Z is the height above the plane formed by the Ag-nuclei. Distances are in Å and the contour lines in electron volts. The shaded dots represent the surface atoms. At the bottom of the figure the surface unit cell is drawn to indicate the two azimuthal orientations. From Lahaye et al. (1995).

2.4. Limiting cases in the scattering dynamics 2.4.1. Full molecular dynamics Quantum effects in gas–surface dynamics are constrained to a few cases. In case of scattering of very light particles such as H, H2 , or He diffraction or quantization of the scattering occurs. In case of molecular scattering the internal degrees of freedom are quantized. All electronic excitation of both lattice and projectile need a quantal description. Finally, in case of activated processes quantum mechanical tunnelling can occur. In many cases by averaging over initial and final states quantum effects are washed out. As a consequence,


35

classical dynamics often give a very good description of gas–surface interactions. The most extensive way of carrying out such calculations is to use molecular dynamics calculations. In this method, the Newtonian or Hamilton’s equations of motion are integrated numerically for specific trajectories, and by integration over initial conditions of many trajectories experimentally observable quantities can be obtained. From the calculations the underlying physics can be recovered, but this is often a tedious process of analyzing many individual trajectories, for instance, by viewing movies of these trajectories. Therefore, it remains insightful to derive simple models and check to what extent the results of full molecular dynamics calculations can be interpreted in terms of the simple models. 2.4.2. Hard cube model Figure 2.2 clearly demonstrates that at low energies the interaction potential can be almost completely flat. This strongly suggests, that parallel momentum in a particle– surface collision is approximately conserved, whereas the normal component of the momentum interacts with a potential normal to the surface, that is composed of a number of specific particle–target atom interactions. Thus a very simple model suggests itself: – Parallel momentum is conserved. – Normal momentum is reversed, observing conservation of momentum and energy.

Fig. 2.3. Schematic diagrams of prototypes of gas–surface interactions as can be probed by molecular beams, presented as side views of the surface atoms or cubes. (A) molecular scattering in which parallel momentum is conserved and the surface is represented by hard cubes. (B) molecular scattering from individual surface atoms. From Bonn et al. (2002).

36

A.W. Kleyn

The situation is depicted in the upper panel of Fig. 2.3. The model, originally by Yamamoto and Stickney, has been reviewed extensively (Yamamoto and Stickney, 1970; Logan, 1973; Goodman and Wachman, 1976; Grimmelmann et al., 1980). Assuming that the target ‘cube’ with effective mass M is stationary before the collision, conservation of energy and momentum imply: E =

4μE (1 + μ)2

with μ =

m , M

where E is the energy transferred from a particle with mass m incident with energy E on a cube with mass M. The conservation of parallel momentum implies that the scattering angle is completely determined by the normal momentum transfer. Transfer from the surface to the atom will result in scattering away from the specular angle and towards the surface normal and viceversa. Introducing thermal motion to the moving cube will allow such energy transfer. One expects that atoms scattered towards the surface normal have gained energy, and ones scattered towards the surface have lost energy. This has been observed for many systems, and an example is shown in Fig. 2.4. Here angular distributions and average final energies are shown for Ar scattering from Ag(111). The line for parallel momentum conservation is plotted in the upper panels, and it seen to be observed quite well for the lowest energy collisions. At higher energies, the parallel momentum conservation is no longer observed. This region will be discussed later. 2.4.3. Hard sphere scattering The high energy limit of Fig. 2.2 shows circular potential energy contours. This implies that essentially the scattering dynamics is reduced to repulsive binary collisions, as is depicted in the lower panel of Fig. 2.3. All such binary collision cross sections decrease monotonously as a function of scattering angle. However, the strictly forward, small angle scattering is not possible due to the presence of other atoms in the surface or the bulk of the solid. Hence, the scattering cross section cannot easily be retrieved from the data. However, a dominant characteristic of the energy transfer can be observed. In binary collisions, there is again a one to one relationship between scattering angle and energy transfer, as in the case of the cube model:

2 E 1 2 − sin2 (θ ) , 1/μ = 1− cos(θ ) ± E (1 + 1/μ)2 with θ the total deflection angle. The + and − signs refer to the two different collisions possible leading to the same total scattering angle but different energy transfer. The minus sign is only relevant in case of strong back scattering in the centre of mass system for the colliding partners, and is usually irrelevant in atom–surface scattering. In this case the energy transfer shows a dependence contrary to that of the cube model: the energy transfer increases with the total scattering angle. This trend can clearly be seen in Fig. 2.4 in the upper panels (Lahaye et al., 1995). Although at 1 eV the potential is far from binary, the trend in the energy loss as a function of angle is reversed, but the energy transfer is much less than for a binary collision between Ar and an Ag atom. For scattering


37

Fig. 2.4. Results from classical trajectory calculations and scattering experiments for in-plane scattering of Ar from Ag(111) with an incidence angle of 40◦ measured with respect to the surface normal. The studies have been carried out for Ei = 0.2, 0.5 and 2.6 eV. In the upper panels results for the relative final energy Ef /Ei are shown, where Ei is the initial energy. Lines indicate the energy transfer computed with the cube model (parallel momentum conservation) and a binary collision model. In the lower panels angular distributions are shown. The left panels are calculated for a zero temperature, initially static lattice; the right panels for Ts = 600 K. From Lahaye et al. (1995).

38

A.W. Kleyn

between O2 and a Ag(111) surface, the binary collision limit is seen in the energy transfer (Raukema et al., 1995). 2.4.4. Rainbow scattering In Fig. 2.2 the potential energy contours shift from flat, via somewhat sinusoidal to circular. The sinusoidal structure allows another simple approximation to the calculation of the scattering dynamics: rainbow scattering. Assuming particles normally incident on a one-dimensional sinusoidal hard wall, having infinite mass, the scattering pattern can be easily estimated. Scattering implies simple specular reflection around the local surface normal. As a consequence, particles incident on the tops or valleys of the sinusoidal corrugation will be completely backwards reflected. Particles incident at the inflection point of the sine wave, will undergo the maximal deflection. Illustrative diagrams can be found elsewhere (Boyer, 1987; Kleyn, 1987; Kleyn and Horn, 1991; Winter and Schuller, 2005). Since trajectories incident around the inflection point, all will give rise to the same, extremal scattering angle Θr , the classical scattering cross section around that scattering angle Θr will exhibit a singularity. This is referred to as rainbow scattering. Detailed accounts of rainbow scattering are given elsewhere, but results for scattering calculations for Ar reflection from Ag(111) show the effect (Lahaye et al., 1995). In Fig. 2.5 the rainbow structure is clearly seen in the calculation for 1 eV and a static surface. The peaks at 35◦ and 55◦ are surface rainbow peaks. At 10 eV these two surface rainbow peaks move to a larger separation, due to an increase of the corrugation at 10 eV and the centre peak also demonstrates two rainbow features. At 100 eV penetration of the surface starts to occur, and some of the rainbows disappear, because the rainbow trajectories lead perhaps to implantation. However, one can ask what is the relevance of the rainbow scattering in the case of Fig. 2.5, because at Ts = 600 K the rainbow structure is washed out. The range of impact point at the surface leading to the rainbow scattering is small, and the thermal roughening of the surface at 600 K effectively inhibits the sharp focusing into rainbow scattering. For rainbow scattering to be observed, there may be two options: take a surface for which the thermal vibration is less severe, and/or increase the energy to better resolve the rainbows. The former has been done for a number of systems, see e.g. Kleyn and Horn (1991), Schweizer et al. (1991), Kondo et al. (2001). The latter has been done by scattering experiments at hyperthermal energies. An example is shown in Fig. 2.6. The system studied is K+ scattering at an energy of 35 eV and normal incidence to a W(110) surface (Tenner et al., 1984, 1986a, 1986b). Shown are the scattered intensity as a function of final scattering angle and final energy for two different orientations of the detection plane, as shown in the figure. Along the more open [1, −1, 0] azimuth shown in the upper panel, a single peak is shown, both for energy and final angle. This intense peak is a surface rainbow peak. The corresponding rainbow trajectory is shown in Fig. 2.7 as number I (Kleyn and Horn, 1991). Its high symmetry counterpart is trajectory VII, for which the final azimuthal angle has changed by 180◦ . It is seen that one surface atom deflects the incoming K+ along the [1, −1, 0] direction, where it just does not hit another surface atom. This gives rise to an extremum in the scattering angle. Changing the impact parameter of trajectory I both towards or away from the second surface atom will decrease the scattering angle. Along the more compact [0, 0, 1] azimuth more structure is seen. The peak labelled b in Fig. 2.6


39

Fig. 2.5. Simulations for in-plane scattering of Ar from Ag(111) with an incidence angle of 40◦ , measured with respect to the surface normal. The panels a and b display results for an initially non-vibrating lattice, c and d for a lattice at 600 K. Calculations for 0.1, 1, 10 and 100 eV are shown. From Lahaye et al. (1995).

is the analogue of the rainbow peak of the upper panel, as is evident from trajectory III. However, two more features are seen. Peak a is due to a collision hitting a surface atom such that it is deflected towards its nearby neighbour. In the collision with this neighbour

40

A.W. Kleyn

Fig. 2.6. Experimental results of K scattering from W(110) at normal incidence and 35 eV. The scattered intensity is shown as a function of the final scattering angle, measured from the surface normal, and the final energy E . Both a 3D and a contour representation of the data are shown. The results are shown for two different azimuthal orientations ([1, −1, 0] top, [001] bottom) of the crystal, as shown in the insets. From Tenner et al. (1986a).

the parallel momentum is reversed and it is scattered back to the first surface atom in the collision sequence, as shown in trajectory V. The trajectory is such, that again an extremum in the scattering angle occurs. In fact, this second rainbow peak is seen in the static surface panel Fig. 2.5 as one of the two inner rainbows near 45◦ and 10 eV. The energy of rainbow peak a is much lower than that of peak b. This is easy to understand as in the corresponding trajectory V two fairly hard collisions with considerable energy transfer occur, in contrast to trajectory I, that shows essentially only a single collision. A third peak is seen as peak c in Fig. 2.6. Contrary to the trajectories discussed so far, this peak is due to a trajectory that is not confined to a single scattering plane, but only exists asymptotically along the same scattering plane, as can be seen in trajectory II. The scattering patterns observed in Fig. 2.6 could be very well reproduced in molecular dynamics simulations, from which the nature of the rainbows could be resolved, and from which a detailed interaction potential has been derived. This study demonstrated that rainbow scattering is a very sensitive


41

Fig. 2.7. 3-dimensional picture of a few representative trajectories of K scattering from W(110) at 35 eV and the equipotential surface of 35 eV. The numbers refer to the impact regions discussed in the text. From Tenner et al. (1986b).

tool for unravelling scattering dynamics. The use of rainbow scattering for this purpose has been discussed and reviewed at several places in the literature (Kleyn and Horn, 1991; Winter and Schuller, 2005; Kleyn, 1994; Pfandzelter et al., 1998; Schuller et al., 2005). 2.4.5. Washboard model The rainbow scattering shown in Figs. 2.5 and 2.6 has been computed by full molecular dynamics simulations. However, since the simplest models of rainbow scattering involve only a hard corrugation function, the rainbow scattering model can be connected to the hard cube model discussed earlier. This is done in the washboard model, introduced by Tully (Tully, 1990; Yan et al., 2004). The idea is that corrugation is added to the cube model, and that the idea of an effective mass of the surface is retained. The resulting model is quite simple, and is depicted in Fig. 2.8 (Tully, 1990). The name of the model should be evident for older readers: washboards were rippled surfaces used for washing of laundry before washing machines were introduced. The model contains an attractive force close to the surface, starting at the end of the vector labelled by −uz , and assumes reflection around the local surface normal when the hard wall of the surface is hit. Parallel momentum is

42

A.W. Kleyn

Fig. 2.8. Schematic representation of the washboard model, showing the corrugation parameter α in the model and the reflection around the local surface normal of the ‘local’ cube. From Tully (1990).

conserved in that collision. At the end of the collision the attractive well region is left, leading to additional deflection, or refraction. Tully has applied his model to the scattering of Ar from Pt(111). Overall, good agreement with experiments and full molecular dynamics simulations is found. In some finer details the washboard model breaks down. Berenbak et al. used the washboard model to study the interaction between Ar and Ru(0001) (Berenbak et al., 2002). Some of their results are reproduced in Fig. 2.9. In the angular distributions the presence of rainbows is obvious. The rainbows are more separated if the surface corrugation (characterized by the angle α) is increased, upper panel. The rainbows are slightly shifted by the effective surface mass (middle panel), and only marginally dependent on the surface temperature in the range shown (lower panel). For higher surface temperature the structure gets washed out in the model calculations, but remains clearly visible at Ts around 500 K (not shown). The behaviour of the energy transfer curves exhibit the characteristics of both the cube model and the significant surface corrugation, contained in the model. The upper panel shows that between the rainbow peaks the energy transfer follows the binary collision model very well. It is impossible to get scattering outside the regions confined by the two rainbow peaks without introduction of surface motion. This is clearly observed. Outside the rainbows, for θf < 35.8◦ or θf > 61.2◦ in the lower panel of Fig. 2.9, the energy transfer changes from binary collision like to that given by the cube model. Changing parameters changes this behaviour in an entirely intuitive way, as shown in the three panels. Although the washboard model has several powerful ‘knobs’ satisfactory agreement with their experimental data could not be obtained by Berenbak et al. (2002). However, performing molecular dynamics simulations with ab-initio pair potentials did not yield more satisfactory results. In fact, the angular dependence of the energy transfer is described better by the washboard model than by the


43

Fig. 2.9. Calculations using the washboard model for Ar scattering from Ru(0001). In the left hand panels the energy transfer is shown, to the right the angular intensity distributions, both as a function of the outgoing angle θf . Results with a variation of three input parameters of the washboard model the maximum angle in the washboard ripple αm (top panel), the cube mass Mcube (middle panel) and the surface temperature Ts (bottom panel). The increased effect of surface temperature on lower energy scattering is also shown in this panel (Ei = 0.5 eV, Ts = 10 K). Fixed parameters are the well depth (W = 0.1 eV) and the incoming angle θi = 40◦ . From Berenbak et al. (2002).

44

A.W. Kleyn

molecular dynamics calculations. The origin of the discrepancy is not clear, perhaps correlation between the motion of surface atoms, somewhat implicitly present in the washboard model, is underestimated in the molecular dynamics simulations. This has been studied in detail by Manson and coworkers (Hayes and Manson, 2006; Ambaye and Manson, 2006; Dai and Manson, 2002; Iftimia and Manson, 2001), as also discussed elsewhere in this volume. Concluding, the washboard model, containing the essential physics from the cube models and the binary collision model can explain a number of observations with the simple physics of these two elementary models. 2.4.6. Different surface regions In the discussion so far, we have assumed that the surface is unstructured, or that the surface exhibits a smooth corrugation coupled to the unit cell. In the case that the surface consists of regions, for instance consisting of islands of different adsorbates, the scattering patterns are the average of the patterns for each of the regions. In case of regional differences within the unit cell this will make the simple models no longer valid, but to a certain extend it can be taken care of by introducing a free parameter in the models. The logical choice is to make the mass ratio of the surface atom, or cube mass a free parameter. Thus, an effective mass is introduced, which was already done at the time of the introduction of the cube models, see e.g. Goodman and Wachman (1976). Lahaye et al. found a rationale for the varying surface mass (Lahaye et al., 1994). These authors noticed that the effective mass can be dependent on the impact point in the unit cell. Top sites differ from hollow sites. In addition these authors noticed that the effective mass of a threefold hollow site of Pt(111) changes as a function of the collision energy. The reason is simple, at low energies the classical turning point is very far from the surface atoms and the projectile is repelled by more than one surface atom at the same time. This effectively increases the mass. At high energies at high symmetry sites, the projectile interacts again with more than one atom, but now the surface atoms are repelled in very different directions. Now the energy transfer is additive, which leads to a lowering of the effective mass of the surface. Also the stiffness of the surface comes into play. This has been observed recently by scattering Xe from graphite. In spite of the enormous mass difference between Xe and C, the Xe is reflected from the graphite surface. A surface effective mass of around 300 is found from modelling of the collisions by sophisticated molecular dynamics calculations involving refined potentials. Scattering from the surface in this case resembles that of scattering from a trampoline. Hard cube models are shown to work quite well, provided the cube mass is increased by a factor of more than 25 with respect to that of the individual surface atoms (Watanabe et al., 2006, 2005; Tomsic et al., 2003; Nagard et al., 1998). Also in scattering of Ne, Ar and Xe from ordered 1-decanethiol monolayers adsorbed on Au(111) very distinct direct inelastic scattering is observed, which should be impossible if the effective mass of the surface would be that of a single end group of the thiol (Gibson et al., 2006, 2003; Isa et al., 2004). The simple models from above extended with an effective mass are capable of explaining some observed dynamics. This is expected to be true as long as there are no major


45

Fig. 2.10. Angular distributions measured for scattering of NO with a translational energy of about 2.1 eV from clean (open diamonds) and H-covered (full circles) Ru(0001). The sharp distribution around the specular angle of 60◦ indicates the angular width of the primary beam. Note the remarkable decrease of the angular width in case of H-coverage. Hydrogen turns the surface into a molecular mirror. From Butler et al. (1997).

non-uniformities at the level of the unit cell. However, if the structure of the surface becomes very different for different regions of the unit cell, this is obviously no longer true. We have observed a peculiar case for scattering of NO or CO from H-covered Ru(0001) (Berenbak et al., 2004, 2001, 2000; Butler et al., 1997; Riedmüller et al., 2000). A measured angular distribution is shown in Fig. 2.10 (Butler et al., 1997). It is seen that covering the surface by hydrogen, which forms an ordered 1 × 1 overlayer, sharpens up the angular distribution quite dramatically. It strongly suggests that the adsorbed hydrogen removes the corrugation and turns the reactive surface into an inert mirror, to which the cube model should be applicable. However, it was determined that the sticking coefficient of NO on HRu(0001) for the same experimental conditions is about 0.5! This suggests that the surface within a unit cell changes from almost flat to absorbing (Berenbak et al., 2004).

2.5. Limiting cases in adsorption dynamics 2.5.1. Cube models Adsorption implies that an incoming particle transfers all of its momentum to the lattice such that it equilibrates with it and is adsorbed. A sharp definition may be hard to give in particular at high surface temperatures because there is a distinct probability that

46

A.W. Kleyn

Fig. 2.11. Schematic representation of adsorption in the cube model (left), and a simple square well potential describing the interaction.

the particle is driven off by a thermal fluctuation in the motion of the surface atoms. The discussion will be started by using the cube model again, depicted schematically in Fig. 2.11 (Yamamoto and Stickney, 1970; Logan, 1973; Goodman and Wachman, 1976; Grimmelmann et al., 1980). A particle with mass m approaches the surface cube with mass M. The interaction potential is approximated by a simple square well potential. We assume that the particle approaches the cube with relative energy E. When it enters the square well the relative energy is increased by the well depth W . With this energy, particle and cube collide. In this collision, due to energy and momentum conservation the particle transfers an amount of: m 4μ(E + W ) with again μ = , E = 2 M (1 + μ) to the cube and will be reflected back. When the outer passage of the well is passed, again energy is transferred and particle and cube recede. Because it is a binary collision with no third body to take away momentum, the particle can never get stuck in the well, irrespective of the well depth. However, in the case of a surface this is not realistic. The cube is embedded in the solid and will lose most if not all of the energy gained by transfer to the lattice. In this case the situation is entirely different. As soon as the particle has transferred more to the cube than its energy in the well E + W it cannot escape. In this way we have derived a simple expression for the instantaneous adsorption coefficient S in the zero temperature limit. If E < E: S = 1, if E > E: S = 0. This leads to the general result that in a classical model the sticking coefficient at Ts = 0 for E approaching zero is unity. In all interactions there is a small well depth with allows for this general conclusion. This requires very rapid dissipation of the normal energy, which has indeed been observed by Head-Gordon et al. (1991). There are two important exceptions: 1) the energy transfer is quantized so that even at very low temperatures there is a finite probability for zero energy transfer (zero phonon excitation). This can be observed at cryogenic temperatures (Schlichting et al., 1988, 1992; Andersson et al., 2002, 2006; Linde and Andersson, 2006).


47

2) the wavelength of the particle becomes so large that within the uncertainty principle the wave function of the incoming particle does not ‘see’ the attractive well. This is the true quantum mechanical limit of sticking for which S = 0. This limit is reached in the nano Kelvin regime. In all practical situations these exceptions do not play a role and S = 1 when E ≈ 0 and Ts ≈ 0. What the meaning of S is depends on the experimental situation. For finite surface temperature the residence time of a particle is limited and can be given by τ = τ0 · eE/kT where τ0 is a pre-exponential constant of typically 10−13 s, E the surface binding energy of the adsorbate, k is Boltzmann’s constant and T the surface temperature. In the case that the residence time is short (often in the case of physisorption) on an experimental timescale, S is generally referred by as the ‘trapping probability’. In the case that the residence time is long (often in the case of chemisorption) on an experimental timescale, S is referred to as the ‘sticking probability’. Trapping can be monitored by looking at the desorbing particles, which usually are in time and by final energy, quite distinct from scattered particles. In Fig. 2.12 the probability for adsorption of Ar at Ag(111) is shown (Lahaye et al., 1995); note the logarithmic energy axis. It is referred to by sticking coefficient because

Fig. 2.12. The initial adsorption or sticking probability of Ar on Ag(111) for several surface temperatures as a function of the collision energy. The incidence angle is 40◦ . From Lahaye et al. (1995).

48

A.W. Kleyn

Fig. 2.13. The sticking probability for NO-Pt(111), K-W(110) and Ar-Pt(111) as a function of the impact energy scaled by the well depth. The lines represent simple model calculations. From Kleyn (1997).

of the short duration of the computational runs used to make this data. As expected the sticking coefficient approaches unity for Ts and E getting very low. The effect of the decrease of the sticking probability with increasing Ts is clearly visible. Due to desorption of the Ar during the time frame of the calculation on a picosecond time scale and direct reflection due to energy transfer from the surface to the Ar the sticking probability decreases with increasing Ts . The fall off of the sticking coefficient with energy is quite distinct at an energy of about 0.1 eV, which approximately equals the well depth for this system. This behaviour is quite general, in Fig. 2.13 the sticking probability S0 is plotted for three different systems (Kleyn, 1997): Ar-Pt(111), W = 0.08 eV (Rettner et al., 1990; Mullins et al., 1989); NO-Pt(111), W = 1 eV (Brown and Luntz, 1993); K-W(110), W = 2 eV (Trilling and Hurkmans, 1976; Hurkmans et al., 1976). The mass ratio, determining the energy transfer in the well is similar for these systems, and the energy dependence of S0 is as well. It should be noted that this is a favourable mass ratio for this demonstration. For nearly equal masses of surface atoms and atoms, the sticking coefficient is expected to be very large for a broad energy range and strongly dependent on the cohesive energy of the surface atoms. For very light atoms impacting on heavy surface atoms, no sticking is expected in the first place and again it is determined by more detailed properties of the surface, such as the probability of phonon or electron–hole pair excitation. For molecular projectiles internal excitation and an orientation dependence of the well depth come in addition, which may be the reason that S0 for NO on Pt(111) does not approach unity for lower energies in Fig. 2.13 and that the fall off of S0 with energy is more gradual than for the atomic projectiles (Lahaye et al., 1996).


49

The cube model predicts scaling of the sticking with the normal energy. For the ArPt(111) system, for which parallel momentum conservation should be a quite reasonable approximation, Rettner and co-workers studied to what extend this is true. They found that sticking was observed to scale with cosn (Θ), where n = 2 at Ts = 0 K, as expected, whereas at Ts = 300 K n is less than one half (Rettner et al., 1990; Mullins et al., 1989). This is attributed to increasing parallel momentum dissipation with increasing Ts . The expression for energy transfer in the cube model predicts a very strong dependence of sticking on the mass ratio. For relatively light atoms hitting surfaces consisting of heavy atoms, e.g. Ar–Ag or Ar–Pt, the predicted trends are indeed observed. For the heavy noble gases Kr and Xe lower sticking coefficients are observed with Ni(111) than with Pt(111) or Pd(111), despite the lower mass of the Ni atoms (Kao et al., 2004). Here the cube model clearly breaks down, due to the strong cohesion of the Ni-lattice. 2.5.2. Displacement trapping and implantation At low energies the effective range of the potential between the projectile and a atom surface atom is large, and the interaction with the surface can be seen as adding a number of displaced large spheres to get the total interaction (Lahaye et al., 1994). This was already demonstrated with Fig. 2.2. Addition of the interaction of the atom with all these large spheres, effectively gives rise to the cube models and an effective decrease of the energy transfer. If the energy is higher and the spheres have effectively shrunk, the simultaneous interaction is still possible, but now the onset of this is very close to the threshold for implantation (Horn et al., 1987, 1988). This can be clearly seen in Fig. 2.12, where for energies exceeding 20 eV the sticking probability is increasing again. Once the atoms start penetrating the surface, the energy transfer and hence the energy deposition increase rapidly (Lahaye et al., 1994), ultimately approaching unity, when the atom penetrates deeply into the solid and is ultimately stopped. Using very thin freestanding films and multi MeV energies, the atoms start to appear exiting at the back of the sample, but this realm of scattering dynamics is completely outside of the scope of this volume. Winters et al. have studied the average energy deposition into Au and Pt films as a function of energy and for several projectiles (Winters et al., 1990). A summary of their data is shown in Fig. 2.14. It should be noted that low temperature sticking was not studied in this figure. Therefore, the amount of energy transfer at low kinetic energy is low. One can directly compare the data of Fig. 2.14 to that of Fig. 2.12 in the high Ts limit. But it should be realised that different quantities are shown. In Fig. 2.14 it is seen that for Ar at 10 eV around 0.2 of the initial energy is transferred to the lattice. But in most cases this energy transfer will not lead to sticking or implantation. The prominent role of the mass ratio is also evident in Fig. 2.14. However, it is seen that the energy transfer for He exceeds that of Ar around a few eV. This is due to the much smaller size of He that can more easily penetrate and as a consequence transfer more energy. A very peculiar way of displacement trapping has been observed recently in the group of Sibener (Gibson et al., 2006). Xe atoms can penetrate deeply between the chains of self assembled 1-decanethiol on Au(111). The atoms can subsequently desorb from the channels and their angular distribution is determined by the inclination of the decanethiol molecules on the surface.

50

A.W. Kleyn

Fig. 2.14. Summary of measurements of the fraction of kinetic energy loss to Au or Pt by He, Ar and Xe. The curves are fits to data obtained by the authors using a simple analytical model. From Winters et al. (1990).

For atomic projectiles adsorption is in most cases non-activated; i.e., there is no repulsive barrier in the potential blocking the way to a deep binding potential. For molecules this is not the case. In particular for dissociative chemisorption there often is a barrier to dissociation that can be overcome by translational or internal energy of the molecule. In these cases the models presented above are still useful to obtain a physical picture of the dynamics, but need to be extended. As the molecular interactions have been discussed in detail in other chapters, it will not be discussed here. Finally, it should be pointed out that collision induced processes at surfaces may drive a number of processes via the impact of an energetic atom. In the simplest descriptions of these processes the simple models introduced in this chapter reappear, as reviewed recently by Asscher and Zeiri (2003).

2.6. Conclusion This chapter has attempted to review some simplified models for atom surface interactions. Also in the time when large scale molecular dynamics calculations are widely available, these simple models remain extremely useful to interpret the results of these calculations. The opportunity to envision a surface as consisting out of a simple cube or a collection of spheres often gives the researcher simple methods to interpret more complex interactions. This chapter is based upon lectures given by the author on several occasions for the gas–surface dynamics community. His interest was triggered by years of work at the IBM Research laboratory in San Jose, CA, the FOM Institute of Atomic and Molecular Physics


51

in Amsterdam and the Leiden Institute of Chemistry. He thanks all colleagues and coworkers for their constructive interactions over the years. Michael Gleeson and Hirokazu Ueta are thanked for their reading of the manuscript. This work is part of the research programme of FOM and is supported financially by NWO.

References Ambaye, H., Manson, J.R., 2006. J. Chem. Phys. 125, 84919-1–84919-8. JCP 125 (2006) 176101-1–196101-2. Andersson, T., Althoff, F., Linde, P., Andersson, S., Burke, K., 2002. Phys. Rev. B 65, 45409-1–45409-12. Andersson, T., Linde, P., Hassel, M., Andersson, S., 2006. J. Chem. Phys. 124, 114703-1–114703-7. Arumainayagam, C.R., Madix, R.J., 1991. Progr. Surf. Sci. 38, 1–102. Asscher, M., Zeiri, Y., 2003. J. Phys. Chem. B 107, 6903–6919. Barker, J.A., Auerbach, D.J., 1984. Surf. Sci. Rep. 4, 1–99. Berenbak, B., Riedmüller, B., Butler, D.A., Rettner, C.T., Auerbach, D.J., Stolte, S., Kleyn, A.W., 2000. Phys. Chem. Chem. Phys. 2, 919–923. Berenbak, B., Riedmüller, B., Rettner, C.T., Auerbach, D.J., Stolte, S., Kleyn, A.W., 2001. Phys. Chem. Comm. 16, 1–6. Berenbak, B., Riedmüller, B., Stolte, S., Kleyn, A.W., 2004. Chemical Physics 301, 309–313. Berenbak, B., Zboray, S., Riedmüller, B., Papageorgopoulos, D.C., Stolte, S., Kleyn, A.W., 2002. Phys. Chem. Chem. Phys. 4, 68–74. Bonn, M., Kleyn, A.W., Kroes, G.J., 2002. Surf. Sci. 500, 475–499. Boyer, C.B., 1987. The Rainbow: From Myth to Mathematics. Princeton University Press, Princeton. Brenner, D.W., 1990. Phys. Rev. B 42, 9458. Brown, J.K., Luntz, A.C., 1993. Chem. Phys. Lett. 204, 451–454. Butler, D.A., Berenbak, B., Stolte, S., Kleyn, A.W., 1997. Phys. Rev. Lett. 78, 4653–4656. Campargue, R. (Ed.), 2001. Atomic and Molecular Beams: The State of the Art 2000. Springer Verlag, Berlin Heidelberg. Dai, J., Manson, J.R., 2002. Nuclear Instrum. Methods Phys. Res. Sect. B 193, 497–503. Gibson, K.D., Isa, N., Sibener, S.J., 2003. J. Chem. Phys. 119, 13083–13095. Gibson, K.D., Isa, N., Sibener, S.J., 2006. J. Phys. Chem. A 110, 1469–1477. Goodman, F.O., Wachman, H.Y., 1976. Dynamics of Gas–Surface Scattering. Academic Press, New York. Gou, F., Gleeson, M.A., Kleyn, A.W., 2006. Phys. Chem. Chem. Phys. 8, 5522–5534. Graves, D.B., Humbird, D., 2002. Appl. Surf. Sci. 192, 72–87. Grimmelmann, E.K., Tully, J.C., Cardillo, M.J., 1980. J. Chem. Phys. 72, 1039–1043. Gross, A., 2003. In: Woodruff, D.P. (Ed.), Surface Dynamics. Elsevier, Amsterdam, pp. 1–26. Hayden, B.E., Mormiche, C., 2003. In: Woodruff, D.P. (Ed.), Surface Dynamics. Elsevier, Amsterdam, pp. 177– 222. Hayes, W.W., Manson, J.R., 2006. Phys. Rev. B 74, 73413-1–73413-4. Phys. Rev. B 75 (2007) 113408-1–1134084. PRB 77 (2008) 89904 (Erratum). Head-Gordon, M., Rettner, C.T., Mullins, C.B., Auerbach, D.J., Tully, J.C., 1991. J. Chem. Phys. 94, 1516–1527. Hodgson, A., 2003. In: Woodruff, D.P. (Ed.), Surface Dynamics. Elsevier, Amsterdam, pp. 143–176. Horn, T.C.M., Haochang, P., Kleyn, A.W., 1987. J. Vacuum Sci. Technol. A 5, 656–657. Horn, T.C.M., Haochang, P., VanDenHoek, P.J., Kleyn, A.W., 1988. Surf. Sci. 201, 573–602. Hurkmans, A., Overbosch, E.G., Los, J., 1976. Surf. Sci. 59, 488–508. Iftimia, I., Manson, J.R., 2001. Phys. Rev. Lett. 87, 93201-1–93201-4. Isa, N., Gibson, K.D., Yan, T., Hase, W., Sibener, S.J., 2004. J. Chem. Phys. 120, 2417–2433. Kao, C.L., Carlsson, A., Madix, R.J., 2004. Surf. Sci. 565, 70–80. Kirchner, E.J.J., Kleyn, A.W., Baerends, E.J., 1994. J. Chem. Phys. 101, 9155–9163. Kleyn, A.W., 1987. Comment At. Mol. Phys. 19, 133–155. Kleyn, A.W., 1994. Surf. Rev. Lett. 1, 157–173.

52

A.W. Kleyn

Kleyn, A.W., 1997. Progr. Surf. Sci. 54, 407–420. Kleyn, A.W., 2003b. In: Woodruff, D.P. (Ed.), Surface Dynamics. Elsevier, Amsterdam, pp. 79–108. Kleyn, A.W., 2003a. Chem. Soc. Rev. 32, 87–95. Kleyn, A.W., Horn, T.C.M., 1991. Phys. Rep. 199, 191–230. Kondo, T., Tomii, T., Yagyu, S., Yamamoto, S., 2001. J. Vacuum Sci. Technol. A 19, 2468–2470. Kroes, G.J., 1999. Progr. Surf. Sci. 60, 1–85. Kroes, G.J., Gross, A., Baerends, E.J., Scheffler, M., McCormack, D.A., 2002. Acc. Chem. Res. 35, 193–200. Lahaye, R.J.W.E., Kleyn, A.W., Stolte, S., Holloway, S., 1995. Surf. Sci. 338, 169–182. Lahaye, R.J.W.E., Stolte, S., Holloway, S., Kleyn, A.W., 1996. J. Chem. Phys. 104, 8301–8311. Lahaye, R.J.W.E., Stolte, S., Kleyn, A.W., Smith, R.J., Holloway, S., 1994. Surf. Sci. 307–309, 187–192. Linde, P., Andersson, S., 2006. Phys. Rev. Lett. 96, 86103-1–86103-4. Logan, R.M., 1973. In: Green, M. (Ed.), Solid State Surface Science. Marcel Dekker, New York, pp. 1–103. McClure, S.M., Reichman, M.I., Seets, D.C., Nolan, P.D., Sitz, G.O., Mullins, C.B., 2003. In: Woodruff, D.P. (Ed.), Surface Dynamics. Elsevier, Amsterdam, pp. 109–142. Mullins, C.B., Rettner, C.T., Auerbach, D.J., Weinberg, W.H., 1989. Chem. Phys. Lett. 163, 111–115. Nagard, M.B., Andersson, P.U., Markovic, N., Pettersson, J.B.C., 1998. J. Chem. Phys. 109, 10339–10349. Nieto, P., Pijper, E., Barredo, D., Laurent, G., Olsen, R.A., Baerends, E.J., Kroes, G.J., Farias, D., 2006. Science 312, 86–89. Pfandzelter, R., Kowalski, T., Igel, T., Winter, H., 1998. Surf. Sci. 411, L894–L899. Raukema, A., Dirksen, R.J., Kleyn, A.W., 1995. J. Chem. Phys. 103, 6217–6231. Rettner, C.T., Auerbach, D.J., Tully, J.C., Kleyn, A.W., 1996. J. Phys. Chem. 100, 13021–13033. Rettner, C.T., Mullins, C.B., Bethune, D.S., Auerbach, D.J., Schweizer, E.K., Weinberg, W.H., 1990. J. Vacuum Sci. Technol. A 8, 2699–2704. Riedmüller, B., Ciobîcã, I.M., Papageorgopoulos, D.C., Berenbak, B., van Santen, R.A., Kleyn, A.W., 2000. Surf. Sci. 465, 347–360. Scoles, G., Bassi, D., Buck, U., Lainé, D. (Eds.), 1988a. Atomic and Molecular Beams Methods I. Oxford University Press, Oxford. Scoles, G., Lainé, D., Valbusa, U. (Eds.), 1988b. Atomic and Molecular Beams Methods II. Oxford University Press, Oxford. Schlichting, H., Menzel, D., Brunner, T., Brenig, W., 1992. J. Chem. Phys. 97, 4453–4467. Schlichting, H., Menzel, D., Brunner, T., Brenig, W., Tully, J.C., 1988. Phys. Rev. Lett. 60, 2515–2518. Schuller, A., Wethekam, S., Mertens, A., Maass, K., Winter, H., Gartner, K., 2005. Nuclear Instrum. Methods Phys. Res. B 230, 172–177. Schweizer, E.K., Rettner, C.T., Holloway, S., 1991. Surf. Sci. 249, 335–349. Tenner, A.D., Gillen, K.T., Horn, T.C.M., Los, J., Kleyn, A.W., 1984. Phys. Rev. Lett. 52, 2183–2186. Tenner, A.D., Gillen, K.T., Horn, T.C.M., Los, J., Kleyn, A.W., 1986a. Surf. Sci. 172, 90–120. Tenner, A.D., Saxon, R.P., Gillen, K.T., Harrison Jr., D.E., Horn, T.C.M., Kleyn, A.W., 1986b. Surf. Sci. 172, 121–150. Tomsic, A., Andersson, P.U., Markovic, N., Pettersson, J.B.C., 2003. J. Chem. Phys. 119, 4916–4922. Trilling, L., Hurkmans, A., 1976. Surf. Sci. 59, 361–372. Tully, J.C., 1990. J. Chem. Phys. 92, 680–686. Vattuone, L., Savio, L., Rocca, M., 2003. In: Woodruff, D.P. (Ed.), Surface Dynamics. Elsevier, Amsterdam, pp. 223–246. Watanabe, Y., Yamaguchi, H., Hashinokuchi, M., Sawabe, K., Maruyama, S., Matsumoto, Y., Shobatake, K., 2005. Chem. Phys. Lett. 413, 331–334. Watanabe, Y., Yamaguchi, H., Hashinokuchi, M., Sawabe, K., Maruyama, S., Matsumoto, Y., Shobatake, K., 2006. Eur. Phys. J. D 38, 103–109. Weaver, J.F., Carlsson, A.F., Madix, R.J., 2003. Surf. Sci. Rep. 50, 107–199. Winter, H., Schuller, A., 2005. Nuclear Instrum. Methods Phys. Res. B 232, 165–172. Winters, H.F., Coufal, H., Rettner, C.T., Bethune, D.S., 1990. Phys. Rev. B 41, 6240–6256. Yamamoto, S., Stickney, R.E., 1970. J. Chem. Phys. 53, 1594–1604. Yan, T.Y., Hase, W.L., Tully, J.C., 2004. J. Chem. Phys. 120, 1031–1043.

CHAPTER 3

Energy Transfer to Phonons in Atom and Molecule Collisions with Surfaces

J.R. MANSON Department of Physics and Astronomy Clemson University Clemson, SC 29634, USA



Contents 3.1. 3.2. 3.3. 3.4.

Introduction . . . . . . . . . . . . . . . . . . Basic model for multiphonon excitation . . Experimentally measured quantities . . . . . Surface scattering theory . . . . . . . . . . . 3.4.1. General treatment . . . . . . . . . . . 3.4.2. Atom–surface scattering . . . . . . . 3.4.3. Rigid molecular rotator . . . . . . . . 3.4.4. Molecular internal vibration modes . 3.4.5. Interaction potential . . . . . . . . . . 3.5. Comparisons with experiment . . . . . . . . 3.5.1. Angular distributions . . . . . . . . . 3.5.2. Translational energy resolved spectra 3.5.3. Rotational energy resolved spectra . . 3.5.4. Internal vibrational mode excitation . 3.6. Conclusions . . . . . . . . . . . . . . . . . . Acknowledgement . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . .

Abstract Experiments measuring the scattering of molecular beams from surfaces are capable of providing important information on the gas–surface interaction. However, such experiments are often carried out under conditions in which a substantial amount of the energy transfer is through excitations of phonons at the surface. This paper reviews theoretical methods of treating energy transfer to the surface from the limit of purely quantum mechanical elastic diffraction and single phonon transfer to the classical limit of large numbers of phonon excitations. In the multiphonon limit, theories of molecule–surface scattering are discussed which are classical in the translational and rotational degrees of freedom but quantum mechanical in the treatment of internal molecular vibrational modes. Through direct comparison with experiments it is shown that qualitative and quantitative explanations, as well as important physical information, can be obtained for many phenomena involving multiple phonon excitation observed in molecule–surface collisions. Examples are discussed that include both situations in which multiphonon excitation is the primary object of study and cases in which it is a secondary effect that must be accounted for. These examples include scattered angular distributions, translational energy-resolved measurements, rotational energy resolved spectra and internal vibrational mode excitation probabilities.

54

55 56 60 62 62 63 66 67 70 72 72 77 82 87 88 90 90

55

3.1. Introduction This paper concerns the exchange of energy in the collision of molecules with a surface. In particular, it is intended as a useful review for describing energy transfer in experiments in which a well-defined molecular beam is directed towards a surface and the final states of the molecules eventually leaving the surface are measured in a detector. Typical experiments are carried out under ultra-high vacuum conditions with projectiles ranging from the smallest possible masses such as H2 molecules or He atoms to large mass gases, and energies are usually in the thermal to hypersonic range which corresponds to translational energies from a few meV to several eV. A collision with the surface can lead to a variety of different outcomes, for example the molecules leaving the surface may be directly scattered, they can be briefly trapped at the surface, or may be adsorbed for longer times before eventually desorbing. In all cases an important channel for exchange of energy is phonon annihilation or creation, and in most cases this is by far the dominant mechanism for energy transfer although excitation of electron–hole pairs or other elementary electronic excitations may play a role for high energies and larger mass gas particles. One of the earliest experiments to attempt to characterize the nature of atom surface collisions was that of Roberts who measured energy transfer between rare gases and a tungsten surface (Roberts, 1930). These experiments stimulated theoretical calculations by Mott and Jackson using the methods of quantum mechanics which were being developed at the time (Jackson and Mott, 1932; Jackson, 1932). Their calculations were the first to describe the energy transfer in surface scattering by what is now known as a single-phonon distorted wave Born approximation. Also during the early years of quantum mechanics Stern and co-workers started a long series of experiments, primarily scattering of He beams from LiF(001) surfaces, in which the apparent motivating objective was to confirm the de Broglie matter wave hypothesis through observation of diffraction peaks (Frisch, 1979; Stern, 1920a, 1920b, 1926; Knauer and Stern, 1929; Estermann and Stern, 1930; Estermann et al., 1931). This seminal work prompted a series of publications by LennardJones and his students that pointed out the importance of trapping in the physisorption potential well and the importance of energy transfer to the phonons (Lennard-Jones and Devonshire, 1936a, 1936b, 1936c, 1936d, 1937a, 1937b, 1936b). Their calculations were also done in the distorted wave Born approximation, and they even investigated multiple phonon transfers, although we know today that the multiple phonon excitations they calculated (Strachan, 1935) are not the dominant contribution to multiple phonon energy exchange (Manson and Tompkins, 1977). With the advent in the 1960s and 1970s of improved ultra-high vacuum techniques, together with the development of highly mono-energetic molecular jet beams, there was a renewed interest in gas–surface experiments. This led to measurements that were capable of detecting single surface phonon creations and annihilations and the first measurements of surface phonon dispersion relations (Brusdeylins et al., 1980, 1981). These experiments have developed to the point that today He atom scattering from surfaces is a standard method of obtaining surface sensitive information. The scattering of many other atomic and molecular particles is used to obtain information on both the surfaces, the molecule– surface interaction potential, and on surface chemical reactions.

56

J.R. Manson

In all such experiments, energy transfer to the phonons is important. In many cases, a significant portion of the incident translational energy can be lost upon collision, for example in the case of NO or N2 scattering from typical metal surfaces, a beam with incident energy of order 1 eV can lose 25–75% of its energy into phonons, which essentially means transferring heat to the surface. Such large energy transfers will have a substantial effect on other processes such as adsorption and excitation of internal molecular modes and thus must be understood. Even in the case of He atom scattering at thermal energies, where the dominant inelastic processes tend to be single phonon excitation of Rayleigh and other surface modes, there is a background in the energy-resolved scattering spectrum that is in large part due to multiphonon events. Identifying and subtracting this background from the single phonon contributions is an essential problem because under certain circumstances the multiphonon background can produce peaks in the scattering spectra that resemble single phonon features (Celli et al., 1991). However, background subtraction is far from the only reason for needing an understanding of multiphonon events. Important features of the molecule–surface interaction potential are revealed in the behavior of multiphonon spectra and, since the study of multiple phonon events ranges from small to large quantum numbers of excitation, it provides an interesting example of the transition from purely quantum phenomena to classical physics at large quantum numbers. Much of the study of phonon excitation in molecule–surface collisions has been treated in earlier reviews, a partial list of which is given in Hulpke (1992), Bortolani and Levi (1986), Toennies (1991), Rettner and Ashfold (1991), Barker and Auerbach (1984), Celli (1984), Toennies (1987), Scoles (1988) Campargue (2001) Benedek and Toennies (1994), Manson (1994), Gumhalter (2001). The organization of this paper is as follows. The next section (Section 3.2) presents a simple and elegant elementary model for describing scattering events in which large numbers of phonons are transferred. This model serves as a basis for understanding and developing a variety of theoretical approaches that have been used for multiphonon scattering from surfaces by atoms and molecules. A brief description of the quantities actually measured in experiments and how they relate to the calculated scattering probabilities is in Section 3.3. In Section 3.4 a general approach to the theory of surface scattering that can be applied to multiple phonon energy transfers is developed. This theory begins with atomic projectiles and then extends the treatment to include the internal degrees of freedom of molecules. The theoretical approach is illustrated and discussed through comparisons with several different types of experimental scattering intensity measurements in Section 3.5. Further discussion and some conclusions are presented in Section 3.6.

3.2. Basic model for multiphonon excitation Although it will be seen in the sections below that the multiphonon contributions to scattering spectra can be manifest in many different forms, the basic aspects of multiphonon energy transfer in many molecule–surface scattering experiments can often be understood and analyzed within the framework of a classical model for energy transfer between an atomic projectile and a target consisting of discrete atoms. This model is perhaps best expressed mathematically in terms of the transition rate w(pf , pi ) which is the probability


57

per unit time that a particle initially in a state of momentum pi will make a transition to the final state pf . This is the same transition rate that is used as the fundamental starting point in the quantum mechanical description of scattering theory (Rodberg and Thaler, 1967) and it provides a convenient description that is independent of any effects imposed by the constraints of the particular experiment used to measure the scattering. The ways in which actual experimentally measured quantities can be determined from the transition rate w(pf , pi ) is discussed below in Section 3.3. The basic transition rate, which can be calculated from purely classical mechanics, for an atomic projectile of mass m colliding with a target of atomic scattering centers with mass MC and whose initial velocity distribution is determined by the equilibrium condition of equipartition of energy is 1 (EfT − EiT + E0T )2 2 1 π 2 w(pf , pi ) = |τf i | exp − , h¯ kB TS E0T 4kB TS E0T

(3.1)

where the translational energy is EfT = p2f /2m, TS is the surface temperature, kB is Boltzmann’s constant, and the recoil energy is given by E0T = (pf − pi )2 /2MC . |τf i |2 is a form factor that depends on the interaction potential between projectile and target atoms, for example, it is a constant for hard sphere collisions. The quantity h¯ is a constant having dimensions of action and the first principles quantum mechanical derivation given below in Section 3.4 shows that this is properly identified to be Planck’s constant divided by 2π. Equation (3.1) is quite old, having been used to describe multiphonon scattering in neutron scattering (Sjölander, 1959) and ion scattering (Micha, 1981). It actually gives a very good quantitative description of the single collision peak observed in certain low energy ion scattering experiments (Muis and Manson, 1996; Powers et al., 2004). At first glance, Eq. (3.1) appears to be a Gaussian-like function of the energy difference EfT − EiT but in actual practice under many initial conditions it can be highly skewed and appear quite different from Gaussian-like behavior. For example, in the limit of very small incident energy Eq. (3.1) goes over to exponential behavior in the final energy 1 EfT (1 + μ)2 2 1 π 2 w(pf , pi ) = |τf i | exp − , 4kB TS μ h¯ kB TS μEfT

(3.2)

where the mass ratio is μ = m/MC . Under many initial conditions, Eq. (3.1) does look Gaussian like. For example, for the incident energy EiT large compared to kB TS and μ < 1 it has only a single peak when plotted as a function of final energy EfT at fixed incident and final angles. The position of the peak, or the most probable final energy, lies very nearly at the zero of the argument of the exponential which is given by the Baule condition EfT = f (θ )EiT in which θ is the total scattering angle (the angle between pf and pi ) and f (θ ) is given by 1 − μ2 sin2 θ + μ cos θ 2 . f (θ ) = 1+μ

(3.3)

58

J.R. Manson

The width of this peak, expressed as the mean-square deviation from the most probable energy is approximately (3.4) (E T )2 ≈ 2g(θ )EiT kB TS , where g(θ ) = with

gT A (θ ) 1+μ−

μ √cos θ f (θ)

,

gT A (θ ) = μ 1 + f (θ ) − 2 f (θ ) cos θ .

(3.5)

(3.6)

The function gT A (θ ) is the approximation to g(θ ) that is obtained in the so-called trajectory approximation (Burke et al., 1991; DiRubio et al., 1994). The intensity at the position of the most probable final energy peak is dictated by the prefactor, or envelope function, which goes as IMAX ∝

1 1 ≈ . (kB TS E0T )1/2 (kB TS gT A (θ )EiT )1/2

(3.7)

The fact that the width increases with the square root of temperature and energy while the maximum intensity decreases with the same square root behavior is indicative of the unitarity property of the differential reflection coefficient of Eq. (3.1), i.e., it is normalizable which indicates that the number of scattered particles equals the number incident. Eqs. (3.3) through (3.6) are recognized as the familiar expressions associated with the properties of collisions of two rigid spheres and generally attributed to the work of Baule dating from the early 1900s (Baule, 1914; Goodman, 1974). The general behavior of Eq. (3.1) can become more complex, for example if μ > 1 it can exhibit double peaks or no well-defined peaks at all depending on the scattering angles. An important, and perhaps somewhat surprising property of Eq. (3.1) is that it depends in no way on the vibrational properties of the target, in fact it will be shown below that it depends only on a distribution of velocities in the target that obeys the equilibrium property of equipartition of energy, examples being thermal vibrations in solids or the distribution of speeds in a gas. This is the same as saying that even though Eq. (3.1) describes energy transfer with the vibrational modes it contains no information about the phonons or any other elementary excitations, such as electron–hole pairs, that may have been created. It describes a collision event in which the projectile atom collides with a target atom, the target recoils, but the projectile then leaves the scene of the collision before the recoiling target atom has an opportunity to create vibrations or other excitations in the target. The recoiling atom eventually does create a cloud of multiple phonon excitations, but by this time the original projectile is far away and carries no detailed information about the nature of the excitations created and destroyed. In spite of its simplicity, and the paucity of information about the properties of the target contained in Eq. (3.1), it and its many related forms are very useful in describing multiphonon energy transfer observed in particle–surface scattering experiments. Specifically, Eq. (3.1) and its properties described in Eqs. (3.3) through (3.7) have been shown to explain the temperature and energy dependence of single scattering events in low energy ion


59

scattering, specifically for Na+ and K+ scattering from Cu(100) (Muis and Manson, 1996; Powers et al., 2004). It has also been used to describe the energy resolved scattering spectra observed in experiments of rare gases scattering from metal surfaces (Muis and Manson, 1997; Dai and Manson, 2003). A straightforward way of deriving Eq. (3.1) above is to recognize that in a collision of an atomic projectile with a discrete target core the transition rate will depend on the essential conditions of conservation of momentum and energy between target and projectile, i.e., the transition rate will be of the form

w(pf , pi ) ∝ (3.8) F δ EfT + εfT − EiT − εiT δ(pf + kf − pi − ki ) kf

where ki and kf are the translational momentum of the target atom before and after the collision, respectively, εiT and εiT are the corresponding target atom energies, and F is a function depending on all final and initial momenta which will be related below to the scattering form factor |τf i |2 . Because the initial and final states of the target particles are not measured in a typical experiment, the transition rate must be averaged over all initial states of the many-body target as signified by the symbol and also must be summed or integrated over all final momenta of the target. The summation over final target momenta is trivially carried out leading to p2 ki · p w(pf , pi ) ∝ F δ EfT − EiT + (3.9) − , 2MC MC where p = pf − pi is the momentum scattering vector. Next, the energy δ-function is represented by its Fourier transform over time t, and additionally making the assumption that F is independent of the initial target momentum, the transition rate becomes k ·p p2 F (pf , pi ) +∞ −i(EfT −EiT + 2M )t/h¯ −i Mi ht¯ C C w(pf , pi ) ∝ (3.10) e , dt e 2π h¯ −∞ where as mentioned above h¯ is a constant having dimensions of mechanical action. (In a quantum mechanical treatment, h¯ becomes Planck’s constant divided by 2π as is indicated in the treatments of Section 3.4 below (Sjölander, 1959; Brako and Newns, 1982; Brako, 1982; Manson, 1991).) The average over initial target states appearing in Eq. (3.10) is readily carried out assuming three dimensional isotropy of initial motion and using a Maxwell–Boltzmann distribution of momenta. ki · p t t2 t2 p2 T exp −i (3.11) kB TS 2 = exp −E0 kB TS 2 . = exp − MC h¯ 2MC h¯ h¯ Now, when the result of Eq. (3.11) is used in Eq. (3.10), the Fourier transform is a simple Gaussian integral and the result is precisely the discrete target transition rate of Eq. (3.1), after making the identification F (pf , pi ) ∝ |τf i |2 for the form factor. This derivation of Eq. (3.1) is not only straightforward and direct, it also points out very clearly all assumptions and approximations which are used in its derivation. In particular it is derived from purely classical physics, its form is dictated by the energy and momentum conservation

60

J.R. Manson

laws, and the statistical mechanics depends only on the equilibrium equipartition of initial momenta in the target. In the sections below, the theoretical treatment of inelastic molecule–surface scattering with energy transfer to the phonons of the target will be reviewed. This will cover the range from purely quantum mechanical processes with only a single phonon excited to the classical limit of many phonons as manifest in the example equation of Eq. (3.1). In the process a series of expressions related to Eq. (3.1) will be developed, all of which have shown applicability to explaining molecule surface scattering experiments. Eq. (3.1), which is the limiting case for a collection of discrete target atoms, will be extended to scattering from smooth surface barriers and to include excitation of internal rotational and vibrational degrees of freedom of the projectile.

3.3. Experimentally measured quantities The transition rate for atomic scattering w(pf , pi ) depends only on the final and initial translational momenta, while for molecular scattering it will depend also on the internal modes, i.e., the transition rate becomes w(pf , lf , αjf ; pi , li , αj i ) where lf or li are the angular momenta for the final and initial states, respectively, and αjf or αj i are the excitation quantum numbers for the j th internal vibrational mode in the final or initial state. The transition rate is a useful quantity on which to base the calculations because it gives the probability per unit time for the transition between two well defined molecular states and is independent of the experimental configuration. Quantities which are actually measured in experiments, such a cross sections or differential reflection coefficients, are proportional to w(pf , lf , αjf ; pi , li , αj i ) with the proportionality given by an appropriate Jacobian density of states that depends on the nature of the experimental apparatus. Since experimental configurations, and detector devices, can vary considerably it is worthwhile to give a brief description of the relationship between the transition rate and quantities that are measured. In an idealized surface scattering experiment what is often measured is the differential reflection coefficient, essentially a differential cross section per unit surface area, given by the transition rate normalized to the incident beam flux with the phase space differential volume element converted to spherical coordinates d3 R(pf , lf , αjf ; pi , li , αj i ) dEfT

dΩf

=

L4 m2 |pf | w(pf , lf , αjf ; pi , li , αj i ), (2π h) ¯ 3 piz

(3.12)

where L is a quantization length and piz = pi cos(θi ) is the incident perpendicular momentum and this appears because it is proportional to the incident particle flux crossing a plane parallel to the surface. The factor of pf arises from expressing the element of phase space volume in spherical coordinates d3 pf = pf2 dpf dΩf = mpf dEfT dΩf . Equation (3.12) gives the probability per unit final translational energy EfT and per unit final solid angle Ωf that a particle initially prepared in a state denoted by quantum numbers i will make a transition to the final state denoted by quantum numbers f . Equation (3.12) is a good representation of what is actually measured by a large number of atomic and molecular surface scattering experiments. It describes an experiment in


61

which all particles scattered from the surface in a particular angular direction can enter the detector, and in practical terms this means that the acceptance angle of the detector is large enough to “see” the entire spot on the surface illuminated by the incident beam. Also, in practical terms, the differential energy element dEfT is the smallest element of energy resolvable by the detection apparatus and dΩf is the solid angle subtended by the detector aperture measured relative to the sample. The type of detector used in the experiment can introduce additional corrections to the differential reflection coefficient of Eq. (3.12). In many experiments, the final molecule is detected as it passes through a short detection length and the overall probability of detection is small. In this case, the detection probability is proportional to the time spent traversing the detector length, which is inversely proportional to the translational speed. Thus for this type of detector, often referred to as a density detector, the differential reflection coefficient must be divided by a factor of pf to account for the detector correction. As examples of other types of detectors, a stagnation detector for atomic scattering requires no correction to Eq. (3.12) while for ion scattering a typical detector is an electrostatic analyzer which requires a correction of 1/pf2 . Another variant of detector correction occurs in experiments utilizing a very large distance between the sample and the detector, such as very long time-of-flight arm used to obtain high translational energy resolution (Brusdeylins et al., 1980, 1981). In such cases the acceptance angle of the detector may be reduced to the point where the detector “sees” only a small portion of the illuminated spot on the sample. In this case, the factor 1/ cos(θi ) associated with the factor 1/piz appearing on the right hand side of Eq. (3.12) must be replaced by 1/ cos(θf ). The reason for this is that in Eq. (3.12) the factor 1/ cos(θi ) accounts for the increase in total area of the spot on the sample surface that is illuminated by the incident beam. However, if the detector does not accept the intensity coming from the whole spot but “sees” only a small portion of this spot then the total area of illuminated surface seen by the detector is dictated by the final scattering angle through a factor of 1/ cos(θf ). Clearly, these two possibilities represent two extremes, and there are many experimental situations in which the illuminated beam spot and the area subtended by the detector are comparable in size. In such situations the geometrical correction factors can get quite complicated. The above remarks on detector corrections represent only a small and idealized sampling of situations that can arise in actual experimental configurations. However, determining the corrections that must be used in each situation can be quite important in comparing theoretical calculations with measurements. For example, in measurements of translational energy-resolved spectra the intensity of scattered particles is plotted as a function of final translational energy. Narrow single surface phonon peaks in such a plot will not be appreciably shifted by a detector correction such as a factor of 1/pf but their relative intensities will be affected. On the other hand, a broad multiphonon background peak will have both its intensity altered and peak position shifted by such a correction, and at low energies such shifts can amount to a substantial fraction of the incident beam energy. Thus, it becomes important to take into account detector corrections in comparing calculations with data.

62

J.R. Manson

3.4. Surface scattering theory 3.4.1. General treatment A starting point for describing molecular scattering from a many-body target is the quantum mechanical state-to-state transition rate for a molecular projectile prepared in a well-defined incident state denoted by the set of quantum numbers {pi , li , αj i } making a transition to a final state denoted by {pf , lf , αjf } This is given by the generalized Fermi golden rule (Rodberg and Thaler, 1967) 2π |Tf i |2 δ(Ef − Ei ) , w(pf , lf , αjf ; pi , li , αj i ) = (3.13) h¯ {n } f

where the average over initial translational and rotational states of the target crystal is denoted by and the sum is over all unmeasured final states {nf } of the crystal which can scatter a projectile into its specified final state. The energies Ei and Ef refer to the total energy of the system of projectile molecule plus the crystal before and after collision, respectively. The Tf i are the matrix elements of the transition operator T taken with respect to unperturbed initial and final states of the system. The transition rate is the fundamental quantity for describing a scattering process, because all measurable quantities in a scattering experiment can be calculated from it. In the semiclassical limit, the transition rate can be expressed as the Fourier transform over all times of a generalized time-dependent correlation function. In this same level of approximation, it is useful to assume the decoupling approximation which in this instance means that the elastic part of the interaction potential is assumed to commute with the inelastic part, and the transition rate is expressed as (Bortolani and Levi, 1986; Manson et al., 1994) w(pf , lf , αjf , pi , li , αj i ) ∞ 1 = 2 |τf i |2 (3.14) e−i(Ef −Ei )t/h¯ exp{−2W} exp Q(t) dt, h¯ −∞ where exp{−2W} is a generalized Debye–Waller factor and Q(t) is a generalized timedependent correlation function. |τf i |2 is the scattering form factor which becomes the square modulus of the off-energy-shell transition matrix of the elastic part of the interaction potential. There can be several mechanisms for energy transfer in the collision process, such as phonons, rotational excitations, and internal mode excitations, each of which is considered here. If each of these processes is considered as independent, then the transition rate can be written in this separability limit as w(pf , lf , αjf , pi , li , αj i ) ∞ V V T T R R 1 = 2 |τf i |2 e−i(Ef −Ei +Ef −Ei +Ef −Ei )t/h¯ h¯ −∞ × KT (t, TS )KR (t, TS )KV (t, TB ) dt,

(3.15)


63

T is the translational energy of the final (f ) or initial (i) projectile where as in Eq. (3.1) Ef,i R is the corresponding rotational energy of the projectile, and E V is the energy of state, Ef,i f,i the projectile’s internal vibrational state. KT (t, TS ) is the scattering kernel for translational motion and phonon excitation, KR (t, TS ) is the scattering kernel for rotational excitation, and KV (t, TB ) is the kernel for internal vibrational mode excitation. Equation (3.15) is self consistent in the following sense: although each of the three energy exchange mechanisms is treated as being independent, each operates taking into consideration the energy losses or gains caused by the other mechanisms. Starting from Eq. (3.15), the problem now becomes one of choosing models for the scattering kernels for each of the energy exchange processes and this is done in the following. In the next subsection energy transfers between the translational energy of a simple atomic projectile and the phonon modes are developed. Section 3.4.3 gives a discussion of the molecular rotational degrees of freedom by treating a rigid rotating molecule colliding with the surface. In Section 3.4.4 the full problem of a molecule with both rotational and vibrational degrees of freedom is discussed.

3.4.2. Atom–surface scattering In order to begin with the phonon modes it is convenient to temporarily ignore the internal degrees of freedom of the molecule and consider the problem of a pseudo-atomic projectile interacting with the surface. For the interaction with phonons, an extension of the semiclassical model originally introduced by Brako and Newns for inelastic scattering of ions and atoms from smooth surfaces is a useful approach (Bortolani and Levi, 1986; Brako and Newns, 1982; Brako, 1982). They showed that in the semiclassical limit the phonon scattering kernel can be expressed in terms of a general exponentiated correlation function QT (R, t), where R is the position vector parallel to the surface, as ∞ KT (t, TS ) = (3.16) dR eiK·R e−2WT (pf ,pi ) eQT (R,t) , −∞

where 2WT (pf , pi ) = QT (R = 0, t = 0) is the contribution to the total Debye–Waller factor due to phonon exchange. If the semiclassical limit is now extended to the limit of rapid collisions in which the semiclassical force exerted on the scattering particle can be replaced by the momentum impulse, then the correlation function simplifies to the time dependent displacement correlation function QT (R, t) = p · u(0, 0)p · u(R, t) h¯ 2 , (3.17) where u(R, t) is the phonon displacement at the position R on the surface. The argument of the Debye–Waller factor is given by the standard form, which for TS greater than the Debye temperature ΘD is

2 2 WT (pf , pi ) = p · u(0, 0) h¯ =

3p 2 TS . 2 2MC kB ΘD

(3.18)

In addition to the straight-forward choice of Eq. (3.17) there are other possibilities for the correlation function QT (R, t) that have been proposed. One choice that has been

64

J.R. Manson

very successful in explaining multiphonon transfers in He atom scattering from surfaces is the exponentiated Born approximation which in its simplest formulation consists of replacing the impulse p appearing in Eq. (3.17) with the distorted wave Born approximation matrix element of the first derivative of the interaction potential with respect to the phonon displacement vectors (Burke et al., 1993; Gumhalter et al., 1994; Gumhalter and Langreth, 1999). Equation (3.16) provides a standard and very convenient method for developing the scattering transition rate into an ordered series in numbers of phonons transferred in the collision process. This is by simply expanding the exponentiated correlation function in powers of its argument QT (R, t). The zero order term gives the elastic scattering contribution and results in

2π w(pf , pi ) = (3.19) |τpf ,pi |2 e−2W (p) δP,h¯ G δ EfT − EiT , h¯ G

where P is the parallel component of the momentum transfer p, G is a reciprocal lattice vector of the surface and τpf ,pi is the transition matrix element of the static elastic interaction potential taken between final state pf and initial state pi . Eq. (3.19) is in fact the exact quantum mechanical expression for diffraction from a periodic surface and it is seen that scattered intensity is observed only if both energy is conserved and if momentum parallel to the surface is conserved modulo a surface reciprocal lattice vector multiplied by h. ¯ To obtain the single phonon contribution exp{QT (R, t)} is expanded to first order which leads to the following approximation to the expression for the transition rate 2π w(pf , pi ) = (3.20) |τpf ,pi |2 e−2W (p) n(ω) + 1 p · ρ(P + h¯ G, ω) · p h¯ G where n(ω) is the Bose–Einstein distribution function for a phonon of frequency ω and ρ(P + h¯ G, ω) is the phonon spectral density tensor, i.e., the function that completely specifies the phonon spectrum of the surface as a function of momentum and frequency (Maradudin et al., 1963). Equation (3.19) is the expression for single phonon creation. The expression for single phonon annihilation is obtained by replacing n(ω)+1 by n(ω). Equation (3.19) is a simple version of one of many approximations proposed for treating single phonon transfers in atom–surface scattering that have proven to be very useful in describing aspects of single phonon He atom scattering (Hulpke, 1992; Bortolani and Levi, 1986; Celli, 1984; Gumhalter, 2001; Hofmann et al., 1994, 1997). A form of the well-known distorted wave Born approximation is obtained if the transition matrix τpf ,pi is replaced by a matrix element of the elastic term of the scattering potential taken with respect to states pf and pi corresponding to energies that differ by the phonon energy h¯ ω. The classical limit of multiple phonon exchange is obtained from Eqs. (3.17) and (3.16) by making an expansion of the correlation function QT (R, t) over small times and small position vectors around the point of collision, leading to

i p · u(0, 0)p · u(R, t) h¯ 2 = 2WT (pf , pi ) − tE0T h¯ E0T kB TS R 2 t2 − 2 E0T kB TS − , 2h¯ 2 vR2 h¯

(3.21)


65

where vR is a weighted average over phonon velocities parallel to the surface (Brako and Newns, 1982; Brako, 1982). If phonons are the only mechanism for energy transfer, such as is often the case in atom scattering, then the scattering kernel developed in Eqs. (3.16)–(3.21) leads to a transition rate that is Gaussian-like in both the translational energy transfer EfT − EiT and the parallel component P of the momentum transfer (Brako and Newns, 1982; Brako, 1982; Manson, 1991; Meyer and Levine, 1984) 3/2 (EfT − EiT + E0 )2 + 2vR2 P2 2h¯ vR2 π |τf i |2 exp − . w(pf , pi ) = Su.c. kB TS E0 4kB TS E0 (3.22) This is the smooth surface scattering model for classical multiphonon transfers in an atom– surface collision. It takes into account the broken symmetry in the perpendicular direction caused by the presence of the surface, i.e., for every phonon exchanged only momentum parallel to the surface is conserved. The perpendicular momentum exchange is not conserved as a consequence of the broken symmetry in that direction. The Debye–Waller factors, normally present in quantum mechanical theory, have disappeared because they were canceled by the first term in Eq. (3.21) leaving the Gaussian-like behavior with an envelope factor that varies as the negative 3/2 power of the recoil energy and surface temperature. There are also other classical expressions for describing classical multiphonon exchange in atomic collisions. If the surface is regarded as a collection of isolated scattering centers in thermodynamic equilibrium, then the scattering kernel in the classical limit becomes simpler than Eq. (3.21): i t2 KT (t, TS ) = exp − tE0T − 2 E0T kB TS , (3.23) h¯ h¯ which leads to the expression of Eq. (3.1) described above in Section 3.2. The smooth surface model of Eq. (3.22) differs from the discrete model of Eq. (3.1) in that it contains the additional Gaussian-like function of the parallel momentum transfer P and the envelope function (i.e., the prefactor) varies as the −3/2 power of TS E0T rather than the −1/2 power. The physical difference between the two expressions of Eqs. (3.22) and (3.1) is the corrugation of the surface. Equation (3.22) describes a surface that is smooth except for the vibrational corrugations caused by the time dependent motions of the underlying atoms, while Eq. (3.1) describes a surface that is highly corrugated, so highly corrugated that each scattering center is distinct. Other models for the classical multiphonon limit have been proposed for surfaces that are corrugated in a manner intermediate between these two extreme limits (Manson, 1998). These intermediate models all have the common feature of Gaussian-like behavior in energy transfer, and in addition show that the mean square corrugation of the surface can be directly related to the temperature and recoil energy dependence of the envelope factor. It is of interest to compare expressions of the type represented by Eqs. (3.22) and (3.1) to the results obtained by carrying out quantum mechanical perturbation theory to high orders in numbers of phonons transferred. The lowest order, single-phonon transfer term in perturbation theory is, for example, given by the distorted wave Born approximation of

66

J.R. Manson

Eq. (3.20) but in general each term in the perturbation series contributes to all orders of numbers of phonons excited. However, in atom–surface scattering, the dominant contributions to multiphonon transfer come from higher order terms in the perturbation series. For example, the double phonon contribution arising from the first order perturbation theory term is negligible compared to the double phonon contribution arising in second-order perturbation theory (Manson and Tompkins, 1977; Armand and Manson, 1984). It can readily be shown that expressions of the type of Eqs. (3.22) and (3.1) can be related to summations to all orders of the perturbation series in which only the smallest numbers of phonon excitations are retained in each term (Manson, 1994). The perturbation series can be calculated term by term, and such calculations have sometimes been useful in the study of multiphonon excitation in atom–surface scattering (Armand and Manson, 1984; Armand et al., 1986), but this approach rapidly leads to lengthy and cumbersome calculations for large numbers of phonons excited. 3.4.3. Rigid molecular rotator The next task is to develop a scattering kernel for the rotational motion of the molecular projectile. In order to be consistent with the smooth surface model of Eq. (3.22) for the translational motion, a model is developed that preserves the correct angular momentum conservation for a rotating molecule interacting with a smooth surface, i.e., angular momentum will be conserved in the direction perpendicular to the surface but not in the directions parallel to the surface. Starting from Eq. (3.13) in the semiclassical limit, but with the proper angular momentum conservation for a smooth surface, the scattering kernel for rotational motion is (Dai and Manson, 2002) ∞ dθz eilz θz /h¯ e−2WR (lf ,li ) eQR (θz ,t) , KR (t, TS ) = (3.24) −∞

where QR (θz , t) is a generalized rotational correlation function and the rotational contribution to the Debye–Waller factor is 2WR (lf , li ) = QR (θz = 0, t = 0). In the limit of a quick collision, where the angular forces are given by the angular impulse, the correlation function becomes a correlation function of the angular displacement Θ(θz , t): QR (θz , t) = l · Θ(0, 0)l · Θ(θz , t) h¯ 2 , (3.25) where l = lf − li is the angular momentum transfer. At this point, the calculation of the angular scattering kernel is still fully quantum mechanical, although it is in the semiclassical limit. The extension to the classical limit of exchange of large numbers of rotational quanta is again similar to Eq. (3.21):

i t2 l · Θ(0, 0)l · Θ(θz , t) /h¯ 2 = 2WR (lf , li ) − tE0R − 2 E0R kB TS h¯ h¯ R E0 kB TS θz2 − , 2 2h¯ 2 ωR

(3.26)

c + l 2 /2I c + l 2 /2I c is the rotational recoil energy, the I c where E0R = lx2 /2Ixx y yy z zz xx,yy,zz are the principal moments of inertia of a surface molecule and ωR is a weighted average


67

of libration frequencies of the surface molecules in the z-direction. The constant ωR plays a similar role for rotational transfers as the weighted average of parallel phonon velocities vR in Eq. (3.22). Both of these quantities can be computed if the complete dynamical structure function of the surface is known although they are usually treated as parameters. The principal moments Iiic are normally expected to be those of a surface molecule in the case of a molecular target. However, if the projectile molecules are large and strike more than one surface molecule simultaneously, then Iiic is expected to become an effective moment of inertia, larger than that of a single molecule. In the case of monatomic solids, an effective surface molecule consisting of two or more surface atoms must be chosen. In 2 I c can be regarded as an alternative choice of the parameter ω . either case, the product ωR R ii The result of combining the rotational kernel of Eq. (3.26) with the phonon kernel of Eq. (3.17) is a transition rate for the scattering of a rigid rotator molecule given by w(pf , lf ; pi , li ) =

1 2 2 (2π)3 vR4 ωR h¯ 2 |τf i |2 T R 2 3 Su.c (E0 ) E0 (kB TS ) 1 2 2P2 vR2 π × exp − kB TS (E0T + E0R ) 4kB TS E0T 2 2lz2 ωR × exp − 4kB TS E0R (EfT − EiT + EfR − EiR + E0T + E0R )2 × exp − . (3.27) 4kB TS (E0T + E0R )

This result is similar to the atomic scattering expression of Eq. (3.22) but with additional terms contributed by the rotational degrees of freedom. In particular, the Gaussian-like term in energy transfer includes the difference between final and initial rotational energies as well as the rotational recoil, and there is an additional Gaussian-like expression in the surface-normal component of the rotational angular momentum. Additional prefactors have also appeared. 3.4.4. Molecular internal vibration modes The remaining task is to include in the transition rate the contributions from vibrational excitations of the internal modes of the molecular projectile. If these modes are treated in the harmonic limit, and consistently with the semiclassical approximations used in obtaining the translational and rotational scattering kernels, the problem becomes that of a collection of forced harmonic oscillators (Mahan, 1990). The general result has been worked out for the case of surface scattering (Manson, 1988), and the internal mode scattering kernel can be written in the following form: KV (t, TB ) =

NA κ,κ =1

f

e

p

p

i(pf ·rκ,κ −pi ·riκ,κ )/h¯ −WVp,κ (pf ,pi ) −WV ,κ (pf ,pi ) QV ,κ,κ (pf ,pi ,t)

e

e

e

,

(3.28)

68

J.R. Manson

where NA is the number of atoms in the molecule. The position of the κth atom of the molecule just before the collision (i) or just after (f ), is given by expressions of the form f f f riκ (t) = riκ + uiκ (t) so that, for example, rκ,κ = rκ − rκ . The vibrational displacement uiκ (t) relative to riκ due to the internal mode, decomposed into cartesian components denoted by β, is Nν h¯ uβκ (t) = (3.29) e(κ |β)[aj e−iωj t + aj+ eiωj t ], 2Nν mκ ωj j j =1

where Nν is the total number of internal modes and mκ is the mass of the κth molecular atom. aj and aj+ are, respectively, the annihilation and creation operators for the j th mode of frequency ωj , and e(κj |β) is the polarization vector which is obtained from a normal modes analysis of the molecule. The displacement correlation function for internal vibration modes of the projectile molecule is then written as p

QV ,κ,κ (pf , pi , t) =

3

Nν

1 e(κj |α)e∗ (κj |α ) √ 2Nν h¯ mκ mκ ωj j =1 α,α =1 × nM (ωj )eiωj t + nM (ωj ) + 1 e−iωj t , pα pα

(3.30)

where nM (ωj ) is the Bose–Einstein function for an assumed equilibrium distribution of initial molecular vibrational states. The Debye–Waller factor associated with the κth atom of the projectile molecule becomes p

WV ,κ (pf , pi ) =

1 p Q (pf , pi , t = 0). 2 V ,κ=κ

(3.31)

p

In the internal mode correlation function QV ,κ,κ (pf , pi , t) the normal modes commute with each other since they are independent, and Eq. (3.30) can be further expanded to (Wyld, 1993) p

e

QV ,κ,κ (pf ,pi ,t)

Nν

Q

p

(pf ,pi ,t)

= e j =1 jV ,κ,κ Nν ∞

nM (ωj ) + 1 αj /2 −iα ω t = e j j , (3.32) I|αj | bκ,κ (ωj ) nM (ωj ) α =−∞ j =1

j

where I|αj | (z) is the modified Bessel function of integer order αj and argument z. The argument of the modified Bessel function of Eq. (3.32) is given by bκ,κ (ωj ) =

3

pα pα

α,α =1

×

1 e(κνj |α)e∗ (κνj |α ) √ Nν h¯ mκ mκ ωj

nM (ωj ) nM (ωj ) + 1 .

(3.33)

Equations (3.28) through (3.33) define the scattering kernel for excitation of internal molecular modes.


69

The three scattering kernels for translation, rotation and internal vibrations can now be inserted back into Eq. (3.15) and all integrals can be readily carried out. The general result for the transition rate is w(pf , lf , pi , li ) 2 1/2 2π h¯ 2 vR2 2π h¯ 2 ωR 1 2 = 2 |τf i | E0T kB TS E0R kB TS h¯ 1/2 2 2lz2 ωR 2P2 vR2 π h¯ 2 × exp − exp − (E0T + E0R )kB TS 4E0T kB TS 4E0R kB TS NA f p p i(p ·r −p ·ri )/h −W (p ,p ) × e f κ,κ i κ,κ ¯ e−WV ,κ (pf ,pi ) e V ,κ f i κ,κ =1

×

Nν ∞ j =1 αj =−∞

I|αj |

nM (ωj ) + 1 αj /2 bκ,κ (ωj ) nM (ωj )

ν 2 (EfT − EiT + EfR − EiR + E0T + E0R + h¯ N s=1 αs ωs ) × exp − . 4(E0T + E0R )kB TS (3.34) The transition rate of Eq. (3.34) is actually expressed, for compactness, as a product over all normal modes labeled by j and a summation over the excitation quantum number denoted by αj . To obtain the discrete transition rate to a particular internal mode final state or combination of states, one takes the corresponding (j, αj )th term of Eq. (3.34). The result of Eq. (3.34) retains many of the features of the simpler expressions of Eqs. (3.22) and (3.1) for atomic scattering with the exchange of only phonons. The dominant feature is the Gaussian-like function containing the three different modes of energy exchange (phonons, rotations and internal molecular vibrations) together with the recoil terms from phonons E0T and from rotational exchange E0R . The width of the Gaussianlike function varies as the square root of the temperature and the sum of the two recoil energies. This is not a true Gaussian because of the momentum dependencies of the recoil energies. There are also Gaussian-like functions in the exchange of parallel momentum P and perpendicular angular momentum lz that arise from retaining the correct momentum conservation conditions for a smooth surface. The envelope factors vary as negative powers of the temperature multiplied by recoil energies. These envelope factors guarantee the overall unitarity of the total scattered intensity, i.e., as the temperature and/or incident energy is increased, the maximum intensity of the Gaussian-like function decreases in order that the total integral over final states remains constant. In Eq. (3.34) the quantum behavior of the internal mode excitations is expressed differently than the classical behavior for translational and rotational motion. The strength of the αj quantum excitation of the j th mode is proportional to the modified Bessel function I|αj | (bκ,κ (ωj )). Because these are quantum features the vibrational contribution to the Debye–Waller factor is still present, as are quantum phase factors involving the positions rκ of the individual molecular atoms before and after the collision. The presence of these quantum phase factors can cause interference effects, but because in an actual experiment

70

J.R. Manson

the orientation of a molecule is not measured, the transition rate must be averaged over molecular orientations in order to compare directly with experiments. Recoil effects due to the excitation of internal modes are not explicitly apparent in Eq. (3.34), but they are included within the modified Bessel function. In many cases, such as where the incident molecular energy and the surface temperature are not large compared to the energy of internal molecular vibrational excitations, the expansion of Eq. (3.34) to only single quantum excitations is sufficient. This expansion is w(pf , lf , pi , li ) 1/2 2π h¯ 2 vR2 1 π h¯ 2 2 = 2 |τf i | (E0T + E0R )kB TS E0T kB TS h¯ 1/2 2 2 2lz2 ωR 2π h¯ 2 ωR 2P2 vR2 × exp − exp − E0R kB TS 4E0T kB TS 4E0R kB TS ×

NA

f

e

p

i(pf ·rκ,κ −pi ·riκ,κ )/h¯ −WVp,κ (pf ,pi ) −WV ,κ (pf ,pi )

e

e

κ,κ =1

(EfT − EiT + EfR − EiR + E0T + E0R )2 × exp − 4(E0T + E0R )kB TS +

3 γ ,γ =1

pγ pγ

Nν j =1

1 e(κj |γ )e∗ (κj |γ ) √ 2h¯ Nν mκ mκ ωj

(EfT − EiT + EfR − EiR + E0T + E0R − h¯ ωj )2 × nM (ωj ) exp − 4(E0T + E0R )kB TS

+ nM (ωj ) + 1 (EfT − EiT + EfR − EiR + E0T + E0R + hω ¯ j )2 . × exp − (3.35) 4(E0T + E0R )kB TS Of the three terms in Eq. (3.35) the one proportional to nM (ωj )+1 gives the single quantum creation rate, the term proportional to nM (ωj ) is for single quantum annihilation, and the third term is the rate for scattering with no internal mode excitation. 3.4.5. Interaction potential In obtaining the molecular scattering transition rates of Eqs. (3.34) and (3.35) the major input so far has been the statistical mechanics and the general conservation laws governing the various degrees of freedom. The interaction potential determines the scattering form factor |τf i |2 . In the semiclassical limit of interest here τf i has been identified as the transition matrix for inelastic scattering, calculated from the elastic part of the potential, i.e., the transition rate for the elastic potential extended off of the energy shell (Manson et al., 1994). A very useful expression is one suggested by Mott and Jackson in their very early work, and is the same as is used for most applications of the distorted wave Born approximation. This is to let τf i be the off-energy-shell matrix element of the elastic part of the


71

molecule–surface potential taken with respect to its own eigenfunctions between the initial and final state. An example of this is the Mott–Jackson matrix element of an exponentially repulsive potential V 0 (z) = V0 e−βz .

(3.36)

This is a one-dimensional potential, and its matrix elements are given by (Goodman and Wachman, 1976) πqi qf (qf2 − qi2 ) sinh(πqf ) sinh(πqi ) 1/2 h¯ 2 β 2 vJ −M (pf z , piz ) = , m cosh(πqf ) − cosh(πqi ) qi qf (3.37) where qi = piz /h¯ β and qf = pf z /h¯ β. In the semiclassical limit of a hard repulsive surface, β → ∞, the Jackson–Mott matrix element, as well as matrix elements for other 1-D potentials with a hard repulsive part, becomes vJ −M (pf z , piz ) → 2pf z piz /m.

(3.38)

Since this is a quite general limiting case for a large class of potentials having a strong repulsive term to represent the surface barrier it is reasonable to use it as an approximation for τf i . In the comparisons of calculations with experimental data discussed in the sections below, this is the form factor that is applied. The Van der Waals force between the surface and the incident molecular projectile gives rise to an attractive well in front of the surface. This is the physisorption well and it can have important effects at low incident translational energies. In the case of classical scattering, the main consequence of the Van der Waals attractive potential is to enhance the energy of the incoming particle associated with the direction normal to the surface. This is a refractive effect quite similar to that which occurs for light waves in optical media. Since refraction is the dominant effect, the potential can be modeled by a one-dimensional potential well and for a given well depth |D|, the refraction does not depend on the functional shape of the attractive part of the potential. Thus, this attractive force can be simulated by an attractive one-dimensional square-well potential in front of the repulsive barrier, and the width of the well is unimportant. The effect of the collision process is to replace the perpendicular component of the momentum pqz near the surface and inside the well by a larger value , which includes the depth D pqz 2 p qz = pqz + 2m|D|. 2

(3.39)

This refracts all projectiles at the leading edge of the well and causes them to collide with the repulsive barrier with a higher normal energy. The expressions for projectile translational energy and for scattering angle inside of the potential well become, respectively T E f,i = Ef,i + |D|, T

and cos(θf )

=

EfT cos2 (θf ) + |D| EfT + |D|

(3.40) 1/2 .

(3.41)

72

J.R. Manson

Transition rates calculated for molecular scattering inside the well can then be projected to the asymptotic region outside the well by multiplication with an appropriate Jacobian function, which can be easily determined from Eqs. (3.40) and (3.41).

3.5. Comparisons with experiment In the following a number of comparisons of calculations with experimental data are presented for several different types of experiments that illustrate the importance of multiphonon energy transfer in molecule–surface collisions. There are four different categories of experiments considered, (1) angular distributions for which the measured quantity is the total scattered intensity regardless of energy or internal state, (2) translational energy resolved spectra taken at fixed incident and detector angles, usually measured by time-offlight methods, (3) rotational energy resolved spectra, and (4) probabilities for excitation of internal molecular vibrational modes. 3.5.1. Angular distributions There is a long history of measurements of angular distributions in atomic and molecular scattering from surfaces and two recent examples are shown here, O2 scattering from Al(111) and CH4 scattering from Pt(111). The O2 /Al(111) system is interesting because of recent work that has exhibited an anomaly in the adsorption probability associated with the incident beam angle. Although it is well-known that aluminum oxidizes readily, the Al(111) surface is surprisingly resistant to oxide formation upon bombardment with oxygen molecules. Recent molecular beams scattering experiments provide evidence for the existence of a weakly bound molecular state, one whose binding energy is less than 0.1 eV (Weiße et al., 2003). An extremely interesting feature observed in this work was a clearly pronounced dip in the observed backscattered O2 intensity in the neighborhood of incident beam angles of 25◦ with respect to the surface normal, and at energies in the range between 90 and 300 meV. This feature is interesting because the authors of Ref. (Österlund et al., 1997) found an enhancement of the sticking probability at nearly the same incident conditions, raising the intriguing question of why should the sticking probability be substantially larger, not at normal incidence as expected, but at an incident polar angle where the normal translational momentum is significantly reduced. The observed increase in sticking associated with a concomitant decrease in scattering intensity occurring over a narrow angular range implies a very specific mechanism may be responsible. Molecular dynamics simulations utilizing a potential energy surface containing three molecular degrees of freedom support the proposition that this feature is caused by steering into a shallow molecular adsorption well located above the same position in the surface unit cell as the maximum in the barrier towards dissociative adsorption (Weiße et al., 2003). Conventional spectroscopic measurements such as HREELS and UPS have not observed molecular O2 in a precursor state. However, the steering mechanism of Ref. (Weiße et al., 2003) provides an explanation for the absence of observable adsorbed O2 because such a state would have a lifetime too short to be measurable by conventional spectroscopy.


73

Early STM experiments indicated that exposing Al(111) to a room temperature gas of O2 led to isolated adsorbate atoms at very low coverage, while at coverages above 3% 1 × 1 islands of oxygen were formed (Brune et al., 1993). More recently Österlund et al. (1997) showed, in a molecular beams study, that the chemisorption reaction probability was below 1% for O2 beams of very low translational energy, at or just above 25 meV, but rapidly rises to 90% as the incident energy is increased to 1 eV. They interpreted the reaction as direct dissociative desorption. A series of experiments using X-ray photoelectron spectroscopy (XPS) and high resolution electron energy loss spectroscopy (HREELS) were interpreted as indicating that the dissociative desorption is precursor mediated (Zhukov et al., 1999). Very recent molecular beams experiments, corroborated by STM measurements and supported by theoretical molecular dynamics simulations, gave evidence for the existence of an abstraction channel in the dissociative adsorption (Komrowski et al., 2001; Binetti et al., 2003). On the other hand, independent STM studies have pointed out that there are alternative interpretations of the observed oxygen adsorption features (Schmid et al., 2001). Thus, the picture emerging from these widely differing experimental investigations is that the oxidation of Al(111) is a complicated process, probably consisting of several simultaneous and overlapping channels. Clearly, it is of interest to examine the available scattering data for the O2 /Al(111) system in order to investigate the role that energy exchange to the phonon field of the surface may be playing and how such energy exchange may affect the sticking process. Examples of the experimentally measured O2 scattering angular distributions from clean Al(111) are shown in Figs. 3.1 and 3.2 and compared to the theoretical calculations using Eq. (3.27). The calculations are averaged over an incident beam with a Maxwell– Boltzmann distribution of rotational energies with rotational temperature 35 K. The solid curves are calculations for a potential with well depth D = 50 meV and the dash-dotted curves are for D = 0. There are two parameters in the theoretical expression of Eq. (3.27), the weighted parallel phonon speed vR and frequency ωR . For vR we have chosen the value 2300 m/s for calculations used to fit the data shown in this paper. In general, vR is expected to be of the order or somewhat smaller than the Rayleigh velocity (Brako and Newns, 1982; Brako, 1982), so the value chosen here compares favorably with the Rayleigh speed measured at 3200 m/s for the 110 and 112 symmetry directions of Al(111) (Lock et al., 1988). The value of ωR is taken to be 1010 s−1 , but the calculated results are essentially independent of ωR for values of this small order of magnitude. In Fig. 3.1 it is seen that the theory predicts broad angular distributions with FWHMs of about 20◦ and peak maxima located near the specular positions or slightly subspecular, in reasonable agreement with experiment. The effect of including a well in the interaction is to shift the calculated angular distribution slightly in the supraspecular direction. This shift can be understood on the basis of the larger average energy losses caused by the molecule colliding with the repulsive surface at a higher effective energy and a more normal collision angle inside the well. Overall, the calculations shown in Fig. 3.1 with a 50 meV well seem to agree somewhat better with the experiment than those without a well, although such a statement cannot be made unambiguously because the agreement also depends on the choice of vR . Nevertheless, the agreement shown in Fig. 3.1 indicates that the results are not inconsistent with the presence of a small attractive well such as proposed in Weiße et al. (2003).

74

J.R. Manson

Fig. 3.1. Angular distributions of O2 scattering from Al(111) for different incident angles as shown with an incident energy Ei = 90 meV and surface temperature TS = 298 K. The experimental data points are shown as symbols (Ambaye et al., 2004). The calculated results are shown as solid curves for a potential with well depth D = 50 meV, and as dash-dotted curves for D = 0.

It is noticeable in Fig. 3.1 that the experimental points appear to lie below the calculated curves at final angles in the neighborhood of the normal direction, and this is especially the case when the incident angle is also near normal. This disagreement may be an artifact of correcting the scattered intensity for the small fraction of the incident beam that can enter the detector from the rear, before striking the surface, when the detector and incident beam directions are close to each other. In the case of incident angles smaller than 20◦ an over-compensation of the correction for this effect may account for much of the apparent discrepancy between experiment and theory (Ambaye et al., 2004). Figure 3.2 shows the most interesting behavior of the measured scattering data. For Ei = 90 meV and a fixed incident angle θi = 20◦ the temperature dependent evolution of the angular distribution lobes is exhibited for 98 < TS < 300 K. At the lowest temperatures the most probable intensity occurs at an angle of about 25◦ , slightly larger than specular. However, as the temperature is increased up to room temperature the angular distribution lobe undergoes a shift towards the normal direction and even becomes slightly subspecular. Near specular or subspecular angular distribution lobes are not expected on the basis of predictions from simple theoretical models that do not allow for parallel momentum


75

Fig. 3.2. Angular distributions for O2 /Al(111) as a function of surface temperature. The incident angle and energy are θi = 20◦ and Ei = 90 meV. The data are shown as symbols (Ambaye et al., 2004), and the calculated curves are for D = 50 meV (solid curve) and D = 0 (dash-dotted curve) as in Fig. 3.1.

exchange with the surface. In situations where the incident energy is large compared to the surface temperature the molecule will lose a significant fraction of its incident translational energy. Thus, if the parallel momentum of the projectile is not allowed to change, the perpendicular momentum will become much smaller which would predict a distinctly supraspecular angular distribution lobe. However, the current theoretical models of Eq. (3.34) allows for the correct transfer of momentum parallel to the surface, i.e., the parallel momentum of the scattered particle is equal to the parallel momentum of the incident particle plus whatever parallel momentum is gained or lost to the phonons exchanged. Perpendicular momentum, on the other hand, is not conserved due to the broken symmetry presented by the surface. Perpendicular momentum is indeed exchanged with the surface, but there is no conservation law in that direction. Thus, the projectile’s final perpendicular momentum is determined by the combined laws of energy conservation and parallel momentum conservation. Clearly, the agreement between theory and experiment shown in Fig. 3.2 indicates that the temperature dependence of the angular distribution data can be explained by a theory containing the correct conservation laws, i.e., the present theory explains the subspecular shift with increasing temperature. As the temperature is increased, proportionately less energy is lost on average by the scattered particles (i.e., they are heated up). Because of the constraint of conservation of parallel momentum, more of this increase in energy goes into the increase of final normal momentum, thus making the angular distribution shift towards the normal.

76

J.R. Manson

As a second example of angular distributions, recently available data for the scattering of methane from clean and ordered Pt(111) (Yagu et al., 1999, 2000, 2001; Tomii et al., 2000; Kondo et al., 2002a, 2003a; Kondo, private communication) are presented and compared with calculations (Moroz and Manson, 2005). The experimental apparatus is constrained to a fixed angle of 90◦ between the beam incident at the angle θi and the final detector direction θf , thus for the measured angular distributions each final angle corresponds to a different incident angle according to the relation θf = 90◦ − θi . The energy resolution of the incident beam was E/E = 0.06 and the angular dispersion was less than 0.5◦ (Yagu et al., 2000). Pt(111) is highly reactive and over a period of time under bombardment by the CH4 beam the surface becomes contaminated with adsorbed molecules as well as dissociation products. In the experiment careful studies were made to assess the rate of build-up of contaminant products on the surface (Kondo et al., 2002a) and it was found that for initial periods of several minutes after routine cleaning protocols were completed the surface remained clean and the quality of the scattered spectra did not degrade. Thus all measurements considered are for scattering from a clean and uncontaminated surface. For the calculations the incident molecular beam is chosen to be in a state that approximates an equilibrium distribution of rotational and vibrational states at low temperatures. The incident beam is averaged over a Boltzmann distribution of rotational states at a temperature of 30 K, approximately the estimated experimental conditions. The vibrational temperature is estimated to be significantly less than 100 K and for the calculations presented here a value of 10 K was used. The calculations are essentially unchanged by variations of more than a factor of two in either of these parameters. The calculated results are averaged over angular orientation of the molecules, and over angular orientation of the angular momentum. The detector used in these experiments is a velocity-dependent density detector, and consequently the calculated spectra must be corrected by a factor of 1/pf in order to compare with the measurements. The calculations require the polarization vectors for the normal modes of the CH4 molecule. These were obtained using a standard classical normal modes analysis in the harmonic approximation (Wilson et al., 1955; Nakamoto, 1970; Woodward, 1972). For CH4 there are a total of nine normal modes that have four distinct frequencies, three of which are degenerate. Calculations indicate that only the two lowest frequency modes with energies hω ¯ = 190.2 and 161.9 meV are appreciably excited under the experimental conditions considered here, with the maximum excitation probability at Ei = 500 meV of approximately 4% and 5%, respectively. Angular distributions for the CH4 /Pt(111) system were measured at two incident energies, Ei = 190 and 500 meV, and for surface temperatures ranging from 400 to somewhat over 800 K. The crystal azimuthal direction was 112. Figure 3.3 shows the experimental data compared to calculations for an angular distribution taken at a temperature of 827 K. The calculated curve is obtained by integrating the differential reflection coefficient of Eq. (3.34) over all final energies and final angular momenta, averaging over an initial Boltzmann rotational distribution at a temperature of 30 K, summing over all excitations of internal vibrational modes, and averaging over molecular orientations. The well depth is taken to be D = 40 meV and the crystal mass MC is that of a single Pt atom.


77

Fig. 3.3. Angular intensity distributions for methane CH4 scattered from a Pt(111) surface compared to the calculations shown as solid curves. The incident translational energies are: (a) 500 meV; (b) 190 meV. The surface temperature is 827 K. Data is from Kondo et al., (2002b, 2003b). The vertical line marks the specular position.

It is seen that the most probable angle of the angular distributions are slightly supraspecular at the lower incident energy and approximately 5◦ larger than the specular position at the higher energy. The specular position θS = 45◦ is indicated by the vertical line. The experimental data and in particular the supraspecular shift with increasing energy are well described by the calculations. In addition the small decrease in full width at half maximum (FWHM) with the larger energy is matched by the calculations. The temperature dependence of the angular distributions is shown in Fig. 3.4 which gives a series of graphs at the lower incident translational energy of 190 meV ranging from T = 400 to 700 K. Also shown is the data at T = 827 K from Fig. 3.3 which was taken on a different day. All other conditions are the same as in Fig. 3.3. It is seen that with increasing temperature there is a small subspecular shift of the most probable scattering angle towards the specular position and an increase in the full with at half maximum, both of which are well matched by the calculations. This subspecular shift is similar to the case for O2 /Al(111) discussed above in Fig. 3.2 and also was observed for methane scattering from LiF(001) (Moroz and Manson, 2004). 3.5.2. Translational energy resolved spectra Multiphonon energy transfer is perhaps most clearly manifest in translational energy resolved spectra and this will be discussed using examples of time-of-flight measurements of methane scattering from both Pt(111) and LiF(001). The experimental measurements on scattering of CH4 from a clean ordered LiF(001) surface that are considered here were taken by the same experimental group as the data for the angular distributions in Figs. 3.3

78

J.R. Manson

Fig. 3.4. CH4 /Pt(111): Angular intensity distributions for different surface temperatures TS = 827 (a), 700 (b), 600 (c), 500 (d) and 400 K (e) as marked. The incident energy is 190 meV and the vertical line marks the specular position. Data is from Kondo et al. (2002b, 2003b).

and 3.4. They were carried out with an incident molecular beam whose translational energy varied between 190 and 500 meV by heating the supersonic jet nozzle and seeding with helium gas (Yagu et al., 2000). Surface temperatures were varied from 300 K to 700 K. All measurements were carried out in the scattering plane, which contains the incident beam and the surface normal. The detector was positioned at a fixed angle of 90◦ from the incident beam, meaning that the incident and final angles at which all measurements were taken are related by θf = 90◦ − θi . An example of TOF energy-resolved spectra compared with calculations based on Eq. (3.34) is shown in Fig. 3.5. The experimental data are exhibited as points for the incident energy of 350 meV. The surface temperature is 300 K and the crystal azimuthal direction is 110. The incident angle goes from θi = 30◦ to θi = 50◦ implying final angles from 60◦ to 40◦ which gives a good sampling ranging across the maxima observed in the angular distributions which were also measured for this system. The vertical lines in each figure indicate the TOF time for elastic scattering (dashed line) and the time corresponding to the Baule estimate of recoil energy loss for hard sphere scattering at these angles (dashed-dotted line) assuming masses of methane and LiF. The Baule estimate is identical for all of the incident angles because the fixed-angle constraint between incident beam and detector makes the total scattering angle always the same. The calculations from Eq. (3.34) shown as solid curves are the sum of the differential reflection coefficient over all final rotational angular momenta and all molecular internal modes, and are for a potential well depth of zero, a rotational temperature of the incident beam of 30 K, and an incident beam vibrational temperature of 10 K. The principal mo-


79

Fig. 3.5. Time of flight distributions of CH4 molecules scattered from a LiF(001) surface in the 110 azimuthal direction for different incident angles θi = 50◦ (a); 40◦ (b); 37.5◦ (c) and 30◦ (d). The incident translational energy is 350 meV and the surface temperature is 300 K. The experimental measurements are shown as data points (Yagu et al., 2000; Tomii et al., 2000) and the calculations are shown as solid lines. The vertical dashed line is the position of elastic scattering and the vertical dash-dot line indicates the hard-sphere energy loss position.

ments of inertia of the CH4 and LiF molecules were calculated classically with the known atomic masses and molecular bonding distances leading to values of the principal moments of inertia of IM = 5.26 × 10−47 kg m2 for CH4 and IC = 3.41 × 10−46 kg m2 for LiF. The velocity vR was chosen to be 1500 m/s which provides a reasonable fit to all data for both TOF spectra and angular distributions. Interestingly, decreasing vR by a factor of two has very little effect on the position of the maximum in the calculated TOF distribution but it increases the width by about 10%, particularly on the large time side. Conversely, increasing vR by a factor of two narrows the distribution by about 10% but again leaves the maximum in nearly the same position. The corresponding weighted rotational speed parameter ωR was chosen to be 1010 s−1 but for values of this small magnitude the calculated results are completely independent of ωR . For the value of MC , the effective mass of the surface molecules, we have used three times the mass of an LiF molecule. A larger effective mass indicates a collective effect in which the incident projectile is colliding with more than a single LiF molecule. The initial analysis using the washboard model carried out by the authors of the experimental study also required surfaces masses several times larger than that of a single LiF (Yagu et al., 2000). The angular dependence of the energy resolved measurements is clearly shown in Fig. 3.5. It is interesting to note that the translational energy losses are greater for a beam with near-normal incidence than for one with more grazing incidence. For θi = 30◦ (and

80

J.R. Manson

Fig. 3.6. Time of flight spectra of CH4 molecules scattered from a Pt(111) surface for different translational energies: 500 meV (a) and 190 meV (b). The incident angle is θi = 45◦ and the surface temperature is 827 K. The two vertical lines indicate the Baule recoil energy loss (dashed-dotted line) and elastic scattering (dashed line) as in Fig. 3.5. The data is from Kondo et al. (2002b, 2003b).

θf = 60◦ ) very few molecules gain energy from the surface and emerge with final energies larger than the incident energy. However, for the more grazing incident beam having θi = 50◦ (and θf = 40◦ ) a significant fraction of the molecules actually gain translational energy from the surface. The calculations follow nicely the incident angular trends observed in the experimental data. Both the data and calculations illustrate the limitations of the simple Baule estimate for the energy loss. For near-normal incident beams the average final energy is substantially less than the Baule estimate, while for more grazing angular incidence the average energy is larger than the Baule estimate. An example of translational energy-resolved spectra for methane scattering from the Pt(111) surface is shown in Fig. 3.6 for EiT = 190 and 500 meV and a surface temperature of 827 K. The incident beam and detector angles are both 45◦ . The calculations shown as solid curves are with a well depth of D = 40 meV. As in Fig. 3.5 the TOF time corresponding to elastic scattering is indicated by the vertical dashed line and the Baule estimate for energy transfer is indicated by the dash-dotted vertical line which at this specular scattering angle is about 15% of the incident energy. It is seen that under these scattering conditions the measured most probable final energy shows an energy loss of approximately the same as the expected recoil loss with a single surface atom, but there is a significant fraction of scattered CH4 molecules that leave the surface at energies larger than that of the incident beam. The FWHM of the scattered distribution becomes significantly smaller with increasing incident energy. The calculated curves match the peak position and the high energy (small TOF time) of the data, but again as in the case of CH4 /LiF(001) of Fig. 3.5 they do not explain the long tail at low energies.


81

Fig. 3.7. CH4 /LiF(001): Time of flight distributions for different values of well depth |D|. The incident energies are: 500 meV (a); 350 meV (b); 190 meV (c). The incident angle is θi = 50◦ and the surface temperature is 300 K. The experimental measurements are shown as data points (Yagu et al., 2000; Tomii et al., 2000; Kondo et al., 2002a) and the calculations for D = 0, 25 and 65 meV shown as solid, long-dashed and dot-dashed lines, respectively. The vertical lines indicate elastic and hard sphere scattering as in Fig. 3.5.

The next example shows one way in which important physical information about the interaction potential can be obtained from the translational energy resolved spectra, in this case information about the physisorption well. Figure 3.7 shows the TOF data for CH4 /LiF(001) at the three incident energies of 190, 350 and 500 meV for θi = 50◦ compared with calculations with well depths D = 0, 25 and 65 meV. At lower incident energies, for the TOF measurements better agreement between data and theory are obtained when an attractive well is included in the potential. At the highest incident energy of 500 meV the presence of a well, which in this case is significantly smaller than the energy, has little effect on the calculated results. However, at the lowest energy of 190 meV there is a significant effect. At all incident energies, the effect of the well is not significant at small TOF times, but it increases the calculated intensity at large TOF times (small final energies) and gives a small shift of the most probable final energy towards smaller final energies. Both of these latter effects bring the calculations into better agreement with experiment, with the best agreement corresponding to the larger well. The comparisons between experiment and calculations in Fig. 3.7 indicate that not only does the physisorption well strongly influence the multiphonon energy transfer at lower incident energies, but also in some cases can be used to estimate the well depth.

82

J.R. Manson

3.5.3. Rotational energy resolved spectra A number of experimental studies have made direct measurements of the exchange of translational energy to rotational energy (Mortensen et al., 2003; Sitz et al., 1988; Janda et al., 1983; Budde et al., 1987), usually presented as plots of the scattered average translational energy EfT as a function of rotational energy EfR . These plots generally show a negative correlation, that is to say the average translational energy tends to decrease with increasing rotational energy. Intuitively, such behavior would be suggested by the law of conservation of energy as applied to these two degrees of freedom, but this is not completely valid because there is always the possibility of energy transfer to and from the surface and the internal vibrational modes. Discussed here is one of these experiments for NO scattering from Ag(111) (Rettner et al., 1991). A series of graphs of EfT versus EfR is shown in Fig. 3.8. The surface temperature is 450 K, incident translational energies range from 99 meV to 963 meV, and the measurements were made under specular conditions with incident angles ranging from 15◦ to 60◦ as marked. In all cases except for the lowest energy incident beam, the average final energy decreases by 10–20% over the range of measured final rotational energies,

Fig. 3.8. NO/Ag(111): Final average translational energy as a function of final rotational energy for several incident energies and angles taken at the specular scattering angle. In each panel the experiment and theory for a given incident energy and angle are compared. Open circles are data at θi = 60◦ , solid circles are data at 45◦ , solid triangles are data at 30◦ and solid squares are data at 15◦ . Calculations are the curves in each panel. TS = 450 K. The solid curve in the panel for EiT = 99 meV is the calculated most probable final translational energy. Data are taken from Rettner et al. (1991).


83

which in some cases extends to as large as 400 meV. Our calculations, shown as curves, also reproduce this same behavior rather well except for the lowest energy. The case of the low energy EiT = 99 meV beam requires further examination. As opposed to the higher energy measurements, there is essentially no decrease in measured average final energy as EfR increases and the final energy is about 60% of the incident energy. This would indicate that a large fraction of the incident translational energy is transferred to the surface lattice and relatively little is being converted to molecular rotational energy. The theoretical calculations, on the other hand, predict a 10% decrease over the measured EfR energy range, and additionally they predict that the average final translational energy should be larger than the incident energy rather than smaller as measured. The answer to both of these discrepancies appears to lie in the manner in which the average final translational energy is defined. It appears to be customary to report experimental measurements as average final translational energies but, however, what is actually measured is closer to the most probable final energy corresponding to the peak position in the intensity versus final translational energy (Luntz, private communication; Sitz, private communication). For large incident beam energies, there is usually little discrepancy between the two definitions but this is not necessarily the case for low energy beams. At low energies comparable to the surface temperature there can be a significant fraction of the scattered particles that gain energy from the surface and this causes the energy-resolved spectra to become quite skewed and non-symmetric about the most probable final energy. Thus the average and most probable energies can become quite different, whereas for high energy incident beams they usually are nearly the same. The solid curve shown in the lower panel of Fig. 3.8 for EiT = 99 meV is the most probable final translational energy as a function of EfR . This appears at a position of net energy loss and shows less dependence on EfR in much better agreement with the experimental points. Thus it appears that, with this reinterpretation of the experimental points at low incident beam energy, the theory predicts relatively well the exchange of energy between translational and rotational degrees of freedom of the molecule, in addition to predicting the overall energy loss or gain to the surface phonon field. A second type of experiment involving rotational energy resolution consists of measurements of the scattered intensity as a function of final rotational energy. Several examples of such spectra for the scattering of NO by Ag(111) are shown in Fig. 3.9 in which the scattered intensity with the detector in the scattering plane and positioned at the specular angle (θf = θi ) is plotted as a function of final rotational energy in a semi-logarithmic graph (Rettner et al., 1991; Kleyn et al., 1981). In Fig. 3.9(a) the surface temperature for all measurements was 650 K, the upper three sets of data were measured at an angle θi = 15◦ with three different incident translational energies (Kleyn et al., 1981), 1000 meV (open circles), 750 meV (filled circles) and 320 meV (open diamonds); the fourth data set (filled diamonds) was taken for EiT = 320 meV and θi = 40◦ . Figure 3.9(b) shows a second set of measurements for the same system taken by the same group at a later date. All those measurements were for θi = 15◦ and two different energies and temperatures: EiT = 850 meV and TS = 273 K (open squares), EiT = 850 meV and TS = 520 K (filled squares), and EiT = 90 meV and TS = 520 K (open triangles) (Rettner et al., 1991). The intensities are presented in arbitrary units, and for clarity the various data sets are separated by arbitrary constant amounts.

84

J.R. Manson

Fig. 3.9. NO/Ag(111): Final rotational energy resolved intensity at specular scattering angles for several incident energies and surface temperatures. (a) Data taken from Kleyn et al. (1981): TS = 650 K; open circles are data for Ei = 1000 meV and θi = 15◦ , solid circles are data Ei = 750 meV and θi = 15◦ , open diamonds are data for Ei = 320 meV and θi = 15◦ and solid diamond is data Ei = 320 meV and θi = 40◦ . (b) Data taken from Rettner et al. (1991). θi = 15◦ ; open squares are data for Ei = 850 meV and TS = 273 K, solid squares are data for Ei = 850 meV and TS = 520 K and triangles are data for Ei = 90 meV and TS = 520 K. Curves are calculations.

These experimental measurements exhibit several evident characteristics; the intensity decreases strongly with increasing rotational energy and there is a steep initial decrease for small EfR followed by a large range of EfR for which the intensity decreases nearly exponentially (i.e., nearly a straight line in the logarithmic plot). For some of the measurements, in particular for those taken at higher incident beam energies, there is a second pronounced decrease in intensity for very large EfR . This high rotational energy feature is termed the rotational rainbow and its onset marks the classical limit of angular momentum transfer to a molecule in a single scattering collision.


85

Also shown in Fig. 3.9 are curves calculated from Eq. (3.34). For each final rotational energy these calculations are the sum of the differential reflection coefficient over the internal vibrational stretch mode excitations, final translational energies, and an average is carried out over all possible molecular orientations. The differential reflection coefficient is also averaged over an assumed Maxwell–Boltzmann distribution of initial rotational states with a temperature TR = 35 K which is in approximate agreement with experimental conditions (Kleyn et al., 1981). A distribution of incident molecular vibrational frequencies at a temperature of 125 K, somewhat larger than the rotational temperature, is also assumed in agreement with estimated experimental conditions. The calculations were found to depend only weakly on the distribution of initial rotational states as long as the average rotational energy was small compared to the incident translational energy, and this is discussed further in connection with Fig. 3.11 below. The velocity parameter vR is given a value of 1000 m/s and ωR is taken to be 1010 s−1 , but as in previous cases the results are not strongly dependent on these parameters, and in this case are not affected even by changes of a factor of two or more. Relatively good quantitative agreement is seen between calculations and experiment. The initial steep decline in intensity is somewhat overemphasized by the calculations but it occurs over the same energy range as the observations. The long plateau of exponential decay for intermediate energies is well matched. The calculations do not reproduce the rotational rainbow behavior observed at higher incident energies, but this is understandable because the simple interaction potential of Eq. (3.38) used to calculate the scattering form factor does not include the possibility of a rainbow. The data of Fig. 3.9(a) have in an earlier study been analyzed using molecular dynamics simulations that included a relatively sophisticated interaction potential energy surface (Kimman et al., 1986). This study obtained a good description of the experimental data and in particular obtained a correct prediction of the rotational rainbow. The same molecular dynamics study was used to explain the data for final translational energy as a function of rotational energy shown above in Fig. 3.8 (Muhlhausen et al., 1985). The present work demonstrates that most of the observed features, with the exception of the rotational rainbow, are not strongly dependent on details of the potential energy landscape but instead arise largely from the statistical mechanics of the scattering process. A more recent and quite extensive experimental examination of a different system, N2 scattering from Cu(110), is shown in the rotational energy spectra of Fig. 3.10. These measurements cover a range of incident beam translational energies from 90 to 1000 meV, surface temperatures from 100 to 700 K as marked and the measurements were made for incident and final angles nearly normal to the surface (Siders and Sitz, 1994). Each of these sets of experimental points is compared to calculations carried out with similar incident beam parameters as in the previous graphs. In this case the calculated curves agree quantitatively with the measurement for the whole rotational energy range except in some cases for the highest energy points measured. The experimental points at small EfR values for most incident conditions display the rather sharp decrease exhibited in the calculations, but at large EfR there is little indication of rotational rainbow cut-off behavior. One interesting point that arises from the comparisons of calculations with the rotational energy spectra data is that for nearly all incident conditions reported, the calculations show a very sharp decrease at small rotational energy, and this decrease is observed in much of

86

J.R. Manson

Fig. 3.10. The final rotational energy distribution under specular geometry conditions for several incident energies and surface temperatures for N2 /Cu (110). (a) TS = 300 K; Ei = 90 meV, 350 meV, 640 meV and 940 meV as marked. (b) Ei = 90 meV; TS = 300 K and TS = 693 K. Symbols are experimental data from Siders and Sitz (1994) and curves are theory.

the available data. However, this data was obtained using molecular jet beams and these are known to produce very cold rotational distributions, typically of a few tens of K and relatively independent of translational energy. Our calculations indicate that the low EfR energy behavior is due to the cold incident rotational distribution and that if an incident beam with a rotational energy distribution comparable to the translational energy were used the effect would disappear. This is demonstrated in Fig. 3.11 which shows several calculations compared to the N2 /Cu(110) data for EiT = 640 meV and TS = 300 K from Fig. 3.10. The dotted curve is the same calculation shown in Fig. 3.10 for an incident beam with a rotational temperature of 35 K. The dashed curve shows a calculation, not for a Boltzmann distribution of rotations, but for a beam with a single rotational energy of 13.4 meV for all N2 molecules. This energy would correspond approximately to the J = 7 rotational quantum level. It is clear that the sharp decrease at low rotational energy is somewhat damped but the behavior at large EfR is essentially unchanged. The solid curve in Fig. 3.11 is for an incident beam having a fixed rotational energy of 26.4 meV (approximately the same as the J = 10 rotational state) and it is seen that


87

Fig. 3.11. The final rotational energy distribution for Ei = 640 meV and surface temperature 300 K for N2 /Cu(110). Symbols are experimental data from Siders and Sitz (1994), the dash curve is theory with a cold initial rotational distribution of TR = 35 K, the long dash curve is theory with initial rotational state with EiR = 13.4 meV (J = 7), solid curve is theory with initial rotational state at 26.4 meV (J = 10) and dot dashed curve is theory with initial rotational state at 100.8 meV (J = 20). θi = θf = 0◦ .

the low energy decrease is significantly reduced. For the dash-dotted curve, the rotational energy is fixed at 100.8 meV, about the same as for the J = 20 quantum level, and the calculated rotational energy spectrum becomes nearly exponential over the entire energy range. However, all curves regardless of initial rotational state saturate to nearly the same behavior at large EfR . Thus, it appears clear that the anomalous behavior of the rotational energy spectra at small EfR is due to the cold rotational distribution of the incident beam. Although we did not show in Fig. 3.11 calculations that average over incident Boltzmann rotational distributions with very high temperatures, our calculations show that the effect is similar and that for high incident rotational temperatures the effect tends to disappear. This, in fact, could become a very useful effect. It indicates that careful measurements of the small EfR behavior of the spectra, when compared with calculations such as those presented here, can be used to determine the rotational energy distribution of the incident beam. 3.5.4. Internal vibrational mode excitation Calculations for the excitation probability of the NO molecular stretch mode in collisions with an Ag(111) surface are presented here. At the incident energies involved in the present experiments the N–O stretch mode, which in the gas phase has a value of 233 meV, is only weakly excited. However, shown in Fig. 3.12 for the case of NO/Ag(111) is a comparison of calculation and experiment (Rettner et al., 1985) of the excitation probability for the first excited state of this mode as a function of incident translational energy. As for the previous calculations, a rotational temperature of 35 K is assumed for the incident beam,

88

J.R. Manson

Fig. 3.12. NO/Ag(111): Probability for excitation of a single quantum of the internal stretch mode of the NO molecule as a function of incident energy compared with measured values (Rettner et al., 1985) shown as open circles. The vertical line indicates the gas-phase stretch mode energy of 233 meV. The surface temperature is 760 K and θi = θf = 15◦ .

and the incident and final angles are θi = 15◦ and θf = 15◦ . The surface temperature is 760 K. The results have been averaged over all collisional orientations of the molecule and over angular directions of the angular momentum. It is seen that the internal mode excitation probability does not become appreciable until the incident translational energy is well above the mode excitation energy. However, due to coupling with the phonons, the energy supplied by the surface vibrations gives a small but non-zero probability of excitation even for incident translational plus rotational energy less than the threshold energy for internal mode excitation of NO. This is seen in both experiment and theory in Fig. 3.12. Excitation probabilities of 7–8% are predicted by the theory for incident translational energies of up to 1 eV. An alternative theoretical model, which calculates the excitation probability assuming that the forced harmonic oscillator behavior is due to electron–hole pair creation has been proposed by Newns (1986), and produces excitation probabilities which also agree well with the data of Fig. 3.12.

3.6. Conclusions In experiments involving the collisions of molecular particles with a surface a major contributor to energy exchange is the excitation of phonon modes of the target. At the quantum level, which implies annihilation or creation of single phonons, observation of such processes can provide fundamental physical information on the dynamics of the surface and on the molecule–surface interaction potential. When multiple quanta of phonons are


89

excited a reduced amount of detail can be obtained about the surface dynamics, but multiphonon transfer needs to be understood because it presents a background that must be subtracted, or otherwise accounted for, in order to reveal features in the scattering spectra due to other mechanisms. In the classical limit of large kinetic energies, heavy molecular masses and high temperatures the multiphonon energy transfers become classical in nature, but even in that limit the multiphonon scattering spectra can reveal important information about the gas–surface collision process. Presented in this paper is a brief review of theoretical descriptions of molecular beam scattering from surfaces. The theory is sufficiently rigorous that it reduces to an exact result for the simple problem of atomic elastic diffraction from surfaces, which involves only the three spatial degrees of freedom. Then, in a straight-forward manner, and initially with atomic projectiles, the theory is first developed to include single-phonon excitations, next to include excitations of a few phonons, and then it is extended to the classical limit of many-phonon transfers. The results obtained for atomic projectiles are then extended to molecular particles within the decoupling approximation. Expressions are first developed for the scattering including classical rotational excitation of a rigid molecule. Finally, the internal molecular vibrational modes are included via a semiclassical quantum mechanical theory. At each of these levels it is possible to obtain expressions for the scattering transition rate that are written as closed-form analytic equations. In the multiphonon limit the transition rates are mixed quantum-classical in form. They are essentially classical in the projectile’s translational and rotational degrees of freedom while quantum mechanical in the excitation of internal molecular vibrational modes. The theory presents a clear and straight-forward picture of the multiphonon excitation and how it affects the dynamics of the other degrees of freedom involved in the molecular collision process. Although one important need for understanding multiphonon effects is for background subtraction in order to clearly reveal the features observed in the data caused by the other degrees of freedom, it is shown that measurements of the multiphonon spectra can reveal important physical information. Among these are the determination of the physisorption well depth, sensitivity to collective effects in the target through determination of effective masses, it can reveal the smoothness of the surface and produce mean-square corrugation amplitudes, it can provide information about the distribution of rotational energies in the incident molecular beam, energy transfer and accommodation coefficients can be measured, and it provides a beautiful example of the transition from the purely quantum mechanical regime of elastic and single-phonon scattering to the classical regime of multiquantum excitation. A number of experimental measurements are analyzed in order to exhibit examples of the influence of multiphonon excitation on molecular scattering from surfaces. The first of these is a description of the angular distribution lobes observed when a molecular beam strikes the surface and the only measured quantity is the total number of particles scattered as a function of final angles. For molecules substantially heavier than hydrogen, the shapes of these lobes are well predicted as functions of the experimentally controllable parameters such as surface temperature, incident kinetic energy, incident angle, molecule species and surface composition.

90

J.R. Manson

A second type of scattering experiment involves making translational energy-resolved measurements at fixed angles. Classical multiphonon theory is shown in several examples to explain the measured properties of the energy transfer. A third class of molecular scattering experiments are measurements of scattered intensity and final average translational energy as functions of rotational energy. Again, for the case of molecules that are larger than hydrogen, classical theories in the translational and rotational degrees of freedom appear to give a good description of the observed results. The final example considered here is the excitation of internal molecular vibrational modes. This provides a good example of how multiphonon effects can be incorporated into a calculation primarily intended to explain the consequences of a different mechanism activated during the scattering process. The approach described here has the advantage that, at least for single collision events, it can be expressed in terms of closed-form mathematical expressions, as opposed to other more numerically intensive methods such as molecular dynamics simulations or trajectory calculations. The analytic form of the results also means that this method of describing multiphonon transfers can readily be incorporated into scattering theories for other excitation processes, at least within the decoupling approximation. This is readily accomplished by expressing the scattering theory for the desired process as a time-dependent Fourier transform, and then convoluting with the multiphonon kernel as is done in Eq. (3.34). An example of this convolution process is the influence of multiphonon energy transfers on internal molecular vibrational mode excitation as discussed above in Section 3.5.4 in connection with Fig. 3.12. If, in the calculations, multiphonon transfers are ignored the internal mode excitation probabilities are substantially overestimated. The addition of multiphonon degrees of freedom, and the concomitant reduced amount of incident energy available because of energy loss to the surface modes, reduces the calculated probabilities and brings them into much better agreement with measurement. It also shows that phonon excitations can cause the appearance of non-zero internal mode excitation probabilities at incident translational energies well below the threshold at which the incident kinetic energy equals the mode energy. The comparisons between theory and experiment exhibited here show that many of the features introduced into molecule–surface scattering spectra as a result of multiphonon excitation can be qualitatively, and often quantitatively, explained within the framework of a mixed quantum-classical theoretical treatment.

Acknowledgement This work was supported by the National Science Foundation under grant number 0089503 and the U.S. Department of Energy under grant number DE-FG02-98ER45704.

References Ambaye, H., Manson, J.R., Weiße, O., Wesenberg, C., Binetti, M., Hasselbrink, E., 2004. J. Chem. Phys. 121, 1901.


91

Armand, G., Manson, J.R., 1984. Phys. Rev. Lett. 53, 1112. Armand, G., Manson, J.R., Jayanthi, C.S., 1986. Phys. Rev. B 34, 6627. Barker, J.A., Auerbach, D.J., 1984. Surf. Sci. Rep. 4, 1. Baule, B., 1914. Ann. Physik 44, 145. Benedek, G., Toennies, J.P., 1994. Surf. Sci. 299, 587. Binetti, M., Weiße, O., Hasselbrink, E., Katz, G., Kosloff, R., Zeiri, Y., 2003. Chem. Phys. Lett. 373, 366. Bortolani, V., Levi, A.C., 1986. Rivista del Nuovo Cimento 9, 1. Brako, R., 1982. Surf. Sci. 123, 439. Brako, R., Newns, D.M., 1982. Phys. Rev. Lett. 48, 1859. Brune, H., Wintterlin, J., Trost, J., Ertl, G., Wiechers, J., Behm, R.J., 1993. J. Chem. Phys. 99, 2128. Brusdeylins, G., Doak, R.B., Toennies, J.P., 1980. Phys. Rev. Lett. 44, 1417. Brusdeylins, G., Doak, R.B., Toennies, J.P., 1981. Phys. Rev. Lett. 46, 437. Budde, F., Mödl, A., Hamza, A.V., Ferm, P.M., Ertl, G., 1987. Surf. Sci. 192, 507. Burke, K., Gumhalter, B., Langreth, D.C., 1993. Phys. Rev. B 47, 12852. Burke, K., Jensen, J.H., Kohn, W., 1991. Surf. Sci. 241, 211. Campargue, R. (Ed.), 2001. Dynamics of Gas–Surface Interactions. Springer, Heidelberg. Celli, V., 1984. In: Langreth, D., Suhl, H. (Eds.), Many Body Phenomena at Surfaces. Academic Press, New York, p. 315. Celli, V., Himes, D., Tran, P., Toennies, J.P., Wll, Ch., Zhang, G., 1991. Phys. Rev. Lett. 66, 3160. Dai, J., Manson, J.R., 2002. Nucl. Instrum. Methods Phys. B 193, 497. Dai, J., Manson, J.R., 2003. J. Chem. Phys. 119, 9842. For a discussion of the trajectory approximation with numerous references, see: DiRubio, C.A., Goodstein, D.M., Cooper, B.H., Burke, K., 1994. Phys. Rev. Lett. 73, 2768. Estermann, I., Frisch, R., Stern, O., 1931. Z. Phys. 73, 348. Estermann, I., Stern, O., 1930. Z. Phys. 61, 95. Frisch, O.R., 1979. What Little I Remember. Cambridge University Press, Cambridge. Goodman, F.O., 1974. Prog. Surf. Sci. 5, 261. Goodman, F.O., Wachman, H.Y., 1976. Dynamics of Gas–Surface Scattering. Academic Press, New York, p. 255. Gumhalter, B., 2001. Phys. Rep. 351, 1. Gumhalter, B., Burke, K., Langreth, D.C., 1994. Surf. Rev. Lett. 1, 133. Gumhalter, B., Langreth, D.C., 1999. Phys. Rev. B 60, 2789. Hofmann, F., Toennies, J.P., Manson, J.R., 1994. J. Chem. Phys. 101, 10155. Hofmann, F., Toennies, J.P., Manson, J.R., 1997. J. Chem. Phys. 106, 1234. Hulpke, E. (Ed.), 1992. Helium Atom Scattering from Surfaces. Springer Series in Surface Sciences, vol. 27. Springer, Berlin. Jackson, J.M., 1932. Proc. Cambridge Phil. Soc. 28, 136. Jackson, J.M., Mott, N.F., 1932. Proc. Roy. Soc. A 137, 703. Janda, K.C., Hurst, J.E., Cowin, J.P., Wharton, L., Auerbach, D.J., 1983. Surf. Sci. 130, 395. Kimman, J., Rettner, C.T., Auerbach, D.J., Barker, J.A., Tully, J.C., 1986. Phys. Rev. Lett. 57, 2053. Kleyn, A.W., Luntz, A.C., Auerbach, D.J., 1981. Phys. Rev. Lett. 47, 1169. Knauer, F., Stern, O., 1929. Z. Phys. 53, 766. Komrowski, A.J., Sexton, J.Z., Kummel, A.C., Binetti, M., Weiße, O., Hasselbrink, E., 2001. Phys. Rev. Lett. 87, 246103. Kondo, T., Private communication. Kondo, T., Sasaki, T., Yamamoto, S., 2002a. J. Chem. Phys. 116, 7673. Kondo, T., Sasaki, T., Yamamoto, S., 2002b. J. Chem. Phys. 116, 7673. Kondo, T., Sasaki, T., Yamamoto, S., 2003a. J. Chem. Phys. 118, 760. Kondo, T., Sasaki, T., Yamamoto, S., 2003b. J. Chem. Phys. 118, 760. Lennard-Jones, J.E., Devonshire, A.F., 1936a. Nature 137, 1069. Lennard-Jones, J.E., Devonshire, A.F., 1936b. Proc. Roy. Soc. (London) A 156, 6. Lennard-Jones, J.E., Devonshire, A.F., 1936c. Proc. Roy. Soc. (London) A 156, 29. Lennard-Jones, J.E., Devonshire, A.F., 1936d. Proc. Roy. Soc. (London) A 156, 37. Lennard-Jones, J.E., Devonshire, A.F., 1937a. Proc. Roy. Soc. (London) A 158, 253. Lennard-Jones, J.E., Devonshire, A.F., 1937b. Proc. Roy. Soc. (London) A 158, 269.

92

J.R. Manson

Lock, A., Toennies, J.P., Wöll, Ch., 1988. Phys. Rev. B 37, 7087. Luntz, A.C., Private communication. Mahan, G.D., 1990. Many-Particle Physics. Plenum Press, New York and London. Manson, J.R., 1988. Phys. Rev. B 37, 6750. Manson, J.R., 1991. Phys. Rev. B 43, 6924. Manson, J.R., 1994. Comput. Phys. Commun. 80, 145. Manson, J.R., 1998. Phys. Rev. B 58, 2253. Manson, J.R., Celli, V., Himes, D., 1994. Phys. Rev. B 49, 2782. Manson, J.R., Tompkins, J., 1977. Proceedings of the 10th International Symposium on Rarefied Gas Dynamics. Prog. Astronautics and Aeronautics 51, 603. Maradudin, A.A., Montroll, E.W., Weiss, G.H., 1963. Theory of Lattice Dynamics in the Harmonic Approximation. Academic Press, New York. Meyer, H.D., Levine, R.D., 1984. Chem. Phys. 85, 189. Micha, D.A., 1981. J. Chem. Phys. 74, 2054. Moroz, I., Manson, J.R., 2004. Phys. Rev. B 69, 205406. Moroz, I., Manson, J.R., 2005. Phys. Rev. B 71, 113405. Mortensen, H., Jensen, E., Diekhöner, L., Baurichter, A., Luntz, A.C., Perunin, V.V., 2003. J. Chem. Phys. 118, 11200. Muhlhausen, C.W., Williams, L.R., Tully, J.C., 1985. J. Chem. Phys. 83, 2594. Muis, A., Manson, J.R., 1996. Phys. Rev. B 54, 2205. Muis, A., Manson, J.R., 1997. J. Chem. Phys. 107, 1655. Nakamoto, K., 1970. Infrared Spectra of Inorganic and Coordination Compounds. John Wiley and Sons, Inc., New York. Newns, D.W., 1986. Surf. Sci. 171, 600. Österlund, L., Zorić, I., Kasemo, B., 1997. Phys. Rev. B 55, 15452. Powers, J., Manson, J.R., Sosolik, C., Hampton, J., Lavery, A.C., Cooper, B.H., 2004. Phys. Rev. B 70, 115413. Rettner, C.T., Ashfold, M.N.R. (Eds.), 1991. Dynamics of Gas–Surface Interactions. The Royal Society of Chemistry, Cambridge. Rettner, C.T., Fabre, F., Kimman, J., Auerbach, D.J., 1985. Phys. Rev. Lett. 55, 1904. Rettner, C.T., Kimman, J., Auerbach, D.J., 1991. J. Chem. Phys. 94, 734. Roberts, J.K., 1930. Proc. Roy. Soc. (London) A 129, 146. Rodberg, L.S., Thaler, R.M., 1967. Quantum Theory of Scattering. Academic Press, New York. Schmid, M., Leonardelli, G., Tcheließnig, R., Biedermann, A., Varga, P., 2001. Surf. Sci. 478, L355. Scoles, G. (Ed.), 1988. Atomic and Molecular Beam Methods. Oxford University Press, Oxford. Siders, J.L.W., Sitz, G.O., 1994. J. Chem. Phys. 101, 6264. Sitz, G.O., Private communication. Sitz, G.O., Kummel, A.C., Zare, R.N., 1988. J. Chem. Phys. 89, 2558. 2572, 6947. Sjölander, A., 1959. Ark. Fys. 14, 315. Stern, O., 1920a. Z. Phys. 2, 49. Stern, O., 1920b. Z. Phys. 3, 27. Stern, O., 1926. Z. Phys. 39, 751. Strachan, C., 1935. Proc. Roy. Soc. (London) Ser. A 150, 456. Toennies, J.P., 1987. Physica Scripta T 19, 39. Toennies, J.P., 1991. In: Kress, W., de Wette, F.W. (Eds.), Springer Series in Surface Sciences, vol. 21. Springer, Heidelberg, p. 111. Tomii, T., Kondo, T., Hiraoka, T., Ikeuchi, T., Yagu, S., Yamamoto, S., 2000. J. Chem. Phys. 112, 9052. Weiße, O., Wesenberg, C., Binetti, M., Hasselbrink, E., Corriol, C., Darling, G.R., Holloway, S., 2003. J. Chem. Phys. 118, 8010. Wilson, E.B., Decius, J.C., Cross, P.C., 1955. Molecular Vibrations. The Theory of Infrared and Raman Vibrational Spectra. McGraw-Hill. Woodward, L.A., 1972. Introduction to the Theory of Molecular Vibrations and Vibrational Spectroscopy. Oxford University Press, Oxford. Wyld, H.W., 1993. Methods for Physics. Addison-Wesley, New York.


Yagu, S., Hiraoko, T., Kino, Y., Yamamoto, S., 2000. Appl. Surf. Sci. 165, 217. Yagu, S., Kino, Y., Hiraoos, T., Sasaki, M., Yamamoto, S., 2001. Appl. Surf. Sci. 169–170, 122. Yagu, S., Kino, Y., Ozeki, K., Yamamoto, S., 1999. Surf. Sci. 433–435, 779. Zhukov, V., Popova, I., Yates, J.T. Jr., 1999. Surf. Sci. 441, 251.

93


CHAPTER 4

Physisorption Dynamics at Metal Surfaces

Mats PERSSON Surface Science Research Centre and Department of Chemistry, The University of Liverpool Liverpool L69 3BX, United Kingdom E-mail: [email protected]

Stig ANDERSSON Department of Physics, University of Göteborg SE-412 96, Göteborg, Sweden



Contents 4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. The physisorption interaction . . . . . . . . . . . . . . . . . . . . 4.2.1. van der Waals attraction and Pauli repulsion . . . . . . . . 4.2.2. Selective adsorption and bound levels . . . . . . . . . . . . 4.2.3. Molecular anisotropy . . . . . . . . . . . . . . . . . . . . . 4.3. Sticking, trapping and energy transfer . . . . . . . . . . . . . . . . 4.3.1. Single- and multi-phonon regime: Forced Oscillator Model 4.3.2. One-phonon regime: Distorted wave Born approximation . 4.3.3. Mean free paths of quasi-bound states . . . . . . . . . . . . 4.3.4. Sticking of positive-energy trapped particles . . . . . . . . 4.3.5. Resonant sticking . . . . . . . . . . . . . . . . . . . . . . . 4.4. Thermal desorption . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Photodesorption of physisorbed species . . . . . . . . . . . . . . . 4.5.1. Dipole transitions in the physisorption well . . . . . . . . . 4.5.2. Direct infrared photodesorption of physisorbed H2 . . . . . 4.5.3. Indirect infrared photodesorption of H2 . . . . . . . . . . . 4.5.4. Photodesorption of He . . . . . . . . . . . . . . . . . . . . 4.6. Concluding remarks . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

97 98 99 101 102 104 107 113 117 120 123 124 127 128 130 135 135 136 137 137

97

4.1. Introduction The atmospheric pressure in just one of numerous physical phenomena present around us, but rarely reflected upon. It is a standard example in early physics courses to show that the pressure is simply the consequence of momentum transfer in impulsive collisions between a surface and the molecules of a surrounding gaseous atmosphere. The collisions can of course be treated in more detail. The pressure is hard to notice, it is stable and natural to us, but the temperature of the air affects us in obvious ways. The sensations of heat and cold cannot be mediated by purely elastic collisions, but require that there is some energy transfer in the collisions between the air molecules and the skin surface. The energy transfer can be violent and spectacular under more extreme conditions, like in cases with glowing meteorites and space shuttles. The collisions including energy transfer may be treated by classical physics concepts (see e.g. Harris, 1991), in fact pure elastic scattering does not even exist classically. However, observations of specular elastic scattering, diffraction and resonant scattering phenomena in gas–surface scattering experiments show that the use of quantum mechanical concepts is not an academic question, but are essential for an accurate description of gas–surface collisions. The interaction between an atom or a molecule and a surface at large distance is determined by the van der Waals attraction VvdW (z) = −CvdW /z3 where z is the normal distance between the center-of-mass of the particle and a surface reference plane close to the outermost layer of ion-cores in the solid. The van der Waals constant CvdW depends on the dielectric properties of the solid and the frequency dependent polarizability of the particle. Hence, the approaching particle will at such distances be accelerated in a direction normal to the surface and the effective angle of incidence with respect to the shorter range part of the mutual interaction will decrease. This effect will be particularly pronounced at incident velocities in the thermal energy range. The van der Waals forces also polarize the incident particle resulting in a finite probability for it losing energy by emissions of photons. The final outcome of the particle–surface encounter depends on the details of the mutual interaction which in turn depends on the chemical nature of the particle and the surface, and possible internal degrees of freedom of the particle. If the incident atom or molecule is chemically inert with respect to the specific surface, so that no rearrangement of the particles electronic structure occurs during the collision, the relevant short range interaction will be repulsive and the particle will reflect from this repulsive potential, and its lateral modulation given by the surface ion core positions, frequently called a “corrugated wall”. A number of factors determine the way in which the particle reflects: the steepness of the wall, the strength of the lateral corrugation, the energy transfer between the particle and the solid, influence of quasi-bound states and the coupling, via the interaction, of translational and possible internal degrees of freedom (rotations and vibrations) of the incident particle. The energy transfer to the solid lattice depends in particular on the ratio of adsorbate mass to substrate atom mass and the time scale of the collision. Molecular rotations are vitally important in the thermal energy range (see Sitz, Chapter 10 in this volume) and there exists an anticorrelation between rotational excitation and energy transfer to the lattice (Kimman et al., 1986). If the particles total energy, comprising kinetic plus potential energy, becomes negative upon collision it is trapped in the poten-

98

M. Persson and S. Andersson

tial well. At low surface temperature, it is unlikely that the particle will gain enough energy from the solid lattice and escape and we can assume that it is permanently stuck on the surface. For physisorbed species, atoms and in many cases molecules, the sticking process is reversible, for example raising the temperature of the substrate will result in thermal desorption of the stuck particles. However, physisorbed molecules may also act as precursors to chemisorbed states, a situation often considered in the formulation of adsorption kinetics (see e.g. King, 1978). Spectroscopic measurements of the transfer of physisorbed oxygen molecules on the Pt(111) surface into a superoxo chemisorbed oxygen state provide a good example of this phenomenon (Luntz et al., 1989; Gustafsson and Andersson, 2004). Light weakly bound species like H2 and He photodesorb quite efficiently via absorption of infrared photons in transitions from the bound particle ground state to free translation continuum states. Energy exchange with sufficiently energetic gas phase particles may result in collision-induced desorption (Beckerle et al., 1989; Kulginov et al., 1997). In this chapter we specifically consider atoms and molecules that physisorb on metal surfaces. No significant change in the electronic configuration takes place upon adsorption and coupling to electronic excitations is expected to be very weak (Schönhammer and Gunnarsson, 1980). The adsorption is to a very good approximation electronically adiabatic and the energy transfer occurs through the phonon system of the solid lattice (Brenig, 1987). These conditions are expected to hold for rare gases adsorbing on all solids and for hydrogen molecules on simple or noble metals, and may also be valid in other cases involving comparatively inert molecules like nitrogen and methane. The molecules possess internal degrees of freedom which add further details to the particle–surface interaction. In order to establish contact between theory and experiment if is crucial that the potential energy surface governing the gas–surface collision process be know as well as possible. Molecular beam scattering experiments can provide such information and also important data regarding energy transfer and sticking for well-defined impact conditions.

4.2. The physisorption interaction Chemically inert atoms and molecules physisorbed on cold metal surfaces are usually identified by their low desorption temperature. The adsorption energies are in the range of a few meV for He to of order 100 meV for Ar, Kr, N2 with desorption temperatures from a few K to a few tens of K. Characteristic spectroscopic signatures are weakly perturbed adsorbate electronic and vibrational excitation spectra including almost free rotations for physisorbed H2 . The adsorption energies may be determined from desorption or isosteric heat of adsorption measurements. For light physisorbed species like He and H2 , the most accurate and detailed measurements of the adsorbate-surface interaction potential are obtained in gas–surface scattering experiments and involve the resonance structure of the elastic backscattering (see e.g. Hoinkes, 1980; Celli and Evans, 1982) which yield sequences of bound levels in the physisorption well. Complementary isotope measurements (3 He, 4 He, H2 , D2 ) give precise determinations of the potential well depths. Even finer details of the interaction like the rotational sub-level splitting for H2 can be determined. Such measurements will be discussed in further detail below. Comprehensive listings and


99

discussions of measured and calculated binding energies including detailed potential parameters can be found in the literature (Nordlander et al., 1985, 1986; Chizmeshya and Zaremba, 1989, 1992; Vidali et al., 1983, 1991). 4.2.1. van der Waals attraction and Pauli repulsion In the current theory for physisorption (Zaremba and Kohn, 1977; Harris and Liebsch, 1982a) the ad-particle metal surface interaction involves the superposition of a short-ranged repulsive term, VR , arising from the overlap of the ad-particle and surface electron clouds (Pauli repulsion) and a long-range attractive van der Waals term, VvdW , arising from adparticle metal electron correlation. These two terms contribute additively to the laterally averaged isotropic interaction potential, V0 (z), as V0 (z) = VR (z) + VvdW (z),

(4.1)

where VR (z) = V0 exp(−αz)

(4.2)

and VvdW (z) = −

CvdW f 2kc (z − zvdW ) . (z − zvdW )3

(4.3)

Here V0 and α determine the strength and inverse range of the repulsive potential, respectively, CvdW is the strength of the asymptotic van der Waals attraction and zvdW is the position of the van der Waals reference plane. The function f (x) = 1−(1+x +x 2 /2) exp(−x) in Eq. (4.3) describes the saturation of the van der Waals attraction as the distance z − zvdW becomes comparable to the extent of the ad-particle (Nordlander and Harris, 1984). The vital importance of this cut-off factor indicates that a non-negligible part of the interaction is not well understood theoretically. This complication may be overcome by ongoing efforts to find within density functional theory an exchange-correlation functional that includes the van der Waals interaction in an accurate way (see e.g. Hult et al., 1999). The repulsive branch is determined from the shifts of the metal one-electron energies by the ad-particle and may be calculated by perturbation theory in a pseudo-potential description of the ad-particle and a jellium model representation of the metal surface. The local density of metal-electron states falls off exponentially away from the surface, which results in an exponentially repulsive potential VR (z) as given by Eq. (4.2). The physical origin of the asymptotic −z−3 dependence of the van der Waals attraction is the dynamic polarization interaction between the ad-particle and the solid substrate. The strength, CvdW , is accordingly determined by the dynamic dipole polarizability, α(ω), of the ad-particle and the bulk dielectric function, (ω), of the metal by an integral over imaginary frequencies iu ∞ 1 (iu) − 1 du α(iu) . CvdW = (4.4) 4π 0 (iu) + 1

100


Fig. 4.1. Calculated lateral average physisorption potentials, V0 (z), for He, Ne and Ar on Cu(110). The distance z is given with respect to the jellium edge. The potential parameters are from Chizmeshya and Zaremba (1989, 1992).

The position of the van der Waals reference plane, zvdW , depends via the dynamic image plane position dIP (ω) on the microscopic screening properties of the metal surface (Zaremba and Kohn, 1976; Liebsch, 1986). ∞ 1 (iu) − 1 du α(iu) zvdW = (4.5) dIP (iu) 4πCvdW 0 (iu) + 1 which involves a specific surface quantity: the centroid of the surface charge d(ω) induced by an external frequency dependent field. Accordingly the physisorption potential, V0 (z) in Eq. (4.1), depends on the details of the surface electron structure via both the electron spill out (VR ) and the centroid of induced surface charge (VvdW ) which will result in a crystal face dependence of V0 (z) for a given ad-particle (Andersson et al., 1996). Figure 4.1 displays typical examples of physisorption potentials for He, Ne and Ar on a Cu(110) surface calculated from potential parameters in the literature (Chizmeshya and Zaremba, 1989, 1992) and listed in Table 4.1. The distance z is given with respect to the jellium edge. A characteristic feature of these potentials is that they are very anharmonic with a slow fall off of the van der Waals attraction. The potential well depth increases by about an order of magnitude from ∼6 meV for He to ∼60 meV for Ar because the attraction increases more than the repulsion does. The bound levels at the bottom of these potential wells are separated by a few meV and transitions between such levels can be measured by suitable spectroscopic methods like inelastic He atom scattering (IHAS) (Gibson and


101

Table 4.1 Physisorption parameters for X-Cu(110) (Chizmeshya and Zaremba, 1989, 1992). The cut-off parameter kc = α/2 X

V0 (eV)

α (a−1 0 )

CvdW (eVa30 )

zvdW (a0 )

He Ne Ar

5.512 8.242 22.36

1.280 1.295 1.264

1.529 3.059 10.39

0.3237 0.3146 0.3754

Sibener, 1985; Kern et al., 1986; Braun et al., 1997) and high-resolution electron-energyloss spectroscopy (HREELS) (Gruyters and Jacobi, 1994; Svensson and Andersson, 1997). 4.2.2. Selective adsorption and bound levels The bound level sequences of the potentials shown in Fig. 4.1 have been investigated in detail by resonance scattering experiments using monochromatic, well-collimated nozzle beams of He, Ne and Ar colliding with a 20 K Cu(110) surface (Andersson et al., 2006; Linde and Andersson, 2006). The resonance structure in the elastic backscattering arises because of a degeneracy between the incident state and an intermediate state involving a bound level of the gas–surface interaction potential. The particle may then be diffracted into this state of selective adsorption where it is trapped in the surface well with enhanced parallel kinetic energy. The kinematical condition for a resonance associated with a diffraction condition involving a surface reciprocal lattice vector G is i = n +

h¯ 2 (Ki + G)2 2mp

(4.6)

where n is the bound-state energy, mp the particle mass, i and Ki the energy and wavevector component parallel to the surface of the incident beam. When the resonance condition is fulfilled the weak periodic lateral corrugation of the basic interaction induces large changes in the diffracted beam intensities. The resonances are conventionally observed as narrow features in plots of the diffracted beam intensities when the experimental incidence conditions are varied. The resonances are intrinsically very sharp in angular and energy space. The line width depends on the lifetime of the particle in the intermediate bound level which is limited by elastic and phonon inelastic processes, phenomena that will be discussed in further detail below. While the separation between the lower lying levels is a few meV the lifetime broadening is only a fraction of a meV. Figure 4.2 shows two sharp resonances in the (10) and (20) diffraction beams of 20 Ne scattered from a 20 K Cu(110) surface (Linde and Andersson, 2006). The resonances involve two bound levels at n = −5.00 and −11.91 meV and the G10 and G20 vectors. A number of levels from −16.66 to −1.61 meV can be determined in this manner. Use of a 22 Ne beam permits a unique assignment of the levels and a single accurate gas– surface potential curve to be constructed according to the Rydberg–Klein–Rees method of molecular physics, as discussed by LeRoy (1976). Resonance measurements using 4 He and 3 He nozzle beams (Andersson et al., 2006) have also established the He–Cu(110)

102


Fig. 4.2. Angular distribution measurements for 20 Ne scattered from a 20 K Cu(110) surface for three incident beam energies, i , at an incident angle of 78.9◦ from the surface normal.

potential. These potentials can be compared with the calculated potentials in Fig. 4.1; for example the calculated well depths are 5.6 meV for He–Cu(110) and 14.7 meV for Ne– Cu(110) (Chizmeshya and Zaremba, 1992) somewhat more shallow than the measured values 6.1 meV and 18.2 meV (Andersson et al., 2006; Linde and Andersson, 2006). The progression of measured and calculated levels agree very well which means that the general shape of the potentials are compatible with the experimental observations and a slight adjustment of, for instance, the repulsive strength parameter V0 in Eq. (4.2) brings the calculated levels in close agreement with the experimental data. The 4 He–Ag(110) system is also well-characterized experimentally (Luntz et al., 1982; Dondi et al., 1986); the bound state energies and the estimated well depth of 6.0 meV are very close to the measured data for He–Cu(110). 4.2.3. Molecular anisotropy For molecules, the anisotropy of the polarizability tensor will introduce a rotational dependence in the molecule–surface interaction. Detailed mapping of the bound level spectrum and the gas–surface interaction potential by resonance scattering measurements has only been performed for H2 (Perrau and Lapujoulade, 1982; Yu et al., 1985; Chiesa et al., 1985; Harten et al., 1986; Andersson et al., 1988; Andersson and Persson, 1993a; Andersson et al., 1996). The availability of the two isotopes H2 and D2 of widely different masses and the different rotational populations of para-H2 (p-H2 ) and ortho-D2 (o-D2 ) and the normal species (n-H2 , n-D2 ) simplifies the data analysis greatly. For instance the rotational


103

Fig. 4.3. Physisorption interaction potentials for H2 (D2 ) on Cu(111) (circles), Cu(100) (squares) and Cu(110) (triangles). The potential functions V0 (z), V1 (z) and V2 (z) are defined in the text. The position z of the molecular center of mass is given with respect to the classical turning point zt at i = 0.

anisotropy of the interaction has been determined via analysis of resonance structure resulting from the rotational (j, m) sub-level splittings observed for n-H2 and p-H2 beams. (Chiesa et al., 1985; Wilzén et al., 1991). Results of this kind of analysis for H2 interacting with the low-index Cu(111), Cu(100) and Cu(110) surfaces (Andersson and Persson, 1993b) are summarized in terms of the three potential functions depicted in Fig. 4.3. V0 (z), V1 (z) and V2 (z) represent respectively, the lateral and angular average of the potential (see Eq. (4.1)), the min-to-max variation of the rotationally averaged, lateral periodic corrugation and the min-to-max variation of the lateral average of the rotational anisotropy. The potential V1 (z) is modeled with an amplitude function (Harris and Liebsch, 1982b) V1 (z) = V1 exp(−βz)

(4.7)

where the exponent β is related to the exponent α of V0 (z) via β = α/2 + (α/2)2 + G210 . The strength parameter V1 is adjusted so that the calculated intensities of the first-order G10 diffraction beams agree with measured values from experimental data. The laterally averaged anisotropic component V2 (z, cos θ ) can be expressed as a linear combination of the repulsive and attractive parts of V0 (z) (see Eq. (4.1))

V2 (z, cos θ ) = βR VR (z) + βvdW VvdW (z) P2 (cos θ ) (4.8) where P2 (cos θ ) is a Legendre function and βR and βvdW are coefficients for the repulsive and attractive anisotropy, respectively. The anisotropy of the polarizability tensor of the free H2 molecule gives βvdW = 0.05 (Harris and Feibelman, 1982). Resonance scattering measurements (Wilzén et al., 1991) of the rotational (j, m) sub-level splittings for n-H2 and p-H2 beams scattering from, for instance, Cu(100) show that, for βvdW = 0.05, the

104


repulsive contribution to V2 is very weak, βR = −0.002. This means that the van der Waals attraction dominates the anisotropy and that an upright molecular orientation is favored. The anisotropy is rather weak and rotational inelastic scattering is a comparatively weak process at low incident particle energies. In this context we note that the anisotropy is similarly weak for N2 , but in this case the rotational inelastic scattering is quite strong simply due to the much smaller rotational transition energies for N2 relative to those for H2 and D2 (Andersson et al., 2000). The simple message of the three potential functions shown in Fig. 4.3 is that the V1 and V2 corrugation terms are rather weak compared to the V0 average, confirming that the basic particle–surface interaction is predominantly one-dimensional.

4.3. Sticking, trapping and energy transfer Sticking and trapping of an inert particle in the physisorption regime is all about energy conversion among various particle degrees of freedom and dissipation of its energy to the phonon excitations. For a cold surface corresponding to the zero temperature limit, the particle will stick irreversibly to the surface as soon it has lost its incident energy. At nonzero coverage, energy transfer to pre-adsorbed particles becomes important as can be seen in Fig. 4.4, which shows the measured sticking coefficient, S, for D2 on a 10K Cu(100) surface versus fractional adsorbate coverage θ (Andersson et al., 1989). The incident D2 beam hits the surface in the normal direction with a kinetic energy of 20 meV. S increases linearly with θ and can be expressed as a sum of sticking via impact on the bare surface and

Fig. 4.4. Measured sticking coefficient, S, versus fractional adsorbate coverage, θ , for D2 on a 10 K Cu(100) surface. Incident D2 beam: 20 meV kinetic energy, 0◦ angle with respect to surface normal.


105

via collisions with pre-adsorbed molecules having probabilities S0 and S1 , respectively, as S = S0 (1 − θ ) + S1 θ.

(4.9)

The data in Fig. 4.4 give S0 = 0.094. By comparison, S1 ≈ 0.75 illustrating the strong influence of collisions with pre-adsorbed molecules. Here we limit ourselves to the zero-coverage limit for sticking. The key physical factors influencing the sticking of a light particle in the physisorption well under these conditions will here be illustrated and discussed with specific reference to sticking of hydrogen and deuterium molecules on the (111), (100) and (110) crystal faces of Cu. The detailed knowledge obtained about the molecule–surface interaction in this case as demonstrated in Section 4.2 has enabled a quantitative comparison between molecular beam sticking data and results from theoretical model calculations. As will be demonstrated in this section a physical picture of the physisorption dynamics of hydrogen molecules on low-index surfaces of Cu has emerged from this comparison. We believe that this picture provides a conceptual basis for the physical understanding of sticking of a light particle on a solid surface and also to a wider class of systems with a weakly adsorbed molecular “precursor”. The reader who is interested in more details than given in this presentation is referred to the original papers (Persson and Harris, 1987; Andersson et al., 1989; Persson et al., 1990; Andersson and Persson, 1993b). The measurements of the zero-coverage sticking coefficient, S0 , of hydrogen molecules on the cold Cu surfaces were carried out in two different ways using collimated nozzle beams of H2 and D2 including p-H2 and o-D2 as detailed in Andersson et al. (1989): (i) kinetics for formation of a full monolayer of physisorbed H2 and D2 molecules; (ii) partial monolayer desorption. Both methods give very similar results. Figure 4.5 displays experimental sticking data for H2 on Cu(100) obtained at normal incidence. The peak at low energy is due to a corrugation mediated selective adsorption resonance involving a bound state level at −1.2 meV. Away from this resonance S0 falls off monotonically from a value about 0.15. In general, we find that S0 for a fixed angle of incidence θi consists of a normal contribution that falls off smoothly with increasing i and resonant contributions that give rise to relatively sharp peaks at energies that disperse strongly with θi . In all cases, the resonant contributions derive from selective adsorption resonances. The observed behavior of S0 raises several questions regarding (i) a quantum or classical mechanical description of the lattice motion; (ii) the physical mechanisms underpinning the incident energy and angle dependence; (iii) influence of surface phonon structure; (iv) role of internal degrees of freedom. A simple classical model of sticking at normal incidence that is based on an impulsive and collinear collision of the incident particle with a single substrate atom illustrates some key concepts for sticking in a physisorption well (Rettner et al., 1989). In this model the particle sticks to the surface when its incident energy i is less than the energy transfer to the surface atom. For an impulsive collision is given by the Baule result (Baule, 1914) for the energy transfer of the incident particle with mass mp to a surface atom with mass ms as, =

4μ (i + D) (1 + μ)2

(4.10)

106


Fig. 4.5. Experimental initial sticking coefficient S0 vs incident energy i at normal incidence for H2 on a 10 K Cu(100) surface.

where μ = mp /ms . The increase of kinetic energy in the potential well is accounted for by adding the well depth D to i . This results in a sticking coefficient S0 which is unity for i < c , where c =

4μ D, (1 − μ)2

(4.11)

and zero at higher i . This model results in a behavior of S0 for hydrogen on copper at normal incidence that is qualitatively different from the observed one shown in Fig. 4.5. The observed non-resonant background of S0 saturates at about 0.2 in the limit of i → 0 and has a slowly decaying tail stretching above 30 meV, whereas Eq. (4.11) and D ≈ 30 meV gives a unity sticking coefficient below c ≈ 2 meV. The main shortcoming of this simple classical model as applied to sticking of the hydrogen molecule on a cold copper surface is not the simplified description of the particle motion and its collision with the surface but rather the classical description of the energy transfer to the lattice motion. That a quantum description is needed can be understood from the forced oscillator model (FOM) for energy transfer and sticking based on the trajectory approximation (Brenig, 1979; Meyer, 1981; Brako and Newns, 1982; Sedlmeir and Brenig, 1980; Persson and Harris, 1987; Gumhalter, 2001). Henceforth this model will be referred to as FOM. Another distinct quantum effect is the quantum reflection of a particle from the potential well (see e.g. Kohn, 1994). However, this effect is only important at extremely low incident energies that cannot be reached by conventional molecular beam scattering methods.


107

4.3.1. Single- and multi-phonon regime: Forced Oscillator Model In the FOM, the lattice motion is treated as a set of quantized harmonic oscillators – phonons – driven by the time-dependent lattice forces from a classical trajectory of the incident particle in the rigid physisorption potential. The incident beam of particles is represented by a swarm of trajectories with the same incident momentum and different impact parameters. The resulting phonon energy loss distribution function P () for each trajectory r(t) can then be obtained to all orders of multi-phonon excitations at zero temperature (T = 0) as (Müller-Hartmann et al., 1971), ∞ ∞

i t it dt P () = (4.12) exp Ps exp − , exp − 1 d 2π h¯ h h ¯ ¯ −∞ 0 where Ps () =

∗ 1 f .C .fj i ij h¯ h¯ h¯ h¯ 3

(4.13)

i,j

is the one-phonon loss function. In Eq. (4.13), fi (ω) is the Fourier transform of the force fi (r(t)) exerted on ion-core i fixed at its equilibrium position and h¯ λ λ ei ej δ(ω − ωλ ) Cij (ω) = (4.14) 2ms ω λ

is the phonon spectral tensor at T = 0. The sum in Eq. (4.14) is over the normal modes λ of the lattice with normalized displacement fields eλi and phonon energies hω ¯ λ . The sticking coefficient at normal incidence is now obtained as an average of the probability for the phonons to absorb more energy of the particle than its incident energy over all trajectories representing the incident beam, ∞ S0 = (4.15) P () d. i

The calculation of S0 requires a quantitative model for the physisorption potential V0 , the lattice forces fi (r(t)) and the phonon spectral tensor Cij (ω). As discussed in Section 4.2 for hydrogen on metal surfaces, there exists an accurate model of the physisorption potential V0 (z) based on experimental data and theory. This model can be used to construct a quantitative model for the lattice forces based on the standard recipe (Celli, 1991) that the range of these forces are assumed to be limited to the outermost surface lattice plane and are assumed to derive from a central pair potential. The functional form of this potential is uniquely determined from the requirement that the total perpendicular component of force exerted on the surface should be equal to dV0 (z)/dz. In contrast to the simple classical model discussed above this model accounts for the fact that the particle interacts with several neighboring surface atoms. For our purpose Cij (ω) is sufficiently well-described by a surface lattice dynamical model based on a central nearest neighboring force constant model truncated at the surface with no modifications in the surface region (Black et al., 1983). This model contains a single parameter that is fitted to the observed maximum bulk phonon energy of about 30 meV for Cu. Details of surface

108


Fig. 4.6. Phonon spectral function, Czz (ω) of the Cu(111) (circles), Cu(100) (squares) and Cu(110) (triangles) crystal faces of Cu as a function of phonon energy hω. ¯

lattice dynamics calculations of phonon spectral functions based on this force constant model can be found in Persson (1987), Persson and Harris (1987). The phonon structures of the three low-index crystal faces of Cu that are relevant for the energy transfer are illustrated in Fig. 4.6 by the spectral function Czz (ω) of the perpendicular motion of a single surface atom. In the low energy region Czz (ω) has essentially a linear dependence on ω as dictated by elastic continuum theory and is not sensitive to the crystal face. The structure sensitive features show up first at higher phonon energies above 10 meV. The peaks and steps in this energy region are derived from the stationary points of the surface phonon dispersions at the surface Brillouin zone boundaries. The results for S0 at normal incidence of H2 and D2 on Cu(100), as obtained in the FOM using the proposed model for the interactions and the surface lattice dynamics, are, as shown in Fig. 4.7, in near-quantitative agreement with the observed S0 . The saturation of S0 at a value well below unity at low i and its slowly decaying tail at high i are both understood from a multi-phonon expansion of P (), as obtained from a Taylor expansion of the second exponential factor in Eq. (4.12) (Gumhalter, 2001),

1 ∞

P () = exp −2W (0) δ() + Ps () + Ps − Ps d + · · · , (4.16) 2 0 where

∞

2W (T ) = 0

1 + 2n Ps () d. h¯

(4.17)

In the last equation we have included explicitly the substrate temperature dependence through the Bose–Einstein distribution function n(ω) = 1/(exp(h¯ ω/kB T ) − 1). The first term in Eq. (4.16) gives the probability exp(−2W (0)) for no energy loss corresponding to elastic scattering and is strictly a quantum mechanical effect. Furthermore, 2W (0) gives


109

Fig. 4.7. Sticking coefficients at normal incidence for (a) H2 and (b) D2 on Cu(100). FOM (solid lines) and experiments (squares).

the mean number of phonons excited. exp(−2W (0)) is usually referred to as the Debye– Waller factor in analogy with a similar quantity in neutron scattering and X-ray diffraction intensities from crystals. Thus the probability 1−exp(−2W (0)) for energy-loss will be less than unity resulting in a saturation of S0 below unity at low i . The relatively large value of about 0.8 for the calculated Debye–Waller factor at low i for H2 –Cu(100) accounts for the observed saturation of S0 . The associated small value of 2W (0) of about 0.2 makes the one- and two-phonon processes corresponding to the second and third terms in Eq. (4.16) dominate P () and S0 . The importance in this case of one-phonon processes at low i is evident from a comparison between S0 calculated using P () and Ps () in Eqs. (4.12) and (4.13), respectively, shown in Fig. 4.8. Note that the one-phonon loss distribution does not contain the Debye– Waller factor and corresponds to the result obtained using first order perturbation theory. The large drop of the one-phonon result for S0 at i ∼ 15 meV is just a reflection of the corresponding behavior of the phonon spectral function Cij (ω) in Fig. 4.6. The observed isotope dependence in the one-phonon regime, corresponding to an increase of S0 with a factor of about 1.4 when going from H2 to D2 , is rationalized from the isotope dependence of the Fourier-transformed lattice forces in Eq. (4.13) defining Ps (). Changing the particle mass mp to m p without changing its incident energy and direction results in a simple mass-scaling of the time-dependence of its trajectory in the rigid

110


Fig. 4.8. Sticking of H2 on Cu(100), comparison between the DWBA (dotted line) and the FOM in the single phonon approximation (dashed line) and the full multi-phonon approximation (solid line).

physisorption potential (Jackson and Persson, 1992), m p m p fi ω . f i (ω) = mp mp

(4.18)

Thus according to Eqs. (4.18) and (4.13), the twice larger mass of D2 than for H2 increases Ps ( = 0) by a factor of two but slows down the motion so that Ps () decays faster with . The net result is an isotope effect of less than a factor of two for S0 . It is also possible, within the FOM, to obtain S0 corresponding to a classical treatment of the lattice motion. The classical energy loss cl for each trajectory is given by the average energy loss, ∞ ∞ cl = (4.19) P () d = Ps () d. 0

0

As shown in Fig. 4.9, the classical result for S0 as obtained from Eq. (4.19) using the calculated set of P () from the trajectories representing the incident beam is close to the result obtained in the simple classical model. The spread of the calculated S0 comes from the variation of the energy loss with the impact parameter. This result demonstrates clearly the importance of a quantum mechanical treatment of the lattice motion for sticking of a light particle in the physisorption well. Note that cl in Eq. (4.19) is independent of T but the result for S0 is only valid at T = 0 K. Furthermore, the FOM with a classical treatment of the lattice motion is not a fully classical treatment because there is no back reaction to the particle trajectory from the energy loss to the lattice motion. The FOM indicates also under which conditions a classical description of the lattice motion is valid for energy transfer and sticking. A small elastic scattering probability,


111

Fig. 4.9. Calculated sticking coefficient of H2 on Cu(100) at normal incidence in the classical (dashed line) and the quantum (solid line) limits of the forced oscillator model.

exp(−2W ), corresponding to a large mean number 2W of phonon excitations is obviously a necessary condition for a classical description. At zero temperature, an increase of mp at fixed incident energy and physisorption potential does not simply decrease the Debye– Waller factor to zero but rather to a finite value as first pointed out by Burke and Kohn (1991). This can be understood from the behavior of the lattice forces and the phonon spectral function with energy, which determines 2W in Eq. (4.17) through Ps () in Eq. (4.13). The zero-frequency limit of the total lattice force is determined by the momentum transfer √ which increases with mp for fixed i (Eq. (4.18)). However, this increase is compensated by the heavier particle moving slower and exciting lower energy phonons with a vanishing contribution to the phonon spectral function, which results in a finite value for 2W in the heavy mass limit. This prediction was confirmed by elastic scattering data of Ar and Kr beams from a cold Cu(111) surface (Althoff et al., 1997). The experiments were carried out even at a low target temperature around 10 K, using thermal desorption induced by short laser pulses to prevent stuck particles from accumulating on the surface. For the weakly corrugated Cu(111) surface 2W is to a good approximation given by the logarithm of the specular reflectivity, ln I00 , displayed in Fig. 4.10 for He, Ne, Ar and Kr versus target temperature (Andersson et al., 2002). At low temperature 2W saturates at values for Ne, Ar and Kr which does not depend on the particle mass; the increasing value of 2W is a consequence of the increasing strength of the particle–surface interaction (see Fig. 4.1). In sticking of Ne on cold metal surfaces (Persson and Harris, 1987; Schlichting et al., 1988), this effect is manifested by a saturation of S0 at a value well below unity at low energies and a dominance of one-phonon events in the sticking process. However, the predominance of excitation of low-energy phonons by a heavy particle makes the Debye–Waller factor exponentially small for increasing substrate temperature T . For instance, an increase of T from 10 to 80 K for scattering of Ar from Cu(111) increases the mean number 2W of phonon excitations from about 2.3 to about 5.9 which reduces

112


Fig. 4.10. Logarithm of measured specular reflectivity, ln I00 , versus target temperature T for He, Ne, Ar and Kr nozzle beams scattered from Cu(111). Incidence conditions: 36 meV beam energy, 70◦ from surface normal. The straight lines are fits to the linear decay of ln I00 .

the Debye–Waller factor dramatically from about 0.1 to 0.003. So in this case, already at Ts = 80 K, the energy transfer is within the classical limit. An increase of i or D also increases 2W , and brings the energy transfer closer to the classical limit, because the magnitude of the Fourier transformed forces fi (ω) is determined by the momentum transfer in the potential well and the decay of fi (ω) becomes slower with frequency for a faster collision. The approach to the classical limit of energy transfer at large 2W can also be directly seen from an asymptotic expansion of P () in Eq. (4.12), which gives rise to a Gaussian shape of the energy-loss function (Gumhalter, 2001), 1 ( − cl )2 P () ≈ √ (4.20) exp − , 2σ 2 2πσ centered around the classical energy loss cl in Eq. (4.19) and with a broadening σ determined by the second moment of Ps (), ∞ σ2 = (4.21) 2 1 + 2n Ps () d. h ¯ 0 Note that the zero-point fluctuations will always give a finite contribution to the broadening but at finite temperature it will be rapidly overwhelmed by the temperature fluctuations.


113

The FOM has a few important shortcomings in describing S0 of H2 physisorption on metal surfaces and more generally regarding sticking of light particles in the physisorption well. The first limitation concerns the crystal face dependence of S0 . The FOM model is not able to account for the enhanced sticking by almost a factor of two on the (110) face of Cu compared to the more close packed (100) and (111) faces (Andersson and Persson, 1993b). The second limitation concerns the dependence of S0 on incident angle. Eq. (4.15) gives the probability that at least i is lost during the first round trip in the physisorption well. This condition gives only a lower bound for S0 at off-normal incidence because some of the particles that lose energy remains trapped at the surface with a total energy of the normal motion less than zero but with an energy larger than zero. These particles will eventually stick by losing their energy on subsequent round trips or reemerge into the gas phase by momentum transfer from the lateral motion to the normal motion via the surface corrugation and this process is not described by the FOM. The third limitation concerns resonant sticking that involves selective adsorption resonances, which require a quantum mechanical description of the particle motion. As we will show these limitations can be more or less handled by the distorted wave Born approximation (DWBA) for sticking in the one-phonon regime and its extensions to handle trapped particles and resonant processes. Note that the FOM has also some obvious intrinsic limitations in describing energy transfer and sticking that have been discussed in several papers (Brenig, 1987; Jensen et al., 1989; Gumhalter et al., 1994). For instance, the trajectory of the particle is inconsistent with the energy loss to the phonons and in particular a non-sticking trajectory is used for the calculation of sticking. However, the FOM based on the trajectory approximation has been justified in the limit of small energy transfer (Gumhalter et al., 1994) and the calculation of sticking at low incident energies compared to the potential well depth is accordingly justified. 4.3.2. One-phonon regime: Distorted wave Born approximation The DWBA for sticking dates back to Lennard-Jones pioneering work in the thirties (Lennard-Jones and Strachan, 1937; Lennard-Jones and Devonshire, 1937) and has been described and applied in many papers (Goodman, 1980, 1981; Stutzki and Brenig, 1981; Stiles and Wilkins, 1985; Andersson et al., 1989; Gumhalter, 2001). This theory amounts to a quantum mechanical treatment of both the particle and lattice motions and handles the particle-phonon coupling by first order perturbation theory so that the energy transfer occurs only through one-phonon excitations. Here we will introduce the key ingredients of this model, including its extension to handle trapped particles and resonances, and discuss its physical content with specific reference to sticking of hydrogen in the physisorption well on a copper surface. The first step in the application of the DWBA to sticking is to find an appropriate zero order Hamiltonian and the associated basis states of the quantum mechanical motion of the hydrogen molecule in the physisorption well of the rigid surface. As demonstrated in Section 4.2, the physisorption interaction is dominated by the one-dimensional potential V0 (z) and the remaining parts of the interaction such as the surface corrugation V1 and the rotational anisotropy V2 are weak. Thus an appropriate set of basis states for the DWBA is then given by |K|u|j, m where h¯ K is the lateral momentum, and |j, m is a free rotor

114


Fig. 4.11. Bound state and scattering wave functions in the H2 physisorption potential V0 (z) on Cu(100). The distance refers to the outermost surface layer of ion cores. Reprinted with permission from Andersson et al. (1989). © 1989 by the American Physical Society.

state with angular momentum j and azimuthal component m. The state |u is either a bound state |n with energy n or a scattering state |kz labeled by its incident, perpendicular momentum kz as is illustrated in Fig. 4.11, which shows the sequence of bound levels, n , in V0 (z) for H2 on Cu(100). For the lateral motion we are using periodic boundary conditions over a large surface box. In this zero-order picture, a particle can be trapped in a bound state at the surface with a total positive energy via a change in the lateral or in the rotational motion. The corrugated and the rotationally anisotropic parts of the physisorption interaction will turn these states into quasi-bound states, which show up as selective adsorption resonances in beam scattering as discussed in Section 4.2. These quasi-bound states turn out to play a profound role in the sticking. In the following, we will primarily focus on processes for the hydrogen molecule where the rotational motion is a spectator and the rotational state will be suppressed if not stated otherwise. An incident particle in a scattering state |p = |K|kz can now trap at the surface by making a transition into a bound state |ν = |K |n by the perturbation Vph from the interaction of the particle with the substrate phonons. For light particles the interaction term Vph is commonly described, as in the FOM, by the lattice forces fi (r), resulting in a linear coupling model of the form:

fiz (r).uiz Vph r, {ui }i = − (4.22) i

1 dV0 = −√ (z)uz (Q) N G Q dz 1 G+Q 2 + iQ.r . × exp − 2 Qc

(4.23)

Here we have kept only the dominant perpendicular component uiz of the displacement of the ith substrate atom. In the second and third lines, we have made use of the surface peri-


115

odicity and introduced collective coordinates for the displacements labeled by the phonon wave vectors Q in the surface Brillouin zone where N is the number of surface atoms used in the “surface box” normalization. A key parameter in Eq. (4.23) is the effective wave√ vector cut-off Qc = (α/zt ) set by the repulsive exponent α of V0 (z) in Eq. (4.2) and the distance zt of the classical turning point of the particle in V0 (z) at i = 0 from the outermost surface layer of ion cores. Using the real space form for Vph in Eq. (4.22), the transition probability Pνp from |p to |ν involves only one-phonon processes and is given by first order perturbation theory as 4mp s − f Pνp = 3 (4.24) ν|fiz |p.Cij .p|fj z |ν. h¯ h¯ kz i,j Here, the initial state |p has been normalized to unit incident amplitude and the phonon spectral function Cij (ω) is defined as in Eq. (4.14). At normal incidence of the hydrogen molecule, the parallel energy transfer is limited by −1 the phonon wave vector cut-off Qc ≈ 1 Å to be about 1 meV so that essentially all final bound states have a total energy less than zero. S0 is then essentially given by the trapping probability T defined as the sum of transition probabilities to all final states ending up as bound states of V0 (z) as, Pνp . T = (4.25) ν

As shown in Fig. 4.8, the resulting S0 is in reasonable agreement with the result obtained in the FOM using the one-phonon loss distribution Ps (). The close agreement between these two seemingly different models can be understood from the following observations. Equations (4.13) and (4.24) show that the one-phonon transition probability Pνp in DWBA is closely related to the one-phonon loss distribution Ps () in FOM. In the semi-classical limit, the lattice force matrix elements ν|fz |p between an incident scattering state and a bound state in Eq. (4.24) reduces directly to the Fourier-transformed lattice force compo − nent fz ( p h¯ ν ) in Eq. (4.13) (Landau and Lifshitz, 1965). This result shows that the use of a non-sticking classical trajectory in FOM to calculate the sticking coefficient is less severe than one would anticipate. Furthermore, the sum over the discrete set of bound states in Eq. (4.25) for T reduces to an energy integral in the classical limit corresponding to the energy integral for S0 in Eq. (4.15). However, the FOM does not respect the finite well depth of the physisorption potential and energy losses larger than i + D are included. Furthermore, the weak structure at and below 10 meV in the DWBA calculation of S0 in Fig. 4.8 is due to resonant-like transitions to bound states involving zone-boundary phonons, which gives rise to sharp peaks in the phonon spectral function (Persson, 1987). A one-phonon theory such as DWBA gives a good account of the sticking coefficient at low energies up to about 10 meV, where one-phonon excitations according to the FOM are dominant. As shown in Fig. 4.12, the DWBA results for S0 of D2 on all the three low-index crystal faces of Cu at normal incidence is in good agreement with the measured S0 in the one-phonon regime. In particular, the enhanced sticking on Cu(110) compared to the other two crystal faces is well reproduced. This effect is caused by structure specific differences in the particle-phonon coupling including umklapp processes. V0 (z) varies only weakly with the crystal face and affects

116


Fig. 4.12. (a) Measured and (b) calculated sticking coefficient at normal incidence of D2 on Cu(111) (circles), Cu(100) (squares) and Cu(110) (triangles).

the sticking probability marginally. The phonon coupling via −1

dV0 dz (z)

in Eq. (4.23) and the

wave-vector cut-off Qc ≈ 1 Å , which are both determined by V0 (z), is then very similar in all three cases. The (110) crystal face differs from the other two faces by having significantly smaller Q and G vectors in one of the surface directions. These smaller wave vectors are favored in the particle-phonon coupling by the wave-vector cut-off as illustrated by the effective spectral function that essentially determines the phonon contribution to the energy transfer, 1 G+Q 2 exp − Czz (Q, ω) Ceff (ω) = (4.26) N Qc G

Q

where Czz (Q, ω) is the wave-vector resolved spectral function for perpendicular displacements of the surface atoms. When neglecting umklapp contributions from G = 0 reciprocal lattice vectors and taking the limit of infinite wave vector cut-off, Ceff (ω) reduces to Czz (ω) for a single atom in Eq. (4.14). Figure 4.13 shows that Ceff (ω) for Cu(110) is appreciably larger than for the other two surfaces over the whole phonon band width. A comparison with the results for Czz (ω) in Fig. 4.6 indicates that the difference in the linearly increasing low-energy range derives from umklapp processes. At off-normal incidence, the trapping probability in Eq. (4.25) gives only an upper bound for the sticking coefficient because the trapped particle can have sufficient parallel energy to reemerge to the gas-phase through converting its parallel energy back into normal energy by scattering from the corrugated part V1 (z) of the physisorption potential. This back scattering will compete with the energy transfer to substrate phonons that will ultimately result in sticking when the particle ends up in a bound state with total energy less than zero. The fate of these positive-energy trapped particles will determine the angle


117

Fig. 4.13. Calculated effective phonon spectral function Ceff (ω) (see Eq. (4.26)) for the (111) (circles), (100) (squares), and (110) (triangles) crystal faces of Cu as a function of the phonon energy h¯ ω.

dependence of sticking. In the extreme limit when the energy transfer rate dominates, all trapped particle will stick resulting essentially in a normal energy scaling of the sticking coefficient with incidence angle. Whereas in the opposite extreme limit when the back scattering rate dominates, all positive-energy trapped particles will reemerge into the gasphase resulting essentially in total energy scaling of S0 with incidence angle. These elastic and inelastic processes limits the life-time (or the mean-free path) of a trapped particle in a quasi-bound state and show up as a life-time broadening of selective adsorption resonances. The relative importance of the elastic and inelastic processes can be unraveled by studying the temperature dependence of life-time broadenings (Persson et al., 1990) of selective adsorption resonances. 4.3.3. Mean free paths of quasi-bound states The intrinsic broadening of a selective adsorption resonance is illustrated in Fig. 4.14 by specular reflectivity I00 /I0 data and sticking S0 data obtained for o-D2 beams scattered from a cold Cu(100) surface (Persson et al., 1990). The lower dip at about 16 meV in the reflectivity data is due to a corrugation-mediated selective adsorption resonance (CMSA) derived from the bound state n = 5 and a reciprocal lattice vector G10 . The lower panel in Fig. 4.14 shows that the sticking is a decay mode of the resonance since it displays a resonance peak that mirrors the dip in the reflectivity. The intrinsic width is largely exaggerated in Fig. 4.14 because of kinematical enhancement from the free-particle dispersion of the resonance. By taking account of this enhancement, the observed line shape could be fitted with a convolution of a Lorentzian line shape, describing the intrinsic life time broadening, with a Gaussian energy resolution function and a function describing the angular spread

118


Fig. 4.14. Resonant sticking S0 and specular scattering probabilities I00 /I0 of o-D2 on Cu(100) vs beam energy i at the angle of incidence θi = 51◦ and azimuthal angle φi = 47.2◦ . The inset shows the measured (open circles) and calculated (solid line) line shapes.

of the beam. This convolution procedure reproduces the observed line shape accurately as shown in the inset of Fig. 4.14 and allows extraction of the intrinsic resonance widths Γν (full width half maximum of the Lorentzian line shape) of CMSA resonances derived from quasi-bound states ν both for H2 and D2 . These widths are plotted as a function of substrate temperature in Fig. 4.15. The intrinsic width Γν of the CMSA resonance derived from the quasi-bound state ν ≡ (n, K) is determined predominantly by reemergence back to the gas-phase by scattering from the surface corrugation and emission or absorption of single phonons. To first order in the electron phonon coupling Vph and the surface corrugation V1 , these elastic and inelastic processes contribute additively to Γν as Γν = Γνel + Γνinel .

(4.27)

The elastic contribution Γνel = h¯ Wνel is given as a sum over elastic transition rates from the quasi-bound state ν to all final open channels |f = |kf z |K + G10 as, Wνel =

2mp kf z |V1 |n2 3 h k f ¯ fz

(4.28)


119

Fig. 4.15. Substrate temperature dependence of the intrinsic line widths Γ of corrugation-mediated selective adsorption resonances. The solid and open symbols are data for Γ derived from experimental resonance line widths for p-H2 (circles) and o-D2 (triangles) observed in specular scattering and sticking on Cu(100), respectively. The solid (p-H2 ) and dashed (o-D2 ) lines are calculated from theory.

where hk ¯ f z = 2mp n (K) − h¯ 2 (K + G10 )2 , z|n is the normalized bound-state function, z|kf z is the scattering wave function for the motion normal to the surface with unity incident amplitude and V1 (z) is the corrugated potential in Eq. (4.7). The inelastic contribution Γνinel = h¯ Wνinel is given by the sum Wfinel Wνinel = (4.29) ν f

over inelastic transition rates Wfinel ν from the quasi-bound state ν to all final bound and continuum states |f accessible via emission (+) or absorption (−) of a single phonon with energy h¯ ω and lateral wave vector Q = ±(K − Kf ). Wfinel ν has a similar form to Pνp in Eq. (4.24) for a transition to a continuum state and is given by, 2π ν − ν inel Wν ν = 2 (4.30) ν |fiz |νCizj z .ν |fj z |ν∗ h¯ h¯ ij for a transition to a final bound state ν . At non-zero substrate temperature T the factors 1 + n(ω) and n(ω) for phonon emission and absorption, respectively, are included in the phonon spectral function. As shown in Fig. 4.15, the close agreement between the calculated and the experimentally inferred substrate temperature dependence of Γν provides strong evidence that the life times of the trapped molecules are determined predominantly by these elastic and inelastic processes. The dependence on T is strong and reduces the observed mean free paths

120


l = h¯ 2 K/(mp Γν ) from l = 140 (60) Å at 10 K to 30 (16) Å at 280 K for H2 (D2 ). The crossover of Γν from a constant to a linear dependence with increasing T is a simple consequence of the behavior of n(ω) with T in the inelastic transition rates. The elastic contribution Γ el is independent of T and is a substantial fraction of the total zero-temperature width. Thus the elastic and inelastic rates are comparable in magnitude and both processes have to be accounted for in determining whether a positive-energy trapped particle will stick or not. 4.3.4. Sticking of positive-energy trapped particles So as to determine the sticking coefficient at off-normal incidence, it is necessary to treat the partitioning of particles trapped at positive energy into stuck and backscattered fractions. A simple way to handle this partitioning is to use a master equation approach for the probability distribution of a particle in trapped states with elastic and inelastic transition rates obtained by first order perturbation theory (Gortel et al., 1980; Böheim, 1984; Brenig, 1987; Andersson et al., 1989; Brivio and Grimley, 1993). This approach is justified by the minute broadenings of the bound state levels compared to their level spacings for hydrogen molecules in a physisorption well. For non-resonant processes and in the one-phonon regime, the initial collision of a particle in an incident state p results in a probability distribution for the particle Pν (t = 0) = Pνp in trapped states ν that are given in DWBA by the inelastic transition probabilities Pνp in Eq. (4.24). The direct sticking probability S0 (t = 0) at zero substrate temperature in the initial collision is given by the probability for the particle to be trapped with negative energy. The probability distribution of a positive-energy trapped particle PT (t) will decay in time either by reemergence of the particle to the gas phase by scattering from the surface corrugation or by climbing down the “ladder” of bound state levels by one-phonon transitions. The resulting time evolution of PT (t), P˙T (t) = −WD PT (t) + WT PT (t),

(4.31)

is described by a rate matrix −WD + WT . The first and second term on the right hand side of Eq. (4.31) describes depletion and accumulation processes, respectively. The matrix WD has only diagonal elements WDν , which are determined by the rate Wνel + Wνinel obtained from Eqs. (4.28) and (4.29). The matrix WT has only off-diagonal elements WT ν ν , which are determined by the one-phonon transition rates Wνinel ν in Eq. (4.30). The one-phonon transitions from positive-energy trapped states to negative-energy trapped states will increase the probability for the particle to stick as, S˙0 (t) = WS PT (t)

(4.32)

where the row vector WS is determined from the sum of one-phonon transition rates in Eq. (4.30) between a positive-energy trapped state to all negative-energy trapped states. Note that at zero temperature, transitions among stuck states do not change S0 (t). The final sticking probability in the limit t → ∞ is then obtained by an integration of Eqs. (4.31) and (4.32) as


121

Fig. 4.16. (a) Measured and (b) calculated sticking coefficient at off-normal incidence (∼60◦ ) of D2 on the (111) (circles), (100) (squares) and (110) (triangles) surfaces of Cu.

S0 = S0 (0) − WS (−WD + WT )−1 PT (0) ∞

p = S0 (0) + WS WD−1 W inel WD−1 PT (0).

(4.33) (4.34)

p=0

The first term on the right hand side arises from direct sticking on the initial collision and the second term via intermediate positive-energy trapping that have been expanded in the second line into the number of one-phonon transitions among the bound state levels. The importance of partitioning of positive-energy trapped particles into stuck and backscattered fractions is nicely illustrated by the sticking at off normal incidence of D2 molecules on the three crystal faces of Cu. As shown in Fig. 4.3, these surfaces exhibit a varying degree of corrugation of the physisorption potential V1 resulting in different elastic backscattering rates into the gas-phase. Figure 4.16 shows a comparison of the calculated sticking coefficients based on Eq. (4.34) of D2 on these three crystal faces at an angle of incidence of about 60◦ with the corresponding measured sticking coefficients. At this incidence angle and low-energy range, the maximum normal energy is about 11 meV, so that one-phonon processes dominate the energy transfer and the DWBA is justified. The calculated results for S0 are in good agreement with the experimental data over all incident energies and crystal faces. The trend of the decrease of S0 over the three crystal faces, being slowest for Cu(111) and fastest for Cu(110), is nicely reproduced. This behavior is a manifestation of the intermediate positive-energy trapping contribution to sticking. On the weakly corrugated Cu(111) surface the result is very close to the pure trapping situation. The back scattering into the gas-phase of positive-energy trapped particles is sufficiently weak so that sticking via further one-phonon emissions is the dominant process. On the more strongly corrugated Cu(110) surface backscattering from the surface corrugation of positive-energy trapped molecules is enhanced so that the deviation from pure trapping

122


is substantial; S0 is reduced by about a factor of 2 at i = 40 meV. The enhanced S0 on Cu(110) at low i has the same physical origin as discussed in relation to the normal incidence data, that is, the specific coupling to the substrate phonons including umklapp. The importance of the partitioning of positive-energy trapped particles on the dependence of S0 on incidence angle is not limited to the quantum regime and has a close analog in the classical regime of sticking of inert particles in the physisorption well (Head-Gordon et al., 1991; Smith et al., 1993; Kulginov et al., 1996). This has been demonstrated from a comparison of sticking probabilities S0 for Ar beams on a Pt(111) surface (Mullins et al., 1989) measured at a substrate temperature of 80 K and classical molecular dynamics calculations of S0 (Kulginov et al., 1996) based on an empirical interaction potential that was constructed from the inversion of molecule beam scattering data. At this temperature the atoms desorb on a relative short but macroscopic time scale so that the sticking probabilities were measured by molecular beam scattering from the time delay of the scattered fraction that was stuck at the surface on a macroscopic time scale. As discussed in Section 4.3.1 for the similar system Ar on Cu(111) with about the same well depth, the energy transfer and sticking should be in the classical regime at T = 80 K in this case. Figure 4.17 shows that the agreement between the calculated and measured S0 is remarkable and that S0 follows closely the scaling i cos1.5 θi . This scaling is intermediate between normal energy scaling i cos2 θi and total energy scaling i and is a consequence of the fact that positive-energy trapped particle trajectories may eventually lose its energy to the substrate or leave the surface by energy conversion from the normal motion to the lateral motion or by gaining energy from the phonon system at T = 80 K. Finally, the saturation of S0 at low energies to a value below unity is not a quantum effect but a consequence of energy gain

Fig. 4.17. Energy scaling of calculated (open symbols) and measured (solid symbols) sticking coefficient S0 for Ar on Pt(111) at a substrate temperature of 80 K and different incidence angles θi . The experimental data and their scaling are taken from Mullins et al. (1989).


123

by the particle at finite-temperature. This gain is important at T = 80 K for Ar, because Ar interacts predominantly with low-energy phonons at these low particle energies. 4.3.5. Resonant sticking The sticking in the physisorption well can also involve resonant processes as illustrated in Figs. 4.5 and 4.14. An analysis of the dispersion of these resonant peaks appearing both in elastic scattering and in sticking data shows that they can be attributed to CMSA resonances. A proper theoretical description of these resonant processes is a daunting task since the coupling to substrate phonons and the elastic perturbation giving rise to a selective adsorption resonance has to be treated on an equal footing and no fully satisfactory treatment has yet been given. In the approach to resonant sticking by Stiles and Wilkins (1985), who considered specifically rotation mediated selective adsorption (RMSA) resonances for H2 on Cu(100), the elastic perturbation was included non-perturbatively in the zero order Hamiltonian and the basis states. The decay of a resonance by elastic back scattering is then automatically included but the calculation of the sticking using first order perturbation theory resulted in very narrow resonance lines with a strong violation of unitarity. In a further development by Stiles et al. (1986) based on a self-consistent onephonon approximation, the inelastic contribution to the resonance width was included and there was no violation of unitarity. However, this theory did not include the repeated emission of phonons down the “ladder” of bound levels for a positive-energy trapped particle. As demonstrated in Section 4.3.4, these processes are important for sticking when the phonon coupling and the elastic perturbation are of the same order. However, the elastic and inelastic perturbations and the “ladder climbing” down to a stuck state can be handled by a semi-phenomenological theory (Böheim, 1984). In this theory, the selective adsorption resonance originating from a quasi-bound state, ν = (n, j, K) is modeled in scattering to have a Breit–Wigner form with a total resonance width Γν and a partial width Γνiel due to decay through the incident channel. The quantity, PT (i ) =

Γνiel Γν , (i − ν )2 + Γν2 /4

(4.35)

is then interpreted as the probability that the resonance is populated as a result of the initial collision. Γν will have both elastic Γνel and inelastic Γνinel contributions as in Eq. (4.27) that can be calculated by perturbation theory. In the case of a CMSA resonance and perturbation theory, Γνiel is given by the contribution from the incident channel in Eq. (4.28). The integrated strength Ttot of the trapping probability at fixed angle of incidence θi is given by, 2πΓiel . (4.36) κ Here κ is the same kinematical factor introduced by the dispersion of the resonance that enhances the observed intrinsic width of the resonance discussed in Section 4.3.3. The corresponding integrated strengths of the resonance dip in the elastic scattering probability, Ptot , and the resonance peak in the sticking, Stot , are now determined by the integrated strength Ttot to populate the resonance, the branching probability (Γ − Γi )/Γ for Ttot =

124


Table 4.2 Observed and calculated resonance strengths in sticking and scattering of corrugation mediated selective adsorption resonances of H2 and D2 on Cu(100). The values in the parentheses are experimental data. The resonances involve a G10 reciprocal lattice vector and the bound states n = 4 and n = 5 for H2 and D2 . θi = 42◦ (50◦ ) for H2 (D2 )

H2 D2

κ

Γ (µeV)

Ttot (meV)

PS

Γ inel Γ

Γ −Γiel Γ

0.34 0.17

55 101

0.41 1.5

0.49 0.55

0.56 0.55

0.80 0.80

Stot (meV)

Ptot (meV)

0.15 (0.1) 0.8 (0.3)

0.33 (0.2) 1.2 (0.4)

backscattering into all open elastic and inelastic channels except the incident channel, and the branching probability PS for sticking as Γ − Γi Ttot , Γ = PS Ttot .

Ptot =

(4.37)

Stot

(4.38)

The branching probability PS is simply given by the probability for sticking via intermediate positive-energy trapping as described by the second term in Eq. (4.34). The observed and calculated resonance strengths, Ptot and Stot of some typical selective CMSA resonances for H2 and D2 on Cu(100) are tabulated in Table 4.2. The main reason why a rather weakly corrugated surface nevertheless gives rise to prominent sticking lines such as in Fig. 4.14 is the strong kinematic enhancements κ −1 of such CMSA resonances by almost an order of magnitude. The calculated values of the resonance strengths are overestimates, in particular for D2 . Given the experimental uncertainties and the simple theory, which, for instance, neglects interference effects giving rise to Fano-like line shapes rather than Lorentzian line shapes, the agreement is rather reasonable. The values for PS inel are close to their upper bounds given by the branching ratios ΓΓ into all inelastic channels and shows that sticking is the dominant inelastic decay mode of these resonances. The Γ −Γ el

inel

difference between Γ i and ΓΓ gives the branching ratio into elastic channels other than the incident channel. Whereas only a single RMSA resonance has unequivocally been identified in the sticking data, several combined corrugation and rotational mediated selective adsorption (CRMSA) resonances have been identified. The anomalous strengths of the CRMSA resonances compared to RMSA resonances suggest that a substantial rotational inelasticity requires participation of lattice modes (Andersson et al., 1989).

4.4. Thermal desorption The dynamics of adsorption and thermal desorption of inert particles in the physisorption well are intimately related. As detailed in Section 4.3 sticking and trapping in the physisorption regime is all about energy conversion among various particle degrees of freedom and dissipation of its energy to the phonon excitations. In the low temperature regime,


125

specifically considered in Section 4.3, it is unlikely that the particle will gain enough energy from the solid lattice and escape and the particle can be assumed to be permanently stuck on the surface. However, at elevated surface temperatures this process becomes increasingly likely and thermal desorption has to be included in the description of the gas–surface interaction. Thermal desorption is in different adaptations, a well established experimental tool for studies of adsorbates on surfaces. The process is thermally activated, the desorption rate shows an Arrhenius dependence with an activation energy, d , and a method like temperature-programmed desorption gives information about d and other details of the desorption kinetics. Depending on initial conditions the desorption dynamics can be quite complex and shortcomings of simple laws regarding distributions of directions, velocities and internal energies of desorbing atoms and molecules have been critically scrutinized in the literature (see e.g. Comsa and David, 1985; Brivio and Grimley, 1993). Thermal desorption of physisorbed inert atoms in thermal equilibrium with the surface and in the limit of zero coverage is the simplest prototype situation. Assuming that the atoms are in thermal equilibrium with the surface, one obtains from the flux of desorbing atoms a simple Arrhenius expression for the desorption rate, d Wd0 = ν exp − (4.39) . kB T In the classical limit kB T h¯ Ω, where Ω is the vibrational frequency in the physisorption potential, the activation energy is equal to the well depth, d = D, and the characteristic Ω frequency factor ν = 2π . In general the energy exchange between the particles and the phonon system is not efficient enough to keep the adsorbed particles with positive energies in thermal equilibrium with surface. The desorption rate can be corrected to account for this effect by considering the global equilibrium of the adsorbed phase with the gas phase. In this situation detailed balance demands that the net flux leaving the surface should be balanced by a net flux incident on the surface. For S < 1, some of the incident particles will scatter back into the gas phase and should be excluded from the incident flux. Correcting for this effect gives a desorption rate (see e.g. Zangwill, 1988). d Wd = S(T )Wd0 = S(T )ν exp − (4.40) . kT Here S(T ) is the sticking coefficient of incident particles averaged over the gas-phase thermal distribution of energies and angles of incidence. As shown in Section 4.3, S0 decreases rapidly with increasing energy for physisorbed particles resulting in a rapid decrease of S(T ) with T . This behavior is nicely illustrated by the measured initial sticking coefficients of Ne, Ar, Kr and Xe on Ru (001) (Schlichting et al., 1992) shown in Fig. 4.18. Thus the sticking dynamics influences also the desorption dynamics and has a significant effect on the desorption rate. A striking consequence of the sticking dynamics resulting in the energy and angle dependence of S0 is observed in the velocity distribution of the desorbing particles (Rettner et al., 1989; Head-Gordon et al., 1991). Detailed balance dictates that a Maxwell distribution of particles should pass in both directions through a plane parallel to the surface just

126


Fig. 4.18. Measured initial sticking coefficients S0 as a function of the gas temperature, Tg , for Ne, Ar, Kr and Xe on a Ru(001) surface kept at 6.5 K. Reused with permission from Schlichting et al. (1992). © 1992 American Institute of Physics.

outside it and in the gas phase (Rettner et al., 1989). Since S0 decreases with increasing incident energy, high velocity particles incident on the surface will scatter directly back into the gas phase with higher probability than particles with lower velocities, which implies that fewer particles with high velocity will desorb thermally in order to maintain a correct Maxwell distribution. This “cooling” effect can be directly observed in experiments with rare gas atom beams incident on metal surfaces kept at different temperatures. The velocity distributions of atoms in the directly scattered fraction and the trapping-desorption fraction can be separated in time-of-flight measurements as demonstrated for example for Ar beams interacting with Pt(111) (Head-Gordon et al., 1991). The velocity distributions of the trapping-desorption fraction can be reproduced closely by a Maxwell–Boltzmann function yielding an effective temperature, Teff . Figure 4.19 displays experimental data of Teff versus desorption direction for Ar atoms scattered from Pt(111) at three surface temperatures Ts . The incident Ar atoms have a kinetic energy of 0.13 eV and hit the surface at an angle of 60◦ relative to the surface normal at a velocity that is much faster than the trapping-desorption fraction. Ar atoms trapped and desorbing in a direction close to the surface normal (0◦ final angle) at the surface temperatures 100 K, 190 K and 273 K exhibit effective temperatures of about 90 K, 150 K and 220 K respectively. These distributions are clearly colder than the expected Maxwell–Boltzmann distributions at Ts an observation which is consistent with the detailed balance argument presented above. The data in Fig. 4.19 also reveal another important aspect of the gas–surface dynamics. At larger final angles the effective temperature of the trapping-desorption fraction increases and Teff even becomes larger than Ts in violation of the detailed balance argument above. The effect is small at Ts = 100 K with Teff ∼ 100 K at 40◦ final angle, but becomes substantial at Ts = 273 K with Teff ∼ 400 K. The basic mechanism that gives rise to Teff > Ts refers to slow accommodation of the parallel component of the incidence velocity on the


127

Fig. 4.19. The effective temperature Teff of the trapping-desorption fraction versus desorption direction for an Ar beam interacting with a Pt(111) surface at three surface temperatures Ts . The incident Ar atoms have a kinetic energy of 0.13 eV and hits the surface at 60◦ angle with respect to the surface normal. Reused with permission from Head-Gordon et al. (1991). © 1991 American Institute of Physics.

very weakly corrugated Pt(111) surface resulting in higher velocities for particles desorbing at larger final angles. The effect is more pronounced at higher surface temperature because the Ar residence time is then so short that the parallel component of the incident velocity does not have time to equilibrate with the surface and the detailed balance argument does not longer apply. The residence time can be estimated from Eq. (4.39) using ν = 1012 /s and D = 0.080 eV which gives half-lives ∼10−11 s at 273 K and ∼10−7 s at 100 K. The normal component of the incidence velocity equilibrates rapidly via the strong repulsive backwall interaction and particles desorbing close to the direction of the surface normal will show the expected behavior discussed above.

4.5. Photodesorption of physisorbed species The conventional description of the physisorption interaction is, as discussed in Section 4.2, expressed in terms of a competition between a long-range van der Waals attraction and a short-range Pauli repulsion. The adsorbate-substrate electron correlation gives, for example, a van der Waals potential which falls off with distance like z−3 . The van der Waals forces also polarize the adsorbate so that the valence electrons are displaced towards the metal surface. The induced dipole moment varies like z−4 (Antoniewicz, 1974; Zaremba, 1976). Density functional calculations, intended for closer distances, also reveal

128


a rapidly varying polarization with distance (Lang, 1983). Both models lead to the prediction that vibrational motion in the potential well should result in a fluctuating dipole and excitations among the well states will be dipole allowed. The strong nonlinear dependence of the induced dipole moment function will result in large probabilities for dipole excited vibrational transitions from the ground vibrational state to high-lying bound levels in the physisorption well but also to free continuum states. These phenomena have been manifested experimentally for the light physisorbed species H2 , and D2 in spectroscopic measurements and in observations of direct infrared photodesorption of these adsorbates and He. Current theory can account quantitatively for these observations as discussed below. 4.5.1. Dipole transitions in the physisorption well The dipole excited transitions between the bound states in the physisorption well have been measured for H2 , D2 and HD adsorbed on the Cu(100) surface using high-resolution electron energy-loss spectroscopy (HREELS) (Svensson and Andersson, 1997). Figure 4.20 displays the low-energy region of the H2 spectrum obtained for an uncompressed monolayer physisorbed at about 10 K on the Cu(100) surface. The spectrum shows loss peaks at 8.9, 15.3 and 20.0 meV corresponding to the n = 0 → 1, n = 0 → 2 and n = 0 → 3 vibrational transitions in the physisorption well, in excellent agreement with the transition energies 8.8, 15.4 and 20.4 meV derived from the H2 –Cu(100) physisorption potential determined via selective adsorption measurements discussed in Section 4.2. All the transitions in Fig. 4.20 are dipole excited as revealed by angular distribution measurements. Examples of such distributions for H2 are displayed in Fig. 4.21 and include the elastic peak, the n = 0 → 1 and n = 0 → 2 vibrational peaks and the j = 0 → 2 rotational peak at 44 meV (not shown in Fig. 4.20). The vibrational angular distributions peak sharply in the specular direction which is characteristic for dipole excited transitions. The inelas-

Fig. 4.20. Electron energy-loss spectrum (HREELS) from a monolayer of H2 adsorbed on the Cu(100) surface at about 10 K. The spectrum is measured in the specular direction for a 3 eV electron beam incident at 48◦ from the surface normal.


129

Fig. 4.21. HREELS angular distributions for H2 adsorbed on Cu(100) at 10 K; elastic peak intensity (solid curve) and inelastic peak intensities vs. detector angle (0◦ = specular direction, b by substrate phonons at the Cu(100) surface can be directly calculated from their cross section for dipole excitation as measured in the electron energy-loss measurements (Andersson et al., 1984; Hassel et al., 1998, 2002). This rate is 2 × 10−5 s−1 per surface unit cell at a blackbody temperature of 296 K. The interaction, via the physisorption potential, between the longitudinal phonons, propagating in a direction normal to the surface and a single adsorbed H2 molecule is well represented by −u0 dV0 (z)/dz. The high frequency phonons with energy >b are, however, weakly coupled to the translational coordinate of the physisorbed molecule and the resulting phonon-induced desorption rate is 5×10−8 s−1 i.e. several orders of magnitude smaller than the direct photodesorption rate. The latter argument can be qualitatively understood considering the small one-phonon sticking probability in the range 26–30 meV for H2 on Cu(100) (see Section 4.3.1). 4.5.4. Photodesorption of He As discussed in Section 4.2, He adsorbs on metal surfaces with considerably lower binding energies than H2 does. On noble metal surfaces the potential well depth is in the range 6–8 meV involving a low number of bound levels which suggests that direct infrared photodesorption could be an effective process also in this case. The van der Waals induced dipole will be weaker than for H2 , since He is less polarizable and the potential minimum is further away from the metal surface, pz ∼ z−4 , but the much shallower potential well may compensate for these effects. Photodesorption of He adsorbed on the Pt(111) surface has been observed in thermal desorption measurements (Niedermayer et al., 2002). These are demanding experiments, which has to be performed at a temperature below 3 K, since the He adsorbate desorbs at temperatures around 4 K. The temperature programmed desorption spectra (TPD) of

136


4 He

at 295 K and 90 K surrounding temperatures show that at 90 K, thermal desorption is the dominating process while at 295 K infrared photodesorption clearly dominates below 3.5 K. At 295 K the photodesorption rate has been determined to be around 0.01 s−1 for 4 He and about 1.5 times higher for 3 He. Theoretical analysis suggests that photodesorption by far-infrared photons coupling to the transition dipole induced by the adsorbate is responsible for the experimental observations (Niedermayer et al., 2002). 4.6. Concluding remarks In this chapter we have considered elementary dynamical processes in the physisorption regime involving inert gas particles interacting with low-index metal surfaces. In order to make contact between experimental observations and theoretical models it is crucial that the particle–surface interaction is known with sufficient precision. In this context the simplest physical situation refers to light physisorbates like H2 , He and Ne in the limit of small coverage of ad-particles and low substrate temperature. For these adsorbates, resonance scattering and diffraction experiments yield in conjunction with current physisorption theory accurate knowledge about the adsorbate–surface interaction potential, including also finer details like the rotational anisotropic interaction for H2 . For heavier ad-particles, reasonable interaction potentials may be obtained from the theory tuned to, for example, measured adsorption energies. Ongoing developments of a seamless physisorption potential based on density functional theory may result in improved possibilities to obtain an accurate a priori calculated potential energy surface. Considering sticking and trapping in the physisorption regime the dynamics is all about energy conversion among the particle degrees of freedom and dissipation of its energy to the phonon system of the solid lattice. Equipped with a detailed knowledge about the potential-energy surface governing the gas–surface collision process and a realistic description of the substrate phonons provides a firm basis for a comparison, even on a quantitative level, between measured and calculated sticking coefficients. This comparison can only be accomplished, in particular for the light particles, by having a quantum mechanical theory for the energy transfer to the phonon system, like the distorted wave Born approximation (DWBA) or the forced oscillator model (FOM). These two theoretical approaches are complementary. The DWBA amounts to a quantum mechanical treatment of both the particle and the lattice motion and the particle-phonon coupling is treated by first order perturbation theory resulting in energy transfer only via one-phonon processes. The FOM on the other hand treats the particle-phonon coupling non-perturbatively allowing for energy loss via emission of an arbitrary number of phonons but at the expense of a classical trajectory approximation of the particle motion. Sticking and trapping in the one-phonon regime and the observed dependence on the particles incident energy and angle is accurately described by the DWBA. This model also accounts for observed differences related to the substrate surface structure due to the surface corrugation and the particle-phonon coupling. The following picture of the sticking of a light particle as a two-step process has emerged: (i) the particle is trapped in the physisorption potential well during the initial collision either by losing its energy via phonon emission or by entering a selective adsorption resonance;


137

(ii) particles trapped in quasi-bound states with positive energy partition into a fraction that eventually sticks by phonon emission and a fraction that reverts to the gas phase by diffraction via the surface corrugation. A particle trapped in a quasi-bound state moves essentially ballistically along the surface with a mean free path being orders of magnitude larger than the lattice spacing and provides a prototype example of a precursor state. For normal incidence, the initial sticking probability calculated by FOM agrees well both with the DWBA results in the range of low incident energies where the energy transfer is dominated by one-phonon processes and with experimental data over a wider energy range involving multiphonons processes. At off-normal incidence, the sticking criterion is ill defined in FOM and has to be judged with respect to experimental observations. The development of a fully quantum model for particle phonon energy transfer remains a challenging issue. Trapping, sticking and thermal desorption at higher substrate temperatures poses further challenging theoretical problems. However, for light inert particles and in the energy range where the energy transfer is dominated by one-phonon processes, the DWBA permits a determination of state-to-state transition rates for realistic models. Regarding experimental developments we note that scattering, trapping and sticking measurements can be performed at the same low substrate temperature and in the limit of low adsorbate coverage by desorbing the stuck particles using short laser pulses. Thermal desorption measurements using short laser pulses permit rapid temperature jumps, which can be exploited for a controlled variation of time scales relative to characteristic desorption times. Regarding direct infrared photodesorption, current physisorption theory lacks a complete description of how to calculate the induced dipole moment function. Fortunately, this information can be obtained from spectroscopic measurements like high-resolution electron energy-loss spectroscopy (HREELS). This simplifies the situation significantly, regarding a quantitative comparison of measured and calculated photodesorption rates. For the light adsorbates of concern, re-adsorption of the particle is a process of rather low probability and can be neglected. The rate of photodesorption via single-photon absorption can then be calculated in a straightforward manner from the adsorbate dipole spectral function. The calculated and measured rates agree well for these conditions.

Acknowledgements We thank Ann-Christine Lindbom for typewriting part of this manuscript and Krister Svensson and Petter Linde for help with the figure material. Many coworkers have contributed over the years and their names are given in the references. We owe special thanks to John Harris for all amusing and stimulating discussions over the years. Financial support from the Swedish Research Council (VR) is gratefully acknowledged.

References Althoff, F., Andersson, T., Andersson, S., 1997. Phys. Rev. Lett. 79, 4429. Andersson, S., Persson, B.N.J., Persson, M., Lang, N.D., 1984. Phys. Rev. Lett. 52, 2073.

138


Andersson, S., Persson, M., 1993a. Phys. Rev. B 48, 5685. Andersson, S., Persson, M., 1993b. Phys. Rev. Lett. 70, 202. Andersson, S., Persson, M., Harris, J., 1996. Surf. Sci. 360, L449. Andersson, S., Wilzén, L., Persson, M., 1988. Phys. Rev. B 38, 2967. Andersson, S., Wilzén, L., Persson, M., Harris, J., 1989. Phys. Rev. B 40, 8146. Andersson, T., Althoff, F., Linde, P., Andersson, S., Burke, K., 2002. Phys. Rev. B 65, 045409. Andersson, T., Althoff, F., Linde, P., Hassel, M., Persson, M., Andersson, S., 2000. J. Chem. Phys. 113, 9262. Andersson, T., Linde, P., Hassel, M., Andersson, S., 2006. J. Chem. Phys. 124, 114703. Antoniewicz, P.R., 1974. Phys. Rev. Lett. 32, 1424. Baule, B., 1914. Ann. der Physik 44, 145. Beckerle, J.D., Johnson, A.D., Ceyer, S.T., 1989. Phys. Rev. Lett. 62, 685. Benvenuti, C., Calder, R.S., Passardi, G., 1976. J. Vac. Sci. Technol. 13, 1172. Black, J.E., Shanes, F.C., Wallis, R.F., 1983. Surf. Sci. 133, 199. Böheim, J., 1984. Surf. Sci. 148, 463. Brako, R., Newns, D.M., 1982. Surf. Sci. 117, 42. Braun, J., Fuhrmann, D., Graham, A.P., Toennies, J.P., Wöll, C., Bilic, A., Gumhalter, B., 1997. J. Chem. Phys. 106, 9911. Brenig, W., 1979. Z. Phys. B 36, 81. Brenig, W., 1987. Phys. Scr. 35, 329. Brivio, G., Grimley, T., 1993. Surf. Sci. Rep. 17, 1. Burke, K., Kohn, W., 1991. Phys. Rev. B 43, 2477. Celli, V., 1991. Surface Phonons. Springer, Berlin, p. 167. Celli, V., Evans, D., 1982. Dynamics of Gas–Surface Interactions. Springer-Verlag, Berlin. Chiesa, M., Mattera, L., Musenich, R., Salvo, C., 1985. Surf. Sci. 151, L145. Chizmeshya, A., Zaremba, E., 1989. Surf. Sci. 220, 443. Chizmeshya, A., Zaremba, E., 1992. Surf. Sci. 268, 432. Chubb, J.N., Gowland, L., Pollard, I.E., 1968. J. Phys. D 1, 361. Comsa, G., David, R., 1985. Surf. Sci. Rep. 5, 145. Cui, J., Fain, S.C., Liu, W., 1989. J. Vac. Sci. Technol. A 7, 1850. Dondi, M.G., Mattera, L., Ferreni, S., Tommasini, F., Linke, U., 1986. Phys. Rev. B 34, 5897. Ferm, P.M., Kurtz, S.R., Pearlstine, K.A., McClelland, G.M., 1987. Phys. Rev. Lett. 58, 2602. Gibson, K.D., Sibener, S.J., 1985. Phys. Rev. Lett. 55, 1514. Goodman, F.O., 1980. Surf. Sci. 92, 185. Goodman, F.O., 1981. Surf. Sci. 111, 279. Gortel, Z.W., Kreuzer, H.J., Teshima, R., 1980. Phys. Rev. B 22, 5655. Gruyters, M., Jacobi, K., 1994. Chem. Phys. Lett. 225, 309. Gumhalter, B., 2001. Physics Reports 351, 1. Gumhalter, B., Burke, K., Langreth, D.C., 1994. Surf. Rev. Lett. 1, 133. Gustafsson, K., Andersson, S., 2004. J. Chem. Phys. 120, 7750. Harris, J., 1991. Dynamics of Gas–Surface Interactions. The Royal Society of Chemistry. Harris, J., Feibelman, P.J., 1982. Surf. Sci. 115, 133. Harris, J., Liebsch, A., 1982a. J. Phys. C 15, 2275. Harris, J., Liebsch, A., 1982b. Phys. Rev. Lett. 49, 341. Harten, U., Toennies, J.P., Wöll, C., 1986. J. Chem. Phys. 85, 2249. Hassel, M., Svensson, K., Bellman, J., Andersson, S., Persson, M., 2002. Phys. Rev. B 65, 205402. Hassel, M., Svensson, K., Persson, M., Andersson, S., 1998. Phys. Rev. Lett. 80, 2481. Head-Gordon, M., Tully, J.C., Rettner, C.T., Mullins, C.B., Auerbach, D.J., 1991. J. Chem. Phys. 94, 1516. Hoinkes, H., 1980. Rev. Mod. Phys. 52, 933. Hult, E., Rydberg, H., Lundqvist, B.I., Langreth, D.C., 1999. Phys. Rev. B 59, 4708. Jackson, B., Persson, M., 1992. J. Chem. Phys. 96, 2378. Jensen, J.H., Chang, L.D., Kohn, W., 1989. Phys. Rev. A 40, 1198. Kern, K., David, R., Palmer, R., Comsa, G., 1986. Phys. Rev. Lett. 56, 2823. Kimman, J., Rettner, C.T., Auerbach, D.J., Barker, J.A., Tully, J.C., 1986. Phys. Rev. Lett. 57, 2053. King, D.A., 1978. CRC Critical Rev. Solid State Mater. Sci. 7, 167.


Kohn, W., 1994. Surf. Rev. Lett. 1, 129. Kulginov, D., Persson, M., Rettner, C.T., Bethune, D., 1996. J. Phys. Chem. 100, 7919. Kulginov, D., Persson, M., Rettner, C.T., 1997. J. Chem. Phys. 106, 3370. Landau, L.D., Lifshitz, E.M., 1965. Quantum Mechanics. Pergamon. p. 165. Lang, N.D., 1983. Surf. Sci. 127, L118. Lennard-Jones, J.E., Devonshire, A.F., 1937. Proc. R. Soc. London, Ser. A 158, 253. Lennard-Jones, J.E., Strachan, C., 1937. Proc. R. Soc. London, Ser. A 150, 442. LeRoy, R.J., 1976. Surf. Sci. 59, 541. Liebsch, A., 1986. Phys. Rev. B 33, 7249. Linde, P., Andersson, S., 2006. Phys. Rev. Lett. 96, 086103. Luntz, A., Mattera, L., Rocca, M., Tommasini, F., Valbusa, U., 1982. Surf. Sci. 120, L447. Luntz, A.C., Grimblot, J., Fowler, D.E., 1989. Phys. Rev. B 39, 12903. Meyer, H.D., 1981. Surf. Sci. 104, 117. Müller-Hartmann, E., Ramakrishnan, T.V., Toulouse, G., 1971. Phys. Rev. B 3, 1102. Mullins, C.B., Rettner, C.T., Auerbach, D.J., Weinberg, W.H., 1989. Chem. Phys. Lett. 163, 111. Niedermayer, T., Schlichting, H., Menzel, D., Payne, S.H., Kreuzer, H.J., 2002. Phys. Rev. Lett. 89, 126101. Nordlander, P., Harris, J., 1984. J. Chem. Phys. 17, 1141. Nordlander, P., Holmberg, C., Harris, J., 1985. Surf. Sci. 152, 702. Nordlander, P., Holmberg, C., Harris, J., 1986. Surf. Sci. 175, L753. Pearlstine, K.A., McClelland, G.M., 1983. Surf. Sci. 134, 389. Perrau, J., Lapujoulade, J., 1982. Surf. Sci. 121, 341. Persson, B.N.J., 1977. Solid State Comm. 24, 573. Persson, M., 1987. Phys. Rev. B 36, 7870. Persson, M., Harris, J., 1987. Surface Science 187, 67. Persson, M., Stroscio, J.A., Ho, W., 1987. Phys. Scr. 36, 548. Persson, M., Wilzén, L., Andersson, S., 1990. Phys. Rev. B 42, 5331. Rettner, C.T., Schweizer, E.K., Mullins, C.B., 1989. J. Chem. Phys. 90, 3800. Schiff, L.I., 1968. Quantum Mechanics. McGraw-Hill, New York. Schlichting, H., Menzel, D., Brunner, T., Brenig, W., 1992. J. Chem. Phys. 97, 4453. Schlichting, H., Menzel, D., Brunner, T., Brenig, W., Tully, J.C., 1988. Phys. Rev. Lett. 60, 2515. Schönhammer, K., Gunnarsson, O., 1980. Phys. Rev. B 22, 1629. Sedlmeir, R., Brenig, W., 1980. Z. Phys. B 36, 245. Smith, R.J., Kara, A., Holloway, S., 1993. Surf. Sci. 281, 296. Stiles, M.D., Wilkins, J.W., 1985. Phys. Rev. Lett. 54, 595. Stiles, M.D., Wilkins, J.W., Persson, M., 1986. Phys. Rev. B 34, 4490. Stutzki, J., Brenig, W., 1981. Z. Phys. B 45, 49. Svensson, K., Andersson, S., 1997. Phys. Rev. Lett. 78, 2016. Vidali, G., Cole, M.W., Klein, J.R., 1983. Phys. Rev. B 28, 3064. Vidali, G., Ihm, G., Kim, H., Cole, M.W., 1991. Surf. Sci. Reports 12, 133. Wilzén, L., Althoff, F., Andersson, S., Persson, M., 1991. Phys. Rev. B 43, 7003. Yu, C.-F., Whaley, K.B., Hogg, C., Sibener, S., 1985. J. Chem. Phys. 83, 4217. Zangwill, A., 1988. Physics at Surfaces. Cambridge University Press. Zaremba, E., 1976. Phys. Lett. 57A, 156. Zaremba, E., Kohn, W., 1976. Phys. Rev. B 13, 2270. Zaremba, E., Kohn, W., 1977. Phys. Rev. B 15, 1769.

139


CHAPTER 5

Intra-molecular Energy Flow in Gas–Surface Collisions

G.R. DARLING Surface Science Research Centre Department of Chemistry The University of Liverpool Liverpool L69 3BX, UK

© 2008 Elsevier B.V. All rights reserved DOI: 10.1016/S1573-4331(08)00005-X


Contents 5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Probing the potential energy surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. Is the motion adiabatic? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. Basic interactions in the PES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Quantum or classical descriptions of the dynamics . . . . . . . . . . . . . . . . . . . . . . 5.4. Changing direction – diffraction, physisorption and steering . . . . . . . . . . . . . . . . 5.5. Rotational excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1. Excitation of rotations and parallel translations . . . . . . . . . . . . . . . . . . . . 5.6. Changing the vibrational state of the molecule . . . . . . . . . . . . . . . . . . . . . . . . 5.6.1. Surface site and rotational state dependence of vibrational excitation/de-excitation 5.7. Dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1. Direct, trapping or steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.2. Incidence angle dependence of dissociation . . . . . . . . . . . . . . . . . . . . . . 5.7.3. Molecular rotation and dissociation . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.4. Molecular vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.5. Classical mechanics versus quantum mechanics in dissociation . . . . . . . . . . . 5.8. Substrate excitations and intra-molecular energy flow . . . . . . . . . . . . . . . . . . . . 5.8.1. Molecule–phonon coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8.2. Electron–hole pair interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . .

145 147 147 150 150 153 159 163 165 167 171 172 173 175 178 182 183 185 188 191 192 192

Abstract The flow of energy between the internal modes of molecules scattering from surfaces is presented with particular focus on the basic phenomenology, in particular how the details of the dynamics informs us about the topography of the potential energy surface. The issues of diabatic versus adiabatic dynamics and classical versus quantum approaches are introduced. The description of the dynamics follows a reductionist approach, outlining how motion in each individual molecular degree-of-freedom changes on interaction with a surface, before considering combined transitions (e.g. rovibrational). Changes in molecular translation are presented in terms of diffraction and the classical picture of rainbow scattering, physisorption and steering. This also forms the basis of the discussion of rotationally inelastic scattering. Vibrational excitation is discussed in terms of the ‘elbow’ potential. Adding rotations and translations, the implications of the surface and molecular orientation dependence of the elbow potential are discussed. Molecular vibration leads into dissociation, with the focus mainly on direct dissociation and how this is affected by changes in angle of incidence and rovibrational state. The role of substrate vibrations in inelastic scattering and dissociation is introduced with simple models. Finally, the possible influence of substrate electronic excitations in inelastic scattering is discussed.

142

Keywords: molecule–surface scattering, molecule diffraction, rotationally inelastic scattering, vibrationally inelastic scattering, molecular dissociation

143


145

5.1. Introduction The development of surface science has generally been driven (and justified to grant funding agencies!) by the twin technologies of heterogeneous catalysis and microelectronics (Kolasinski, 2001). This is also true of gas–surface dynamics; a molecule striking a metal surface is related to catalytic processes, while interactions with a semiconductor substrate are ‘filed under’ computer chip technology. Yet this link is not always as straightforward as we might like, in particular, a typical catalytic environment involves high temperatures, high gas-pressures and a complex multi-component catalyst material chosen not only on the basis of reactivity, but also for a variety of structural and engineering properties. In contrast, our ideal system for accurate study of the dynamics would be a single molecule impacting on a perfectly characterised surface in an ultra high vacuum environment. While kinetics, with its more averaged, thermodynamic description of motion in the region of a (the) transition state, may provide a better prospect of quantitative match with catalysis experiments, dynamics provides a uniquely accurate test of our description of the bonding of the molecule to the surface. Dissociation probabilities and inelastic scattering crosssections calculated and measured on a molecule-by-molecule basis are sensitive to the whole energy landscape through which the molecule passes. If the computed probabilities are to match those of experiment, the electronic structure must be accurately determined for many molecular orientations and bond lengths, surface sites and molecule–surface distances. This is a very stringent test of the approximations used to obtain the electronic structure, the obvious fundamental approximations such as the choice of exchange-correlation potential and also the less obvious choices such as the size of the basis set and the choice of pseudopotentials. In this review, we shall focus on one aspect of gas–surface dynamics, namely the exchange of energy between molecular degrees of freedom occasioned by the interaction with the surface, as outlined in the schematic in Fig. 5.1. This may be an exchange simply between translational degrees of freedom (e.g. from motion parallel to the surface to motion perpendicular to the surface) or it may be more radical, resulting in the dissociation of the molecule. We shall concentrate on what theory and computer modelling can tell us, when compared to experimental results (discussed in the chapter by Sitz in this volume), about the dynamics of the molecule–surface interaction. Much of the basic phenomena have been outlined and explained with reference to diatomic molecules on metal surfaces (Darling and Holloway, 1995, 2003; Diño et al., 2000; Gross, 1998; Kroes and Somers, 2005; Kroes, 1999), and this will therefore dominate our presentation here. Although not conceptually difficult, the theory is more problematic for larger molecules, because a much larger phase-space has to be explored in the electronic structure calculations and also we have more molecular degrees-of freedom requiring accurate treatment of the dynamics. The latter point is critical if the molecular motion requires a quantum description, but can be accommodated fairly easily in classical mechanics. Historically, the understanding in this field developed in a piecewise fashion; pictures of the dynamics involving only a few molecular degrees-of-freedom were built up and then stuck together to make a patchwork of the full dynamics. For instance, separate explanations of the trends in vibrational state changes and in rotational state changes when molecules scatter from surfaces could be pieced together to obtain a description of rovibra-

146

G.R. Darling

Fig. 5.1. A schematic of energy flow within a molecule occasioned by scattering from a surface. The translational energy changes from Ei to Ef , with different entrance, θi , and exit angles, θf , and changes in the internal energy (εv and εR ). The surface temperature, Ts , may also play a role.

tional effects (Wang et al., 2002) (i.e. coupled changes of both the rotational and vibrational states). Although the computational methods for dealing with the quantum mechanics of all six of the molecular degrees-of-freedom (each requiring its own dimension) at once have been around for some time (Gross et al., 1995; Kroes et al., 1997), and at least for H2 surface systems are part of the standard toolkit (Busnengo et al., 2002b; Dianat et al., 2005; Farías et al., 2005; Kroes and Somers, 2005; McCormack et al., 1999a, 1999b; Pijper et al., 2002; van Harrevelt and Manthe, 2004), it is still often necessary to resort to lower dimensional and classical studies to get an understanding of the results. In this review, we shall follow this reductionist approach, describing the basic ideas of motion in a few dimensions, and then where possible seeing how these can be employed to build a picture of the full dynamics. In the next section, we shall discuss the potential energy surface (PES), a vital component in a description of the dynamics. The other vital part of dynamics theory is how to treat the motion, i.e. must it be quantum or can classical mechanics be employed with sufficient accuracy. This is an important question, and in Section 5.3 we shall discuss the basic issues, which are then explored further in the sections devoted to individual degreesof-freedom. These follow in Sections 5.4 to 5.6. Although molecular dissociation will be covered in other chapters of this volume, the intra-molecular energy flow is often intimately connected with the dissociation process, since molecules scattering off the surface can also be thought of as those that failed to dissociate. We shall also therefore briefly discuss the influence of intra-molecular degrees-of-freedom on this process. Finally we shall make some


147

comments on the emerging understanding of the role of the substrate degrees-of-freedom in the flow of energy to the molecular vibrations and rotations.

5.2. Probing the potential energy surface No description of dynamics can be complete without some discussion of the potential energy surface. At the most fundamental level, this is the energy of the molecule–surface system for all possible configurations of the molecule and surface. Put simply, we compute the total energy of the system for a range of choices of all of the inter-atomic distances and bond angles in the system, then interpolate between/fit these energy points with a smooth functional form giving us an energy for all possible positions, orientations and shapes of the molecule and the surface. Of course this bald statement conceals a great deal of effort; repeated calculation of total energies is computationally expensive, and the fitting/interpolation of these results is a difficult process. Although existing approaches can provide extremely accurate results for flat, static surfaces (Busnengo et al., 2000; Olsen et al., 2002), their accuracy is dependent on a large dataset to fit. There is no good, accurate procedure to fit total energies computed for a non-perfect surface, i.e. one which has vacancies, adatoms, or even just thermal distortion from an ideal lattice. 5.2.1. Is the motion adiabatic? Following a formal development of the dynamics, we must start with a Hamiltonian describing the motion and interaction of all of the electrons and nuclei in the system, i.e. H = KN + Ke + VN (R) + Ve (r) + Vn–e (R, r)

(5.1)

where the first two terms represent the kinetic energies of all of the nuclei (KN ) and all of the electrons (Ke ) at positions R and r respectively, the third and fourth terms are the inter-nuclear and inter-electron Coulomb repulsion and the final term is the Coulomb attraction between electrons and nuclei. An exact quantum description requires us to solve the Schrödinger equation for wavefunctions dependent on all of the nuclear and all of the electronic coordinates, clearly an impractical task. Writing this total wavefunction as a combination of products of functions, ψn , dependent only on nuclear coordinates and functions, un , dependent on both electron and nuclear coordinates, ψn (R, t)un (r; R), Ψ (R, r, t) = (5.2) n

the Schrödinger equation becomes a set of coupled equations for the nuclear wavefunctions, ψn , ∂ψn i (5.3) un |Ke + Ve + VN−e |um ψm + Cnm ψm . = (KN + VN )ψn + ∂t m m The first term represents the nuclear motion in the inter-nuclear potential. The second describes primarily the electronic energy levels, while the last term, deriving from the kinetic energy of the nuclei, explicitly couples the electron and nuclear motions through the R

148

G.R. Darling

dependence of the electronic functions un : 1 1 2 Cnm = − un |∇RN |um ∇RN + un |∇RN |um MN 2

(5.4)

N

where MN and RN are the mass and position vector for nucleus N . Expressing the wavefunction in product form as in Eq. (5.2), effectively allows us to deal with the electrons separately from the nuclei. The common choice is to take the electronic part to be solutions of an eigenvalue equation involving the nuclear positions only parametrically through the R dependence of the electron-nucleus attraction, i.e., we find electronic states for fixed nuclear positions,

Ke + Ve (r) + VN−e (R, r) un (r; R) = En (R)un (r; R) (5.5) Equation (5.5) is what we are really solving with our electronic structure codes, and the position dependent eigenvalue En (R) is the potential energy surface. However, there are an infinite number of eigenvalues for each set of nuclear coordinates R, which will give an infinite number of potential energy surfaces of increasing energy (i.e., the lowest energy eigenvalue at every R forms one PES irrespective of the details of the bonding, the 1st excited state forms the next, etc.). The Cnm couple the nuclear motion on the different PESs. These off-diagonal matrix elements will be small if the electronic states vary only slowly (adiabatically) with nuclear position, in which case the molecule will remain on one PES. This is adiabatic dynamics. Nonadiabatic behaviour occurs when a molecule incident in one electronic configuration, say the third lowest eigenstate, finds itself at some point in say the second or fourth lowest eigenstate. In principle, the dynamics should include all electronic configurations that are energetically accessible and symmetry allowed, however, only the ground-state PES formed from the lowest energy eigenvalue at each R can be reliably computed (using density functional theory, DFT) for molecule–surface systems. There is therefore still considerable debate as to when a nonadiabatic description is required. The electrons at the Fermi energy of a metal surface are very likely to behave nonadiabatically since these can be excited with infinitesimal energy cost. These (electron–hole pair) excitations produce a friction, which dissipates the initial energy of the molecule (Bird et al., 2004; Brako and Newns, 1980; Hellsing and Persson, 1984; Mizielinski et al., 2005; Persson and Persson, 1980; Trail et al., 2003, 2002). As we shall discuss in Section 5.8, electron–hole pair mechanisms have also been implicated in vibrational de-excitation of molecules interacting with metal surfaces (Gross and Brenig, 1993; Gross et al., 1991; Huang et al., 2000; Rettner et al., 1985c, 1987). Recent experiments (Gergen et al., 2001a; Nienhaus, 2002; Nienhaus et al., 1999, 2000; Nienhaus et al., 2002) have provided direct, quantitative evidence of the importance of electron–hole pair excitation in gas–surface dynamics by measuring the chemicurrent of electrons and holes in a Schottky diode induced by the adsorption of hydrogen and deuterium atoms at Ag and Cu surfaces and even of NO and O2 interacting with Ag (Gergen et al., 2001b). Nonadiabatic filling of the molecular levels (Citri et al., 1996; Katz et al., 1999; Kosloff and Citri, 1993) is somewhat more contentious. Very rapid change in the molecular affinity level of O2 molecules approaching the surfaces of simple metals, such as Cs (Böttcher et


149

Fig. 5.2. Potential energy of an O2 molecule as a function of height above an Al(111) surface with occupation frozen in either the triplet or singlet state. The triplet potential increases with energy while the singlet decreases. In order to dissociate, a molecule incident in the triplet state would have to transfer onto the singlet state, thus there is a barrier to dissociation, which is absent in a standard spin adiabatic calculation. Adapted from Behler et al. (2005).

al., 1990), leads to electron emission, clearly a nonadiabatic effect. The possibility of such behaviour for the O2 /Al system has been less easy to pin down (Kasemo, 1996; Kasemo et al., 1979; Österlund et al., 1997). Computation of the PES using standard DFT gives a result with no barrier to the dissociation of the molecule (Honkala and Laasonen, 2000; Yourdshahyan et al., 2002), however, experimental studies clearly show activated dissociation (Österlund et al., 1997). This implies a failure of the adiabatic description, which has been investigated with semi-empirical, multiple diabatic PESs (Binetti et al., 2003). Recent calculations indicate that the electronic spin on the molecule can also produce nonadiabaticity (Behler et al., 2005). Far from the surface, the molecule should have a total electronic spin of 1, i.e. it should be in a triplet state, and only when the molecule–surface interaction is substantial should the spin be quenched. In the standard DFT description, the electrons are distributed according to the filling (in energetic order) of the energy levels of the system. The net spin (i.e. the magnetization) can either be fixed by fixing the number of spin up and spin down electrons, or (more usually) the magnetization is minimized along with the electronic energy during the self-consistency cycle, both procedures give a PES with no barrier to dissociative adsorption (Yourdshahyan et al., 2002). Instead, if the spin is constrained to remain in the orbitals derived from the O2 triplet state a spin-diabatic potential curve is obtained that increases as the distance, Z, to the surface decreases, as shown in Fig. 5.2. In contrast, diabatically occupying the molecular spin singlet state gives a potential curve decreasing rapidly with Z, and there is a curve crossing between singlet and triplet energies as the occupation of the molecular 2π ∗ orbital begins to increase. If the molecule remained on the triplet potential until close to the surface (i.e. behaved nonadiabatically) there would be a dissociation barrier, as observed in experiment. However, on the adiabatic potential, where the electronic spin can spread freely as in standard DFT, there is

150

G.R. Darling

no barrier, instead there is clear avoided crossing behaviour, as can be seen in Fig. 5.2. The implication of these calculations is that the electronic spin of the molecule does not relax quickly enough as the molecule approaches the metal surface and it remains in a triplet state. 5.2.2. Basic interactions in the PES In general, when a molecule or atom first approaches a surface there is an attractive interaction due to the long-range Van der Waals dispersion forces. Only at a distance of ∼3 Å will the molecule and surface begin to interact chemically. When the electronic wavefunctions of the molecule and surface begin to overlap, they must orthogonalize to each other to prevent more than two electrons being in the same energy level. For filled levels, this introduces an energy cost, giving a repulsive force (Pauli repulsion). The Van der Waals and Pauli repulsion combined give a well in the PES, the physisorption well, into which molecules can trap. The Van der Waals interaction is much weaker than a chemical interaction, and so the physisorption well tends to be relatively shallow (the molecule is not substantially perturbed from its gas-phase configuration while physisorbed). For H2 on unreactive metal surfaces (Andersson et al., 1988), it is only ∼30 meV, rising to ∼0.2 eV for C2 H4 on a noble metal surface (Vattuone et al., 1999). The Van der Waals interaction is present in all systems, however the effectiveness of Pauli repulsion is dependent on the electronic structure of molecule and surface. For example when a hydrogen molecule approaches a Cu surface, the interaction between the delocalized metal s-states that spill out into the vacuum region and the molecular σg -orbitals gives Pauli repulsion, which is only overcome when there is a radical change in the bonding leading to dissociation (Harris and Andersson, 1985). This gives a barrier to the dissociation of the molecule. In contrast, for metals with partially filled d-bands, the Pauli repulsion can be reduced or even completely overcome, resulting in no barrier to dissociation. We can view the role of the d-states either in terms of direct hybridization between the molecular states and the d-states (Hammer and Nørskov, 1995), or as providing an outlet for s-electrons (Harris and Andersson, 1985), which can transfer at no energy cost into the more localized d-states, thus removing their contribution to the Pauli repulsion. The PES is not only dependent on the species of molecule and surface, but also on the precise disposition of the atoms and molecules. The molecular orientation can be important, for example CO molecules tend to bind C-end down on transition metal surfaces. Site dependence will be important, even on flat, close-packed metal surfaces, molecules will interact differently at atop or bridge or hollow sites. We cannot easily visualize a full PES, since even for a diatomic molecule, ignoring the substrate motion, it is six-dimensional. In discussing changes of the internal energy of the molecule, it is therefore useful to consider the variation of the PES in a subset of the molecular coordinates and to study the dynamics in this reduced dimensionality to understand how it can be affected by the PES topography. 5.3. Quantum or classical descriptions of the dynamics Once the relevant PES (or PESs for nonadiabatic motion) has been computed, we must decide on how to treat the nuclear motion. Quantum mechanics would give the correct and


151

accurate description, but can prove impossibly expensive. Including only the molecular degrees-of-freedom for a diatomic still results in a six-dimensional description, the cost of which will depend on how many states can be populated in scattering. For H2 (Kroes and Somers, 2005), both time-dependent and time-independent quantum approaches have been developed and are now commonly in use. For heavier molecules, the greater mass leads to much smaller energy quanta, requiring more states for a quantum description, but perhaps making such cases more amenable to a classical description. The one case where a quantum description cannot be avoided is when the dynamical phenomena being studied arises from tunnelling of light atoms such as H; here no classical description will suffice. Coherence should also require a quantum description, although it has been found in recent work that even diffraction intensities can be reproduced fairly well with the correct scheme for totalling the classical results (Díaz et al., 2005b), as shown below. Tunnelling apart, the most obvious non-classical phenomenon is the quantization of the internal modes of the molecule. Clearly, in a purely classical description, the vibrational and rotational energies can be given and can acquire any values at all, subject to conservation of the total energy. For the initial molecular state in the gas-phase, we can solve this simply by choosing the classical vibrational energy to be the same as the appropriate quantum eigenenergy. Similarly, the rotational state can be initialized by following a prescription based on the quantum description of the angular momentum vector (Darling et al., 2000; Kay et al., 1998; McCormack and Kroes, 1998, 1999). The resultant, so-called quasi-classical, dynamics is found to be largely adequate in the description of activated molecular dissociation. We can also use the quantum states to partition the trajectories scattered from the surface. For example, to determine the populations of the vibrational states we make energy bins centred on the quantum vibrational eigenvalues, ending midway between successive eigenvalues. Trajectories can then be counted into these bins. Figure 5.3 shows that this procedure can yield results comparing very favourably with full quantum results. Similar binning of the final rotational state can also be employed. For non-activated dissociation, the lack of a full quantum treatment of the molecular vibration introduces significant errors. When the molecule approaches the surface, the vibrational energy leaks out into translations, resulting in an overestimation of the dissociation probability by quasi-classical dynamics in the H2 /Pd(100) system (Gross and Scheffler, 1998). The severity of this problem is system specific, for example, the problem is much less severe for the H2 /Pd(111) system (Busnengo et al., 2003; Diaz et al., 2005a), affecting dissociation only at the lowest energies, where excessive (compared to quantum dynamics) vibration–rotation coupling leads to higher reflection probabilities, i.e. to less dissociation than in quantum dynamics. In addition to failing to retain the internal energy adequately in all cases, classical dynamics has the drawback that quantization cannot ‘appear’ during the trajectory. When the molecule approaches the surface, the potential it sees depends on surface site and molecular orientation, i.e. it is corrugated. The corrugation restricts the rotations and translations across the surface producing librations that must have energy quanta associated with them. The appearance of these new quantized modes can be clearly illustrated in a simple two-dimensional model of a molecule striking a corrugated Gaussian barrier. Figure 5.4a shows the nuclear wavepacket for a molecule striking a corrugated potential at an energy just sufficient to overcome the barrier to dissociation. We can see that the fraction of the

152

G.R. Darling

Fig. 5.3. Vibrationally inelastic scattering probabilities computed for H2 dissociating at a single Cu site (Darling et al., 2000). Quantum results are indicated by lines, classical by symbols. The excellent agreement between the two is typical for an activated dissociation system.

wavepacket traversing the barrier is squashed into the immediate vicinity of the lowest dissociation barrier. Perpendicular to the reaction path, the wavefunction has only a single peak corresponding to the ground-state of the potential slice in this direction, in contrast, just immediately above this region, we can see clear nodal structure corresponding to the first symmetry-allowed excited state. At higher energy, shown in Fig. 5.4b, this state can be populated right up to the barrier, and the next accessible state is visible ∼1 au in front of the barrier. Computing eigenstates and eigenfunctions perpendicular to the reaction path gives a more quantitative local analysis of the dynamics when we project the full wavefunction onto the eigenfunctions (Darling et al., 1997). In Fig. 5.4c we can clearly see that the internal ground-state is indeed the only state populated at the barrier, but just above this


153

Fig. 5.4. (a) Wavefunction traversing a corrugated Gaussian barrier PES (shown underneath in white) with just enough energy to overcome the barrier at its lowest point. Exactly at the barrier, the wavefunction has no nodal structure in the molecular orientation coordinate, showing that only the first of the frustrated rotational modes at the barrier is populated. At about 4 au in the reaction path, nodal structure appears indicating that an excited state can also be populated. (b) Same as (a), but for higher incident energy. Now the excited state can also be populated at the barrier. (c) At each point on the reaction path, eigenfunctions and eigenvalues can be obtained for the frustrated rotational mode. Projecting the wavefunction from (a) onto the eigenfunctions yields a local internal state population analysis (populations indicated by dot size plotted onto the eigenvalues at each point on the reaction path) of the motion (Darling and Holloway, 1998; Darling et al., 1997), which confirms that in part (a), molecules can only traverse the barrier in the lowest internal state. (d) Same as (c) but for the same energy as panel (b). Now the molecules can be transmitted over the barrier in more than one state.

the first excited state has a high population. At the higher energy, Fig. 5.4d, the first two states are populated at the barrier, while the second excited state is energetically allowed just in front of the barrier. 5.4. Changing direction – diffraction, physisorption and steering For changes of direction of a molecular trajectory, i.e., momentum transfer between components normal and parallel to the surface, we are interested in the dependence of the PES on surface site. The surface is of course never completely flat, atop sites protrude

154

G.R. Darling

Fig. 5.5. Variation of the dissociation barrier height with position in the unit cell for H2 /Cu(100). T indicates the atop site, B the bridge, H the hollow. Adapted from White et al. (1994).

further than hollow sites, while sites which are more reactive are likely to be less repulsive, i.e. it is easier for the molecule to approach such sites because the barrier to reaction is lower. An example of the variation of the dissociation barrier magnitude with surface site is shown in Fig. 5.5 for H2 on the Cu(100) surface (Hammer et al., 1994; White et al., 1994). PES corrugation as in Fig. 5.5 will couple molecular motions parallel and perpendicular to the surface. At normal (to the plane of the surface) incidence, in a classical treatment of the scattering, we will see a maximum in momentum transfer for trajectories striking the surface exactly at the inflection points of the potential curve, as indicated in Fig. 5.6a. This gives rise to a singularity in the differential scattering cross-section, a phenomenon known as rainbow scattering, as shown in Fig. 5.6b. In a real system, the singularity will be smeared by energy loss to the substrate and the inevitable averaging over the other molecular degrees-of-freedom. In a quantum description of the molecule there is no rainbow singularity, but for initial momentum K parallel to the surface, the final momentum parallel to the surface is restricted by the periodicity of the surface to the set {K + G} where the G are reciprocal lattice vectors. Thus the quantum scattering gives rise to a series of distinct peaks, as in Fig. 5.6b. There is a peak in intensity in the vicinity of the rainbow angle, but there are clearly far more oscillations in intensity of the peaks, oscillations which are extremely sensitive to the detailed structure of the PES. Good agreement between theoretical and experimental diffraction intensities is a very precise test of the accuracy of the PES at low incident energies. (At higher incident energies and higher surface temperatures, more energy is exchanged with the substrate, leading to a decrease in the coherent scattering responsible for the discrete diffraction peaks.) It has been found that much of the diffraction is out-of-plane (Farías et al., 2005), i.e., not in the plane containing the incident beam direction and the surface normal, see Fig. 5.7a. On taking account of this, a good level of agreement between computed and measured diffraction intensities can be obtained for some hydrogen/metal systems. This is even the case when the surface is reactive (Farías et al., 2005), as in the case of H2 /Pd(111) illus-


155

Fig. 5.6. (a) Classical trajectories incident on a simple corrugated surface, the thick grey line shows potential contour with V = Ei . The trajectory incident at the inflection point of the potential has maximal deflection. (b) Scattering probabilities obtained using classical mechanics (grey line with shading) and quantum mechanics (black lines) for the example shown in (a). Trajectories scattering from the vicinity of the inflection point of the PES give rise to a singularity in the scattering at the rainbow angles, indicated by arrows. The quantum results do not show a clear rainbow feature, but rather have oscillations in intensity which contain detailed information on the PES.

Fig. 5.7. (a) Schematic indicating in-plane and out-of-plane scattering. (b) Measured reflection probabilities (noisy lines) for H2 /Pd(111) compared to theory (smooth lines), showing substantial out-of-plane diffraction. Results from a classical computation are indicated by a dashed line. Adapted from Farías et al. (2005).

trated in Fig. 5.7b, for which there are non-activated paths to dissociative adsorption (the dissociation probability does not go to zero with the molecular translational energy). Although the occurrence of discrete diffraction peaks requires a description including constructive and destructive interference, i.e. a quantum description, the results in Fig. 5.7b show that judicious binning of classical scattering can yield a reasonably accurate approximation to the full quantum results (Díaz et al., 2005b). To obtain the dashed line, the authors have divided the reciprocal space into 2-D Wigner–Seitz cells centred on the reciprocal space lattice points (which give the diffraction spots in a quantum treatment), as indicated in Fig. 5.8a. The classical scattering probability into each diffraction beam is then the fraction of trajectories scattering into the Wigner–Seitz cell centred on that lattice point. This is similar in philosophy to dividing up the vibrational degrees of freedom

156

G.R. Darling

Fig. 5.8. (a) Reciprocal space bins used for partitioning classical trajectories to obtain a quasi-classical approximation to diffraction. (b) Quantum (dark bars) and classical (grey bars) diffraction intensities summed over final rotational state for two different incidence conditions. Adapted from Diaz et al. (2005a).

into energy bins centred on the quantum vibrational eigenvalues, as in the quasi-classical approach. Summed over final rotational state, the quantum and quasi-classical diffraction probabilities follow similar trends (Fig. 5.8b), and this is also true for state resolved diffraction, but the quasi-classical treatment cannot yield the detail – the oscillations in diffraction intensities as a function of energy/angle – characteristic of an interference phenomenon. At the very lowest energies, the combination of diffraction and a physisorption well can lead to trapping, in a process known as selective adsorption. Motion normal to the surface is exchanged for motion parallel to the surface as in diffraction, but the translational kinetic energy in the trapped molecules is greater than the initial total energy, Ei . The excess is made up by the energy, εn (negative), of the nth bound-state in the physisorption well: 1 |Ki + G|2 + εn (5.6) 2M where the first term gives the translational kinetic energy of the trapped molecules, with initial momentum Ki parallel to the surface. In essence, the molecules are diffracting into a channel that is closed (not energetically accessible) far from the surface, and simultaneously dropping into one of the bound states of the physisorption potential, as illustrated in Fig. 5.9. For simple, light molecules on smooth metal surfaces, selective adsorption shows up as dips in the specular reflectivity at energies satisfying Eq. (5.6), as shown in Fig. 5.10. Since the bound-state energies are very sensitive to the details of the physisorption well, this is an extremely accurate probe of the PES far from the surface where density functional theory fails due to its inaccurate (in present formulations) treatment of dispersion forces. In principle, selective adsorption can be completely elastic in that the molecule need not lose any energy to the substrate, the trapping can be reversed by diffraction into states of lower parallel momentum, allowing the molecule to escape from the surface. Heavier molecules Ei =


157

Fig. 5.9. A schematic representation of selective adsorption. The left-hand side shows the potential energy as a function of distance from the surface with the initial wavefunction and bound state wavefunctions shown. The left-hand side shows the total energy in each state plotted against the parallel momentum, for a very weakly corrugated surface (the energy increases quadratically). Selective adsorption involves diffraction into a bound state of the physisorption potential. Adapted from Celli (1984), Darling and Holloway (1990).

Fig. 5.10. The experimental signature of a selective adsorption resonance is a dip in the specular reflectivity, from Andersson et al. (1989).

can also be trapped into physisorption wells, but selective adsorption features cannot be observed because the bound states in the physisorption well are too close together to be resolved, and because there is greater energy loss to the substrate vibrations which broadens the selective adsorption features.

158

G.R. Darling

Fig. 5.11. (a) Classical trajectories (white) incident on a stepped surface (PES indicated by black lines) showing strong steering into a chemisorption well at the step edge. The trajectories are scrambled so that information on the angle of incidence is lost. (b) If the steering in (a) is strong enough, there is no dependence of the sticking probability on the angle of incidence. When the steering is less effective, the sticking shows a minimum at normal incidence to the step terraces. (c) Experimental results for C2 H4 incident on Ag(410). In this system the binding to the step terrace is only 0.2 eV, while that at the step edge is 0.4 eV, as in the model PES in (a). Adapted from Savio et al. (2003).

For small molecules on flat, unreactive surfaces, the attractive part of the physisorption well has little variation with surface site causing changes of direction primarily from the increase in the momentum normal to the surface (analogous to refraction). For larger molecules on rougher surfaces, e.g. stepped surfaces, the physisorption well can vary apprecia-


159

bly across the surface, leading to significant steering of the incoming molecules. Steering occurs to some extent for all systems, it arises simply from differences in the PES producing forces in certain directions. With attractive potentials, the steering pulls the molecules towards specific surface sites (and into particular orientations) (Gross et al., 1995; Kay et al., 1995). They do not come to a dead stop at these sites (this would require infinite friction), but overshoot. The molecules will trap if the momentum normal to the surface becomes less than zero (as in selective adsorption), and will be stuck once the total energy becomes less than zero, as indicated in the example shown in Fig. 5.11a. In this model of the sticking of ethylene on the stepped Ag(410) surface (Savio et al., 2003), the molecules preferentially adsorb at the top of the steps, where they are bound by 0.4 eV compared to 0.2 eV on the step terraces. If the steering to the step is sufficiently strong, the molecular motion is so scrambled that the initial angle of incidence is irrelevant (Fig. 5.11b), and the sticking/trapping in the physisorption well is approximately independent of the angle of incidence of the molecules, as observed in the experimental results shown in Fig. 5.11c. By sufficiently strong we mean that the variation in well depth cannot be small, but also the potential well must extend reasonably far from the surface so that there is sufficient time for the steering forces to produce a noticeable change in the trajectory of the molecule. When these conditions are not met, the sticking probability has a minimum where the largest amount of momentum must be given to the surface for the first bounce to lead to trapping, i.e. for incidence normal to the step terraces, as shown by the grey curve in Fig. 5.11b.

5.5. Rotational excitation There are many similarities between the excitation of rotations of a molecule and the excitation of parallel momentum discussed above. The rotational states of molecules are changed by PESs corrugated in the molecular orientation coordinates, i.e., they change when the molecule–surface interaction is dependent on the orientation of the molecule. This will be true for all molecules, most obviously for long chain polyatomics and heteronuclear diatomics, but also for H2 (a broadside approach is favoured over an end-on approach because the molecule can dissociate more readily in the former orientation) and even CH4 , although we might expect the effects to be smaller for this almost spherical molecule. Observing changes in rotational states of molecules is less straightforward than determining changes of direction because we must be capable of measuring the rotational state distributions of the molecules before and after scattering. This limits the range of molecules for which such a detailed analysis can be done. NO is a particularly favourable case as it can even be oriented using hexapole electric fields prior to striking the surface (Kleyn, 2003). We can explore the rotational excitation mechanism in great detail using classical molecular dynamics (McCormack and Kroes, 1999; McCormack et al., 1999a; Wang et al., 2000). Figure 5.12 shows classical trajectories for H2 molecules constrained to approach a single site of a non-reactive metal such as Cu. As a function of time, the molecular bond length oscillates about the equilibrium value, only lengthening slightly at the point of closest approach of the molecule to the surface. Initially the molecule is not rotating. For H2 the potential energy is lowest when the molecule is broadside to the surface, i.e. the molecular

160

G.R. Darling

Fig. 5.12. Molecular degrees-of-freedom versus time for a classical trajectory of H2 undergoing rotational excitation. The torque has peaks producing rotational motion when the bond reaches its maximal extent, rmax , where the PES is most corrugated. In the top panel, the molecule emerges rotationally excited, whereas in the lower panel, one end of the molecule strikes the surface inducing rotations, but when the molecule rotates around the second end strikes the surface quenching the rotation (Wang et al., 2000).

bond is parallel to the surface plane, because this is the orientation most favourable for dissociation. For molecules approaching end-on, the potential is simply repulsive, and as the molecular bond extends, one atom is pushed further into the surface increasing the potential energy for this orientation. Therefore as a function of the molecular bond orientation angle, θ , we can expect the potential to be more corrugated for longer molecular bonds, giving rise to more efficient rotational excitation with longer bond lengths (Wang et al., 2000). Examining the classical trajectory in the top panel of Fig. 5.12, we can see that there is a peak in the torque when the bond is at its greatest extent, rmax , and this repeats each vibrational period. The bond orientation then changes in response to this torque, i.e. the molecule rotates. On the outward path, the torque changes sign, i.e. is there is a slight tendency for collisions with the other end of the molecule to hinder the rotation (cf. Fig. 5.13). On increasing the initial translational energy, we see largely the same picture, however, on the outward part of the trajectory shown the negative torque is now completely effective in removing the induced rotation, and the molecule emerges rotationally cold.


161

Fig. 5.13. The classical trajectories from Fig. 5.12 overlaid on the PES at rmax as a function of molecule surface distance and molecular orientation. At low energies the molecules exhibit rainbow scattering as in Fig. 5.6, but at high energy the molecule chatters, striking the surface with both ends, and the rotational motion is quenched (Wang et al., 2000).

Figure 5.13 shows sets (swarms) of these trajectories with very similar incidence conditions projected onto the Z, θ-plane, overlaid on the potential calculated at rmax . We can see that the trajectories are reflected from a sloping, repulsive region of the potential, and as Ei is increased, they can penetrate deeper into the PES, into more repulsive and more steeply sloping regions of the PES. At low energies we see behaviour largely similar to that in Fig. 5.6a, and we can expect rainbow scattering behaviour in this case too, with rainbow maxima at particular final rotational states. However, if the deflection is too great, as shown by the trajectories for 0.3 eV, the molecules scatter across the region of low potential occurring when the bond axis is parallel to the surface, and strike the opposing slope of the PES. In this second collision, the molecular rotation is quenched. This behaviour is known as chattering – the molecule strikes the surface with one end, and is spun around to strike again with the other end. Although we can produce very exact pictures of the dynamics, problems remain in tying up the most accurate computational results with state-to-state scattering experiments (Somers et al., 2004; Watts et al., 2000). The upper panel of Fig. 5.14 shows a comparison between the results of six-dimensional quantum calculations and experimental measurements for the elastic reflectivity (scattering into the same molecular state as the initial state) of H2 scattered from the Cu(100) surface. For two PESs, differing in the accuracy of the DFT calculation and of the fit, the results are in broad agreement with the experiment, following the same trend, if a little uncertain in the overall magnitude. The bottom

162

G.R. Darling

Fig. 5.14. Results of six-dimensional quantum calculations (lines) of state-selective reflectivity of H2 scattering from Cu(100) compared to experiment (symbols). For both variants of the PES (IV and V), the top panel shows reasonable accuracy for the internally elastic channel, but rotationally inelastic scattering is greatly overestimated in theory (bottom panel). Adapted from Somers et al. (2004).

panel of Fig. 5.14 shows the case of scattering into a rotationally excited state. Here we can see that there is very poor agreement between theory and experiment. The reasons for this are presently unclear. It may be that the PES is simply not accurate enough in the key regions where this rotational excitation is occurring, or interaction with the excitations of the substrate, electron–hole pairs or phonons, may lead to extremely efficient quenching of the rotationally excited state. Orientational and rotational effects in molecule–surface scattering have perhaps been best studied for NO/metal systems (see the chapter by Sitz in this volume). An example of a PES typical of such systems is shown in Fig. 5.15 for the NO/Pt(111) system (Lahaye et al., 1996). NO generally binds N end down to metal surfaces leading to a PES having a well for the N down orientation, while for O down the well is shallower, or the PES could even be completely repulsive, as in Fig. 5.15. A molecule moving on such a PES will be steered around to the N down orientation, i.e. it will be rotated. Comparing Fig. 5.15 with Figs. 5.6a and 5.13, it is clear that the potential contour lines as a function of molecular orientation have inflection points. We should, therefore, expect to see rotational rainbow scattering, i.e., there will be a molecular orientation, θ, for which there is a maximal deflection of the trajectory leading to a singularity in the differential scattering probability as a function of rotational state. On surfaces with shallow molecular chemisorption wells, rotational rainbows are indeed seen, but only for molecules incident O-end down (Guezebroek et al., 1991), i.e. it is the repulsive part of the PES giving the rainbow effect not the attractive well. For orienta-


163

Fig. 5.15. Model PES for NO interacting with a Pt surface (Lahaye et al., 1996). NO prefers to bind N end down while approach with O end down experiences repulsion, therefore molecules steer into the N down orientation.

tions too close to N-down, attraction into the well induces chattering (as in the trajectories for 0.3 eV in Fig. 5.13), randomizing the rotations, which then follow a Boltzmann distribution, shown in the top panel of Fig. 5.16. Since chattering occurs more readily for molecules emerging at small scattering angles, i.e. closest to the surface normal, the rainbow is only seen in the rotational distribution measured close to grazing incidence. When the chemisorption well is deep, as in the NO/Pt(111) system, there still appears to be a rainbow feature, as shown in the bottom panel of Fig. 5.16. This is not, however, directly associated with the inflection point of the potential, rather it indicates a rotational energy cutoff (smeared out by thermal motion of the surface). Molecular dynamics simulations show that greater rotational excitation does occur as the orientation angle increases towards the inflection point in the PES contours (Lahaye et al., 1996), but if too much rotation is induced, the molecules are pulled into the chemisorption well where chattering quenches the rotational state. The rainbow-like feature is thus associated with molecular orientations closer to the O-down orientation than the inflection point of the potential. 5.5.1. Excitation of rotations and parallel translations Rotational excitation can also give rise to selective adsorption, i.e., the molecular rotational energy is higher than its incident energy and to balance this, the molecule drops into a bound state of the physisorption well (Andersson et al., 1988, 1989; Wilzén et al., 1991). More generally a combination of rotational and parallel momentum excitation gives rise to

164

G.R. Darling

Fig. 5.16. Measured intensities for scattering of NO from Pt(111). For small exit angle, top panel, the results follow Boltzmann statistics at the temperatures indicated for Ei of 1.6 eV (triangles), 0.53 eV (circles) and 0.34 eV (squares). The lower panel shows results for more grazing scattering, with a rainbow-like peak at low energies. Adapted from Lahaye et al. (1996).

selective adsorption according to: 1 1 (5.7) |Ki + G|2 + J (J + 1) + εn , 2M 2I where I is the moment of inertia of the molecule and J its final rotational state (we have assumed an initial state of J = 0, for simplicity). The quantum nature of this requires a light molecule, but it is difficult to observe with H2 due to the usually weak orientation dependence of the H2 -surface PES in the physisorption well. Features are stronger for HD (Cowin et al., 1983, 1985; Rettner et al., 1985a, 1985b), because although both ends of the molecule are chemically identical, the centre-of-mass is displaced towards the D end. For a given height (of the centre-of-mass) above the surface, the H down orientation would therefore have an atom closer to the surface than the D down orientation, giving rise to a stronger corrugation of the PES in the Z, θ-plane. The site dependence of the PES is also a site dependence of the orientational dependence of the PES. For instance, CO molecules preferentially bond C end down on top of metal atoms in a surface, while for the O end down, there can be a repulsive interaction, as in Fig. 5.15 for NO. On top of a hollow site, however, the C-down orientation might not be as strongly attractive, nor the O-down so strongly repulsive. This will lead to changes of rotational state accompanying diffraction, i.e. to rotationally inelastic diffraction. Even for H2 molecules this can be a strong effect (Gross and Scheffler, 1996), especially when scattering from a reactive metal, where some surface sites have no barrier to molecular dissociation for molecules oriented broadside to the surface. As shown in Fig. 5.17, the Ei =


165

Fig. 5.17. Coupled rotational and parallel momentum excitation for the H2 /Pd(100) system leads to rotationally inelastic diffraction as strong as the rotationally elastic diffraction. Adapted from Gross and Scheffler (1996).

rotationally inelastic contribution to the diffraction channels is as strong as the rotationally elastic contribution (at least in theory). When the lateral corrugation of the PES depends strongly on the bond orientation, the diffraction also depends on mJ . Miura et al. (2002) have shown that this leads to a difference in the rotational alignment of molecules scattered on or off-specular.

5.6. Changing the vibrational state of the molecule Molecular vibration is the motion most obviously connected with dissociation. In this section, we shall discuss how the PES topography can affect vibration, but we leave discussion of dissociation to Section 5.7. The role of molecular vibration has been most clearly worked out for H2 /metal systems, and therefore this will dominate the discussion, however, the results will be relevant to other molecule–surface combinations. For a particular surface site and molecular orientation the PES as a function of molecule–surface distance and molecular bond length takes the form of the ubiquitous ‘elbow’ PES shown schematically in Fig. 5.18, where at the top left of each panel we are describing the vibration of

166

G.R. Darling

Fig. 5.18. Schematic elbow potentials (functions of molecular bond length and molecule–surface distance) for the dissociation of a diatomic at a surface. Molecules incident on the right-hand PES could show vibrational excitation and de-excitation in the scattered fraction and vibration enhanced dissociation because the barrier is late. For the early barrier PES shown in the left-hand PES, vibrations and translations would not couple.

the molecule far from the surface, while the bottom right represents the vibrational mode between the dissociated fragments and the surface. The potential varies smoothly between these two extremes, resulting in a softening of the intra-molecular vibration in the curved region of the elbow. A key feature of the PES determining whether there are vibrational effects is the location of the dissociation barrier with respect to the bend of the elbow. If the barrier occurs before the bend, termed an early barrier, the potential is increasing by the same amount for all bond lengths. Although it is increasing, the potential as a function of bond length is not changing shape appreciably, and so there is no coupling between different vibrational states. In contrast, when the barrier occurs after the bend in the elbow, the molecule can traverse the bend before being reflected. Motion in Z couples strongly to motion in r, resulting in changes in the vibrational state. A coordinate transformation is often applied to the elbow potential (Brenig et al., 1993). We define a reaction path coordinate, s, as the path of steepest descent from the barrier maximum, marked by the dashed line the left-hand panel of Fig. 5.18. Vibration is then taken to be locally orthogonal to this line (in a direction ρ), in the gas-phase, this vibration exactly corresponds to the intra-molecular vibration, while at large r and small Z it corresponds to the vibration of the dissociated fragments. This replaces the vibrational coordinate with a series of vibrationally adiabatic potential energy curves, which are the vibrational eigenvalues computed in the ρ-direction at each s. Coupling between the adiabatic potential curves is determined largely by the curvature of the reaction path. In this representation, an early barrier PES will have the same barrier height in every vibrational state (Halstead and Holloway, 1990), as shown in Fig. 5.19. The eigenvalues for the vibration of the dissociated fragments on the surface depend on the total mass of the molecule, whereas the vibrational energy of the intact molecule depends on the reduced mass, which is considerably smaller. This means that in the absence of a barrier, the vibrational levels decrease in the curve of the elbow, indicated by the dashed curves in Fig. 5.19. For a late barrier, this decrease subtracts from the barrier height, the barrier is effectively lower and


167

Fig. 5.19. Vibrationally adiabatic potentials as a function of reaction path on early and late barrier elbow PESs for v = 0 and v = 1. For the early barrier, vibrationally energy (shown by the dashed line) is not released until after the barrier maximum, so both vibrational states have the same barrier height. A late barrier occurs after the bend in the elbow and is greatly reduced by the vibrational energy release. The curvature of the reaction path couples the vibrational states, it is indicated by the shaded region.

by an amount which increases with vibrational state (Halstead and Holloway, 1990), as shown in Fig. 5.19. The coupling between vibrational states by the PES curvature mechanism is symmetrical; given the same total energy we can expect the same amount of vibrational relaxation from, say, the v = 1 state to the v = 0 state as there is excitation from v = 0 to v = 1. Both have been observed in experiments on the scattering of H2 and D2 molecular beams from Cu surfaces (Hodgson et al., 1992; Rettner et al., 1992). Figure 5.20 shows results obtained by Hodgson and coworkers (Hodgson et al., 1992) for the reflectivity of the v = 1, J = 4 state of D2 relative to that of the v = 0, J = 4 state. At energies up to the vibrational excitation threshold, the reflectivity decreases due to both vibrational relaxation and dissociation of the vibrationally excited molecules. Above the threshold energy of ∼0.36 eV, the population of the v = 1 state can be replenished at the expense of the v = 0 state. The relative reflectivity then increases sharply as the population in v = 1 grows while that in v = 0 falls (the decrease in the v = 0 population at the higher energies will also be due to dissociation of these molecules). Vibrational relaxation of H2 on metals has been extensively studied by Sitz and coworkers using state-to-state scattering methods (Gostein et al., 1995, 1997; Shackman and Sitz, 2005; Watts and Sitz, 2001; Watts et al., 2000). Their results indicate energy flow to and from the substrate, as well as between the molecular degrees-of-freedom. We shall discuss these results further in Section 5.8. 5.6.1. Surface site and rotational state dependence of vibrational excitation/de-excitation The bonding between a molecule and a surface will depend on the precise disposition of the molecule and surface atoms. Thus, for some molecular orientations at cer-

168

G.R. Darling

Fig. 5.20. Reflectivity of the v = 1 state relative to that of the v = 0 state measured for H2 scattering from Cu(111), adapted from Hodgson et al. (1992).

tain surface sites the interaction may be attractive while for others it may be repulsive, as for the NO/Pt example shown in Fig. 5.15. The topography of the elbow PES will also alter according to surface site and molecular orientation (Hammer et al., 1994; White et al., 1994), as shown by the example in Fig. 5.21 for the H2 /Cu(111) system. After the molecule dissociates, the H atoms prefer to adsorb in the three-fold hollow site. If the approach is centred at a bridge site, where the barrier is lowest, the barrier is late, at a bond length of ∼1.1 Å. The reaction path shows some curvature, so we should expect some vibrational effects in the scattering from this site. In fact, solving the dynamics on the two-dimensional slice of the PES shown in Fig. 5.21a yields a negligible amount of vibrational excitation – the bend in the elbow is simply not sharp enough (Kinnersley et al., 1996). However, when the molecules approach the atop site, the PES has a much more sharply bent elbow (Fig. 5.21b), and two-dimensional calculations (Kinnersley et al., 1996) show vibrational excitation to be very efficient here. Such results are borne out by higher dimensional calculations of H2 interacting with Cu surfaces (Kroes, 1999; Kroes et al., 1996). In these systems, therefore we should find that the vibrational excitation probability is dominated by molecules in certain orientations and at certain surface sites. This “selection” of the surface site by a particular process (here vibrational excitation) can also be found in the (initial) state-resolved dissociation probabilities, discussed below. As can be seen in Fig. 5.19, the vibrational energy release for a late barrier is much greater for the v = 1 state than for the v = 0 state. This will occur at all surface sites/ molecular orientations where there is a late barrier. When roughly the same amount of vibrational energy is released for each molecule–surface conformation, the difference between maximum and minimum barriers will be the same for both vibrational states, but all the barriers will be at much lower energy for the v = 1 state. This makes the corrugation of the vibrationally effective PES higher at low energies for vibrationally excited molecules than for vibrationally cold molecules, implying that for the same translational energy, the


169

Fig. 5.21. Elbow PESs show considerable surface site dependence for H2 /Cu(111). On the bridge site (a), where the dissociation barrier is lowest, the barrier is late enough to give vibration enhanced dissociation, but there is no vibrationally inelastic scattering because the curvature is low. At the atop site (b), the barrier is much higher and later, and the reaction path is more curved. Although dissociation is less favourable, there is strong vibrationally inelastic scattering. Adapted from Kinnersley et al. (1996).

vibrationally excited molecules should show a greater amount of diffraction and (given that our descriptions the underlying PESs and phenomena in Sections 5.4 and 5.5 are so similar) also of rotational excitation in the scattering (Darling and Holloway, 1992), provided the translational energy is high enough for the molecules to reach the curvature of the elbow. To demonstrate this for diffraction may require higher energies than is feasible due to the Debye–Waller reduction in the diffraction intensities. However, Hodgson et al. (1997) measured a rapid increase in the rotational excitation probability of the H2 (v = 1, J = 0) state scattered from Cu(111) surfaces at low energy, but could find no evidence of rotational excitation in the vibrational ground-state (although subject to a much lower precision due to the thermal occupation of both initial and final states). It should be borne in mind, of course, that there are still unresolved, large discrepancies between theory and experiment (Somers et al., 2004) in the strength of rotational transitions for H2 scattering from Cu, as evidenced by Fig. 5.14. Combined rotational-vibrational changes (Watts et al., 2000) in the H2 /Cu(100) system have been shown to be more likely for cartwheel states, with angular momentum parallel to the surface, than for helicopters, with angular momentum perpendicular to the surface (McCormack et al., 1999b). Vibrational excitation and de-excitation occur preferentially at the atop site where the azimuthal corrugation (i.e. the corrugation found on rotating the molecule about the surface normal) of the PES is weaker than at other sites, while the variation with polar angle is still strong because it ranges from the broadside orientation to the highly repulsive end-on orientation. Consequently, although J → J transitions can be strong, transitions between magnetic sublevels of the J states, i.e. mJ → m J transitions, are relatively weak. If mJ ≈ 0 (cartwheel states) the molecules can rotationally de-excite

170

G.R. Darling

Fig. 5.22. Probabilities for rovibrationally inelastic scattering as a function of the orientation of H2 molecules at the classical turning point (Wang et al., 2002).

while simultaneously vibrationally exciting, i.e. there is rotation–vibration conversion. For helicopter molecules, mJ ≈ J , de-excitation is not possible if mJ is approximately conserved, since this would lead to a reduction in J , but not in mJ (mJ J ). The overall effect of rotation–vibration energy transfer is that the vibrationally excited molecules are rotationally cooler than the incident beam. Classical trajectories can be used to provide a very detailed picture of precisely where on the PES vibration–rotation coupling occurs (Wang et al., 2002). It is found that the transitions are strongest for molecules in particular orientations. This can be seen in Fig. 5.22, which shows the transition probability as a function of the molecular bond orientation at the classical turning point (where the molecule has stopped moving towards the surface and is turning around to return to the gas-phase). The coupled rotation–vibration changes occur predominantly for molecules in very precise orientations. The origin of this is clear from the ‘typical’ classical trajectory illustrated in Fig. 5.23. The top panel shows the time dependence of the molecular centre-of-mass, Z, and bond length, r. In the region of the turning point, Z ∼ 0.6 au, there is a slight extra elongation of the molecular bond. The lower panel shows the same trajectory overlaid on a contour plot of the PES in r and θ . The lengthening of the bond can be seen to be a result of the late barrier, i.e. the molecule is attempting to dissociate, but because the bond is not parallel to the surface (i.e. θ = 90◦ ) dissociation cannot happen. Rather the trajectory reflects from a region of the PES curving round towards the dissociation barrier. Reflecting from an angled barrier, converts motion along r to motion along θ, i.e. there is vibration–rotation conversion. Reversal of this trajectory gives rotational to vibrational energy transfer, the process labelled VEARC in Fig. 5.22. The classical trajectories show that only when the trajectory hits the PES on a region of r–θ curvature can vibration–rotation coupling occur.


171

Fig. 5.23. A trajectory showing rotational excitation accompanying vibrational de-excitation (i.e. a vibrational to rotational energy transfer) (Wang et al., 2002). The top panel shows the evolution in the Z (molecule–surface distance) and r (molecular bond length) coordinates. In the lower panel, the motion is projected onto the r–θ (molecular bond orientation) plane. Coupling of vibrations and rotations occurs because the molecule attempts to dissociate at an unfavourable bond angle.

Although the vibration–rotation coupling can apparently be understood in terms of mechanical couplings in certain regions of the PES, there is still a problem in comparing the magnitude of the effects found in theory with those measured in experiment. As for the pure rotational excitation, there is again a far smaller amount of vibration–rotation coupling in the experimental results (Shackman and Sitz, 2005).

5.7. Dissociation So far we have considered transfer of energy between molecular degrees-of-freedom such that the molecule remains intact and (mostly) returns to the gas-phase. Interrogation of the scattered molecules to determine the internal states is not always possible, limiting the systems we can study in this fashion. Dissociation, in contrast, can be measured for any system as long as the barrier is not too high. The path to dissociation will take the molecule through many of the same configurations it would traverse when scattering, so the dissociation will also depend on the interplay between the internal motions and the PES. The dissociation probability can be measured by comparing the reflectivity of the surface with

172

G.R. Darling

that of an inert surface (King and Wells, 1972), or by directly measuring the amount of the dissociation product on the surface, or could be obtained from measurements of associative desorption from adsorbed fragments by assuming detailed balance (Hodgson, 2000; Michelsen et al., 1993a; Rettner et al., 1993b). The dissociation products are thermalized on the surface, so the final state in this case does not itself contain information about the dissociation process. To obtain such information, we must analyze how the dissociation is affected by the initial rovibrational state of the molecule (if possible), by its angle of incidence and also by the surface temperature. Detailed modelling is then required to convert these experimental results into the topography of the PES. 5.7.1. Direct, trapping or steering In the grossest terms, the dissociation can be classified in terms of three paradigms: direct dissociation, trapping (or precursor) mediated dissociation and steering dominated dissociation. In direct dissociation, the molecule will simply fall apart if it has enough energy, while those molecules failing to dissociate will return to the gas-phase. A signature of this process would be an increase of the dissociation probability, S, with increasing translational energy, Ei , as more molecules become able to overcome the dissociation barrier. As its name suggests, in trapping mediated dissociation all molecules that dissociate have first been trapped intact in a precursor, molecularly adsorbed, state. This implies a well in the PES occurring before the molecule overcomes the (final) barrier to dissociation. As Ei increases, it becomes harder and harder for the molecule to lose enough energy to the surface to trap, and consequently the dissociation probability decreases with increasing Ei . Finally in steering dominated dissociation, strong attractive forces pull the molecule into particular surface sites and orientations before dissociation, which is usually non-activated at such sites. Increasing Ei counteracts the steering forces and again dissociation decreases. In reality, these paradigms are not completely independent. Access to a molecularly chemisorbed dissociation precursor can be activated, as is the case for O2 on Ag(100) (Vattuone et al., 1994a, 1994b). In this case the dissociation probability would (at least initially) increase with molecular translational energy. The precursor need not be a static entity stably chemisorbed at one site. In the dissociation of N2 on the W(110) surface DFT calculation showed a barrier to a molecular precursor state (Corriol and Darling, 2004). However molecular dynamics simulation with a static surface gives trajectories as in Fig. 5.24, where the molecule cannot settle into the chemisorbed state (the static surface cannot absorb the molecular energy), but also cannot escape from the surface because the transition state between gas-phase and precursor is sterically hindered (Corriol and Darling, 2004). The molecule dissociates at a bridge site after a few bounces on the surface. Steering and trapping are also linked, because the existence of attractive steering forces implies wells in the PES into which molecules can trap (Busnengo et al., 2001a; Crespos et al., 2001; Darling et al., 1998). It can be possible in detailed modelling to separate these, but since both lead to the same dependence of dissociation on translational energy, there are no unambiguous experimental signatures, unless a stable adsorbed precursor state can be identified by, say, electron spectroscopy.


173

Fig. 5.24. (a) Snapshots of a classical trajectory showing the trapping of N2 molecules on W(110) prior to dissociation (Corriol and Darling, 2004). In this PES, access to the precursor state is activated. The molecule cannot settle into the molecular chemisorption site, shown in (b), and cannot escape the surface, because this is sterically hindered. Eventually the molecule dissociates at a bridge site leaving the atoms in quasi 3-fold hollow sites, as in (c).

5.7.2. Incidence angle dependence of dissociation If the PES is the same at every surface site, there is nothing to couple momentum normal and parallel to the surface. We would expect trapping and/or dissociation to depend only on the component of momentum normal to the surface. For a given energy, therefore, we would find a minimum in trapping (and hence precursor mediated dissociation) at normal incidence to the surface for which the normal component of momentum is greatest (cf. Fig. 5.11b), and hence the most energy must be lost for trapping to occur. Corrugation of the surface makes the trapping less efficient because molecules can scatter back into the gas-phase off the surface corrugation. The angular dependence of sticking is then more generally quantified in terms of the ‘energy scaling’

S(Ei , θ) = S Ei cosn θ, 0 (5.8) for energy E and incidence angle θ (with respect to the surface normal) the scaling exponent is n, i.e. for normal incidence, the scaling exponent is 2. It is common in the field to talk about the ‘normal energy’ En = Ei cos2 θ, rather than the momentum, thus for a flat surface the sticking exhibits normal energy scaling. For trapping on a corrugated surface, the angular dependence is less pronounced than on a flat surface and the exponent generally ranges between 0 (total energy scaling, i.e. the sticking is angle independent) and 2 (Arumainayagam and Madix, 1991). Note this implies that plotted against total energy,

174

G.R. Darling

the sticking/dissociation probability at off-normal incidence lies above that at normal incidence by an amount decreasing with n. If the sticking at off-normal incidence was below that at normal incidence when plotted against total energy, this would correspond to a negative value of n. This could occur if the molecule is scattering from a very rough/highly corrugated surface (D’Evelyn et al., 1987). For activated dissociation, at the simplest level, the dissociation barrier changes in magnitude with surface site, or it changes in location or width, as shown in Fig. 5.25a and b (Darling and Holloway, 1994a, 1998; Gross, 1995). These are termed energetic and geometric corrugation, and have opposing influences on the dissociation probability. In energetic corrugation, molecules at normal incidence are funnelled towards the lowest barrier to dissociation, however, this site is shadowed for molecules approaching at an angle, and consequently there is less dissociation for a given En at off-normal incidence. In contrast, for geometric corrugation, molecules at off-normal incidence can more effectively attack the inclined edge of the PES facing the incidence direction. A combination of the normal and parallel momentum can be utilized to traverse the barrier, making the dissociation for a given En greater at off-normal incidence, as in Fig. 5.25d. Of course these corrugations are both present to some extent in a real PES. An approximate balance can be obtained resulting in approximate normal energy scaling even for a corrugated PES (Darling and Holloway, 1994a), as observed in many experiments. When n = 2, we have normal energy scaling of the dissociation, while for n < 2 offnormal incidence favours dissociation as in geometric corrugation, and for n > 2, offnormal incidence hinders dissociation, like energetic corrugation. At energies below the dissociation barrier, dissociation must occur via tunnelling. Since the tunnelling increases exponentially with incidence energy, parallel momentum will always aid dissociation (Gross, 1995), i.e. n < 2 for both types of surface corrugation, as indicated in the inset of Fig. 5.25c (Darling and Holloway, 1998). This has been experimentally demonstrated to be so for the H2 /Cu(111) system (Murphy and Hodgson, 1998), where the scaling exponent increases with energy, before saturating at a value of 2 at high energy. Although theory can reproduce this result, the scaling exponent is not a trivial number to extract, particularly in the region of resonant features in the dissociation curve. A typical result can be seen in Fig. 5.26: at high energies n oscillates about a value of 2, but as the energy decreases into the tunnelling regime, so n decreases. Examination of Fig. 5.25c shows a number of step-like features in the dissociation curves, which are caused by resonances (Chatfield et al., 1991a, 1991b; Gross, 1999; Kinnersley et al., 1996). As discussed in Section 5.3, the restriction of motion at the dissociation barrier has associated with it a quantization (a greater quantization in the case of the already quantized rotations). In the simple example in Fig. 5.4, we could imagine a series of potential barriers corresponding to different modes of the quantized libration. When the incident energy reaches one of the barrier maxima, the dissociation jumps because another transmission channel has opened (Kinnersley et al., 1996). Although for frustrated rotations these levels can be widely spaced, they are largely overlapping for the frustrated translations across the surface. However, the individual levels reveal themselves in the energy derivative of the dissociation curve, as shown in Fig. 5.27. These resonant features have not (yet) been identified in experiment, possibly because they get substantially smeared out as a result of interaction with the substrate phonons and electronic excitations.


175

Fig. 5.25. Simple surface site dependent PESs. In energetic corrugation, the barrier height varies with surface site, causing shadowing of the lowest barriers, and less dissociation, at off-normal incidence. In energetic corrugation, varying the barrier position or width gives a slope in the PES open to attack by molecules incident at an angle. In this case, parallel momentum assists dissociation at low energies. Adapted from Darling and Holloway (1998).

Combinations of wells and barriers can produce strong angular effects fitting neither of the above models. Specifically, a potential well at one surface site can refract off-normally incident trajectories incident onto and over a barrier at another surface site (Corriol et al., 2003; Weiße et al., 2003). Crucially, if the well occurs at a site with the higher barrier, then at normal incidence, molecules are steered into the well, but strike off the high barrier and return to the gas-phase, while molecules incident at an angle are refracted onto and over the lower barrier at the other site. In consequence, the dissociation at off-normal incidence is actually higher than that at normal incidence, corresponding to n < 0 in Eq. (5.8). This situation is illustrated in Fig. 5.28. 5.7.3. Molecular rotation and dissociation Molecular rotation can both enhance and hinder dissociation (Beauregard and Mayne, 1993; Dai et al., 1994; Darling and Holloway, 1993, 1994b; McCormack and Kroes, 1999; McCormack et al., 1999a; Michelsen et al., 1992, 1993b; Rettner et al., 1993a). Enhance-

176

G.R. Darling

Fig. 5.26. Calculated angular scaling exponent for H2 /Cu(111) along two azimuths, from a five-dimensional model (Darling and Holloway, 1998). At high energies, there is approximate normal energy scaling, but at low energies when molecules are tunnelling through the barrier, n decreases (Gross, 1995).

Fig. 5.27. Transition state resonances can be seen in the derivative of the sticking probability (Kinnersley et al., 1996).

ment is due to mechanical transfer of the rotational energy into translation over the barrier. As the molecule dissociates, its bond length increases, decreasing the moment of inertia. The energy of each rotational state will then decrease, and the excess energy feeds into


177

Fig. 5.28. Angular dependence of the dissociation probability for a molecule incident on a surface at which dissociation is activated, but there is also a shallow precursor well in at the surface site where the dissociation barrier is highest. Classical results show a peak in dissociation at off normal incidence, as trajectories are refracted onto the lowest dissociation barrier. Quantum molecules have a more restricted passage over the barrier and the results follow the classical trend only when the mass, m, is high (for the same incident energy and angle, the classical results are mass independent).

translations/vibrations. Although it was first identified as important for a late barrier PES (cf. Fig. 5.18), this effect is actually much more common, because the molecular bond softens as the molecule approaches the surface, and this simple centrifugal enhancement can also be seen in the non-activated dissociation of helicoptering molecules, i.e. those with their molecular axis parallel to the surface (Gross and Scheffler, 1995). Countering the centrifugal effect is an orientational hindrance similar in essence to energetic corrugation discussed above. The higher barrier to dissociation for molecules in the end-on orientation effectively shadows the lower barriers in the broadside orientation. Increasing rotational state is equivalent to increasing the angle of incidence in Fig. 5.25c, i.e. the dissociation decreases with increasing rotational energy. The PES can also select out certain molecular orientations in dissociation (McCormack and Kroes, 1999). If a molecule approaches at an atop site, then if the PES shows little azimuthal corrugation (as for H2 on metals), the dissociation will occur preferentially for helicoptering molecules because dissociation of the cartwheels is rotationally hindered by the highly repulsive PES in the end-on geometry. At a bridge site the azimuthal corrugation is likely larger, introducing hindering also for helicopters. Overall, the helicopters generally have higher dissociation probabilities and the cartwheels are preferentially scattered back into the gas-phase. This should lead to appreciable differences in the dissociation probabilities of rotationally aligned molecular beams. However, not all molecules can be prepared in aligned beams. In such cases, we can invoke microscopic reversibility arguments to say that desorption and adsorption are basically the same dynamical phenomenon but running in opposite directions, and extract from the desorption a measure of the orientation dependence of the dissociation probability (Gulding et al., 1996; Hodgson, 2000; Rettner et al.,

178

G.R. Darling

Fig. 5.29. Measured rotational alignment for D2 molecules desorbing from Cu(111). Adapted from Hou et al. (1997).

1993b; Schröter and Zacharias, 1989; Wetzig et al., 1996; Zacharias, 1988, 1990). Specifically, the rotational alignment (the quadrupole moment of the orientational distribution) 2 3Jz − J 2 (2) A0 = (5.9) J2 can be both measured and computed. If cartwheels dominate the distribution, A(2) 0 < 0, (2) while if A0 > 0 more of the molecules are helicopters. The rotational alignment is dependent on the molecular energy, Ei , as shown in Fig. 5.29 (Hou et al., 1997), because with increasing energy molecules can more easily overcome the dissociation barriers in less favourable orientations. For non-activated reactions, there should also be a quadrupole alignment favouring helicopter molecules, however, although calculation gives A(2) 0 > 0, the experiments show little variation and the alignment is close to, or even slightly below 0 (Gross, 1998). 5.7.4. Molecular vibrations Figure 5.19 shows that for a late barrier PES, the effective potential barrier for vibrationally excited molecules is lower than for molecules in the vibrational ground-state, whereas for an early barrier PES, the dissociation barriers are roughly independent of the molecular rotational state. Therefore when the dissociation barrier is late, we should expect vibrational enhancement of reaction, but we should expect no vibrational enhancement for an early barrier. The extent to which vibration increases the dissociation is expressed in terms of


179

Fig. 5.30. Computations of dissociation and vibrational excitation for H2 molecules in a low dimensional model showing that dissociation at a single site with a single elbow PES cannot match both the vibrational efficacy for dissociation and the vibrational excitation threshold for H2 /Cu. In the top panel, molecules can traverse the elbow region at low energies. There is efficient de-excitation from v = 1, leading to dissociation in the v = 0 state. This gives too high a vibrational efficacy. In the bottom panel, the barrier is thicker preventing the molecules from rounding the elbow until higher energy. The vibrational efficacy is close to the experimental value of 0.5, but the threshold energy for vibrational excitation from v = 0 is too high.

the vibrational efficacy, Ξ . If the dissociation curves for v = 1 and v = 0 are separated by an energy E (cf. Fig. 5.30) then E (5.10) ε01 where ε01 is the spacing between the vibrational levels. For the H2 /Cu system, Ξ ≈ 0.5 (Michelsen et al., 1993a). However, vibrational excitation/de-excitation is intimately connected with vibrationally enhanced dissociation on a late barrier PES (Darling and Holloway, 1994c; Kroes et al., 1996; McCormack et al., 2000). Figure 5.30 shows results from single site models of dissociation on an elbow PES with the same barrier height and highly curved reaction path, but with two different barrier widths. In the top panel, the barrier is narrower than in the bottom panel and consequently molecules can get into the curved part of the elbow at lower energies than in the bottom panel, where the molecules are repelled from the surface before the elbow bend. When the molecules can access the elbow region, efficient T–V coupling leads to vibrational deexcitation of molecules incident in the v = 1 state (Darling and Holloway, 1994c). The vibrational energy released assists the molecules over the barrier, but this results in E ≈ Ξ=

180

G.R. Darling

Fig. 5.31. McCormack et al. (2000) solved the vibrational thresholds problem by finding that the dissociation in the v = 0 state occurs preferentially at the bridge site (top panel), while the dissociation in the v = 1 state occurs at the atop site where the barrier is higher, but also later and accompanied by a higher curvature.

ε01 , i.e. Ξ ≈ 1. If the molecules are prevented from accessing the curvature until higher energies, then the onset of de-excitation can be delayed to occur near the vibrationally adiabatic barrier (cf. Fig. 5.19), and the vibrational efficacy is close to the experimental value. In this case, however, the vibrational excitation from v = 0 to v = 1, is pushed to higher energy, although experiment shows the threshold for excitation is close to the dissociation threshold for molecules in the v = 0 state. The difficulty of fitting experimental thresholds within a single surface site model, suggests we should look for different processes to occur at different surface sites. McCormack et al. (2000) have found a particularly elegant solution to the problem of coupled excitation and dissociation thresholds. On the Cu(100) surface, the lowest barrier to dissociation is at the bridge site with the H atoms going towards the 4-fold hollow sites. Calculating position resolved dissociation flux, they find that for the v = 0 state, molecules do indeed dissociate preferentially at the bridge site, less easily at the hollow site and poorest of all at the atop site where the barrier is highest, as shown in the top panel of Fig. 5.31. However, the PES at the atop site has much higher curvature than at the bridge site (cf. Fig. 5.21), and this causes molecules in the v = 1 site to vibrationally de-excite in front of the barrier using


181

Fig. 5.32. Vibrational enhancement of dissociation can also occur for non-activated dissociation. In the H2 /Pd(100) system, the elbow PES opens out as the molecular bond softens when it approaches the surface. This causes a decrease in the vibrational frequency (ω in the left-hand panel) and consequently the v = 1 adiabatic potential is lower than the v = 0 potential. This occurs also for sites where the dissociation is activated, with the net result that there is a very high vibrational efficacy (∼0.7) for this system. Adapted from Gross and Scheffler (1995).

the released energy to enhance dissociation. This then changes the order of site preference for dissociation; molecules in the v = 1 state dissociate most readily at the top site, followed by the bridge site, with the hollow site contributing least. The top site is also where the strong vibrational excitation occurs, i.e. the vibrational excitation and dissociation of v = 0 molecules will be dominated by different sites. The vibrational efficacy acquires a reasonable value because although the v = 1 molecules can exploit all their vibrational energy to dissociate, they do so at the atop site where the barrier is higher than at the bridge site, where the v = 0 molecules preferentially dissociate. The division into early and late barrier PESs, although useful, is clearly an oversimplification. Vibrationally enhanced dissociation can even occur in a non-activated system, as demonstrated by Gross and Scheffler (1995). The dissociation of H2 on Pd(100) shows steering (and trapping), with the molecules drawn into regions of attractive PES towards dissociation. Not only is the PES attractive, but the H–H bond is also softening, with the result that both the potential energy and the H–H vibrational frequency decrease as the molecule dissociates, leading to a more attractive vibrationally adiabatic potential for vibrationally excited states, as in Fig. 5.32. This also occurs at less favoured dissociation sites on this surface with the result that vibrationally excited molecules dissociate very readily, yielding a high vibrational efficacy, as shown in Fig. 5.32. Vibrationally enhanced dissociation has also been observed for polyatomic molecules, particularly for methane (Beck et al., 2003; Juurlink et al., 2005; Maroni et al., 2005; Schmid et al., 2002; Smith et al., 2004), using laser excitation of the molecules in the gasphase to pump particular vibrational modes prior to scattering. Indeed, it has even been demonstrated that pumping two quanta into one C–H vibration is more effective in promoting dissociation than pumping one quantum into two separate C–H vibrations, indicating that the mechanism for vibrational enhancement is similar to that discussed here.

182

G.R. Darling

Fig. 5.33. Comparison of quantum, quasi-classical (circles) and classical (triangles) reaction probabilities for the H2 /Cu(100) system (McCormack and Kroes, 1998). The quasi-classical can be seen to perform extremely well in this case.

5.7.5. Classical mechanics versus quantum mechanics in dissociation We have outlined in Section 5.3 the factors to consider in the quantum v’s classical comparison. For dissociation, it is generally found that quasi-classical calculations perform extremely well for activated dissociation, such as for the H2 /Cu system. Figure 5.33 shows a comparison of the dissociation computed for six-dimensional models, using quantum mechanics, classical mechanics and quasi-classical mechanics (McCormack and Kroes, 1998). We can see that although the pure classical underestimates dissociation because of the lack of initial molecular vibrational energy, the quasi-classical performs very adequately, missing only the many resonance states (due to a combination of the transition state resonances discussed in Section 5.7.2 with H–H vibrational resonances in the vicinity of the barrier) close to 0.5 eV, and the jump in the quantum dissociation centred on 0.7 eV, which has the appearance of a rotational transition state resonance channel opening, although this has not been clearly identified in this case. For non-activated dissociation, the situation appears much less favourable for the classical approach. Eichler et al. (1999) have shown very good agreement between quantum calculation and the results of experiment for H2 /Pd(100) (Fig. 5.34), however both classical mechanics and quasi-classical mechanics overestimate the dissociation, as shown in the right panel of Fig. 5.34 (Gross and Scheffler, 1998). The quasi-classical actually performs the poorer in this case because the zero-point energy leaks out promoting reaction. The vibrational zero-point energy can also be lost to molecular rotation (Busnengo et al., 2002a). Rotational hindrance (Section 5.7.3) then causes the quasi-classical method to underestimate the dissociation probability, although the error is much smaller than that in Fig. 5.34. Classical methods can also fail in more subtle ways. Figure 5.28 shows the angular dependence of dissociation for an activated model system having a well in front


183

Fig. 5.34. Comparison of theory (Eichler et al., 1999) and experiment (Rendulic et al., 1989; Rettner and Auerbach, 1996) for the dissociation probability of H2 on Pd(100) (left-hand panel), the agreement is excellent. The left-hand panel shows that in this system, the quasi-classical approach actually performs worse for the dissociation than the classical (Gross and Scheffler, 1998). This is due leakage of the vibrational energy in the quasi-classical calculation.

of the highest dissociation barrier, as discussed in Section 5.7.2 (Corriol et al., 2003; Weiße et al., 2003). Vibrations are treated adiabatically, yet there is still marked difference between quantum and classical results, which gradually diminishes as the molecular mass is increased (when the system should behave more classically). This behaviour is due to the quantum system being more restricted in its motion over the barrier (Darling et al., 1997; Kay et al., 1998), i.e. the quantum molecule can only be transmitted through the quantized frustrated modes, the transition state resonances, as in Fig. 5.4. For molecules incident at an angle, the restriction in open channels leads to less transmission for a quantum molecule. The presence of the well in the model amplifies this slight difference in transmission leading to the marked difference in trends in Fig. 5.28 (Corriol, 2003; Corriol et al., 2003).

5.8. Substrate excitations and intra-molecular energy flow Although this chapter concerns the flow of energy within the molecule, that can also be affected by the state of the substrate. This has been well known for some time for certain systems, in particular the NO/Ag system (Rettner et al., 1985c) and for H2 reacting with Si surfaces, where the covalent nature of the H-Si bonds leads to large distortion of the substrate during adsorption/desorption, and a description requiring substrate dynamics is absolutely vital even at the lowest level (Dürr and Höfer, 2006). In recent years, the temperature of the substrate has also been found to be important for H2 interacting with metals (Gostein et al., 1997; Murphy and Hodgson, 1998; Shackman and Sitz, 2005; Watts and Sitz, 2001; Watts et al., 2000). Admittedly, we might not expect the effects to be dominant for H2 on metals since quantum calculations with a static surface can give accurate estimates of the dissociation probability, cf. Fig. 5.34. Nevertheless, there can be large

184

G.R. Darling

Fig. 5.35. (A) Arrhenius plots of the dissociation probability for H2 /Cu(111) at the translational energies indicated (Murphy and Hodgson, 1998). Clearly the lower the energy, the higher the apparent activation energy. (B) The apparent activation energy for dissociation decreases linearly as the molecular translational energy increases (Murphy and Hodgson, 1998). (C) Arrhenius plots for rotational excitation in the scattering of H2 from Cu(100) for the first two molecular vibrational states (Watts and Sitz, 2001).

discrepancies between theoretical and experimental estimates of state-resolved reflectivity (Somers et al., 2004; Watts et al., 2000), as shown in Fig. 5.14. Some experimental results for the dissociation and scattering of H2 at Cu surfaces are shown in Fig. 5.35. In panel A, we can see that the dissociation probability on the Cu(111) surface follows Arrhenius behaviour very well (over this limited temperature range), but that the surface temperature decreases in importance as we approach the dissociation barrier, i.e. the surface contribution to the dissociation is most evident in the tunnelling regime. In panel B, the surface contribution is quantified in terms of an apparent activation energy, Ea , derived from the


185

Arrhenius plots of panel B. Ea decreases linearly with translational energy to a value close to zero at 600 meV. In panel C, we show Arrhenius plots of the 1 → 3 rotational excitation probability for H2 scattered from Cu(100). Again, the surface temperature dependence can be quantified in terms of an activation energy, which is clearly different for different quantum states. 5.8.1. Molecule–phonon coupling Viewed as a simple classical binary collision, conservation of energy and momentum between the molecule and a surface atom gives the Baule formula (Baule, 1914) for the energy transferred to the (initially static) surface atom EB =

4μ (Ei + W ) (1 + μ)2

(5.11)

where μ is the ratio of the molecular to the surface mass. If we consider H2 compared to transition metals, then this ratio is small, thus justifying neglect of substrate vibrations on the impinging molecule. Hooking the surface mass to a harmonic oscillator potential gives a more realistic model of a phonon. Hand and Harris (1990) were the first to implement this model with a reactive elbow PES. They coupled an elbow PES to a harmonic oscillator representing the vibration of a surface atom and investigated the effect of surface recoil. Results indicated a broadening of dissociation curves with increasing surface temperature for H2 /Cu, although more substantial changes could be discerned for a better match of

Fig. 5.36. Arrhenius plots of the rotational excitation of H2 scattering from Pd(100) at the energies indicated. The experimental results (Watts and Sitz, 1999) are indicated by large symbols, classical theory (Busnengo et al., 2001b) by small symbols. The lines are a guide to the eye.

186

G.R. Darling

masses, such as N2 /Fe. The lasting legacy of the model has often been taken to be that the thermal effects in the H2 /Cu system are weak to the point of being negligible. From Fig. 5.35, we can see that this is definitely not the case near or below the threshold for dissociation or inelastic scattering. Extended versions of the Hand–Harris cube model has been implemented by two groups to address the problem posed by the data in Fig. 5.35, and similar data for H2 on Pd surfaces. Busnengo et al. (2001a, 2001b, 2002a, 2002b, 2003), Crespos et al. (2001), Díaz et al. (2005b) have made extensive studies of non-activated dissociation. Their results, based on classical calculations sampling the surface vibration from a Boltzmann distribution, indicate that steering and trapping are intimately linked – the majority of molecules scattering into rotationally inelastic channels do so only after briefly trapping. This leads to efficient energy exchange with the surface oscillator, hence the surface can contribute a sizeable fraction of the energy required for the excitation. For incident energies below the gas-phase excitation threshold for rotational excitation, the surface supplies the balance of energy required, i.e. the amount of energy supplied by the surface, the apparent activation energy, decreases as the translational energy approaches the transition threshold. Arrhenius plots of the probability of rotational excitation show excellent agreement with the experimental results, as can be seen in Fig. 5.36. Quantum wavepacket results for the H2 /Cu system (Wang et al., 2001, 2004), introducing surface temperature via summation of the final probabilities Boltzmann weighted according to the initial surface oscillator state, also show good qualitative agreement with the experimental trends, as can be seen from Fig. 5.37. There is a strong influence of surface temperature which follows an Arrhenius dependence (over a similar temperature range to that explored in the experiment) for all dissociation and scattering probabilities. Thermal ‘activation energies’ extracted from these are found to decrease linearly with translational energy, in agreement with experiment. The magnitude of the ‘activation energy’ also depends on precisely which process is considered: it differs between rotational transitions, Ea is different for the 0 → 6 and 0 → 4 transitions, the activation energy for the latter even depends on whether the molecular vibrational state is v = 0 or v = 1. Similar sensitivity of the activation energy to the quantum state has been found in experiment (Watts and Sitz, 2001), as can be seen in Fig. 5.35C. There is a negligible amount of molecular trapping at these energies for the H2 /Cu system, so the Arrhenius behaviour does not come from full or partial thermalization of the molecules. Rather, a simple semi-analytical model (Wang et al., 2004) illustrates that the role of the surface oscillator is to provide some of the energy to assist in the process, i.e. increasing the vibrational energy of the surface results in a downshift of the translational energy threshold for the dissociation/inelastic scattering process, i.e. for initial surface oscillator state n, the threshold is ETn = ET0 − nE, where E is a parameter. A Boltzmann weight of these shifted transition probabilities over the substrate vibrational states reproduces exactly the trends observed in Fig. 5.37, in particular, the linear translational energy dependence of the activation energy below threshold and the sharp change and much weaker variation above threshold. The Arrhenius dependence of the scattering and dissociation probabilities thus emerges as a consequence of the transition threshold downshifting on average with increasing thermal (Boltzmann) population of the phonon states.


187

Fig. 5.37. Computed (quantum), thermally averaged rotational excitation probability as a function of translational energy for D2 scattering from a single surface site model of Cu(111) at the temperatures indicated (Wang et al., 2004). Replotting this data in Boltzmann plots, we can extract apparent activation energies for each incidence energy relative to the threshold energy, as shown in the right-hand panel.

Using a linear approximation to the energy dependence of the transition probability, fn (Ei ), for surface oscillator state n provides insight into the translational energy dependence of Ea . As the energy progressively decreases below the threshold, the lower surface oscillator states drop out of the Boltzmann summation, as they have insufficient energy to effect the transition. If the lowest oscillator state contributing is n , the sum over initial oscillator states can be written fn+n (Ei ) exp(−nω/kTs ) S(Ei , TS ) ∝ e−n ω/kTs (5.12) n=0

where ω is the frequency of the surface oscillator. As Ei decreases, so n increases, i.e. we have to go to progressively higher oscillator states to shift the threshold down far enough to reach Ei , hence the exponent in the prefactor also increases, i.e. the activation energy increases approximately linearly as Ei decreases. In this model, the most important parameter in determining the translational energy dependence of the activation energy is E. If we reduce E, then the slope of Ea (Ei ) increases, a feature found in the results of the full quantum dynamics calculations: the slope is greater for the 0 → 4 rotational transition than for 0 → 6. E is the shift in the threshold due to the thermal motion, why this should vary from one particular transition to another is not presently clear. Simplifying to a one-dimensional model of a barrier coupled to an oscillator (Hudson, 2007) shows that Ea is not necessarily the energy required to reach the threshold, but is related to the coupling between molecule and oscillator, and to whether a quantum or classical treatment of the dynamics is employed. In spite of the uncertainties in interpretation, the quantum dynamics model also obtains the correct trends in the energy exchange between molecule and substrate (Darling et al., 2002) (this is different from Ea ). If molecules gain internal energy in the scattering (an activated process), there is a net increase in molecular energy from the substrate for translational energies below threshold, as observed in experiment, since below threshold the

188

G.R. Darling

Fig. 5.38. Final translational energy of hydrogen molecules scattered from Cu surfaces into rotationally excited states as a function of the initial translational energy. Results from theory (Darling et al., 2002) of D2 scattering from the J = 0 to the J = 4 state are shown in the left-hand panel, experimental results (Watts and Sitz, 2001) for H2 scattering J = 1 to the J = 3 state are shown in the right-hand panel. Both showing an energy gain from the surface that decreases with increasing initial translational energy.

substrate must supply some energy for the transition to occur at all. The computed results are surprisingly similar to experimental measurements (Watts and Sitz, 2001), as shown in Fig. 5.38. The results are insensitive to the transition considered, i.e. the energy exchange is much the same whether we consider the J = 0 → 4 or J = 0 → 6 transition, or even the v = 0 → 1 transition, all that matters for the energy gain is how far below threshold is the translational energy. In other words, the surface makes up the energy deficit in a fashion almost independent of the threshold energy, but, unlike the H2 /Pd system modelled by classical mechanics, the energy deficit and the activation energy are not identical. For processes in which the molecular energy is decreased in scattering, the experimental trends are also reproduced: there is a greater energy loss to the substrate for transitions in which there is a greater decrease in the internal energy. For example, the energy loss to the substrate increases in vibrational de-excitation as the final J state of the molecules decreases from J = 6 (internal energy change 214 meV) to J = 0 (internal energy change 371 meV). We can interpret this as resulting from the harder collision of the faster moving molecules following de-excitation with higher internal energy change, i.e. the internal energy is converted to molecular translations and the molecule simply hits the surface harder. 5.8.2. Electron–hole pair interactions In Section 5.2, we discussed the issue of adiabatic versus nonadiabatic behaviour of the electrons. We might expect this issue to be most important when molecules interact with metal surfaces, since we can excite electrons across the Fermi energy with practically zero energy cost. The excited electron created and the hole it leaves behind combine to give


189

bosons that we can treat with standard many-body theory (Bird et al., 2004; Gunnarsson and Schönhammer, 1982; Hellsing and Persson, 1984; Mizielinski et al., 2005; Persson and Persson, 1980; Schönhammer and Gunnarsson, 1984; Trail et al., 2003, 2002). We can use a forced oscillator model (the bosons are treated quantum mechanically, while the molecule–surface interaction is replaced with a time-dependent driving force derived from a classical trajectory) to obtain the probability of transferring energy E with the electron– hole pairs, viz. 2 ∞ λj

1 1 − e−iεj t dt eiEt exp − P (E) = (5.13) ε 2π −∞ j j

where the index j sums over the electron–hole pair states of energy εj . The coupling, λj , can be written 1 ∞ iεj t ∂ λj = (5.14) e δEF (t) dt π −∞ ∂t The derivative of the generalized instantaneous phase-shift, δEF (t), gives the rate at which electronic states drop below the Fermi energy, εF , the higher the rate, the greater the interaction with the electron–hole pairs. The states of interest are, of course, molecule induced. In particular, when a molecular affinity level crosses EF , there should be efficient excitation of electron–hole pairs (Trail et al., 2002). When H2 interacts with metals, just such a situation appears to obtain. The initially unoccupied 2σu∗ state broadens, drops in energy and gradually fills as the molecular bond stretches (Hammer and Nørskov, 1995). From the standard formalism, we should expect a large loss of energy to the electronic excitations in the substrate. Certainly, there is substantial energy exchange with the substrate. For the Pd(111) surface 120 ± 34 meV is exchanged in the transition v = 1, J = 1 to v = 0, J = 5 (Shackman and Sitz, 2005), however, this could be accounted for by phonon excitation alone, bearing in mind the success of the modelling shown in Fig. 5.36. Figure 5.38 shows reasonable agreement also for Cu surfaces. However, we should bear in mind that the theories for both Pd and Cu surfaces use a rather crude representation of the surface vibrations, which contains unknown error margins; the agreement could simply be fortuitous. So far, one puzzle remains for H2 /Cu, namely that theory and experiment massively disagree on the strength of rotational inelasticity (Somers et al., 2004; Watts et al., 2000), cf. Fig. 5.14. This could perhaps be resolved with the inclusion of electron–hole pair damping of the internal states of the molecule. Electron–hole pairs are believed to play a more major role in the interaction of NO molecules with metal surfaces (Huang et al., 2000; Rettner et al., 1985c, 1987; White et al., 2005, 2006; Wodtke et al., 2002, 2003), particularly with noble metal surfaces, and low workfunction surfaces. The top panel of Fig. 5.39 shows the vibrational excitation probability (v = 0 to v = 1) for NO scattering from Ag(111) (Rettner et al., 1985c, 1987). It is strongly surface temperature dependent, and also dependent on the incident energy. Some of this behaviour can definitely be accounted for within the simple cube models outlined in Section 5.8.1 (Gates and Holloway, 1994), yet some is also ascribed to electron–hole pair coupling (Gross and Brenig, 1993; Gross et al., 1991). Scattering from the very similar Au surface, we can still see temperature and energy dependent vibrational

190

G.R. Darling

Fig. 5.39. Vibrationally inelastic scattering of NO from noble metals. In the top panel the population of v = 1 on scattering v = 0 molecules from Ag(111) is shown in an Arrhenius plot at two incident energies (Rettner et al., 1985c). In the bottom panel, both excitation and de-excitation of molecules incident in v = 2 on the Au(111) surface are shown as a function of molecular translational energies at the surface temperatures indicated (Huang et al., 2000).

excitation (bottom panel of Fig. 5.39), but we also have energy dependent vibrational deexcitation, which is insensitive to the surface temperature. White et al. (2005) have made a more direct measurement that points to electronic nonadiabaticity. Scattering NO from a Cs covered Au(111) surface (the Cs lowers the workfunction of the Au) they found electron emission. As shown in Fig. 5.40, the electron emission is efficient once the vibrational energy exceeds the workfunction of the surface, indicating that this is a direct conversion of molecular vibrational energy into electronic energy. In this


191

Fig. 5.40. The probability of electron emission from a Cs promoted Au surface as a function of the vibrational state of impinging NO molecules (White et al., 2005). Clearly, for highly excited states the molecular vibrational energy pumps into the electronic energy allowing electrons to escape the surface. The shaded region indicates the workfunction of the surface on the molecular vibration scale.

case, the Born–Oppenheimer approximation is clearly invalid. A full numerical description would be extremely complex, coupling molecular motion to that of electrons. The recent results, shown in Fig. 5.40, present an exciting new challenge to the theory of gas–surface dynamics.

5.9. Conclusions Since the end of the 1980s, theory has been particularly successful in gas–surface dynamics. Undoubtedly the increase of computer power has played a role, making it possible to solve the quantum dynamics of a molecule striking the surface including all of the molecular degrees-of-freedom. This opens up the possibility of making detailed quantitative comparison with experiment, and at least for H2 interacting with reactive transition metals such as Pd and Pt, the agreement is remarkably good. However, even for the much-studied H2 /metal systems there are still gaps, notably in the magnitude of inelastic scattering probabilities from Cu surfaces, theory simply predicts too much. This might point to important physics we have so far omitted from our description, or it might require an extremely accurate description of the PES in some critical region. The resolution of this will come with more and better experiment and theory. Ten years ago we would have had the same worries about diffraction intensities being much too great in calculation. This has now been resolved by computing and measuring the out-of-plane component, and excellent agreement between theory and experiment is found (Farías et al., 2005). Moving beyond hydrogen to heavier molecules and polyatomics without sacrificing the high level of accuracy obtainable for hydrogen, while be a considerable challenge for the

192

G.R. Darling

future because of the greatly increased dimensionality. Also we will have to pay more attention to the vibrational and electronic modes of the substrate. While we have some understanding of how these are involved in inelastic scattering of hydrogen, the models employed to date are clearly simplistic. Electronic nonadiabaticity might not play such an important role for H2 , but is definitely important for O2 and NO, and likely for many other molecules. Major, new theoretical approaches will be required to address this. Gas– surface dynamics will certainly keep theoretical chemists and physicists in gainful employ for many years to come.

Acknowledgements Thanks to Stephen Holloway for encouraging comments on the manuscript, and to Kurt Kolasinski for his very careful reading and helpful suggestions for improvements.

References Andersson, S., Wilzén, L., Persson, M., 1988. Phys. Rev. B 38, 2967. Andersson, S., Wilzén, L., Persson, M., Harris, J., 1989. Phys. Rev. B 40, 8146. Arumainayagam, C.R., Madix, R.J., 1991. Prog. Surf. Sci. 38, 1. Baule, B., 1914. Ann. der Physik 44, 145. Beauregard, J.N., Mayne, H.R., 1993. Chem. Phys. Lett. 205, 515. Beck, R.D., Maroni, P., Papageorgopoulos, D.C., Dang, T.T., Schmid, M.P., Rizzo, T.R., 2003. Science 302, 98. Behler, J., Delley, B., Lorenz, S., Reuter, K., Scheffler, M., 2005. Phys. Rev. Lett. 94, 036104. Binetti, M., Weisse, O., Hasselbrink, E., Katz, G., Kosloff, R., Zeiri, Y., 2003. Chem. Phys. Lett. 373, 366. Bird, D.M., Persson, M., Trail, J.R., Holloway, S., 2004. Surf. Sci. 566, 761. Brako, R., Newns, D.M., 1980. Solid State Comm. 33, 713. Brenig, W., Brunner, T., Gross, A., Russ, R., 1993. Z. Phys. B 93, 91. Busnengo, H.F., Salin, A., Dong, W., 2000. J. Chem. Phys. 112, 7641. Busnengo, H.F., Crespos, C., Dong, W., Salin, A., Rayez, J.C., 2001a. Phys. Rev. B 63, 041402. Busnengo, H.F., Dong, W., Sautet, P., Salin, A., 2001b. Phys. Rev. Lett. 87, 7601. Busnengo, H.F., Crespos, C., Dong, W., Rayez, J.C., Salin, A., 2002a. J. Chem. Phys. 116, 9005. Busnengo, H.F., Pijper, E., Somers, M.F., Kroes, G.J., Salin, A., Olsen, R.A., Lemoine, D., Dong, W., 2002b. Chem. Phys. Lett. 356, 515. Busnengo, H.F., Pijper, E., Kroes, G.J., Salin, A., 2003. J. Chem. Phys. 119, 12553. Böttcher, A., Imbeck, R., Morgante, A., Ertl, G., 1990. Phys. Rev. Lett. 65, 2035. Celli, V., 1984. In: Langreth, D., Suhl, H. (Eds.), Many Body Phenomena at Surfaces. Academic Press, Orlando, p. 315. Citri, O., Baer, R., Kosloff, R., 1996. Surf. Sci. 351, 24. Chatfield, D.C., Friedman, R.S., Truhlar, D.G., Garrett, B.G., Schwenke, D.W., 1991a. J. Am. Chem. Soc. 113, 486. Chatfield, D.C., Friedman, R.S., Truhlar, D.G., Schwenke, D.W., 1991b. Faraday Discuss. Chem. Soc. 91, 289. Corriol, C., 2003. PhD Thesis. The University of Liverpool. Corriol, C., Darling, G.R., Holloway, S., 2003. Surf. Sci. 532–535, 198. Corriol, C., Darling, G.R., 2004. Surf. Sci. 557, L156. Cowin, J.P., Yu, C.F., Sibener, S.J., Wharton, L., 1983. J. Chem. Phys. 79, 1155. Cowin, J.P., Yu, C.F., Wharton, L., 1985. Surf. Sci. 161, 221. Crespos, C., Busnengo, C., Dong, W., Salin, A., 2001. J. Chem. Phys. 114, 10954. D’Evelyn, M.P., Steinrück, H.P., Madix, R.J., 1987. Surf. Sci. 180, 47.


193

Dai, J.Q., Sheng, J., Zhang, J.Z.H., 1994. J. Chem. Phys. 101, 1555. Darling, G.R., Holloway, S., 1990. J. Chem. Phys. 93, 9145. Darling, G.R., Holloway, S., 1992. Chem. Phys. Lett. 191, 396. Darling, G.R., Holloway, S., 1993. Faraday Discuss. 96, 43. Darling, G.R., Holloway, S., 1994a. Surf. Sci. 304, L461. Darling, G.R., Holloway, S., 1994b. J. Chem. Phys. 101, 3268. Darling, G.R., Holloway, S., 1994c. Surf. Sci. 307–309, 153. Darling, G.R., Holloway, S., 1995. Rep. Prog. Phys. 58, 1595. Darling, G.R., Kay, M., Holloway, S., 1997. Phys. Rev. Lett. 78, 1731. Darling, G.R., Holloway, S., 1998. Faraday Discuss. 110, 253. Darling, G.R., Kay, M., Holloway, S., 1998. Surf. Sci. 400, 314. Darling, G.R., Wang, Z.S., Holloway, S., 2000. Phys. Chem. Chem. Phys. 2, 911. Darling, G.R., Wang, Z.S., Holloway, S., 2002. Chem. Phys. Lett. 365, 157. Darling, G.R., Holloway, S., 2003. In: Woodruff, D.P. (Ed.), The Chemical Physics of Solid Surfaces, vol. 11. Elsevier, Amsterdam, p. 27. Dianat, A., Sakong, S., Gross, A., 2005. Eur. Phys. B 45, 425. Diaz, C., Somers, M.F., Kroes, G.J., Busnengo, H.F., Salin, A., Martin, F., 2005a. Physical Review B 72. Díaz, C., Somers, M.F., Kroes, G.J., Busnengo, H.F., Imbeck, R., Salin, A., Martín, F., 2005b. Phys. Rev. B 72, 035401. Diño, W.A., Kasai, H., Okiji, A., 2000. Prog. Surf. Sci. 63, 63. Dürr, M., Höfer, U., 2006. Surf. Sci. Rep. 61, 465. Eichler, A., Hafner, J., Gross, A., Scheffler, M., 1999. Phys. Rev. B 59, 13297. Farías, D., Díaz, C., Rivière, P., Busnengo, H.F., Nieto, P., Somers, M.F., Kroes, G.J., Salin, A., Martín, F., 2005. Phys. Rev. Lett. 93, 246104. Gates, G.A., Holloway, S., 1994. Surf. Sci. 307–309, 132. Gergen, B., Nienhaus, H., Weinberg, W.H., McFarland, E.W., 2001a. Science 294, 2521. Gergen, B., Weyers, S.J., Nienhaus, H., Weinberg, W.H., McFarland, E.W., 2001b. Surf. Sci. 488, 123. Gostein, M., Parhikhteh, H., Sitz, G.O., 1995. Phys. Rev. Lett. 75, 342. Gostein, M., Watts, E., Sitz, G.O., 1997. Phys. Rev. Lett. 79, 2891. Gross, A., Chiba, Y., Brenig, W., 1991. Prog. Theor. Phys. Suppl. 106, 335. Gross, A., Brenig, W., 1993. Surf. Sci. 289, 335. Gross, A., 1995. J. Chem. Phys. 102, 5045. Gross, A., Scheffler, M., 1995. Chem. Phys. Lett. 256, 417. Gross, A., Wilke, S., Scheffler, M., 1995. Phys. Rev. Lett. 75, 2718. Gross, A., Scheffler, M., 1996. Chem. Phys. Lett. 263, 567. Gross, A., 1998. Surf. Sci. Rep. 32, 291. Gross, A., Scheffler, M., 1998. Phys. Rev. B 57, 2493. Gross, A., 1999. J. Chem. Phys. 110, 8696. Guezebroek, F.H., Wiskerke, A.E., Tenner, M.G., Kleyn, A.W., Stolte, S., Nimiki, A., 1991. J. Phys. Chem. 95, 8409. Gulding, S.J., Wodtke, A.M., Hou, H., Rettner, C.T., Michelsen, H.A., Auerbach, D.J., 1996. J. Chem. Phys. 105, 9702. Gunnarsson, O., Schönhammer, K., 1982. In: Vanselow, R., Howe, R. (Eds.), Chemistry and Physics of Solid Surfaces IV. Springer Verlag, Berlin, p. 363. Halstead, D., Holloway, S., 1990. J. Chem. Phys. 93, 2859. Hammer, B., Scheffler, M., Jacobsen, K.W., Nørskov, J.K., 1994. Phys. Rev. Lett. 73, 1400. Hammer, B., Nørskov, J.K., 1995. Surf. Sci. 343, 211. Hand, M.R., Harris, J., 1990. J. Chem. Phys. 92, 7610. Harris, J., Andersson, S., 1985. Phys. Rev. Lett. 55, 1583. Hellsing, B., Persson, M., 1984. Physica Scripta 29, 360. Hodgson, A., Moryl, J., Traversaro, P., Zhao, H., 1992. Nature 356, 501. Hodgson, A., Samson, P., Wight, A., Cottrell, C., 1997. Phys. Rev. Lett. 78, 963. Hodgson, A., 2000. Prog. Surf. Sci. 63, 1. Honkala, K., Laasonen, K., 2000. Phys. Rev. Lett. 84, 705.

194

G.R. Darling

Hou, H., Gulding, S.J., Rettner, C.T., Wodtke, A.M., Auerbach, D.J., 1997. Science 277, 80. Huang, Y., Wodtke, A.M., Hou, H., Rettner, C.T., Auerbach, D.J., 2000. Phys. Rev. Lett. 84, 2985. Hudson, A.D., 2007. PhD Thesis. The University of Liverpool. Juurlink, L.B.F., Smith, R.R., Killelea, D.R., Utz, A.L., 2005. Phys. Rev. Lett. 94. Kasemo, B., Törnqvist, E., Nørskov, J.K., Lundqvist, B.I., 1979. Surf. Sci. 89, 554. Kasemo, B., 1996. Surf. Sci. 363, 22. Katz, G., Zeiri, Y., Kosloff, R., 1999. Surf. Sci. 425, 1. Kay, M., Darling, G.R., Holloway, S., 1995. Chem. Phys. Lett. 245, 311. Kay, M., Darling, G.R., Holloway, S., 1998. J. Chem. Phys. 108, 4614. King, D.A., Wells, M.G., 1972. Surf. Sci. 29, 454. Kinnersley, A.D., Darling, G.R., Holloway, S., Hammer, B., 1996. Surf. Sci. 364, 219. Kleyn, A.W., 2003. In: Woodruff, D.P. (Ed.), The Chemical Physics of Solid Surfaces, vol. 11. Elsevier, Amsterdam, p. 79. Kolasinski, K.W., 2001. Surface Science: Foundations of Catalysis and Nanoscience. J.W. Wiley, Chichester. Kosloff, R., Citri, O., 1993. Faraday Discuss. Chem. Soc. 96, 175. Kroes, G.-J., Somers, M.F., 2005. J. Theor. Comput. Chem. 4, 493. Kroes, G.J., Wiesenekker, G., Baerends, E.J., Mowrey, R.C., 1996. Phys. Rev. B 53, 10397. Kroes, G.J., Baerends, E.J., Mowrey, R.C., 1997. J. Chem. Phys. 107, 3309. Kroes, G.J., 1999. Prog. Surf. Sci. 60, 1. Lahaye, R.J.W.E., Stolte, S., Holloway, S., Kleyn, A.W., 1996. J. Chem. Phys. 104, 8301. Maroni, P., Papageorgopoulos, D.C., Sacchi, M., Dang, T.T., Beck, R.D., Rizzo, T.R., 2005. Phys. Rev. Lett. 94. McCormack, D.A., Kroes, G.J., 1998. Chem. Phys. Lett. 296, 515. McCormack, D.A., Kroes, G.J., 1999. Phys. Chem. Chem. Phys. 1, 1359. McCormack, D.A., Kroes, G.J., Olsen, R.A., Baerends, E.J., Mowrey, R.C., 1999a. J. Chem. Phys. 110, 7008. McCormack, D.A., Kroes, G.J., Olsen, R.A., Baerends, E.J., Mowrey, R.C., 1999b. Phys. Rev. Lett. 82, 1410. McCormack, D.A., Kroes, G.J., Olsen, R.A., Groeneveld, J.A., van Stralen, J.N.P., Barends, E.J., Mowrey, R.C., 2000. Chem. Phys. Lett. 328, 317. Michelsen, H.A., Rettner, C.T., Auerbach, D.J., 1992. Phys. Rev. Lett. 69, 2678. Michelsen, H.A., Rettner, C.T., Auerbach, D.J., 1993a. In: Madix, R.J. (Ed.), Surface Reactions. Springer, Berlin, p. 123. Michelsen, H.A., Rettner, C.T., Auerbach, D.J., Zare, R.N., 1993b. J. Chem. Phys. 98, 8294. Miura, Y., Kasai, H., Diño, W.A., 2002. J. Phys.: Condens. Matter 14, L479. Mizielinski, M.S., Bird, D.M., Persson, M., Holloway, S., 2005. J. Chem. Phys. 122. Murphy, M.J., Hodgson, A., 1998. J. Chem. Phys. 108, 4199. Nienhaus, H., Bergh, H.S., Gergen, B., Majumdar, A., Weinberg, W.H., McFarland, E.W., 1999. Phys. Rev. Lett. 82, 446. Nienhaus, H., Bergh, H.S., Gergen, B., Majumdar, A., Weinberg, W.H., McFarland, E.W., 2000. Surf. Sci. 445, 335. Nienhaus, H., 2002. Surf. Sci. Rep. 45, 3. Nienhaus, H., Gergen, B., Weinberg, W.H., McFarland, E.W., 2002. Surf. Sci. 514, 172. Olsen, R.A., Busnengo, H.F., Salin, A., Somers, M.F., Kroes, G.J., Barends, E.J., 2002. J. Chem. Phys. 116, 3841. Persson, B.N.J., Persson, M., 1980. Surf. Sci. 97, 609. Pijper, E., Kroes, G.J., Olsen, R.A., Baerends, E.J., 2002. J. Chem. Phys. 117, 5885. Rendulic, K.D., Anger, G., Winkler, A., 1989. Surf. Sci. 208, 404. Rettner, C.T., DeLouise, L.A., Cowin, J.P., Auerbach, D.J., 1985a. Chem. Phys. Lett. 118, 355. Rettner, C.T., DeLouise, L.A., Cowin, J.P., Auerbach, D.J., 1985b. Faraday Discuss. Chem. Soc. 80, 127. Rettner, C.T., Fabre, F., Kimman, J., Auerbach, D.J., 1985c. Phys. Rev. Lett. 55, 1904. Rettner, C.T., Kimman, J., Fabre, F., Auerbach, D.J., Morawitz, H., 1987. Surf. Sci. 192, 107. Rettner, C.T., Auerbach, D.J., Michelsen, H.A., 1992. Phys. Rev. Lett. 68, 2547. Rettner, C.T., Michelsen, H.A., Auerbach, D.J., 1993a. J. Vac. Sci. Technol. A 11, 1901. Rettner, C.T., Michelsen, H.A., Auerbach, D.J., 1993b. Faraday Discuss. 96, 17. Rettner, C.T., Auerbach, D.J., 1996. Chem. Phys. Lett. 253, 236. Savio, L., Vattuone, L., Rocca, M., Corriol, C., Darling, G.R., Holloway, S., 2003. Chem. Phys. Lett. 382, 605. Schmid, M.P., Maroni, P., Beck, R.D., Rizzo, T.R., 2002. J. Chem. Phys. 117, 8603.


195

Schröter, L., Zacharias, H., 1989. J. Chem. Soc., Faraday Trans. 2, 1361. Schönhammer, K., Gunnarsson, O., 1984. In: Langreth, D., Suhl, H. (Eds.), Many-Body Phenomena at Surfaces. Academic Press, New York, p. 421. Shackman, L.C., Sitz, G.O., 2005. J. Chem. Phys. 123, 064712. Smith, R.R., Killelea, D.R., DelSesto, D.F., Utz, A.L., 2004. Science 304, 992. Somers, M.F., Olsen, R.A., Busnengo, H.F., Baerends, E.J., Kroes, G.J., 2004. J. Chem. Phys. 121, 11379. Trail, J.R., Graham, M.C., Bird, D.M., Persson, M., Holloway, S., 2002. Phys. Rev. Lett. 88, 166802. Trail, J.R., Bird, D.M., Persson, M., Holloway, S., 2003. J. Chem. Phys. 119, 4539. van Harrevelt, R., Manthe, U., 2004. J. Chem. Phys. 121, 3829. Vattuone, L., Boragno, C., Pupo, M., Restelli, P., Rocca, M., Valbusa, U., 1994a. Phys. Rev. Lett. 72, 510. Vattuone, L., Rocca, M., Boragno, C., Valbusa, U., 1994b. J. Chem. Phys. 101, 703. Vattuone, L., Valbusa, U., Rocca, M., 1999. Phys. Rev. Lett. 82, 4878. Wang, Z.S., Darling, G.R., Holloway, S., 2000. Surf. Sci. 458, 63. Wang, Z.S., Darling, G.R., Holloway, S., 2001. Phys. Rev. Lett. 87, 226102. Wang, Z.S., Darling, G.R., Holloway, S., 2002. Surf. Sci. 504, 66. Wang, Z.S., Darling, G.R., Holloway, S., 2004. J. Chem. Phys. 120, 2923. Watts, E., Sitz, G.O., 1999. J. Chem. Phys. 111, 9291. Watts, E., Sitz, G.O., McCormack, D.A., Kroes, G.J., Olsen, R.A., Groenveld, J.A., Van Stralen, J.N.P., Barends, E.J., Mowrey, R.C., 2000. J. Chem. Phys. 114, 495. Watts, E., Sitz, G.O., 2001. J. Chem. Phys. 114, 4171. Weiße, O., Wesenberg, C., Binetti, M., Hasselbrink, E., Corriol, C., Darling, G.R., Holloway, S., 2003. J. Chem. Phys. 118, 8010. Wetzig, D., Dopheide, R., Rutkowski, M., David, R., Zacharias, H., 1996. Phys. Rev. Lett. 76, 463. White, J.A., Bird, D.M., Payne, M., Stich, I., 1994. Phys. Rev. Lett. 73, 1404. White, J.D., Chen, J., Matsiev, D., Auerbach, D.J., Wodtke, A.M., 2005. Nature 433, 503. White, J.D., Chen, J., Matsiev, D., Auerbach, D.J., Wodtke, A.M., 2006. J. Chem. Phys. 124, 064702. Wilzén, L., Althoff, F., Andersson, S., Persson, M., 1991. Phys. Rev. B 43, 7003. Wodtke, A.M., Huang, Y., Auerbach, D.J., 2002. Chem. Phys. Lett. 364, 231. Wodtke, A.M., Huang, Y., Auerbach, D.J., 2003. J. Chem. Phys. 118, 8033. Yourdshahyan, Y., Razaznejad, B., Lundqvist, B.I., 2002. Phys. Rev. B 65, 75416. Zacharias, H., 1988. Appl. Phys. A 47, 37. Zacharias, H., 1990. Int. J. Mod. Phys. B 4, 45. Österlund, L., Zoric, I., Kasemo, B., 1997. Phys. Rev. B 55, 15452.


CHAPTER 6

Inelastic Scattering of Heavy Molecules from Surfaces

Greg O. SITZ Department of Physics University of Texas Austin, TX 78712, USA



Contents 6.1. Introduction . . . . . . . . . . . . . . . . . . . . . 6.2. Experimental techniques . . . . . . . . . . . . . . 6.2.1. Molecular beams . . . . . . . . . . . . . . 6.2.2. Quantum state specific detection . . . . . . 6.3. Diatomic molecules . . . . . . . . . . . . . . . . . 6.3.1. Translation . . . . . . . . . . . . . . . . . . 6.3.2. Rotation . . . . . . . . . . . . . . . . . . . 6.3.2.1. Nitric oxide . . . . . . . . . . . . 6.3.2.2. Nitrogen and carbon monoxide . . 6.3.2.3. Angular momentum polarization . 6.3.3. Vibration . . . . . . . . . . . . . . . . . . . 6.3.3.1. Vibrational survival and relaxation 6.3.3.2. Vibrational excitation . . . . . . . 6.3.3.3. Highly excited vibrations . . . . . 6.4. Polyatomics . . . . . . . . . . . . . . . . . . . . . 6.4.1. Rotation . . . . . . . . . . . . . . . . . . . 6.4.2. Vibration . . . . . . . . . . . . . . . . . . . 6.5. Conclusion and outlook . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

198

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .

199 202 203 203 204 204 205 207 210 215 219 220 220 222 222 223 224 226 226 226

199

6.1. Introduction A complete understanding of the interaction of a molecule with a surface necessarily involves consideration of the role of the internal degrees-of-freedom (DOF) of the molecule. These include the rotational and vibrational motions as well as electronic excitations and center of mass translation. Experimental studies of energy transfer amongst these DOF and a surface is the subject of this chapter. In a classic review article, now 20 years old but still valuable, Barker and Auerbach surveyed both theoretical and experimental aspects of atom and molecule scattering (Barker and Auerbach, 1985). The present article will focus on molecules, and I will restrict this discussion to molecules heavier than D2 , that is to molecules for which quantum effects are relatively unimportant. The large separation of rotational and vibrational states together with the small mass of H2 (and D2 ) leads to a rich variety of quantum scattering phenomena worthy of a separate article. On the other hand, for heavier molecules a classical description usually captures all the essential details. A current theoretical view of inelastic scattering can be found in the article by Manson in this volume (Manson, 2008). For light molecules like H2 and its isotopic variants, internal state changing collisions accompanying scattering from a surface can be observed in high resolution measurements of the scattered angular and velocity distributions. For heavier molecules, this approach becomes problematic as the requirements on the resolution become extreme, and scattering involving multiple surface phonons tends to wash out discrete energy transfer. To provide internal state resolution, then, optical spectroscopic techniques have been developed and will be discussed here in some detail. Perhaps the earliest detailed studies of molecule–surface scattering were natural extensions of atom-surface experiments, where, even for the heaviest rare gas atoms, direct scattering in the form of quasi-specular reflection and incomplete energy accommodation were observed. For molecular scattering, two additional factors are added that raise new questions. First, to what extent do the internal molecular degrees-of-freedom wash out the non-thermal characteristics of the scattered flux? Second, how big of an effect would the typically much stronger interaction with the surface for a molecule (including such channels as dissociative adsorption) have on the scattering? Pioneering experiments showed clearly that direct scattering was not only possible but common. In some of the earliest work, Saltsburg and Smith showed a non-diffuse angular distribution for the scattering of CH4 from an epitaxial Ag(111) film (Saltsburg and Smith, Jr., 1966). An example of their measurements is shown in Fig. 6.1. For NH3 scattering from the same film, the figure shows that the angular distribution was diffuse, characteristic of complete accommodation with the surface. The difference between the two was interpreted in terms of a much deeper potential well for NH3 compared to CH4 . Complex characteristics of the scattered intensities and angular distributions as a function of incident angle and energy were attributed to energy transfer involving internal DOF’s. In somewhat later work, the scattering of N2 from tungsten showed a direct inelastic channel that looked very much like the scattering of argon (Janda et al., 1980), that is, much like the similarity between CH4 and neon shown in Fig. 6.1. The first and most basic question to address concerning the internal DOF’s in this discussion one of equilibrium: are the distribution functions describing the internal states of a collection of molecules travelling away from a surface characterized by a temperature?

200

G.O. Sitz

Fig. 6.1. Angular distributions of Ne, CH4 , and NH3 scattered from silver. The source temperature was 300 K, the surface temperature was 560 K, and the incident angle was 50◦ . Data from Saltsburg and Smith, Jr. (1966).

If so, is it the temperature of the surface? If the answer to these questions is yes, then the problem is solved, and this chapter is finished. However it turns out in many cases the answer is no, and a consideration of the dynamics is in order. An alternative way to phrase the question that translates easily into experiment is: does the state of a scattered molecule depend on its state prior to interaction with the surface? The first class of experiments which showed that the flux of molecules leaving a surface need not be in equilibrium with the surface were measurements of scattered angular and velocity distributions. As the data in Fig. 6.1 show, for a beam of molecules incident on the surface at a specified angle, in some cases the angular distribution of the scattered molecules showed a peak near the specular angle. Furthermore, the scattered velocity distribution was found to depend on the incident velocity. These results were a clear indication of incomplete equilibration, and generated substantial research in the field. The terminology used to describe the limiting scattering regimes for molecules are carried over from atom-surface studies: direct-inelastic (DI) and trapping-desorption (TD). These are illustrated in Fig. 6.2. For direct-inelastic scattering the motion of the centerof-mass of the molecule has a single turning point. In this regime, the final state distributions depend strongly on the initial conditions (incident angle and velocity). For trappingdesorption, the molecule loses enough energy on its initial encounter with the surface to become trapped in the molecule–surface well. The molecule spends a period of time in the well that is long enough to scramble any memory of its initial state, and then is ther-


201

Fig. 6.2. Illustration of Direct-Inelastic (DI) and Trapping-Desorption (TD) scattering mechanisms. Plotted are results from a classical molecular dynamics simulation of CH4 scattered from Ir(110) (Sitz and Mullins, 2002). The dark line in the position of the carbon atom and the lighter line the position of one of the hydrogen atoms.

mally desorbed. In this case, the final molecular state depends on the surface temperature (but need not be fully specified by that temperature). Most of the scattering phenomena discussed in this chapter will fall into one of these two categories. A third mechanism, intermediate between DI and TD, has been invoked to explain observations in a few cases. This mechanism is sometimes termed indirect inelastic or multibounce. Here the molecule undergoes more than one turning point in the well, but still retains some memory of its initial state. Examples will be given below where this distinction is warranted. Several comments are in order at this point to aid in the utilization of Fig. 6.2. First, the most important factor governing the interaction between the molecule and the surface is the relative energy of the two when the molecule encounters the repulsive wall of the potential. This ratio largely determines the extent to which energy is transferred amongst the various degrees-of-freedom of the system. An extension of this idea is that the primary influence of the depth of the attractive well is that the molecule is accelerated as it passes over the well, and hits the repulsive wall with an effective energy equal to the incident translational energy plus the well depth (Ed ). For a system where the well depth is relatively large (e.g., NO/Pt(111), where Ed = 1.4 eV), this effect dominates the branching between scattering channels. Second, the interaction of the molecule with the surface is fairly local, that is it directly involves only a few surface atoms. These atoms are, of course, coupled to the rest of the surface and bulk, but the initial interaction is effectively short range. Therefore, the mass ratio between the incident molecule and a few (1–4) surface atom masses can be used for a rough estimate of the extent and direction of energy transfer. This idea holds independent of the well depth (after accounting for the acceleration just mentioned). So, for example, a given system might have a deep well but massive surface atoms, and the scattering may have a significant direct-inelastic channel. On the other hand, a system with a shallow well depth may be dominated by trapping-desorption if the molecule mass is close to that of the effective surface particle.

202

G.O. Sitz

6.2. Experimental techniques Two key aspects are needed in an experimental apparatus to provide meaningful measurements of inelastic scattering of molecules: (1) a source which is capable of generating a beam with a well characterized (and hopefully small range) of incident parameters (translational energy, angle of incidence, internal state distribution) and (2) a detection system which can resolve the angle, translational energy and internal state of the scattered molecule. These two are in addition to an ultrahigh vacuum chamber in which a well characterized target can be prepared and maintained. General features of beam generation and detection will be considered below; sample preparation and characterization are common enough in surface science that they will not be discussed in any detail here. A schematic diagram of a generic beam–surface scattering apparatus is shown in Fig. 6.3.

Fig. 6.3. Illustration of a generic molecular beam–surface scattering apparatus with features typically of the many of the experiments discussed in this chapter. The chambers pumped by three pumps to left makeup the differentially pumped source, while the chamber to the right is the target chamber containing the sample surface. Particular items are labelled: A – the supersonic beam source; B – a rotatable sample manipulator shown in two positions, one facing the beam source and two facing a diagnostic instrument; C – a surface analysis instrument such as an Auger electron spectrometer; D – a quadrupole mass spectrometer; E – ion or photon detector.


203

6.2.1. Molecular beams Studies of dynamics at surfaces in general and internally inelastic scattering of molecules in particular took off with the application of the supersonic molecular beam source (Scoles, 1992). The source is simple: a pressurized gas expands through a small aperture into a vacuum chamber. Typically the pressure in the source is in the range of a few bar and the aperture size varies from a few 10s of microns for continuous beam sources to several hundred microns for low duty factor pulsed sources. The figure of merit in terms of beam intensity and supersonic cooling is the product of the source pressure and aperture size. A number on the order of 1 bar-mm is typical. This type of beam source provides a flux of molecules with a narrow range of velocity (or translational energy) and a rotational state distribution characterized by a temperature that is typically a few percent of the source gas temperature. The expansion is adiabatic in the sense that no heat flows into or out of the expanding gas. The narrow velocity distribution is a result of collision processes occurring in the free expansion of a pressurized gas into a vacuum. This is considered as a cooling of the gas in a frame of reference flowing with center of mass of the expanding gas. The rotational degree (or degrees for a polyatomic molecule)-of-freedom cools as rotational energy is transferred into the directed center of mass motion. The gas flow from a supersonic beam source is typically collimated by one to three apertures separating vacuum chambers that are separately pumped (see Fig. 6.3). This differential pumping serves to reduce the background flux of molecules on the target surface of molecules not in the direct beam. A typical source will deliver a flux of molecules at the target surface on the order of 1013 –1015 cm−2 s−1 . 6.2.2. Quantum state specific detection A large fraction of the experiments related to the inelastic scattering of molecules use some type of optical spectroscopy to determine the internal state distribution of the scattered molecules. The two most common techniques are (a) laser induced fluorescence (LIF, sometimes more accurately termed laser fluorescence excitation, or LFE) and (b) resonance enhanced multiphoton ionization (REMPI). These are illustrated in Fig. 6.4 as applied to the nitric oxide molecule. Both of these are well suited to the low number density typical in a molecular beam experiment (roughly 105 –108 /cm3 per quantum state). The REMPI technique has evolved as the technique of choice since suitable transitions have been identified and characterized for a number of molecules of interest (NO, N2 , CO, NH3 , etc). In addition the ions resulting from the absorption are easier to collect and detect than are fluorescent photons. The wavelength of the radiation required to detect stable molecules like those mentioned is usually in the ultraviolet region of the spectrum. These wavelengths are routinely generated through nonlinear frequency conversion (doubling and tripling) from tunable lasers operating at visible wavelengths. The high intensity required for efficient nonlinear optics necessitates the use of pulsed lasers. In addition, REMPI processes are intrinsically nonlinear and require high intensity light sources. The use of pulsed lasers is often coupled with application of pulsed molecular beam sources to provide high number density during the limited duty cycle of the probing laser. One advantage of the pulsed-beam/pulsed-laser

204

G.O. Sitz

Fig. 6.4. Laser induced fluorescence and 1 + 1 resonance enhanced multiphoton ionization in nitric oxide. Note that the ground state of NO is actually split by the spin orbit interaction into two electronic states, the 2 1/2 and 2 3/2 states separated by 15 meV.

combination is that translational energy measurements are readily carried out by simply varying the delay between the beam pulse and the detection probe. A few experiments have been done in which direct absorption of infrared radiation is used to determine internal state distributions of surface scattered molecules (Francisco et al., 1996; Bronnikov et al., 1996). Either the attenuation of the laser is measured directly or a bolometer is used to detect the increased internal energy of the molecules which have absorbed IR radiation. The primary focus of these developments has been on detection of methane, since no excited states suitable for LIF or REMPI schemes are available for this molecule. For measurements of angular and velocity distributions where internal state resolution is not required, electron impact quadrupole mass spectrometry (QMS) is routinely employed. Not much will be said here about this topic, since QMS is a standard tool in much of surface science. 6.3. Diatomic molecules 6.3.1. Translation The velocity distribution of a sample of molecules can be measured in a non-state-resolved manner using a quadrupole mass spectrometer and a molecular beam which is either pulsed


205

or modulated with a mechanical chopper. An early survey of a number of atomic and molecular species scattered from an epitaxial Ag(111) film was performed by Asada and Matsui (1982). They chose a particular set of conditions (Tsource = 300 K, Ts = 500 K, and θi = 50◦ ) where the incident normal kinetic energy (En = Ei cos2 θ) was quite close to the one-dimensional value of the average surface atom kinetic energy kB Ts /2.1 This resulted in the average kinetic energy of the molecules scattered at the specular angle being nearly equal to the incident kinetic energy, and made the trends in scattered energy as a function of scattered angle easy to see. What was found was that molecules scattered at angles greater than specular (i.e., leaving the surface at more grazing angles), lost kinetic energy, whereas those scattered closer to the normal gained kinetic energy. This is just what is expected if the interaction is predominately one-dimensional. If energy exchange is with the motion normal to the surface and energy parallel to the surface is conserved then molecules that gain energy exit closer to the normal than specular and vice versa. This trend was found to greater or lesser extent for all molecules studied (N2 , O2 , NO, CO, CO2 and CH4 ); it was suggested that differences in the results reflected differences in coupling to rotation. For detection schemes with internal state resolution, the scattered velocity distributions are often strongly coupled with the scattered rotational and/or vibrational state. These situations will be discussion in the subsequent sections according to the internal degree-offreedom measured. 6.3.2. Rotation The rotational state of a molecule scattered from a surface reflects the anisotropy of the interaction potential between the two. In general, variation of a potential with a coordinate is called a force; it case of rotation, the coordinate is an angle (say between the surface normal and a molecular axis), and results in a torque. Measurements of the rotational state distribution thus provide insight into the variation of the interaction potential with the orientation of the molecular framework. These distributions are the most interesting and carry the most dynamical information when they are non-thermal; after all, if the distribution is characterized by the surface temperature, a one dollar thermocouple is just as informative as a spectroscopic measurement utilizing a quarter million dollar laser system. All beam–surface scattering experiments involving diatomic molecules heavier than D2 have shown that transfer of energy into the rotational degree-of-freedom is facile. For direct-inelastic scattering, this is a mostly a result of the fact that the typical ratio between the molecule’s incident translational energy and the its rotational constant is more than 100:1. For trapping-desorption the ratio of the ‘surface energy’, kTs to the rotational constant is similarly large. An example of the incident and scattered rotationally-resolved REMPI spectrum is shown in Fig. 6.5 for the direct-inelastic scattering of N2 from Ag(111) (Sitz et al., 1988a). The supersonic molecular beam produces a rotational distribution at a temperature of a few Kelvin (top panel); the scattered distribution shows a broad range of excited rotational states (bottom panel). The even-odd alternation in the line intensities is a result of the 2:1 nuclear spin degeneracy of the rotational states of 14 N2 . 1 The term “normal energy” is a poor one, but has become common. Obviously, energy being a scaler quantity does not have components. What is meant is: the energy associated with the normal component of the momentum. This is quite a mouthful and I will succumb to the normal usage with this caveat.

206

G.O. Sitz

Fig. 6.5. REMPI spectra of N2 incident on and scattered from Ag(111) showing rotational excitation (Sitz et al., 1988a). The conditions were Ei = 0.3 eV, Ts = 90 K, θi = −15◦ , and θf = 20◦ .

An informative way to reduce and present data of the type shown in Fig. 6.5 is in the form of a so-called Boltzmann plot. This a plot of the natural log of the population of a given rotational state scaled by its degeneracy versus the energy of that state. If the distribution of rotational state populations is characterized by a temperature, the points on this plot will fall on a line. The inverse of the slope of the line will be proportional to the rotational temperature. A linear Boltzmann plot is common, although the temperature is frequently not equal to the surface temperature. A Boltzmann plot for the N2 data shown in Fig. 6.5 is shown in Fig. 6.6. Included on this plot are results for incident beam energies other than that shown in Fig. 6.5. A more detailed discussion of these results will be given below in Section 6.3.2.2. The data shown in Fig. 6.6 deviate from a straight line, particularly at higher incident energy. Specifically, an excess of population is seen at high values of the rotational quantum number J . This effect was first seen for NO scattered from Ag(111) and is interpreted as a rotational rainbow (Kleyn et al., 1981). In the simplest picture, the molecule can be considered to be an ellipsoid and the surface as being flat. For an impulsive collision the energy transferred into rotational will be a function of the angle θ that the internuclear axis of the molecule makes with the surface normal. At 0 and 90 degrees the energy transferred to rotation will be zero and will rise to a maximum at an intermediate angle. Near the angle corresponding to the maximum final rotation, there will be a range of angles resulting in nearly the same amount of rotation. This leads to pileup in the probability of excitation to this level of rotation and an abrupt falloff for higher values. This phenomena is similar


207

Fig. 6.6. Boltzmann plots for the rotational state distribution for N2 scattered from Ag(111) at Ts = 90 K, for 4 different incident translational energies (Sitz et al., 1988a).

to classical rainbow scattering and the picture has been adopted for rotationally inelastic scattering in a surface context (Kleyn and Horn, 1991). The physical picture is one in which the angle that the molecular axis makes with the surface normal at the time of a quasiimpulsive collision (plus the incident momentum) determines the final rotational state of the scattered molecule. This section is not intended to be a comprehensive review of rotationally inelastic scattering of diatomic molecules from surfaces, but rather to provide an overview of the kinds of phenomena which have been observed and to provide some insight into what factors govern the scattering. Enough detailed studies of rotationally inelastic scattering have been done for two diatomic molecules (NO and N2 ) that some general trends have been established; the scattering of these two will be discussed in some detail, with some remarks about other diatomics added where appropriate. 6.3.2.1. Nitric oxide The first examples of a spectroscopic measurement of the non-thermal rotational state distribution for a molecule after scattering from a surface were for NO scattered from a graphitic layer on Pt(111) (Frenkel et al., 1981) and from Ag(111) (McClelland et al., 1981; Kleyn et al., 1981). The NO molecule was selected largely for practical reasons: the wave-

208

G.O. Sitz

lengths necessary to interrogate the rotational state distribution by laser induced fluorescence were accessible (tuneable light near 225 nm), and there were essentially no other stable molecules which were compatible with UHV for which this was the case. These early experiments showed that the NO rotational state distribution was not fully accommodated to the surface and was dependent on the initial kinetic energy and, to a lesser extent, on the surface temperature. This was also the system in which rotational rainbows were first observed (Kleyn et al., 1981). The scattering was described as direct inelastic since the final rotational state distribution depended on the incident translational energy. A detailed study of the dependence of the final rotational state distribution as a function of incident translational energy and surface temperature was performed by the Zare group (McClelland et al., 1981). The average rotational energy of the scattered molecules (Er ) was fit to an equation linear in the incident normal energy (En ) and surface temperature (Es ):

Er = a En + Ew + bEs . (6.1) The term Ew was interpreted as a measure of the laterally averaged well depth of the NO/Ag interaction potential. The effect is to add this energy to the incident energy. The coefficient a is then a measure of the efficacy of energy transfer into rotation; the fitted value was a = 0.088 for the = 1/2 fine structure state and a = 0.132 for = 3/2. The fitted well depths were also slightly different for the two states, but was roughly 0.25 eV. The interaction of NO with metal surfaces is often characterized by a relatively strong interaction (the noble metals being somewhat weaker). This results in a large initial sticking coefficient (for example, 0.65–0.90 for Pt) and a substantial contribution from a trapping/desorption channel. Segner et al. (1983) and Jacobs et al. (1989) investigated this phenomena in detail in the NO/Pt(111) system where the well depth is approximately 1.4 eV. Segner et al. found that the scattered rotational state distribution was always well fit by a Boltzmann distribution. At low surface temperature, Tr was equal to Ts , but at higher Ts , Tr was substantially lower. This is shown in Fig. 6.7. A key point to note is that the scattered rotational temperature does not depend on the incident translational energy, a clear indication of scattering dominated by trapping/desorption. As noted earlier, this shows that trapping this does not mean that desorption will yield distributions characterized by Ts . The detailed interpretation of this data is complicated by the fact that at Ts < 300 K the surface is at least partially covered by adsorbed NO, whereas at higher Ts (> 400 K) the surface is clean. Jacobs et al. employed a pulsed supersonic beam (Segner et al. had used a continuous source) which enabled them to distinguish between the dominant trapping/desorption channel and a minority direct-inelastic channel. This was achieved by varying the time that the scattered molecules were probed relative to the incident beam pulse. By working at relatively low Ts and thereby long surface residence times, the flux from the TD channel could be stretched out over a long period of time and contribute only a minor amount to the measured signal. The direct inelastic channel was found to be non-Boltzmann and to contain scattered molecules with rotational energy well in excess of the incident translational energy, indicating that substantial transfer from surface phonons in to molecule rotation. This energy transfer pathway is opened up by the acceleration of the incident molecule as it passes over the deep attractive well of the potential. The large energy transfer may also


209

Fig. 6.7. Rotational temperature versus surface temperature for NO scattered from Pt(111). The open symbols are for an incident energy of Ein = 80 meV; the filled are for Ein = 210 meV. Data taken from Segner et al. (1983).

involve a process in which the molecule bounces multiple times in the potential well before escaping. The scattered NO was also found to have its rotational angular momentum vector aligned; this will be discussed below in Section 6.3.2.3. While the interaction of NO with platinum is strong, the NO adsorbs and desorbs molecularly, that is the platinum surface is not active for dissociation of the molecule. A second example of a strongly interacting surface and one which is active for dissociation is Ru(0001). Below 200 K, NO adsorbs molecularly, with a binding energy of about 1.5 eV; above this temperature dissociation into Nads plus Oads occurs with a sticking coefficient of 0.7–0.8 (Butler et al., 1997). The reaction probability and scattering dynamics are greatly altered by the presence of atomic hydrogen on the surface. The reaction probability is reduced to 0.2–0.3, presumably by the blocking of active sites. The angular distribution from the clean surface is a broad (≈37◦ FWHM) quasi-specular lobe, typical of NO scattering from a number of surfaces. Remember, this is the distribution of the molecules which do not dissociate. For scattering from the Ru(0001)-(1 × 1)H surface, the angular distribution narrows dramatically (to ≈8◦ FWHM) with a correspondingly large increase in the peak intensity. In addition, at the specular angle, the scattering is essentially elastic with the final translational energy nearly equal to the incident energy (Ef /Ei ≈ 0.9), even for incident energies up to 2.1 eV. Furthermore, and equally puzzling, the scattered rotational state distributions look fairly similar to those found for scattering from the inert Ag(111) surface where the angular distributions are much broader and where much more energy is lost to the surface (Berenbak et al., 2001). No quantitative model of the changes in the scattering produced by the adsorption of hydrogen in this system has been proposed. A molecule with both a permanent electric dipole moment and a nonzero angular momentum projection onto its internuclear axis can be oriented in an inhomogeneous electric field (Bernstein, 1982). Here orientation means a preferred direction of the internuclear axis in a space-fixed reference frame. A prototypical example is again found in the NO molecule with its 2 electronic ground state. This represents a rare opportunity to further study the angular anisotropy in the molecule–surface interaction poten-

210

G.O. Sitz

tial. A pronounced effect was found for NO scattered from Ag(111) (Tenner et al., 1990; Geuzebroeak et al., 1991). It was found that ‘O’-end collisions lead to larger rotational excitation and rotational rainbows whereas ‘N’-end collisions resulted in lower rotational excitation and no rainbows. Since the minimum energy configuration for NO bound on metal surfaces is with the ‘N’-end down, the scattered measurements were interpreted as showing the additional torque on the molecule when it approaches with an unfavorable, high potential energy geometry. Angular distributions also showed the effects of initial orientation for NO (Kuipers et al., 1989) and for several polyatomics (CF3 H and related symmetric tops). The primary cause of these differences is most likely differences in the effective depth of the molecule–surface potential well for different molecular orientations. Additional details can be found in references (Mackay et al., 1989) and (Sitz, 2002). Behavior surprisingly similar to that observed in the NO/Pt(111) system has been seen for NO scattered from germanium (Modl et al., 1985, 1986). At low incident energy (Ei = 100 meV), the scattering is trapping-desorption: the scattered angular distribution varied as cos θ , and the scattered intensity was consistent with a thermally equilibrated velocity distribution. The rotational distribution was Boltzmann and varied with the surface temperature. This variation is almost exactly that found for NO/Pt(111) (shown in Fig. 6.7): Tr = Ts for Ts 300 K, and Tr becomes independent of Ts reaching a limiting value of Tr ≈ 400 K at high Ts . This behavior was the same for incident energies of 100 and 225 meV, just as shown in Fig. 6.7 for Pt. The remarkable aspect of this comparison is that the well depth for NO on Pt is about 1.4 eV, whereas for NO on germanium it is less than 0.25 eV, yet the rotational distributions are nearly the same even when they are clearly not equilibrated at Ts . As the incident energy was increased in the range of 730 to 820 meV, a specular lobe appeared in the angular distribution indicating a growing direct-inelastic channel. This was confirmed by time resolved measurements which clearly showed two channels corresponding to TD and DI scattering. The ratio of scattering into the two channels depended on Ts and on final angle. At the higher incident energies, the rotational distribution was non-Boltzmann with an excess of population at higher J ’s, independent of Ts . It is worth noting that at surface temperatures below 600 K, the germanium surface is oxidized by decomposition of NO2 present as an impurity in the NO. At Ts > 700 K, this GeO desorption is rapid enough that the surface is essentially clean germanium. No differences in the scattering were reported that could be associated with the different surface compositions. 6.3.2.2. Nitrogen and carbon monoxide Interpretation of NO scattering data faces problems arising from the nature and strength of the interaction of the molecule with metal surfaces, and an alternative, simpler system was sought. Experiments with N2 became viable when two sensitive REMPI schemes were developed. In early work a 2 + 2 REMPI process through the a 1 g intermediate state was utilized for a detailed series of experiments in the N2 /Ag(111) system (Sitz et al., 1988a, 1988b; Kummel et al., 1988, 1989). Analysis using this spectroscopic scheme allowed for (and, in fact, required) a detailed determination of the angular momentum polarization of the scattered molecules. This will be discussed in Section 6.3.2.3. Later, a 2 + 1 scheme + through the a 1 g intermediate state was developed (Lykke and Kay, 1989, 1991b); this pathway provided nearly 100 times more sensitive detection and yielded a direct map of


211

the rotational state distribution without the necessity of a polarization analysis, although such an analysis was still possible (Hanisco et al., 1992b). The scattering of N2 from Ag(111) has become something of a paradigm for direct inelastic scattering. The inert character of N2 , the small well depth and the small mass of N2 relative to a silver atom combine to mean that trapping is improbable. The strong dependence of the scattered rotational state distribution on the incident energy shown in Fig. 6.6 is evidence of a direct, impulsive interaction. In addition, the rotational distribution varies dramatically with final scattering angle (Sitz et al., 1988a). For an incident angle of 30◦ , molecules scattered at angles greater than specular show a large population excess at high J -states, while those scattered at subspecular angles show a much smaller population at high J . This can be understood if the rotational energy comes out of the normal component of the incident momentum and the parallel momentum is conserved. Then scattered molecules exciting in high J -states will have used up more of their normal momentum and leave the surface at a more grazing angle. This picture also indicates that the N2 sees a relatively flat, uncorrugated surface. Measurements of the angular momentum polarization of the scattered molecules that support this picture will be discussed below in Section 6.3.2.3. Pronounced rotational rainbows and sensitive dependence of Er on final angle were also found for N2 scattered from Au(111) (Lykke and Kay, 1991a). Equation (6.1) was found to the describe the dependence of the average rotational energy on incident energy and surface temperature for NO scattered from Ag(111). The strong variation of the scattered rotational distribution with exit angle for N2 scattering from Ag and Au makes application of this equation difficult. However, a similar analysis was done for the N2 /Au(111) system using the rotational energy at the peak of the rotational rainbow (Epeak ) in place of the average. A linear dependence of the form Epeak = a(En + Ew )

(6.2)

where a is a measure of the efficacy of T → R coupling and Ew is the depth of the attractive well. For N2 /Au(111), results of a = 0.28 and Ew = 49 meV were obtained. The effect of surface atom mass can is evident when results for N2 scattered from Ag and Au are compared with results for Cu (Siders and Sitz, 1994). For a copper surface temperature of 85 K, time-of-flight spectra clearly showed both direct-inelastic and trapping desorption channels. Trapping is significant for scattering in this case because of the closer match between the N2 and copper atom masses. The TD channel showed a rotational temperature within a few degrees of Ts . The rotational temperature for the DI channel depended on the incident energy is a manner given by Eq. (6.1). Fitted values were 0.1 eV for the well depth and a = 0.07 for the efficacy of T → R energy transfer. The well depth is small and comparable to that found for Ag and Au, therefore the difference in the scattering dynamics is attributed to the relative mass ratios and corresponding efficiency of energy transfer. The value of the b from Eq. (6.1) was large and found to vary with Ei ; this was suggested to be the result of scattering in between TD and DI in which the molecule bounced multiple times in the gas–surface potential well but did not lose complete memory of its initial conditions. An additional dynamical feature can be found when the correlation between final translational (T) and rotational (R) energy is examined. An example of this is shown in Fig. 6.8, for N2 scattered from W(110) (Hanisco and Kummel, 1993b) and Ag(111) (Sitz et al.,

212

G.O. Sitz

Fig. 6.8. Energy distributions as a function of rotational energy for N2 scattered from Ag(111) (circles, Ei = 0.3 eV and Ts = 90 K) and W(110) (diamonds, Ei = 0.5 eV and Ts = 1200 K). The filled symbols are scattered translational energies and the open symbols are translational plus rotational energies. Ag data from Sitz et al. (1988a) and W data from Hanisco and Kummel (1993b).

1988a). This was first investigated for the NO/Ag(111) system (Kimman et al., 1986; Rettner et al., 1987a) and also observed for N2 /Cu(110). The linear decrease in final translational energy with increasing scattered rotational energy is expected for T → R energy transfer. The reason for the increase in final total energy (T + R) is more subtle. The angle between the molecules internuclear axis and the surface normal at impact largely determines the final rotational state. To scatter into the highest J -states, this angle is close to 45◦ for a homonuclear diatomic. At these angles, the mass of the molecule is effectively smaller with regards its impact with the surface, and it is this effective mass which governs the overall energy loss to the surface. At 0◦ and 90◦ the effective mass is the full mass of the molecule, while at 45◦ only one atom is effectively interacting with the surface. Figure 6.8 illustrates another characteristic typical of molecule–surface scattering at energies of a few tenths of an electron volt: about 25% of the incident energy is dissipated to the surface. For the scattering of N2 from the inert Ag(111) surface, the anisotropy leading to rotational excitation is predominately in the repulsive part of the potential. It is interesting to ask if anisotropy in the attractive part of the potential can manifest itself in the rotational state distribution. This anisotropy can take one of two forms. First, for a molecule like NO or CO, binding to the surface usually occurs with the O-end of the molecule up. This implies an anisotropy in the potential as a function of the orientation of the molecule as discussed in Section 6.3.2.1 on experiments with oriented NO. The second possibility is in a reactive system, with the reaction being dissociative adsorption of the incident diatomic molecule. A common characteristic of potential energy surfaces calculated for reactive systems is a strong preference in the reaction pathway for a molecule with its internuclear axis


213

parallel to the surface (Darling and Holloway, 1995). This should result in a measurable influence on the final rotational state of molecules which do not react but scatter from a surface where reactive channels are open. The W(110) surface is one of the few surfaces that is active for dissociation of the very strong bond in N2 ; for this reason, a comprehensive study of inelastic scattering is available for this system (Hanisco and Kummel, 1993b). The scattering was found to be direct inelastic based on strong variations of scattered distributions with incident energy. A strong correlation between total final energy and final rotational energy was also clear and further supports the DI character of the scattering. Energy transfer into rotation was relatively large (roughly 9%), but no rotational rainbows were observed under any conditions. The final rotational state distributions were Boltzmann, and the temperature was independent of incident or final angle. This latter observation was taken to mean that the N2 –W(110) interaction potential is very corrugated. This conclusion is supported by measurements of the angular momentum polarization discussed below. The contribution of the reactive character of the N2 /W(110) interaction potential to the rotationally inelastic scattering could be studied by comparison with scattering from a W(110) surface passivated by adsorption of hydrogen or nitrogen (Hanisco and Kummel, 1993a). For the passivated surfaces, energy transfer decreased by roughly 15%. In addition, for hydrogen passivation (but not for nitrogen), the corrugation is greatly reduced as evidenced by the large variation of the rotational state distributions with final angle. A much more dramatic variation in rotationally inelastic scattering with adsorbate coverage has been observed for N2 scattered from clean and hydrogen covered Pd(111) (Lykke and Kay, 1989). For both surfaces, the scattering was direct-inelastic, but the peak intensity at the specular angle was more than an order of magnitude smaller and the rotational state distribution was much warmer for scattering from the clean surface. Measurements made by varying the surface temperature and monitoring the hydrogen coverage, showed that the scattered intensity started to change before any hydrogen had desorbed. This may indicate transfer of hydrogen between surface and subsurface sites, but this is not understood in detail. It is worth noting that, in contrast to the NO/Ru system discussed in Section 6.3.2.1, palladium is not active for the dissociation of N2 , so closing of reactive channels in the PES by adsorption of hydrogen is not responsible for the changes in the scattering dynamics. Additional insight into the effect of anisotropy in the attractive part of the potential can be obtained by comparing scattering of CO with that of N2 . CO is known to bind in an upright geometry on many metal surfaces with the carbon atom down. In a scattering context, the permanent dipole moment of CO leads to an increased variation of the potential with the angle of the internuclear axis. This will effect the amount of rotational inelastic scattering, however the deeper well depth for CO versus N2 will have a pronounced influence as well. A comparison of CO and N2 scattered from Ag(111) under identical conditions has been reported by Kummel and coworkers (Hanisco et al., 1992a). Similar to N2 , the CO rotational state distributions depend on final angle and shown pronounced rainbows at more glancing exit angles. The CO rotational rainbows are brightest when the CO is incident normally, compared to N2 where they are more pronounced for more grazing angles of incidence. This most likely results from a more corrugated potential for CO. The rotational state distributions for both CO and N2 showed a peak at low J as well as the high J rainbow peak, particularly at normal incidence. This peak reflects the fraction of the large

214

G.O. Sitz

population of low J -states in the incident beam that collide with the surface at orientation angles near 90◦ , and therefore do not undergo much rotational excitation. The survival of this feature for CO indicates the attractive anisotropy in the potential is not large enough to torque the molecules out of this orientation prior to the collision (at least at the fairly large incident energy of 0.75 eV). The deeper well depth accelerates the CO just prior to collision and this results in an overall increase in the energy transfer to the surface as well as into rotation. The net result is that the CO leaves the surface rotationally hotter but with less translational energy compared to N2 for equal incident energies. The J -state-resolved translational energy distribution is also broader for CO than for N2 . While the well depth for CO on silver is greater than that for N2 (roughly 0.25 eV versus 0.1 eV), it is still a relatively weak interaction. Inelastic scattering of CO has been studied for scattering from Ni(111), where the well depth is considerably larger, about 1.3 eV (Hines and Zare, 1993). For the range of incident energies studied (0.26–0.45 eV), roughly 50% of the incident flux was reflected directly and the remainder trapped. By using a pulsed molecular beam and working at surface temperatures above 675 K, trapped CO was completely desorbed between beam pulses, and the surface kept clean. Directly scattered CO was observed with rotational energy up to incident beam energy. This is possible because of the acceleration of the incoming molecule by the deep attractive well. A mechanism of rotational trapping was postulated by which an amount of energy exceeding the incident translational energy was transferred into rotation resulting in a molecule with insufficient energy to escape the well, and leading to a cutoff in the scattered rotational distribution. A weak rotational rainbow was seen as well and was interpreted as arising from the weakly interacting O-end of the molecule. In theory, a heteronuclear molecule like CO could show two rainbows resulting from the difference between the two ends of molecule. It was suggested that the rainbow associated with the strongly interacting C-end of the molecule was beyond the rotational cutoff and thus was in unobserved since those molecules were trapped. In all of studies described so far, and the overwhelming majority of those reported to date, the mass of the incoming molecule has been much less than the mass of a surface atom. This has lead to relatively inefficient energy transfer for kinematic reasons, and to the observation of direct-inelastic scattering in numerous systems. One system in which this is not true is the scattering of N2 from ice (Gotthold and Sitz, 1998). While the binding energy of N2 on ice is small (thermal desorption occurs at a temperature of around 43 K), the mass ratio means that energy transfer in the initial collision is efficient. For a N2 beam with Ei = 90 meV, this leads to a trapping probability of near unity, and to N2 desorbing from the ice in rotational and translational equilibrium at Ts . At higher incident translational energies (350 and 750 meV), a small direct-inelastic channel could be observed by using a pulsed molecular beam and timing the probe laser to detect the earliest arriving scattered molecules. These molecules had a rotational temperature well in excess of the surface temperature that depended on the incident energy and not on Ts . Remarkably, the molecules in the DI channel had lost more than 90% of their incident translational energy to the ice surface.


215

6.3.2.3. Angular momentum polarization The previous section clearly demonstrated that a wealth of information on the dynamics of gas–surface interactions can be obtained from studies of rotationally inelastic scattering. Additional insights are available from measurements which are sensitive to the spatial distribution of direction of the angular momentum vector, J. The forces acting on an initially rotationally cold molecule to produce a change in J are obviously anisotropic, and since these forces are responsible for the torque causing the molecule to change its rotational state, anisotropy in the resulting direction of J is clearly expected. Since the force on the molecule is predominately directed along the surface normal, a preference for J lying in the plane of the surface is expected. Quantum mechanically, angular momentum polarization is described as an unequal population of the (2J +1)-mJ states associated with a given J -state. This has traditionally been cast in terms of the multipole moments of the distribution, where the probability of being in the state (J, mJ ) is given by

{k} {k} P (J, mJ ) = n(J ) (6.3) ξ(k)Aq± (J ) J, mJ |Jq± (J )|J, mJ /Jk k,q

where n(J ) is the population of the state J , ξ(k) is normalization constant, and the term {k} (J, mJ |Jq± (J )|J, mJ )/Jk at high J is proportional to the familiar spherical harmonic {k}

Yq± (θ, φ). Essentially, an individual term in this expansion specifies how much a given spatial distribution resembles the corresponding spherical harmonic. The form of this expansion is natural since it connects readily with experiments using optical absorption with polarized light. For example, for a single photon LIF measurement with linearly polarized excitation where the fluorescence is spectrally unresolved and is detected independent of its polarization, only two moments contribute to the result: the k = 0, q = 0 or monopole, and the k = 2, q = 0 or quadrupole (assuming that the spatial distribution of J is cylindrically symmetric about the surface normal). The quadrupole moment is effectively a measure of the expectation value of cos2 θ , where θ is the angle between J and the surface normal. The first system for which rotational polarization was observed was again NO scattered form Ag(111) (Luntz et al., 1982; Kleyn et al., 1985) and utilized single photon LIF. An example of the polarization in this system is shown in Fig. 6.9. For the normalization used, the limiting values for the quadrupole moment (in the high J limit), are b2 /b0 = 5 for J nˆ and b2 /b0 = −2.5 for J ⊥ n, ˆ where nˆ is the surface normal. (This notation is {2} different than that introduced in Eq. (6.3): b2 corresponds to A0 .) The results indicate a large polarization at intermediate J , with J aligned predominately perpendicular to n, ˆ that is parallel to the surface plane. The small alignment at low J may be a remnant of unpolarized molecules in the incident beam. The decrease in alignment at the highest J values is not entirely understood, but may be correlated with the final scattering angle. As discussed in Section 6.3.2.1, both trapping/desorption and direct-inelastic scattering were observed in the NO/Pt(111) system. In addition to measuring rotational state distributions, Jacobs et al. were able to determine the alignment for the scattered NO (Jacobs et al., 1989). For the DI channel, the results were qualitatively very similar what was found for DI scattering of NO from Ag(111): no alignment at the lowest J ’s, an increasing negative

216

G.O. Sitz

Fig. 6.9. Angular momentum polarization for NO scattered from Ag(111). Plotted is the quadrupole moment normalized to the monopole moment as a function of the scattered rotational energy. The data are for an incident angle of 15◦ , En = 0.44 eV, and Ts = 650 K. Data taken from Kleyn et al. (1985).

quadrupole moment for intermediate J (meaning J ⊥ n), ˆ and then a decrease to near zero alignment for the highest values of J . The largest alignment at intermediate J was about half the maximum possible. For the trapping desorption channel, the NO was aligned preferentially with J n. ˆ The alignment was again zero at low J , and increased steadily with J to a maximum of {2} about A0 = 0.15 at the highest J . The sign of the quadrupole moment means that the molecule is preferentially rotating in a plane parallel to the plane of the surface. This was a somewhat surprising result in view of the fact that it is known that the absorption geometry has the NO bond axis perpendicular to the surface plane. It seems counterintuitive that this configuration would lead to desorbed NO molecules rotating in a plane parallel to the surface. However several interesting points concerning the dynamics of the desorption were advanced to explain the observations. First, the desorption most likely occurs from a weakly bound state near in energy to the top of the molecule–surface potential well, and not from the most stable state in the bottom of the well. The molecule is thought to be a nearly free rotor in this final, pre-desorption, state. Next, a plausible mechanism for the final kick to escape the well is a rotation to translation energy exchange within the molecule. Such a process would favor loss of rotational energy from the component of J parallel to the surface; the perpendicular component of J would be expected to more weakly coupled to translation. This is essentially the flip side of the argument put forth to explain the alignment of J ⊥ nˆ found in the direct inelastic scattering of NO from Ag(111). A considerably more detailed study of angular momentum polarization is available for N2 scattered from Ag(111) (Sitz et al., 1988a, 1988b; Hanisco et al., 1992a) and from clean and passivated W(110) (Hanisco and Kummel, 1993a). The detection scheme in this case + was 2 + 2 REMPI (via either the a 1 g or a 1 g intermediate state). The fact than the ini-


217

tial absorption step involved 2 photons meant that polarization moments up through rank 4 could be measured. For measurements employing linearly polarized light, this means a {4} hexadecapole alignment moment (A0 , rough proportionally to cos4 θ) could be measured as well as the quadrupole and monopole moments. The angular momentum of the scattered N2 scattered from Ag(111) showed a high degree of alignment for intermediate and high values of J , with J ⊥ n. ˆ In some cases {2} {4} the alignment was close to perfect (A0 = −1.0, A0 = +0.375). The alignment was relatively insensitive to the incident energy, incident angle or surface temperature. Interestingly, no deviation from large alignment was found for the highest J -values, as was the case for the NO/Ag(111) system. This system fits neatly into the picture that the scattering is dominated by forces normal to the surface and these forces lead to rotation strongly polarized with J ⊥ n. ˆ In contrast, for N2 scattered from clean or passivated tungsten the alignment was not so readily interpreted. At low incident energy (Ei = 0.25 eV), for the clean W(110) and W(110)-(2 × 2)N surfaces little or no alignment was observed for low and intermediate values of J with a small (for W(110)-(2 × 2)N) to modest amount (for W(110)) of negative alignment at the highest values of J . The qualitative conclusion based these observations is that the interaction potential is highly corrugated resulting in strong in-plane forces. The range of direction of the forces means the corresponding torque also lacks strong directionality and the final angular momentum is not strongly polarized. {2} N2 scattered from W(110)-(1 × 1)H showed positive alignment peaking at A0 = +0.5 {2} for J = 10 and negative alignment peaking at A0 = −0.56 around J = 26. The alignment for scattering from W(110)-(1 × 1)H was strongly dependent on the incident energy. {2} At the highest energy studied (Ei = 1.0 eV), A0 was positive across for all values of J . The final translational and rotational distributions were still dependent on the incident energy, so the scattering was not trapping-desorption (which lead to positive alignment in the NO/Pt(111) system discussed above). No detailed modelling has been done for the N2 /W systems (as has been done for N2 /Ag), and a comprehensive understanding of the alignment is lacking. In addition to measurements with linearly polarized light, for the N2 /Ag system experiments were also done with circularly polarized light; these measurements were sensitive {1} {3} to the dipole (A1− ) and octapole (A1− ) moments of the distribution. These odd moments quantify the orientation of the angular momentum vector. A net orientation represents a difference in population for magnetic sublevels +mJ and −mJ , that is a net helicity of {k} the rotation (this idea is illustrated in Fig. 6.10). The notation in the symbol Aq with q = −1 means that direction of the quantization axis is perpendicular to the scattering plane. Nonzero values of the orientation moments indicate a breakdown of cylindrical symmetry. An example of angular momentum orientation is shown in Fig. 6.11 for N2 scattered from Ag(111). The N2 was incident along the surface normal for this data set, but the scattered flux was probed at an angle 15◦ off normal. For a final scattering angle along {1} the surface normal, symmetry requires that A1− = 0.0. The orientation was found to increase as final scattering angles further from the surface normal were probed. A classical trajectory model was able to reproduce both the dependence on θf and final J ; the calcu-

218

G.O. Sitz

Fig. 6.10. Conceptual illustration of angular momentum orientation for N2 scattered from Ag(111). The probe laser propagates into the plane of the paper and is circularly or elliptically polarized.

{1}

Fig. 6.11. Dipole orientation moment, A1− , as a function of rotational state for N2 scattered from Ag(111) at Ei = 0.3 eV, Ts = 90 K, θi = 0◦ and θf = +15◦ . Filled symbols are the measurements and the open symbols are the theory. Data is taken from Kummel et al. (1988).

lated results are also shown in Fig. 6.11. The conclusion from analysis of the trajectories was that in-plane forces arising from corrugation of the potential were responsible for the orientation and its variation with angle and J -state. For non-normal angle of incidence, the orientation at the specular angle was qualita{1} tively similar to the result shown in Fig. 6.11: A1− was largest and negative for the largest value of J , switched sign for intermediate J ’s, and gradually decreased to zero for the lowest values of J (Sitz et al., 1988b). For final angles off of specular, the orientation showed a complicated dependence on the final scattering angle as well on the value of J . For su{1} perspecular scattering (θf > θi ), A1− actually showed a minus-plus-minus oscillation as a function of J .


219

{1}

The observed variation of A1− with final angle and J could be qualitatively reproduced with a simple hard-cube, hard ellipsoid model with an added in-plane friction. The in-plane friction lead to a splitting of the rotational rainbow into two: one associated with +θ and the other with −θ where θ is the angle between the rotor axis and the surface normal at the collision. The final J ’s associated with ±θ have opposite orientation, but the maximum value of J (i.e., the value of J at the rainbow) for the two angles is different because of the {1} angle of incidence. This leads to an oscillation of A1− with J . In the preceding discussion of the alignment moments for N2 it was assumed that the angular momentum spatial distribution was cylindrically symmetric about the surface nor{2} mal. Thus, for example, the quadrupole moment (A0 ) was the real quadrupole moment of the distribution. In fact, the measurements yield what are termed apparent moments which, in general, can have contributions from a number of terms in Eq. (6.3) (Kummel et al., 1986). It is clear from the observation of odd rank moments (see Fig. 6.11) that the angular momentum distribution does not, in general, have cylindrical symmetry. However, it is likely that the contribution of non-cylindrically symmetric moments to the alignment moments is small (see the Appendix in Sitz et al. (1988a)), and I have tried to avoid confusion by not stressing this distinction in what is essentially a qualitative discussion. 6.3.3. Vibration In contrast to collisions which change the rotational state of a molecule, vibrational state changing collisions for diatomic molecules at a surface are generally much less probable. This result is well known for gas-phase dynamics, and the physical explanation given is often much the same: the average collision energy is small compared to the typical vibrational level spacing. However, in molecular beam–surface scattering experiments, the incident beam energy can easily be a factor of 5 or more times as large as the vibrational quantum, and vibrational excitation probabilities are still often small. In addition, a number of experiments have addressed the process of the relaxation of an initially vibrationally excited molecule, and the probabilities are again found to be small. The explanation then has to address the question of the mechanism for vibrational energy transfer. Three mechanisms have been explored for vibrationally inelastic scattering at surfaces: (1) a direct, mechanical process, essentially pictured as coupled, ball-and-spring oscillators; (2) an electronically non-adiabatic process in which some electron transfer occurs from the surface to the molecule. This extra charge on the molecule drives a change in the bond length and therefore provides a coupling to vibrational motion; (3) curvature in the molecule–surface interaction potential resulting from the presence of an open channel to dissociative adsorption. The applicability of each of these processes will be discussed below for specific cases, but generally simple mechanical coupling can be ruled out for direct inelastic scattering of diatomics because of the large mismatch between the diatomic oscillation frequency and the phonon spectrum of the surface. This is not the case for polyatomic molecules which may have bending modes at low frequency, see the discussion about NH3 in Section 6.4.2.

220

G.O. Sitz

6.3.3.1. Vibrational survival and relaxation A conceptually straightforward way to study vibrationally inelastic scattering at surfaces is to prepare a sample of molecules incident on a surface in a vibrationally excited state, and then to monitor the fate of those molecules. Such studies employing molecular beams and infrared excitation have been performed, and again NO has been the molecule of choice. LIF or REMPI detection of the states involved is then a simple matter of tuning the laser to the appropriate wavelength range. It is often the case that transitions from the ground and first excited vibrational state can be recorded in a single laser scan. The results for NO(ν = 1) scattering from LiF (Misewich et al., 1985), silver (Misewich and Loy, 1986), and graphite (Vach et al., 1989, 1987), were remarkably similar: the survival probability was in the range from 0.7–0.9, and essentially independent of incident energy, incident angle or surface temperature. In addition, the final rotational distributions are very nearly the same for the excited vibrational state as for the ground state. The scattered angular and/or rotational state distributions in these experiments indicated that the scattering was predominately direct inelastic. The conclusion was that the vibration was weakly coupled to other degrees-of-freedom and was basically a spectator in the scattering process. The weak coupling between molecule vibration and surface degrees-of-freedom on an insulator surface was shown in even greater detail for the HCl(ν = 2, J = 1)/MgO(100) system (Korolik et al., 2000). HCl was prepared in ν = 2 by direct overtone pumping in the IR in a molecular beam and detected by 2 + 1 REMPI. At higher incident energy (Ei = 0.90 eV), direct inelastic scattering dominated: the scattered HCl(ν = 2) showed substantial rotational excitation with Tr > Ts and retained roughly half its initial translational energy. At low incident translational energy (Ei = 0.11 eV) a trapping desorption mechanism took over: the scattered rotational and translational temperatures were equal to the surface temperature. Even so, a substantial number of molecules survived and desorbed in ν = 2. Thus a significant fraction (estimated to be at least 10%) of the excited molecules retained their vibrational excitation, even when their estimated residence time on the surface at Ts = 180 K was longer than 1 µsec. For a surface temperature of 120 K, the signal observed for HCl(ν = 2) decreased by an order of magnitude. This was interpreted in terms of a model in which the HCl resided long enough on the surface to diffuse to a step edge where the binding is stronger and vibrational relaxation is expected to be enhanced. For reasonable energetics, a kinetic model of these process was constructed which was consistent with the observations. The combination of full internal state and velocity resolution was essential in this experiment in order to develop a complete picture of the dynamics. A more comprehensive review of the scattering of molecules initially prepared in vibrationally excited states can be found in Sitz (2002). 6.3.3.2. Vibrational excitation Vibrational excitation in a direct inelastic scattering process was first observed for the case of NO scattering from Ag(111) (Rettner et al., 1985, 1987b). The standard signatures of DI scattering were found for the observed NO(ν = 1), namely a quasi-specular angular distribution and a rotational state distribution within ν = 1 that depended strongly on incident beam energy. The vibrational excitation probability (integrated over rotational state)


221

increased linearly with the normal component of the incident kinetic energy (up to about 0.07 at En = 1.24 eV), but exponentially with surface temperature. In addition, measurement of the final translational energy of the NO which had undergone vibrational excitation showed that most of the excitation energy came from the surface and not from the molecules’ incident energy. These last two observations generated a great deal of interest, as they may be indicative of the electron transfer mechanism. Similar phenomena were also observed for the NO/Cu(110) system (Watts et al., 1997). The model suggested for the exponential surface temperature dependence postulated that the NO vibration is coupled to thermally excited electron-hole pairs in the metal. This idea is carried over from gas-phase electron–molecule scattering where it is known that vibrational excitation is enhanced by so-called shape resonances, which are temporary negative ion states. In the case of molecule–surface scattering, the negative molecule ion is stabilized (near a metal surface) by its image in the conduction electrons of the surface. The excited metal electrons participating in the transfer come from the high energy tail of the Fermi–Dirac distribution in the metal hence the exponential factor in the probability. Some additional, indirect, evidence supporting the idea of an electronic mechanism for the vibrational excitation came from an analysis of the scattered rotational state distributions. These were found to be the same for the vibrationally elastic and vibrationally inelastic channels, that is the rotational and vibrational parts of the problem were independent. In a mechanism involving mechanical couplings between these degrees-of-freedom and translation this would likely not be the case. The results just described for vibrational excitation do not rule out electronically adiabatic models which can account for the observations if the molecular bond length and vibrational frequency are allowed to vary as a function of z, the molecule surface distance (Gross and Brenig, 1993a, 1993b). One aspect in which electronically adiabatic and electron-mediated models differ significantly is in their prediction of the kinetic energy dependence of vibrational relaxation. A definitive experiment has been reported in which vibrational excitation and de-excitation have been examined at the same time: the system was NO(ν = 2) scattered from Au(111) (Huang et al., 2000b). NO could be prepared in the ν = 2 state by direct overtone pumping with an efficiency of a few percent. The observation was that both excitation and de-excitation increased strongly with incident kinetic energy. This result is in qualitative agreement with the mechanism involving electron–hole pairs, but not with the electronically adiabatic model. An additional interesting finding in this experiment was that the rotational state distribution of NO that had undergone vibrational relaxation was about 20% less than that for molecules which had undergone vibrationally elastic scattering. This result may be an indication that the orientation of the NO is important in the electron transfer process since, as discussed above in Section 6.3.2, rotational excitation is most sensitive to this orientation angle. An exhaustive search for vibrational excitation in direct-inelastic scattering was performed for the CO/Au(111) system (Rettner, 1993). The result was that the probability for direct vibrational excitation (ν = 0 → ν = 1) was less than 10−3 for kinetic energies up to 1.4 eV (the vibrational constant for CO is 0.27 eV) and surface temperatures up to 800 K. The small amount of vibrational excitation that was observed was consistent with a trapping-desorption channel in which the CO was vibrationally equilibrated at Ts . The upper limit on direct vibrational excitation is at least an order of magnitude less than that

222

G.O. Sitz

found for NO/Ag(111). No clear explanation of this result has been put forward; as Rettner discusses in detail, the electron affinities for CO versus NO and surface work functions for Au versus Ag are not so different that the electron transfer mechanism proposed for NO/Ag should not apply for CO/Au and lead to comparable vibrational excitation. The resolution to this question remains an open topic. 6.3.3.3. Highly excited vibrations The discussion so far has dealt with transitions between molecules with low levels of vibrational excitation, essentially only the v = 0, 1 and 2 states. (Multiple quantum transitions for low lying vibrational levels in polyatomic molecules are discussed in Section 6.4.2.) More recent experiments have probed multiquantum transitions in highly vibrationally excited NO (Hou et al., 1999c, 1999b, 1999a; Huang et al., 2000a). In these experiments, NO is prepared in a molecular beam by the process of stimulated emission pumping (SEP). SEP is a two laser technique in which one laser excites the NO into a + selected (v , J ) level of the A2 electronic state, and a second laser drives the molecule back down into a chosen (v , J ) level in the ground X 2 electronic state. The efficiency of this process is dictated by the Franck–Condon factors between the vibrational levels in the two electronic states. A complicating factor is that the fluorescence lifetime of the + A2 state (200 ns) is short enough compared to the pulse duration of the lasers (roughly 6 ns) that some population is distributed by spontaneous emission over a range of ground state vibrational levels. However, this effect can largely be corrected for through a ‘laseron/laser-off’ subtraction. Direct transitions involving multiple quanta of vibration have been reported for NO scattering from Au(111), LiF, copper and partially oxidized copper. The oxidation was a byproduct of the exposure to the excited NO. For scattering from gold, efficient transfer of up to 10 quanta of vibration was observed, with the largest probability found for v of 7 or 8. In contrast, on LiF, very little vibrational energy transfer was found with vibrationally elastic scattering dominating. Also, the energy lost from vibration did not appear in the final rotation or translation of the reflected molecules, and therefore had to have been dissipated to the surface. These two findings constitute strong evidence that the vibrational relaxation is mediated by electron transfer between the surface and the NO molecule. As just mentioned, an additional result of these studies was the reactivity of highly vibrationally excited NO on a clean copper surface was more than 3 orders of magnitude larger than that of the ground state NO. Clearly the vibrational excitation survives long enough that a molecule can use vibrational energy to overcome a reaction barrier at a surface.

6.4. Polyatomics In contrast to the large number of scattering studies involving diatomic molecules, there has been relatively little work with polyatomics. This is due in part to the added complexity of the necessary spectroscopic analysis, and in part to the fact that the trapping probability for larger molecules is often near unity and thus little dynamical information is anticipated.


223

The bulk of the experiments that have been reported are for small, hydrogen containing molecules where direct inelastic scattering is expected. 6.4.1. Rotation Kay and Raymond used REMPI to measure the rotational state distribution of NH3 scattered from a NH3 saturated W(100) surface (Kay and Raymond, 1986a, 1986b). For all conditions studies, they were able to fit the rotational state distributions of the scattered molecules to a temperature. For Ts in the range from 300 to 380 K, the scattered rotational temperature was equal to Ts , but for surface temperatures above 380 K, Tr was less than Ts . Part of this variation may result from the changing composition of the surface: at temperatures above 300 K, NH3 starts to decompose and desorb on tungsten. At 300 K where the surface composition is known and stable, the scattered angular and translational energy were indicative of a trapping-desorption mechanism. These studies were somewhat limited in that only a single incident translational was used (5.49 kcal/mol, or 230 meV, corresponding to 6% NH3 in helium). Polyatomic molecules have an additional rotational degree-of-freedom compared to diatomics. This DOF is labelled with the quantum number K which indicates the projection of the angular momentum vector J onto a molecular axis (thus |K| J ). For a symmetric top molecule like NH3 , physically K corresponds to the character of the rotation: for high values of K (K ≈ J ) the molecule is spinning about its C3v symmetry axis whereas for low values of K, the molecule is tumbling end over end. Kay and coworkers were able obtain a measure of the K-state distribution for NH3 scattering. The uncertainty associated with the surface composition in the case of a tungsten target was resolved by using a gold sample, where NH3 desorbs with no decomposition for temperatures in the range of 300–800 K. The cooling in the supersonic beam source produced a beam of molecules predominately in low J and therefore low K states. A strong propensity was found in the scattered molecules for low values of K, even for large rotational excitation (that is large J ). This is physically reasonable: it indicates that the gas–surface interaction potential has a relatively small anisotropy for rotation about the NH3 C3v axis (an azimuthal rotation) and a relatively large anisotropy for rotation about an axis perpendicular to the C3v axis. An additional contributing factor is that the rotational constant, and hence the level spacing, is largest for the high K states since the motion involves only motion of the light hydrogen atoms. The rotational motion for the low K tumbling states involves motion of the nitrogen atom as well and the resulting lower value of the rotational constant means these states require less energy to excite. In a significant step forward, Miller and coworkers developed an apparatus that could be used to study inelastic scattering of polyatomic molecules using high resolution infrared spectroscopy to provide internal state resolution (Francisco et al., 1996; Wight and Miller, 1998a). They have reported results for acetylene (C2 H2 ) and methane (CH4 ), both scattering from LiF(100). For C2 H2 the scattered rotational state distributions were well described by a temperature (Tr ), and this temperature was close to the surface temperature. However, the dependence of Tr on the incident energy (En ) clearly indicated that the reflected molecules were not equilibrated with the surface. This is shown in Fig. 6.12 (Francisco et al., 1996). At lower En , Tr depends linearly on En , with a slope

224

G.O. Sitz

Fig. 6.12. Rotational temperature versus normal incident energy for C2 H2 scattered from LiF(100) for an incident angle of 60◦ . The energies of the two lowest vibrations of C2 H2 (the bending modes, ν4 and ν5 ) are indicated with arrows. Data taken from Francisco et al. (1996).

of about 0.2; this is roughly the translation to rotation energy transfer found for diatomics. At higher En , Tr becomes nearly independent of En ; the authors suggest that this happens when vibrationally inelastic channels become energetically accessible. In addition, across the whole range of incident translational energies studied, the scattered angular distributions showed a pronounced peak near the specular angle. The angular distributions become narrower as En is increased, and the peak scattering angle moves to larger angles as measured from the surface normal. Methane is a spherical top with a large rotational constant resulting from the light hydrogen atoms. Nonetheless, significant rotational excitation has also been observed for CH4 scattered from LiF(100). The observation was made at the specular angle for an incident angle of 60◦ , so the scattering mechanism is direct inelastic and not trapping desorption. Again, the rotational state distribution could be fitted with a single temperature (Tr = 240 K, for Ts = 300 K and an incident energy of 75 meV), even though the individual rotational states belong to the three different nuclear spin families present in CH4 . Measurements of the angular distribution at this incident energy imply that the methane–surface interaction potential is relatively uncorrugated. The substantial rotational inelasticity observed was suggested to arise from a long range attractive anisotropy resulting from the ionic character of the surface and the electric octopole moment of the methane molecule. 6.4.2. Vibration Compared to diatomics, polyatomic molecules have a much greater density of vibrational states as well as having excited states at lower energy associated with bending motions. The energy gaps between these vibrations start to overlap with surface phonons leading to the expectation that energy transfer into and out of these modes could become important.


225

Fig. 6.13. Excitation of the ν2 vibration (the umbrella mode) in NH3 as a function of normal incident energy at Ts = 300 K. Data taken from Kay et al. (1987).

For the scattering of ammonia from gold, efficient vibrationally inelastic scattering has been observed (Kay et al., 1987). The lowest lying vibrational mode for NH3 is the umbrella mode, and a direct translation to vibration (T → V) coupling into this mode was observed, with up to 3 quanta being excited. Results are shown in Fig. 6.13 as a function of incident energy and final vibrational level. The excitation probability depended linearly on the incident translational energy, with a sharp threshold. The threshold was slightly below the minimum energy of the corresponding vibrational level. In contrast to the vibrational excitation of NO scattered from Ag and Cu (Section 6.3.3), the umbrella excitation in NH3 did not depend on the surface temperature. The scattering processes was interpreted in terms of an electronically adiabatic, “balls and springs” mechanism. A reduced dimension, classical molecular dynamics calculation gave support to these ideas and showed that long range angular anisotropy in the potential would tend to align the molecule in a geometry favorable for excitation of the bending vibration. The geometry is one in which the plane of the 3 hydrogen atoms is parallel to the surface plane, and the hydrogens are down. Vibrational excitation occurs when the molecule encounters the short range, impulsive part of the potential. The conceptual picture that emerged was supported by the observed linear dependence of the excitation probability on the incident normal kinetic energy, with a nonzero threshold. A higher threshold was found for higher quanta of excitation. Scattering of vibrationally excited acetylene from LiF has been studied using the combination of IR absorption and bolometric detection developed for rotational studies mentioned in Section 6.4.1 (Wight and Miller, 1998b; Wight et al., 1999). The C2 H2 was prepared in the first excited symmetric C–H vibrational state. The results were very similar to what was found for the HCl(ν = 2)/MgO system (see Section 6.3.3). First, angular dis-

226

G.O. Sitz

tributions showed both direct-inelastic and trapping-desorption scattering, and the branching between these channels was primarily determined by the incident translational energy. Second, no measurable vibrational quenching was observed in the direct-inelastic channel, and substantial vibrational survival was found in the trapping desorption channel, even for surface residence times sufficient to fully accommodate the translation. Careful comparison of the angular distributions for ground and vibrationally excited C2 H2 at low incident energy (Ei = 92 meV) where some trapping desorption is apparent showed evidence of vibrational quenching in the TD channel. However, the data were inconsistent with a single decay rate and indicated a sequential relaxation through the manifold of excited vibrational states between the symmetric C–H vibration and ground state. This conclusion is supported by comparison of results for the isotopic variant C2 HD with C2 H2 . The initial decay step in C2 H2 is facilitated by a Fermi resonance between the stretch and bend vibrations in the molecule. This resonance is missing in C2 HD and the relaxation is much slower. Survival results obtained by varying the residence time via the surface temperature provided input for a model of the steps in the quenching. Fitted rate constants for steps in the relaxation (for C2 H2 ) varied from 1.5 × 1011 s−1 to 5.0 × 106 s−1 , and tended to be slower for steps involving smaller energy gaps. The initial step was about two orders of magnitude slower in C2 HD. 6.5. Conclusion and outlook Inelastic scattering from unreactive surfaces involving the translational and rotational degrees-of-freedom show a rich variety of phenomena which are, for the most part, readily understood in terms simple, classical pictures. This situation becomes less clear when comparing clean and modified surfaces where it is not understood in any detail how, for example, the presence of an adsorbate changes the molecule surface interaction potential. For inelastic scattering involving the vibrational degree-of-freedom, experimental results break down into two classes: ones in which the vibration is essentially a spectator and state changing collisions are highly unlikely, and those in which the vibrational state does change but in which the transition appears to be mediated by electron transfer. The former are understood in terms of a mismatch between the molecular and surface-phonon frequencies. In the case of electronically mediated transitions, detailed understanding is still at a rudimentary level.

Acknowledgements The author gratefully acknowledges support for this work under Grant No. CHE-0238224 from the National Science Foundation.

References Asada, H., Matsui, T., 1982. Speed distributions of monatomic, diatomic and polyatomic molecular beams scattered by the (111) plane of silver. Japan. J. Appl. Phys. 21, 259–263.


227

Barker, J.A., Auerbach, D.J., 1985. Gas–surface interactions and dynamics: Thermal energy atomic and molecular beam studies. Surf. Sci. Rep. 4, 1–99. Berenbak, B., Riedmuller, B., Rettner, C.T., Auerbach, D.J., Stolte, S., Kleyn, A., 2001. Rotational excitation of NO on Ru(0001)-(1 × 1)H. Phys. Chem. Comm. 4, 66–71. Bernstein, R.B., 1982. Chemical Dynamics via Molecular Beam and Laser Techniques. Clarendon Press, Oxford, New York. Bronnikov, D.K., Zyrianov, P.V., Kalinin, D.V., Filimonov, Y.G., Kleyn, A., Hilico, J.C., 1996. A diode laser spectrometer for state-resolved experiments on the methane–surface system. Chem. Phys. Lett. 249, 423– 432. Butler, D.A., Berenbak, B., Stolte, S., Kleyn, A., 1997. Elastic scattering in a reactive environment: NO on Ru(0001)-(1 × 1)H. Phys. Rev. Lett. 78, 4653–4656. Darling, G.R., Holloway, S., 1995. The dissociation of diatomic molecules at surfaces. Report Prog. Phys. 58, 1595–1672. Francisco, T.W., Camillone III, N., Miller, R.E., 1996. Rotationally inelastic scattering of C2 H2 from LiF(100): Translational energy dependence. Phys. Rev. Lett. 77, 1402–1405. Frenkel, F., Hager, J., Krieger, W., Walther, H., Campbell, C.T., Ertl, G., Kuipers, H., Segner, J., 1981. Rotationally inelastic gas–surface scattering investigated by laser-induced fluorescence. Phys. Rev. Lett. 46, 152–155. Geuzebroeak, F.H., Wiskerke, A.E., Tenner, M.G., Kleyn, A.W., Stolte, S., Namiki, A., 1991. Rotational excitation of oriented molecules as a probe of molecule–surface interaction. J. Phys. Chem. 95, 8409–8421. Gotthold, M.P., Sitz, G.O., 1998. Inelastic scattering of N2 from water ice. J. Phys. Chem. B 102, 9557–9564. Gross, A., Brenig, W., 1993a. Adiabatic versus non-adiabatic effects in the vibrational excitation in NO/Ag scattering. Chem. Phys. 177, 497–508. Gross, A., Brenig, W., 1993b. Vibrational excitation of NO in NO/Ag scattering revisited. Surf. Sci. 289, 335–339. Hanisco, T.F., Kummel, A.C., 1993a. The effect of surface passivation on rotationally inelastic scattering: N2 scattered from W(110), W(110)-(2 × 2)N, W(110)-(1 × 1)H, and Pt(111). J. Chem. Phys. 99 (9), 7076–7089. Hanisco, T.F., Kummel, A.C., 1993b. Rotationally inelastic scattering of N2 scattered from W(110). J. Vac. Sci. Technol. A 11, 1907–1913. Hanisco, T.F., Yan, C., Kummel, A.C., 1992a. Energy and momentum distributions and projections in the scattering of CO from Ag(111). J. Chem. Phys. 97 (2), 1484–1496. Hanisco, T.F., Yan, C., Kummel, A.C., 1992b. Resonantly enhanced multiphoton ionization of N2 a 1 σg+ (v = v ) → x 1 σg+ (v = 0, 1, 2). 2. Alignment and orientation measurements. J. Phys. Chem. 96, 2982–2993. Hines, M.A., Zare, R.N., 1993. The interaction of CO with Ni(111): Rainbows and rotational trapping. J. Chem. Phys. 98, 9134–9147. Hou, H., Huang, Y., Gulding, S.J., Rettner, C.T., Auerbach, D.J., Wodtke, A.M., 1999a. Direct multiquantum relaxation of highly vibrationally excited NO in collisions with O/Cu(111). J. Chem. Phys. 110, 10660– 10663. Hou, H., Huang, Y., Gulding, S.J., Rettner, C.T., Auerbach, D.J., Wodtke, A.M., 1999b. Enhanced reactivity of highly vibrationally excited molecules on metal surfaces. Science 284, 1647–1650. Hou, H., Rettner, C.T., Auerbach, D.J., Huang, Y., Gulding, S.J., Wodtke, A.M., 1999c. The interaction of highly vibrationally excited molecules with surfaces: Vibrational relaxation and reaction of NO(ν) at Cu(111) and O/Cu(111). Faraday Discuss. 113, 181–200. Huang, Y., Rettner, C.T., Auerbach, D.J., Wodtke, A.M., 2000a. Vibrational promotion of electron transfer. Science 290, 111–114. Huang, Y., Wodtke, A.M., Hou, H., Rettner, C.T., Auerbach, D.J., 2000b. Observation of vibrational excitation and deexcitation for NO(ν = 2) scattering from Au(111): Evidence for electron–hole-pair mediated energy transfer. Phys. Rev. Lett. 84, 2985–2988. Jacobs, D.C., Kolasinski, K.W., Shane, S.F., Zare, R.N., 1989. Rotational population and alignment distributions for inelastic scattering and trapping/desorption of NO on Pt(111). J. Chem. Phys. 91, 3182–3195. Janda, K.C., Hurst, J.E., Becker, C.A., Cowin, J.P., Wharton, L., Auerbach, D.J., 1980. Direct inelastic and trapping-desorption scattering of N2 from polycrystalline W: Elementary steps in the chemisorption of nitrogen. Surf. Sci. 93, 270–286. Kay, B.D., Raymond, T.D., 1986a. Quantum-resolved molecular-beam scattering of ammonia from ammoniasaturated tungsten (100). Chem. Phys. Lett. 130, 79–83.

228

G.O. Sitz

Kay, B.D., Raymond, T.D., 1986b. Rotationally inelastic gas–surface scattering: NH3 from NH3 -saturated W(100). J. Chem. Phys. 85, 4140–4147. Kay, B.D., Raymond, T.D., Coltrin, M.E., 1987. Observation of direct multiquantum vibrational excitation in gas–surface scattering: NH3 on Au(111). Phys. Rev. Lett. 59, 2792–2794. Kimman, J., Rettner, C.T., Auerbach, D.J., Barker, J.A., Tully, J.C., 1986. Correlation between kinetic-energy transfer to rotation and to phonons in gas–surface collisions of NO with Ag(111). Phys. Rev. Lett. 57, 2053– 2057. Kleyn, A.W., Horn, T.C.M., 1991. Rainbow scattering from solid-surfaces. Physics Reports 199, 192–230. Kleyn, A.W., Luntz, A.C., Auerbach, D.J., 1981. Rotational energy transfer in direct inelastic surface scattering: NO on Ag(111). Phys. Rev. Lett. 47, 1169–1172. Kleyn, A.W., Luntz, A.C., Auerbach, D.J., 1985. Rotational polarization in NO scattering from Ag(111). Surf. Sci. 152/153, 99–105. Korolik, M., Suchan, M.M., Johnson, M.J., Arnold, D.W., Reisler, H., Wittig, C., 2000. Survival of HCl(ν = 2) in trapping-desorption from MgO(100). Chem. Phys. Lett. 326, 11–21. Kuipers, E.W., Tenner, M.G., Kleyn, A.W., Stolte, S., 1989. Dependence of the NO/Ag(111) trapping probability on molecular orientation. Chem. Phys. Lett. 138, 451–460. Kummel, A.C., Sitz, G.O., Zare, R.N., 1986. Determination of populations and alignment of the ground state using two-photon nonresonant excitation. J. Chem. Phys. 85, 6874–6897. Kummel, A.C., Sitz, G.O., Zare, R.N., Tully, J.C., 1988. Direct inelastic scattering of N2 from Ag(111). III. Normal incident N2 . J. Chem. Phys. 89, 6947–6955. Kummel, A.C., Sitz, G.O., Zare, R.N., Tully, J.C., 1989. Direct inelastic scattering of N2 from Ag(111). IV. Scattering from high temperature surface. J. Chem. Phys. 91, 5793–5801. Luntz, A.C., Kleyn, A.W., Auerbach, D.J., 1982. Observation of rotational polarization produced in molecule– surface collisions. Phys. Rev. B 25, 4273–4275. Lykke, K.R., Kay, B.D., 1989. Rotationally inelastic scattering of N2 from clean and hydrogen covered Pd(111). J. Chem. Phys. 90, 7602–7603. Lykke, K.R., Kay, B.D., 1991a. Rotational rainbows in the inelastic scattering of N2 from Au(111). J. Phys. Condens. Matter 3, S65–S70. Lykke, K.R., Kay, B.D., 1991b. Two-photon spectroscopy of N2 : Multiphoton ionization, laser-induced fluorescence, and direct absorption via the a 1 g+ state. J. Chem. Phys. 95 (4), 2252–2258. Mackay, R.S., Curtiss, T.J., Bernstein, R.B., 1989. Determination of preferred orientation for sticking of polar molecules in beams incident on a graphite(0001) surface. Chem. Phys. Lett. 164, 341–344. Manson, J.R., 2008. Energy transfer to phonons in atom and molecule collisions with surfaces. In: Hasselbrink, E., Lundqvist, B. (Eds.), Handbook of Surface Science, vol. 3. Elsevier, Amsterdam (this book, Chapter 3). McClelland, G.M., Kubiak, G.D., Rennagel, H.G., Zare, R.N., 1981. Determination of internal-state distributions of surface scattered molecules: Incomplete rotational accommodation of NO on Ag(111). Phys. Rev. Lett. 46, 831–834. Misewich, J., Loy, M.M.T., 1986. Single quantum state molecular beam scattering of vibrationally excited NO from Ag(111) and Ag(110). J. Chem. Phys. 84, 1939–1940. Misewich, J., Zacharias, H., Loy, M.M.T., 1985. State-to-state molecular-beam scattering of vibrationally excited no from cleaved LiF(100) surfaces. Phys. Rev. Lett. 55, 1919–1922. Modl, A., Gritsch, T., Budde, F., Chuang, T.J., Ertl, G., 1986. Dynamics of NO molecular-beam scattering from a Ge surface. Phys. Rev. Lett. 57, 384–387. Modl, A., Robota, H., Segner, J., Vielhaber, W., Lin, M.C., Ertl, G., 1985. Rotational state distributions of NO molecules after interaction with germanium surfaces. J. Chem. Phys. 83, 4800–4807. Rettner, C.T., 1993. The search for direct vibrational excitation in gas–surface collisions of CO with Au(111). J. Chem. Phys. 99, 5481–5489. Rettner, C.T., Fabre, F., Kimman, J., Auerbach, D.J., Morawitz, H., 1985. Observation of direct vibrational excitation in gas–surface collisions: NO on Ag(111). Phys. Rev. Lett. 55, 1904–1907. Rettner, C.T., Kimman, J., Fabre, F., Auerbach, D.J., Barker, J.A., Tully, J.C., 1987a. Dynamics of gas–surface energy transfer: Inelastic scattering of NO from Ag(111). J. Vac. Sci. Technol. A 5, 508–512. Rettner, C.T., Kimman, J., Fabre, F., Auerbach, D.J., Morawitz, H., 1987b. Direct vibrational excitation in gas– surface collisions of NO with Ag(111). Surf. Sci. 192, 107–130.


229

Saltsburg, H., Smith Jr., J.N., 1966. Molecular-beam scattering from the (111) plane of silver. J. Chem. Phys. 45, 2175–2183. Scoles, G. (Ed.), 1992. Atomic and Molecular Beam Methods, vol. 2. Oxford University Press, New York, Oxford. Segner, J., Robota, H., Vielhaber, W., Ertl, G., Frenkel, F., Hager, J., Krieger, W., Walther, H., 1983. Rotational state populations of NO molecules scattered from clean and adsorbate-covered Pt(111) surfaces. Surf. Sci. 131, 273–289. Siders, J.L.W., Sitz, G.O., 1994. Observation and characterization of direct inelastic and trapping desorption channels in the scattering of N2 from Cu(110). J. Chem. Phys. 101, 6264–6270. Sitz, G.O., 2002. Gas surface interactions studied with state-prepared molecules. Report Prog. Phys. 65, 1165– 1193. Sitz, G.O., Kummel, A.C., Zare, R.N., 1988a. Direct inelastic scattering of N2 from Ag(111). I. Rotational populations and alignment. J. Chem. Phys. 89, 2558–2571. Sitz, G.O., Kummel, A.C., Zare, R.N., Tully, J.C., 1988b. Direct inelastic scattering of N2 from Ag(111). II. Orientation. J. Chem. Phys. 89, 2572–2582. Sitz, G.O., Mullins, C.B., 2002. Molecular dynamics simulations of the influence of surface temperature on the trapping of methane on iridium single-crystal surfaces. J. Phys. Chem. B 106, 8349–8353. Tenner, M.G., Kuipers, E.W., Langhout, W.Y., Kleyn, A.W., Nicolasen, G., Stolte, S., 1990. Molecular beam apparatus to study interactions of oriented NO and surfaces. Surf. Sci. 236, 151–168. Vach, H., Hager, J., Walther, H., 1987. Energy transfer processes during the scattering of vibrationally excited NO molecules from a graphite surface. Chem. Phys. Lett. 133, 279–282. Vach, H., Hager, J., Walther, H., 1989. Survival, relaxation, and excitation of vibrational energy during the scattering of NO from a graphite surface. J. Chem. Phys. 90, 6701–6708. Watts, E.K., Siders, J.L.W., Sitz, G.O., 1997. Vibrational excitation of NO scattered from Cu(110). Surf. Sci. 374, 191–196. Wight, A.C., Miller, R.E., 1998a. Rainbow scattering of methane from LiF(001): Probing the corrugation and anisotropy of the gas–surface potential. J. Chem. Phys. 109, 1976–1982. Wight, A.C., Miller, R.E., 1998b. Sequential quenching of acetylene scattered from LiF(001): Trapping desorption versus direct scattering. J. Chem. Phys. 109, 8626–8634. Wight, A.C., Penno, M., Miller, R.E., 1999. Sequential vibrational relaxation of polyatomic molecules at surfaces: C2 HD and C2 H2 scattered from LiF(001). J. Chem. Phys. 111, 8622–8627.


CHAPTER 7

Reaction Dynamics and Kinetics: TST, Non-equilibrium and Non-adiabatic Effects, Lateral Interactions, etc.

Vladimir P. ZHDANOV Department of Applied Physics Chalmers University of Technology S-412 96 Göteborg, Sweden Boreskov Institute of Catalysis Russian Academy of Sciences Novosibirsk 630090, Russia



Contents 7.1. 7.2. 7.3. 7.4. 7.5. 7.6.

Introduction . . . . . . . . . . . . . . . . . . . . Transition-state theory . . . . . . . . . . . . . . Tunneling . . . . . . . . . . . . . . . . . . . . . Non-equilibrium effects . . . . . . . . . . . . . Non-adiabatic effects . . . . . . . . . . . . . . . Lateral interactions . . . . . . . . . . . . . . . . 7.6.1. General equations . . . . . . . . . . . . . 7.6.2. Temperature-programmed desorption . . 7.6.3. Surface diffusion . . . . . . . . . . . . . 7.6.4. Conventional reaction kinetics . . . . . . 7.6.5. Oscillations, chaos and pattern formation 7.7. Surface heterogeneity . . . . . . . . . . . . . . . 7.8. Reaction kinetics on nm-sized catalyst particles 7.9. Conclusion . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . .

233 234 237 237 240 245 247 249 254 256 259 261 262 263 264 264

Abstract As a rule, the dynamics and kinetics of elementary steps of chemical reactions occurring on solid surfaces are complex due to the specifics of the adsorbate–substrate interaction, adsorbate–adsorbate lateral interactions, surface heterogeneity, spontaneous and adsorbateinduced changes in a surface, and/or limited mobility of reactants. In the present chapter, the main concepts of the rate theory of such reactions are illustrated by using the results obtained during the past decade. The goal of the presentation is to show new trends and opportunities in this interdisciplinary field of research.

232

233

7.1. Introduction This volume of “Handbook of Surface Science” is focused on dynamic processes occurring on solid surfaces. This subject should certainly encompass chemical or, more specifically, catalytic reactions running on solid surfaces, because chemical reactions are inherently related to the dynamics. In particular, almost all discussions of the basic principles of the rate theory of chemical reactions begin with the statement that such reactions can be interpreted in terms of the motion of atomic nuclei along the potential energy surface or surfaces representing the electronic and nuclear-repulsion energy of the system as a function of nuclear coordinates. In practice, heterogeneous catalytic reactions usually occur at thermal conditions and can often be described by using a presumably reasonable set of elementary reaction steps and employing the corresponding rate constants. Customarily, the latter is a prerogative of chemical kinetics. Thus, to some extent, the role of the dynamics is complementary and might sometimes appear to be inferior, because the reaction rate constants can often be viewed simply like phenomenological (or empirical) parameters. Despite this reservation, the reaction dynamics does play a central role in construction and validation of the conceptual and mathematical basis of chemical kinetics. In principle, the dynamics and kinetics can be discussed separately. For general readership, it is more reasonable however to articulate the complementary role of the dynamics with respect to the kinetics and to focus the presentation on the points significant for the kinetics. Following this line, we discuss in this chapter the dynamical aspects of the kinetics of heterogeneous catalytic reactions. The kinetics of chemical reactions in gas and liquid phases are usually described by employing the conventional mass-action law equations. The laws governing the kinetics of heterogeneous catalytic reactions are as a rule much more complex due to adsorbate– adsorbate lateral interactions, surface heterogeneity, spontaneous and adsorbate-induced changes in a surface, and/or limited mobility of reactants. The importance of these factors was recognized by the heterogeneous catalysis community in the middle of the previous century. Only since the late 1960s with the development of surface science (Duke, 1993), it has however become possible to study in detail the non-ideality of rate processes on solid surfaces. From the beginning of its development, this branch of surface science attracts appreciable attention. For the academic society, this field is of interest due to richness and complexity of the kinetics of heterogeneous catalytic reactions. From a practical point of view, the understanding of the kinetics of catalytic reactions is important, because heterogeneous catalysis is the mainstay of the chemical industry. More than 90% of the chemical manufacturing processes in use throughout the world utilize catalysis and primarily heterogeneous catalysis (Thomas and Thomas, 1997). In applied chemistry, the utilization of kinetic data occurs at almost all the steps including process development, process optimization, and catalyst development (Bos et al., 1997). During the last two decades, various aspects of the kinetics of heterogeneous catalytic reactions were extensively reviewed in the literature. In particular, the corresponding textbooks were written by Dumesic et al. (1993), Van Santen and Niemantsverdriet (1995), and Masel (1996). The author of this chapter (Zhdanov, 1991a) collected the literature on the dynamical aspects of the kinetics of heterogeneous catalytic reactions. There are also many more specialized reviews (e.g., Kang and Weinberg, 1995; Ertl, 2000;

234

V.P. Zhdanov

Stoltze, 2000; Dumesic, 2001; Zaera, 2002; Zhdanov, 2002a; Liu and Evans, 2005; Christensen and Norskov, 2008, see also references below). To complement already available textbooks and reviews, we outline here general concepts of the rate theory of heterogeneous catalytic reactions and illustrate them using primarily examples borrowed from original articles published during the last fifteen years.

7.2. Transition-state theory As already noted, an elementary chemical rate process can be identified with the motion of atomic nuclei along the potential energy surface or surfaces representing the electronic and nuclear-repulsion energy of the system as a function of nuclear coordinates. This concept is based on the Born–Oppenheimer approximation implying separation of fast motion of electrons and slow motion of nuclei. At thermal conditions, chemical rate processes usually occur along the pathways that lead along the lowest potential energy surface from one stable minimum to another. If this potential energy surface is well separated from higher potential energy surfaces so that the transitions to the latter surfaces are negligible (this is often the case in practically important situations), an elementary process is called adiabatic. Non-adiabatic elementary processes include transitions between different potential energy surfaces. The rates of adiabatic elementary chemical processes can be calculated by using the transition-state theory (TST) worked out by Eyring, Wigner, and Pelzer before World War II. Since its development, TST forms the mainstay of chemical kinetics (for the current status of TST and numerous references, see a recent paper by Schenter et al. (2003)). According to TST, the minimum energy path connecting two stable conformations is identified as the reaction coordinate. The transition (or activated) state is associated with the position of maximum energy along the reaction coordinate, representing a saddle point at the potential energy surface. The reaction rate is identified with the reactant flux along the reaction coordinate across the saddle point in the direction of the final conformation. The flux is calculated assuming thermodynamic equilibrium between the activated and initial states. These approximations make it possible to represent a reaction rate constant as kTST =

kB T FA! exp(−Ea /kB T ), h FA

(7.1)

where FA and FA! are the partition functions of reactants in the initial and activated states, Ea is the energy difference between these states, and kB and h are the Boltzmann and Planck constants. For molecular desorption, for example, FA and FA! are the partition functions of a molecule in the adsorbed and activated states, respectively. For bimolecular (or trimolecular) processes, the initial state includes two (or three) separately located reactants, and accordingly FA can be represented as a product of the corresponding partition functions. The basic assumptions of TST restricting its applicability are as follows: 1. Statistical equilibrium between reactants and activated complexes. 2. Classical motion along the reaction path. 3. Separability of the reaction coordinate from other coordinates.

Reaction Dynamics and Kinetics

235

4. Passage through the transition state from reactant to product side is assumed to be a “moment of decision” for the reacting system, which then will never return to the transition state. During heterogeneous catalytic reactions, adsorption of reactants is usually accompanied by local rearrangement of substrate atoms. Activation of reactants is accompanied by local relaxation of the substrate as well. For this reason, the energy Ea and partition functions FA and FA! should be calculated by taking into account this relaxation (this means that FA and FA! should in general include the substrate degrees of freedom). In calculations of Ea , this prescription is usually fulfilled. The ratio of FA and FA! is however often assumed to be dependent primarily on the reactant degrees of freedom and accordingly the substrate degrees of freedom are ignored. From assumptions 1 and 4 above, it is clear that TST usually yields an upper limit of a rate constant. The corrected rate constants are customarily represented as k = κkTST ,

(7.2)

where κ 1 is the so-called transmission coefficient. Numerous molecular dynamics simulations of adiabatic gas-phase reactions and surfaces diffusion indicate that if the potential energy surface is accurate, TST is fairly accurate (within a factor of two or better) as well. Another advantage of TST is that it automatically takes into account thermodynamics and accordingly the TST rate constants for forward and backward processes satisfy the detailed balance principle. Phenomenologically, a rate constant is usually represented as k = ν exp(−Ea /kB T ),

(7.3)

where ν and Ea are the apparent Arrhenius parameters, determined by the relations d (7.4) ln k and ν = k exp(Ea /kB T ). dT The pre-exponential factor and activation energy defined by Eq. (7.3) are slightly different compared to those in Eq. (7.1). To identify the relation between the two set of the parameters, Eq. (7.1) is usually rewritten as E a = kB T 2

kTST =

kB T exp −G!0 /kB T , h

(7.5)

where G!0 ≡ Ea − kB T ln(FA! /FA ) is the Gibbs (or free) energy of activation. Substituting Eq. (7.5) into Eqs. (7.4) and using the conventional thermodynamic relations, one gets Ea = H0! + kB T

and ν = e

kB T exp S0! /kB , h

(7.6)

where H0! and S0! are the enthalpy and entropy of activation. To illustrate the relation between the parameters in Eqs. (7.1) and (7.3) more explicitly, it is instructive to analyze the situation when the pre-exponential factors in Eqs. (7.1) can be represented as BT n , where n is the exponent, and B is a constant independent of

236

V.P. Zhdanov

temperature. In this case, Eqs. (7.4) yield Ea = Ea + nkB T

and ν = BT n exp(n).

(7.7)

In reality, n may be either positive or negative, and accordingly the parameters in Eq. (7.3) may be somewhat larger or smaller than those in Eq. (7.1). In general, the reactants may possess vibrational, rotational and translational degrees of freedom. The range of possible values of the corresponding partition functions is wide (especially for rotation and translation), and accordingly the range of possible values of the pre-exponential factors (in Eq. (7.1) or (7.3)) is expected to be wide as well (Zhdanov, 1991a, 1991b). For example, the pre-exponential factor for desorption is usually in the range from 1013 up to 1019 s−1 (note that kB T / h = 1.04 × 1013 s−1 for T = 500 K). The pre-exponential factor for the sticking coefficient for adsorption may vary from 1 down to 10−7 . For elementary reaction steps occurring on solid surfaces (e.g. for reaction between two adsorbed particles or for dissociation of adsorbed species) or for diffusion jumps along the surface, the pre-exponential factors may also be in a wide range. Often however the rotational and translational degrees of freedom are either absent or less significant in such processes, and accordingly the pre-exponential factors are closer to 1013 s−1 . The important point is that the transition state for reactions in the forward and backward directions is the same. For this reason, a high (low) value of the pre-exponential factor for one direction should correspond to a high (low) value of the pre-exponential factor in the reverse direction. If for example the sticking coefficient for adsorption is close to unity, the pre-exponential factor for desorption is usually expected to be several orders of magnitude higher than 1013 s−1 . Due to adsorbate–adsorbate lateral interactions, the rate constants of elementary surface processes usually strongly depend on coverage. To accurately measure the Arrhenius parameters for such processes, this dependence and the dependence on temperature should be separated, i.e., the Arrhenius plots should be constructed at fixed coverage. Such experiments are not simple, because the coverage has a tendency to rapidly decrease with increasing temperature. During the 1980s, the Arrhenius parameters for desorption of simple molecules were measured by many groups. There were also a few experimental studies of the Arrhenius parameters for elementary reaction steps (e.g., for reaction between adsorbed CO and O on Pt and Ir). The corresponding data were collected by the author of this chapter (Zhdanov, 1991b). The available experience indicates that the results of the most accurate measurements tend to be close to what one can expect on the basis of TST. More recently, Paserba and Gellman (2001) measured the Arrhenius parameters for desorption of a set of straight chain alkanes [H(CH2 )n H, with 5 n 60] from the surface of single crystalline graphite. According to TST, the pre-exponential factor for desorption is expected in this case to be high, 1019 s−1 or higher. In line with this prediction, the experiment indicates that ν = 1019.6±0.5 s−1 for all the oligomers. The activation energy for desorption was found to increase with increasing n as Ea = A + Bnα , with the exponent α = 0.50 ± 0.01 (a model explained this dependence was proposed by Gellman and Paserba (2002)). Similar measurements were performed for a series of poly(ethylene glycol)-dimethyl ethers on graphite (Paserba et al., 2002) and linear and cyclic alkanes on


237

Cu(111) and Pt(111) (Lei et al., 2004). Theoretically, the dynamics of desorption of large molecules was analyzed by Fichthorn and Miron (2002) and Krylov and Hermans (2002).

7.3. Tunneling TST implies classical motion along the reaction coordinate. According to quantum mechanics, reactants may however tunnel via the potential barrier. Due to tunneling, the transmission coefficient in Eq. (7.2) may be large, κ 1, and the apparent Arrhenius parameters defined by Eqs. (7.3) and (7.4) may decrease with decreasing temperature. For realistic potential barriers, these effects may be observed in reactions with participation of hydrogen or deuterium at temperatures below 300 K (Miller, 1993; Sato, 2005). Heterogeneous catalytic reactions usually occur at higher temperatures. For this reason, tunneling has not attracted appreciable attention of the heterogeneous catalysis community. In the recent literature, one can however sometimes find the claims that tunneling plays a significant role in practically important catalytic reactions, occurring with participation of relatively heavy reactants, provided that a potential barrier is narrow. One of the likely examples seems to be dissociative adsorption of N2 on Fe or Ru at thermal energies (Romm et al., 1997). The detailed analysis performed by Henriksen et al. (2000) and Henriksen and Hansen (2002) indicates that in this case the tunneling appears to be significantly influenced by the interplay of the translational and vibrational degrees of freedom. Sundell and Wahnström (2004a, 2004b) have recently studied hydrogen diffusion on Cu(001) in the quantum regime using first-principles electronic-structure calculations based on the density-functional theory (DFT). Specifically, they have performed for the first time ab inicio calculations of the tunneling matrix element. At low temperatures, the calculated quantum tunneling rates are found to agree with experimental results if the coupling to the lattice and the non-adiabatic response of the electronic degrees of freedom are properly taken into account.

7.4. Non-equilibrium effects One of the key assumption of TST is that the energy distribution of reactants is close to the Boltzmann one. This assumption is often reasonable. Under conditions of slow activation, however, the population of the reactant states with energy exceeding the activation energy, E > Ea , can be exhausted due to reaction, and accordingly a rate constant can be lower than that predicted by TST, as it was first shown in the seminal paper by Kramers (1940) (for numerous relevant references, see a comprehensive review by Hanggi et al. (1990) and more recent articles by Banerjee et al. (2002) and Pollak and Talker (2005)). In his paper, Kramers analyzed the escape of a particle from a one-dimensional potential well by passing over a potential barrier. First, he formulated this problem in terms of the Langevin equation. This equation is however not suitable for analytical calculations. For this reason, Kramers reformulated the problem in terms of the Fokker–Planck equation. Although in general the latter equation cannot be exactly solved either, it can easily be

238

V.P. Zhdanov

integrated accurately in the most interesting cases of low and high friction. In these limits, the escape process is described in terms of diffusion in the energy space and coordinate space, respectively. If the energy distribution is close to equilibrium near the bottom of the potential well and exhausted at E > Ea , the escape rate constant is expressed as (ηEa /kB T ) exp(−Ea /kB T ) for η kB T ω0 /2πEa , k= (7.8) (ω0 ω! /2πη) exp(−Ea /kB T ) for η ω! , where η is the friction coefficient, ω0 the frequency of small oscillations of a particle near the equilibrium position, and ω! the frequency characterizing a potential barrier. The corresponding TST expression for the escape rate constant is k = (ω0 /2π) exp(−Ea /kB T ).

(7.9)

Comparing Eqs. (7.8) and (7.9) indicates that TST is applicable if kB T ω0 /2πEa η ω! .

(7.10)

The simplest estimate of the coefficient of friction due to excitation of phonons is (Zhdanov, 1991a) η ω0 m/M,

(7.11)

where m is the reactant mass, and M is the mass of a lattice atom. Using this or other expressions for the friction coefficient, one can verify that, in the case of chemisorption, condition (7.10) as a rule is fulfilled. For this reason, the non-equilibrium corrections to TST are not too important. The exceptions to this rule are likely for light adsorbates when m/M 1. The Kramers model and its modifications are focused on a single irreversible reaction step. In reality, chemical reactions usually occur via several steps, and the products of one step are often reactants in another step. Just after reaction events, the energy of reaction products is usually appreciably higher than the thermal energy and in principle this extra energy can be used to accelerate other reaction steps and/or for self-acceleration. For example, surface reactions always contain adsorption steps. Just after adsorption, the energy of vibrations of a molecule or atom in the adsorption potential is high (close to the heat of adsorption), i.e., a molecule or atom is “hot”, and their energy can be employed to accelerate other steps. If this happens, one can formally introduce the rate constants of elementary reaction steps as well, and some of these rate constants may be appreciably higher than those predicted by TST. Simple models illustrating the possibility of the use of the energy of one reaction for acceleration of another reaction were constructed about 25 years ago (Zhdanov, 1980). The reactants were represented by two- or three-level systems. The situations treated were as follows: (i) products of the first reaction are reactants in the second, (ii) products of the first reaction exchange energy with reactants of the second, and (iii) reaction products exchange energy with the reactants of the same reaction. A conclusion drawn was that under steadystate conditions for physically reasonable rates of vibrational relaxation it is rather difficult to achieve reaction acceleration. The impact of the results obtained was low, because the models were too schematic and the results were published in the journal circulating mainly in USSR.


239

At the same time, Harris and Kasemo (1981) presented a deep qualitative analysis of the situation when, just after adsorption, “hot” molecules or atoms react with chemisorbed species. They argued that in this case (i) the distinction between the classical Eley–Rideal and Langmuir–Hinshelwood mechanisms is ill-defined, (ii) the reaction rate may be higher than one expects, and (iii) the energy distribution of products may be highly non-thermal. These formative conclusions have attracted appreciable attention and are still often mentioned in the literature (see, e.g., a review by Kleyn (2003)). Recently, the role of “hot” precursors in surface reactions was discussed by Molinari and Tomellini (2002) and Tomellini (2004). The model they used in several articles (Molinari and Tomellini, 2002, and references therein) implies A and B (or B2 ) adsorption and rapid energy exchange between A particles and between B particles and postulates the existence of the quasi-steady-state energy distribution with over-population of the excited states compared to the Boltzmann distribution. Although the model was applied to specific reactions, e.g., to CO oxidation on Pt (Molinari and Tomellini, 2002), its validity remains to be clarified. In CO oxidation, for example, there are two steady-state reaction regimes (Zhdanov and Kasemo, 1994). (i) At CO excess, the surface is primarily covered by CO. Typically, CO is close to adsorption–desorption equilibrium and accordingly the over-population of the excited vibrational states of CO (here and below we bear in mind excitation along the CO–Pt bond) is unlikely. For O atoms, formed after O2 adsorption, the over-population may in principle occur, but in this case the energy exchange between vibrationally excited O atoms does not take place. (ii) At O2 excess, the surface is primarily covered by O, and accordingly the energy exchange between vibrationally excited CO molecules is improbable. For O atoms, the energy exchange may occur, but its rate can hardly be faster than the rate of vibrational energy relaxation. The other model used by Tomellini (2004) is based on the master equations. It takes into account a few vibrational levels. This approach is more general compared to that employed earlier (Molinari and Tomellini, 2002) in the sense that it makes it possible to treat various situations. More recently, Tomellini (2007) has also discussed excitation of electron–hole pairs during atom recombination on metal surfaces. During the past two decades, various aspects of surface reactions occurring under thermal or molecular-beam conditions have been studied by using the molecular dynamics technique (see reviews by Shalashilin et al. (1998) and Gross (1998)). In combination with the DFT calculations of a potential energy surface, this technique is now superior. As a relevant example, it is appropriate to mention a quasi-classical treatment (Shalashilin et al., 1998) of the dynamics of HD formation during the impingement of H or D atoms upon a Dor H-covered Cu(111) surface. In this case, in agreement with the experiment, the theory indicates that a significant fraction of the reaction events occurs via “hot”-atom pathways as opposed to the conventional Eley–Rideal pathway, in which the incident atom reacts directly with the adsorbed atom. Physically, it is clear that with increasing the rate of energy relaxation one should observe a transition from the reaction regime dominated by “hot” precursors to conventional thermal activation. Analytically, this transition was scrutinized (Zhdanov, 2005a) on the basis of the Fokker–Planck equation, describing energy relaxation as biased diffusion in the energy space, or more specifically in the interval from E = 0 to Eb (these two en-

240

V.P. Zhdanov

ergies correspond, respectively, to the bottom of the potential well and to the supply of “hot” reactants). The results of calculations (see, e.g., Fig. 7.1) were found to depend on three dimensionless parameters, Ea /kB T , Eb /kB T , and p ≡ r(kB T )2 /De , where r is the parameter characterizing the reaction rate of reactants with E > Ea , and De is the coefficient of diffusion in the energy space (it can be expressed via the friction coefficient as De ηkB T E). In particular, the role of “hot” precursors was found to be negligible at (Eb − Ea )p/kB T 1. Except the cases when the ratio m/M is very low (e.g., in reactions with participation of H or D), this condition is usually fulfilled. This means that (i) the rate constant k [Fig. 7.1(c)] can hardly be several orders of magnitude higher than kTST and (ii) the inverted energy distributions like that shown in Fig. 7.1(a) for p = 0.3 are unlikely. The distributions like those for p = 0.03 and 0.1 cannot however be excluded. In the cases when the reaction products leave the surface just after reaction, the energy distribution F (E) (Fig. 7.1(b)) of particles passing through the potential barrier is related to their energy distribution observed in the gas phase. To get the latter distribution, one should shift F (E) up due to the existence of the descending part of the potential energy surface. In addition, one should take into account that after passing the activation barrier a part of energy is absorbed by the substrate (for CO oxidation on the Pt-group metals, this aspect of the reaction dynamics was analyzed by Gumhalter and Matsushima (2004)). This part is however usually not expected to be appreciable. For this reason, one can compare F (E) and the energy distribution of the reaction products. Looking through relevant experimental studies collected in a recent review by Matsushima (2003), or more specifically through the data for CO oxidation and NO reduction by CO on Pt (Bald and Bernasek, 1998), CO oxidation on Pd(110) (Fig. 7.2; Moula et al., 1999; Nakao et al., 2005), and NO reduction by CO on Pd(111) and Pd(110) (Nakao et al., 2003), one can conclude that the inverted distributions like that shown in Fig. 7.1(b) for p = 0.3 or the distributions with an appreciable high-energy tail like for p = 0.1 are not observed. This means that as expected the available experimental data indicate that even in rapid reactions with participation of such molecules as CO, O2 and NO the situation is close to conventional activation.

7.5. Non-adiabatic effects In adiabatic elementary rate processes, described by TST, the potential energy surface for nuclear motion is well separated from higher potential energy surfaces so that the transitions to the latter surfaces are negligible. Non-adiabatic rate processes include transitions between different potential energy surfaces. Identification and scrutinity of non-adiabatic rate processes on solid surfaces is one of the central goals of the theory of heterogeneous chemical reactions (Wodtke et al., 2004). At present, the situation in this field is far from clear. Although the breakdowns of the Born–Oppenheimer approximation in reactions on metals are often anticipated and the relevant experimental data appear to be abundant, good specific examples illustrating in detail what may happen are still lacking. Among a few advances in this field, it is appropriate to mention, e.g., direct detection of hot electrons and holes excited by adsorption of atomic H and D on ultrathin Ag and Cu films (Nienhaus et al., 1999; Nienhaus, 2002; see also a review by Gadzuk (2002)). In many other cases, the


241

Fig. 7.1. Specifics of a heterogeneous catalytic reaction, complicated by “hot” precursors, at Ea /kB T = 15 and Eb /kB T = 30: (a) Energy distribution, f (E), of reactants for p = 0.03, 0.1, and 0.3. (b) Energy distribution, F (E), of particles passing through the potential barrier. (c) Reaction rate constant as a function of p (k0 corresponds to the case when f (E) is close to the Boltzmann distribution, i.e., k0 kTST ). (According to Zhdanov (2005a).)

242

V.P. Zhdanov

Fig. 7.1. (Continued)

Fig. 7.2. Specifics of CO oxidation on Pd(110): (a) Rate of CO2 formation as a function of CO pressure (in the “active” and “inhibited” regions, the surface is covered primarily by O and CO, respectively; the insets show the corresponding angular distribution of the CO2 flux). (b) CO2 velocity distribution at Ts = 460 K (the dotted lines indicate two-temperature deconvolutions, and the solid lines represent their summation). (According to Matsushima (2003).)

interpretation of experimental data is often far from straightforward. One of the reasons of this situation is that the conventional software (e.g., the DFT packages) does not allow to accurately treat excited states. In this section, we first describe three generic models, illustrating what may happen in non-adiabatic processes, and then briefly discuss a few specific examples. The rate of non-adiabatic processes is reduced due to the need of jumps from one potential energy surface to another, and accordingly the corresponding rate constant is usually represented as a product of the transmission coefficient and kTST (see Eq. (7.2)). The transmission coefficient can often be calculated by analyzing the 1D nuclear motion along


243

Fig. 7.3. Diabatic potential-energy curves as a function of the reaction coordinate. Panel (a) corresponds to the conventional two-state Landau–Zener model. Panel (b) is for the situations when a rate process is limited either by one-electron transfer or by two-electron exchange between the reactants and metal. Ui is the energy of the initial state. Uf (thick solid line) is the energy after electron transfer or electron exchange with participation of metal electrons with the Fermi energy. The thin solid lines correspond to the formation of excited electrons, holes, or electron–hole pairs.

the reaction coordinate, q, near the crossing of the diabatic potential energy surfaces (at q q0 ). According to this approximation, the transmission coefficient is defined as ! ∞ ∞ P (v)f (v)v dv f (v)v dv, κ= (7.12) 0

0

where P (v) is the transition probability, v is the particle velocity at q = q0 , and f (v) is the Maxwell velocity distribution. Near the crossing of the diabatic potential energy curves, Ui (q) and Uf (q) (Fig. 7.3(a)), their dependence on the reaction coordinate can often be considered to be linear. This approximation corresponds to the famous Landau–Zener model (Nikitin, 1974). In the most interesting case when the process rate is limited by weak coupling of the diabatic states, this model yields P (v) =

4πV 2 , h¯ vF

(7.13)

where V is the transition matrix element, F = |∂Ui /∂q − ∂Uf /∂q|q=q0 the difference of the potential-energy slopes, and q0 the coordinate corresponding to the crossing of the potential energies (note that this expression for P (v) takes into account that this coordinate is passed two times during the particle motion along Ui (q)). The transmission coefficient is accordingly given by κ=

(2π)3/2 m1/2 V 2 , h¯ F (kB T )1/2

where m is the particle mass.

(7.14)

244

V.P. Zhdanov

The Landau–Zener model is widely used to describe non-adiabatic processes occurring in the gas and liquid phases. In the literature, one can also find examples when the conventional version of this model is employed to interpret processes on a metal surface. In the latter case, the two-state approximation may however fail, because the electronic states in a metal form a continuum (Fig. 7.3(b)), and accordingly the Landau–Zener model should be modified. The modifications depend on the specifics of reactions. If a rate process is limited by one-electron jumps, e.g., from the states, located below the Fermi level in a metal, to a vacant adsorbate orbital (this results in the formation of a hole), one has (Zhdanov, 1991a) P (v) =

4πmvρV 2 h¯ Fi

(7.15)

and κ=

(2π)3/2 (mkB T )1/2 ρV 2 , h¯ Fi

(7.16)

where ρ is the density of the electron states, and Fi = |∂Ui /∂q|q=q0 the potential-energy slope. Non-adiabatic processes may also be limited by spin conversion. A likely example here − is dissociative adsorption of O2 . This molecule is well known to be in the triplet state 3 g . +

The lowest excited states are singlets 1 g and 1 g . If the O2 -metal interaction is repulsive in the triplet state and attractive in one of the singlet states, the dissociative adsorption of O2 may be limited by the transition between these states. The spin conversion may results from the spin–orbit interaction. This interaction is however weak, V ∝ 1/c (c is the velocity of light). For this reason, the spin conversion in rate processes on a metal surface seems to occur more likely via spin exchange with the metal. In the case of O2 adsorption, for example, one of the electrons forming the triplet state may jump to a vacant state, located above the Fermi level in the metal, and simultaneously an electron with the opposite spin may jump to O2 from a state, located below the Fermi level in the metal. According to this scenario, the spin conversion is accompanied by excitation of an electron–hole pair. This means that this process is possible in the region to the right from q0 (i.e., at q > q0 ) where Ui > Uf (Fig. 7.3(b)). In this region, the spin-conversion rate can easily be calculated at a given nuclear coordinate by using the golden rule, and then one can calculate P (v) and κ. Following this line, one can get (Zhdanov, 2006) P (v) =

4m2 v 3 ρ 2 V 2 F , 3h¯ Fi2

(7.17)

and κ=

25/2 π 3/2 m1/2 (kB T )3/2 ρ 2 V 2 F , h¯ Fi2

(7.18)

where V is the average spin-exchange matrix element. Equations (7.12)–(7.18) describe various scenarios of non-adiabatic processes occurring on solid surfaces. In particular, Eqs. (7.13), (7.15) and (7.17) can be used to interpret the


245

molecular-beam experiments performed at fixed energy of scattering molecules. In the simplest case, the initial kinetic energy, E, of a molecule and its velocity at q = q0 are related as v = [2(E − Ea )/m]1/2 , where Ea ≡ Ui (q0 ) is the activation barrier. According to the conventional Landau–Zener model, P ∝ 1/v ∝ 1/(E − Ea )1/2 (Eq. (7.13)), i.e., the reaction probability decreases with increasing energy. For one-electron transfer, one has P ∝ v ∝ (E − Ea )1/2 (Eq. (7.15)), i.e., the reaction probability slowly increases with increasing energy. According to Eq. (7.17), P ∝ v 3 ∝ (E − Ea )3/2 , i.e., the reaction probability rapidly increases with increasing energy. Equations (7.14), (7.16) and (7.18), describing reactions occurring at thermal conditions, indicate that the transmission coefficient may be a few orders of magnitude lower than unity. To illustrate non-adiabatic features in specific rate processes, it is appropriate to mention O2 adsorption on Ag(111). The molecular-beam experiments indicate that in this case the dissociation probability rapidly increases with increasing kinetic energy of O2 molecules (Raukerma et al., 1996). In particular, the results obtained can be fitted as P ∝ (E −Ea )3/2 . An attempt to interpret these results on the basis of a model implying one-electron transfer was performed by the author of this chapter (Zhdanov, 1997). The 1D version of this model (Eq. (7.15)) did not allow one to fit the experimental data. To reach agreement with the experiment, it was necessary to take into account the 3D corrections. In contrast, the 1D model implying spin conversion (Eq. (7.17)) makes it possible to describe the experiment. Thus, the rate of O2 adsorption on Ag(111) is likely to be limited by spin conversion. Kato et al. (1998) scrutinized the effect of spin conversion on the rate of O2 adsorption on Si(001). The triplet-singlet transition was described using the conventional Landau– Zener model. The coupling matrix element was considered to be related with the spin–orbit interaction. In analogy with the O2 /Ag(111) system, the probability of O2 dissociation on Al(111) rapidly increases with increasing translation energy (Österlund et al., 1997). To interpret this dependence, Hellman et al. (2003) used a model implying the dissociation to be limited by one-electron transfer from the metal to O2 . More recently, Behler et al. (2005, 2008) analyzed the O2 /Al(111) system by using DFT. The potential barrier for adsorption was found to be related to slow transition between the triplet and singlet states. The dynamics of this transition was however not treated explicitly. For a more detailed discussion of non-adiabatic processes on solid surfaces, the readers are referred to the chapter written by B.I. Lundqvist.

7.6. Lateral interactions In adsorbed layers, at appreciable coverages, the potential energy surface for nuclear motion usually depends not only on the coordinates of particles directly participating in a rate process but also on the arrangement of other particles located in adjacent sites. The lateral interaction of reactants with the latter particles shift the potential energy (Fig. 7.4) and accordingly may influence the reaction rate. This effect can be described in the lattice-gas approximation implying that before activation (in the ground state) and during crossing

246

V.P. Zhdanov

Fig. 7.4. Schematic cross section of the potential energy surface along the reaction coordinate. The thin and thick solid lines are for the cases when the reacting particles have no and have neighbours, respectively. i and i! are the lateral interactions (with neighbours) in the initial (ground) and activated (transition) states. The subscript i characterizes the arrangement of particles.

the potential barrier (in the activated state), the adsorbed species are located on a lattice near, respectively, well defined potential minima and saddle positions. These positions are identified with adsorption sites, and the adsorbed particles are described in terms of occupation of these sites and the corresponding lateral interactions. Specifically, one should distinguish the lateral interactions in the ground and activated states, i and i! (the subscript i characterizes here the arrangement of particles). From the point of view of chemical kinetics, these interactions are equally important. The energetics of adsorption depends however only on the interaction in the ground state. This means that in the situations when an adsorbed overlayer is close to equilibrium this interaction determines the probabilities of different arrangements of adsorbed particles. The lateral interaction in the activated state may explicitly manifest itself, e.g., in the coverage dependence of the energy distribution of the reaction products directly desorbing to the gas phase as shown in calculations (Zhdanov, 1991a) and observed in several reactions (Bald and Bernasek, 1998; Nakao et al., 2005). Lateral interactions (in the ground state) result in ordering of adsorbed particles. With the development of the low-energy electron diffraction (LEED) technique, this phenomenon was observed in thousands of adsorbate/substrate systems (Somorjai, 1994). Analyzing the LEED data obtained at different coverages and temperatures makes it possible to construct adsorbate-substrate phase diagrams (Van Hove et al., 1986). Comparing measured and calculated phase diagrams (for the latter, see e.g. a review by Patrykiejew et al. (2000)), one can obtain values of lateral interactions. Another common way to evaluate lateral interactions is based on the analysis of desorption kinetics (see Section 7.6.2). Quantitative information on lateral interactions can also be extracted from high-resolution scanning tunneling microscopy (STM) data by analyzing distribution of adsorbed particles (Österlund et al., 1999). The values of lateral interactions between two nearest-neighbour (nn) particles are often in the range 1–4 kcal/mol. This means that the lateral interactions are relatively weak com-


247

pared to the adsorption energy. The specifics of lateral interactions depend on their nature (Einstein, 1978; Norskov, 1993). In particular, dipole–dipole and elastic interactions are usually repulsive, (r) ∝ 1/r 3 (r is the adsorbate–adsorbate distance). Indirect substratemediated interaction behaves as (r) ∝ sin(2kF r)/r n , where kF is the Fermi momentum, and n is the exponent (usually n is between 3 and 5). Attractive van der Waals interaction, (r) ∝ −1/r 6 , is usually important only for adjacent particles. In kinetic simulations, the lateral interactions are often assumed to be pairwise. In general, the interactions may however be non-additive, i.e., one should take into account three-body interactions, etc. Although accurate calculation of lateral interactions is hardly possible even with the current computer facilities, there are examples (see Section 7.6.2) indicating that the interactions obtained by using DFT are often in reasonable agreement with the experiment. At low coverages, adsorbed particles have no neighbours. At high coverages, all the nn sites are often occupied. This means that the change of the adsorption energy with increasing coverage is about z1 ∼ 5–30 kcal/mol, where 1 is the nn interaction, and z the number of nn sites. This difference may result in dramatic changes in the reaction kinetics with increasing coverage. This was clear already in the 1930s (Zhdanov, 2002a and references therein). 7.6.1. General equations General equations describing the effect of lateral interactions on the kinetics of adsorption, desorption and simplest reactions can easily be derived by using TST and the lattice-gas approximation (Zhdanov, 1981, 1991a). For a given arrangement of adsorbed particles, the lateral-interaction-related shift of the activation energy of a rate process is i! −i (Fig. 7.4), and accordingly the transition probability is proportional to exp[−(i! − i )/kB T ]. The process rate is given by summation of the contributions corresponding to various arrangements. If an adsorbed overlayer is in a one-phase state, the process rate can be identified with the corresponding phenomenological kinetic equations, and one can obtain an expression for a process rate constant. This strategy is applicable for elementary rate processes occurring on uniform and heterogeneous surfaces. It can be used in the situations when the surface is stable and also often in the cases of adsorbate-induced surface restructuring. On a uniform surface, far example, the kinetics of monomolecular adsorption and desorption is described as dθ/dt = ka P − kd θ, where θ is the coverage, P is pressure, and ka = (1 − θ )ka◦ P0,i exp(−i! /kB T ), kd = kd◦

(7.19)

(7.20)

i

PA,i exp −(i! − i )/kB T

(7.21)

i

are the adsorption and desorption rate constants. In these equations, P0,i is the probability that a vacant site has the environment denoted by index i. PA,i is a similar probability for an adsorbed particle. The constants ka◦ and kd◦ , corresponding to the low-coverage limit, can be

248

V.P. Zhdanov

expressed via the partition functions like in Eq. (7.1). Expressions (7.20) and (7.21) imply that the partition functions are independent of the arrangement of adsorbed particles (if necessary, the corresponding dependence can easily be taken into account (Zhdanov, 1989b)). In addition, the use of expressions (7.20) and (7.21) imply that adsorption and desorption of a particle occur directly to or from an adsorption site. In reality, both processes may be influenced by the presence of precursor states. The effect of precursor states on the kinetics of adsorption and desorption should be analyzed by explicitly treating the adsorbate migration via such states. In real systems, diffusion of adsorbed particles is usually rapid, an adsorption overlayer is close to equilibrium, and accordingly one can employ the statistical mechanics in order to calculate the arrangement probabilities P0,i and PA,i . In particular, using the grand canonical distribution, Eq. (7.21) can be rewritten as (Zhdanov, 1991a)

P0,i exp −i! /kB T , kd = kd◦ (1 − θ )/θ exp(μa /kB T ) (7.22) i

where μa is the chemical potential of adsorbed particles defined so that μa → kB T ln(θ ) at θ → 0, or (cf. Eqs. (7.20) and (7.22))

kd = kd◦ ka /ka◦ θ exp(μa /kB T ). (7.23) Relationships (7.22) and (7.23) are convenient for analytical calculations. In particular, they can be used in order to illustrate that expressions (7.20) and (7.21) satisfy the detailed balance principle. More specifically, relationships (7.22) and (7.23) make it possible to show that Eq. (7.19) in combination with expressions (7.20) and (7.21) correctly describe adsorption–desorption equilibrium. At adsorption–desorption equilibrium, Eq. (7.19) yields ka P = kd θ.

(7.24)

On the other hand, the statistical physics prescribes that at equilibrium the chemical potentials of adsorbed and gas-phase particles have to be equal, μ a = μg .

(7.25)

Substituting Eqs. (7.20) and (7.22) into Eq. (7.24) and employing the standard expression for the chemical potential of the gas-phase particles, μg = kB T ln P + A,

(7.26)

where A is the constant depending on details of the definition of μg , one can easily obtain condition (7.25). This means that Eqs. (7.20) and (7.21) are consistent with the detailed balance principle. An appropriate note here is that Eq. (7.23) can actually be derived from this principle (provided that surface diffusion is relatively fast). Thus, Eq. (7.23) is often applicable beyond the limitations of TST and the lattice-gas approximation. In addition, the latter derivation of Eq. (7.23) does not need the assumption that adsorption and desorption of a particle occur directly to or from an adsorption site. Thus, Eq. (7.23) holds in the presence of precursor states as well.


249

In the case of desorption, the activated complex is often weakly bound to the surface and accordingly the interaction i! is small or negligible compared to i . In general, however, these interactions may be comparable. A priori, there are no rules relating i and i! . Sometimes, however, the interaction i! is assumed to be proportional to i , and accordingly the contribution of lateral interactions to the adsorption and desorption activation energies is represented respectively as i! = (1 − α)i and i! − i = −αi . These expressions are similar to the Bronsted–Polanyi relation between the activation energy and reaction exothermicity (the use of this relation implies that 0 α 1). The general equations outlined above are simple. In practice, their use is however complicated by the need to accurately calculate the probabilities of various arrangements of adsorbed particle or the coverage dependence of the adsorbate chemical potential. This part of the problem can be solved by employing cluster approximations (e.g., the quasi-chemical approximation), the transfer-matrix technique (TMT), or Monte Carlo (MC) simulations. The MC method is especially convenient, because it makes it easily possible to analyze the situations when the lateral interactions are complex and/or a reaction occurs with participation of several reactants. An alternative approach for describing the kinetics of adsorption and desorption was proposed and used by Ward an co-workers in a long series of papers (Ward et al., 1982a, 1982b; Ward and Fang, 1999; Ward, 2002; see also references in the Comment (Zhdanov, 2001) on Ward’s papers). For monomolecular adsorption on a uniform surface, for example, they write

dθ/dt = K exp (μg − μa )/kB T − exp (μa − μg )/kB T , (7.27) where K is the “exchange” constant. The two terms in the right-hand part of this equation represent respectively the rates of adsorption and desorption. Equation (7.27) was also employed by Rudzinski and co-workers (Rudzinski and Aharoni, 1997; Rudzinski et al., 1999; Panczyk and Rudzinski, 2003; see also references in the corresponding Comment (Zhdanov, 2001)). In particular, they tried to generalize Eq. (7.27) to the case of heterogeneous surfaces (Rudzinski et al., 1999). The main difference between Eq. (7.27) and the former equations is that in the latter case the rates of adsorption and desorption depend simultaneously on μa and μg . This results in unusual predictions. For example, substituting expression (7.26) for the chemical potential of gas-phase particles into Eq. (7.27), one can obtain that the ratio of the adsorption and desorption rate is proportional to P 2 . This and other predictions drawn on the basis of Eq. (7.27) make however no sense, because the analysis of the derivation of Eq. (7.27) indicates flaws in the use of the golden rule (Zhdanov, 2001). 7.6.2. Temperature-programmed desorption A common way to evaluate lateral interactions is based on the analysis of desorption kinetics. Due to adsorbate–adsorbate lateral interactions or surface heterogeneity, the desorption rate usually rapidly decreases with decreasing coverage. If temperature is constant, this effect often makes it difficult to study experimentally the desorption kinetics in a wide range of coverage. To overcome this problem, the temperature is often increased in the linear fashion, T (t) = T (0) + βt, where β is the heating rate. The temperature dependence of

250

V.P. Zhdanov

the desorption rate in such temperature-programmed desorption (TPD) regimes is called the TPD spectrum. Lateral interactions result in broadening or splitting of TPD spectra and accordingly can be estimated by comparing the measured and calculated TPD kinetics. Experimental and theoretical studies of the TPD kinetics are numerous (Kang and Weinberg, 1995; Zhdanov, 1991a). In this section, we outline the results of a few recent theoretical works in order to show the current state of the art in this field, or more specifically (i) to illustrate typical TPD spectra (from very simple, like for the CO/Rh(100) system, to rather complex like for the N2 O/Rh(110) system), (ii) to demonstrate the use of various techniques, and (iii) to highlight what can be obtained by using DFT. Relatively simple TPD kinetics are usually observed in the case of physical adsorption. The lateral attraction between physically adsorbed particles is weak. Desorption occurs however at low temperatures and accordingly the influence of lateral interaction on the TPD spectra is appreciable. Due to this interaction, the TPD peaks shift to higher temperature with increasing initial coverage so that the apparent order of the desorption kinetics is close to zero or is between zero and one. This well-known effect was analyzed in detail by Lehner et al. (2003). Specifically, they performed kinetic MC simulations of Xe desorption from Pt(111) (it occurs at T 90 K; the nn lateral interaction was found to be −0.16 kcal/mol) and N2 desorption from Cu(110)-(2×1)O (it occurs at T 43 K; the nn lateral interaction, −0.07 kcal/mol, was assumed to be directed along the Pt rows). Kinetic MC simulations of the TPD spectra for the CO/Rh(100) system were performed by Jansen (2004). In this case, CO adsorbs on top sites and accordingly the substrate was represented by a square lattice. The experimental TPD spectra were described (Fig. 7.5)

Fig. 7.5. TPD spectra for the CO/Rh(100) system: (a) experiment, (b) simulations with (thick lines) and without (thin lines) lateral interactions. The heating rate is 5 K/s. The initial coverages are indicated near the curves. (According to Jansen (2004).)


251

Fig. 7.6. O2 TPD spectra for the O/Pt(111) system: (a) experiment, (b) and (c) simulations with 2 = 2 kcal/mol and 3 = −1 and 0 kcal/mol, (d) scheme illustrating the lateral interactions. The initial coverages are 0.073, 0.093, 0.164 and 0.194 ML, respectively. The heating rate is 8 K/s. (According to Zhdanov and Kasemo (1998).)

taking into account the first three nn interactions, 1 = 5.7 kcal/mol, 2 = 0.26 kcal/mol, and 3 = 0.22 kcal/mol. Van Bavel et al. (2005) presented kinetic MC simulations of the CO TPD spectra for the CO/N/Rh(100) system. The adsorbate–adsorbate lateral interactions, calculated by using DFT, were found to be pairwise (this conclusion is in line with the results obtained by Jansen (2004)). The agreement between the simulations and experiment was however imperfect. The O2 TPD spectra for the O/Pt(111) system were simulated (Zhdanov and Kasemo, 1998) by integrating a kinetic equation expressing the desorption rate via the O chemical potential (like in Eq. (7.23)). The latter quantity was calculated using the MC technique. Oxygen atoms were assumed to be adsorbed on a triangular lattice of fcc sites. The nn O–O lateral interaction, 1 , was considered to be strongly repulsive so that there are no

252

V.P. Zhdanov

occupied nn sites. The second and third nn interactions, 2 0 and 3 0, were chosen to reproduce the p(2 × 2) ordering of oxygen atoms. Comparison between the simulations and experimental TPD spectra (Fig. 7.6) is supporting repulsive O–O interactions, e.g., 2 = 2 kcal/mol and 3 = 0. More specifically, the TPD spectra are hard to reconcile with the assumption that the O–O lateral interactions are partly attractive and that this attraction is sufficiently strong to stabilize the p(2 × 2) islands at low coverages and relatively high temperatures (about or above 400 K) typical for catalytic reactions. More recently, the O2 TPD spectra for the O/Pt(111) system were calculated by Jansen and Offermans (2005) using the kinetic MC technique in combination with DFT. According to DFT, the first two nn lateral interactions are in the range 1 = 4.8–6.2 kcal/mol and 2 = 1.1–1.3 kcal/mol, the third interaction (3 ) is negligible, and the linear three-body interaction is about 1.5 kcal/mol. The lateral interaction in the activated state was taken into account by using the Bronsted–Polanyi relation (specifically, α was varied from 1 to 0.77). With these parameters, the model reproduces the experiment. Jansen (2008) also discussed in detail the O ordering on Pt(111). Stampfl et al. (1999) analyzed the O2 TPD spectra for the O/Ru(0001) system (Fig. 7.7). The desorption kinetics were calculated by integrating a phenomenological kinetic equation. In particular, the desorption rate was expressed (cf. Eq. (7.23)) via the sticking coefficient, obtained from the experiment, and the O chemical potential calculated using TMT. Adsorption was allowed on hcp and fcc sites (the role of the latter sites was found to be mi-



253

Fig. 7.7. O2 TPD spectra for the O/Ru(0001) system: (a) theory, (b) experiment. The initial coverage ranges from 0.1 to 1 ML. The heating rate is 6 K/s. (According to Stampfl et al. (1999).)

Fig. 7.8. O2 TPD spectra for the O/Rh(100) system. The heating rate is 6 K/s. (According to Hansen and Neurock (2000).)

nor, due to low occupation). The adsorption energies and lateral interactions were obtained employing DFT. For adsorption on hcp sites, the first three nn interactions were found to be 1 = 6.1 kcal/mol, 2 = 1.0 kcal/mol, and 3 = −0.6 kcal/mol. The absolute values of the three-body interactions were predicted to be about 1 kcal/mol. With these ingredients, the model reasonably fits the experiment. Hansen and Neurock (2000) presented kinetic MC simulations of the O2 TPD spectra for the O/Rh(100) system (Fig. 7.8). The adsorption energies and lateral interactions were obtained using DFT. In particular, the lateral interactions were described by employing a DFT-parameterized bond order conservation model.

254

V.P. Zhdanov

In the examples described above (Figs. 7.5–7.8), the adsorbed overlayer contains particles of one kind. Under chemically reactive conditions, there are usually two or more adsorbates and accordingly one needs to introduce a lot of parameters in order to simulate the corresponding TPD spectra. For instance, we refer to interesting DFT-based MC simulations of the NO and N2 TPD spectra observed during NO adsorption and decomposition on the (111) and (100) faces of Rh (Hermse et al., 2003; Van Bavel et al., 2004). The former work indicate that on Rh(111) the NO ordering and N2 TPD spectra can be described by using the DFT lateral interactions provided that one takes into account that different species are adsorbed at different sites. According to the latter work, the DFT lateral interactions are significantly higher than those estimated from fitting the experiment. Petrova and Yakovkin (2005) used the MC technique in order to simulate the CO2 TPD spectra for CO oxidation on Pt(111). The adsorbate–adsorbate lateral interactions were chosen to reproduce the ordering of adsorbed particles. Another example of TPD occurring under chemically reactive conditions is N2 desorption observed during N2 O decomposition on Rh(110) (Imamura et al., 2004). In this case, the N2 O decomposition occurs between 60 and 180 K and results in the appearance of five N2 TPD peaks (Fig. 7.9). Experimental studies of the angular distribution of the flux of the reaction products indicate that N2 molecules leave the surface during or after N2 O dissociation events. The corresponding MC simulations (Zhdanov and Matsushima, 2005) take into account both channels of N2 desorption and also N2 O–O lateral interactions stabilizing N2 O adsorption. With these ingredients, the model reproduces the main features of the observed TPD kinetics, including the positions and intensities of the four peaks, registered between 60 and 150 K and related to N2 desorption accompanying N2 O-dissociation acts, and the peak recorded at 160 K and attributed to desorption of adsorbed N2 molecules. A similar model was recently used to simulate N2 desorption observed during N2 O decomposition on Rh(100) (Zhdanov and Matsushima, 2007b). 7.6.3. Surface diffusion The peculiarities of the dynamics and statistics of diffusion of adsorbed particles were comprehensively reviewed by the author of this chapter (Zhdanov, 1991a) and more recently by Zgrablich (1997), Nieto et al. (1998), Barth (2000), Ala-Nissila et al. (2002), Naumovets and Zhang (2002), and Antczak and Ehrlich (2005b). Complementing these reviews, we present here a few recent references. Diffusion of particles on a surface often occurs via jumps to nn vacant sites. The corresponding jump rate constants can be calculated by using TST. If however the energyexchange between adsorbed particles and the substrate is slow and the corrugation of the adsorption potential is weak or if the temperature is sufficiently high, the long-range jumps may be significant as well (Zhdanov, 1991a). In particular, the diffusion coefficient may be inversely proportional to the friction coefficient. The experiments performed by Antczak and Ehrlich (2005a) indicate, e.g., that in the case of diffusion of single Ir atoms on W(110) the long-range jumps become important already at T 370 K. Theoretically, the contribution of long-range jumps to diffusion was discussed by Braun and Ferrando (2002), Vega et al. (2002), Shushin and Pollak (2003), Guantes et al. (2003), and Sancho et al. (2004).


255

Fig. 7.9. N2 TPD spectra for N2 O adsorption on the (1 × 1) Rh(110) surface: (a) angular-integrated experimental data, (b) MC simulations. The initial N2 O coverages are indicated near the curves. The heating rate is 2.5 K/s. (According to Zhdanov and Matsushima (2005).)

As already mentioned in Section 7.3, the tunnel diffusion of hydrogen on Cu(001) was theoretically studied by Sundell and Wahnström (2004a, 2004b). The coefficient of surface diffusion often strongly depends on coverage due to adsorbate–adsorbate lateral interactions. This aspect of chemical diffusion on square and

256

V.P. Zhdanov

triangular lattices was studied by Tarasenko et al. (2000, 2001, 2003) by using the realspace renormalization group and MC techniques. Zaluska-Kotur et al. (2002, 2003, 2004) presented MC simulations of oxygen diffusion on Ru(0001) and W(110). The coverage dependence of hydrogen diffusion on Pt(111) was experimentally studied by Zheng et al. (2004). The specifics of surface diffusion complicated by long-range lateral interactions was discussed by Yakes et al. (2007). The coefficient of chemical diffusion is well known to be proportional to ∂μa /∂θ , where μa is the chemical potential of adsorbed particles. Due to this factor, the diffusion coefficient as a function of coverage or temperature is expected to exhibit power-law or logarithmic singularities near the points corresponding to continuous phase transitions (Zhdanov, 1992). These singularities were studied by using the TM and MC techniques (Myshlyavtsev et al., 1995; Uebing and Zhdanov, 1998a,1998b). Diffusion of particles adsorbed on a lattice with two non-equivalent sites was simulated by Chvoj et al. (1999) and Tarasenko and Jastrabik (2002). The effect of adsorbate-induced surface restructuring on diffusion was analyzed by Tarasenko and Jastrabik (2003). Diffusion on a single-crystal surface containing steps may be suppressed or facilitated if it occurs, respectively, perpendicular or parallel to steps (Zhdanov, 1989a). Recently, the effect of steps on diffusion was theoretically analyzed by Merikoski and Ying (1997, 1998), Masin et al. (2003a, 2003b, 2004) and Chvoj et al. (2006). Diffusion on rough surfaces was discussed by Adler et al. (2004). In all the theoretical studies mentioned above, adsorbed particles are considered to occupy single sites. Diffusion of long chainlike molecules, occupying a few sites, was analyzed as well by using the MC technique (Hjelt and Vattulainen, 2000; Zhdanov and Kasemo, 2000b). 7.6.4. Conventional reaction kinetics In catalytic reactors, reactions usually occur under steady-state conditions. Taking into account the diversity of reaction mechanisms and complexity of elementary reaction steps (see, e.g., the reaction schemes compiled by Stoltze (2000) and Cortright and Dumesic (2001)), one can hardly classify in detail all the possible types of the steady-state reaction kinetics. To facilitate our presentation, we distinguish three general classes of reaction kinetics occurring respectively (i) far from equilibrium, (ii) near adsorption–desorption equilibrium, and (iii) near reaction equilibrium. In this section, we will briefly discuss reaction kinetics belonging to classes (ii) and (iii) and also transient kinetics. Reactions running far from equilibrium are discussed in Section 7.6.5. First, let us consider the simplest irreversible reaction, A + B → C, occurring via the Langmuir–Hinshelwood mechanism, Agas Aads , Bgas Bads ,

(7.28)

Aads + Bads → (AB)gas ,

(7.29)

in the situation when step (7.29) is slow and accordingly the system is close to the adsorption–desorption equilibrium. To calculate the reaction rate, W , one can use in this case the reactant coverages corresponding to adsorption equilibrium. For the Langmuir layer, an


257

elementary analysis based on this approximation yields W = kr KA PA KB PB /(1 + KA PA + KB PB )2 ,

(7.30)

where kr is the rate constant of the Langmuir–Hinshelwood step, KA and KB are the adsorption equilibrium constants, and PA and PB are the reactant pressures. For fixed B pressure, the reaction rate first linearly increases with increasing PA and then after reaching a maximum becomes inversely proportional to PA . To illustrate, the effect of the nonideality of the reaction steps on the steady-state reaction kinetics, we show (Fig. 7.10) the corresponding kinetics calculated with for KB PB = 1 repulsive lateral interactions, AA = BB = AB = 2kB T , between nearest-neighbour adsorbed particles. Comparing the ideal and non-ideal kinetics indicates that the asymptotic dependences of the reaction rate on PA are the same in both cases, but the transition region is much wider in the nonideal case. The second generic case is when a reaction with reversible steps runs near the reaction equilibrium. Under such conditions, the reaction rates in the forward and backward directions are often related by the fairly general Boreskov–Horiuti–Enomoto rules as reviewed in detail elsewhere (Zhdanov, 2002a, 2007). A good classical example of reactions occurring near equilibrium is ammonia synthesis (see recent calculations by Hellman and Honkala (2007) taking lateral interactions into account). The kinetics of reactions occurring under steady-state conditions near adsorption– desorption or reaction equilibrium are often not too sensitive to the details of a reaction mechanism. The transient kinetics are frequently more informative. For example, it is appropriate to mention STM experiments and MC simulations focused on CO oxidation on Pt(111) (Völkening and Wintterlin, 2001). In the experiments, preadsorbed oxygen was titrated by CO at temperatures between 237 and 274 K. On the nm scale, the reaction was observed to be accompanied by initial reordering, compression and subsequent shrinking of the (2 × 2) oxygen islands. The measured kinetics were reproduced using a model taking into account attractive O–O and repulsive O–CO lateral interactions. The fact that no reaction occurred in the interior of the oxygen islands was attributed to the crucial role of highly reactive O–CO configurations at the island boundaries. A similar model was recently used by Nakai et al. (2004, 2005) in order to interpret the transient kinetics of CO oxidation studied with near-edge X-ray absorption fine structure spectroscopy. Another example illustrating the interplay between the experiment and theory is H2 oxidation on Pt(111). Under steady state conditions at relatively high temperatures, this reaction is usually believed to run via steps (Hellsing et al., 1991) Had + OHad → H2 Ogas

or OHad + OHad → H2 Ogas + Oad .

(7.31)

Under transient conditions at low temperatures (below 300 K), the reaction may also occur via steps 2H2 Ogas + Oad → 2OHad + H2 Oad ,

(7.32)

2H2 Ogas + Oad → 3OHad + Had .

(7.33)

According to DFT calculations by Karlberg et al. (2003), the overlayer composition (OH+H2 O) corresponding to step (7.33) is energetically much more favorable compared

258

V.P. Zhdanov

Fig. 7.10. Steady-state kinetics of the A + B → AB reaction occurring on a square lattice via the Langmuir–Hinshelwood (LH) mechanism [steps (7.28) and (7.29)] near the adsorption–desorption equilibrium. Panels (a) and (b) show the reaction rate and reactant coverages as a function of A pressure for fixed B pressure. The solid lines are for the ideal Langmuir layer. The dashed lines are for the model taking into account (in the quasi-chemical approximation) repulsive lateral interactions, AA = BB = AB = 2kB T , between nearest-neighbour adsorbed particles. In the latter case, the reaction rate has been calculated for the two situations when (i) there are no lateral interactions in the activated state for the LH step (long dashed curves) and (ii) the lateral interactions in the activated state are the same as in the ground state (short dashed curves), respectively. kr is the rate constant of the LH step, and KA is the adsorption equilibrium constant (for the kinetics with lateral interactions, kr and KA correspond to the low-coverage limit). (According to Zhdanov (2002a).)


259

to that (2OH + H) corresponding to step (7.32). Employing DFT, Karlberg and Wahnström (2004) calculated adsorbate–adsorbate lateral interactions for the reaction intermediates and performed MC simulations illustrating that the ordered structure corresponding to the former overlayer composition is in good agreement with the structures observed in STM experiments. Nagasaka et al. (2005) presented MC simulations of the propagation of the reaction fronts during titration of adsorbed oxygen by hydrogen. 7.6.5. Oscillations, chaos and pattern formation Heterogeneous catalytic reactions occurring far from equilibrium are especially interesting, because under certain values of the governing parameters such reactions often exhibit kinetic phase transitions, oscillations, chaos, and pattern formation. All these phenomena have already been comprehensively reviewed in the current literature (Schüth et al., 1993; Slinko and Jaeger, 1994; Zhdanov and Kasemo, 1994; Imbihl and Ertl, 1995; Gruyters and King, 1997; Rotermund, 1997; Zhdanov, 2002b; Slinko and Jaeger, 2005). Here, we will only briefly (i) articulate the role of lateral interactions in kinetic oscillations and (ii) note new tendencies in simulations of oscillations. The experience accumulated during the past three decades indicates that kinetic oscillations in heterogeneous catalytic reactions are often observed in systems where a rapid bistable catalytic cycle is combined with a relatively slow “side” process, e.g., with oxide formation, carbon deposition, or adsorbate-induced surface restructuring. Such side processes are usually strongly influenced by or occur due to adsorbate–adsorbate, adsorbate–substrate and substrate–substrate lateral interactions and accordingly should often be described in terms of the theory of phase transitions. Equally often, oscillations in heterogeneous catalytic reactions are related to chemical non-linearities. In this case, the adsorbate–adsorbate lateral interactions are frequently crucial as well, because due to such interactions the kinetic equations describing elementary reaction steps are strongly non-linear, and accordingly oscillations and/or pattern formation become possible even in reactions with relatively simple mechanisms (for example, see recent simulations of oscillations in the N2 O–H2 and N2 O–CO reactions on Ir(110) (Peskov et al., 2005), bistability in the N2 O–CO reaction on Pd(110) (Zhdanov and Matsushima, 2007a), and pattern formation in reactions with promoters and poisons (De Decker et al., 2004; Zhdanov, 2005b)). Customarily, kinetic phase transitions, oscillations, chaos, and pattern formation observed in heterogeneous catalytic reactions are described by using the mean-field reaction or reaction–diffusion equations (for a recent example, see the analysis of oscillations in CO oxidation on Pt(100) by Hoyle et al. (2007)). The applicability of this general approach for interpretation of reactive processes accompanied by phase transitions is however usually open for debate. This is perhaps the key reason why the understanding of complex behaviour of heterogeneous catalytic reactions is often still far from complete. The shortcoming of the mean-field approximation can be overcome by employing the MC technique. During the past decade, this technique was proved to be efficient in simulations of oscillations, chaos, and pattern formation in catalytic reactions (Zhdanov, 2002b). For example, we show (Figs. 7.11 and 7.12) the results of MC simulations of kinetic oscillations and pattern formation on the nm scale in CO oxidation on Pt(100).

260

V.P. Zhdanov

Fig. 7.11. Example of MC simulations of oscillatory kinetics of CO oxidation on Pt(100) due to the interplay of the reaction steps and adsorbate-induced “hex”-(1 × 1) surface restructuring (this process represents a first-order phase transition): (a) CO (A) and O (B) coverages, fraction of Pt atoms in the (1 × 1) state, κ, and (b) reaction rate [per MC step (MCS)] as a function of time. The results were obtained on a (100 × 100) lattice with periodic boundary conditions. One MCS was defined as (100 × 100) attempts to realize the adsorption-reaction-surface-restructuring steps. The rate of these steps was two orders of magnitude lower compared to that of CO diffusion. In reality, CO diffusion is much faster, but the increase of the rate of CO diffusion does not change the results. (According to Zhdanov (1999).)

For a more detailed discussion of kinetic oscillations, chaos, and pattern formation, the readers are referred to the already mentioned reviews and/or to the chapter written by R. Imbihl.


261

Fig. 7.12. Snapshots of the lattice for the MC run shown in Fig. 7.11. Plus signs indicate the (1 × 1) substrate patches. Filled and open circles exhibit CO and O species. (According to Zhdanov (1999).)

7.7. Surface heterogeneity The structure of heterogeneous surfaces may be very complex. The reaction kinetics on such surfaces may be far from those derived for the ideal Langmuir layer. A general concept here is that there is distribution of adsorption sites over binding energies and catalytic activity (see the reviews by Boudart (1997) and Murzin (2005) and a relatively recent example of the use of this concept in kinetic simulations (Dooling et al., 1999)). Specifically, there are active sites or active centers at which reactions primarily occur. Such sites are usually believed to be associated with various defects. For example, single-crystal surfaces contain terraces separated by steps. In this case, active sites may be associated with step or kink sites (kinks are defects of steps). The surface-science based studies confirm that steps are often (but not always) more active than regular terrace sites (Zhdanov, 2002a). The interpretation of reaction kinetics on heterogeneous surfaces is usually difficult. The use of DFT in combination with kinetic MC simulations opens new opportunities in this field (Christensen and Norskov, 2008). For example, the DFT calculations indicate that the steps are much more active than terraces in NO dissociation on Pd(111) (Hammer, 2001) and in ammonia synthesis on Ru (Hellman et al., 2006). The corresponding MC simulations (Olsson et al., 2003) show what may happens under such conditions in the NO–CO reaction. The kinetics of ammonia synthesis occurring on steps was analysed by Hellman and Honkala (2007).

262

V.P. Zhdanov


Another trend which merits to be mentioned here is related to attempts to experimentally and theoretically clarify the specifics of catalytic reactions running on composite surfaces (Shvartsman et al., 1999, 2000; Jansen and Hermse, 1999; Hermse and Jansen, 2000; Zhdanov and Kasemo, 2000a, 2001).

7.8. Reaction kinetics on nm-sized catalyst particles Real catalysts usually consist of small metal particles deposited on the internal surface of a more or less inactive porous support (this design is used to maximize the active surface area per unit weight and volume of the catalyst). In practice, such catalysts are shaped either as porous pellets (2–20 mm in diameter) or so-called monoliths with a porous “washcoat”. The size of pores varies in a large range from 1–2 nm (micropores) to 50 nm (macropores). Often the pores are mesoscopic (2–50 nm). The size of metal particles, d, may vary in a wide range as well. In zeolites (with pores of about 1 nm), metal particles often contain only a few atoms (such particles are called clusters). More typical size is 1–20 nm. Sometimes, however, supported particles are much larger (e.g., the silver particles employed for selective oxidation of ethylene are roughly 103 nm in size, and Pt and Rh particles of car exhaust cleaning catalysts are typically 10–100 nm after some time of use). The turnover rate of catalytic reactions is well known to often depend on d. At the smallest length scales of the order of 1 nm, metal particles have 2D or 3D sizes comparable to


263

the electronic screening length in metals, and the electron structure is significantly different from that of bulk metals. Consequently, their catalytic activities are often also different compared to larger particles or single-crystal surfaces. At somewhat larger sizes, 3–4 nm, the electronic properties of the particles are already close to those of the bulk metal, except at the atoms contacting or very near the support where metal support interaction may modify the catalytic properties. The latter may be significantly perturbed, electronically, and thereby also have different catalytic activities. Since the particles are small, these sites may be important or even dominant for the overall catalytic kinetics. At still larger sizes, above about 10 nm, the particles are electronically essentially identical to bulk metal but may still exhibit remarkably different kinetics, compared to single crystals. In this case, the reasons of structure sensitivity may be purely kinetic. The basic underlying mechanisms for kinetic phenomena which do not occur on single crystals are: (i) the different catalytic activities on different facets of a small supported crystalline particle become coupled in a strongly non-linear fashion due to diffusion occurring over facet boundaries, (ii) equilibrium-shape changes of small particles, caused by adsorbates, induce changes in catalytic behaviour, (iii) different kinetic rate constants at the facet boundaries of a supported particle compared to those for the perfect facets give rise to new kinetics, and (iv) spillover by diffusion of reactants, between the particle and its support, also create new kinetics. All these factors are especially important for catalytic reactions occurring far from adsorption–desorption equilibrium. The specifics of reaction kinetics on nm-sized catalyst particles was comprehensively discussed in several reviews (Henry, 1998; Zhdanov and Kasemo, 2000c, 2003; Libuda and Freund, 2005). The progress in this field is expected to be faster with the use of DFT in combination with kinetic MC simulations. This approach makes it possible to identify active sites, to scrutinize the reaction kinetics, and sometimes to quantitatively predict the catalytic activity. For example, it is appropriate to mention the DFT-based analysis of the kinetics of ammonia synthesis over a nanoparticle ruthenium catalyst (Honkala et al., 2005). This reaction is limited by N2 dissociation. The most active sites for this step were found to be located on the boundaries of the (111) facets. The reaction kinetics was treated taking into account the size distribution of ruthenium particles measured by transmission electron microscopy. The calculated rate was found to be within a factor of 3 to 20 of the experimental one. Such examples illustrate the potential of computer-based methods in the search for catalysts.

7.9. Conclusion In this chapter, the general concepts used to rationalize the dynamics and kinetics of heterogeneous catalytic reactions were illustrated by using the results obtained during the past decade. The goal of the presentation was to show new trends in this interdisciplinary field of research. After reading this chapter, one is expected to feel that despite the long flourishing history this field is still in progress, and many interesting problems here are open for further studies.

264

V.P. Zhdanov

Acknowledgements The author thanks Bengt Kasemo, Tatsuo Matsushima, Christian Uebing, and Kirill I. Zamaraev for collaboration and useful discussions.

References Adler, P.M., Malevich, A.E., Mityushev, V., 2004. Phys. Rev. E 69, 011607. Ala-Nissila, T., Ferrando, R., Ying, S.C., 2002. Adv. Phys. 51, 949. Antczak, G., Ehrlich, G., 2005a. Phys. Rev. B 71, 115422. Antczak, G., Ehrlich, G., 2005b. Surf. Sci. 589, 52. Bald, D.J., Bernasek, S.L., 1998. J. Chem. Phys. 109, 746. Banerjee, D., Banik, S.K., Bag, B.C., Ray, D.S., 2002. Phys. Rev. E 66, 051105. Barth, J.V., 2000. Surf. Sci. Rep. 40, 75. Behler, J., Delley, B., Lorenz, S., Reuter, K., Scheffler, M., 2005. Phys. Rev. Lett. 94, 036104. Behler, J., Reuter, K., Scheffler, M., 2008. Phys. Rev. B 77, 115421. Bos, A.N.R., Lefferts, L., Marin, G.B., Steijns, M.H.G.M., 1997. Appl. Catal. A: Gen. 160, 185. Boudart, M., 1997. In: Ertl, G., Knözinger, H., Weitkamp, J. (Eds.), Handbook of Heterogeneous Catalysis, vol. 1. VCH, Weinheim, p. 1. Braun, O.M., Ferrando, R., 2002. Phys. Rev. E 65, 061107. Christensen, C.H., Norskov, J.K., 2008. J. Chem. Phys. 128, 182503. Chvoj, Z., Chab, V., Conrad, H., 1999. Surf. Sci. 426, 8. Chvoj, Z., Masin, M., Alla-Nissila, T., 2006. J. Stat. Mechan., P10003. Cortright, R.D., Dumesic, J.A., 2001. Adv. Catal. 46, 161. De Decker, Y., Marbach, H., Hinz, M., Günther, S., Kiskinova, M., Mikhailov, A.S., Imbihl, R., 2004. Phys. Rev. Lett. 92, 198305. Dooling, D.J., Rekoske, J.E., Broadbelt, L.J., 1999. Langmuir 15, 5846. Dumesic, J.A., 2001. Adv. Catal. 46, 161. Dumesic, J.A., Rudd, D.F., Aparicio, L.M., Rekoske, J.E., Trevino, A.A., 1993. The Microkinetics of Heterogeneous Catalysis. ACS, Washington, DC. Duke, C.B., 1993. Foreword to “Surface Science: The First Thirty Years”. Surf. Sci. 299/300. Einstein, T.L., 1978. Crit. Rev. Sol. State Mater. Sci. 7, 261. Ertl, G., 2000. Adv. Catal. 45, 1. Fichthorn, K.A., Miron, R.A., 2002. Phys. Rev. Lett. 89, 196103. Gellman, A.J., Paserba, K.R., 2002. J. Phys. Chem. B 106, 13231. Gadzuk, J.W., 2002. J. Phys. Chem. B 106, 8265. Gross, A., 1998. Surf. Sci. Rep. 32, 291. Gruyters, M., King, D.A., 1997. Faraday Trans. 93, 2947. Guantes, R., Vega, J.L., Miret-Artes, S., Pollak, E., 2003. J. Chem. Phys. 119, 2780. Gumhalter, B., Matsushima, T., 2004. Surf. Sci. 261, 183. Hammer, B., 2001. J. Catal. 199, 171. Hanggi, P., Talkner, P., Borkovec, M., 1990. Rev. Mod. Phys. 62, 251. Hansen, E.W., Neurock, M., 2000. Surf. Sci. 464, 91. Harris, J., Kasemo, B., 1981. Surf. Sci. 105, L281. Hellman, A., Razaznejad, B., Yourdshahyan, Y., Ternov, H., Zoric, I., Lundquist, B.I., 2003. Surf. Sci. 532, 126. Hellman, A., Honkala, K., Remediakis, I.N., Logadottir, A., Carlsson, A., Dahl, S., Christensen, C.H., Norskov, J.K., 2006. Surf. Sci. 600, 4264. Hellman, A., Honkala, K., 2007. J. Chem. Phys. 127, 194704. Hellsing, B., Zhdanov, V.P., Kasemo, B., 1991. J. Catal. 132, 210. Hjelt, T., Vattulainen, I., 2000. J. Chem. Phys. 112, 4731. Henriksen, N.E., Hansen, F.Y., Billing, G.D., 2000. Chem. Phys. Lett. 330, 139.


265

Henriksen, N.E., Hansen, F.Y., 2002. Phys. Chem. Chem. Phys. 4, 5995. Henry, C.R., 1998. Surf. Sci. Rep. 31, 231. Hermse, C.G.M., Frechard, F., van Bavel, A.P., Lukkien, J.J., Niemantsverdriet, J.W., van Santen, R.A., Jansen, A.P.J., 2003. J. Chem. Phys. 118, 7081. Hermse, C.G.M., Jansen, A.P.J., 2000. Surf. Sci. 461, 168. Honkala, K., Hellman, A., Remediakis, I.N., Logadottir, A., Carlsson, A., Dahl, S., Christensen, C.H., Norskov, J.K., 2005. Science 307, 555. Hoyle, R.B., Angher, A.T., Proctor, M.R.E., King, D.A., 2007. Phys. Rev. Lett. 98, 226102. Imamura, K., Horino, H., Rzenicka, I., Kobal, I., Kokalj, A., Ohno, Y., Nieuwenhuys, B.E., Hiratsuka, A., Matsushima, T., 2004. Surf. Sci. 566, 1076. Imbihl, R., Ertl, G., 1995. Chem. Rev. 95, 697. Jansen, A.P.J., 2004. Phys. Rev. B 69, 035414. Jansen, A.P.J., 2008. Phys. Rev. B 77, 073408. Jansen, A.P.J., Hermse, C.G.M., 1999. Phys. Rev. Lett. 83, 3673. Jansen, A.P.J., Offermans, W.K., 2005. Lect. Not. Comp. Sci. 3480, 1020. Kang, H.C., Weinberg, W.H., 1995. Chem. Rev. 95, 667. Karlberg, G.S., Olsson, F.E., Persson, M., Wahnström, G., 2003. J. Chem. Phys. 119, 4865. Karlberg, G.S., Wahnström, G., 2004. Phys. Rev. Lett. 92, 136103. Kato, K., Uda, T., Terakura, K., 1998. Phys. Rev. Lett. 80, 2000. Kleyn, A.W., 2003. Chem. Soc. Rev. 32, 87. Kramers, H.A., 1940. Physica 7, 284. Krylov, S.Y., Hermans, L.J.F., 2002. Phys. Rev. B 65, 092205. Lehner, B., Hohage, M., Zeppenfeld, P., 2003. Chem. Phys. Lett. 369, 275. Lei, R.Z., Gellman, A.J., Koel, B.E., 2004. Surf. Sci. 554, 125. Liu, D.J., Evans, J.W., 2005. Multiscale Model. Simul. 4, 424. Libuda, J., Freund, H.-J., 2005. Surf. Sci. Rep. 57, 157. Masel, R.I., 1996. Principles of Adsorption and Reaction on Solid Surfaces. Wiley, New York. Masin, M., Vattulainen, I., Ala-Nissila, T., Chvoj, Z., 2003a. Surf. Sci. 529, L256. Masin, M., Vattulainen, I., Ala-Nissila, T., Chvoj, Z., 2003b. Surf. Sci. 544, L703. Masin, M., Vattulainen, I., Ala-Nissila, T., Chvoj, Z., 2004. Surf. Sci. 566, 143. Matsushima, T., 2003. Surf. Sci. Rep. 52, 1. Merikoski, J., Ying, S.C., 1997. Phys. Rev. B 56, 2166. Merikoski, J., Ying, S.C., 1998. Phys. Rev. B 56, 15912. Miller, W.H., 1993. Acc. Chem. Res. 26, 174. Molinari, E., Tomellini, M., 2002. Chem. Phys. 277, 373. Moula, M.G., Wako, S., Cao, G.Y., Kimura, K., Ohno, Y., Kobal, I., Matsushima, T., 1999. Phys. Chem. Chem. Phys. 1, 3677. Murzin, D.Y., 2005. Ind. Eng. Chem. Res. 44, 1688. Myshlyavtsev, A.V., Stepanov, A.A., Uebing, C., Zhdanov, V.P., 1995. Phys. Rev. B 52, 5977. Nagasaka, M., Kondoh, H., Ohta, T., 2005. J. Chem. Phys. 122, 204704. Nakai, I., Kondoh, H., Amemiya, K., Nagasaka, M., Nambu, A., Shimada, T., Ohta, T., 2004. J. Chem. Phys. 121, 5035. Nakai, I., Kondoh, H., Amemiya, K., Nagasaka, M., Shimada, T., Yokota, R., Nambu, A., Ohta, T., 2005. J. Chem. Phys. 122, 134709. Nakao, K., Hayashi, H., Uetsuka, H., Ito, S., Onishi, H., Tomishige, K., Kunimori, K., 2003. Catal. Lett. 85, 213. Nakao, K., Ito, S., Tomishige, K., Kunimori, K., 2005. Chem. Phys. Lett. 410, 86. Naumovets, A.G., Zhang, Z.Y., 2002. Surf. Sci. 500, 414. Nienhaus, H., 2002. Surf. Sci. Rep. 45, 3. Nienhaus, H., Bergh, H.S., Gergen, B., Majumdar, A., Weinberg, W.H., McFarland, E.W., 1999. Phys. Rev. Lett. 82, 446. Nieto, F., Tarasenko, A.A., Uebing, C., 1998. Def. Diff. Forum 162, 59. Nikitin, E.E., 1974. Theory of Elementary Atomic and Molecular Processes in Gases. Clarendon, Oxford. Norskov, J.K., 1993. In: King, D.A., Woodruff, D.P. (Eds.), Coadsorption, Promoters and Poisons. Elsevier, Amsterdam, p. 1.

266

V.P. Zhdanov

Olsson, L., Zhdanov, V.P., Kasemo, B., 2003. Surf. Sci. 529, 338. Österlund, L., Pedersen, M.O., Stensgaard, I., Legsgaard, E., Besenbacher, F., 1999. Phys. Rev. Lett. 83, 4812. Österlund, L., Zoric, I., Kasemo, B., 1997. Phys. Rev. B 55, 15452. Panczyk, T., Rudzinski, W., 2003. Langmuir 19, 1173. Patrykiejew, A., Sokolowski, S., Binder, K., 2000. Surf. Sci. Rep. 37, 207. Paserba, K.R., Gellman, A.J., 2001. Phys. Rev. Lett. 86, 4338. Paserba, K.R., Vaidyanathan, N., Gellman, A.J., 2002. Langmuir 18, 9799. Peskov, N.V., Slinko, M.M., Carabineiro, S.A.C., Nieuwenhuys, B.E., 2005. Catal. Today 105, 223. Petrova, N.V., Yakovkin, I.N., 2005. Surf. Sci. 578, 162. Pollak, E., Talker, P., 2005. Chaos 15, 02616. Raukerma, A., Butler, D.A., Box, F.M.A., Kleyn, A.W., 1996. Surf. Sci. 347, 151. Romm, L., Katz, G., Kosloff, R., Asscher, M., 1997. J. Phys. Chem. B 101, 2213. Rotermund, H.H., 1997. Surf. Sci. Rep. 29, 265. Rudzinski, W., Aharoni, C., 1997. Langmuir 13, 1089. Rudzinski, W., Borowecki, T., Dominko, A., Panczyk, T., 1999. Langmuir 15, 6386. Sancho, J.M., Lacasta, A.M., Lindenberg, K., Sokolov, I.M., Romero, A.H., 2004. Phys. Rev. Lett. 92, 250601. Sato, S., 2005. Chem. Phys. 315, 65. Schenter, G.K., Garrett, B.C., Truhlar, D.G., 2003. J. Chem. Phys. 119, 5828. Schüth, F., Henry, B.E., Schmidt, L.D., 1993. Adv. Catal. 39, 51. Shalashilin, D.V., Jackson, B., Persson, M., 1998. Faraday Disc. 110, 287. Shushin, A.I., Pollak, E., 2003. J. Chem. Phys. 119, 10941. Shvartsman, S., Shcütz, E., Imbihl, R., Kevrekidis, I.G., 1999. Phys. Rev. Lett. 83, 2857. Shvartsman, S., Shcütz, E., Imbihl, R., Kevrekidis, I.G., 2000. Phys. Rev. Lett. 85, 902. Slinko, M.M., Jaeger, N.I., 1994. Oscillatory Heterogeneous Catalytic Systems. Elsevier, Amsterdam. Slinko, M.M., Jaeger, N.I. (Eds.), 2005. Oscillatory Behaviour of Heterogeneous Catalytic Reactions. Catal. Today 105 (2). Somorjai, G.A., 1994. Introduction to Surface Chemistry and Catalysis. Wiley, New York. Stampfl, C., Kreuzer, H.J., Payne, S.H., Pfnur, H., Scheffler, M., 1999. Phys. Rev. Lett. 83, 2993. Stoltze, P., 2000. Progr. Surf. Sci. 65, 65. Sundell, P.G., Wahnström, G., 2004a. Phys. Rev. Lett. 92, 155901. Sundell, P.G., Wahnström, G., 2004b. Phys. Rev. B 70, 081403. Tarasenko, A.A., Jastrabik, L., 2002. Surf. Sci. 507, 108. Tarasenko, A.A., Jastrabik, L., 2003. Surf. Sci. 532, 588. Tarasenko, A.A., Nieto, F., Jastrabik, L., Uebing, C., 2000. Eur. Phys. J. D 12, 311. Tarasenko, A.A., Nieto, F., Jastrabik, L., Uebing, C., 2001. Phys. Rev. B 64, 075413. Tarasenko, A.A., Nieto, F., Jastrabik, L., Uebing, C., 2003. Surf. Sci. 536, 1. Thomas, J.M., Thomas, W.J., 1997. Principles and Practice of Heterogeneous Catalysis. VCH, Weinheim. Tomellini, M., 2004. Surf. Sci. 556, 184. Tomellini, M., 2007. Surf. Sci. 601, 2260. Uebing, C., Zhdanov, V.P., 1998a. Phys. Rev. Lett. 80, 5455. Uebing, C., Zhdanov, V.P., 1998b. J. Chem. Phys. 109, 3197. Van Bavel, A.P., Ferre, D.C., Niemantsverdriet, J.W., 2005. Chem. Phys. Lett. 407, 227. Van Bavel, A.P., Hermse, C.G.M., Hopstaken, M.J.P., Jansen, A.P.J., Lukkien, J.J., Hilbers, P.A.J., Niemantsverdriet, J.W., 2004. Phys. Chem. Chem. Phys. 6, 1830. Van Hove, M.A., Weinberg, W.H., Chan, C.-M., 1986. Low-Energy Electron Difraction. Springer, Berlin. Van Santen, R.A., Niemantsverdriet, J.W., 1995. Chemical Kinetics and Catalysis. Plenum, New York. Vega, J.L., Guantes, R., Miret-Artes, S., 2002. Phys. Chem. Chem. Phys. 4, 4985. Völkening, S., Wintterlin, J., 2001. J. Chem. Phys. 114, 6382. Ward, C.A., Findlay, R.D., Rizk, M., 1982a. J. Chem. Phys. 76, 5599. Ward, C.A., Rizk, M., Tucker, A.S., 1982b. J. Chem. Phys. 76, 5606. Ward, C.A., Fang, G., 1999. Phys. Rev. E 59, 429. Ward, C.A., 2002. J. Non-Equil. Thermodyn. 27, 289. Wodtke, A.M., Tully, J.C., Auerbach, D.J., 2004. Int. Rev. Phys. Chem. 23, 513. Yakes, M., Hupalo, M., Zaluska-Kotur, M.A., Gortel, Z.W., Tringides, M.C., 2007. Phys. Rev. Lett. 98, 135504.


267

Zaera, F., 2002. Surf. Sci. 500, 947. Zaluska-Kotur, M.A., Gortel, Z.W., Teshima, R., 2002. Phys. Rev. B 66, 165418. Zaluska-Kotur, M.A., Krukowski, S., Turski, L.A., 2003. Phys. Rev. B 67, 155406. Zaluska-Kotur, M.A., Lusakowski, A., Krukowski, S., Turski, L.A., 2004. Surf. Sci. 566, 210. Zgrablich, G., 1997. In: Rudzinski, W., Steele, W.A., Zgrablich, G. (Eds.), Equilibria and Dynamics of Gas Adsorption on Heterogeneous Surfaces. Elsevier, Amsterdam. Zhdanov, V.P., 1980. Proc. Acad. Sci. USSR 254, 392 (in Russian). Zhdanov, V.P., 1981. Surf. Sci. 111, 63. Zhdanov, V.P., 1989a. Phys. Lett. A 137, 409. Zhdanov, V.P., 1989b. Surf. Sci. 209, 523. Zhdanov, V.P., 1991a. Elementary Physicochemical Processes on Solid Surfaces. Plenum, New York. Zhdanov, V.P., 1991b. Surf. Sci. Rep. 12, 183. Zhdanov, V.P., 1992. Phys. Lett. A 161, 556. Zhdanov, V.P., 1997. Phys. Rev. B 55, 6770. Zhdanov, V.P., 1999. Surf. Sci. 426, 345. Zhdanov, V.P., 2001. J. Chem. Phys. 114, 4746. Zhdanov, V.P., 2002a. Surf. Sci. 500, 966. Zhdanov, V.P., 2002b. Surf. Sci. Rep. 45, 231. Zhdanov, V.P., 2005a. Surf. Sci. 575, 313. Zhdanov, V.P., 2005b. Phys. Chem. Chem. Phys. 7, 2399. Zhdanov, V.P., 2006. J. Exp. Theor. Phys. 102, 737. Zhdanov, V.P., 2007. Surf. Rev. Lett. 14, 419. Zhdanov, V.P., Kasemo, B., 1994. Surf. Sci. Rep. 20, 111. Zhdanov, V.P., Kasemo, B., 1998. Surf. Sci. 415, 403. Zhdanov, V.P., Kasemo, B., 2000a. Phys. Rev. B 62, R4849. Zhdanov, V.P., Kasemo, B., 2000b. Proteins, Str. Func. Gen. 39, 76. Zhdanov, V.P., Kasemo, B., 2000c. Surf. Sci. Rep. 39, 25. Zhdanov, V.P., Kasemo, B., 2001. Chaos 11, 335. Zhdanov, V.P., Kasemo, B., 2003. In: Weickowski, A., Savinova, E.R., Vayenas, C.G. (Eds.), Catalysis and Electrocatalysis at Nanoparticle Surfaces. Marcel Dekker, New York, p. 35. Zhdanov, V.P., Matsushima, T., 2005. Surf. Sci. 583, 253. Zhdanov, V.P., Matsushima, T., 2007a. Phys. Rev. Lett. 98, 036101. Zhdanov, V.P., Matsushima, T., 2007b. Surf. Sci. 601, 2373. Zheng, C.Z., Yeung, C.K., Loy, M.M.T., Xiao, X.D., 2004. Phys. Rev. B 70, 205402.


CHAPTER 8

Understanding Heterogeneous Catalysis from the Fundamentals

T. BLIGAARD and J.K. NØRSKOV Center for Atomic-scale Materials Design Department of Physics Technical University of Denmark DK-2800 Lyngby, Denmark

B.I. LUNDQVIST Center for Atomic-scale Materials Design Department of Physics Technical University of Denmark DK-2800 Lyngby, Denmark Department of Applied Physics Chalmers University of Technology SE-412 96 Göteborg, Sweden



Contents 8.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2. Surface-science heritage of understanding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2. Descriptors for metal catalysts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3. Trends in adsorption energies on transition metal surfaces . . . . . . . . . . . . . . . . . . 8.2.4. The d-band model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4.1. One-electron energies and binding-energy trends . . . . . . . . . . . . . . . . . . 8.2.4.2. The Newns–Anderson model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5. Trends in chemisorption energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5.1. Variations in adsorption energies from one metal to the next . . . . . . . . . . . . 8.2.5.2. Ligand effects in adsorption – changing the d-band center . . . . . . . . . . . . . 8.2.5.3. Ensemble effects in adsorption – the interpolation principle . . . . . . . . . . . . 8.2.6. Trends in activation energies for surface reactions . . . . . . . . . . . . . . . . . . . . . . 8.2.6.1. Electronic effects in surface reactivity . . . . . . . . . . . . . . . . . . . . . . . . 8.2.6.2. Geometric effects in surface reactivity . . . . . . . . . . . . . . . . . . . . . . . . 8.2.7. Brønsted–Evans–Polanyi relationships in heterogeneous catalysis . . . . . . . . . . . . . 8.2.7.1. Correlations from DFT calculations . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.7.2. Universal relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.8. Activation barriers and rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.8.1. Transition-state theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.8.2. Variational transition state theory and recrossings . . . . . . . . . . . . . . . . . 8.2.8.3. Harmonic transition state theory (HTST) . . . . . . . . . . . . . . . . . . . . . . 8.3. Variations in catalytic rates – volcano relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1. Dissociation rate-determined model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2. Sabatier analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. Optimization and design of catalysts through modeling . . . . . . . . . . . . . . . . . . . . . . . 8.4.1. The low-temperature water gas shift (WGS) reaction . . . . . . . . . . . . . . . . . . . . 8.4.2. Methanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5. Some catalytic reactions from the fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2. Kinetic simulations of oxidation of some monoxides . . . . . . . . . . . . . . . . . . . . . 8.5.2.1. NO + O → NO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2.2. CO + O → CO2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.3. Ammonia synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.4. Hydrogen evolution reaction on MoS2 and close interplay between theory and experiment 8.6. Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

273 277 277 277 279 280 281 284 286 286 288 293 295 295 297 298 299 302 304 305 306 306 308 310 311 313 314 315 318 318 318 319 324 325 330 333 335

Abstract Catalysis describes the acceleration of a chemical reaction by means of a substance that is itself not consumed by the overall reaction. It is not only important for numerous human

270

activities, but it has also always been a major spur for the development of surface science. Today there is an extensive surface-science heritage of understanding, and there are examples of catalysis phenomena that are now understood from the fundamentals. For instance, modernday theory is able to predict the turnover frequency of an industrially relevant catalytic reaction in a semi-quantitative way. A major part of this chapter sums up such a successful surface-scientific development, pointing out descriptors for metal catalysts and identifying trends in adsorption energies and activation energies for surface reactions on transition metal surfaces by extensive computations. This is done using the density-functional theory (DFT), whose accuracy in this context is secured, and analyzed in electron-structural terms, in particular the d-band model. Via correlations determined from DFT calculations, universal relationships in heterogeneous catalysis are built up, including variations in catalytic rates, volcano relations. The optimization and design of catalysts through modeling is within reach. For instance, experimental verification for pure CO methanation, for CO2 methanation, and for simultaneous CO and CO2 methanation means that a technical methanation catalyst is discovered on the basis of computational screening. To further detail the surface-science approach to catalysis it is natural to supplement this presentation with some other examples of recent work on some catalytic reactions from the fundamentals. Oxidation of some monoxides illustrates the use of kinetic Monte Carlo simulations. The successful prediction of the outcome of the ammonia synthesis from first-principles supports the view that in the future theory will be a fully integrated tool in the search for the next generation of catalysts. The hydrogen evolution reaction on MoS2 is given as an example of successful interplay between theory and experiment. It is concluded that, thanks to the strong development in surface science, the understanding of heterogeneous catalysis from the fundamentals is approaching an advanced stage. Design of new catalyst on the basis of computational screening is today a realistic perspective. The list of issues that need further considerations includes nonadiabaticity, complex reactions, and other classes of catalyst materials than transition metals.

271


273

8.1. Introduction Catalysis is a word coined by Jöns Jakob Berzelius in 1835. In chemistry and biology, it means the acceleration (increase in rate) of a chemical reaction by means of a substance that is itself not consumed by the overall reaction (see Wikipedia). This substance is called a catalyst. Heterogeneous catalysts are present in phases that are different from the reactants. For example, it can be a solid catalyst in a liquid reaction mixture. Homogeneous catalysts are in the same phase, e.g., a dissolved catalyst in a liquid reaction mixture. Biocatalysts are often seen as a separate group and thought of as mixed homogeneous and heterogeneous catalysts, because the enzyme is in solution itself, while the reaction takes place on the surface of the enzyme. Catalysis is important for numerous human activities. For instance, ammonia is a substance made by catalysis. It is used in, e.g., textile processing, water purification, explosive manufacturing, and, most important, as fertilizer, thereby making the increase in world population possible. The energy sector depends critically on heterogeneous catalysis. For instance, the gasoline at a fueling station has seen about ten catalysts on its way through the refinery, and future hydrogen or other energy carriers will also depend on heterogeneous catalysis. The list of heterogeneous catalysts includes vanadium oxide in the contact process, nickel in the manufacturing of margarine, alumina and silica in the cracking of alkanes, platinum, rhodium, and palladium in catalytic converters, and many more. Mesoporous silicates and other support materials have found utility in heterogeneous reaction catalysis, thanks to their large accessible surface areas, which allows for high catalyst loading. In modern cars, catalysts are compulsory. Incomplete combustion of the fuel in their engines produces carbon monoxide, which is toxic. The electric spark and high temperatures also allow oxygen and nitrogen to react and form nitrogen monoxide and nitrogen dioxide, which are responsible for photochemical smog and acid rain. Catalytic converters reduce such emissions by adsorbing CO and NO onto catalytic surfaces, where the gases undergo a redox reaction. Carbon dioxide and nitrogen are desorbed from the surface and emitted as for humans relatively harmless gases: 2CO + 2NO → 2CO2 + N2 . In refineries and in petrochemical applications many of the used catalysts are regenerated and reused multiple times to save costs, energy and reduce the environmental impact from recycling or disposal of spent catalysts. Environmental problems are frequent in news reports, like the one about the city getting bad air from big ferries entering its harbor and using crude oil. Legislative requirements for ultra low sulfur transport fuels give new hydrodesulfurization (HDS) challenges for the refining industry. In addition, the refiners face a growing demand for diesel fuels. In order to produce low sulfur transport fuels, the refiner may choose between different revamp or grassroots options and the most cost-effective solution will depend on the specific refinery situation with respect to configuration, feedstock blends and product-slate. The selection of catalyst types is then an important decision. For surface science there has always been a spur to understand catalysis. Early in the history of catalysis, there were empirical results that transition metals could be important catalysts. Later, at, e.g., the 1974 Gstaad colloquium on the physical basis of heterogeneous catalysis (Drauglis and Jaffe, 1975), many relevant concepts were aired. These include

274

T. Bligaard et al.

electronically adiabatic potential-energy surfaces, nonadiabatic processes, electronic and geometric factors, molecular orbitals, local densities of states, and the Woodward–Hoffman reaction rules. Today we see examples of catalysis phenomena that are really understood from the fundamentals. For instance, modern-day theory is able to predict the turnover frequency of an industrially relevant catalytic reaction in a semiquantitative way (Honkala et al., 2005; Hellman et al., 2006b). A simple model for heterogeneous catalysis involves the catalyst providing a surface, on which the reactants temporarily become adsorbed and have some bonds weakened sufficiently for new bonds to be created. The bonds between the products and the catalyst are weaker, so the products are released. For example, in the Haber–Bosch process to manufacture ammonia, finely divided iron acts as a heterogeneous catalyst. The reactant gas molecules are adsorbed onto the metal surface, get their internal bonds weakened, and are held in close proximity to each other. The nitrogen molecule has a particularly strong triple bond, which in this process is weakened and even broken. Further, the hydrogen and nitrogen molecules and atoms are brought closer together than would be the case in the gas phase, so the rate of reaction increases. In some cases the surface can even open new reaction paths. The catalytic action of Ni surfaces on the CO disproportionation reaction, CO + CO → C + CO2 might illustrate this (Andersson et al., 1977; Lundqvist et al., 1979). The surface here circumvents energy and symmetry restrictions that exist for the corresponding gas-phase reaction. It appears that the chemisorption of the produced carbon is important for the thermodynamics and the partial filling of the 2π ∗ -correlated levels of adsorbed CO for the reaction rate (Andersson et al., 1977), as implied by the Woodward–Hoffman reaction rules (Hoffmann and Woodward, 1970; Pearson, 1979). Understanding of structural and mechanistic details of a catalyzed heterogeneous reaction leads both directly and indirectly to the development of new and better catalysts. For catalyst technology, the most sensitive probe of a catalyst’s performance is the rate and selectivity of a chemical reaction. These macroscopic observations require supplementary microscopic information to remove ambiguity in the deduction of a catalytic mechanism. This information, down to the atomic level, concerning the structure and reactivity of the intermediates, the nature of adsorption sites (and sometimes the active sites) and their number, is the main objective of the science of catalysis. The ultimate goal of catalysis research is to design and tune activity and selectivity of catalysts by controlling their structural properties at the atomic level. Identification of concepts to achieve this goal is still one of the key issues of research in catalysis. A variety of different strategies have been put forward, most of which are based on structure-reactivity relationships, taking different aspects of these exceedingly complex systems into account. To this end, for instance, model systems using metal particles supported on well ordered thin oxide films of appropriate thickness, which allow investigations with the rigor of modern surface science while grasping essential aspects of the complexity of real systems, have proved to give valuable insight into the details of geometric and electronic structure, as well as adsorption and reaction properties. With its successive maturing, surface science has gone from a qualitative science to a quantitative one. Among other things, this means that early thinking and results might come back in a modern dress. For instance, for a dispersed metal, the catalytic activity


275

might be controlled by the thickness of the supporting oxide layer. This concept was introduced some 20 years ago. Today’s experimental challenge is to provide indisputable proof for the proposed mechanisms. This is usually hampered either by the complexity of the samples investigated and/or the lack of appropriate methodology to exclude interference with alternative mechanisms. A timely approach is to develop experimental control together with realistic theoretical modeling, in this case with respect to the thickness and structure of the oxide films. Then these structural properties can be used to control their functional characteristics and thus the catalytic properties of a metal deposited onto them. Here, control may concern transport of species, e.g., hydrogen or oxygen, through the film or it may concern electronic interaction of the interior metal-oxide interface with adsorbed species on top. The latter idea is related to those proposed at the end of the 1940s by Cabrera and Mott in connection with metal oxidation (Cabrera and Mott, 1949; Fromm, 1998). Charge transfer towards the adsorbed species may be induced via defects or by reduced metal ions in the surface. This modifies considerably the chemical activity of the adsorbed species. A rather inert molecule would turn into a reactive species. Similar effects have also been discussed for metal atoms and nanoclusters deposited onto charged oxide defects, which are known to be rather chemically inert, may then be charged negatively, and according to calculations become rather chemically active, e.g., the carbon monoxide to carbon dioxide reaction (Ganesh et al., 2006). Charge transfer may also occur for metal atoms and clusters adsorbed on a supported thin oxide film, provided that the adsorbed metal exhibits a high electronegativity or electron affinity, and the oxide film does not exceed a few monolayers. In particular, Au atoms adsorbed on thin MgO(001) films grown on Mo(001) and Ag(001) are expected to be negatively charged in contrast to their counterparts on the corresponding bulk MgO, where Au atoms are essentially neutral (Yulikov et al., 2006). Individual metal adatoms can even be manipulated to be neutral, negatively, and positively charged, respectively (Olsson et al., 2007). In the early days, catalysis was discussed in terms of electronic and geometric factors. A search for “electronic factor in catalysis” and “geometric factor in catalysis” in Science Citations Index gives many more hits for the “electronic factor”, which tells a clear story. However, rather recently there is a revival for the geometric effect, as a significant correction to the electronic one. Designing a catalyst can be viewed as manipulating the electron structure of the active atom(s). This manipulation can be made in a number of ways, including structuring, charge transfer and alloying. The ability of solid surfaces to make and break bonds of molecules approaching from the surroundings is the basis for the phenomenon of heterogeneous catalysis. Many chemical reactions are catalyzed by solid surfaces; let it be large-scale chemical processes in industry or exhaust-gas cleaning in automobiles. When working as a catalyst for a chemical reaction, a solid surface has four closely coupled functions: (i) It adsorbs the reactants and cleaves the required bonds; (ii) It holds the reactants in close proximity so that they can react; (iii) It might provide new reaction mechanisms and paths; and (iv) It lets the products desorb back into the surrounding phase. Understanding the adsorption bond is therefore crucial to the understanding of the way surfaces act as catalysts. Knowing what factors determine whether a surface is a good catalyst for a given chemical reaction or

276

T. Bligaard et al.

not, we will have the concepts needed to guide us towards better and more efficient catalysts. The activity and selectivity of surfaces can be affected in various ways, for instance, by alloying a metal surface. The relative importance of various factors that are responsible for these variations are then an issue. For a metal active in a certain reaction, two factors can be conceived, (1) a “geometric” or “ensemble-size” effect, where, for instance, catalytic reactions requiring large ensembles of active atoms are suppressed more than those requiring only small ones, and (2) an “electronic” or “ligand” effect, where the electronic structure may be changed by, e.g., alloying. For instance, an infrared-spectroscopic investigation of adsorption of the probe molecule CO on the surfaces of alumina-supported Pt–Cu alloys, unalloyed Al2 O3 -supported metals, and alloys supported on silica have documented a geometric factor, caused by dilution of Pt by Cu, while, within error limits, an electronic or ligand effect has not been observed. The upward shift of the frequency of the CO/Cu band upon alloying Cu by Pt is most probably caused by “individualization” of the Cu atoms in the Pt layer (Toolenaar et al., 1983). Many catalysts are metals, and during the 20th century metal and semiconductor surfaces have formed the dominating class of substrates in Surface Science. However, for instance, thin oxide films both lend themselves as model supports for studies in heterogeneous catalysis, for example, to study the growth and reaction of metal deposits (atoms, clusters and nanoparticles). If the thickness of the film is chosen appropriately, these thin films are reasonable models to mimic the situation on bulk materials. If thin films below a critical thickness are studied, these materials exhibit properties in their own right. Their structural properties may be tuned to control their functional characteristics, with possible implications for heterogeneous catalysis. In the following, a set of simple concepts are developed that allow an understanding of variations in the reactivity from one surface to the next. These variations are essential to the understanding of heterogeneous catalysis, because they hold the key to distinguishing between a good and a bad catalyst. The general idea is to manipulate the electronic structure of the active atom(s). The present chapter concentrates on transition-metal surfaces, but many of the concepts discussed should be applicable to oxides, sulfides and other catalytically interesting materials. In summary, understanding of heterogeneous catalysis from the fundamentals is achieved by analyzing the concept of reactivity and relating it to certain key concepts of adsorption (descriptors). The major method to achieve this is to study trends in atomic atom , molecular chemisorption energies, E molecule , and reactivities, adsorption energies, Eads ads Eact , for certain key surface reactions. In this process, density-functional theory (DFT) is an important tool. Through numerous surface-science measurements, its useful accuracy and limitations in current implementations has been assessed. Now it can be used to calculate values for the quantities needed in the trend studies, in particular for situations where no experimental values are available or are difficult to obtain. DFT is also used as a basis for the theoretical framework for heterogeneous catalysis on transition-metal catalysts and the like.


277

8.2. Surface-science heritage of understanding 8.2.1. Introduction The close interplay between experiment and theory is a characteristic of surface science. This is particularly true for our understanding of bonding on transition metals. A battery of experiments give a very good characterization of the adsorption of simple molecules on most transition metals in terms of geometric structure, vibrational structure, bond properties, electronic structure, kinetics, and thermo-chemistry (Somorjai, 1994; Thomas and Thomas, 1997; Chorkendorff and Niemantsverdriet, 2003). A rich database of high-quality experimental results provides a basis for tests of theoretical results. Also, theoretical methods are well developed today. In particular, this applies to the densityfunctional theory (DFT) (Hohenberg and Kohn, 1964; Kohn and Sham, 1965), with a semilocal description of exchange and correlation (Langreth and Mehl, 1983; Perdew, 1986; Perdew and Wang, 1992; Perdew et al., 1998; Hammer et al., 1999) (see, e.g., Handbook of Surface Science, Series editors: S. Holloway and N.V. Richardson, vol. 2, Volume editors: K. Horn and M. Scheffler, pages 4 and 98 (2000)) which can provide a semiquantitative account of the mentioned adsorption properties (Hammer and Nørskov, 2000; Greeley et al., 2002; van Santen and Neurock, 2006). This makes it possible to base our analysis of catalytic properties on DFT (Drauglis and Jaffe, 1975; Bligaard et al., 2007; Honkala et al., 2005; Chorkendorff and Niemantsverdriet, 2003; Hammer and Nørskov, 2000; Jaramillo et al., 2007). The understanding of heterogeneous catalysis from the fundamentals of surface science was once on the horizon, but it opens up today as a realistic possibility. In this chapter, the present status of development is indicated by a few examples. Most of them concern metallic catalysts, for a variety of reasons. However, the gist of the story is the adsorbate-induced electron structure. Thus much of the thinking should be transferable to other catalysts. The three pillars of surface science are accurate surface-sensitive experiments, insightful and sometimes numerically accurate theory, and the constructive interplay between experiment and theory. Once operating in the qualitative domains, surface science has today reached a certain maturity, including many examples of successful quantitative comparisons between accurate and detailed experimental results and theoretical predictions or confirmations. Out of this, there grows an understanding that can be used in applications, such as catalysis. 8.2.2. Descriptors for metal catalysts Electrons hold molecules and materials together. Thus our understanding is based on electron structure and its changes at contacts between the gas-phase molecules and the catalyst material. Due to their important role as catalysts and the wealth of surface-science knowledge about them, transition metals form a natural group of materials for our examples. The fact that understanding develops out of both successful conceptual frameworks and detailed values of certain well-specified quantities will be illustrated, in particular, by looking at trends. As a result, a general scheme for the design of working catalysts develops.

278

T. Bligaard et al.

Encounters between molecules and metal surfaces occur in many processes, such as adsorption, desorption, recombination, etc. Some of these processes can be activated. Thus it is well worth studying trends in, e.g., (molecular) adsorption energies, (atomic) chemisorption energies, and activation energies for surface reactions on transition metal surfaces. A proper conceptual framework, valuable to put them into a useful context, is provided by the electron structure of the adsorbate-substrate system. As a catalyst breaks and forms chemical bonds, the phenomenon of chemisorption is central. Initially, there were only qualitative models for chemisorption based on a large degree of insight. In particular, the Newns–Anderson model has been important for our thinking (Newns, 1969). DFT has also been very important, as used in the early days on the jellium model for free-electron metals (Lang and Williams, 1976; Gunnarsson et al., 1976; Lang and Williams, 1978) (with estimates of the coupling to d-electrons (Lundqvist et al., 1979; Jacobsen et al., 1987)), and during the last two decades or so on real solids (Hammer and Nørskov, 1997; Scheffler and Stampfl, 2000). Today, numerous DFT calculations have supplied us with numbers for many properties of adsorbate-substrate systems, and somewhat fewer with interpretations of the results in terms of simpler models. The so-called d-band model (Hammer and Nørskov, 1995, 2000; Bligaard et al., 2007) has in its simplicity given bearings to quite extensive implications. This chapter is meant to illustrate how this model inspires us to pronounce some quite far-reaching implications for catalysis, which are then “checked” against results of DFT theory, which during the last twenty years has reached an almost chemical accuracy. This chapter is meant to show the role of Surface Science in heterogeneous catalysis by some illustrative examples. The main example exploits this insight into the nature of adsorption, in particular, for building up an understanding of the variation of adsorptionbond energies from one transition metal to the next (Bligaard et al., 2007). It will be demonstrated that from such an understanding one can extract concepts that allow a determination of what characterizes a good catalyst for a given reaction. One aim is to introduce explicit descriptors useful to characterize the catalytic properties of the surface. Experimental data can be understood in terms of these descriptors. As a consequence, order is brought about to the known literature on catalytic rates of different metal surfaces. Furthermore, the descriptors may potentially be useful in designing new catalysts. The following section describes how an understanding of the differences in reactivity between different elemental metals can be developed. In an extended description, the concepts deduced may be used to describe reactivity of adsorbed species on an alloy of two metals. Part of the understanding is to realize that two general classes of phenomena determine the reactivity in heterogeneous catalysis: the electronic effect (Somorjai, 1994), with focus on the dominant electronic effects, and the influence of surface structure, the socalled geometric effect (Somorjai, 1994). The discussion will start by considering simple adsorption systems and proceed to variations in activation energies for surface reactions. Since long ago, transition metals have been identified as possible active catalysts. The valence electrons in transition metals are of sp and d type, where the d electrons give each transition metal its particular character. In adsorption, the sp electrons are important for the shifting and broadening of adsorbate energy levels, while the d electrons also contribute to the chemisorption bond, but in most cases their role is to fine-tune the bond strength and the electron structure, very important in catalysis, as emphasized early by Newns (1969).


279

While the magnitude of extra coupling Vda was once (Lundqvist et al., 1979) estimated with the help of the Extended Hückel method (Fassaert and van der Avoird, 1976), DFT is the current method for such calculations. 8.2.3. Trends in adsorption energies on transition metal surfaces Understanding of adsorption energies on metal surfaces can be reached along many tracks. One such goes over the study of trends in adsorption energies on a range of late transition metal surfaces and from there to first consider adsorption of the oxygen atom (Hammer and Nørskov, 2000). Calculated oxygen binding-energy values, as functions of the distance of the O atom above the surface, vary in a characteristic way over a part of the d-band metals in the periodic table (Fig. 8.1). The O-metal bond gets stronger towards the left in the transition-metal series and stronger to the 3d- than to the 4d- and 5d-electrons, all in excellent agreement with experimental findings (Hammer and Nørskov, 2000): Ru binds

Fig. 8.1. Calculated adsorption-energy values for the oxygen atom at different separations between O atom and surface for some close-packed transition-metal surfaces (Hammer and Nørskov, 1997). The value of the binding energy of the molecule O2 is shown per O atom in the Ru box for comparison (only metals with an adsorption-energy minimum below this value will be able to dissociate O2 exothermally).

280

T. Bligaard et al.

Fig. 8.2. The density of states for an O atom adsorbed on some transition-metal surfaces (Hammer and Nørskov, 1997). Part of it is projected onto the d states of the surface atoms of the surfaces considered in Fig. 8.1 (light). The oxygen 2px projected density of states is also shown (dark). Differences in the formation of bonding and antibonding states below and above the metal d states are clearly seen.

much more strongly than Pd and Ag, Au is very noble with a bond energy per O atom less than that in O2 , Ag is barely able to dissociate O2 exothermically, and Cu forms quite strong bonds. These variations can be understood in terms of variations in the electronic structure of O adsorbed on different metals (Fig. 8.2).1 The interaction of an atom like O with a transition metal results in the formation of bonding and antibonding states below and above the metal d bands, respectively. Figure 8.2 illustrates how the antibonding states for O on, e.g., Ru are less filled than on Pd and Ag, resulting in a weaker bond in that order according to the bonding-antibonding picture. However, this picture is too simple to explain why the bond to 3d-metals is stronger than that to 4d- and 5d-metals, and how more complex systems behave, and thus calls for an extension, like the one below. 8.2.4. The d-band model Historically, several different models have been used to emphasize the role of the d electrons in chemisorption. However, as general results are aimed at, as general a starting point 1 For colors in figures see the web version of this book.


281

as possible is used in the following. DFT (Hohenberg and Kohn, 1964) is a powerful tool to describe electron structure and bonding. It allows the total energy to be expressed in terms of the electron density as (Kohn and Sham, 1965) Etot [n] = THK [n] + F [n] = THK [n] n(r)n(r ) 1 + + v(r)n(r) + Enn + Exc [n], 2 |r − r | where THK [n] =

ψi − 12 ∇ 2 ψi = εi − veff (r)n(r)

i occ

(8.1)

(8.2)

i occ

is the kinetic energy of a noninteracting electron gas moving in an effective potential veff (r) chosen so that the noninteracting system has the same electron density as the real system. F [n] is the sum of the average electrostatic energy, the interaction of the electrons with the external potential set up by the nuclei, the nucleus–nucleus interaction, Enn , and the exchange-correlation energy, Exc [n]. For a molecular-orbital (MO) description and thus for the development of a model to describe trends in bond energies from one transition metal to another a basis is provided by DFT in the Kohn–Sham version (Gunnarsson and Lundqvist, 1976). However, a model based on the density of one-electron states, so convenient in the simple theory, poses a fundamental question: How can bond energies be obtained directly from the Kohn–Sham one-electron energies, given that Eqs. (8.1) and (8.2) show that the total energy is not equal to the sum of the one-electron energies? This problem is present in all simple one-electron descriptions of bonding in molecules. A working way to address it is to use the fact that changes in the Kohn–Sham one-electron energies can give changes in bond energies. This can be shown to apply for correctly calculated quantities. For analyses of bond making and breaking, which is the major concern in this chapter, this should be enough. In the following, the main lines of an analysis made by Hammer and Nørskov (1997, 2000) are given. 8.2.4.1. One-electron energies and binding-energy trends The change in adsorption energy of an adsorbate a at a metal surface m, when the metal is modified slightly to m, ˜ should be estimated. For instance, the modification could be the adsorption of another atom or molecule on m close to a, but it could also be the replacement of the metal m by another metal close to m in the Periodic Table. The difference in adsorption energy between the two cases is looked for: ˜ − E[m] δEads = E[m]

˜ + a] − Etot [m] ˜ − Etot [a] − Etot [m + a] − Etot [m] − Etot [a] = Etot [m

˜ + a] − Etot [m] ˜ − Etot [m + a] − Etot [m] . = Etot [m (8.3) It is convenient to divide space into two regions, A and M. In the near-adsorbate region, A, the electron density and one-electron potential are affected only slightly by the change of m to m. ˜ Likewise, in the near-metal region, M, the effect of the adsorbate is assumed to be weak. In the generalized variational principle of DFT, which says that changes in the

282

T. Bligaard et al.

electron density and in the one-electron potential give rise to changes in the total energy to second-order (Hohenberg and Kohn, 1964), this can be exploited. In region A, which is dominated by the adsorbate, the same density and potential can be chosen, irrespective of the metal. Similarly, in region M the density and potential are assumed to be independent of the presence of the adsorbate. There are only second-order errors in δEads , when the density and potential are frozen in this way. This is fruitful thinking, and the central ideas for this approach can be broadly applied (for instance, in precursor developments of d-band effects on chemisorption (Lundqvist et al., 1979), in effective-medium theory (Jacobsen et al., 1987), and in DFT), and has bearings on, e.g., the design of catalysts. It is thus worthwhile to air it here. First, the contribution to δEads from the F [n] term in Eq. (8.1) is assessed with respect to its local properties. The leading contributions to the exchange-correlation energy functional, Exc [n], are local functions of position in space, in particular, in the generalizedgradient approximation used in this chapter. For the electrostatic-energy contributions this is not generally true, however. For example, an adsorbate with a dipole moment gives rise to an electrostatic potential in region M. For the present, such nonlocal electrostatic interactions between regions A and M are neglected. With such assumptions, F can be divided into contributions from the two regions F = FA + FM and written

˜ + a] − FA [m] ˜ − FA [m + a] − FA [m] δFlocal = FA [m

˜ + a] − FM [m] ˜ − FM [m + a] − FM [m] + FM [m

˜ + a] − FA [m + a] − FA [m] ˜ − FA [m] = FA [m

˜ + a] − FM [m] ˜ − FM [m + a] − FM [m] . + FM [m Within the frozen-density approximation, each parenthesis in the last equation is zero. The only contribution from F to the adsorption energy difference is therefore the nonlocal electrostatic energy, ρ(r)ρ(r ) δEes,A−M = dr dr . A M |r − r | Similarly, in the kinetic energy difference, the frozen potential and density ansatz renders the net contribution from the nv integrals zero, and only the difference in the oneelectron energies calculated with the frozen potentials outlined above contributes: δTHK = δE1el = ε υM [m], ˜ υA [a] − ε υM+A [m] ˜ " # − ε υM [m], υA [a] − ε υM+A [m] . The adsorption-energy difference is therefore given by the one-electron energy difference plus the difference in electrostatic interaction between the surface and the adsorbate in the two situations, δEads = δE1el + δEes,A−M .

(8.4)

This very interesting result shows that energy differences are given by an inter-atomic electrostatic energy plus the sum of the one-electron energy differences, provided they


283

Fig. 8.3. Schematic illustration of the formation of a chemical bond between an adsorbate valence level and the s and d states of a transition metal surface (Hammer, 1998).

are calculated in the right way, that is, by using frozen potentials. Equation (8.4) gives a theoretical background for using arguments based on the one-electron spectra. It shows that such arguments are particularly useful for making comparisons between similar systems, but not for calculating the total binding energy. Non-self-consistent one-electron energy differences thus do contain information about bonding trends. This simple understanding of the electronic structure of adsorbates on metal surfaces, developed in Hammer and Nørskov (1997, 2000) can be exploited. It is a general fact that adsorbate energy levels that interact with a broad substrate energy band, such as the free-electron-like sp bands of metals, experience shifts and broadenings (Fig. 8.3) (Newns, 1969; Lang and Williams, 1976; Gunnarsson et al., 1976; Lang and Williams, 1978). For transition metals, a subsequent switching on of the coupling to the narrow d-band makes bonding and antibonding states appear (Newns, 1969), schematically shown in Fig. 8.3 and more exactly expressed in Fig. 8.2. The effects of exchanging one metal with the next or of changing the surroundings to the adsorbate-bonding metal atoms can be seen already from Fig. 8.3. All transition metals have a broad, roughly half-filled s band, which for simplicity can be assumed independent of the metal in question. The adsorption energy can therefore be written as E = E0 + Ed ,

(8.5)

where E0 is the binding-energy contribution from the free-electron-like sp electrons and Ed is that from the extra interaction with the transition-metal d electrons. The basic assumption of the d-band model is that E0 is independent of the metal, a good but nonrigorous approximation. For instance, it fails for small metal particles, where the sp levels do not form a continuous spectrum (on the scale of the metal-adsorbate coupling strength). It also fails for metals without any contribution to the bonding from the d states. The other basic assumption is that the d contribution can be estimated as the non-self-consistent one$ electron energy change, Ed ∼ = ε(n (ε) − n(ε)) dε, where n (ε) and n(ε) are the adsorbate-induced densities of states with and without the d coupling, respectively. It applies to the extent described above. In the following it will be shown that the model can describe a large number of trends, in spite of its approximate nature. First the model used to estimate Ed will be introduced.

284

T. Bligaard et al.

8.2.4.2. The Newns–Anderson model The Newns–Anderson model (Anderson, 1961; Newns, 1969; Muscat and Newns, 1978) provides a simple account for adsorption that includes all the essential ingredients to describe the coupling between an adsorbate state and a band of metal states (Grimley, 1967, 1971; Gadzuk, 1974; Lundqvist et al., 1978). The basics of the model and simple methods to utilize it for bond energy estimates are introduced in the following. An adsorption system, with a metal surface that has one-electron states |k with energies εk and an adsorbate with a single valence state |a of energy εa , is considered. With the adsorbate just outside the surface, the coupling between |a and |k is described by the matrix element Vak = a|H|k, where H is the Hamiltonian of the combined system. In terms of the free adsorbate and surface solutions, there are other matrix elements of the Hamiltonian, Haa = εa and Hkk = εk . Essential features of the solution |i to the Schrödinger equation with the Hamiltonian H can be expressed in terms of the projection of the density of states on the adsorbate state, na (ε) = i |i|a|2 δ(ε − εi ), summing over the eigenstates of the full Hamiltonian. Rewriting this as na (ε) = −

a|ii|a 1 1 Im = − Im Gaa (ε), π ε − εi + iδ π i

with = 0+ , and the projection Gaa (ε) of the single-particle Green function, G(ε) = δ|ii| i ε−εi +iδ , defined by the formal equation (ε − H + iδ)G(ε) = 1, an efficient way of expressing the chemisorption problem is obtained. The density of states na (ε) can be obtained from the imaginary part of the |a projection of G(ε), and similar for other quantities. The equations can be solved for Gaa to yield Gaa (ε) = ε−εa1−q(ε) , which expresses the chemisorption as a self-energy, q(ε) = Λ(ε) − i(ε), with the real and imaginary parts, (ε) = k |Vak |2 δ(ε − εk ), related by a Hilbert transform, P (ε ) Λ(ε) = (8.6) dε . π ε − ε This immediately gives the projected density of states as na (ε) =

1 (ε) . π (ε − εa − Λ(ε))2 + (ε)2

(8.7)

Knowledge of (ε) and therefore also Λ(ε) allows the calculation of the change in the sum of one-electron energies associated with chemisorption as εF (ε) 1 Arctan dε − εa . E = 2 (8.8) π −∞ ε − εa − Λ(ε) This expression can give an estimate of the d-contribution to the bonding, Ed . This amounts to using a (fictitious) transition metal with no d electrons as the metal surface m, in the preceding section, and using the real transition metal including the d electrons as the metal m. ˜ The d contribution to the bonding is obtained from the one-electron energy difference, Ed , where unperturbed values for the energies of adsorbate state and metal-d states are implicitly invoked in the simplest form of the Newns–Anderson model, with


frozen density and potential assumed. Introduction of the group orbital, 1 Vak |k, V = |Vak |2 |d = V k

285

(8.9)

k

brings the Newns–Anderson chemisorption function in a particularly simple and useful form, 2 (ε) = πVad nd (ε)

with nd (ε) =

k|d2 δ(ε − εk ),

where the matrix element Vad = a|H |d is introduced. The group orbital is a combination of the metal states that couple directly to the adsorbate state. Therefore the projection of the substrate density of states onto this state defines the chemisorption function. In the description of transition-metal substrates, the group orbital is a single localized d orbital, or a linear combination of a few d orbitals, on the transition-metal atoms with a direct bond to the adsorbate. In the following, the first moment of the relevant group orbital is approximated by the average values for all the d states of the relevant atoms. In some cases an improved description can be obtained by explicitly using the d states with the proper symmetry (Mason et al., 2004). The group-orbital projected metal density of states, nd (ε), can be characterized by its moments: ∞ % ∞ n ε nd (ε) dε nd (ε) dε. Mn = (8.10) −∞

−∞

The first moment is the center, M1 = εd , and the second moment is the width, M2 = W . The moments of nd (ε) and (ε) are the same. The change in the sum of the one-electron energies E, varies with the d-band center, M1 = εd . Solutions to a Newns–Anderson model with a semi-elliptical model for the chemisorption function show this (Fig. 8.4). Different surface-projected densities of states, nd (ε), are shown as functions of the values of the d-band center εd in Fig. 8.4. Conservation of the number of d electrons couples the band width and center for each metal. The figure demonstrates that, as the d-band shifts up in energy, Ed becomes increasingly negative (meaning stronger bonds). This significant result cannot be obtained from simple rules from gas-phase chemistry, as seen by comparing the results of Fig. 8.4 and those of the solution to the similar problem with molecular states with energies εa and εd with a coupling matrix element Vad . With state |a completely filled (as in Fig. 8.4) and state |d partly filled, the binding energy is given by δE = −2(1 − f )ω, where f is the filling fraction and ω the absolute value of the downshift of the bonding state and the upshift of the antibonding one, 2 εd − εa 2 + εd − εa ω = Vad (8.11) − . 2 2

286

T. Bligaard et al.

Fig. 8.4. Change in the sum of the one-electron energies E, calculated with the Newns–Anderson model. The parameters are chosen to illustrate an oxygen 2p level interacting with the d states of palladium with a varying d-band center, εd . In all cases the number of d electrons is kept fixed. The corresponding variations in the metal and adsorbate projected densities of states are shown above. Notice that the adsorbate-projected density of states has only a small weight on the antibonding states since it has mostly metal character. Adapted from Bligaard et al. (2007), Hammer and Nørskov (1997), Hammer (1998). 2

Vad For a weak coupling (|Vad | εd −εa ) the usual perturbation result is obtained, ω ∼ . = εd −ε a Both expressions have ω decreasing as εd shifts up in energy, contrary to the result seen in Fig. 8.4. The real difference between the gas-phase bonding and the bonding at a metal surface is the presence of the Fermi energy εF . The occupancy of the anti bonding state is not determined solely by the filling of the d states, f , but rather by the position of the antibonding states relative to εF . As the d-band shifts up in energy, the number of antibonding states above εF increases. This leads to a stronger bond, as clearly shown by Fig. 8.4. If the adsorbate level lies above the d band (and hence above the Fermi level) then the bond always gets stronger, when the d states move up in energy. From this one therefore concludes that an upshift in the d-band energies should lead to a strengthening of the bonding of the adsorbate to the transition-metal surface. In a transition-metal series, the εd level is shifted upwards towards the left. Thus the adsorption bond is stronger towards the left in the transition-metal series.

8.2.5. Trends in chemisorption energies The trend study starts with chemisorption energies, leaving activation-energy barriers for the next subsection. 8.2.5.1. Variations in adsorption energies from one metal to the next The dissociative chemisorption energy of an atom is the adsorption energy of that atom measured relative to one half of the dissociation energy of the corresponding free diatomic


287

Fig. 8.5. Variations in the O adsorption energy along the 4d transition metal series. The results of full DFT calculations are compared to those from the simple d-band model and to experiments. Below the same data are plotted as a function of the d-band center. From Hammer and Nørskov (2000).

Table 8.1 Parameters describing the electronic structure of the transition metals. From Ruban et al. (1997)

molecule. Its trends for atomic oxygen along a series of 4d-transition metals (Fig. 8.5) illustrate the use of the d-band model. Both experiment and DFT calculations show that the bond gets stronger (E more negative) towards the left in the Periodic Table. The same trend applies in the 3d- and 5d-series (Fig. 8.1). It is also the rule for a number of other adsorbates (Nilsson et al., 2005). These trends are described quite well by the dband model with Ed calculated with the Newns–Anderson model. Clearly they correlate

288

T. Bligaard et al.

Fig. 8.6. Values of atomic O adsorption-energy values calculated with DFT (from Fig. 8.1) of the coupling-matrix element |Vad |2 (from Table 8.1) for Cu, Ag and Au, and their correlation. Adapted from Hammer and Nørskov (2000).

well with εd − εF . The change from one metal to the next along such a series reflects the decreased number of d-electrons and hence the upshifted d-band (in relation to the Fermi energy) towards the left in the transition-metal series (Table 8.1). The coupling matrix element |Vad |2 increases rapidly down the columns in the Periodic Table (Table 8.1), and, as expected from Eq. (8.11), the adsorption-energy values from DFT (Fig. 8.1) scale well with |Vad |2 for Cu, Ag and Au (Fig. 8.6). Figure 8.6 demonstrates nicely a well-known phenomenon, which has had great impact on human lives and traditions: why Au is so inert with respect to oxygen. First, the oxygen bond is weakest for metals with low-lying, filled d-bands, as here the antibonding states mixing adsorbate and metal-d electrons are completely filled. Among metals with filled d-band (f = 1 in Table 8.1), Au has the largest matrix element |Vad |2 , and thus the largest Pauli repulsion and the weakest oxygen chemisorption bond. The incentive to dissociate O2 is weak or absent. Among the elemental metals, Au is the leader in this respect, followed by Hg and Ag. In the other end, among the metals having f = 1 in Table 8.1, Zn has the smallest matrix element |Vad |2 . According to the relation between binding and |Vad |2 , Zn should be the one with the strongest oxygen bond. Indeed, this is found to be the case. In this way, the d-band model, including Pauli repulsion, can be used to understand variations in oxygen binding energies among metals in the periodic table. A similar description can be used for a number of other adsorbates (Hammer and Nørskov, 2000; Gajdos et al., 2004). 8.2.5.2. Ligand effects in adsorption – changing the d-band center The position of the average energy of the d electrons relative to the Fermi level, εd − εF , is an important parameter for the interaction energy. To elucidate that further, a number of


289

Fig. 8.7. Schematic illustration of the coupling between band width and d-band center for a band with a fixed number of d electrons. For decreasing band width, the only way of maintaining the number of d electrons is to shift the center of the band upwards. Adapted from (Hammer and Nørskov, 2000).

systems with essentially the same values for the energy of the adsorbate state(s), εa , and for the coupling matrix element, Vad , are now considered. The focus is on situations, where an adsorbate binds to a specific kind of transition-metal atom, and where the surroundings (or ligands) of the relevant metal atoms contributing to the coupling (cf. Eq. (8.9)), are changed. From the analysis above, we would expect that the position of the average energy of the d electrons relative to the Fermi level, εd − εF , should determine the variations in the interaction energy in these cases to a first approximation. The interaction energy might be affected by other properties, like the shape and width of the projected d density of states nd (ε). Often these variations are coupled to the d-band center variations, however, and can therefore be lumped into that dependence. A situation, where the width (the second moment, W ) of nd (ε) decreased for some reason – it could be because the surface layer is strained so that the coupling, Vdd , of the metal d states to the neighbors is smaller (W ∼ |Vdd |) or because the number of metal neighbors (the coordination number, NM ) is decreased by creating a step or a kink on the surface (W ∼ 0.5 ) – illustrates this point. A change in W for a fixed ε − ε , would change the number NM d F of d electrons. However, for a given kind of metal, it is generally found that the number of d electrons does not change (Hammer and Nørskov, 2000; Bligaard et al., 2007). Rather, the system compensates for this by shifting the d states up in energy as illustrated in Fig. 8.7. Three classes of trends determined largely by variations in εd − εF are considered. 8.2.5.2.1. Variations due to changes in surface structure For one kind of transition-metal atoms, the d-band center can be varied by changing the structure. The band width depends on the coordination number of the metal, so change of local structure can lead to a substantial change in the d-band center energy (Hammer et al., 1997). The coordination number is 9 for an atom in the most close-packed (111) surface of Pt, 8 for the more open (100) surface, 7 for a step or for the (110) surface, and as low as 6 at a kink. This can lead to variations in the d-band center of almost 1 eV (Fig. 8.8), and the chemisorption energy of, e.g., CO varies by a similar amount (Fig. 8.9). The experimental fact that steps bind CO more strongly than flat surfaces is in excellent agreement with the d-band model prediction

290

T. Bligaard et al.

Fig. 8.8. Projected densities of states onto the d states of the surface atoms for different Pt surfaces with decreasing atom density and thus coordination: The hexagonally reconstructed (100) surface, the close-packed (111) surface, the step atoms on a (211) surface and the kink atoms on an (11 8 5) surface (Hammer et al., 1997).

that the adsorption bond is determined largely by variations in εd − εF (Somorjai, 1994; Yates, 1995). Coordination and proximity of the surface Pt atoms is affected by the tendency of Pt surfaces to restructure into overlayers with a density of Pt atoms even higher than in the original close-packed (111) surface (van Hove et al., 1981). Overlap matrix elements and hence the band width are therefore larger on the reconstructed (111) surface than on the unreconstructed one, the d-bands lowered, and consequently the bond to CO even weaker. The reconstructed Pt surfaces are examples of strained overlayers and can be studied theoretically by simply straining a slab and performing the DFT calculation on it. For example, such calculations give continuous changes in the d-band center and in the stability of adsorbed CO (Fig. 8.9). The effect of varying the number of layers of a thin film of one metal lying on another metal can also be described in the d-band model (Hammer, 2006; Roudgar and Gross, 2003a). For atomic chemisorption, similar structural effects are found (see the middle panel of Fig. 8.9). The adsorbate bond to low-coordinated atoms at steps is stronger and the barriers to dissociation are lower than those to high-coordinated atoms and thus lower d-band centers. Consequently, the d-band model explains the many observations of stronger chemisorption bonds on steps than on flat surfaces (Somorjai, 1994; Somorjai and Bent, 1985; Yates, 1995; Henry et al., 1992; van Hardeveld and van Montfoort, 1966; Tripa et al., 1999; Mills et al., 2003). The correlation with the d-band center is found to be present for all adsorbates. This shows the general applicability of the d-band model. 8.2.5.2.2. Variations due to alloying The effect of alloying can also be understood in terms of d-band shifts. This can be deduced from Fig. 8.9 and shown in even more detail, as in Fig. 8.10, where adsorption results are shown for a series of Pt(111) surfaces. Here the second layer has been replaced by a layer of chosen 3d-transition metal atoms for a series of different 3d-metals. The effect on the reactivity of a Pt(111) overlayer is shown


291

Fig. 8.9. Calculated chemisorption energies for CO (top panel) and for different atomic adsorbates (middle panel), together with transition-state energies for the indicated dissociation reactions (bottom panel) are shown as functions of the average energy of the d states projected onto the surface atoms to which the adsorbates form bonds. Adapted from Mavrikakis et al. (1998).

in terms of adsorption-energy values for the different sandwiched 3d-metals. Such nearsurface alloys (Greeley and Mavrikakis, 2004), or “skins”, have been extensively studied as oxygen-reduction catalysts in PEM fuel cells (Markovic and Ross, 2002; Toda et al., 1999; Zhang et al., 2005). The projected DOS’s show the d states of the surface Pt to be shifted down in energy for second-layer metals towards the left in the periodic table. The DFTcalculated O and H adsorption energies show the same trends: the shift of the d-band center up in energy towards the Fermi level give increasingly stronger bonds. For these

292

T. Bligaard et al.

Fig. 8.10. Left: Calculated adsorption-energy values for H and O atoms on a series of Pt(111) surfaces, where the second layer has been replaced by a layer of 3d transition metals. Right: Calculated d-projected densities of states for the Pt surface atoms in the studied near-surface alloys. From Kitchin et al. (2004).

Fig. 8.11. Electrochemically measured changes in the hydrogen adsorption energy for Pd overlayers on a number of metals shown as a function of the calculated shift of the d-band center. From Kibler et al. (2005).

near-surface alloys, the band width is changed by hybridization between d states in the first and second layers. Such an indirect interaction can also be termed a ligand effect – the metal ligands of the surface atoms are changed. For metal overlayers, where a monolayer of one metal is deposited on top of another metal, there are similar effects. However, here one more effect appears, as the overlayer usually takes the lattice constant of the substrate. Metal overlayers thus show a combination of ligand and strain effects. Even in such a case, the d-band centers are found to describe changes in adsorption energies quite well (Løvvik and Olsen, 2003; Roudgar and Gross, 2003b; Filhol et al., 2004; Meier et al., 2004).


293

Fig. 8.12. Calculated shifts of d-band centers for a number of overlayer structures. The values are given relative to the d-band center of the pure overlayer metal surface. The shifts therefore reflect the change in reactivity of the overlayer relative to the pure metal. Adapted from Ruban et al. (1997).

Electrochemically determined variations in hydrogen adsorption on different Pd overlayers illustrate this, when plotted as a function of the calculated d-band shifts (Fig. 8.11) obtained by putting Pt on top of Fe, Co, Ni, Cu, Ru, Rh, or Ir (Kibler et al., 2005), because the Pt d-band is here shifted down relative to pure Pt(111). For anode catalysts for PEM fuel cells, this example is important, as poisoning by CO here is a severe problem, and thus a surface that binds CO more weakly (but still dissociates H2 ) is desirable. The d-band shifting and the subsequent control of the adsorbate binding energy have been directly observed in single-crystal experiments (Behm, 1998; Davies et al., 1998) and in fuel cells (Hoogers and Thompsett, 1999; Igarashi et al., 2001; Strasser et al., 2003). Information about d-band shifts is provided by a number of spectroscopic surface methods (Woodruff and Delchar, 1986) as described in vol. 2. There is synchrotron-based highresolution photoemission spectroscopy to directly measure d-band centers, giving results in good agreement with those of DFT calculations (Mun et al., 2005). In some cases, a shift in the d states can be measured as a core-level shift, as the d states and the core levels shift together (Weinert and Watson, 1995; Hennig et al., 1996). This suggests a possible explanation of observed correlations between surface core level shifts (Rodriguez and Goodman, 1992) for a number of metal overlayers and similar results (Hammer and Morikawa, 1996) for fuel-cell catalysis (Toda et al., 1999). 8.2.5.3. Ensemble effects in adsorption – the interpolation principle Until now, most effects considered have been indirect, with an adsorbate that interacts with one metal atom, for which in turn surroundings or ligands are varied. In another important case a single adsorbate interacts with an ensemble of different metal atoms (Sachtler and Somorjai, 1983), for which it can be shown that the chemisorption energy (i.e. the adsorption energy in the mixed site) to a first approximation is an appropriate average of the

294

T. Bligaard et al.

Fig. 8.13. Illustration of the interpolation principle. For a series of surface alloys, oxygen chemisorption-energy values from DFT calculations are compared with those of two simple interpolation models (Andersson et al., 2006).

contributions from the individual components, an “interpolation” principle that turns out useful in design of new catalysts, as discussed later (Jacobsen et al., 2001). To illustrate the interpolation principle (Liu and Nørskov, 2001; Andersson et al., 2006), oxygen binding-energy values for a large number of alloys from DFT calculations are compared to those from two different levels of the simple model. The interpolation principle is present in both models: (i) An estimate for the bond to a site with two kinds of neighbors, A and B, on the same substrate B, is E(Ax B1−x /B) = xE(B/B) + (1 − x)E(A/B), with typical errors of the order 0.1 eV; (ii) For an overlayer of one metal on another, a cruder model would be to use chemisorption energies calculated for the pure metal surfaces as a start. This neglects any strain and ligand effects and gives E(Ax B1−x /B) = xE(B/B) + (1 − x)E(A/A), which results in a slightly larger error (average for data set in Fig. 8.13 being 0.17 eV). However, the adsorption energy can be estimated on any alloy from a data base of adsorption energies calculated for pure metals (Table 2), thanks to its extreme simplicity (Bligaard et al., 2004). For useful estimates of adsorption energies of molecules on alloys the interpolation model is far from perfect but fast. The usefulness is supported by the d-band model, as shown by applying Eq. (8.5) to adsorption on a surface with several metal components. Under the assumption that E0 in Eq. (8.5) is independent of the metal considered, all effects due to having several metal components are to be found in the Ed term, which is a function of the d-band center. For small variations in εd , the relationship must be linear, Ed (εd ) = Ed (εd0 ) + Ed (εd − εd0 ).

(8.12)

The d-band center is given by the first moment of the chemisorption function, Eq. (8.10). We therefore need to understand qualitatively how (ε) for a multi-component system can be obtained by expanding the metal wave functions in localized basis sets |j and |j


295

consisting of d states on the individual atoms of the surface (Abild-Pedersen et al., 2005a), (ε) = a|H |j j |kk|j j |H |aδ(ε − εk ). j,j

k

Here the d states are assumed to form complete sets for the metal states, an assumption that should be sufficiently accurate for the determination of qualitative trends. The first moment of (ε) is then j,j k a|H |j j |kk|j j |H |aεk εd = j,j k a|H |j j |kk|j j |H |a j,j a|H |j j |Hmetal |j j |H |a = . j,j a|H |j j |j j |H |a Use of the relation j |Hmetal |j = εj =

ε

|j |k|2 δ(ε − εk ) dε

k

gives:

εd =

j

|Vaj |2 εj + j j =j a|H |j j |Hmetal |j j |H |a , 2 j |Vaj | + j j =j a|H |j j |j j |H |a

where |Vaj |2 = a|H |j j |H |a. For sufficiently localized metal d states, the second terms in both the numerator and denominator are small, giving εd ≈ V1 j |Vaj |2 εj ; V 2 = 2 j |Vaj | . Use of this result in Eq. (8.12) shows Ed to be an appropriately weighted average of the contributions from the different metal atoms with bonds to the adsorbate (that is, with a non-negligible matrix element). 8.2.6. Trends in activation energies for surface reactions Catalytic reactions on metal surfaces need for their description adsorption energies of reactants, intermediates, and products, as well as activation energies that separate different intermediate steps. The ammonia synthesis, N2 + 3H2 → 2NH3 , illustrates this by the full potential-energy diagram for this catalytic reaction (Fig. 8.14). The activation energy for a given reaction is defined as the difference between the energy of the transition state and that of the initial state, Ea = ETS − EIS . To understand differences in reactivity on metal surfaces, the variation of the interactions of molecules in their transition states with the metal surface has to be understood. Interestingly enough, the trends are qualitatively the same as for chemisorption energies, as shown in the following discussion. 8.2.6.1. Electronic effects in surface reactivity The arguments behind the d-band model are general enough to apply to the interactions in the transition state, as well as in the initial and final (adsorbed) states of the process.

296

T. Bligaard et al.

Fig. 8.14. Calculated potential-energy diagram for ammonia synthesis over close-packed (0001) and stepped Ru surfaces (Honkala et al., 2005; Hellman et al., 2006b). A∗ denotes an empty site and X∗ an adsorbed species. Insets show the transition-state (TS) configurations for N2 dissociation over the terrace and step sites (Honkala et al., 2005; Dahl et al., 1999).

Correlations between the transition-state energies and the d-band center are therefore expected to be the same as for chemisorption energies. Figure 8.9 illustrates this in its bottom panel, and Fig. 8.15 shows in detail how the activation energy for methane on different Ni surfaces scales with the center of each d band projected onto the metal states, to which the transition state couples. Figure 8.15 shows that both alloying (NiAu) and structuring (compare Ni(111), Ni(211) and the effect of strain) give effects that are described by d-band center variations. In fact, alloying effects can be observed directly in molecular-beam scattering experiments that monitor the methane sticking probability as a function of the Au coverage on a Ni(111) surface (Fig. 8.16). Such an insight has formed the basis for the design of catalysts with new properties (Besenbacher et al., 1998). Also the structural effects can be observed experimentally. For the methane activation over Ni surfaces, Fig. 8.15 shows the stepped Ni(211) surface to have a considerably lower barrier for dissociation than the flat (111) surface, as confirmed in an elegant set of experiments (Besenbacher et al., 1998). Sulfur atoms, known to adsorb preferentially at steps and to increase the barrier for methane activation considerably (Fig. 8.16), can be used to titrate the step sites to block the step sites selectively. Figure 8.17 shows experimental data for the carbon uptake of a Ni(14 13 13) surface, which contains 4% step atoms. If 2% of S is adsorbed, corresponding to a half-covered step, where all step atoms are blocked by having one S neighbor, the rate of methane dissociation is decreased substantially. This is a direct observation of the ability of step atoms to dissociate methane much faster than the terrace atoms. The indirect interaction between two adsorbates can also be described by the d-band model. This is also illustrated by Fig. 8.15 (the effects of pre-adsorbed C atoms). Two adsorbates that interact with the same metal atom often repel each other. For example, C


297

Fig. 8.15. Calculated activation energy for methane dissociation, as it varies over a number of different surfaces. The results are shown as a function of the d-band center energy of the d states that couple to the transition-state methane molecule. From Abild-Pedersen et al. (2005a).

atoms adsorbed on a Ni(211) surface affect the d states of the neighboring Ni atoms making them less reactive for a second atom or molecule. Similar effects have been observed for a number of adsorbates (Hammer, 2001). Additional effects of direct interactions between adsorbates that are not described in the d-band model are also shown in Fig. 8.15. A large adsorbate like S has sizable overlaps with the valence orbitals of the incoming molecule, giving rise to a repulsion, which is larger than what can be readily explained by the indirect interaction through d-band shifts. 8.2.6.2. Geometric effects in surface reactivity Methane activation takes place over a single Ni atom, and at this active site a change in structure has an effect primarily via changes in the electronic structure in the vicinity of the site. Surface structure can affect reactivity of transition metal surfaces also in a purely geometric way, however. A combination of electronic and geometric effects gives the stepped Ru surface a substantially lower barrier for dissociation of N2 than the closepacked (0001) surface (Fig. 8.14) (Dahl et al., 1999). The transition-state energy at the step is lower than that on the close-packed surface thanks to the electronic effect of the Ru step atoms having higher-lying d states at steps. This electronic effect can also be observed for atomic N adsorption (Fig. 8.14). However, to explain the more than 1 eV difference in barrier for dissociation of N2 this is not enough. In addition, there is a purely geometric effect due to the fact that at the step more metal atoms can participate in the stabilization of the transition state than on the flat surface (Dahl et al., 1999; Hammer, 1999). While the transition-state structure on the flat surface engages only four Ru atoms, of which one contributes to the stabilization of the two N atoms (Fig. 8.14), giving extra repulsion (see above), that at the step use five Ru atoms to stabilize the two N atoms (Fig. 8.14), none of them being “shared” (Dahl et al., 1999). The first speculation about a reactive step site involving five atoms to be particularly important in catalysts was made by Van Hardeveld (van Hardeveld and van Montfoort, 1966).

298

T. Bligaard et al.

Fig. 8.16. Measured sticking probability (relative to that of the clean surface) of a methane molecular beam on Ni(111) surfaces with varying amounts of Au alloyed into the surface (Besenbacher et al., 1998). The result of a model (prediction) based on DFT calculations of the change in the activation energy due to the addition of Au atoms is also shown. The beam data primarily measures methane sticking on the facets.

Geometric effects in heterogeneous catalysis has a long history, dating back at least to the work of Taylor in 1925 (Somorjai, 1994; Taylor, 1925; Gwathmey and Cunningham, 1958; Yates, 1995; Zambelli et al., 1996). An idea about the order of magnitude of the effect can be obtained from theoretical calculations, and recent detailed experiments have shown such results to be correct. For N2 dissociation on Ru steps and close-packed surfaces, for instance, the difference in barrier is so large that the small number of steps on even the best (0001) single-crystals should completely dominate the rate. Decoration of the few steps of a Ru(0001) surface with gold atoms show that the reactivity of the step atoms is at least nine orders of magnitude larger than that of the terrace atoms (Fig. 8.18) (Dahl et al., 1999). The step effects are found to be quite general, as discussed later. Another direct observation of the role of steps in dissociation reactions has been made for ethylene activation on Ni surfaces (Fig. 8.19) (Dietrich et al., 1996; Vang et al., 2005). 8.2.7. Brønsted–Evans–Polanyi relationships in heterogeneous catalysis Variations in adsorption energies and transition-state energies are governed by the same basic physics, as shown in the previous chapters. Then the fact that variations in adsorption energies of different molecules and transition-state energies can be found correlated is not surprising. Such relationships are extremely important for (i) building an understanding


299

Fig. 8.17. Experimental data for the thermal dissociation rate of methane (measured by C uptake), as a function of S atom coverage of the 4% of steps on a Ni(14 13 13) surface. From Abild-Pedersen et al. (2005b).

of heterogeneous catalysis from the fundamentals, (ii) estimating activation energies in a simple way on the basis of adsorption energies, and (iii) guidance in building kinetic models to understand trends in catalytic activity. 8.2.7.1. Correlations from DFT calculations In chemistry, linear correlations between activation (free) energies and reaction (free) energies are widespread, dating back to Brønsted (1928) and Evans and Polanyi a decade later (Evans and Polanyi, 1938). In heterogeneous catalysis, such relations have been assumed to hold (Boudart, 1997). However, making reliable predictions based on such linear correlations had to wait for DFT calculations becoming accurate enough to make it possible to establish such correlations between activation energies and reaction energies over a sufficient range. Figures 8.20 and 8.21 show some of the first published such Brønsted– Evans–Polanyi (BEP) relationships (Logadottir et al., 2001). A BEP relation is characterized by its slope, whose value varies from one reaction to another. For dissociative adsorption processes involving simple diatomic molecules, the slope is often close to unity. This implies that the electronic structure of the transition state is similar to that of the final state, indicating a late transition state, as seen in the transition-state structures in Fig. 8.21. By the same token, associative desorption shows very little dependence on the reaction energy, as long as the dissociation process is activated, as observed directly in thermal desorption experiments. On the Fe(111) surface, where dissociation is barely activated, recombinative desorption of N2 occurs at 750 K (Bozso et al., 1977), and

300

T. Bligaard et al.

Fig. 8.18. Measured thermal sticking coefficients of N2 on a clean Ru(0001) surface and on the same surface covered with 0.01–0.02 ML of gold as functions of the reciprocal temperature (Dahl et al., 1999). A result from a similar measurement at room temperature is given for comparison (Dietrich et al., 1996).

Fig. 8.19. a) STM image (200 × 200 Å ) of a Ni(111) surface after exposure to ethylene (10−8 Torr; 100 s) at room temperature (Vang et al., 2005). A brim of decomposed ethylene is formed along the step edges. b) STM 2 image (400 × 400 Å ) of a Ni(111) surface with the step edges blocked by Ag atoms. No decomposition of ethylene is observed on this modified surface. © 2005 Nature Publishing Group 2

on Cu(110), where it is impossible to dissociate N2 thermally, it occurs with peaks at 750 K (Heskett et al., 1988). The geometric effect discussed above for N2 dissociation on Ru surfaces is found for all the metals considered in DFT studies. In general the electronic and geometric effects in the reactivity of molecules are difficult to distinguish. A step on a surface of course exhibits a new geometric arrangement of the surface but it has also an electronic effect due to the higher-lying d states at the low-coordinated atoms at the edge of the step. Such an electronic effect shifts both ETS and E, and the shift is along the BEP line. Therefore


301

Fig. 8.20. Activation barrier for ethyl C–H bond breaking on pseudomorphic Pd overlayers, calculated by DFT, as a function of the ethylene adsorption energy, also calculated by DFT. Adapted from Pallassana and Newrock (2000).

Fig. 8.21. Transition-state energies for N2 dissociation shown, calculated by DFT, as functions of the dissociative N2 chemisorption energy for both close-packed and stepped metal surfaces, also calculated by DFT. Adapted from Logadottir et al. (2001).

the slope of the line is a measure of a purely geometric effect. In principle, a new line can be drawn there for every considered surface geometry, so a family of BEP lines can be imagined. For N2 dissociation, where a large number of geometries have been investigated (Honkala et al., 2005), the close-packed surface and the step (Fig. 8.21) seem to be close

302

T. Bligaard et al.

Fig. 8.22. BEP plot for C–O and C–C bond cleavage reactions on Pt(111), calculated for a number of transition-state (ETS ) and final-state energies (EFS ) that are measured relative to initial state-gas-phase energies (Alcalá et al., 2003).

to the extrema (in principle, it should be investigated for each reaction). Guiding lines can be set up for BEP diagrams. For instance, low-lying lines correspond to surface geometries, where the two fragments of N2 dissociation both can be stabilized without too many “shared” metal atoms. Another type of BEP relation in surface chemistry (Fig. 8.22) shows transition-state vs. final-state energies for one and the same surface (Pt(111)) but for a number of different C–O and C–C bond cleavage reactions that may occur during ethanol reforming (Alcalá et al., 2003). This interesting reaction finds application in hydrogen production from renewable bio-resources (Cortright et al., 2002). 8.2.7.2. Universal relationships Scaling is an example of a universal relation that can be found by systematic DFT calculations. Comparison of transition-state energies for dissociation of a number of similar molecules shows scaling with their reaction energies in much the same way as the relationships discussed previously (Fig. 8.23) (Nørskov et al., 2002). This remarkable result indicates a quite similar nature of the relationship between the final state and the transition state for dissociation of these molecules, a fact also borne out by comparison of structures (Fig. 8.24). For a given surface structure essentially all transition states look the same for the many systems considered in Fig. 8.23. The geometric effect discussed above for N2 dissociation holds for all the adsorbates considered (Fig. 8.24). Thus CO, NO, and O2 dissociation should also be much faster at steps than at the most close-packed surface (Liu and Hu, 2003; Ciobîcã and van Santen, 2003; Mavrikakis et al., 2002). This is in agreement with a growing body of experimental evidence, as noted above (Mavrikakis et al., 2002; Zubkov et al., 2003; Gambardella et al., 2001).


303

Fig. 8.23. BEP plots for dissociation of a number of simple diatomic molecules (Nørskov et al., 2002). For a given surface geometry, the data cluster around the same “universal” line.

Fig. 8.24. Transition-state structures (side and top views) calculated for different diatomic molecules that dissociate on different close-packed (top) and stepped (bottom) metal surface (Nørskov et al., 2002).

304

T. Bligaard et al.

Other systems may follow a BEP relationship, but then a different one than the simple diatomic molecules considered here (Michaelides et al., 2003). Dehydrogenation and C– C bond breaking are examples of such universality classes that may follow other similar relationships. There are exceptions to the BEP relations, most notably H2 dissociation on near-surface alloys (Greeley and Mavrikakis, 2004). Also these deviations from the rules can be described in the d-band model, though (Greeley and Mavrikakis, 2004). 8.2.8. Activation barriers and rates Energetics of adsorption systems can be richly elucidated by electron-structure calculations on relevant species, as shown above. Knowledge about adsorption energies and reaction barriers of the relevant molecular species is a good starting point for determination of the actual rates of catalytic turnover for a given reaction. In order to reliably evaluate absolute rates, however, one still has to go through a series of steps. In principle, electron-structure calculations could be used to calculate reaction rates directly by performing molecular-dynamics (MD) simulations on the relevant system for sufficiently long times to obtain reliable statistics (Jónsson et al., 1998). Only rarely this is a practical approach, however, as many interesting reactions (if not all) occur on time scales that are much longer than the periods for the atomic vibrations, a well-known major problem with full MD simulations (Landau and Binder, 2000). Reliable integration of the equations of motion for the atomistic system in question requires sampling with time intervals in the dynamics that are smaller than an atomic vibration period, as information about the dynamics is simply lost if longer time-steps are used. Fortunately, for many important processes, reaction rates can be obtained without a full MD simulation (Hellman et al., 2006a). Most reaction rate theories used for elementary processes build upon the ideas introduced in the so-called transition-state theory (TST) (Eyring, 1935; Wigner, 1938; Eyring, 1938). Here focus is on TST, because it (and its harmonic approximation, HTST) has been shown to yield reliable results for many elementary processes at surfaces. With access to rate constants for the key elementary processes of a catalytic reaction, the total catalytic rate can be calculated. The coupling between elementary and over-all rates of a catalytic reaction can also be performed in different ways, depending on the detail in which the elementary reaction rates are known, the importance of adsorbate–adsorbate interactions, and the degree to which the adsorbate coverage structure can be expected to be important for the total rate. The full solution of the reaction equations including neighbor interactions up to some predetermined neighbor shell is usually obtained by lattice kinetic Monte Carlo (KMC) approaches (Voter, 1986). Successfully and routinely applied to surfaces-growth kinetics (see, e.g., this volume, chapters by H. Brune and K.W. Kolasinski), KMC has also been used for heterogeneous catalytic systems (Neurock and Hansen, 1998; Ovesson, 2002; Reuter et al., 2004; Ovesson et al., 2005; Honkala et al., 2005; Hellman et al., 2006b). To describe trends in catalysis, however, it is often adequate to use simple mean-field models, where surface coverages are averaged over the entire surface (Boudart and DjégaMariadassou, 1984; Dumesic, 1999; van Santen and Niemantsverdriet, 1995; Wierzbicki and Kreuzer, 1991). Here often adsorbate–adsorbate interactions are neglected, thereby ignoring most details about the adsorbate-structure, islanding phenomena, etc. However,


305

the mean-field methods have had a strong impact on the understanding of heterogeneous catalytic processes (Dumesic et al., 1993) and give the benefit to often provide analytic expressions, thereby easing interpretation and insights into the mechanisms of the catalytic surfaces. The mean-field methods can also to varying degree be extended to include lateral interactions (Dumesic et al., 1993). After a brief introduction of TST, some applications of these mean-field models will be described. 8.2.8.1. Transition-state theory In TST (Eyring, 1935; Wigner, 1938; Eyring, 1938; Horiuti, 1938) a reaction rate for a rarely occurring elementary reaction is calculated. The problem is approached by separating space into two regions called the reactant region (RR), where the system can be found before reacting, and the product region (PR), the region for the product of the elementary reaction in question. The border between the two regions is referred to as the transition state (TS). The corresponding low-energy configurations are often referred to as the initial state (IS) and final state (FS), respectively. A TS assumed to have zero thickness has one dimension less than the configuration space of the system, naturally chosen at the energy ridge between RR and PR, particularly if entropy effects were to be neglected. In TST, the division of configuration can also be formulated in terms of a more general configuration space. In typical applications of TST to potential-energy surfaces (PES’s), quantum tunneling effects are assumed negligible, and the Born–Oppenheimer approximation is invoked. In addition, two key assumptions are made, (i) Boltzmann-distributed reactants, valid for a thermally equilibrated system and a thermally equilibrated incoming flux of reactants, and (ii) once in the TS, the system will never reenter the IS region, an approximation that has to be seriously considered and that can be effectively corrected for. TST states that the rate constant of an elementary reaction is given by kTST = PTS · rc,TS ,

(8.13)

where PTS is the probability to find the system in the TS region, and rc,TS is the rate by which the TS region is crossed. Here a finite width of the TS region is assumed, with the option to let it approach zero later. The assumed Boltzmann distribution gives the probability of finding the system in a given region of configuration space and thus the probability to be in the infinitesimal vicinity of thickness, δx, around the TS, as $ PTS

= δx · $ TS

e−V (x )/kB T d x

−V ( x )/kB T d x RR e

= δx ·

ZTS , ZRR

(8.14)

where V (x) is the potential of one particle, and ZTS and ZRR are called the configuration integrals over the TS (dividing surface) and the reactant region, respectively. The rate for crossing the infinitesimal region of thickness, δx, around the TS towards the FS is rc,TS =

v⊥ , δx

(8.15)

306

T. Bligaard et al.

where v⊥ is the velocity perpendicular to the TS. This velocity can be evaluated directly from the Boltzmann distribution: $∞ − i 12 mi vi2 /kB T v · e dv kB T ⊥ ⊥ = v⊥ = 0$ ∞ , − i 12 mi vi2 /kB T 2πμ e dv⊥ −∞

where μ is an effective mass of the system in motion at the transition state. It can be evaluated for specific applications (Jóhannesson and Jónsson, 2001), but often it is apparent already from the context of the application which mass is to be used. The central result of TST is then the definition of the TST rate constant, which is obtained by combining Eqs. (8.13)–(8.15), kB T ZTS . · kTST = PTS · rc,TS = (8.16) 2πμ ZRR 8.2.8.2. Variational transition state theory and recrossings The TST assumes that configurations that appear in the TS and have a velocity towards the PR will eventually end up in the PR, thus miscounting recrossing. It is natural to choose the TS in such a way that the rate constant is minimized (Keck, 1960, 1967). In general, however, there are considerable difficulties in representing the dividing surface for systems of many atoms (Jóhannesson and Jónsson, 2001; Carter et al., 1989; Sprik and Ciccotti, 1998). By going beyond TST, further improvements can be obtained by including dynamical corrections to the TST rate constant (Keck, 1962; Yamamoto and Chem, 1960; Bennett et al., 1977; Chandler, 1978), which gives superior estimates of the rate constant, but can computationally be rather demanding (Voter, 1986). 8.2.8.3. Harmonic transition state theory (HTST) The TST is very often used in its harmonic approximation (HTST), which is applicable under the normal assumptions of TST but further demands that the PES is smooth enough for a harmonic expansion of the potential energy to be well represented by its second-order Taylor expansion around both the IS and in a first-order saddle point on the PES. The procedure for determining the HTST rate constant thus follows a series of welldefined and well-known steps. Future developments attract a special interest in methods that do not require information about the FS of the reaction, and which, in principle, allow for long-time simulations without predefined event tables (Olsen et al., 2004). The majority of current studies in theoretical catalysis concerns predefined reactions, for which both the IS and FS are known, and the objective of the saddle point search is to find the saddle point in between. For problems of this type one most often applies the Nudged Elastic Band (NEB) method (Jónsson et al., 1998; Henkelman and Jónsson, 2000; Henkelman et al., 2000) or one of the many algorithms derived from the NEB (Maragakis et al., 2002; Chu et al., 2003). The NEB algorithm establishes an ensemble of “images” of the system along the minimal energy path (MEP) from the IS to the FS of the reaction. An a priori determination of the FS is needed as well. The saddle point is determined as the maximum energy configuration along the MEP. The TS in HTST is chosen as the uniquely defined dividing surface, which is the hyper-plane going through the saddle point, and which is perpendicular to the reaction coordinate in the saddle point.


307

Normal-mode analyses in the IS and in the TS saddle point (SP) make it then possible to obtain the harmonic expansion of the potential in the RR and the TS. The HTST rate constant then becomes (at kB T hυi for all i): &D i=1 υi,IS −Ebarrier /kB T kHTST = &D−1 (8.17) e , i=1 υi,SP 1 √ xSP ) − V ( xIS ), frequency υi = 2π · ki /μi , of eigenmode i correwhere Ebarrier = V ( sponding to an effective mass, μi , and force constant ki obtained from the normal-mode analysis by diagonalizing the mass-weighted Hessian matrix (Atkins and Friedman, 1997). The expression thus has an Arrhenius form, kArrh = υe−E/kB T . Since Eq. (8.17) has one frequency more in the numerator than in the denominator, it is often interpreted as an attempt frequency of the reactant system multiplied by a Boltzmann factor corresponding to the energy barrier between the initial and TS saddle-point states. The TST result is often written on the form k=

kB T −G/kB T kB T S/kB −H /kB T = ·e , e e h h

(8.18)

where G, S, and H are differences in Gibbs free energy, entropy, and enthalpy between the initial and transition states, respectively. This form is often very convenient in applications, since it allows the inclusion of calculated partition functions or tabulated entropies in a straight-forward fashion. The change in entropy between reactant and transition states is relatively small, particularly true for systems without changes in the freetranslational degrees of freedom between these states (as free translations give the largest entropy contribution (as in, e.g., gases)). The factor kBhT is of the order of 1013 s−1 at catalytically relevant temperatures. In the absence of larger entropic effects, the prefactor is thus of the order of 1013 s−1 , which is also what is being observed experimentally (van Santen and Niemantsverdriet, 1995). A trend study in catalysis is typically made on a given reaction, searching for systematical variation with respect to the catalytic surface. Variations in entropy from system to system are typically very small, affecting the rate by at most a couple of orders of magnitude (Bligaard et al., 2003). This variation is very small compared to the changes in rates caused by varying energy barriers. For example, a change in barrier by 1 eV leads at room temperature to a change in the Boltzmann factor by approximately 1020 . Since activation barriers between neighboring metals in the Periodic Table vary by amounts of the order of perhaps half an eV, it often suffices for trend studies of in heterogeneous catalysis to ignore effects of varying prefactor and to concentrate on only the variations in the reaction energetics. Many adsorbate and bulk systems fulfill the application criteria for HTST. The PES is often sufficiently harmonic in the initial state and in the saddle point, at least for strongly bonded systems. The rather high energy barriers involved in most surface processes is perhaps the cause of this. The most important processes are often slow, which might be considered natural, as strong bonds lead to attempt frequencies around 1013 s−1 . For elementary processes with non-substantial energy barriers, the rate automatically turns out very high, and the IS and FS would then turn out to be in thermal equilibrium.

308

T. Bligaard et al.

Fig. 8.25. Rates for CO oxidation over a RuO2 (110) surface, comparison of calculated (bottom curve) and experimental results (top curve) (Reuter et al., 2004).

Certainly, for some important systems, including the class of gas-phase reactions, the assumptions of HTST are not fulfilled. In the gas phase there are zero-frequency modes, such as translation and rotation, and these lead to totally different configuration integrals than those obtained from a normal-mode analysis. For these systems the HTST rates can be modified in a simple manner by incorporating the molecular partition functions into the formula (Evans and Polanyi, 1937; Chorkendorff and Niemantsverdriet, 2003). Both the kinetic Monte Carlo (KMC) and mean-field models for solving the reaction equations of a heterogeneously catalyzed system are usually built upon elementary rates obtained from the HTST approach. For CO oxidation over a RuO2 (110) surface, an application of the KMC method based on activation energies calculated with DFT (Reuter et al., 2004) shows a very good agreement with detailed surface-science experiments (Fig. 8.25) (Reuter et al., 2004). The same is true for a similar attempt to calculate the rate of the ammonia synthesis over Ru (Fig. 8.26), showing a surprisingly good agreement with highpressure data (Honkala et al., 2005).

8.3. Variations in catalytic rates – volcano relations For a detailed description of the rate of a given heterogeneous reaction, elaborate kinetic methods are available, as described above and exemplified below. For trends in catalysis, a more general description can be made, as described in this section. Mean-field microkinetic models are in many cases adequate for a quantitative description of the reaction rate (Linic and Barteau, 2003a; Kandoi et al., 2006), as exemplified by methanol decomposition over Pt (Fig. 8.27). When studying trends, the mean-field models have some distinct advantages, since the introduction of a few additional assumptions (such as inclusion of a rate-determining reaction and the steady-state approximation) will often result in the model becoming entirely analytical.


309

Fig. 8.26. Rates for the ammonia synthesis on a nanoparticle Ru catalyst under industrial conditions, comparison of calculated and experimental values (Hellman et al., 2006b).

Fig. 8.27. Methanol decomposition over Pt, as determined from a microkinetic model (Kandoi et al., 2006).

The microkinetic models in this section, built upon BEP relations of the type described above, will be used to show that an underlying BEP relation in general leads to the existence of a volcano relation. They will be used also in combination with the universal BEP relation to explain, why for a wide range of reactions, good catalysts lie in a surprisingly narrow interval of dissociative chemisorption energies. Connections like these should be very useful in the design of catalysts. To provide a conceptual framework for analyzing microkinetic models of heterogeneous reactions a simple tool, referred to as the “Sabatier Analysis”, is described. The Sabatier Analysis of the microkinetic models developed in this section suggests that the clustering of good catalysts can be explained by the combination of the universal BEP relation and activated readsorption of synthesis products onto the catalyst.

310

T. Bligaard et al.

Fig. 8.28. Energy diagram for the microkinetic Model 1 (Bligaard et al., 2007).

8.3.1. Dissociation rate-determined model A large number of important heterogeneously catalyzed reactions proceed from a diatomic reactant and necessitates the cleavage of a strong molecular bond. A simple model can be created for this category. Model 1: Dissociative chemisorption as the rate-determining step: One of the simplest such reactions is of the type A2 + 2B → 2AB

{1}

where the dissociation of the diatomic molecule, A2 , occurs at the same time as the molecule is adsorbed on the surface. The other reactant, B, reacts with the adsorbed species, A, without prior adsorption of B on the surface, and with direct desorption of the product, AB, to the gas phase. The reaction scheme of elementary steps is thus A2 + 2∗ → 2A∗ ∗

A + B → AB +

{2} ∗

{3}

where an asterisk represents an active surface site. The turnover frequency (TOF), which is the frequency of net creation of product molecules per active site on the catalyst, can be expressed by the microkinetic model for this system (Somorjai, 1994; van Santen and Niemantsverdriet, 1995; Dumesic et al., 1993), r(T , Px ) = 2 · k1 · PA2 · θ∗2 (1 − γ ).

(8.19)

Here k1 is the temperature-dependent rate constant for the forward direction of reaction step 1, which is assumed to follow an Arrhenius expression with activation energy of ET of Fig. 8.28, PA2 is the pressure of the reactant A2 , θ∗ is the fraction of the active sites that are free, and γ is the approach to equilibrium (sometimes also referred to as the reversibility of the reaction) (de Donder, 1927; Dumesic et al., 1993; Dumesic, 2001). For the expression to be valid in general, γ should be the approach to equilibrium for reaction step 1, but if the dissociative chemisorption is the rate-determining step, which we shall assume for now (later this assumption will be removed), then the γ in Eq. (8.19) is the over all gas phase


311

approach to equilibrium, γ =

2 PAB

Keq · PA2 · PB2

,

(8.20)

where Keq is the equilibrium constant for the reaction, PB is the pressure of the reactant, B, and PAB is the pressure of the product, AB. Under the assumption that dissociative chemisorption is the rate-determining step, the coverage of free sites can be determined analytically, θ∗ =

1 1+

θA θ∗

=

1 1+

PAB K2 PB

=

1+

1 , K1 PA2 γ

(8.21)

where K1 = exp(−G1 /kB T ) is the equilibrium constant for reaction step 1 with standard reaction Gibbs energy G1 = E1 − T S1 . Under the given assumptions it is thus possible to obtain an analytical expression for the turnover frequency, which is the primary reason for assuming that a given reaction step is rate-determining. The dissociative chemisorption energy E1 determines how very reactive surfaces (very negative E1 ) poison the reaction in the sense that there will be very few free sites to dissociate onto. The turnover frequency will thus decrease as E1 → −∞. The BEP relation relates the TS energy of dissociation to the dissociative chemisorption energy on a given surface. The activation barrier is large on less reactive surfaces (E1 numerically small or even positive), and the turnover frequency thus also decreases, as E1 → ∞. In the intermediate range, the turnover frequency goes through a maximum that often resembles a volcano, and the turnover frequency is thus said to follow a volcano curve. Such volcano curves can be analytically determined for some simple model reactions, provided that some further assumptions are made (realistic at least for the case of NH3 synthesis) (van Santen and Niemantsverdriet, 1995; Evans and Polanyi, 1937; Kolasinski, 2002; Eichler et al., 1999; Eichler et al., 2000). There is also a Model 2 that is discussed elsewhere (Bligaard et al., 2004). 8.3.2. Sabatier analysis The above analysis is primarily valid for heterogeneously catalyzed reactions that are dissociation-rate determined with an optimal catalyst. The ammonia synthesis is among the known such reactions (Stoltze and Nørskov, 1985; Dahl et al., 2000). The rate-determining step is less well identified for other less studied reactions. To find the exact numerical solution of the given microkinetic model and then as a post-treatment calculate whether the dissociation step is rate-determining or not is then impractical. Instead, an analytical solution based on an assumption about some rate-determining step is preferred, since such an analytical tool stimulates the subsequent development in the understanding of the reaction in question, available for analysis when the assumption of rate determination breaks down. The microkinetic Model 1 is first evaluated numerically exactly, now without any assumption about a rate-determining step, with the goal to develop such tools. The resulting volcano curves (Fig. 8.30) are on their right-hand sides identical to the solutions in Model 1, when a dissociation step is assumed rate-determining (Fig. 8.29). On the lefthand side, they also closely follow the behavior in Fig. 8.30, when reaction conditions

312

T. Bligaard et al.

Fig. 8.29. Volcano plots (normalized turnover frequencies vs. E1 ) for Model 1 at various approaches to equilibrium (Bligaard et al., 2004). For equilibrium-limited reactions (such as ammonia synthesis) the approach to equilibrium will vary from 0 at the inlet of the reactor to almost 1 at the outlet. That the optimal catalyst depends on the approach to equilibrium indicates that the optimal catalyst for the reaction will depend on the position in the reactor bed (Jacobsen et al., 2002).

Fig. 8.30. Sabatier volcano-curve: The limiting case of the exact numerical solution of the microkinetic Model 1 (Bligaard et al., 2004).

are close to equilibrium. Far from equilibrium, however, they show a drastically different behavior. In Fig. 8.30 the solution for γ = 10−15 is actually also shown, however, indistinguishable from the volcano-curve with γ = 10−11 . Maximal desorption and maximal dissociation rates are calculated simply as the rates of desorption or dissociation assuming an optimal coverage for the given reaction step, and the lines representing them clarify the situation. The turnover frequency is automatically bounded by the line corresponding to the maximal possible dissociation rate for a microkinetic model assumed to be dissociation rate determined. The other exact bound on the turnover frequency, that the turnover frequency cannot be larger than the maximal desorption rate, will eventually be violated


313

for very reactive surfaces or very small approaches to equilibrium. The exact solution looks like the solution obtained with the rate-determining step, until that solution approaches the maximal desorption rate, to which it will join smoothly (through a suitable adjustment of the coverages). Several important conclusions can be drawn from Fig. 8.30: In general a simple catalytic reaction which includes the dissociation of a diatomic molecule will have this dissociation as the rate-determining step, when the reaction takes place under conditions close to equilibrium. The ammonia synthesis, the prototype of an equilibrium-limited reaction, is dissociation-rate determined and agrees well with Fig. 8.29 (de Donder, 1927). For a reaction taking place far from equilibrium, the actual approach to equilibrium becomes unimportant, and the volcano plot very closely follows the volcano defined by the minimum value among the maximal possible rates for all reaction steps. For the general case, the limiting Sabatier volcano curve can be defined as TOF Sabatier = min(max R1 , max R2 , . . . , max RN ).

(8.22)

Here max Ri means the maximal rate of reaction step i, which is calculated by assuming optimal coverages for that reaction step. This (usually multi-dimensional) volcano-curve will be referred to as the Sabatier volcano-curve, as it is intimately linked to the original Sabatier principle (Sabatier, 1911; Balandin, 1969). This principle states that desorption from a reactive metal catalyst is slow and will increase on less reactive metals. On very noble metals, however, the large energy barrier for dissociation decreases the dissociation rate. The best catalyst must be a compromise between the two extremes. The optimal compromise is not necessarily obtained exactly where the maximal desorption and dissociation rates compete, as shown above. That is only the case far from equilibrium. Close to equilibrium the maximum is often attained, while dissociation is the rate-determining step, and the maximum of the volcano-curve will then be reached due to a lack of free sites to dissociate into. 8.4. Optimization and design of catalysts through modeling Given the simplicity of the kinetics arguments in Section 8.3, the agreement with observations is surprisingly good. It indicates that we have access to the key concepts for an understanding of trends in overall reactivity for this class of surface-catalyzed reactions. So far this chapter is an attempt to explain the experimentally observed variations in adsorption energetics. It also attempts to introduce modeling of reaction rates of catalytic reactions on the basis of the adsorption energetics. However, attempts to predict what is not already known are the real tests of the quality of a theory. Is there today a real competition with modern experimental-design approaches (Cong et al., 1999) in usefulness as a tool for the design of new heterogeneous catalysts, or will we have to move the modeling precision much further towards “chemical accuracy” before we can even contemplate design? One argument pointing in favor for design by “Catalysis Informatics” is certainly the constantly increasing availability of computational power. A power that we can translate directly into better accuracy as well as larger databases of computed materials. Below two examples are given, which could lead to some optimism for the role of theory as a supplementary tool in future catalyst design.

314

T. Bligaard et al.

8.4.1. The low-temperature water gas shift (WGS) reaction The water gas shift (WGS) reaction (Schumacher et al., 2005), H2 O(g) + CO(g) → H2 (g) + CO2 (g),

{WGS}

is employed in several industrial processes. An important WGS process is the production of hydrogen from hydrocarbons in combination with steam reforming. High-purity hydrogen produced this way is used for, e.g., the ammonia synthesis. The reaction is moderately exothermic, and thus low temperatures favor a high equilibrium conversion of CO. The WGS reaction is also relevant for the methanol synthesis and fuel cell-powered vehicles. The proposed reaction mechanism for the WGS reaction is rather complex, H2 O(g) + ∗ → H2 O∗ ∗

{WGS1}

H2 O + → H + OH

{WGS2}∗

2OH∗ → H2 O∗ + O∗

{WGS3}

∗

∗

∗

∗

∗

∗

OH + → O + H

∗

{WGS4}∗

2H∗ → H2 (g) + 2∗ ∗

∗

∗

CO∗2

CO(g) + → CO ∗

CO + O →

{WGS5} {WGS6} +

∗

{WGS7}∗

CO∗2 → CO2 (g) + ∗ H

∗

+ CO∗2

{WGS8} ∗

∗

→ HCOO + ,

{WGS9}

where ∗ represents a surface site consisting of typically two copper atoms. Steps WGS2, WGS4, and WGS7 are considered to be rate-controlling. The theory needed for the kinetic modeling has been published (Ovesen et al., 1992). Variations in reactivity of such a complex reaction through the transition metals might be expected to need a large set of parameters for its description. However, thanks to an analysis based on DFT, BEP relations and adsorption-energy correlations, it is possible to reduce the number of descriptors to only two, the adsorption energies of oxygen and carbon monoxide on each transition-metal surface. The rate of the WGS reaction over transition-metal surfaces as a function of the binding energies of atomic O and CO, based on chemisorption-energy values on step sites (Bligaard et al., 2004), is a three-dimensional volcano plot (Fig. 8.31). The results with chemisorption data on terrace sites (Gokhale, A.A., Kandoi, S., Grabow, L.C., Dumesic, J.A., Mavrikakis, M., unpublished observations) look qualitatively similar (Schumacher et al., 2005). The model, which assumes a simple redox mechanism to be dominant over all metal catalysts, gives a good qualitative estimation of the relative activity order of the different transition-metal catalysts, when compared to experiment. However, it fails quantitatively over the broad range of metals. It shows that copper is the most active of the late and noble transition metals (Fig. 8.31). Even better, a way to improve the industrial catalyst is predicted. The modeling of the trends in reactivity behind the WGS reaction predicts a slightly higher reactivity with respect to both oxygen and carbon monoxide adsorption increases.


315

Fig. 8.31. The turnover frequencies for the low-temperature water-shift (WGS) reaction as a function of adsorption energies of oxygen and carbon monoxide (Schumacher et al., 2005). The positions of the step sites on noble and late transition metals are shown. As observed experimentally only copper appears to be a suitable pure metal catalyst for the process.

8.4.2. Methanation The methanation reaction is primarily used to remove traces of CO and CO2 in the hydrogen feed gas for the ammonia synthesis. It has been known for more than a century (Sabatier and Senderens, 1902) and it has found renewed interest in connection with the transformation of coal to natural gas. The hydrogen for ammonia synthesis is here normally produced by steam reforming with subsequent water gas shift as described above. Unreacted CO is then removed by the carbon monoxide methanation process: CO(g) + 3H2 (g) → CH4 (g) + H2 O(g)

{Methanation}.

This process proceeds by dissociative chemisorption of CO and H2 and the subsequent recombination of adsorbed species to form CH4 and H2 O, which finally desorb (Goodman et al., 1980). The simplified (contracted) energy diagram for the process over Ni, Ru, and Re in Fig. 8.33 shows that the barrier for dissociating CO is small over Re compared to those over Ni and Ru. Over Re the barriers for desorbing methane and water are high, however. The opposite is the case for Ni, the barrier for dissociating CO being high, but those for desorbing water and methane being lower than on Ru and Re. Ru constitutes the best compromise, as on this metal none of the barriers are very high. All this is well in line with the Sabatier principle. CO dissociation obeys a BEP relation, when the dissociation barrier is correlated against the dissociative chemisorption energy (Fig. 8.23). In Fig. 8.33 (right-top), such a BEP relation is shown, taking the strongly adsorbed CO precursor as reference instead of CO in the gas phase. When the experimentally measured activity is plotted against this dissociation energy, a very well-behaved volcano appears (Fig. 8.33 (right-bottom)).

316

T. Bligaard et al.

Fig. 8.32. Energy diagram for CO methanation over Ni (Andersson et al., 2008).

Fig. 8.33. Contracted energy diagrams for CO methanation over Ni, Ru, and Re (left). BEP relation for CO dissociation over transition metal surfaces (right-top) and the corresponding volcano-relation for the turnover frequency (right-bottom). From Andersson et al. (2006, 2008).


317

Fig. 8.34. The measured rate for CO removal from a feed gas containing 2% CO in 1 bar of H2 as a function of the Ni content in FeNi alloy catalysts (Andersson et al., 2006). Results are shown for two different temperatures and two metal concentrations.

The quality of this volcano turns out to be so high that it is possible to predict alloys which show a higher methanation activity and at a lower price of the constituents than the industrially used (Ni) catalyst. It should be used together with the interpolation principle (see Section 8.5.3) (Andersson et al., 2006). From the “Pareto-optimal” (Pareto, 1906) set of interpolated alloys with respect to low cost and short distance to the optimal dissociative chemisorption energy (Fig. 8.35) one notes that some Fe–Ni alloys are predicted to yield better activity than Ni and that they do so at a reduced cost, since Fe is a much cheaper material than Ni. Subsequent experiments corroborate these theoretical predictions. Curiously, there was no success for a simultaneous state-of-the-art high-throughput experimentation study in predicting the utility of Fe–Ni alloys as improved low-cost catalysts for the methanation reaction (Yaccato et al., 2005). This experimental verification for pure CO hydrogenation has recently been supplemented by verifications for CO2 hydrogenation as well as for simultaneous CO and CO2 hydrogenation (Sehested et al., 2007). This means that a technical methanation catalyst is discovered on the basis of computational screening (Sehested et al., 2007).

318

T. Bligaard et al.

Fig. 8.35. Pareto (Pareto, 1906) plot of interpolated catalysts predicted to be good compromises with respect to cost and activity for methanation. The positions of the interpolated catalysts are determined by the cost of their constituent elements vs. their distance from the optimal dissociative chemisorption energy for CO with respect to the experimentally observed optimum (see Fig. 8.33 right-bottom). From Andersson et al. (2006), Sehested et al. (2007).

8.5. Some catalytic reactions from the fundamentals 8.5.1. Introduction Most of this chapter is devoted to a conceptual picture of heterogeneous catalysis, including its surface-science basis and its successful applications. To further detail the surfacescience approach to catalysis it is natural to supplement this presentation with some other examples of recent development. Some monoxide oxidation reactions, the ammonia synthesis, and the hydrogen evolution reaction are used to illustrate the DFT-KMC method, Monte Carlo simulations, and the close interplay between theory and experiment, typical for surface science. 8.5.2. Kinetic simulations of oxidation of some monoxides Catalytic reactions develop with time. However, typical process time scales differ drastically from typical atomic and electronic time scales. To follow the many particles of the system from the fundamentals over process times is computationally very demanding. One way to bridge from the fundamentals to real processes is by the DFT-KMC approach, where the kinetic Monte Carlo (KMC) provides rates for reactions and reaction steps, based on input parameters calculated in DFT, which in turn provides an extensive mapping of the energetics that is typically in agreement with available results from accurate experimental surface-science methods. This approach is proven successful in the description of, e.g., epitaxial growth of single-component metal (see, e.g., chapter by H. Brune in this volume)


319

and semiconductor systems (see, e.g., chapter by K.W. Kolasinski in this volume). Application of the DFT-KMC method to catalytic reactions requires generalization to many components. Here the first two such studies are briefly reviewed, both concerning oxidation of monoxides, NO (Ovesson, 2002; Ovesson et al., 2005) and CO (Reuter et al., 2004; Reuter and Scheffler, 2006), respectively. 8.5.2.1. NO + O → NO2 Platinum is a well-known oxidation catalyst for many surface reactions. As such, it is currently being used to convert NO to NO2 for absorption in so-called NOx traps under excess oxygen conditions, where direct conversion of NO to nitrogen gas is prohibitive. The system and reaction are suitable for kinetic simulations. The many reaction parameters can be derived from DFT (Bogicevic and Hass, 2002), and kinetic Monte Carlo (KMC) simulations can be performed (Ovesson, 2002; Ovesson et al., 2005). In this way, this oxidation process is shown not to be an inherent property of the platinum catalyst itself. In fact, the intrinsic NO + O → NO2 reaction is found to be inhibited (endothermic) rather than promoted on Pt(111), due to strong oxygen-platinum bonds. The platinum becomes an efficient oxidation catalyst only at a sufficiently high oxygen chemical potential, as its oxygen bonds are weakened with increasing coverage, and the NO2 formation reaction becomes exothermic. At that point, Pt catalyzes the reaction by lowering the activation barrier for the kinetic reaction. As detailed and essential surface-science results are lacking, the computed results can only be described as congruent with those of flow-reactor experiments. A strong temperature dependence is found for the turn-over frequency, which hopefully should encourage further UHV studies. More stringent emissions and fuel-economy requirements spur the introduction of leanburn gasoline and diesel engines, where hydrocarbons are combusted in an excess atmosphere of oxygen (Shelef and McCabe, 2000). Conventional three-way catalysts operate very well in traditional power trains but do not remove nitrogen oxides (NOx , x = 1–2) from the exhaust gas efficiently under such conditions. Selective catalytic reduction of NOx by means of injection of urea or some other reductant into the catalytic converter (Parvulescu et al., 1998; Cant and Liu, 2000) and temporary chemical trapping of NOx under lean conditions and subsequent release during brief excursions to fuel-rich conditions (Takahashi et al., 1996; Fridell et al., 1999) are two attractive approaches to remedy this that both rely on the ability to catalytically oxidize NO, the primary component of NOx , from the combustion process, into NO2 . Noble-metal catalysts are typically used, in spite of unclear points on the limits of their activity and on their interaction with oxide supports. Today such information can in principle be provided by theory, energetics and reaction barriers provided with chemical accuracy by DFT calculations, and simulations by KMC should inform about relevant size and time scales. A hybrid DFT-KMC approach that combines the advantages of either method might give significant new insight into the oxidation reaction and the means to separate intrinsic catalytic properties from the effects of varying thermodynamic conditions (Ovesson, 2002; Ovesson et al., 2005). In heterogeneous catalysis, reaction rates depend on (i) activation energies, (ii) residence times in precursor states that are very sensitive to adsorbate interactions, and to some extent diffusion limitations. To quantitatively assess complex reaction kinetics, theory must be able to account for these effects over relevant time and length scales. The magnitude

320

T. Bligaard et al.

and extent of both direct and indirect adsorbate interactions can be described by computationally intensive first-principles DFT calculations (Stampfl et al., 1999; Li et al., 2002; Bogicevic et al., 2000), now for a three-component system and incorporation into the simulations. The combination of DFT and KMC simulations is used to meet the challenge to extend the realm of first-principles investigations from the time and length scales of atom dynamics to those of technologically relevant chemical processes like epitaxial growth (Ruggerone et al., 1997; Bogicevic et al., 1998; Ovesson et al., 1999; Kratzer and Scheffler, 2002) and catalytic reactions. An extensive mapping of the elementary processes involved and an accurate calculation of their activation energies is required, and an instep into the complex area of catalysis sets extra demands (Ovesson, 2002; Ovesson et al., 2005). For the oxidation of NO on Pt(111), this means a detailed assessment of the energetics, mass transport, and reaction kinetics of O, NO, and NO2 over Pt(111). The direction of the NO oxidation reaction is found to be influenced by the oxygen coverage. The lateral adsorbate interactions are found possible to express in terms of using pair-wise models, a valuable simplification for a system of this complexity. In a KMC simulation, the key elementary steps have first to be identified. Such a list, containing the steps adsorption, diffusion, reaction, desorption, and more, is obtained by combining existing data on surface intermediates with detailed balance and chemical intuition. The rate of each step is then calculated within TST, using activation energy values derived from DFT with, e.g., the PW91 implementation of the generalized-gradient approximation (GGA) (Perdew, 1991; Hammer et al., 1999), as described in (Bogicevic and Hass, 2002). Transition states (TS) for the diffusion and reaction processes are calculated with the nudged elastic band (NEB) method (Jónsson et al., 1998). Four system replicas are used between the initial- and final-state geometries to achieve a smooth minimum-energy path upon relaxation, with activation energies determined via spline fits. For adsorbate diffusion (O and NO), the TS’s are quite simple, located near the geometric saddle points, while for surface-mediated chemical reactions, the minimum-energy paths and TS’s are highly complex. On the Pt(111) surface, O and NO adsorb preferentially at fcc three-fold hollow sites, the latter oriented normal to the surface, with N down (Bogicevic and Hass, 2002; Huang and White, 2003). Adsorption energies are calculated at θ = 1/16 monolayer (ML) (where unit coverage (1 ML) is defined as one adsorbate per surface Pt atom), using the 4-layer slab model (Ovesson et al., 2005) corresponding well with previous calculations at θ = 1/9 ML (Bogicevic and Hass, 2002), using 5-layer slabs. Adsorption structures of O and NO can be determined relatively straight-forwardly. For the more complex and less well understood NO2 adsorption, a number of geometries can be conceived, like (i) NO-down atop along 110, where the second O atom points up and away from the surface, (ii) N-down atop with two O atoms in bridge sites along 110 toward the vacuum, (iii) N-down atop with the O atoms in fcc and hcp sites along 112, and (iv) the O atoms down in top sites along 110 and N above in the bridge site in between. The calculated minimum-energy geometry is (i) (Fig. 8.36), conventionally labeled µ-N,O-nitrito, in excellent agreement with HREELS data (Bartram et al., 1987). The low-coverage KMC simulations for rates of all elementary processes, including diffusion, chemical reactions, and desorption, use parameters computed within DFT (Bogicevic and Hass, 2002). A common prefactor of 1013 Hz is adopted for all processes,


321

Fig. 8.36. Top view of the minimum-energy path for NO2 formation from NO and O, as computed within DFT-GGA and NEB (Ovesson et al., 2005). The O atoms are black and the N atom lies underneath the right O atom in the two figures to the left and, when moving to the right, it is successively more discernible as a medium-gray shadow stretching from this position towards the left O atom.

a reasonable value according to rigorous computations of the diffusion prefactor from first principles and TST (Ratsch and Scheffler, 1998). A general use of a value like this can be justified, as the rates depend exponentially on the activation energies and linearly on the prefactors. Lean exhaust is characterized by high O2 concentrations and thus a significant oxygen coverage of the noble metal. Lateral NO–O interactions are therefore of key interest under such conditions, and values for pair-interaction energies between O, NO and NO2 on Pt(111) have been calculated (with adsorbates typically in threefold fcc hollows) (Bogicevic and Hass, 2002). The first nearest-neighbor (1-NN) interactions are in general repulsive and of the order of 0.2–0.3 eV (except O–NO2 ), i.e. considerable. Already in next-nearest (2-NN) configurations, adsorbate interactions are much weaker, by a factor of 2–4, and in 3-NN configurations and beyond, they are less than or about 0.05 eV. The relaxations induced by adsorbate interactions are all found to be 0.2–0.3 Å at the shortest separations but only a tenth of that for 2-NN. The interaction energies for the O–NO2 entries are not strictly comparable with those of the other molecule pairs, as the designations 1-NN, 2-NN, etc., are made with respect to the fcc site closest to the NO2 . The O–NO2 interactions are quite weak in all configurations, as expected, however, given the stability of NO2 . The pair-interaction energetics (Vi {XY} for species X and Y at ith nearest neighbor separation) determines the ground-state adsorbate patterning of the surface, if interactions beyond the 3-NN separation and many-body contributions can be neglected. For the O/Pt(111) system, the p(2 × 2) structure is then predicted at θ = 25% (Kaburagi and Kanamori, 1978), as the inequalities V3 > 0, V1 > 3V3 , and V2 > 2V3 hold, in good agreement with experimental findings (Stipe et al., 1997). In the gas phase, the reaction NO + 1/2 O2 → NO2 is exothermic by 1.20 eV. On the clean Pt(111) surface, the O2 dissociates at 150 K (Wintterlin et al., 1996), and the oxidation reaction NO + O → NO2 is instead endothermic by 0.7 eV. Thus Pt(111) by itself does not catalyze NO oxidation! However, the energetics changes in favor of NO2 with increasing O coverage, as O repels O and NO strongly. For an NO in a saturated O p(2 × 2) overlayer (i.e. removing an O from the p(2 × 2) structure to form NO2 , from configuration 12 to 13 in Fig. 8.37), the reaction is nearly thermo-neutral. The NO2 becomes favored by 0.28 eV, when the initial O coverage is increased to 5/16 (from configuration 14 to 15, i.e. before adding NO). Thus the NO → NO2 conversion on Pt(111) is driven by lateral adsorbate interactions. The kinetic consequences of these interactions can be predicted accurately, only if a

322

T. Bligaard et al.

Fig. 8.37. Sample of adsorption geometries considered in KMC simulation (Ovesson et al., 2005). All adsorption occurs in threefold fcc hollow (Bogicevic and Hass, 2002). The black dots denote the adsorbates, mostly O atoms, but NO being the upper left black dot in 6), the lower left one in 8), and the upper right one in 11), the middle one in 13), and the extreme right, left, and bottom ones in 14), and NO2 the upper and bottom ones in 13) and 15).

larger fraction of the full configuration space (beyond the sample calculations displayed in Fig. 8.37) were explored, however. Both reaction energetics and kinetics can be assessed in KMC simulations, if the interaction energetics can be properly accounted for, in addition to clean-surface activation energies. A tractable solution apt for KMC implementation would be a pair-wise summation of interaction energies. For more complex adsorbate configurations the expansion of the energy in terms of pair interactions that are truncated at the 3-NN separation. For the configurations in Fig. 8.37, a pair-wise summation with 3NN interactions works rather well, errors being smaller than 0.10 eV per atom, even at the highest coverages and/or most frustrated geometries (i.e. those allowing the least lateral relaxation). The kinetic effects of lateral interactions and O coverage on the NO + O → NO2 conversion in oxygen-rich environments can be assessed by a set of first-principles KMC simulations (Voter, 1986) of NO deposition on the O-precovered Pt(111) surface, with activation energies calculated by DFT (Bogicevic and Hass, 2002). The relevant (i.e. not too rare) processes are O diffusion (activation energy = 0.56 eV), NO diffusion (0.24 eV; diffusion barrier for N is found a little high (0.81 eV)), which values together with energies for NO2 formation (1.25 eV), NO2 decomposition (0.90 eV), and NO2 desorption (1.28 eV) go into the KMC simulation. The complex reaction NO + O → NO2 starts with an upright N-down NO molecule adsorbed in an fcc site with an oxygen atom in a neighboring fcc site and ends in the µ-N,O-nitrito NO2 configuration, i.e. with all fragments in their energetically preferred sites. The reaction proceeds with the NO molecule diffusing towards the adsorbed O atom via a bridge site (Fig. 8.36), because of its higher mobility (Bogicevic and Hass, 2002). The adsorbed O atom has already moved about 0.8 Å to accommodate the formation of a chemical bond, as the NO crosses the bridge site and the two fragments are about 2.5 Å away. The activation energy for this process is found to be about 1.25 eV at low coverage,


323

noting that this value is likely to be sensitive to the local adsorption environment and that there may be lower-energy reaction pathways. Coverage effects are accounted for by summing pair interactions in initial and final configurations up to 3-NN separations and then adjusting barrier energies with a simple 0 + 1 (E − E ), where E 0 interpolation formula (Fichthorn and Scheffler, 2000), EA = EA f i A 2 is the low-coverage limit of the activation energy, and Ef and Ei are the total interaction energies of the final and initial states of the motion, respectively. All prefactors are set to 1013 Hz (Zhdanov, 1991; Gomer, 1990). To reduce CPU-time consumption, adsorbate diffusivities are suppressed by a common factor at high temperatures, carefully checking convergence with respect to this parameter (Ovesson et al., 2005). The O adatom is 0.41 eV more stable in the fcc hollow site than in the hcp site, which implies a relatively high diffusion barrier. This preference is found to increase with a precoverage of a p(2 × 2) O overlayer on the surface (Ovesson et al., 2005), in contrast to what is found by thermal desorption (Jerdev et al., 2002). For NO, the fcc-hcp difference is smaller (0.10 eV), but NO diffusion appears to be of minor importance for the reaction kinetics. Finally, for NO2 the kinetics is not so much influenced by the fact that it does not bind in registry with the substrate. The reason is that there is always one fcc site close to the NO2 , where adsorption is impossible, and which can then be treated as occupied by the NO2 . Hence, only fcc sites are considered in the KMC simulations. The adsorbates move on a triangular lattice with periodic boundary conditions, containing 40 × 40 sites. On the clean surface, O atoms are first deposited at random, secondly the system is equilibrated for 1 s, then exactly one NO molecule is deposited, and finally the system is annealed for another 2 s, with focus on the dilute limit of NO abundance. NO2 may form, decompose, and desorb within this time. The final fraction of NO2 (both at the surface and in the gas phase) is computed as a function of temperature and initial oxygen coverage. There are unexpected features (Fig. 8.38) (Ovesson et al., 2005): (i) For the highest temperature considered the NO oxidation reaction turns on at about 0.2 ML O coverage, but at 300 K it requires up to 0.25 ML more oxygen. This is mainly a kinetic effect, as the opposite reaction does not take place for any coverage above 30%. The oxidation reaction does not occur to an appreciable extent for any temperature at 0.20 ML O coverage – as expected from the sample DFT results discussed above – but the enhancement of the reactivity with increasing θO is strongly temperature-dependent. Thus, with slight changes in the O coverage around 0.25 ML, the reaction can be driven back and forth at high temperatures, whereas large changes are required at low temperatures, for kinetic reasons. The NO2 formation reaction is thus activated by both temperature and accumulation of O atoms around the NO. Hopefully, this strong sensitivity of the oxidation efficiency to the oxygen coverage should stimulate experimental or theoretical studies of the external conditions for bringing the system slightly above or below the critical coverage. In summary, DFT-KMC simulations (Ovesson, 2002; Ovesson et al., 2005) show that the NO → NO2 reaction on an O-covered Pt(111) substrate has a catalytic activity that depends strongly on the oxygen coverage. In the low-coverage limit, the surface reaction is highly endothermic, but it turns exothermic at θO ∼ 25%, due to repulsive lateral O–O and O–NO interactions. At room temperature, an O coverage of 0.45 ML is required for the reaction to be activated, however, for kinetic reasons. Flow-reactor experiments (Olsson et

324

T. Bligaard et al.

Fig. 8.38. Calculated efficiency of the NO oxidation process on Pt(111), as a function of initial oxygen coverage and temperature. Statistical errors of two standard deviations are indicated (Ovesson et al., 2005).

al., 2001) show similar results. Hopefully, future ultra-high-vacuum studies are encouraged by this. 8.5.2.2. CO + O → CO2 A first-principles statistical mechanics approach enabling us to simulate the steady-state situation of heterogeneous catalysis has been developed also for the catalytic oxidation of CO at RuO2 (110) (Reuter et al., 2004; Reuter and Scheffler, 2006). All-electron DFT is used together with transition-state theory to obtain the energetics of the relevant elementary processes (incl. gas-phase molecules, dissociation, adsorption, surface diffusion, surface chemical reactions, and desorption), and then the statistical mechanics problem is solved by the KMC method (correlations, fluctuations, and spatial distributions of the chemicals at the surface of the catalyst under steady-state conditions). The results are in quantitative agreement with all existing experimental data. The turnover frequency (TOF) of the reaction (the CO2 formation rate) is calculated as a function of temperature and partial pressures (Fig. 8.25), the resulting CO2 formation rate (in the full (T , pCO , pO2 ) space) can be displayed in movies of the atomic motion and reactions over times scales from picoseconds to seconds, and the statistical analyses provide insight into the concerted actions ruling heterogeneous catalysis, in particular, and open thermodynamics, in general (Reuter et al., 2004). A narrow region of highest catalytic activity is clearly shown by the TOF values in (T , pCO , pO2 ) space. Kinetics here builds an adsorbate composition found nowhere in the thermodynamic surface phase diagram. Among the surprising results revealed by the statistical analysis of the surface dynamics and of the various processes is the fact that the chemical reaction with the most favorable energy barrier happens a factor of 0.30 less frequently than the energetically second most favorable reaction (Reuter et al., 2004). The conditions for how and when the pressure gap between UHV studies and realistic pressure conditions is bridged are also illuminated: at first, an O2 -rich environment changes the material from Ru to RuO2 , and after that the correct pressure conditions appear to be of key importance, as also on RuO2 (110) there is a low-pressure surface phase that has little in common with the catalytically active situation. The obtained agreement between


325

the theoretical results and experimental data confirms furthermore that CO2 is primarily formed from the adsorbed CO and O, and that the metal oxide, once it was created, does not play an active role, i.e. there is no indication of significant bulk diffusion (Reuter et al., 2004). The high catalytic activity of this system is intimately connected with a disordered, dynamic surface “phase” with significant compositional fluctuations. In this active state the catalytic function results from a self-regulating interplay of several elementary processes. 8.5.3. Ammonia synthesis The ammonia synthesis is one of the most studied catalytic reactions in surface science. The present prominent status of such studies is indicated by the facts that (i) its rate over a nanoparticle ruthenium catalyst can be calculated directly on the basis of a quantum-physical treatment of the problem using DFT, (ii) the results from such a calculation can be directly compared to measured rates over a ruthenium catalyst supported on magnesium aluminum spinel, and (iii) a theoretical treatment on ruthenium particles with a size distribution as measured by transmission electron microscopy gives a rate within a factor of 3 to 20 of the experimental rate (Honkala et al., 2005; Hellman et al., 2006a, 2006b). Such results of course offer hope for computer-based methods in the search for catalysts. Today, DFT calculations provide detailed theoretical descriptions of the way in which solid surfaces act as catalysts for chemical reactions, in particular, the relevant activation energies. For example, selective oxidation of ethylene on an Ag catalyst can have its experimental data described by a mean-field kinetic model developed on the basis of DFT calculations (Linik and Barteau, 2003b). However, the interactions between adsorbed surface species are complex, and many different possible reaction paths and their consequences are implicitly neglected by the mean-field description, a shortcoming overcome by DFT-KMC descriptions, like those in Section 8.5.2 for the oxidations of NO and CO over Pt(111) and RuO2 (110) surface, respectively (Ovesson, 2002; Reuter et al., 2004; Ovesson et al., 2005). To develop a kinetic description that includes the full complexity of interactions and reaction paths for a complete catalytic reaction under industrial conditions and over a packed bed of a high-surface-area nanoparticle catalyst, further steps can be taken. For the ammonia (NH3 ) synthesis, a natural example, DFT calculations can be used to get the reactionrate value for a supported nanoparticle Ru catalyst that is in good agreement with values measured over a wide range of industrially relevant synthesis conditions (Honkala et al., 2005; Hellman et al., 2006a, 2006b). Only the particle size distribution for the Ru catalyst, determined from transmission electron microscopy (TEM), is used as experimental input. The prototype reaction of NH3 -synthesis has helped to develop many key concepts in the field (Boudart, 1994). Almost 100 years ago, large-scale screening experiments revealed Ru and Fe as the best elementary metal catalysts (Haber, 1966; Bosch, 1966; Mittach, 1950). As early as in 1933, the nature of the rate-determining step for Fe-based catalysts, N2 dissociation, was pinpointed (Emmett and Brunauer, 1933; Emmett and Brunauer, 1934). About 30 years later, surface-science studies revealed a detailed picture of the N2 dissociation process (Boszo et al., 1977; Ertl et al., 1979; Paal et al., 1981; Ertl et al., 1982; Ertl, 1983) (cf. Nobel-Prize Award interview with Gerhard Ertl in

326

T. Bligaard et al.

Fig. 8.39. Activation energies (Ea ) and TS configurations for different local environments, as calculated in Ref. (Honkala et al., 2005). The B5 site is shown with (A) no adsorbates in the neighboring sites, (B) adsorbed N on the upper step, (C) adsorbed N on the lower step, (D) adsorbed H on the lower step, (E) adsorbed H on the upper step, (F) adsorbed NH on the upper step, (G) adsorbed NH2 on the upper step, and (H) adsorbed H on both the upper and lower steps. N atoms are shown in blue.

the introduction of this volume). The direct link between ultra-low-pressure surfacescience results and NH3 synthesis data at elevated pressure and temperature has been shown experimentally (Spencer et al., 1982; Strongin et al., 1987; Strongin and Somorjai, 1988) and theoretically (Stoltze and Nørskov, 1985; Dumesic and Triviño, 1989; Boudart, 1994). The complete reaction mechanism on Ru, with all elementary steps, can be quantitatively outlined by use of DFT calculations (Logadottir and Nørskov, 2003). Trends in reactivity can be predicted and understood by DFT calculations (of activation barriers and stabilities of intermediates, for a series of different catalysts) combined with a simple microkinetic model (Logadottir et al., 2001). The “final” step in a complete first-principles description of the Ru-catalyzed NH3 synthesis has now also been taken (Honkala et al., 2005): The calculation starts by finding the potential-energy diagram for the full reaction (Dahl et al., 1999) (Fig. 8.14), which shows that step sites are much more reactive for N2 dissociation than the close-packed (0001) surface (Honkala et al., 2005), a fact that is verified experimentally (Dahl et al., 1999). Under all realistic reaction conditions, N2 dissociation is by far the slowest step, as found by combining the results for step sites in Fig. 8.39 with harmonic transition-state theory (Logadottir and Nørskov, 2003). These findings are exploited to treat N2 dissociation as rate limiting and only consider dissociation along step sites where the active (B5) sites exist (Dahl et al., 1999). Low-coverage results, like in Fig. 8.14, might have fewer adsorbates than the catalytic reality, with other atoms and molecules adsorbed in the vicinity of the react-


327

ing molecule. The activation energy for dissociation, Ea,i , and thus the rate constant, ki = ν exp(−Ea,i /kT ), where ν is the prefactor, k the Boltzmann constant, and T the temperature, depends on the local environment (Fig. 8.14). In a model with each local environment i contributing with a weight Pi , assumed equal to the probability of finding this envi3 2 ronment, the total rate r is r(T , pN2 , pH2 , pNH3 ) = (1 − pNH3 /(pH2 pN2 Kg )) i Pi ki pN2 , where Kg is the gas-phase equilibrium constant, and pN2 , pH2 , and pNH3 are the pressures of N2 , H2 , and NH3 , respectively. The gas-phase equilibrium is established by the factor 2 /(p 3 p K )). (1 − pNH H2 N 2 g 3 The probability Pi is given by the equilibrium between H2 and NH3 in the gas phase and adsorbed H, N, NH, NH2 , and NH3 – all of the steps after the rate-limiting N2 dissociation in Fig. 8.14 are in equilibrium. For all adsorbates the free energy is calculated, with a complete harmonic normal-mode analysis. Possible nearest-neighbor interactions along the step are also calculated. On the lower step, only H atoms appear, and thus only H–H interactions. Grand-canonical Monte Carlo simulations on the ensemble of the system in equilibrium give Pi , where environment i is defined by configurations with simultaneous empty upper and lower step sites, N2 dissociation giving one N atom each above and below the step, and by the occupancy of the four neighboring sites. To test that the predicted rate applies for a real catalyst under industrially relevant conditions, the activity is measured for a well-characterized Ru catalyst supported on magnesium-aluminum spinel Ru (Dahl et al., 2000), used as a plug flow reactor, properly loaded variation with the input partial pressures, the flow, and the temperature (Honkala et al., 2005). The integrated NH3 synthesis rate down through the reactor is the calculated NH3 productivity (Honkala et al., 2005; Hellman et al., 2006a, 2006b). Only the number of active sites per gram of catalyst is now needed to compare calculated and experimental results. This characterization is obtained by TEM, which gives the particle size distribution for about 103 particles. The active sites for N2 dissociation, B5, are found from the TEM image of the Ru particle with its clearly shown steps (Fig. 8.40A). As direct counting is not possible, a Wulff construction is used to give the basic particle shape, relying on calculated surface-energy values of all the low-energy facets of Ru. In this way the fraction of B5 sites is estimated as a function of particle size (Honkala et al., 2005). For the (001)/(101) edge, a reconstruction lowers the energy, according to DFT calculations (Honkala et al., 2005). The edge row of atoms should then be removed, which creates the steps that contain B5 sites along the edge. This quite crude estimate of the particle morphology neglects interaction with the support and changes in the surface energies caused by adsorption, but it gives a shape quite similar to that found in the TEM images. The fraction of B5 (step) sites appears insensitive to details. The smallest Ru particles should lack active sites, a fact in full agreement with the observed increase in experimental activity at sintering of small Ru particles in a supported Ru catalyst (Dahl et al., 1999). The counted active-site density folded with the measured size distribution gives the number of active sites per Ru mass, probably a lower bound, due to the uncertain claim that the Wulff polyhedra are complete, i.e. that all layers are filled (Honkala et al., 2005; Hellman et al., 2006a, 2006b). In direct comparison (Fig. 8.41A), the calculated and measured NH3 productivities at various conditions agree excellently (overall rate is too small by a factor of 3 to 20, and the

328

T. Bligaard et al.

Fig. 8.40. (A) TEM image of a supported Ru particle with a step. (B) Particle size distribution function obtained from the TEM experiments, where d is the particle diameter. (C) A typical calculated Ru particle, with an average diameter of 2.9 nm. Atoms that belong to active B5 sites are shown in black. (D) Density of active sites as a function of particle diameter, as calculated through analysis of the atomistic Wulff construction. From Honkala et al. (2005).

calculated inhibition by NH3 tends to be slightly too weak), given that there are no fitted parameters. The calculations also give some insight into the exact nature of the active sites for the NH3 synthesis reaction over Ru. Among the local environments that contribute to the synthesis in one specific slice of the reactor (Fig. 8.41B), the one with the lowest activation energy (Fig. 8.14) is not the main contributor to the total rate, which instead is dominated by the configuration with adsorbed H on the upper step. This is due to the state of the surface during synthesis conditions (Honkala et al., 2005; Hellman et al., 2006a, 2006b). The agreement between theory and experiment (Fig. 8.41A) might look surprising, in view of the inherent accuracy of DFT calculations (of the order 0.2 to 0.3 eV (Hammer et al., 1999)). For instance, an error of 0.25 eV in the activation energy gives a 14,800 per cent error in the rate of an elementary reaction at 600 K. The dilemma is resolved in terms of the considerably smaller sensitivity to the absolute error of the total reaction rate than of the rate of an individual step (Honkala et al., 2005). The sensitivity of the results to the uncertainty emanating from the main approximation in the DFT calculations, the exchangecorrelation energy functional, is tested by performing all calculations with both the RPBE (Hammer et al., 1999) and PW91 functionals (Perdew and Wang, 1992). There appears to be a compensation between the different steps in the total reaction (Bligaard et al., 2003), and the interpolation between the two results, E(x) = xERPBE + (1 − x)EPW91 , gives a plot of the rate with a very weak dependence (Fig. 8.41C): The barrier for N2 dissociation decreases substantially (by 0.6 eV), making the dissociation of N2 much faster, while the


329

Fig. 8.41. (A) Ammonia productivity, when results from model and experiment are compared (results on the diagonal line would mean complete agreement) (Honkala et al., 2005). Parameter settings: total pressure = 100 bar; ratio of N2 to H2 = 1:3; total flow range over 0.2 g of catalyst lying between 40 to 267 ml/min (standard temperature and pressure); conversion range ranging from 0 to 20% of equilibrium; and temperature from 320 to 440 ◦ C. (B) Relative contributions to the rate from different local environments in one slice of the reactor. On the x axis, the local environment is represented symbolically in a way that shows the dissociation sites and four neighboring sites on the upper and lower steps. (C) Results for the reaction rate (turnover frequency) with different mixing x of results obtained with PW91 and RPBE, two different treatments of exchange and correlation in the generalized gradient approximation (GGA). As indicated, there are large variations in the rate for N2 dissociation on free sites, pN2 kN2 , and the coverage of free sites at the step, θfrs , which compensate to give a small variation in the total rate that is proportional to both pN2 kN2 and θfrs .

330

T. Bligaard et al.

dissociative chemisorption-energy values of H2 and N2 change from −0.36 (RPBE) to −0.52 (PW91) eV and from −0.8 (RPBE) to −1.4 (PW91) eV, respectively, resulting in increased coverage (through the equilibrium with H2 and NH3 in the gas phase) and decreased number of free sites for dissociation. The net rate is only slightly affected, as the barrier for dissociation and the stability of the intermediates on the surface vary together, the so-called compensation effect. In the catalytic activity as a function of the bond strength a volcano results, thanks to this compensation between two opposing effects (Bligaard et al., 2003), which is largest near to the maximum of the volcano (that is, for the best catalysts). A discrepancy remains between the measured and calculated results (Fig. 8.41A). It might come from too few active sites or erroneous differences between adsorption energies. One test of the latter is to calculate points that match the measured ones completely at all temperatures and flows and find a stabilization of adsorbed H relative to the NHx species by 0.06 eV to be enough, which would indicate that the underestimated rates in Fig. 8.41A come from small relative errors. Another test turns the coin around, showing calculations able to predict the absolute rate to within a factor of 3 to 20, suggests relative errors in the exchange-correlation functional to be of the order 0.06 eV, which corresponds to kT at reaction conditions (Honkala et al., 2005). Also other reactions should show compensation effects, as colinear variations in activation energies and bond strengths are found quite generally (Logadottir et al., 2001). There is thus good hope for DFT calculations to give good overall descriptions of catalytic activities in general, thanks to this built-in insensitivity to absolute errors and to improved accuracy of the DFT methods for relative energies (Honkala et al., 2005). 8.5.4. Hydrogen evolution reaction on MoS2 and close interplay between theory and experiment In heterogeneous catalysis the active sites are important and should be identified (Somorjai, 1994; Zambelli et al., 1996). While for homogeneous catalysts they in general are clearly defined and quantified, solid catalysts offer more challenges. There is often a variety of sites, and multiple sites might work together. Each require special efforts to identify and quantify, necessary steps to develop improved catalysts. In situ and ex situ experiments and computational theory (Dahl et al., 1999; Campbell, 2001; Jaeger, 2001) work hand in hand on this challenging task (Jaramillo et al., 2007). For the hydrogen evolution reaction (HER), 2H+ + 2e− → H2 (Bockris and Kahn, 1993; Conway and Tilak, 2002), such an approach is used in a recent determination of the active site of nanocatalyst MoS2 (Jaramillo et al., 2007). The HER process is important in electrochemistry, for fuel cells, and for solar H2 production (water splitting), and the replacement of precious metal catalysts like Pt (Conway and Tilak, 2002; Hinnemann et al., 2005) is at issue. While bulk MoS2 poorly catalyzes HER (Jaegermann and Tributsch, 1988), the edges of nanoparticles of MoS2 should be active for HER (Hinnemann et al., 2005) according to DFT calculations. MoS2 is used as an industrial catalyst for hydrodesulfurization (HDS) (Topsøe et al., 1996; Chianelli et al., 2006). Experimental and theoretical surface-science interplay (Bollinger et al., 2001; Helveg et al., 2000; Kibsgaard et al., 2006; Bollinger et al., 2003)


331

Fig. 8.42. A series of STM images of MoS2 nanoparticles on Au(111) (Jaramillo et al., 2007). The particles exhibit the typical polygon morphology with conducting edge states and are dispersed on the Au surface irrespective of coverage and annealing temperature (400 ◦ C or 550 ◦ C). (A) Low coverage (0.06 nm2 MoS2 /nm2geom. ),

annealed to 400 ◦ C (470 Å by 470 Å, 1.2 nA, 4 mV). (B) High coverage (0.23 nm2 MoS2 /nm2geom. ), annealed to 550 ◦ C (470 Å by 470 Å, 1.2 nA, 1.9 V). (C) Atomically resolved MoS2 particle, from a sample annealed to 550 ◦ C, showing the predominance of the sulfided Mo-edge (Jaramillo et al., 2007; Perez et al., 1998) (60 Å by 60 Å, 1.0 nA, 300 mV).

and a combination of reactivity and ex situ characterization of industrial samples (Prins et al., 1989; Topsøe et al., 1996; Chianelli et al., 2006) have together given a detailed insight into this catalyst and reaction. The MoS2 structure consists of flat polygons of S–Mo–S trilayers stacked in a graphite-like manner (dependent on synthesis conditions) (Topsøe et al., 1996). There are terrace and edge sites on a single trilayer, according to DFT (Hinnemann et al., 2005) and STM (Lauritsen et al., 2004). A unique link between the well-defined model-system structures and catalytic activity under standard reaction conditions had been lacking, however (Hinnemann et al., 2005). Recently, such a link is provided for HER on the nanocatalyst MoS2 (Jaramillo et al., 2007). On an Al(111) surface, MoS2 samples (Helveg et al., 2000) with deliberately chosen nanoscale properties are prepared by PVD (of Mo in H2 S) and characterized by STM, all in UHV. Immediately after deposition, each sample is vacuum transferred to a second UHV chamber for STM imaging (Fig. 8.42). The dominant edge structure of MoS2 is that of a sulfided Mo edge, and that edge is particularly favored by larger-sized particles (Hinnemann et al., 2005; Lauritsen et al., 2004). By controlled sintering, the ratio of terrace and edge sites is changed (Jaramillo et al., 2007). According to DFT this sulfided Mo edge provides the active sites for HER (Chianelli et al., 2006). The HER activity measured after imaging in an electrochemical cell is described in terms of polarization curves and Tafel plots (Fig. 8.43). The exchange current density, the most inherent measure of activity for HER (Bockris and Kahn, 1993; Conway and Tilak, 2002; Trasatti, 1972; Nørskov et al., 2005), give for each sample data points that fall on a straight line (Fig. 8.44), when plotted as a function of the MoS2 edge-state length (Fig. 8.44B). With a rate of the reaction directly proportional to the number of edge sites for all samples, regardless of particle size, the edge site must be the active one (Jaramillo et al., 2007).

332

T. Bligaard et al.

Fig. 8.43. Polarization curves (A), showing H2 evolution on all samples, and Tafel plots (B) (log current versus potential) in a cathodic potential window for the five different MoS2 , as well as a blank sample (Jaramillo et al., 2007). Samples annealed to 400 ◦ C are dark blue, samples annealed to 550 ◦ C light blue. All of the MoS2 samples have Tafel slopes of 55 to 60 mV per decade irrespective of annealing temperature and coverage. Sweep rate: 5 mV/s.

Fig. 8.44. Exchange current density, extracted from the Tafel plot in Fig. 8.43, versus (A) MoS2 area coverage and (B) MoS2 edge length, measured on all imaged particles and normalized by the imaged area (Jaramillo et al., 2007). Samples are annealed to 400 ◦ C (open circles) and 550 ◦ C (filled circles). The exchange current density does not correlate with the area coverage of MoS2 , whereas it shows a linear dependence of the MoS2 edge length.

The turnover frequency (TOF) per active site, 0.02 s−1 , compares favorably with other catalysts (0.9 s−1 for Pt(111) and about 10−9 s−1 for Hg) (Jaramillo et al., 2007). In the volcano plot for HER, with the Gibbs energy for atomic adsorption of hydrogen as abscissa (Trasatti, 1972; Nørskov et al., 2005) (Fig. 8.45), nanoparticulate MoS2 has an interesting position. The plot reflects the Sabatier principle that says optimal surfaces are those that exhibit moderate binding energies of reaction intermediates, here the DFT value for adsorption of atomic hydrogen (Nørskov et al., 2005). “This agreement validates the predictive


333

Fig. 8.45. Volcano plot of the exchange current density as a function of the DFT-calculated Gibbs free energy of adsorbed atomic hydrogen for nanoparticulate Mo2 and the pure metal (Jaramillo et al., 2007). As seen, MoS2 follows the same trend as the pure metals. The MoS2 exchange current density is normalized to the atomic site density of Pt for comparison. Samples are polycrystalline unless otherwise noted.

capability of this DFT model as well as its applicability beyond metal catalysts” (Jaramillo et al., 2007). DFT calculations suggest a hydrogen coverage of only one per four edge atoms under operating conditions (Chianelli et al., 2006). So, while Pt(111) operates at an H coverage of about 1 ML (Conway and Tilak, 2002; Markovic et al., 1997; Skúlason et al., 2007), only every fourth MoS2 -edge atom evolves molecular H2 at a given time (Hinnemann et al., 2005). A factor of 4 would be gained, if all MoS2 -edge site could be made to adsorb H. Conceivably, this could be accomplished by tuning the electronic structure of the edge towards a stronger adsorption of H (Nørskov et al., 2005). If at the same time the inherent turnover per edge site were improved, a catalyst with an overall activity comparable to that of the Pt-group metals would be designed (Jaramillo et al., 2007).

8.6. Conclusions and outlook Surface bonding and catalytic activity are closely related. An attempt is being made in this chapter to illustrate this. One of the main conclusions is that in a surface catalyzed reaction adsorption energies of the main intermediates often are very good descriptors of catalytic activity. The underlying reason is the presence of correlations between activation barriers and reaction energies for a number of surface reactions, known as Brønsted-Evans-Polanyi relations. When combined with simple kinetic models, such correlations lead to volcanoshaped relationships between catalytic activity and adsorption energies. A model that describes the coupling between the adsorbate states and the transitionmetal d states provides the basis for understanding both the variation in adsorption energies from one transition metal to the next and the variation from one surface geometric structure

334

T. Bligaard et al.

to the next. One important finding is about variations in the reactivity of a given metal at changed surroundings. To a large extent, they are governed by the local value of the average energy of the d states. In this way notions are provided to understand some of the electronic and geometric factors governing catalytic rates. It is now possible to understand the basic descriptors well enough for the simplest catalytic reactions on transition metal surfaces and to use the insight to help identifying new catalysts. Our fundamental understanding of transition-metal catalysis has reached a useful level, according to this evidence. The prospects for catalyst development are enormous. Design of new catalysts will be substantially simpler than today, if insight and computational methods can be used to narrow down the number of possible catalysts for a given reaction. A large number of challenges remain, however. Up to now most work has been concentrating on transition-metal surfaces and on very simple reactions. The analysis in this chapter draws heavily on the Born–Oppenheimer (or adiabatic) approximation (BOA). Nonadiabaticity is since long well-established in many subfields of surface science, such as ion-neutralization spectroscopy, surface chemiluminescence, and exoelectron emission (cf. chapter on Electron Transfer and Nonadiabaticity by B.I. Lundqvist, A. Hellman, and I. Zoric in this volume). As a contrast, the possible breakdown of the BOA in some thermal surface reactions is still sometimes described as a controversial issue (Wodtke, 2006). This debate is spurred by accounts for experimental results for, e.g., state-selected, highly vibrationally excited molecules (Hou et al., 1999; Huang et al., 2000; White et al., 2005) and initial sticking (Nørskov and Lundqvist, 1979; Österlund et al., 1997; Zhdanov, 1997; Hellman et al., 2003; Behler et al., 2005). Wherever there is a real competition between electronic and nuclear time scales, the validity of BOA should be questioned. Commonly, the energy released in low-energy chemisorption or physisorption of molecules on metal surfaces is expected to be dissipated by surface vibrations (phonons). As a contrast, chemically induced electronic excitations have been observed at metal surfaces in an experiment that uses large-area, ultrathin-film, Schottky diode devices (Gergen et al., 2001), with results for gas interactions with polycrystalline silver for a variety of species with adsorption energies, Eads , between 0.2 and 3.5 eV. In particular, the number of detected electrons per incident reactant of Ag/Si Schottky diodes to electrons produced during adsorption is found to correlate strongly with the Eads (Fig. 8.46). For the various adsorbates studied, a power law relation in the form ∼ (Eads )n is observed, with an exponent n ∼ 2.7. Such correlations look similar to those in Section 8.2, which form the basis of the scaling used to propose new catalysts. Maybe one in the future will find scaling laws that include nonadiabaticity? To understand trends for complex reactions with many reaction steps, new methods need to be developed. Since it will be extremely difficult to perform experiments or DFT calculations for all systems of interest, this should preferentially be done by developing models to understand trends. Many catalysts are not metallic, and we need to develop the concepts that have allowed us to understand and develop models for trends in reactions on transition-metal surfaces to other classes of surfaces: oxides, carbides, nitrides, and sulfides. The relationships between heterogeneous catalysis and homogeneous catalysis or enzyme catalysis would also be extremely interesting to have conceptually developed in order to understand them. Finally, the theoretical methods need further development. The level of accuracy is now sufficient to describe some trends in reactivity for transition metals,


335

Fig. 8.46. Initial electron detection sensitivity (number of detected electrons per incident reactant ) for various gases of Ag/Si Schottky diodes as a function of Eads (Gergen et al., 2001).

but to describe the finer details including possibly catalyst selectivity a higher accuracy is desirable. To attain reliable descriptions of some oxides and other insulators might not be possible unless the theoretical methods to treat exchange and correlation effects are further improved.

References Abild-Pedersen, F., Greeley, J., Nørskov, J.K., 2005a. Catal. Lett. 105, 9. Abild-Pedersen, F., Lytken, O., Engbæk, J., Nielsen, G., Chorkendorff, I., Nørskov, J.K., 2005b. Surf. Sci. 590, 127. Alcalá, R., Mavrikakis, M., Dumesic, J.A., 2003. J. Catal. 218, 178. Anderson, P.W., 1961. Phys. Rev. 124, 41. Andersson, M.P., Bligaard, T., Christensen, C.H., Nørskov, J.K., 2008. Submitted for publication. Andersson, M.P., Bligaard, T., Kustov, A., Larsen, K.E., Greeley, J., Johannessen, T., Christensen, C.H., Nørskov, J.K., 2006. J. Catal. 239, 501. Andersson, S., Lundqvist, B.I., Nørskov, J.K., 1977. In: Proc. 7th Intern. Vacuum Congr. and 3rd Intern. Conf. on Solid Surfaces, Vienna, p. 815. Atkins, P.W., Friedman, R.S., 1997. Molecular Quantum Mechanics, third ed. Oxford University Press, Oxford, Section 10.13. Balandin, A.A., 1969. Adv. Catal. 19, 1. Bartram, M.E., Windham, R.G., Koel, B.E., 1987. Surf. Sci. 184, 57. Behler, J., Delley, B., Lorenz, S., Reuter, K., Scheffler, M., 2005. Phys. Rev. Lett. 94, 036104. Behm, R.J., 1998. Acta Phys. Pol. 93, 259. Bennett, C.H., 1977. Exact defect calculations in model substances. In: Nowick, A.S., Burton, J.J. (Eds.), Algorithms for Chemical Computation, ACS Symposium Series, vol. 46, p. 63. Besenbacher, F., Chorkendorff, I., Clausen, B.S., Hammer, B., Molenbroek, A., Nørskov, J.K., Stensgaard, I., 1998. Science 279, 1913. Bligaard, T., Honkala, K., Logadottir, A., Nørskov, J.K., Dahl, S., Jacobsen, C.J.H., 2003. J. Phys. Chem. B 107, 9325. Bligaard, T., Nørskov, J.K., 2007. In: Nilsson, A., Pettersson, L.G.M., Nørskov, J.K. (Eds.), Heterogeneous Catalysis.

336

T. Bligaard et al.

Bligaard, T., Nørskov, J.K., Dahl, S., Matthiesen, J., Christensen, C.H., Sehested, J.S., 2004. J. Catal. 224, 206. Bockris, J.O.M., Kahn, S.U.M., 1993. Surface Electrochemistry: A Molecular Level Approach. Plenum, New York. Bogicevic, A., Hass, K.C., 2002. Surf. Sci. 506, L237. Bogicevic, A., Ovesson, S., Hyldgaard, P., Lundqvist, B.I., Brune, H., Jennison, D.R., 2000. Phys. Rev. Lett. 85, 1910. Bogicevic, A., Strömquist, J., Lundqvist, B.I., 1998. Phys. Rev. Lett. 81, 637. Bollinger, M.V., Jacobsen, K.W., Nørskov, J.K., 2003. Phys. Rev. B 67, 085410. Bollinger, M.V., Lauritsen, J.V., Jacobsen, K.W., Nørskov, J.K., Helveg, S., Besenbacher, F., 2001. Phys. Rev. Lett. 87, 196803. Bosch, C., 1966. Nobel Prize Lecture 1931. In: Nobel Lectures, Chemistry 1922–1941. Elsevier, Amsterdam. Boszo, F., Ertl, G., Grunze, M., Weiss, M., 1977. J. Catal. 49, 18. Boudart, M., 1994. Top. Catal. 1, 405. Boudart, M., 1997. In: Ertl, G., Knözinger, H., Weitkamp, J. (Eds.), Handbook of Heterogeneous Catalysis. Wiley-VCH, Weinheim, p. 1. Boudart, M., Djéga-Mariadassou, G., 1984. Kinetics of Heterogeneous Catalytic Reactions. Princeton University Press, Princeton, New Jersey. Bozso, F., Ertl, G., Grunze, M., Weiss, M., 1977. J. Catal. 49, 18. Brønsted, N., 1928. Chem. Rev. 5, 231. Cabrera, N., Mott, N.F., 1949. Rep. Prog. Phys. 12, 163. Campbell, C.T., 2001. Science 294, 1471. Cant, N.W., Liu, I.O.Y., 2000. Catal. Today 63, 133. Carter, E.A., Ciccotti, G., Hynes, J.T., Kapral, R., 1989. Chem. Phys. Lett. 156, 472. Chandler, D., 1978. J. Chem. Phys. 68, 2959. Chianelli, R.R., Siadati, M.H., De la Rosa, M.P., Berhault, G., Wilcox, J.P., Bearden, R., Abrams, B.L., 2006. Catal. Rev. 48, 1. Chorkendorff, I., Niemantsverdriet, J.W., 2003. Concepts of Modern Catalysis and Kinetics. Wiley-VCH, Weinheim. Chu, J.W., Trout, B.L., Brooks, B.R., 2003. J. Chem. Phys. 119, 12708. Ciobîcã, I.M., van Santen, R.A., 2003. J. Phys. Chem. B 107, 3808–3812. Cong, P.J., Doolen, R.D., Fan, Q., Giaquinta, D.M., Guan, S.H., McFarland, E.W., Poojar, D.M., Self, K., Turner, H.W., Weinberg, W.H., 1999. Ang. Chem. Int. Ed. 38, 484. Conway, B.E., Tilak, B.V., 2002. Electrochim. Acta 47, 3571. Cortright, R.D., Davda, R.R., Dumesic, J.A., 2002. Nature 418, 964. Dahl, S., Logadottir, A., Egeberg, R.C., Larsen, J.H., Chorkendorff, I., Törnqvist, E., Nørskov, J.K., 1999. Phys. Rev. Lett. 83, 1814. Dahl, S., Sehested, J., Jacobsen, C.J.H., Törnqvist, E., Chorkendorff, I., 2000. J. Catal. 192, 391. Davies, J.C., Hayden, B.E., Pegg, D.J., 1998. Electrochim. Acta 44, 1181. de Donder, T., 1927. L’Affinité. Gauthier-Villers. Dietrich, H., Geng, P., Jacobi, K., Ertl, G., 1996. J. Chem. Phys. 104, 375. Drauglis, E., Jaffe, R.L. (Eds.), 1975. Physical Basis of Heterogeneous Catalysis; See, in particular, the discussion on p. 476. Dumesic, J.A., 1999. J. Catal. 185, 496. Dumesic, J.A., 2001. J. Catal. 204, 525. Dumesic, J.A., Rudd, D.F., Aparicio, L.M., Rekoske, J.E., Treviño, A.A., 1993. The Microkinetics of Heterogeneous Catalysis. Am. Chem. Soc. Dumesic, J.A., Triviño, A.A., 1989. J. Catal. 116, 119. Eichler, A., Hafner, J., Gross, A., Scheffler, M., 1999. Phys. Rev. B 59, 13297. Eichler, A., Mittendorfer, F., Hafner, J., 2000. Phys. Rev. B 62, 4744. Emmett, P.H., Brunauer, S., 1933. J. Am. Chem. Soc. 55, 1738. Emmett, P.H., Brunauer, S., 1934. J. Am. Chem. Soc. 56, 35. Ertl, G., 1983. J. Vac. Sci. Technol. A 1, 1247. Ertl, G., Lee, S.B., Weiss, M., 1982. Surf. Sci. 114, 527. Ertl, G., Weiss, M., Lee, S.B., 1979. Chem. Phys. Lett. 60, 391.


337

Evans, M.G., Polanyi, M., 1937. Trans. Faraday Soc. 33, 448. Evans, M.G., Polanyi, M., 1938. Trans. Faraday Soc. 34, 11. Eyring, H., 1935. J. Chem. Phys. 3, 107. Eyring, H., 1938. Trans. Faraday Soc. 34, 41. Fassaert, D.J.M., van der Avoird, A., 1976. Surf. Sci. 55, 291. Fichthorn, K.A., Scheffler, M., 2000. Phys. Rev. Lett. 84, 5371. Filhol, J.S., Simon, D., Sautet, P., 2004. J. Am. Chem. Soc. 126, 3228. Fridell, E., Skoglundh, M., Westerberg, B., Johansson, S., Smedler, G., 1999. J. Catal. 183, 2993. Fromm, E., 1998. Kinetics of Metal–Gas Interactions At Low Temperatures. Springer, New York, p. 79. Gadzuk, J.W., 1974. Surf. Sci. 43, 44. Gajdos, M., Eichler, A., Hafner, J., 2004. J. Phys. (Cond. Mat.) 16, 1141. Gambardella, P., Sljivancanin, Z., Hammer, B., Blanc, M., Kuhnke, K., Kern, K., 2001. Phys. Rev. Lett. 87, 056103. Ganesh, R., Pala, S., Liu, F., 2006. J. Chem. Phys. 125, 144714. Gergen, B., Nienhaus, H., Weinberg, W.H., McFarland, E.W., 2001. Science 294, 2521. Gomer, R., 1990. Rep. Prog. Phys. 53, 917. Goodman, D.W., Kelly, R.D., Madey, T.E., Yates, J.T., 1980. J. Catal. 63, 226. Greeley, J., Mavrikakis, M., 2004. Nature Materials 3, 810. Greeley, J., Nørskov, J.K., Mavrikakis, M., 2002. Ann. Rev. Phys. Chem. 53, 319. Grimley, T.B., 1967. Proc. Phys. Soc. 90, 751. 92 (1967) 776. Grimley, T.B., 1971. J. Vac. Sci. Technol. 8, 31. 9 (1971) 561. Gunnarsson, O., Hjelmberg, H., Lundqvist, B.I., 1976. Phys. Rev. Lett. 37, 292. Gunnarsson, O., Lundqvist, B.I., 1976. Phys. Rev. B 13, 4274. Gwathmey, A.T., Cunningham, R.E., 1958. Adv. Catal. 10, 57. Haber, F., 1966. Nobel Prize Lecture 1918. In: Nobel Lectures, Chemistry 1901–1921. Elsevier, Amsterdam. Hammer, B., 1998. Faraday Discuss. 110, 323. Hammer, B., 1999. Phys. Rev. Lett. 83, 3681. Hammer, B., 2001. Phys. Rev. B 63, 205423. Hammer, B., 2006. Top. Catal. 37, 3. Hammer, B., Hansen, L.B., Nørskov, J.K., 1999. Phys. Rev. B 59, 7413. Hammer, B., Morikawa, Y., 1996. Phys. Rev. Lett. 76, 2141. Hammer, B., Nielsen, O.H., Nørskov, J.K., 1997. Catal. Lett. 46, 31. Hammer, B., Nørskov, J.K., 1995. Nature 376, 238. Hammer, B., Nørskov, J.K., 1997. Theory of Adsorption and Surface Reactions. In: Lambert, R., Pacchioni, G. (Eds.), NATO ASI Series E, vol. 331. Kluwer Academic Publishers, Dordrecht. Hammer, B., Nørskov, J.K., 2000. Adv. Catal. 45, 71. Hellman, A., Barends, E.J., Biczysko, M., Bligaard, T., Christensen, C.H., Clary, D.C., Dahl, S., van Harrevelt, R., Honkala, K., Jonsson, H., Kroes, G.J., Luppi, M., Manthe, U., Nørskov, J.K., Olsen, R.A., Rossmeisl, J., Skúlason, E., Tautermann, C.S., Varandas, A.J.C., Vincent, J.K., 2006a. J. Phys. Chem. B 110, 17719. Hellman, A., Honkala, K., Remediakis, I.N., Logadottir, A., Carlsson, A., Dahl, S., Christensen, C.H., Nørskov, J.K., 2006b. Surf. Sci. 600, 4264. Hellman, A., Razaznejad, B., Yourdshayan, Y., Ternow, H., Zoric, I., Lundqvist, B.I., 2003. Surf. Sci. 532, 126. Helveg, S., Lauritsen, J.V., Lægsgaard, E., Stensgaard, I., Nørskov, J.K., Clausen, B.S., Topsøe, H., Besenbacher, F., 2000. Phys. Rev. Lett. 84, 951. Henkelman, G., Jónsson, H., 2000. J. Chem. Phys. 113, 9978. Henkelman, G., Uberuaga, B.P., Jónsson, H., 2000. J. Chem. Phys. 113, 9901. Hennig, D., Ganduglia-Pirovano, M.V., Scheffler, M., 1996. Phys. Rev. B 53, 10344. Henry, C.R., Chapon, C., Goyhenex, C., Monot, R., 1992. Surf. Sci. 272, 283. Heskett, D., Baddorf, A., Plummer, E.W., 1988. Surf. Sci. 195, 94. Hinnemann, B., Moses, P.G., Bonde, J., Jørgensen, K.P., Nielsen, J.H., Horch, S., Chorkendorff, I., Nørskov, J.K., 2005. J. Am. Chem. Soc. 127, 5308. Hoffmann, R., Woodward, R.B., 1970. In: Conservation of Orbital Symmetry. Academic Press, New York. Hohenberg, P., Kohn, W., 1964. Phys. Rev. B 136, 864.

338

T. Bligaard et al.

Honkala, K., Hellman, A., Remediakis, I.N., Logadottir, A., Carlsson, A., Dahl, S., Christensen, C.H., Nørskov, J.K., 2005. Science 307, 555. Hoogers, G., Thompsett, D., 1999. CATTECH 3, 106. Horiuti, J., 1938. Bull. Chem. Soc. Jpn. 13, 210. Hou, H., Huang, Y.H., Gulding, S.J., Rettner, C.T., Auerback, D.J., Wodtke, A.M., 1999. J. Chem. Phys. 110, 10660. Huang, Y.H., Rettner, C.T., Auerback, D.J., Wodtke, A.M., 2000. Science 290, 111. Huang, W.X., White, J.M., 2003. Surf. Sci. 529, 455. Igarashi, H., Fujino, T., Zhu, Y., Uchida, H., Watanabe, M., 2001. Phys. Chem. Chem. Phys. 3, 306. Jacobsen, C.J.H., Dahl, S., Boisen, A., Clausen, B.S., Topsøe, H., Logadottir, A., Nørskov, J.K., 2002. J. Catal. 205, 382. Jacobsen, C.J.H., Dahl, S., Clausen, B.S., Bahn, S., Logadottir, A., Nørskov, J.K., 2001. J. Am. Chem. Soc. 123, 8404. Jacobsen, K.W., Nørskov, J.K., Puska, M.J., 1987. Phys. Rev. B 35, 7423. Jaeger, N.I., 2001. Science 293, 1601. Jaegermann, W., Tributsch, H., 1988. Prog. Surf. Sci. 29, 1. Jaramillo, T.F., Jørgensen, K.P., Bonde, J., Nielsen, J.H., Horch, S., Chorkendorff, I., 2007. Science 317, 100. Jerdev, D.I., Kim, J., Batzill, M., Koel, B.E., 2002. Surf. Sci. 498, L91. Jóhannesson, G.H., Jónsson, H., 2001. J. Chem. Phys. 115, 9644. Jónsson, H., Mills, G., Jacobsen, K.W., 1998. In: Berne, B.J., Ciccotti, G., Coker, D.F. (Eds.), Classic and Quantum Dynamics in Condensed Phase Simulations. World Scientific, Singapore. Kaburagi, M., Kanamori, J., 1978. J. Phys. Soc. Jpn. 44, 718. Kandoi, S., Greeley, J., Sanchez-Castillo, M.A., Evans, S.T., Gokhale, A.A., Dumesic, J.A., Mavrikakis, M., 2006. Top. Catal. 37, 17. Keck, J.C., 1960. J. Chem. Phys. 32, 1035. Keck, J.C., 1962. Disc. Faraday Soc. 33, 173. Keck, J.C., 1967. Adv. Chem. Phys. 13, 85. Kibler, L.A., El-Aziz, A.M., Hoyer, R., Kolb, D.M., 2005. Angew. Chem. Int. Ed. 44, 2080. Kibsgaard, J., Lauritsen, J.V., Laegsgaard, E., Clausen, B.S., Topsøe, H., Besenbacher, F., 2006. J. Am. Chem. Soc. 128, 13950. Kitchin, J.R., Nørskov, J.K., Barteau, M.A., Chen, J.C., 2004. J. Chem. Phys. 120, 10240. Kohn, W., Sham, L.J., 1965. Phys. Rev. A 140, 1133. Kolasinski, K.W., 2002. Surface Science. Wiley. Kratzer, P., Scheffler, M., 2002. Phys. Rev. Lett. 88, 036102. Landau, D.P., Binder, K., 2000. A Guide to Monte Carlo Simulations in Statistical Physics. Cambridge University Press, Cambridge, UK. Lang, N.D., Williams, A.R., 1976. Phys. Rev. Lett. 37, 212. Lang, N.D., Williams, A.R., 1978. Phys. Rev. B 18, 616. Langreth, D.C., Mehl, M.J., 1983. Phys. Rev. B 28, 1809. Lauritsen, J.V., Nyberg, M., Nørskov, J.K., Clausen, B.S., Topsøe, H., Laegsgaard, E., Besenbacher, F., 2004. J. Catal. 224, 94. Li, W.X., Stampfl, C., Kreuzer, H.J., Payne, S.H., Pfnuer, H., Scheffler, M., 2002. Phys. Rev. B 65, 075407. Linic, S., Barteau, M.A., 2003a. J. Catal. 214, 200. Linik, S., Barteau, M.A., 2003b. J. Am. Chem. Soc. 125, 4034. Liu, Z.-P., Hu, P., 2003. J. Am. Chem. Soc. 125, 1958–1967. Liu, P., Nørskov, J.K., 2001. Phys. Chem. Chem. Phys. 3, 3814. Logadottir, A., Nørskov, J.K., 2003. J. Catal. 220, 273. Logadottir, A., Rod, T.H., Nørskov, J.K., Hammer, B., Dahl, S., Jacobsen, C.J.H., 2001. J. Catal. 197, 229. Løvvik, O.M., Olsen, R.A., 2003. J. Chem. Phys. 118, 3268. Lundqvist, B.I., Gunnarsson, O., Hjelmberg, H., Nørskov, J.K., 1979. Surf. Sci. 89, 196. Lundqvist, B.I., Hjelmberg, H., Gunnarsson, O., 1978. Adsorbate-induced electronic states. In: Feuerbacher, B., Fitton, B., Willis, R.F. (Eds.), Photoemission and the Electronic Properties of Surfaces. Wiley, New York. Maragakis, P., Andreev, S.A., Brumer, Y., Reichman, D.R., Kaxiras, E., 2002. J. Chem. Phys. 117, 4651. Markovic, N.M., Grgur, B.N., Ross, P.N., 1997. J. Phys. Chem. B 101, 5405.


339

Markovic, N.M., Ross, P.N., 2002. Surf. Sci. Rep. 45, 121. Mason, S.E., Grinberg, I., Rappe, A.M., 2004. Phys. Rev. B 69, 161401. Mavrikakis, M., Bäumer, M., Freund, H.J., Nørskov, J.K., 2002. Catal. Lett. 81, 153. Mavrikakis, M., Hammer, B., Nørskov, J.K., 1998. Phys. Rev. Lett. 81, 2819. Meier, J., Schiøtz, J., Liu, P., Nørskov, J.K., Stimming, U., 2004. Chem. Phys. Lett. 390, 440. Michaelides, A., Liu, Z.-P., Zhang, C.J., Alavi, A., King, D.A., Hu, P., 2003. J. Am. Chem. Soc. 125, 3704. Mills, G., Gordon, M.S., Metiu, H., 2003. J. Chem. Phys. 118, 4198. Mittach, A., 1950. Adv. Catal. 2, 81. Mun, B.S., Lee, C., Stamenkovic, V., Markovic, N.M., Ross, P.N., 2005. Phys. Rev. B 71, 115420. Muscat, J.P., Newns, D.M., 1978. Progr. Surf. Sci. 9, 1. Neurock, M., Hansen, E.W., 1998. Comp. Chem. Eng. 22, 1045. Newns, D.M., 1969. Phys. Rev. 178, 1123. Nilsson, A., Pettersson, L.G.M., Hammer, B., Bligaard, T., Christensen, C.H., Nørskov, J.K., 2005. Catal. Lett. 100, 111. Nørskov, J.K., Bligaard, T., Logadottir, A., Bahn, S., Hansen, L.B., Bollinger, M., Bengaard, H., Hammer, B., Sljivancanin, Z., Mavrikakis, M., Xu, Y., Dahl, S., Jacobsen, C.J.H., 2002. J. Catal. 209, 275. Nørskov, J.K., Bligaard, T., Logadottir, A., Kitchin, J.R., Chen, J.G., Pandelov, S., Stimming, U., 2005. J. Electrochem. Soc. 152, 123. Nørskov, J.K., Lundqvist, B.I., 1979. Surf. Sci. 89, 251. Olsen, R.A., Kroes, G.J., Henkelman, G., Arnaldsson, A., Jonsson, H., 2004. J. Chem. Phys. 121, 9776. Olsson, F.E., Paavilainen, S., Persson, M., Repp, J., Meyer, G., 2007. Phys. Rev. Lett. 98, 176803. Olsson, L., Persson, H., Fridell, E., Skoglundh, M., Andersson, B., 2001. J. Phys. Chem. B 105, 6895. Österlund, L., Kasemo, B., Zoric, I., 1997. Phys. Rev. B 55, 15452. Ovesen, C.V., Stoltze, P., Nørskov, J.K., Campbell, C.T., 1992. J. Catal. 134, 445. Ovesson, S., 2002. Ph.D. thesis, Chalmers University of Technology. Ovesson, S., Bogicevic, A., Lundqvist, B.I., 1999. Phys. Rev. Lett. 83, 2608. Ovesson, S., Lundqvist, B.I., Schneider, W.F., Bogicevic, A., 2005. Phys. Rev. B 71, 115406. Paal, Z., Ertl, G., Lee, S.B., 1981. Appl. Surf. Sci. 8, 231. Pallassana, V., Newrock, M., 2000. J. Catal. 191, 301. Pareto, V., 1906. Manuale di Economia Politia. Societa Editrice Libraria, Milano. Parvulescu, V.I., Grange, P., Delmon, B., 1998. Catal. Today 46, 233. Pearson, R.G., 1979. Acc. Chem. Rev. 4, 152. Perdew, J.P., 1986. Phys. Rev. B 33, 8822. Perdew, J.P., 1991. In: Ziesche, P., Eschrig, H. (Eds.), Electronic Structure of Solids. Akademi Verlag, Berlin. Vol. 11. Perdew, J.P., Burke, K., Ernzerhof, M., 1998. Phys. Rev. Lett. 80, 891. Perdew, J.P., Wang, Y., 1992. Phys. Rev. B 45, 13244. Perez, J., Gonzalez, E.R., Villullas, H.M., 1998. J. Phys. Chem. B 108, 10931. Prins, R., de Beer, V.H.J., Somorjai, G.A., 1989. Catal. Rev. Sci. Eng. 31, 1. Ratsch, C., Scheffler, M., 1998. Phys. Rev. B 58, 163. Reuter, K., Frenkel, D., Scheffler, M., 2004. Phys. Rev. Lett. 93, 116105. Reuter, K., Scheffler, M., 2006. Phys. Rev. B 73, 045433. Rodriguez, J.A., Goodman, D.W., 1992. Science 257, 897. Roudgar, A., Gross, A., 2003a. J. Electronanal. Chem. 548, 121. Roudgar, A., Gross, A., 2003b. Phys. Rev. B 67, 33409. Ruban, A., Hammer, B., Stoltze, P., Skriver, H.L., Nørskov, J.K., 1997. J. Mol. Catal. A 115, 421. Ruggerone, P., Ratsch, C., Scheffler, M., 1997. In: King, D.A., Woodruff, D.P. (Eds.), Growth and Properties of Ultrathin Epitaxial Layers. Elsevier Science, Amsterdam, p. 490. Sabatier, P., 1911. Hydrogénations et déshydrogénations par catalyse. Ber. Deutsch. Chem. Gesellshaft 44, 1984. Sabatier, P., Senderens, J.B., 1902. Acad. Sci. Paris C.R. 134, 514. Sachtler, J.W.A., Somorjai, G.A., 1983. J. Catal. 81, 77. Scheffler, M., Stampfl, C., 2000. Theory of adsorption on metal substrates. In: Horn, K., Scheffler, M. (Eds.), Handbook of Surface Science. In: Electronic Structure, vol. 2. Elsevier, Amsterdam, pp. 286–356.

340

T. Bligaard et al.

Schumacher, N., Boisen, A., Dahl, S., Gokhale, A.A., Kandoi, S., Grabow, L.C., Dumesic, J.A., Mavrikakis, M., Chorkendorff, I., 2005. J. Catal. 229, 265. Sehested, J., Larsen, K.E., Kustov, A.L., Frey, A.M., Johannessen, T., Bligaard, T., Andersson, M.P., Nørskov, J.K., Christensen, C.H., 2007. Topics in Catalysis 45, 9. Shelef, M., McCabe, R.W., 2000. Catal. Today 62, 35. Skúlason, E., Karlberg, G.S., Rossmeisl, J., Bligaard, T., Greeley, J., Jonsson, H., Nørskov, J.K., 2007. Phys. Chem. Chem. Phys. 9, 3241. Somorjai, G.A., 1994. Introduction to Surface Chemistry and Catalysis. Wiley, New York. Somorjai, G.A., Bent, B.E., 1985. Prog. Colloid Polym. Sci. 70, 38. Spencer, N.B., Schoonmaker, R.C., Somorjai, G.A., 1982. J. Catal. 74, 129. Sprik, M., Ciccotti, G., 1998. J. Chem. Phys. 109, 7737. Stampfl, C., Kreuzer, H.J., Payne, S.H., Pfnuer, H., Scheffler, M., 1999. Phys. Rev. Lett. 83, 2993. Stipe, B.C., Rezaei, M.A., Ho, W., 1997. J. Chem. Phys. 107, 6443. Stoltze, P., Nørskov, J.K., 1985. Phys. Rev. Lett. 55, 2502. Strasser, P., Fan, Q., Devenney, M., Weinberg, H.W., Liu, P., Nørskov, J.K., 2003. Phys. Chem. B 107, 11013. Strongin, D.R., Carrazza, J., Bare, S.R., Somorjai, G.A., 1987. J. Catal. 103, 213. Strongin, D.R., Somorjai, G.A., 1988. J. Catal. 109, 51. Takahashi, N., Shinjoh, H., et al., 1996. Catal. Today 27, 63. Taylor, H.S., 1925. Proc. R. Soc. London Ser. A 108, 105. Thomas, J.M., Thomas, W.-J., 1997. Principle and Practice of Heterogeneous Catalysis. Wiley-VCH, Weinheim. Toda, T., Igarashi, H., Uchida, H., Watanabe, M., 1999. J. Electrochem. Soc. 146, 3750. Toolenaar, F.J.C.M., Stoop, F., Ponec, V., 1983. J. Catal. 82, 1. Topsøe, H., Clausen, B.S., Massoth, F.E., 1996. In: Anderson, J.R., Boudart, M. (Eds.), Catalysis Science and Technology. Springer/Verlag, New York. Trasatti, S., 1972. J. Electroanal. Chem. 39, 163. Tripa, C.E., Zubkov, T.S., Yates, J.T., Mavrikakis, M., Nørskov, J.K., 1999. J. Chem. Phys. 111, 8651. van Hardeveld, R., van Montfoort, A., 1966. Surf. Sci. 4, 396. van Hove, M.A., Koestner, R.J., Stair, P.C., Biberian, J.P., Kesmodel, L.L., Bartos, I., Somorjai, G.A., 1981. Surf. Sci. 103, 189. van Santen, R.A., Neurock, M., 2006. Molecular Heterogeneous Catalysis. Wiley-VCH, Weinheim. van Santen, R.A., Niemantsverdriet, J.W., 1995. Chemical Kinetics and Catalysis. Plenum Press. Vang, R.T., Honkala, K., Dahl, S., Vestergaard, E.K., Schnadt, J., Lægsgaard, E., Clausen, B.S., Nørskov, J.K., Besenbacher, F., 2005. Nature Materials 4, 160. Voter, A., 1986. Phys. Rev. B 34, 6819. Weinert, M., Watson, R.E., 1995. Phys. Rev. B 51, 17168. White, J.D., Chen, J., Matsiev, D., Auerback, D.J., Wodtke, A.M., 2005. Nature (London) 433, 503. Wierzbicki, A., Kreuzer, H.J., 1991. Surf. Sci. 257, 417. Wigner, E., 1938. Trans. Faraday Soc. 34, 29. Wintterlin, J., Schuster, R., Ertl, G., 1996. Phys. Rev. Lett. 77, 123. Wodtke, A.M., 2006. Science 312, 64. Woodruff, D.P., Delchar, T.A., 1986. Modern Techniques of Surface Science. Cambridge University Press, Cambridge. Yaccato, K., Carhart, R., Hagemeyer, A., Lesik, A., Strasser, P., Volpe, A.F., Turner, H.W., Weinberg, W.H., Grasselli, R.K., Brooks, C., 2005. App. Catal. A: Gen. 296, 30. Yamamoto, T., Chem, J., 1960. Phys. 33, 281. Yates, J.T. Jr., 1995. J. Vac. Sci. Technol. A 13, 1359. Yulikov, M., Sterrer, M., Heyde, M., Rust, H.P., Freund, H.-J., Pachioni, G., Scagnelli, A., 2006. Phys. Rev. Lett. 96, 146804. Zambelli, T., Wintterlin, J., Trost, J., Trost, G., Ertl, G., 1996. Science 273, 1688. Zhang, J.L., Vukmirovic, M.B., Sasaki, K., Nilekar, A.U., Mavrikakis, M., Adzic, R.R., 2005. J. Am. Chem. Soc. 127, 12480. Zhdanov, V.P., 1991. Surf. Sci. Rep. 12, 183. Zhdanov, V.P., 1997. Phys. Rev. B 55, 6770. Zubkov, T., Morgan, G.A. Jr., Yates, J.T. Jr., Kühlert, O., Lisowski, M., Schillinger, R., Fick, D., Jänsch, H.J., 2003. Surf. Sci. 526, 57–71.

CHAPTER 9

Non-linear Dynamics in Catalytic Reactions

R. IMBIHL Institut für Physikalische Chemie und Elektrochemie Leibniz-Universität Hannover Callinstrasse 3-3a D-30167 Hannover, Germany



Contents 9.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2. Observation of rate oscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3. Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1. Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2. Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3. Activator-inhibitor models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3.1. Point models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3.2. Simple wave patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.4. Mathematical modeling of surface reactions . . . . . . . . . . . . . . . . . . 9.4. Oscillation mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1. Catalytic CO oxidation on Pt . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1.1. Bistability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.1.2. The surface phase transition model . . . . . . . . . . . . . . . . . . 9.4.1.3. Mathematical modeling of the SPT mechanism . . . . . . . . . . . . 9.4.1.4. Facetting and oscillations on high-index planes . . . . . . . . . . . . 9.4.1.5. The oxide model and the pressure gap . . . . . . . . . . . . . . . . . 9.4.2. Catalytic CO oxidation on Pd surfaces . . . . . . . . . . . . . . . . . . . . . 9.4.3. Catalytic NO reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3.1. NO reduction on Pt and Rh surfaces . . . . . . . . . . . . . . . . . . 9.4.3.2. Pt(100)/NO + CO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.3.3. The NO + H2 and NO + NH3 reaction on Pt(100) . . . . . . . . . . 9.4.3.4. The NO + H2 reaction on Rh surfaces . . . . . . . . . . . . . . . . . 9.4.3.5. NO reduction on Ir surfaces . . . . . . . . . . . . . . . . . . . . . . 9.5. Chemical wave patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1. Basic considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1.1. Differences to patterns in the liquid phase . . . . . . . . . . . . . . . 9.5.1.2. Spatially resolving in situ methods . . . . . . . . . . . . . . . . . . . 9.5.1.3. Survey of experimentally studied systems . . . . . . . . . . . . . . . 9.5.1.4. Mathematical modeling . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2. Simple chemical wave patterns . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2.1. Bistability and front nucleation . . . . . . . . . . . . . . . . . . . . . 9.5.2.2. Target patterns, spiral waves . . . . . . . . . . . . . . . . . . . . . . 9.5.3. Patterns modified by surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3.1. Global coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.3.2. State-dependent anisotropy . . . . . . . . . . . . . . . . . . . . . . . 9.5.4. Complex patterns due to structural or chemical modifications of the substrate 9.5.4.1. Subsurface oxygen formation . . . . . . . . . . . . . . . . . . . . . 9.5.4.2. Reaction-induced substrate changes . . . . . . . . . . . . . . . . . . 9.5.5. Patterns driven by energetic interactions . . . . . . . . . . . . . . . . . . . . 9.5.5.1. General remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.5.2. Reactive phase separation . . . . . . . . . . . . . . . . . . . . . . . . 9.5.5.3. Mass transport with pulses . . . . . . . . . . . . . . . . . . . . . . .

342

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

345 346 349 349 351 352 352 353 355 356 356 356 357 361 362 366 368 370 370 371 373 374 377 377 377 377 378 381 381 383 383 384 386 386 388 390 390 396 396 396 398 399

9.5.6. Atomic scale experiments and fluctuations . . . 9.5.6.1. Scanning tunneling microscopy . . . . 9.5.6.2. Field electron and field ion microscopy 9.5.6.3. Fluctuations in surface reactions . . . . 9.6. Chaos and turbulence . . . . . . . . . . . . . . . . . . . 9.6.1. Temporal chaos . . . . . . . . . . . . . . . . . . 9.6.2. Turbulence . . . . . . . . . . . . . . . . . . . . . 9.7. Controlling wave patterns and oscillations . . . . . . . 9.7.1. Parameter forcing . . . . . . . . . . . . . . . . . 9.7.2. Controlling turbulence . . . . . . . . . . . . . . 9.7.3. Microstructured surfaces . . . . . . . . . . . . . 9.7.3.1. Basic concepts . . . . . . . . . . . . . . 9.7.3.2. Unreactive boundaries . . . . . . . . . 9.7.3.3. Reactive boundaries . . . . . . . . . . . 9.7.4. Manipulation by local laser heating . . . . . . . 9.8. Conclusions and outlook . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

401 401 402 404 409 409 410 411 411 413 413 413 415 417 419 419 420 420

Abstract Chemical wave patterns and oscillatory kinetics of heterogeneously catalyzed reactions have been intensely studied in the past two decades. The profound understanding of these phenomena which has been reached today can to a large part be attributed to the use of surface science methods allowing to obtain detailed mechanistic information. The progress which has been achieved in this field is reviewed here with the emphasis being laid on single crystal studies. Keywords: catalysis, non-linear dynamics, kinetic oscillations, chemical waves, self-organization, non-equilibrium phenomena, spatio-temporal patterns, pattern formation

343


345

9.1. Introduction Heterogeneously catalyzed reactions are systems far from thermodynamical equilibrium. Consequently one finds in those systems structures which are not allowed at thermodynamical equilibrium and which result from self-organization processes (Mikhailov, 1991; Nicolis, 1989; Pismen, 2006): rate oscillations, chemical wave patterns, stationary concentration patterns, chaos, etc. To demonstrate the different types of self-organization on a catalytic surface is the purpose of this chapter. When rate oscillations in a heterogeneously catalyzed reaction were first discovered by the group of Wittke in 1970 (Beusch et al., 1972; Hugo, 1970) oscillatory reactions have already been known for decades in homogeneous liquid phase reactions – the most famous example being the well-known Belousov–Zhabotinskii reaction discovered about 50 years ago (Field and Schneider, 1989). Less well known is the fact that the first report of oscillatory behavior dates back to a study of electrochemical reactions published by Fechner in 1828 (Fechner, 1828). Self-organization phenomena are of course present in many different areas ranging from physics, chemistry, biology, geology to economy and social sciences (Haken, 1983; Murray, 1989). It was soon realized that the mathematical description of all these phenomena has a common denominator which is that the underlying equations have to be non-linear. This mathematical requirement earned the whole field its name which is “non-linear dynamics” (NLD). Soon after the discovery of rate oscillations in catalytic CO oxidation on platinum the new phenomenon sparked a number of experimental studies but progress was hampered by ill defined catalysts and by the fact that these early studies were typically conducted under atmospheric pressure conditions thus ruling out most surface analytical techniques which typically relied on electrons. The breakthrough in this field came with the surface science approach using low pressure (p < 10−3 mbar) single crystal experiments in combination with an in situ characterization with surface analytical techniques. This approach, which was started in the group of Ertl, led to a wealth of detailed mechanistic information which could serve as a reliable basis for the formulation of realistic mathematical models (Eiswirth and Ertl, 1994; Ertl, 1990; Imbihl and Ertl, 1995). While initially mainly the mechanistic aspects were investigated the focus of attention soon turned to chemical wave patterns which were studied with various spatially resolving techniques down to atomic resolution (Rotermund, 1997; Sachs et al., 2001). A natural consequence of the success of these studies was the idea to control patterns via micro- or nano-structured surfaces or to use a laser beam in order to steer chemical wave patterns by locally perturbing a reaction system (Graham et al., 1994; Mikhailov and Showalter, 2006). The hope was that taking advantage of the non-linear properties of catalytic reactions, it should be possible to improve yield and selectivity of these reactions (Schütz et al., 1998). This leads back to the initial question of the impact of all these studies of oscillations and wave patterns on catalysis. The catalysts itself under working conditions represents a dissipative structure, i.e. a structure which only exists under non-equilibrium conditions. The extent to which a surface deviates from its equilibrium shape and equilibrium chemical composition depends of course on how strong the forces are which drive the system away from thermodynamical equilibrium. Under the conditions of so-called “real catalysis” the structure of a catalyst in

346

R. Imbihl

operation will be determined largely by dynamic processes. The area of non-linear dynamics and self-organization processes thus becomes connected with the pressure and material gap problem in heterogeneous catalysis1 (Dwyer and Hoffmann, 1992). The study of nonlinear dynamics on catalytic surfaces is therefore not only a matter of understanding more or less exotic phenomena like rate oscillations or chemical waves but also becomes vital for explaining real catalysis. In this chapter the focus will be on the contribution of surfaces science type studies to the field of pattern formation and oscillations on catalytic surfaces. Phenomena like kinetic oscillations and chemical wave patterns have been found not only in low pressure single crystal studies but from UHV to atmospheric pressure and over a vast range of materials from monolithic catalysts like foils, ribbons, wires, field emitter tips to supported catalysts, monodispersed nanosized particles in a zeolite matrix, and micro- and nanostructured composite catalysts. For the high pressure studies the reader is referred to a number of well written reviews. Since the early reviews by Sheintuch and Schmitz (1977) and by Slinko and Slinko (1978), a number of reviews of oscillations and wave patterns on surfaces have appeared (Eiswirth and Ertl, 1994; Ertl, 1990; Imbihl, 1989, 1993; Imbihl and Ertl, 1995; Schüth et al., 1993). A rather broad overview of oscillatory catalytic reactions comprising more than 20 oscillatory reactions and more than 100 reaction systems is the one published by Schüth et al. (1993). The results of single crystal studies can be found in (Eiswirth and Ertl, 1994; Ertl, 1990; Imbihl, 1993; Imbihl and Ertl, 1995). The area of non-linear dynamics on catalytic surfaces has been treated in several focus issues of journals.2 A monography by Slinko and Jaeger connects the results of UHV single crystal studies with investigations conducted in the mbar to atmospheric pressure range (Slinko and Jaeger, 1994). The vast majority of all experimental studies concentrates on a small group of reactions and these are the reactions of the automotive catalytic reactor: catalytic CO oxidation and NO reduction and the O2 + H2 reaction on the noble metals of Pt, Pd, Rh, and Ir. It is this group of reactions on single crystal surfaces on which we focus in this report because for these reactions well established mechanisms are available allowing the formulation of realistic mathematical models.

9.2. Observation of rate oscillations Figure 9.1a displays an example of rate oscillations in catalytic CO oxidation measured with a Pt(110) sample in the 10−4 mbar range (Eiswirth et al., 1989). For determining the existence range for oscillations one typically only varies one parameter corresponding to a cut through a multidimensional parameter space. By plotting the steady states and the oscillation amplitude vs. the parameter which is varied one obtains so-called bifurcation diagrams a typical example of which is shown in Fig. 9.1b (Wicke et al., 1980). The term 1 See the special of PCCP devoted to the pressure and material gap in catalysis: PCCP 2007 9, 3449–3660. 2 See the focus issues: Catal. Today 70/4 (2001), Chaos 12/1 (2002), N. J. Phys. 5 (2003), J. Phys. Chem. 100/49 (1996).


347

Fig. 9.1. Rate oscillations in catalytic CO oxidation on platinum. (a) Kinetic oscillations in catalytic CO oxidation on Pt(110) measured via the variation of the CO2 partial pressure, pCO2 , and the work function variation φ. (From Eiswirth et al., 1989.) (b) Bifurcation diagram showing the steady state behavior and the development of rate oscillations (hatched areas) at various temperatures in catalytic CO oxidation on a Pt/Al2 O3 , SiO2 supported catalyst at atmospheric pressure. The CO partial pressure pCO is varied as bifurcation parameter. (From Wicke et al., 1980. Reproduced with permission of the authors.)

bifurcation simply means a change in the qualitative behavior of a system as one parameter, the bifurcation parameter, is varied. In this example the transitions from a steady state to oscillatory behavior occurs continuously at the low pCO side and discontinuously at the high pCO side. Due to the exothermicity of the reaction the oscillations in the reaction rate are accompanied by periodic temperature variations which at atmospheric pressure can reach up to 100 K amplitude. At low pressure (p < 10−3 mbar) the reaction still produces heat but in relation to the heat capacity of the sample and to the heat which is supplied by electric heating and also to the heat which is radiated off, the contribution of the reaction heat is so small that the reaction can be considered as truly isothermal. With few exceptions the experimental studies of oscillations and wave patterns can be divided into two groups: (i) Single crystal experiments typically conducted under isothermal conditions at low pressure (p < 10−3 mbar) in a UHV chamber which is operated as a continuous flow reactor. The molecular flow rapidly eliminates concentration gradients in the chamber so that we can assume perfect mixing in the gas-phase.

348

R. Imbihl

(ii) Experiments with polycrystalline catalysts carried out at high pressure in the mbar to atmospheric pressure range. The reaction is non-isothermal and in most cases a concentration profile exists in the gas-phase. Clearly, the first group of experiments has the advantage of great conceptual simplicity, not only because the surfaces are well defined with respect to structure and chemical composition but also because the low pressure allows the use of electrons on which most surface analytical techniques are based upon. Most importantly, heat and mass transport problems are absent which complicate the analysis of the data at high pressure (Slinko and Jaeger, 1994). For obtaining oscillations in the reaction rate it is essential that the different oscillating parts of the catalytic surface are synchronized via an efficient coupling mode because otherwise the contributions from different parts of the surface average out to a stationary reaction rate. For coupling together different regions of a surface three basic mechanisms exist: (i) Diffusional coupling via mobile adsorbates. This coupling mode is local in nature and can in general not synchronize oscillations over the macroscopic distances of a single crystal surface. Diffusion is essential for chemical wave patterns because fronts and pulses etc propagate through diffusional coupling.3 (ii) Global coupling via the gas-phase. As shown in Fig. 9.1a the oscillations in the CO2 production rate are accompanied by small variations in pCO of about 1% which arise due to mass balance in the reaction. Since the reaction rate integrates over the whole catalytic surface and since all parts of the surface in a gradient-free reactor are affected in the same way and practically without any delay, the partial pressure variations of the gases represent global coupling (Eiswirth et al., 1989; Imbihl and Veser, 1994). (iii) Thermal coupling. This coupling mode is only present at higher pressure (>10−1 mbar) when the reaction heat produces measurable variations of the surface temperature (Luss, 1992; Luss and Marhawa, 2002; Luss and Sheintuch, 2005). If present, this coupling mode usually dominates due to the fast spreading of heat and due to the strong influence of the temperature on the reaction kinetics. In industrial catalysis heat patterns are highly important due to the formation of “hot spots” which can lead to the destruction of a reactor. Experimentally, the surface changes during oscillatory behavior can be followed with integral methods or spatially resolving methods. The latter will be discussed in connection with the observation of chemical waves in Section 9.5. Under the low-pressure conditions of single crystal experiments, in principle, an arsenal of surface analytical tools is available but due to the destructive interaction of electrons with sensitive adsorbates the use of techniques like Auger Electron Spectroscopy is in general problematic. A non-destructive and very sensitive method is work function (WF) measurements with the Kelvin probe yielding about 1 mV sensitivity. In the catalytic CO oxidation on Pt the WF changes are mainly determined by the oxygen coverage and since under oscillatory conditions the reaction rate, r(CO2 ), is proportional to the oxygen coverage the WF variations reflect r(CO2 ) as demonstrated in Fig. 9.1a (Eiswirth et al., 1989). 3 Chemical waves without diffusional coupling are of course possible in so-called phase waves.


349

With the use of photons instead of electrons as surface probe in principle no pressure limit exists and methods like Fourier transform infrared spectroscopy (FTIR), X-ray diffraction (XRD), and ellipsometry are applicable as in situ techniques over the whole range from UHV to atmospheric pressure (Rotermund et al., 1995). Solid state potentiometry has been introduced by Vayenas et al. to monitor directly the oscillatory activity of oxygen in the electrochemical oxidation of CO or ethylene on the Pt electrode of a solid state electrolyte (Vayenas et al., 1981; Vayenas and Michaels, 1982).

9.3. Theoretical background 9.3.1. Basic definitions The evolution of the spatially and temporally varying concentrations on the surface is governed by reaction-diffusion (RD) equations ∂ci ∂ 2 ci = Fi (μ, c) + Di 2 ∂t ∂x in which the first term, Fi , contains the reaction kinetics of the species i and the second term describes the diffusion of the species i. In this equation c is a vector standing for the concentrations of the various chemical species and μ denotes a set of parameters such as temperature, pressure etc. For diffusion, we assume simple Fickian diffusion representing the diffusion constant of species i by Di . When we neglect the spatial degrees of freedom corresponding to a spatially uniformly reacting surface, then the temporal evolution of a system is governed by a set of coupled ordinary differential equations (ODE’s) dxi = Fi (x, μ) dt where we have replaced the concentrations ci by the general variables, xi . The number of components of the vector x, n, is equal the dimensions of the system. For representing the dynamics of the system, we choose a so-called phase space representation in which the n state variables xi span a coordinate system which contains all possible states of the system. The temporal evolution of the system is described by a trajectory in this n-dimensional phase space. Trajectories are not allowed to cross because this would violate the uniqueness of the vector field F (x, μ). Representing the temporal evolution of a system by trajectories offers the advantage that the different solutions can be nicely visualized. In the following we will restrict ourselves first to 2-dimensional and then to 3-dimensional systems with the state variables x1 , x2 , and x1 , x2 , x3 , respectively. It turned out that most cases of interest can be explained by use of only two or at most three variables. For classifying the behavior of trajectories it is convenient to introduce some mathematical definitions. The subsets of the phase space on which the trajectories reside as t approaches +∞ are so-called limit sets. These limit sets organize the vector flow in their

350

R. Imbihl

Fig. 9.2. Basic type of bifurcations which can occur in a two-variable system, i.e. in a system described by two ODE’s. Shown are the phase portraits as the bifurcation parameter μ is swept across the bifurcation point, μc , and the experimentally observable behavior of a signal A during this sweep. In the phase diagrams filled circles denote a stable node, half-filled circles a saddle point, and open circles an unstable node. The following abbreviations are used for bifurcations: sn = saddle node, h = Hopf, snp = saddle node of periodic orbits, sl = saddle loop, sniper = saddle node of infinite period.


351

vicinity. A limit set is called an attractor if all trajectories in the neighborhood move towards this limit set and a repellor if the vector flow is directed away from the limit set. In the first case the limit set is called stable while in the second case it is unstable. The simplest kind of limit set is fixed point. Since a fixed point is formally obtained by setting the time derivatives equal to zero, Fi (x, μ) = 0, such a fixed point corresponds to a stationary solution. If a single stable fixed point exists, the system is monostable and all trajectories in the neighborhood converge towards this stable fixed point as depicted in the top diagram of Fig. 9.2a. If the fixed point is unstable, it acts as a repellor and the trajectories in its vicinity are moving away from the fixed point. Finally, a third kind of fixed point exists, which is attractive in one direction and repelling in another direction and this fixed point is called saddle-point. A saddle-point is present if two stable fixed points coexist as depicted in the third plot from top in Fig. 9.2a. In such a bistable system the saddle-point acts as a so-called separatrix, dividing the basis of attraction for the two stable fixed points. In two dimensions only two types of limit sets exist which are fixed points and limit cycles. A limit cycle is a closed curve in phase space which in time corresponds to a periodically oscillating system as indicated in Fig. 9.2b. Chaotic oscillations are not possible in 2D just for topological reasons. It is impossible to draw a chaotic trajectory in 2D without violating the requirement that trajectories are not allowed to cross (s.a.). 9.3.2. Bifurcations A qualitative change in the behavior of a system upon a parameter variation is called bifurcation (Guckenheimer and Holmes, 1983; Pismen, 2006; Thompson and Stewart, 1987). As an example we may consider the so-called Hopf bifurcation depicted in Fig. 9.2b. In this bifurcation type oscillations are generated as a stable steady state becomes unstable upon changing a parameter. The trajectories which represent the dynamical behavior of the system in a so-called phase space representation are no longer attracted by the fixed point which represents the steady state solution. Instead the trajectories circulate around this point such that a stable limit cycle is generated, i.e. the system exhibits now sustained oscillations. The Hopf bifurcation predicts a characteristic behavior of the oscillations that develop beyond the bifurcation point which can be compared with the experiment. The oscillation amplitude grows continuously as μ is swept past the bifurcation point, μc , following a square-root dependence. Since only the vector field in the vicinity of a fixed point is considered in a Hopf bifurcation this type of bifurcation is called local bifurcation. Such local bifurcations can be detected with a mathematical tool called linear stability analysis. One linearizes the equations around the steady state and examines the behavior of a small perturbation which drives the system away from the fixed point. Depending on whether the perturbation decays or grows the point is stable or unstable, respectively. The stability of a fixed point in an n-variable system is characterized by n eigenvalues and n eigenvectors. For a two-variable system only a small number of bifurcations are possible which are all depicted in Fig. 9.2 by their phase portraits and the experimentally observed bifurcation diagrams. The simplest and most important bifurcation is the saddle-node (sn) bifurcation in Fig. 9.2a discussed above. Such a bifurcation separates monostability from bistability

352

R. Imbihl

and in the experiment one observes a hysteresis upon parameter variation. A bifurcation analogous to the sn bifurcation is obtained as one replaces the stable/unstable fixed point by a stable/unstable limit cycle. As in a sn bifurcation a stable limit cycle may collide with an unstable limit cycle leading to annihilation of both. This type of bifurcation which is called saddle node of periodic orbits (snp) or global (subcritical) Hopf bifurcation is depicted in Fig. 9.2c. In contrast to the (supercritical) Hopf the oscillations in the subcritical Hopf do not develop gradually but the system jumps immediately to large amplitude oscillations at the bifurcation point. In the remaining two bifurcations a limit cycle is created by connecting the repellent direction of a saddle point with its attracting direction such that a so-called homoclinic orbit generated. In a saddle loop (sl) bifurcation shown in Fig. 9.2d one observes an annihilation of oscillations as a stable limit cycle which surrounds an unstable node collides with the saddle point. Finally, in a saddle node of periodic orbit (sniper) bifurcation the homoclinic orbit is destroyed through a sn bifurcation taking place on the limit cycle itself. In this type of bifurcation sketched in Fig. 9.2e only a stable node is left beyond μc . Characteristic for the two homoclinic bifurcations discussed above is that the oscillation period tends towards infinity as the control parameter approaches the bifurcation point. The oscillation amplitude collapses to zero in a discontinuous transition beyond μc . Analogous to first and second order phase transitions thus also the bifurcations exhibit continuous and discontinuous changes in the system variables. For more than two variables complex dynamical behavior is possible. One may observe mixed-mode oscillations and deterministic chaos is possible (Thompson and Stewart, 1987). In a phase space representation, mixed-mode oscillations are visualized by motion on a torus and deterministic chaos by motion on a so-called “strange attractor” (see Section 9.6). Depending on the number of parameters which have to be varied to generate a certain type of bifurcation, one speaks of codimension-1, codimension-2 bifurcations, etc. The dynamic behavior of a system can be conveniently summarized in bifurcation diagrams analogous to phase diagrams in equilibrium thermodynamics. Such bifurcation diagrams can be constructed by monitoring the stability of the steady state solution as one (or several) parameters are varied. Typically such bifurcation analyses are conducted today with the help of program packages (e.g. AUTO, see Ref. (Doedel, 1986)). A systematic analysis that allows one to predict whether a proposed sequence of chemical reactions allows oscillatory solutions is the so-called stochiometric network analysis (SNA) developed by Clarke (1980). As shown by Eiswirth et al. this method which analyses the feedback loops inherent in a reaction scheme can also be applied to isothermal surface reactions if terms like “surface phase transition”, “subsurface oxygen”, etc. are translated into a “chemical” language (Eiswirth et al., 1996, 1991b, 1991a). 9.3.3. Activator-inhibitor models 9.3.3.1. Point models A large part of the dynamic behavior of a chemical reaction system can already be understood in terms of simple models involving only two species with antagonistic effects on the reaction which are called “activator” and “inhibitor”.


353

The most well known of these activator-inhibitor models is the so-called FitzHugh– Nagumo (FHN) model whose reaction part is described by the ordinary differential equations (ODE’s) (FitzHugh, 1961; Winfree, 1991) du 1 u3 (9.1) = u− − v = f (u, v), dt ε 3 dv (9.2) = u + b − av = g(u, v) dt in which u denotes the activator and v the inhibitor. a and b are constants. The activator is produced in an autocatalytic reaction and together with the activator also the inhibitor is generated which limits the activator production. The inhibitor kinetics take place on a much slower time scale than the activator kinetics as expressed by the factor 1/ε in the first equation. First only the solution of the ordinary differential equations (ODE’s) or point model is considered. The stationary solutions are given by the intersections of the two nullclines resulting from setting f (u, v) = 0 and g(u, v) = 0. One obtains, besides the trivial case of monostability, three types of solutions – bistability, excitability, and oscillatory behavior – which are visualized by trajectories around the nullclines in Fig. 9.3a. In a monostable system only one stable fixed point exists, but in bistable system three intersections are present: two stable and one unstable fixed point. In the oscillatory system again only one fixed point appears but this point is unstable. If a system is excitable then a stable fixed point exists but only a finite perturbation is required to drive the solution away from the stable fixed point. Only after a large excursion through phase space the system returns to the stable resting state. Due to the separation of the time scales for u and v the trajectory follows in large parts the nullcline for u, f (u, v). 9.3.3.2. Simple wave patterns Chemical wave patterns are obtained as solutions of so-called reaction-diffusion (RD) equations obtained formally by complementing the reaction terms described by ODE’s with diffusion terms for the mobile species. In the FHN model the different types of solutions of the ODE’s correspond to different types of media categorized as bistable, excitable or oscillatory (Mikhailov, 1991). These different media support different types of chemical waves depicted schematically in Fig. 9.3b. These basic types of chemical waves are obtained by integration of the PDE formulation of the FHN model: du (9.1) = f (u, v) + Du ∇ 2 u, dt dv (9.2) = g(u, v) + Dv ∇ 2 v. dt Before we discuss these solutions we first take a look at the mechanism which leads to the most elementary type of chemical wave which is a reaction front. It was already recognized by Luther in 1906 that reaction fronts can arise if an autocatalytic reaction exists and if the autocatalytic component X is allowed to diffuse (Luther, 1906). An autocatalytic reaction, for example, is given by A + X → 2X, or more generally, by a reaction of the type A + mX → nX

with n > m.

354

R. Imbihl

Fig. 9.3. Basic solutions of an activator–inhibitor model given here by the FitzHugh–Nagumo equations. (a) Different types of stationary points in the ODE model obtained by the intersection of the two nullclines, f (u, v) = 0 and g(u, v) = 0: monostable, bistable, excitable and oscillatory behavior. Shown are the nullclines and the phase portraits of the trajectories in the u, v-plane. (b) Basic 1D-solutions of the RD system showing fronts, pulses, and stationary (Turing) patterns.

If the reaction is ignited at a given point, then the concentration of the autocatalytic component X will soon rise to a very high level. As the species X diffuses to a neighboring region, the reaction will also be ignited there. Repetition of this process creates a propagating reaction front. In a bistable system transitions between the two stable states occur via such reaction fronts as depicted in the upper left diagram of Fig. 9.3b. The reaction fronts travel with a constant velocity and with a constant concentration profile and their velocity


355

√ cf can be approximated as cf ∼ DX K in which K represent a pseudo first order reaction constant and DX is the diffusion constant of species X. In an excitable system the perturbation generates a pulse which propagates through the medium and leaves the medium after its passage in the same state as before. As shown in the upper right diagram of Fig. 9.3b the activator concentration rises steeply at the leading edge whereas the inhibitor concentration builds up slowly. When the inhibitor reaches a certain level the activator falls steeply again while the inhibitor concentration decays slowly in the so-called refractory period of the pulse. Only when the refractory period has passed the excitation of a new pulse is possible. In 2D-systems the excitation of pulses generates target patterns and rotating spiral waves. These types of chemical wave patterns are also supported by oscillatory media which in contrast to excitable media have their own characteristic oscillation period. Finally, if the inhibitor diffuses much more rapidly than the activator a stationary concentration pattern may evolve in which activator and inhibitor exhibit an oscillatory variation along the spatial coordinate as depicted in the bottom diagram of Fig. 9.3b. Such stationary concentration patterns which are only possible in a non-equilibrium system are commonly known as Turing structures after A.M. Turing who predicted this type of pattern in a pioneering paper in 1952 (Turing, 1952). 9.3.4. Mathematical modeling of surface reactions In the formulation of surface reaction models with ODE’s, it is implicitly assumed that the adparticles are uniformly distributed like in an ideal 2D-gas. Such a mean-field treatment is strictly justified only if the mobility of the adparticles is very high and if no repulsive or attractive interactions between adparticles are operating. For any realistic system, however, such energetic interactions exist leading to island formation and inducing spatial correlations. In order to take these effects into account lattice gas models have been formulated which typically employ a Monte Carlo algorithm (Evans, 1991; Evans et al., 2002). While the original Monte Carlo algorithm applies to an equilibrium situation time-dependent processes are typically simulated with a kinetic Monte Carlo scheme. In the field of statistical physics such lattice gas models have found widespread interest mainly because the kinetic transitions observed in these systems bear many analogies to phase transitions in classical thermodynamics. One of the most popular models of this type, the ZGB model, has been introduced by Ziff, Gulari and Barshed to describe catalytic CO oxidation (Ziff et al., 1986). The ZGB model is, however, quite unrealistic since it neglects energetic interactions and contains an oxygen poisoned state of the surface which has never been observed experimentally with Pt catalysts. Albeit simulations with lattice gas models are in principle exact, such models suffer from the fact that due to computational limitations they have to use unrealistic low surface mobilities (Evans, 1991; Evans et al., 2002). High surface mobilities, on the other hand, tend to restore the validity of the mean field equations. Hybrid models have been developed which treat the fast diffusing species as a mean field variable while the full lattice gas algorithm is applied to the slow diffusing adsorbate (Tammaro et al., 1995) (see also Section 9.5.1.4).

356

R. Imbihl

9.4. Oscillation mechanisms 9.4.1. Catalytic CO oxidation on Pt 9.4.1.1. Bistability A catalytic effect of Pt wires on the reaction of CO with oxygen was already observed by Faraday and since then catalytic CO oxidation has been the subject of numerous papers including the “classical” work of Langmuir (Langmuir, 1921, 1922). Meanwhile, it has been well established that the mechanism of this reaction follows a Langmuir–Hinshelwood scheme described by the following three equations in which * denotes a vacant adsorption site (Engel and Ertl, 1979): CO + * COad O2 + 2* → Oad Oad + COad → CO2 + 2* A mathematical model derived from this scheme contains just two variables, the CO coverage θCO and the oxygen coverage, θO : dθCO (9.1) = sOH k1 pCO [1 − θCO ] − k2 θCO − k3 θCO θO dt dθO (9.2) = sO2 k4 pO2 (1 − θCO − θO )2 − k3 θCO θO dt where the term with k1 represents CO adsorption, with k2 CO desorption, with k3 the surface reaction and with k4 O2 adsorption. Integration of this model yields bistable behavior, but no rate oscillations (Bär et al., 1992b). The kinetics are characterized by two different branches: a high rate branch at low pCO in which nearly every CO molecule which hits the surface reacts and the CO2 production therefore rises linearly with pCO , and a low rate branch at high pCO in which the surface is poisoned by a large CO coverage which inhibits oxygen adsorption. The two rate branches are well visible in Fig. 9.1b where, in addition to the model, rate oscillations occur in the transition region between the two branches. Bistability is a general property of catalytic CO oxidation on Pt found also with all those systems which do not display oscillatory behavior. Pure bistability is observed, at p < 10−3 mbar in the system Pt(111)/CO+O2 (Berdau et al., 1999). The existence of the two rate branches can be traced back to an asymmetric inhibition of the reaction by the reactants. With oxygen forming a very open adlayer structure CO can still adsorb and react while a fully CO covered surface completely inhibits the adsorption of oxygen and hence poisons the reaction. Under conditions where no oscillations occur the existence of the two kinetic branches leads to bistability in a certain parameter range, i.e. one observes a clockwise hysteresis in the reaction rate upon variation of pCO (Bär et al., 1992b; Evans et al., 2002; Zhdanov and Kasemo, 1994b). At high enough temperature (T > 600 K for atmospheric pressure) the CO coverage does not reach the coverage necessary for complete inhibition of the reaction, and hence the range of bistability vanishes in a so-called cusp point. The bistable behavior of CO oxidation has been analyzed with differential equations and in numerous lattice gas models the most well known being the so-called ZGB model


357

(Evans, 1992; Evans and Miesch, 1991; Ziff et al., 1986). In order to generate oscillations an additional feedback mechanism is required activating and deactivating the surface. How such a mechanism works is shown in the subsequent sections. 9.4.1.2. The surface phase transition model The key observation leading to the formulation of the surface phase transition (SPT) or reconstruction model was that under low pressure conditions (p < 10−3 mbar) only those Pt orientations displayed oscillatory behavior which were reconstructed in their clean state whereas the structurally stable Pt(111) surface merely exhibited bistability but no rate oscillations (Cox et al., 1983; Eiswirth and Ertl, 1986; Ertl et al., 1982). Structural models of the three low-index planes of Pt are displayed in Fig. 9.4a. Of the three low-index planes only the close-packed Pt(111) surface is stable in its bulk-like 1 × 1 termination, while the more open (100) and (110) surfaces reconstruct into a quasihexagonal (“hex”) (Heilmann et al., 1979; van Hove et al., 1981) and a 1 × 2 “missing row” geometry (Fery et al., 1988; Jackman et al., 1982; Kellog, 1985; Niehus, 1984), respectively. The reconstruction of both, Pt(100) and Pt(110), can reversibly be lifted by certain adsorbates such as CO, NO etc. (Bare et al., 1984; Gardner et al., 1990a, 1990b; Thiel et al., 1983). This constitutes an adsorbate-induced phase transition which is controlled by critical adsorbate coverages (Thiel et al., 1983). The driving force for adsorbate-induced surface phase transitions can be rationalized on the basis of simple thermodynamic considerations illustrated with the CO-induced 1 × 1 hex phase transition of Pt(100) in Fig. 9.4b. The clean Pt(100) surface is reconstructed because of the lower surface energy of the hex phase as compared to the 1 × 1 phase. The relative stability of the two phases may, however, switch if an adsorbate is more strongly bound on the 1 × 1 phase than on the hex phase. As soon as the gain in adsorption energy overcompensates the loss in reconstruction energy, the reconstruction is lifted by the adsorbate. The difference in adsorption energy was shown to be quite substantial for Pt(100)/CO where values of 155 kJ/mol and 115 kJ/mol have been determined for the 1 × 1 and hex phase, respectively (Brown et al., 1998; Thiel et al., 1983). Rate oscillations occur under conditions where oxygen adsorption is rate-limiting and since the oxygen sticking coefficient, sO2 , is structure sensitive on Pt surfaces, the phase transition can cause a periodic switching between two states of different catalytic activity. On Pt(100) sO2 differs drastically between the 1 × 1 and the hex phase with sO1×1 = 0.3 and 2 −4 –10−3 (Griffiths et al., 1984; Guo et al., 1994; Pasteur et al., 1996), while the ≈ 10 sOhex 2 corresponding difference for Pt(110) is much smaller with sO1×2 ≈ 0.3–0.4 and sO1×1 ≈ 0.6 2 2 (Brown et al., 1998; Freyer et al., 1986). How the phase transition can cause oscillatory behavior on Pt(110) is illustrated by Fig. 9.5. Starting with a CO covered 1 × 1 phase the adsorption rate of oxygen and hence the catalytic activity will be high. As a consequence more adsorbed CO will be consumed by reaction than is supplied by adsorption and the CO coverage decreases. Below a critical value the 1 × 1 structure can no longer be maintained and the surface will reconstruct into the 1 × 2 phase. On this surface sO2 is low and consequently the CO coverage will rise. Above θCO,crit ≈ 0.2 the reconstruction is lifted and the initial situation of a CO covered 1 × 1 surface is established again.

358

R. Imbihl

Fig. 9.4. (a) Reconstructed and non-reconstructed surfaces of the three low-index planes of Pt. (b) Energy diagram for the CO-induced lifting of the hex reconstruction on Pt(100). (From Thiel et al., 1983.)

Direct evidence for the operation of this mechanism was obtained by in situ LEED experiments with Pt(100) and Pt(110) which demonstrated that the oscillations of the reaction


359

Fig. 9.5. Ball model illustrating the CO induced 1 × 1 1 × 2 surface phase transition of Pt(110). The different oxygen sticking coefficients, sO2 ’s, of the two phases are responsible for rate oscillations during catalytic CO oxidation. The model also demonstrates how the necessary mass transport of Pt atoms creates an atomic step on the surface.

rate were in fact accompanied by periodic structural changes of the substrate (Cox et al., 1983; Eiswirth et al., 1989; Imbihl et al., 1986). For Pt(100) the corresponding LEED measurements together with the WF trace which reflects the CO2 production rate (see Fig. 9.1a) are displayed in Fig. 9.6a. Although the basic mechanism is identical, the oscillations on Pt(100) and Pt(110) exhibit quite different properties. In contrast to Pt(100), where one typically finds irregular oscillations, the oscillations on Pt(110) are usually very regular and one observes a variety of different wave-forms ranging from rapid harmonic oscillations (periods τ of the order of seconds) at high temperature (T > 500 K), over mixed-mode oscillations at intermediate temperature to slow (τ > 1 min) harmonic oscillations at low temperature (T < 450 K) (Eiswirth and Ertl, 1986). The regularity and irregularity of the oscillations on Pt(110) and Pt(100), respectively, essentially reflects a different degree of spatio-temporal organization on the two surfaces (see Section 9.5) (Eiswirth et al., 1989). On Pt(110) highly efficient gas-phase coupling ensures a synchronized oscillating surface while this coupling mode plays no role on Pt(100). The different efficiency of gasphase coupling is a consequence of the differently wide existence ranges for oscillations in pCO -parameter space (Eiswirth et al., 1985). As demonstrated in Fig. 9.7 which shows the results from measurements with a cylindrical Pt single crystal the oscillatory range on Pt(100) is rather wide but shrinks to about a few per cent of pCO on the (110) orientation (Sander et al., 1992a). Consequently the pCO variations of typically 1% which arise due to mass balance in the reaction (see Fig. 9.1a) will have a strong influence on Pt(110) but very little impact on the Pt(100) surface. As demonstrated by mathematical modeling, the different width of the oscillatory range for Pt(100) and Pt(110) is a consequence of how strongly sO2 differs between the 1 × 1 and the reconstructed surface (Eiswirth et al., 1989). On Pt(100) sO2 differs by two to three orders of magnitude between the 1 × 1 and

360

R. Imbihl

Fig. 9.6. Experimental and simulated oscillations in catalytic CO oxidation on Pt(100). (a) Experiment. In-situ LEED measurements and WF measurements showing the coupling between the oscillations in the reaction rate and periodic structural changes via the 1 × 1 hex phase transition of Pt(100). The reaction rate rCO2 has been measured here via the WF variation which is proportional to rCO2 under oscillatory conditions. The amount of hex reconstruction is given by the intensity of one of the hex spots while the half-order spot of the c(2 × 2)-CO structure represent the amount of non-reconstructed 1 × 1 phase and the CO coverage. (From Cox et al., 1983.) (b) Simulation. A 4-variable model simulates the oscillations in (a). θCO should be compared with the c(2 × 2) ¯ in (a). (From Imbihl et al., intensity, θhex with the hex intensity in (a) and θO with the intensity of the 1¯ 1-beam 1985.)


361

Fig. 9.7. Existence diagram for the occurrence of kinetic oscillations measured on a cylindrical Pt surface which exhibits all orientations of the [001]-zone at T = 480 K and pO2 = 5 × 10−5 Torr. The hatched area marks the oscillatory parameter range. Due to the narrowness of the existence range (in pCO ) in between (110) and (210) the width had to be represented by a single line in the plot. The bars encircled by dotted lines represent the existence range for oscillations on plane single crystal surfaces. (From Sander et al., 1992a.)

the reconstructed surface while the corresponding ratio for Pt(110) is only about 1.5 (see above). 9.4.1.3. Mathematical modeling of the SPT mechanism For mathematical modeling the two equations which represent the bistable behavior of catalytic CO oxidation on a stable surface have to be complemented by a third variable describing the change in the surface properties brought about by the surface phase transition Such models have been formulated for Pt(100) and Pt(110) and they reproduce rather well the experimental observations as demonstrated with the oscillations on Pt(100) in Figs. 9.6a and 9.6b (Eiswirth and Ertl, 1994; Imbihl et al., 1985; Hoyle et al., 2007). The three-variable model for Pt(110) by Krischer, Eiswirth, Ertl (KEEmodel) was analyzed in depth (Krischer et al., 1992a). Denoting the fraction of the surface which is in the 1 × 1 state by θ1×1 with the amount of 1 × 2 then given by (1 − θ1×1 ), one obtains the following set of equations in which θCO and θO represent the CO and the oxygen coverage, respectively: dθCO = k1 pCO (1 − θCO /θCO,sat )3 − k2 θCO − k3 θCO θO dt dθO = sO2 k4 pCO (1 − θCO /θCO,sat − θO /θO,sat )2 − k3 θCO θO dt

(9.1) (9.2)

362

R. Imbihl

with + θ1×2 · sO1×2 sO2 = θ1×1 · sO1×1 2 2 ⎧ for θCO 0.2 −k5 θ1×1 ⎨ dθ1×1 3 3 i = k5 i=O qi θCO − θ1×1 , for 0.2 θCO 0.5 ⎩ dt for θCO > 0.5 k5 (1 − θ1×1 ),

(9.3)

The terms with k1 and k2 describe CO adsorption and desorption, respectively, oxygen adsorption is represented by the k4 term, and the reaction between CO and O in the term containing k3 . The 1 × 1 1 × 2 phase transition in Eq. (9.3) is described as an activated process in which the surface structure relaxes to that amount of 1×1 phase that is stabilized by the CO coverage. A polynomial expression is used here to reproduce the experimental result that 1×1 nuclei only form beyond a critical CO coverage of 0.2 on the 1×2 phase and completion of the lifting of the reconstruction is achieved at θCO = 0.5 (Gritsch et al., 1989; Imbihl et al., 1988a). This three-variable model can be mapped onto the two variable model of FitzHugh– Nagumo (see Section 9.3.3) because, coupled through the reaction, the oxygen coverage usually varies anticorrelated to the CO coverage so that only two real variables remain. Catalytic CO oxidation on Pt(100) and Pt(110) thus falls into the class of activator-inhibitor models with the oxygen coverage representing the fast activator and the reconstructed phase the slow inhibitor. The 4-variable model for the oscillations on Pt(100) describes the lifting of the hex reconstruction by an island growth process (Imbihl et al., 1985). A Ginzburg–Landau functional for the SPT in this model has been proposed by Andrade et al. (1989, 1994). A modified treatment of the SPT using the experimentally determined power law dependence hex )4.5 , was of the 1 × 1 growth on the CO coverage on the hex phase, rgrowth 1×1 ∼ (θCO introduced (Gruyters et al., 1995; Irurzun et al., 2003; Mola et al., 2004). Various lattice gas models based on Monte Carlo algorithms have been formulated to simulate the oscillations on both surfaces, Pt(100) and Pt(110) (Gelten et al., 1998; Kuzovkov et al., 1998; Moeller et al., 1986; Rose et al., 1994; Wu and Kapral, 1992). As alternative to the SPT model the existence of a second type of oxygen species which is less reactive than chemisorbed oxygen has been proposed but this model lacks experimental support (Vishnevskii et al., 1995). In the group of Vicente et al. various modifications of the SPT model for Pt(100)/CO + O2 have been introduced to make the model more realistic (Chavez et al., 1998, 2000a, 2000b). Their modifications included the effect of lateral interactions between the adparticles and the influence of inert sites on the bifurcation behavior. 9.4.1.4. Facetting and oscillations on high-index planes In the phase transition model the surface is implicitly treated as a strict 2-dimensional system, i.e. all structural changes take place within the surface plane. However, already with a simple ball model one can show that this strict 2-dimensionality is not obeyed as demonstrated in Fig. 9.5 with the 1 × 1 1 × 2 SPT of Pt(110). Since the two surface phases differ by 50% in their density of surface Pt atoms a mass transport of half of the surface Pt atoms is required each time the SPT takes place. This mass transport necessarily


363

creates steps as shown in Fig. 9.5. The surface therefore becomes roughened and if steps accumulate even a facetting of the surface may take place. Both processes are observed and they have consequences for the oscillatory properties of a surface (Falta et al., 1992; Imbihl, 1992; Ladas et al., 1988b, 1988a). A reaction-induced facetting of the initially flat Pt(110) surface can be observed for p > 10−5 mbar and for not too high temperature. For pO2 = 1.5 × 10−4 mbar this high T-limit lies around 530 K (Ladas et al., 1988b; Sander and Imbihl, 1991). In LEED the facetting process can be followed as a continuous process by the splitting of the integral order LEED beams whose separation marks the steepness of the facets. Almost exclusively facets of the [001]-zone are formed which can all be built up by varying the density of (100) steps. The facetting is accompanied by a slow rise of the reaction rate until the high rate branch is reached (Ladas et al., 1988b). As depicted by the rate vs. pCO plot in Fig. 9.8a facetting takes place in the range between the rate maximum and the low rate branch which is exactly the parameter range where the phase transition proceeds. In the 10−4 mbar range facetting takes place on a time scale of ≈ 10–30 min. The facets that are formed are actually microfacets since their size is in the range of ≈ 100 Å. The increase in catalytic activity which is reflected by a shift of the rate maximum of the facetted surface towards higher pCO could be traced back to an increase of the oxygen sticking coefficient caused by the introduction of (100) steps. Consequently, the limiting case in facetting is reached with the Pt(210) orientation where (100) steps and (110) terrace units alternate and which exhibits the highest oxygen sticking coefficient with a value of 0.6 (Ladas et al., 1988b; Sander et al., 1991). Remarkably, for a given pCO value the facetting only progresses so far until the position of the rate maximum has shifted to the chosen pCO value. Thus a kind of self-optimization process of the catalyst exists in which the surface adapts its structure dynamically to the reaction conditions to generate a maximum in catalytic activity. Facetting increases the surface energy and therefore a thermal reordering process exists which tends to keep the surface flat and whose rate determines the upper (p-dependent) T-limit for facetting (Sander and Imbihl, 1991). The facetting of a surface is of course a quite common phenomenon for many adsorbate systems but usually a thermodynamical driving force exists provided by a higher adsorption energy of the adsorbates on the restructured surface (Flytzani-Stephanopoulos et al., 1980; Wei and Phillips, 1992). Thermal desorption experiments with the facetted Pt(110) surface did up to now not reveal such an energetic stabilization of the facetted surface (Imbihl et al., 1988b). The fact that facetting of Pt(110) was only observed with an on-going surface reaction suggested the interpretation of the facetted surface as a dissipative structure, i.e. of a structure which is stabilized by the dynamics of the reaction and not by a hidden classic thermodynamical driving force. Two Monte Carlo simulations which were entirely based on the LH mechanism of catalytic CO oxidation and on the properties of the 1 × 1 1 × 2 surface phase transition supported this interpretation because they were able to reproduce practically all essential findings of the experiment (Imbihl et al., 1991; Monine and Pismen, 2001; Monine et al., 2004). As shown by Fig. 9.9 the microfacets which develop form a regular sawtooth-like arrangement of facets of equivalent orientation. Such a periodic structure with a spatial wavelength of 20 nm has also been found in experiment (Falta et al., 1990, 1992). As observed in experiment also the simulated facetted surface flattens slowly as the reacting gases are shut off (Falta et al., 1992; Ladas et al., 1988b).

364

R. Imbihl

Fig. 9.8. (a) Conditions for facetting and increase in catalytic activity due to facetting for catalytic CO oxidation on Pt(110). The full line indicates the rate curve for the non-faceted surface, while the dashed curve indicates the increase in catalytic activity after strong facetting of the surface. The different regions indicated on top of the rate curve all refer to the non-facetted Pt(110) surface. The arrow denotes the increase of the reaction rate during facetting. (From Imbihl, 1992.) (b) Induction period in the development of kinetic oscillations on Pt(110) due to reaction-induced facetting. (From Eiswirth and Ertl, 1986. Reproduced with permission of the authors.)

The reaction-induced facetting clarifies a number of at first sight rather puzzling observations with oscillations on Pt(110). The appearance of an induction period i.e. the gradual development of large amplitude oscillations shown in Fig. 9.8b at low temperature was traced back to facetting (Ladas et al., 1989). As the change in the surface structure slowly shifts the oscillatory window in parameter space the initially constant reaction rate starts to oscillate. In terms of dynamics reversible facetting introduces a second slower time scale besides the fast 1 × 1 1 × 2 phase transition leading to mixed-mode oscillations in an intermediate T-range (Eiswirth and Ertl, 1986). At low temperature faceting can cause a strong increase of the oscillation period up to a factor of 100 as compared to the unfacetted surface. Remarkably, also the degree of facetting or, more precisely, the inclination angle of the facets can undergo oscillatory variations parallel to the oscillations of the reaction rate (Ladas et al., 1988a). The consistency of the concept of reversible structural changes causing oscillatory behavior was tested with Pt(210), and with a Pt single crystal with cylindrical shape whose


365

Fig. 9.9. Monte Carlo simulation of the formation of regularly spaced facets during catalytic CO oxidation on a Pt(110) surface. (From Monine et al., 2004.) (a) Facetted surface of 150 × 150 atoms2 dimension with the elevated parts being marked by a brighter gray level. (b) Changes in the surface profile during the development of ¯ facetting. The surface sections at different time moments were taken in the [110]-direction as shown by the line ¯ in (a). The periodicity of 50–60 lattice units in the [110]-direction corresponds to 14–17 nm.

axis was oriented in the [001]-direction (Sander et al., 1991, 1992a). Its surface therefore contained all orientations of the [001]-zone. On Pt(100) no facetting was observed which appears plausible with regard to the fact that the 1×1 hex phase transition only involves a mass transport of 20% of the surface atoms and not of 50% as on Pt(110). Under reaction conditions the Pt(210) facets into (110) and (310) orientations and after this induction period oscillations develop (Sander et al. 1991, 1992b). The SPT mechanism can thus also be extended to the high-index planes of Pt but now a two-stage mechanism is required in which first the high-index plane facets into low-index planes on which then the SPT

366

R. Imbihl

mechanism sketched above can operate. Based on the similarity of the existence diagram for oscillations on Pt(210) with that of the Pd(110) system and on the observation of more strongly bonded oxygen states at high O2 exposure the phase transition model for Pt(210) has been disputed and instead a subsurface oxygen mechanism analogous to Pd(110) (see Section 9.4.2) was suggested (Berdau et al., 1997). However, no further evidence for the validity of this mechanism was provided. 9.4.1.5. The oxide model and the pressure gap One of the earliest models formulated to explain the occurrence of rate oscillations at atmospheric pressure in catalytic CO oxidation on Pt, Pd, Ir catalysts is the so-called oxide model proposed in a paper by Sales, Turner, and Maple (Sales et al., 1982). The model is based on the plausible hypotheses that at atmospheric pressure noble model catalysts can become oxidized and that these oxides are less reactive than the pure metal catalyst. The oxide model explains the rate oscillations as being due to a periodic oxidation and reduction of the surface. In the active state covered by oxygen the surface is oxidized thus becoming inactive but in the inactive state the surface is CO covered so that the oxide is reduced ad the active state is established again. For supported Pt catalysts an experimental proof was provided by in situ X-ray diffraction (Hartmann et al., 1994a). The analysis of the diffraction profiles showed that the small Pt particles of 1–1.5 nm diameter undergo a periodic oxidation and reduction parallel to the rate oscillations reaching a maximum degree of oxidation of 20–30% with the oxides PtO and Pt3 O4 . An analogous experimental proof for small Pd particles was obtained with in situ X-ray absorption spectroscopy (Ressler et al., 1997). In a series of recent experiments with an STM that allows an in situ imaging of the reacting surface at atmospheric pressure some of the basic assumptions of the oxide model were challenged by the group of Frenken (Hendriksen et al., 2005a, 2005b; Hendriksen and Frenken, 2002) These in situ experiments at high p were feasible by constructing a very small reaction cell of about 0.5 ml volume where only the STM tip was inside the reaction cell while the temperature sensitive piezo drive was outside the cell. When they exposed a Pt(110) sample at p = 1 bar to a CO/O2 gas atmosphere of varying composition the surface switched between a surface oxide of roughly 1 monolayer thickness and a metallic state (Hendriksen and Frenken, 2002). As shown in Fig. 9.10 they saw in fact an oxidation and reduction of the surface as proposed in the oxide model but the surprising result was that, opposite to what Sales at al. had assumed, the oxide was the reactive state of the surface and not the inactive state. The atomically thin Pt oxide layer observed in the STM experiments will have properties different from a bulk oxide but since these results were obtained at atmospheric pressure they describe the oscillatory behavior of massive Pt catalysts at high pressure. The Pt surface accordingly oscillates between a reactive oxidized state and a less reactive metallic state. This picture is clearly incomplete because what is missing is the feedback mechanism which causes the surface to switch between the two states. In addition, also the role of heat and mass transport in the very small reactor needs to be analyzed. Mainly from the quantum chemical side much effort has been devoted in recent years in showing that intermediate phases exist between chemisorbed oxygen and a bulk oxide (Reuter and Scheffler, 2002, 2004). DFT calculations of very thin oxide layers revealed


367

Fig. 9.10. Kinetic instabilities in catalytic CO oxidation on P(110) at 0.5 bar and T = 425 K. Shown are the STM images and the mass spectrometric signals recorded simultaneously at T = 425 K. Images (a), (b), the lower part of (c) and (e) show the metallic state in which the surface is mainly CO covered. Images (d) and the upper part of (c) display the oxide. The arrows mark the transitions between oxidized and reduced surface. Size of the STM images: 210 nm × 210 nm; flow rate: 3.0 ml/min. (From Hendriksen et al., 2005b. Reproduced with permission of the authors.)

the existence of reaction paths for CO oxidation with very low activation barriers (Gong et al., 2004). In a recent XPS study of the interaction of oxygen with a stepped Pt surface, it was shown that even at 10−7 mbar a 1D-oxide can develop along the step edges of Pt(322) (Wang et al., 2005). This 1D-oxide is quite reactive despite its oxide character which is evidenced by a shift in the Pt 4f level. Evidently the interaction of oxygen with noble metals is much more complex than a simple distinction between bulk oxide and chemisorbed oxygen would indicate. An additional complication arises with the so-called subsurface oxygen species. This species which has been observed in many pattern forming systems

368

R. Imbihl

with PEEM is, however, poorly characterized (Schaak and Imbihl, 2000; von Oertzen et al., 2000) (see Section 9.5.4.1). The previous results have shown that on Pt surfaces the oscillation mechanism switches from a phase transition/facetting mechanism at low pressure to an oxidation/reduction mechanism at high pressure. This switching in the mechanism with pressure is nicely illustrated by the fact that beyond 10−2 mbar also the structurally stable Pt(111) surface exhibits rate oscillations caused by an oxidation/reduction of the surface (Colen et al., 1998). A comparison of a Pt(100) single crystal with Pt supported on SiO2 in the mbar range demonstrated that kinetic oscillations in catalytic CO oxidation occur on the supported catalyst in a much wider parameter range than on Pt(100) – a phenomenon attributed to coupling effects between the particles and facet orientations (Lauterbach et al., 1999; Lele and Lauterbach, 2002). In contrast to the conceptually simple low pressure oscillations a full understanding of the oscillation mechanism at high pressure still needs to be established. 9.4.2. Catalytic CO oxidation on Pd surfaces Kinetic oscillations under low pressure conditions were found on Pd(110) but as demonstrated by Fig. 9.11b, in contrast to the rate oscillations on Pt single crystal surfaces a minimum pressure of 10−3 mbar is required for these oscillations (Ehsasi et al., 1989; Ladas et al., 1993; Ehsasi et al., 1993). Since clean Pd surfaces do no reconstruct the phase transition mechanism from the platinum surfaces cannot be operating here. Moreover, in contrast to Pt surfaces oxygen adsorption on Pd surfaces is only weakly structure sensitive. A clue of what could be the driving force for the oscillations on Pd(110) was provided by two observations. Firstly, it was shown that several monolayers of oxygen can penetrate into the subsurface region of Pd bulk and segregate back to the surface forming again chemisorbed oxygen (Ladas et al., 1989). The reversible formation of subsurface oxygen, Osub , described by Oad Osub does not alter the metallic state of Pd. Secondly, the occurrence of rate oscillations was found to be bound to a reversal of the usual clockwise hysteresis in reaction rate vs. pCO into a counterclockwise (ccw) hysteresis depicted in Fig. 9.11a. Based on the assumption that subsurface oxygen lowers the oxygen sticking coefficient an oscillation mechanism can be constructed in which a periodic filling and depletion of the subsurface oxygen reservoir modulates the catalytic activity. The same mechanism – deactivation by filling of the sub-O reservoir at low pCO and restoration of the active state by depletion of the sub-O reservoir at high pCO – explains the ccw hysteresis. A realistic mathematical model which was formulated with this mechanism could reproduce the essential results of the experiment (Bassett and Imbihl, 1990; Hartmann et al., 1994b). Studies of the system Pd(110)/O2 with a differentially pumped XPS which could be operated up to 1 mbar revealed that at 10−2 mbar Pd oxide, PdO, already forms at 400 K while at 10−4 mbar no PdO is detected (Bondzie et al., 2000). Furthermore, it was demonstrated that this oxide is quite reactive and that its formation and reduction proceeds relatively fast, on a time scale of minutes. A participation of PdO in the oscillation mechanism is therefore very likely. This conclusion found support by an in situ STM study of the system Pd(100)/CO + O2 conducted under atmospheric pressure by the group of


369

Fig. 9.11. Kinetic oscillations in catalytic CO oxidation on Pd(110). (a) Connection between the occurrence of a ccw hysteresis in the dependence of the reaction rate (pCO2 ) on pCO and the existence of kinetic oscillations in catalytic CO oxidation on Pd(110). (From Ladas et al., 1989). (b) Cross-shaped bifurcation diagram for catalytic CO oxidation on Pd(110) showing the various regions of monostability, bistability and oscillatory behavior in pCO , pO2 parameter space at T = 350 K. τ1 and τ2 represent the boundaries of the bistable region. (From Ehsasi et al., 1993. Reproduced with permission of the authors.)

Frenken (Hendriksen et al., 2005b). The results they obtained were very similar to those with Pt(110) discussed in the previous section. An oxide was formed which was more reactive than the metallic state and they observed autonomous oscillations between these states as depicted in Fig. 9.12. In contrast to Pt(110), however, a ccw hysteresis was observed which is of course reminiscent of the hysteresis in Fig. 9.11a. Again as with Pt(110) at high pressure, the question which feedback mechanisms drive the reaction between these two branches remains open. Apparently, over the whole pressure range from 10−3 mbar to atmospheric pressure the oscillations on Pd surfaces are driven by an oxidation reduction mechanism. To what extent Pd bulk oxides, surface oxides and/or subsurface oxygen contribute to the oscillatory cycles

370

R. Imbihl

Fig. 9.12. Reaction rate for the oxidized and metallic Pd(100) surface measured during catalytic CO oxidation at T = 408 K and pO2 = 1.25 bar. Oscillatory transitions between the two branches in the bistable region are indicated. (From Hendriksen et al., 2005b. Reproduced with permission of the authors.)

and what feedback mechanisms operate at atmospheric pressure are open questions to be clarified in future experiments. The surface oxides on Pd(100) and Pt(110) are not the only case where the oxide surface was proven to be more reactive than the metallic surface. Most well known is the extensively studied system Ru(0001)/CO + O2 where the increase in catalytic activity at higher pressure was shown to be due to the formation of the oxide RuO2 (Over et al., 2000; Over and Muhler, 2003; Reuter et al., 2004; Reuter and Scheffler, 2003). An oscillatory system where the periodic change between two oxides, CuO and Cu2 O is visible by eye is methanol oxidation over copper at atmospheric pressure (Werner et al., 1997). 9.4.3. Catalytic NO reduction 9.4.3.1. NO reduction on Pt and Rh surfaces Oscillatory behavior in catalytic NO reduction has been found with CO, NH3 , H2 and hydrocarbons as reducing agent on Pt, Rh, Pd, and Ir catalysts (Imbihl and Ertl, 1995; Janssen et al., 1997). With single crystal surfaces rate oscillations and wave patterns in catalytic NO reduction have been extensively studied on Pt and Rh surfaces, and to a smaller degree also on Ir surfaces (de Wolf and Nieuwenhuys, 2001). Studies with Rh and Pt field emitter tips were conducted with the NO + H2 reaction and the NO + NH3 reaction (de Wolf and Nieuwenhuys, 2001; Van Tol et al., 1992b; Voss and Kruse, 1998). The main difference between Pt and Rh surfaces is the higher efficiency of Rh surfaces in dissociating NO and this is of course the reason why Rh is present in the automotive catalytic converter (Egelhoff, 1982; Lox and Engler, 1999). On Pt surfaces NO dissociation is highly structure sensitive. Whereas about 60% of molecularly adsorbed NO dissociates upon heating on Pt(100), the corresponding numbers are only about 10% for Pt(110) and


371

about 1% for Pt(111) (Banholzer et al., 1983; Gohndrone and Masel, 1989; Gorte et al., 1981). Consequently rate oscillations were only found on Pt (100) and its vicinals but not on Pt(110) and Pt(111). All three NO reducing reactions, NO + CO, NO + H2 and NO + NH3 display rather similar dynamical behavior on Pt(100) (Imbihl, 1993; Imbihl and Ertl, 1995). These reaction systems can be considered as well understood as underlined by realistic mathematical simulations. The NO + H2 reaction has been investigated in detail on Rh(110) and Rh(111) but mainly in connection with chemical wave patterns. The excitation mechanism for wave patterns in Rh(110)/NO + H2 which with sufficiently strong gas-phase coupling can also generate rate oscillations will be presented below. 9.4.3.2. Pt(100)/NO + CO Kinetic oscillations in the NO + CO reaction were observed in the 10−4 mbar range on a Pt ribbon by Lintz et al. and on supported Pt catalysts at atmospheric pressure by Schüth and Wicke (Adlhoch et al., 1981; Schüth and Wicke, 1989). On Pt(100) oscillatory behavior was discovered by King et al. at extremely low pressure, in the 10−9 mbar range (Singh-Boparai and King, 1980). Oscillatory behavior in the NO + CO reaction at 10−6 mbar occurs in two separated T-windows as shown in Fig. 9.13 (Fink et al., 1991b; Veser and Imbihl, 1994a, 1994b). At 10−4 mbar and T = 470 K oscillatory behavior was found in a single broad existence range where in contrast to the conditions at 10−6 mbar CO is always in excess (Magtoto and Richardson, 1998). In the following we discuss the oscillations at 10−6 mbar which can be considered as well understood. In the lower-lying T-window no sustained oscillations exist but the surface displays chemical wave patterns which means that the surface is oscillating on a local scale (Veser and Imbihl, 1994b). By applying a small synchronizing T-jump rate oscillations can be excited which decay within a small number of cycles due to the lack of an efficient synchronization mechanism (Dath et al., 1992). In contrast to the oscillations in the upper T-window which are coupled to the 1 × 1 hex phase transition the oscillations in the lower-lying range proceed on a substrate which remains in a 1 × 1 state all the time. This observation shows that the surface phase transition of Pt(100) cannot be essential for the oscillation mechanism. In the upper T-window one observes sustained rate oscillations which proceed on a largely hex reconstructed surface. A defect free hex phase can be considered as more or less inert with respect to NO dissociation whereas the 1 × 1 phase is very active. The rate oscillations are coupled to the phase transition but in contrast to what one expects from the different activity of the two surface phases the maximum in the rate does not coincide with a minimum in hex intensity but with a maximum in hex intensity (Fink et al., 1991b; Hopkinson and King, 1993a). As demonstrated with FTIR measurements the structural defects which are generated as a consequence of the 20% different densities of surface Pt atoms in the surface phases determine to a large degree the catalytic activity in this T-range (Miners and Gardner, 2000). Neglecting N2 O formation, the mechanism of the NO + CO reaction can be described by the following sequence of steps (Fink et al., 1991b). (R1) CO + * COad (R2) NO + * NOad

372

R. Imbihl

Fig. 9.13. Hysteresis in the NO + CO reaction rate on Pt(100) due to the 1 × 1 hex surface phase transition. Shown are the variations in the reaction rate, in the work function, and in the LEED intensity of one of the hex beams which represents the degree of reconstruction. The hatched areas mark the existence ranges for oscillatory behavior during cooling down. (From Fink et al., 1991b.)

(R3) NOad + * → Nad + Oad (R4) 2Nad → N2 + 2* (R5) COad + Oad → CO2 + 2* The slow rate-determining step in this sequence is the dissociation of NO in R3 which requires an additional vacant site in order to proceed. Since more vacant sites are liberated in subsequent product forming steps than are consumed by the dissociation of NO an autocatalytic behavior with respect to the production of vacant sites results. This au-


373

tocatalysis explains the existence of a so-called “surface explosion”, i.e. the formation of extremely narrow product peaks (FWHM ≈ 2–5 K) in temperature programmed reaction (TPR) experiments with NO and CO coadsorbed on Pt(100) (Fink et al., 1991a; Lesley and Schmidt, 1985). The above scheme was modified by Makeev to include the reversible step R3 (Makeev, 1996). The autocatalysis in the above scheme is also the main driving force for the rate oscillations as was shown by formulating a three-variable model based on steps R1–R5 (Fink et al., 1991b; Imbihl et al., 1992). Structural transformations play no role in this model and therefore it describes the oscillations in the lower-lying T-window. A detailed bifurcation analysis has been conducted for this three-variable model (Imbihl et al., 1992). The surface phase transition is not the driving force for the rate oscillations but the SPT is essential for synchronizing the surface via gas-phase coupling resulting in a spatially uniformly oscillating surface. As shown in a molecular beam study of Pt(100)/CO by King et al. the kinetics of the CO-induced lifting of the hex reconstruction obeys a power law, hex )4.5 (Hopkinson et al., 1993; Hopkinson and King, 1993b, 1993a). As a rgrowth 1×1 ∼ (θCO consequence of the large exponent, small variations in pCO which arise due to mass balance in the reaction can very efficiently synchronize the oscillating system. By incorporating the adsorbate-driven 1 × 1 hex phase transition the three-variable model was extended to six variable-models which successfully reproduce the two existence ranges for oscillations and the stationary branches of the reaction but fail to simulate the T-dependence of the oscillation and the correct phase relationship between hex phase and reaction rate (Fink et al., 1991b; Hopkinson and King, 1993a). Apparently the role of structural defects which are dynamically created in the phase transition needs to be taken properly into account. Various lattice–gas models have been formulated to simulate the oscillatory behavior in the NO + CO reaction with and without participation of the 1 × 1 hex surface phase transition (Kortluke et al., 1998, 1999, Kortluke and von Niessen, 1996; Perera and Vicente, 2003; Tammaro and Evans, 1998c; Zhdanov, 1999a). With kinetic Monte Carlo simulations based upon the same mechanism it was demonstrated by Kevrekidis et al. that a bifurcation analysis can also be carried out with a lattice gas model yielding to a good approximation the same results as an ODE model (Makeev and Kevrekidis, 2004). 9.4.3.3. The NO + H2 and NO + NH3 reaction on Pt(100) In the NO + H2 reaction both N2 and NH3 are formed according to 2NO + 2H2 → N2 + 2H2 O 2NO + 5H2 → 2NH3 + 2H2 O and, at low pressure with a small yield, also N2 O appears as byproduct (Siera et al., 1991). In the NO + NH3 reaction the main product is N2 besides some N2 O: 6NO + 4NH3 → 5N2 + 6H2 O 8NO + 2NH3 → 5N2 O + 3H2 O Rate oscillations in the NO + NH3 reaction on Pt have been studied at high pressure, in the mbar range, by Takoudis and Schmidt on a Pt wire and by Katona and Somorjai on a Pt foil (Katona and Somorjai, 1992; Takoudis and Schmidt, 1983).

374

R. Imbihl

On Pt(100) the two reaction systems have been studied in the 10−6 and 10−5 mbar range. Both systems exhibit quite similar behavior, i.e. a rate hysteresis caused by the SPT and rate oscillations which take place on a largely hex reconstructed surface (Cobden et al., 1992, Lombardo et al., 1992b, 1992a; Siera et al., 1991; Slinko et al., 1992; Van Tol et al., 1992b). In contrast to the NO + CO reaction no oscillations are found on a surface which remains in a 1 × 1 state all the time. In situ LEED experiments showed that the oscillations in the NO + H2 reaction are coupled to the SPT (Lombardo et al., 1992a). The different steps in the NO + NH3 reaction could be made directly visible with PEEM showing a cyclic transformation of islands as depicted in Fig. 9.14 (Lombardo et al., 1993; Veser et al., 1992). 1 × 1-NOad /Oad islands imaged as dark area nucleate on a nearly adsorbate free hex surface but as they expand secondary fronts nucleate insides the dark islands which transform the surface into a 1 × 1 surface covered with NHx (x = 0–3) fragments which are imaged as bright area. The NHx fragments cannot stabilize the 1 × 1 surface which relaxes back into the initial state of the hex reconstructed surface. Mathematical modeling with ODE’s, PDE’s and lattice gas models could reproduce both, the integral behavior of the reaction rate as well as the cyclic transformation of the surface in NO + NH3 via subsequent reaction fronts (Gruyters et al., 1996; Lombardo et al., 1993; Makeev and Nieuwenhuys, 1998a, 1998b; Rafti et al., 2006; Uecker et al., 2003; Zhdanov, 1999b). 9.4.3.4. The NO + H2 reaction on Rh surfaces Rate oscillations in this reaction have been observed on Rh(110) (Heinze et al., 1995; Mertens and Imbihl, 1996b) as well as on Rh(111) and Rh(533) (Cobden et al., 1996, 1998; Janssen et al., 1995, 1994, 1996; Makeev et al., 1996) but the focus in the vast majority of these studies was on chemical wave patterns which have been exhaustively investigated on Rh(110) (see Section 9.5.3.2). Here we present the excitation mechanism for Rh(110) which with some smaller modifications probably also applies for Rh(111). The oscillation/excitation mechanism consists essentially of a switching between an oxygen covered and a nitrogen covered surface. The crucial element in the excitation cycle is the repulsion between chemisorbed oxygen, Oad , and chemisorbed nitrogen, Nad , leading to a displacement of the weaker bonded nitrogen by the more strongly bonded oxygen (Comelli et al., 1992; Mertens et al., 1997). In thermal desorption (TD) spectra this repulsion shows up by a downward shift of the N2 peak by nearly 100 K in the presence of coadsorbed oxygen as compared to a TDS peak of a pure nitrogen adlayer on Rh(110). On Rh(100) nitrogen is bonded much more strongly than on Rh(110) and Rh(111) and consequently no rate oscillations or wave patterns have been found on this orientation. With scanning photoelectron spectroscopy (SPEM) as chemically resolving in situ method the concentration profile of a propagating pulse in the NO + H2 reaction can be obtained and the result is displayed in Fig. 9.15a. The excitation of a propagating pulse on the oxygen covered surface can be described by the following sequence of steps shown schematically in Fig. 9.15b. (Mertens et al., 1997; Schaak et al., 1999a): (i) Starting with a completely oxygen covered surface hydrogen can only adsorb at some defects. Adsorbed oxygen is reactively removed by hydrogen and the auto-


375

Fig. 9.14. (a) PEEM image showing the formation of fluctuating adsorbate islands during kinetic oscillations in the NO+NH3 reaction on Pt(100). Experimental conditions: pNO = 1.3×10−6 mbar, pNH3 = 2.1×10−6 mbar and T = 438 K. The different gray levels are assigned to the following phases: medium gray – adsorbate free hex surface, dark area – NOad /Oad covered 1 × 1 surface, bright area – NHx (x = 1–3) covered 1 × 1 area. (From Veser et al., 1992.) (b) The scheme shows the mechanistic interpretation of the cyclic transformations the three identifiable surface phases undergo in PEEM.

376

R. Imbihl

Fig. 9.15. Mechanism of pulse propagation in the NO + H2 reaction on Rh(110). (a) Concentration profile of a propagating pulse as measured by SPEM. The pulse is moving from right to left. The panel in the middle indicates the corresponding overlayers as observed with LEED. The resting state is a surface fully covered by oxygen, in the excited state the surface is nitrogen covered and the refractory tail is given by a mixed N,O-overlayer. Experimental conditions: T = 530 K, pNO = 1.7 × 10−7 mbar, pH2 = 6.4 × 10−7 mbar. (From Schaak et al., 1999a.) (b) Scheme of excitation mechanism (see text.)


377

catalytic increase in the number of vacant adsorption sites leads to a propagating reaction front. (ii) On the largely adsorbate free surface thus created hydrogen and NO can adsorb and dissociate uninhibitedly. Since oxygen is removed through reaction with hydrogen nitrogen accumulates. (iii) On the nitrogen covered surface NO can still adsorb and dissociate. The oxygen coverage thus increases leading through repulsive interactions to the destabilization of nitrogen which desorbs. The initial state of an oxygen covered surface is finally restored. The Rh(110) surface is characterized by an enormous structural variability exhibiting a whole zoo of different N,O-induced reconstructions (Comelli et al., 1998; Kiskinova, 1996). By selecting a diffraction beam which belongs to an ordered adsorbate layer the different stages in the excitation cycle of a pulse (see Fig. 9.3b) can be made directly visible as LEEM images (Schmidt et al., 2000). 9.4.3.5. NO reduction on Ir surfaces The clean surfaces Ir(100) and Ir(110) exhibit reconstructions similar to Pt(100) and Pt(110) but the analogy to Pt surfaces only holds partially because the details of the surface phase transition are different and because the Ir surfaces have a much higher tendency for oxide formation than Pt surfaces. Oscillatory behavior in the NO + H2 reaction was found at 10−6 mbar on Ir field emitter tips, on Ir(110), Ir(510) and Ir(210) (de Wolf and Nieuwenhuys, 2001; de Wolf et al., 1998). On Ir(110) rate oscillations were also observed in the catalytic reduction of N2 O with H2 or CO (Carabineiro et al., 2002). From in situ XPS measurements of Ir(110)/NO + H2 which showed that the rate hystereses are associated with large hystereses in the N- and O coverage, it was concluded that the oscillatory mechanism is similar to Rh(110)/NO + H2 where the mutual displacement of oxygen and nitrogen adlayers plays the key role (de Wolf et al., 2001).

9.5. Chemical wave patterns 9.5.1. Basic considerations 9.5.1.1. Differences to patterns in the liquid phase Many of the wave patterns found on catalytic surfaces like fronts, spiral waves, etc. can be explained in terms of simple two-variable activator-inhibitor models (see Section 9.3.3) (Kapral and Showalter, 1994). Except for an elliptical deformation by diffusion on anisotropic surfaces, in principle, no difference to patterns in the liquid phase exists for these patterns. The properties of surfaces give, however, also rise to new types of patterns which have no counterpart in the liquid phase. Leaving aside heat waves which arise at higher pressure due to the exothermicity of reactions we can distinguish between the following new effects introduced by surfaces: (i) Global coupling. As already discussed at the beginning (see Section 9.2) the partial pressure variations which arise due to mass balance in the reaction introduce a global coupling which leads to new types of patterns such as standing waves,

378

R. Imbihl

cellular patterns etc. in which the reaction is synchronized over the whole single crystal surface (Eiswirth et al., 1989; Mertens et al., 1994). (ii) Anisotropy and state-dependent anisotropy. With exception of the isotropic (100) and (111) surfaces of fcc metals diffusion on surfaces is in general anisotropic. Due to adsorbate-induced reconstructions or energetic interactions of the adparticles the anisotropy may vary depending on the coverage and thus be state-dependent. Novel type of patterns like rectangular patterns or traveling wave fragments are possible in such media (Gottschalk et al., 1994; Mikhailov, 1994). (iii) Energetic interactions between adparticles. With finite coverages these interactions are in principle always present on surfaces. They lead to coverage dependent kinetic constants, island formation and non-Fickian diffusion. Nearly atomically sharp interfaces in reaction fronts, the formation of stationary or traveling nanosized islands and reactive phase separation are some of the effects seen in catalytic reactions. (iv) Reaction-induced substrate modification by structural or chemical changes. All catalytic reactions cause in principle a substrate modification. Depending on the degree of restructuring the surface may just be roughened or facetting may occur leading to real morphological changes. Chemical changes may involve penetration of chemisorbed oxygen into deeper layers of the metal or oxide phases may form. New degrees of freedom and an additional slow time scale are thus introduced leading to memory effects and complex patterns. In the liquid phase all substances diffuse about equally fast but quite in contrast diffusion constants for surface diffusion vary over many orders of magnitude. This property has important consequences for one particular type of chemical wave patterns which relies on differences in the diffusivity, the so-called Turing patterns (Kapral and Showalter, 1994; Mikhailov, 1991; Murray, 1989). Stationary concentration patterns in reaction–diffusion systems were predicted more than 50 years ago in a landmark paper by A.M. Turing (Turing, 1952) but it was only about 15 years ago that these structures were finally realized in experiments, in liquid phase with a variant of the BZ reaction, the CIMA (=(chlorite, iodide, malonic acid) reaction and on catalytic surfaces by the microfacetting of a Pt(110) surface during catalytic CO oxidation (see Section 9.4.1.4) (Castets et al., 1990; Falta et al., 1990). The reason why it took so long to the experimental realization of Turing patterns lies in the mechanism for a Turing pattern which requires a large difference in the diffusivity of the activator and inhibitor species. In the liquid phase where all substances diffuse about equally fast this condition was realized by adding starch to the iodine containing CIMA solution. Alternatively, microemulsions might be used to generate large differences in diffusivity as demonstrated in recent experiments (Vanag and Epstein, 2004). Since diffusion constants on surfaces vary over many orders of magnitude surface reactions should represent ideal candidates for discovering Turing structures. On surfaces, however, energetic interactions between the diffusing adspecies are present and these interactions, as will be shown below, provide a competing mechanism for the formation of stationary patterns. 9.5.1.2. Spatially resolving in situ methods While in the first study of chemical wave patterns in Pt(100)/CO + O2 spatial resolution was achieved by simply deflecting the electron beam of a conventional LEED sys-


379

Table 9.1 Spatially resolving in situ techniques for imaging reaction dynamics Method

p-range [mbar]

Resolution

Contrast mechanism

Information

PEEM

UHV-10−3

0.1–1 µm

local difference in work function

reactions dynamics

LEEM

UHV-10−5

15 nm

diffraction contrast as main mechanism

surface topography, ordered structures

MEM

UHV-10−5

30 nm

reflectivity differences due to surface topography and local work function variation

surface topography, reaction dynamics

SPEM

UHV-10−6

0.1 µm

local abundance of elements

elemental distribution, chemical status

EMSI/RAM

UHV-atm

>1 µm

differences in optical properties

concentration in surface and near-surface region

STM

UHV-10−2 100 -atm

atomic

density of states near EF

atomic structure

tem across the surface by means of a pair of Helmholtz coils (Imbihl et al., 1986), in the past two decades a number of powerful methods have been developed which allow an in situ imaging of chemical wave patterns with high resolution (Günther et al., 2002a; Rotermund, 1997). The methods summarized in Table 9.1 cover different length scales, are applicable in different pressure ranges and, most importantly, they always provide only part of the information which is required to obtain a complete picture of the surface processes in a pattern forming system. For example, the most widely used technique, PEEM, images primarily the work function and is capable of imaging wave patterns but it provides no or only rather indirect information about the chemical identity of the imaged species and about structural or chemical modifications of the substrate. In particular, for studying chemically complex systems, complementary information from different techniques is required. The following methods summarized in Table 9.1 have been used for observing chemical wave patterns: • Photoelectron emission microscopy (PEEM) – a technique which was developed already in the 1930s – is based on the photoelectronic effect (Günther et al., 2002a). If one illuminates the sample with photons whose energy is just above the threshold for excitation of photoelectrons then the yield of photoelectrons depends sensitively on the local work function. The instrument used in most laboratory studies today was developed by W. Engel and yields a resolution of about 0.1–1 µm (Engel et al., 1991). • Scanning photoelectron microscopy (SPEM). Chemical information with a spatial resolution about 0.1 µm and a temporal resolution of a few seconds sufficient to image dynamic processes is provided by SPEM (Günther et al., 2002a; Kiskinova et al., 1999). This technique utilizes photons in the energy range of a few hundred eV so that photoelectrons from (element specific) core levels are ejected.

380

R. Imbihl

•

•

•

•

Photons from a synchrotron source are focused by means of a zone plate into a very narrow spot of ≈ 0.1 µm diameter. By rastering the sample while tuning the energy of a hemispherical energy analyzer to a specific core level one obtains a map of the elemental distribution on the surface. Low energy electron microscopy (LEEM)/mirror electron microscopy (MEM). A high spatial resolution down to ≈10–15 nm can be achieved with low energy electron microscopy (LEEM) (Bauer, 1994; Schmidt et al., 1998). In LEEM one of the diffracted beams of a LEED experiment with energies of ≈ 0–200 eV is used for imaging the surface. Since the contrast in LEEM is primarily based on diffraction, surface topographical features, ordered adsorbate layers and substrate modifications can be imaged. By tuning the energy of the incoming electrons to zero a variant of electron microscopy, mirror electron microscopy (MEM) is obtained whose main contrast mechanism, similar to PEEM, also relies on work function differences but, in addition, also on topographical differences (Swiech et al., 1993). In combination with a synchrotron source the detection part of the LEEM instrument can be used to generate PEEM images with photoelectrons from core levels (XPEEM) (Günther et al., 2002a). These so-called spectromicroscopic techniques combine the imaging capabilities of a microscope with the option to obtain local spectroscopic information. New developments (e.g. the so-called SMART project at BESSY with a resolution of 1–2 nm) are underway pushing the frontiers further towards higher resolution and better spectroscopic information. Scanning tunneling microscopy (STM). With STM atomic resolution with nearly video rates (25 frames/s) can be achieved today but the method suffers from two inherent disadvantages. Firstly, chemical waves in most cases involve a much larger length scale which is in the macroscopic range so that atomic scale imaging is not very useful. Secondly, the tip shields part of the surface from the incoming gas particles so that the reacting system is locally perturbed. Despite these difficulties in a spectacular experiment discussed below atomic scale reaction fronts could be imaged (see Section 9.5.6.1) (Sachs et al., 2001; Völkening et al., 1999). As shown in the previous section by constructing a specially designed STM attached to a reaction cell an in situ imaging of the reacting surface at atmospheric pressure had been made possible (Hendriksen and Frenken, 2002). Field emission microscopy (FEM)/field ion microscopy (FIM). Close to atomic resolution and atomic resolution can be achieved with two techniques invented by E.W. Müller more than 50 years ago, field emission microscopy (FEM) and field ion microscopy (FIM) (Müller and Tsong, 1969). Both techniques are practically obsolete today and with exception of some phenomenological studies of oscillatory systems and some fluctuation studies these techniques play little role in the study of chemical wave patterns (see Section 9.5.6.3). Ellipsometry for surface imaging (EMSI)/reflection anisotropy microscopy (RAM). At pressures beyond 10−3 mbar the range of available analytical tools for an in situ characterization of the catalytic surface shrinks drastically and the methods of choice are based on the use of photons. The change in the polarization of light upon reflection from a surface can be utilized to image patterns with submonolayer sensitivity and a spatial resolution of about 1 µm as was demonstrated


381

in ellipsometric studies of catalytic CO oxidation on Pt(110) (Rotermund, 1997; Rotermund et al., 1995). The EMSI method and its variant RAM which is more sensitive to substrate changes have the advantage that practically no pressure limit exists for their application (Dicke et al., 2000). The disadvantage is that ellipsometric measurements yield no direct surface information but require simulations with a layer model in order to extract surface quantities from the measured polarization angles. Furthermore, at high pressure unpolarized stray light from rough surfaces will limit potential applications. At high pressure the reactions are in general no longer isothermal and one may observe heat waves traveling across the surfaces of catalyst foils, wires or supported catalysts. Such heat waves can be followed with infrared imaging thermography allowing to resolve temperature differences of about 0.1 K (Luss, 1992; Luss and Sheintuch, 2005; Yamamoto et al., 1995). 9.5.1.3. Survey of experimentally studied systems Practically all studies of chemical wave pattern focus on simple reactions from the automotive catalytic converter, catalytic CO oxidation, catalytic NO reduction and the O2 + H2 reaction which were investigated on Pt, Rh, Ir and Pd surfaces. The majority of investigations even concentrated on one single reaction system, catalytic CO oxidation on Pt(110). This system exhibits firstly, an enormous variety of different patterns and is, secondly, chemically very simple so that mathematical modeling successfully reproduced nearly all of the observed phenomena. Spiral waves, target patterns, soliton-like behavior, cellular patterns, Turing structures, turbulence and complex patterns on a locally roughened surface have all been found on Pt(110) (Eiswirth and Ertl, 1994; Imbihl and Ertl, 1995; Jakubith et al., 1990). For comparison, catalytic CO oxidation on Pt(100) was a much less attractive target of studies since due to the absence of global coupling irregular wave patterns prevailed (Imbihl et al., 1986; Rotermund et al., 1989). Due to a number of rather peculiar wave patterns the system Rh(110)/NO + H2 has been studied exhaustively: rectangularly shaped spiral wave and target patterns, and traveling wave fragments were among the patterns found there (Mertens and Imbihl, 1994, 1996a). In the O2 + H2 reaction which was studied on Pt(111), Rh(110) and Rh(111) only bistable behavior was observed, but subsurface oxygen formation, parameter-dependent anisotropy of front propagation and atomic scale experiments added interesting features to these systems. 9.5.1.4. Mathematical modeling In the simulation of chemical wave patterns we can distinguish between (i) realistic models which refer to a specific system, and are based on an experimentally determined mechanism and use experimentally determined reaction constants, (ii) general models like the FitzHugh–Nagumo model which use a simplified general reaction scheme and do not refer to a specific reaction systems (see Section 9.3.3) and (iii) universal models like the complex Ginzburg–Landau (CGL) equation which describe the behavior in the vicinity of a certain bifurcation and are therefore applicable to all physical and chemical systems which exhibit this particular type of bifurcation.

382

R. Imbihl

Realistic models are available for practically all pattern forming reactions studied on single crystal surfaces. A problem arises typically with the values for the diffusion constants because the experimental diffusion data are very limited and depending on the experimental method the values may vary over several orders of magnitude for the same adsorbate system (Bauer and Allen, 1995). A possible way out of this dilemma are DFT calculations which can provide the diffusion parameters with good enough accuracy to be used in simulations. Energetic interactions between adparticles which influence all kinetic constants and, in particular, also diffusion are typically neglected. Diffusion is usually treated as Fickian diffusion. Diffusion on surfaces, however, is insofar different from diffusion in the liquid phase since always a vacant site is required before a particle can hop to a neighboring site. Coadsorbates therefore influence the diffusion of adsorbates not only through energetic interactions but also through pure site blocking. Corresponding correction terms for Fickian diffusion have been suggested by Evans et al. (Evans et al., 2002; Tammaro and Evans, 1998a, 1998b). Universal models. It was shown that in the vicinity of a Hopf bifurcation the behavior of an oscillatory medium can be represented by the so-called complex Ginzburg–Landau equation (Aranson and Kramer, 2002; Cross and Hohenberg, 1993; Kuramoto, 1984; Pismen, 2006): ∂A = (1 − iω)A − (1 + iβ)|A|2 A + (1 + iε)∇ 2 A ∂t The amplitude A(r, t) is complex and oscillations are represented by motion on a unit circle in the complex plane. ω is the oscillation frequency, β and ε are real and denote the non-linear shift of the oscillation frequency and the dispersion, respectively. This equation describes an ensemble of small amplitude harmonic oscillators which are coupled via diffusion. The equation is thus universally valid. For any real system as long as it is close enough to a Hopf bifurcation, at least, in principle, a mathematical transformation can be found which maps the limit cycle of the real system onto the unit circle of the CGLE. This CGL equation which has been extensively analyzed in the literature proved to be highly useful for the analysis of spatio-temporal chaos and for the effects of parameter forcing. Modified by an appropriate term the CGLE allowed a systematic analysis of the different solutions that arise in oscillatory surface reactions with global coupling (Battogtokh and Mikhailov, 1996; Mertens et al., 1994). The simulations of chemical wave patterns on surfaces are typically conducted by integration of PDE’s and not with lattice gas models for two simple reasons: (i) Even with very fast computers the hopping rates of adsorbed adparticles would still be far below the values required to simulate realistic diffusion of mobile adsorbates. (ii) If we interpret a lattice site as an atomic adsorption site, then a simulation with a 1000×1000 lattice would correspond roughly to an area of 100 nm×100 nm which is by a factor of 102 –104 below the macroscopic length scale of wave patterns which typically have a dimension in a range from 10 to 1000 µm. The vastly different length scales from microscopic adsorbate islands to macroscopic wave patterns represent in fact a challenging problem in modeling and various concepts have been developed to solve this problem (e.g. the two-tier-model (Pismen et al., 1998), the hybrid model (Liu and Evans, 2005) etc.).


383

9.5.2. Simple chemical wave patterns 9.5.2.1. Bistability and front nucleation Bistability is frequently encountered in catalytic reactions because a strong inhibition by one adsorbed reactant on the adsorption of the other reactant suffices to create two stable states, a reactive one in which both reactants can adsorb and react and an unreactive one in which a high adsorbate coverage inhibits the adsorption of one of the reactants (Zhdanov and Kasemo, 1994b). In catalytic CO oxidation on Pt the reactive state is an oxygen covered or nearly adsorbate free surface and the unreactive state is a surface fully covered with CO (Bär et al., 1992b). In the O2 + H2 reaction on Rh surfaces the poisoned state is given by the oxygen covered surface whereas in the reactive state the surface is nearly adsorbate free (Mertens and Imbihl, 1995). In a bistable system transitions between the two stable states can be initiated by reaction fronts (Mikhailov, 1991). Both states can coexist only at the equistability point. At this point the interface has zero velocity and for any other parameter value the less stable state is pushed out by the more stable state via a propagating reaction front which moves with constant velocity and a constant profile across the surface (see Fig. 9.3b). The front √ velocity thus changes sign at the equistability point. The front width lc is given by lc ∼ DX τ with τ being the life time of a diffusing particle X in the front region and DX is the diffusion constant. An experimental study of the bistable system Pt(111)/CO + O2 has been conducted by Berdau et al with PEEM (Berdau et al., 1999); a theoretical analysis with PDE’s was carried out by Bär et al. and by Zhdanov et al. (Bär et al., 1992b; Zhdanov and Kasemo, 1994a); lattice gas simulations have been conducted by Evans et al. (2002) The front velocity in CO oxidation on Pt is determined by the diffusion rate of CO since the more strongly bonded oxygen can be considered as almost immobile compared to CO. Such qualitative arguments which hold well for CO oxidation on Pt have to be considered with some care as was demonstrated with the bistable O2 + H2 reaction on Rh(110) (Mertens and Imbihl, 1995). On an anisotropic surface a circular reaction front becomes elliptically deformed and the ratio of the diffusion constants yields the velocity ratio, √ vx : vy ∼ Dx : Dy . A front elliptically deformed in the [110]-direction, i.e. the direction along the (110) troughs was observed on Rh(110) but with increasing temperature the anisotropy changed to a front which was elliptically elongated along the [001]direction. The explanation was that at elevated temperatures the diffusion of oxygen can no longer be neglected because using Fick’s first law the high coverage and steep gradient of oxygen in the front region compensates partially the much higher diffusivity of hydrogen which exhibits a quite low coverage and therefore a smooth gradient. The switching of the velocity-limiting adsorbate is associated with a change in anisotropy thus explaining the experimental results (Makeev and Imbihl, 2000). Due to adsorbate interactions which lead to ordered adsorbate structures even simple reaction fronts can be quite rich in detail. As demonstrated in a LEEM study of the titration of a CO saturated Pt(100) surface with oxygen the front profile exhibits a sequence of ordered CO overlayers (Swiech et al., 1994; Tammaro et al., 1998). Such a substructure not only reflects a cross section through the phase diagram Pt(100)/CO but it also contains the coverage dependence of the CO diffusion constant which can be extracted from a front

384

R. Imbihl

profile with a so-called Metano analysis. Besides the coverage also the surface topography influences the diffusion rate. CO diffusion on Pt, for example, slows down across atomic steps. On stepped surfaces reaction fronts therefore do not propagate with constant speed but display a kind of stop-and-go motion as demonstrated in a MEM/LEEM study of Pt(100)/CO + O2 (Rausenberger et al., 1993). Based on elementary considerations, it can be shown that the curvature dependence of front velocity, cf , obeys the relation cf = cf0 ± D R (Mikhailov, 1991). In this formula 0 cf represents the velocity of a plane wave front, D the diffusion constant and 1/R the curvature with the ± sign depending on whether the front is concave (+) or convex (−). This relation basically ensures the stability of planar reaction fronts against perturbations of its shape. For the macroscopic patterns observed with PEEM the curvature effect influences the front velocity only slightly. From the above equation a very elementary property can be deduced which is crucial for understanding pattern formation on catalytic surfaces. Since the front velocity of an expanding circular nucleus shrinks with decreasing radius R because the curvature 1/R increases, a critical size for front nucleation exists defined by cf (Rcrit ) = 0. This yields Rcrit = D/cf0 and with typical values for Pt(111)/CO + O2 , i.e. DCO = 10−8 cm2 s−1 and cf0 = 1 µm/s, one calculates Rcrit ≈ 1 µm (Bär et al., 1992b). This means that only macroscopic defects like scratches or large impurities can ignite front nucleation but not microscopic defects such as atomic steps. Experiments show that chemical waves typically nucleate at macroscopic defects given, for example, by scratches or large impurities. Only in very few cases homogeneous nucleation due to statistical fluctuations in the density of adparticles has been made responsible for initiating fronts (Reichert et al., 2001). With the high resolution of LEEM/MEM it could be shown for CO oxidation on Pt(110) that the nucleation of a chemical wave at a surface impurity is actually not a simple process but may involve localized excitations at the interface between defect and surroundings, breathing modes of the nucleus, and unsuccessful nucleation events where the wave cannot escape from the neighborhood of the defect (Wei et al., 2006, 2007). It was shown that not only the defect itself but also the modified surface properties in the vicinity of the defect have to be taken into account. 9.5.2.2. Target patterns, spiral waves Pulses in excitable and oscillatory 2D-systems give rise to target patterns and spiral waves (Mikhailov, 1991). Spiral waves which are ubiquitous in such systems can develop either by nucleation at a defect or, more simply, they are formed as a planar chemical wave breaks up into two parts, through collision with a large defect, for example. The free ends curl in forming a pair of counterrotating spiral waves. Accordingly, spirals can be considered as a kind of defect in a chemical wave pattern and a certain type of turbulence in oscillatory media is in fact characterized by a large density of spiral wave fragments. In simulations by Bär and Eiswirth it was shown that under certain conditions spiral waves may become unstable (Bär and Eiswirth, 1993). The spirals break up into small fragments until finally a state of chemical turbulence is reached. Spiral waves have attracted considerable theoretical interest but it turned out that most of their properties can already be well described within the so-called kinematic approximation worked out by Mikhailov and by Tyson and Keener which neglects the internal processes


385

in a pulse and just considers its propagation properties (Mikhailov, 1991; Mikhailov and Zykov, 1991; Tyson and Keener, 1988). Spiral waves and target patterns are found in surface reactions displaying excitable and/or oscillatory behavior, i.e. in Pt(110)/CO + O2 (Jakubith et al., 1990; Nettesheim et al., 1993), in the NO + CO reaction on Pt(100) and its vicinals (Graham et al., 1995b; Veser and Imbihl, 1992, 1994b, 1994a), in Pt(100)/NH3 + NO (Veser et al., 1992) and in the NO + H2 reaction on Rh(110) (Mertens and Imbihl, 1994, 1996a, 1996b), Rh(533) (Schaak et al., 1999b) and Rh(111) (Janssen et al., 1996; Schaak and Imbihl, 2002). For simulating the wave patterns the ODE’s describing the oscillatory/excitable point model just had to be complemented by corresponding diffusion terms. As demonstrated by extensive mathematical modeling with a term added for CO diffusion the 3-variable (KEE) model for Pt(110)/CO + O2 could rather well reproduce the simple wave patterns found in this system. In catalytic CO oxidation on Pt(110) in the parameter range adjacent to the oscillatory range both CO pulses traveling on the oxygen covered surface and oxygen pulses propagating on a CO covered surface are found (Bär et al., 1994; Falcke et al., 1992). A particular kind of bistability called dynamic bistability is sometimes encountered in systems where one also observes oscillatory/excitable behavior. This type of behavior was observed on Pt(110)/CO + O2 (Bär et al., 1995b) and Rh(110)/NO + H2 (Gottschalk et al., 1994). In dynamic bistability each of the two states is stable in a point model, i.e. in a system described by ODE’s but adding diffusion renders the two states unstable against a diffusive perturbation. Pulses and spiral waves can develop in such a system but in contrast to a truly excitable system the spiral waves exhibit a less regular shape (Bär et al., 1995b). For combining the macroscopic length scale of the wave patterns with the microscopic scale of adsorbate islands a two-tier model for catalytic CO oxidation on Pt(110) was developed by Pismen et al. (Pismen et al., 1998; Pismen and Rubinstein, 1999). As an extension of the KEE model Verdasca et al. replaced Fickian diffusion of CO by a model in which the different binding strength of CO to the two surface phases governs the mass transport of CO (Verdasca et al., 2002). On surfaces spiral waves are typically pinned to defects the diameter of which determines the rotational period. As demonstrated by the PEEM images in Fig. 9.16 the differently sized defects on the Pt(110) surface lead to spirals with different rotational period (Nettesheim et al., 1993). The elongation of the spirals in the [110]-direction is due to anisotropic CO diffusion which is fast along the [110]-oriented troughs. Both, single-armed as well as multi-armed spirals have been observed on surfaces. The wave trains occurring in the lower-lying temperature range for oscillatory behavior in the NO + CO reaction on Pt(100)-1 × 1 were reproduced by complementing the threevariable system for this reaction by terms for CO and NO diffusion (Evans et al., 1992; Tammaro and Evans, 1998c). Due to averaging effects these wave trains correspond to a stationary reaction rate and a stimulating small T-jump is required to generate rate oscillations which then decay rapidly as target patterns again spread out over the uniformly reacting surface. In the NO + CO reaction on Pt(100) the breaking up of wave trains in spiral fragments was used to image surface defects which occur during the plastic deformation of a Pt(100) single crystal (Hartmann et al., 2004).

386

R. Imbihl

Fig. 9.16. PEEM images demonstrating the temporal evolution of spirals with strongly different rotation periods and wave-lengths during CO oxidation on Pt(110): T = 448 K, pCO = 4.3 × 10−5 mbar and pO2 = 4 × 10−4 mbar. The spiral with the largest wavelength rotates around a core of 25 × 14 µm2 , while the size of the core region for the fast rotating spiral visible in the foreground is only 5 × 3 µm2 . (From Nettesheim et al., 1993. Reproduced with permission of the authors.)

Various lattice–gas models have been proposed to simulate the oscillatory kinetics of the NO + CO reaction on Pt(100) and the formation of reaction fronts caused by the autocatalytic decomposition of a layer of coadsorbed NO + CO (Perera and Vicente, 2003; Tammaro and Evans, 1998c, 1998a; Zhdanov, 1999a). 9.5.3. Patterns modified by surfaces 9.5.3.1. Global coupling The partial pressure variations of a reactant gas i which arise due to a catalytic reaction in a well mixed flow reactor can be expressed as p 0 − pi dp = i + pi∗ (rdes − rad ) dt τ


387

Fig. 9.17. Standing wave patterns accompanying rapid kinetic oscillations in CO oxidation on a Pt(110) surface. The PEEM images which represent an area of 0.3 × 0.5 mm2 were recorded in intervals of 0.5 s. Experimental conditions: T = 550 K, pO2 = 4.1 × 10−4 mbar, pCO = 1.75 × 10−4 mbar. (From Jakubith et al., 1990.) Reproduced with permission of the authors.

where pi0 denotes the partial pressure without reaction, τ the reactor residence time, pi∗ the pressure increase caused by desorption of one monolayer (ML) of the gas i and rdes and rad represent the desorption and adsorption rate, respectively, of the gas i in ML/s. The reactor residence time τ is determined by the pumping rate S and the chamber volume V by τ = V /S. For a typical UHV experiment with the oscillatory catalytic CO oxidation on Pt(110) the amplitude of the pCO variations is about 1%. Global coupling does not always lead to a synchronized oscillating surface but depending on the kind of feedback that exists between gas-phase and the surface reaction the homogeneous state can also be destabilized leading to (Ising-like) domains which oscillate with opposite phases (Sheintuch, 1981). A systematic investigation of the different cases has been conducted by Mikhailov et al. (Mertens et al., 1994). If the system without global coupling exhibits turbulence, then global coupling may suppress turbulence leading to the formation of standing waves or modify the turbulence such that for example spatiotemporal intermittency results. Simulations with the PDE model for Pt(100)/NO + NH3 demonstrated that very fast diffusion of the reactants establishes a kind of global coupling, i.e. it leads to the same type of wave patterns (Uecker, 2005). A striking example of synchronization via the gas-phase are the standing waves displayed in Fig. 9.17 which develop in catalytic CO oxidation on Pt(110) at T > 500 K (Jakubith et al., 1990). At lower temperature, due to gas phase coupling cellular patterns

388

R. Imbihl

Fig. 9.18. Bifurcation diagram for the NO + CO reaction on Pt(100) showing the occurrence of sustained rate oscillations at elevated temperature (range I in Fig. 9.13). The diagram shows the range where the CO2 production rCO2 is stationary (filled squares) and where the reaction rate exhibits kinetic oscillations. The oscillation amplitude, i.e. the upper and lower turning points are marked by open squares. The inset shows the Feigenbaum scenario which is found at the upper temperature boundary of the oscillatory range. The different abbreviations on top refer to different types of spatio-temporal patterns which can be observed below the oscillatory range. Experimental conditions: pNO = pCO = 4 × 10−6 mbar. (From Veser et al., 1993.)

appear as demonstrated in a LEEM/MEM study (Rose et al., 1996). Theoretically, the influence of gas phase coupling in catalytic CO oxidation on Pt(110) was investigated by Falcke et al. who used the 3-variable KEE-model (Falcke and Engel, 1994b, 1994a; Falcke et al., 1995). In the upper T-window of oscillations in the NO + CO reaction on Pt(100) the surface exhibits a synchronized oscillating state in which the rate oscillations are coupled to the 1 × 1 hex surface phase transition (Veser and Imbihl, 1994a). As depicted in Fig. 9.18 one finds at both ends of the oscillatory window interesting transitions to a stationary rate caused by the break-down of global coupling. At the lower T-boundary a discontinuous transition to a stationary rate takes place initiated by the formation of a target pattern as defects become supercritical and start to emit waves (Veser et al., 1993). At the upper Tboundary a stationary rate is reached via a Feigenbaum scenario, i.e. a sequence of perioddoublings leading to small amplitude chaotic oscillations which then die out (Veser and Imbihl, 1994a). A similar break-down of global coupling associated with a transition from a spatially uniformly oscillating state to chemical wave patterns and, finally, spatio-temporal chaos (turbulence) is observed in the NO + NH3 reaction on Pt(100) (Veser et al., 1992). 9.5.3.2. State-dependent anisotropy On anisotropic surfaces one find elliptically distorted wave patterns caused by a different diffusivity of adsorbates along the crystallographic axes. This anisotropy can be


389

Fig. 9.19. PEEM images demonstrating different types of chemical wave patterns which arise due to the state-dependent anisotropy of the Rh(110) surface in the NO+H2 reaction. The diameter of the imaged area varies between 300 and 400 µm. The crystallographic axes indicated in (c) refer to all images. Experimental conditions: T = 620 K, pNO and pH2 vary in the 10−6 mbar range. (From Gottschalk et al., 1994; Mertens et al., 1995; Mertens and Imbihl, 1996a.) (a) Rectangular target patterns. (b) Rectangular spiral wave. (c) Simultaneous presence of elliptical and rectangular front geometries in the region of dynamic bistability. (d) Traveling wave fragments.

removed by simply rescaling the coordinates (simple anisotropy). Such a rescaling no longer works if two adsorbates with different anisotropies are involved or if the anisotropy varies depending on the coverage. This may happen either due to energetic interactions between the adparticles or due to adsorbate-induced reconstructions. With such a statedependent anisotropy new types of patterns are possible which are not allowed in isotropic media or in media with simple anisotropy. This was demonstrated experimentally with the NO + H2 reaction on a Rh(110) surface and theoretically both with general models as well as with realistic models (Bär et al., 2002; Gottschalk et al., 1994; Makeev et al., 2001; Meron et al., 2001; Mertens and Imbihl, 1994; Mikhailov, 1994). As shown in Fig. 9.19 one finds on Rh(110) rectangularly shaped target patterns (a) and spiral waves (b), the simultaneous presence of rectangular and elliptical front geometries (c), and wave fragments traveling along certain crystallographic directions (d). The unusual patterns have their origin in the enormous structural variability of the Rh(110)

390

R. Imbihl

surface leading a state-dependent anisotropy (Comelli et al., 1998; Kiskinova, 1996). As discussed earlier (see Section 9.4.3.4) the pulses seen as traveling white bands in PEEM represent essentially nitrogen covered surface area propagating on the oxygen covered Rh(110) surface (Mertens et al., 1997; Schaak et al., 1999a; Schmidt et al., 2000). Since the Rh–N–Rh chains in the (2 × 1)/(3 × 1) reconstructions of adsorbed nitrogen are oriented perpendicular to the troughs of the oxygen-induced-missing row type reconstructions in the c(2 × 6)/c(2 × 8)-O phases, the anisotropy changes along the profile of a pulse. Not only the rectangular front geometries but also the traveling wave fragments have their origin in this state-dependent anisotropy which prevents a spiraling in of the free ends of the fragments (Mertens et al., 1995). Despite the structural complexity of the system the experimentally determined existence diagram of wave patterns could be reproduced with a realistic model in almost every detail as demonstrated by Fig. 9.20 (Makeev et al., 2001; Schaak and Imbihl, 1997). As shown in Fig. 9.21a traveling wave fragments have also been observed in catalytic CO oxidation on Pt(110) where they exhibit one additional remarkable property (Rotermund et al., 1991). In some cases when two wave fragments collide, they do not annihilate as expected but after some interaction time continue their journey unchanged. This soliton-like behavior is the result of a particular situation when they collide near a surface defect as demonstrated in the simulation in Fig. 9.21b (Bär et al., 1992a). When the two fragments collide in the vicinity of a surface defect which was mimicked by increasing the oxygen sticking coefficient, s(O2 ), in a certain region then the excitation survives at the defect sending out two new traveling wave fragments. The wave fragment itself should not be called soliton because the soliton-like behavior is the result of a particular situation and not an intrinsic property of the waves. 9.5.4. Complex patterns due to structural or chemical modifications of the substrate 9.5.4.1. Subsurface oxygen formation Figure 9.22 shows in an x–t-plot of PEEM images how the collision of two reaction fronts in the O2 + H2 reaction on Rh(111) leads to the formation of a very bright area in the collision region (Monine et al., 2001; Schaak and Imbihl, 2000). The bright area appearing as reaction intermediate in this experiment represents an area with reduced WF and has accordingly been attributed to subsurface oxygen, i.e. an oxygen species located in a region underneath the surface plane. Quite surprisingly, the same phenomenon that areas with reduced WF develop as intermediates in pattern forming surface reactions has been observed with quite a number of catalytic reactions on noble metals involving oxygen: Pt(100)/CO + O2 (Lauterbach et al., 1994; McMillan et al., 2005; Rotermund et al., 1993), Pt(110)/CO + O2 (von Oertzen et al., 1996; Rotermund et al., 2002; von Oertzen et al., 1998, 2000), Rh(111)/NO + H2 (Janssen et al., 1996; Schaak and Imbihl, 2002), Rh(111)/O2 + H2 (Schaak and Imbihl, 2000). WF decreases of more than 1 eV below the level of the clean surfaces have been reported. Mathematical models which assume a reversible formation of a subsurface oxygen species, Osub , according to Oad Osub could rather well reproduce the corresponding experiments (McMillan et al., 2005; Monine et al., 2001; von Oertzen et al., 1998, 2000).


391

Fig. 9.20. Comparison between experimental and simulated bifurcation diagram for Rh(110)/NO + H2 . The regions where different types of spatial patterns formation occur are shown in the (pH2 , T )-plane at fixed pNO = 1.6 × 10−6 mbar. Abbreviations: RP = rectangular patterns, WF = wave fragments, DM = double metastability, el = equistability line. For a complete description see Makeev et al. (2001). (a) Experimental bifurcation diagram. (From Schaak and Imbihl, 1997.) (b) Calculated bifurcation diagram for the reaction-diffusion model. (From Makeev et al., 2001.)

392

R. Imbihl

Fig. 9.21. Soliton-like behavior of oxygen pulses during catalytic CO oxidation on a Pt(110) surface. (a) Experimental PEEM image demonstrating the propagation of stable oxygen islands visible as black elongated bars on a CO cover surface. The islands propagate with a velocity of ≈ 3 µm/s along the [001]-direction as indicted by the arrows. The experimental conditions were T = 485 K, pCO = 1 × 10−4 mbar and pO2 = 3.5 × 10−4 mbar. (From Rotermund et al., 1991.) Reproduced with permission of the authors. (b) Model calculations demonstrating the soliton-like behavior that results when two oxygen pulses interact near a surface defect. (From Bär et al., 1992a.) Reproduced with permission of the authors.


393

Fig. 9.22. Formation of subsurface oxygen caused by the collision of two reduction fronts in the O2 +H2 reaction on Rh(111). The x–t-diagram was constructed by taking PEEM intensity profiles in a direction perpendicular to the front line. Deep dark areas represents oxygen covered surface, gray area the oxygen freed surface. The formation of a low WF area in between the two colliding fronts is interpreted as being due to subsurface oxygen formation. Experimental conditions: pretreatment of the sample with oxygen for 40 h; titration at T = 450 K with pH2 = 4 × 10−7 mbar. (From Schaak and Imbihl, 2000.)

Chemically the formation of subsurface oxygen represents a partial oxidation of the surface. With respect to dynamics sub-O introduces an additional variable with a slower time scale which affects pattern formation because its presence modifies the adsorption properties of a surface and influences the surface phase transition. The chemical modification of the surface solved a longstanding problem of explaining the standing waves in Pt(110)/CO + O2 . The three-variable KEE-model with global coupling exhibited standing waves but these waves were not robust and they exhibited in disagreement with the experiment a small amplitude superimposed on a uniformly oscillating background (Falcke et al., 1995). As demonstrated by Fig. 9.23 including the sub-O species leads to standing waves which agree very well with the experiment (von Oertzen et al., 2000). The essential mechanism is the reflective collision of pulses, i.e. the subsurface oxygen survives the collision long enough to trigger the nucleation of new pulses. Global coupling via the gas phase is required to ensure a robust pattern which otherwise would be destroyed by unavoidable inhomogeneities of the surface. A phenomenon which is also caused by a partial oxidation of the surface is the appearance of triangular reaction fronts (Schaak and Imbihl, 1998). After exposing the Rh(111) surface to a large dose of oxygen ( 1 superharmonic entrainment and k/ l < 1 subharmonic entrainment. These cases are nicely illustrated by the particularly rich “dynamical phase diagram” of catalytic CO oxidation on Pt(110) shown in Fig. 9.34 (Eiswirth and Ertl, 1988; Krischer

412

R. Imbihl

Fig. 9.34. Dynamic phase diagram for periodically forced oscillations in CO oxidation on Pt(110). The experimental data were recorded at T = 525 K and T = 530 K in the 10−5 mbar range. The axes A and Tex /To denote the amplitude with which pO2 was modulated and the period length of the modulation Tex expressed with respect to the period To of the autonomous oscillations. The shaded areas in the diagram indicate regions of quasiperiodic behavior in between the entrainment bands. (From Eiswirth and Ertl, 1988.) Reproduced with permission of the authors.

et al., 1992b). This diagram which was obtained by forcing the fast harmonic oscillations exhibits different entrainment bands (Arnold tongues) separated by regions of quasiperiodic behavior. Other examples of forced single crystal systems are the catalytic CO oxidation on Pt(100) and the NO + CO and NO + H2 reaction on Pt(100) (Dath et al., 1992; Schütz and Imbihl, 2002; Schwankner et al., 1987). In recent years the interest in forcing experiments strongly decayed. As was shown quite generally by Vance, Tsarouhas and Ross the skeleton bifurcation diagram of a forced system does not depend on the specific characteristics of a reaction system, but only on the type of bifurcation (of the autonomous system) around which the system is perturbed (Tsarouhas and Ross, 1989; Vance et al., 1989). The hope to have a kind of “kinetic spectroscopy” with which kinetic parameters of a dynamical system can be relatively easily extracted is thus not fulfilled. In contrast to the forcing of an autonomously oscillating system represented by ODE’s the forcing of a spatially extended system has only partially been explored. Experiments with a light sensitive variant of the BZ reaction and a digital video projector with which the different parts of an oscillating medium could be addressed separately revealed a variety of new patterns (Lin et al., 2000; Petrov et al., 1977). In catalytic CO oxidation on Pt(110) subjected to a 1:1 resonant forcing so-called phase kinks were found as predicted by theory (Punckt and Rotermund, 2007). Forcing experiments in a different direction were conducted in which systems were subjected to external noise. With random forcing sustained oscillations were obtained in the NO + CO reaction on Pt(100) which at lower temperature only displays excitable behavior (Dath et al., 1992). The influence of external noise was studied quite extensively with the


413

bistable system Ir(111)/CO + O2 (Wehner et al., 2005). With PEEM as spatially resolving method they explored systematically how the nucleation of the new phase depends on the forcing parameters. 9.7.2. Controlling turbulence For controlling chaotic systems a number of methods have been developed (Mikhailov and Showalter, 2006). The signal of a chaotic system is monitored and subsequently corrective perturbations are computed which stabilize an unstable periodic orbit, i.e. the irregular oscillations are transformed into regular oscillations. For spatially extended systems a similar method has been developed based on a delayed global feedback. This functioning of this method has been successfully demonstrated with catalytic CO oxidation on Pt(110) as depicted by Fig. 9.35 (Bertram et al., 2003; Beta et al., 2004; Kim et al., 2001). A global signal I obtained by integrating the PEEM intensity over the imaged area was used to modulate the CO partial pressure after an adjustable delay time τ such that pCO (t) = p0 + μ I (t − τ ) − I0 where p0 and I0 represent the average level of pCO and the PEEM intensity and the parameter μ controls the intensity of the feedback. Starting with a state of spiral turbulence (or defect mediated turbulence) which integrated yields a stationary reaction rate switching on the feedback loop generates a spatially uniformly oscillating surface associated with regular oscillations in the reaction rate (a). Depending on the reaction conditions, the choice of the feedback parameters τ and μ, various types of response behavior are obtained: intermittent turbulence (b), so-called phase clusters in which the surface separates into domains oscillating with ant-phase relationship (c), and standing waves (d). Thus delayed global feedback not only suppresses turbulence but also produces new types of patterns. If not the averaged PEEM intensity but only a certain Fourier component is selected, then the corresponding pattern can be stabilized (Beta et al., 2004). 9.7.3. Microstructured surfaces 9.7.3.1. Basic concepts The influence of boundary conditions on pattern formation in RD systems has been studied in simulations as well as in experiments (Mikhailov and Showalter, 2006). The selection of patterns by the domain size, the propagation of chemical waves through narrow channels or slits, and the propagation in a medium with obstacles have been explored. With catalytic surfaces a number of lithographic techniques like electron beam lithography or optical lithography and chemical methods are available which allow the generation of micro- and nano-structured composite surfaces. The great advantage of this concept is that the size and the geometry and the chemical composition of domains and their surroundings can be varied at will. Dynamic coupling effects between domains of different reactivity can thus be systematically explored in their size and geometry dependence. The size dependence of the activity of supported catalysts is of course a classical theme in heterogeneous catalysis and a large body of empirical data exists. Nanolithography has

414

R. Imbihl

Fig. 9.35. Four spatio-temporal patterns observed in catalytic CO oxidation on Pt(110) with global delayed feedback. (From Kim et al., 2001.) Reproduced with permission of the authors. (a) Suppression of spiral wave turbulence (b) Intermittent turbulence (c) Phase clusters (d) Standing waves. In each part the upper row displays three subsequent PEEM images with a field-of-view of 500 µm in diameter, images in the middle row are space–time diagrams showing the evolution along the line ab indicated in the first image. Dark areas are O covered, brighter regions are mainly CO covered. The diagrams in the bottom row display the temporal variation of the CO partial pressure (dark line) and the variation of the integral PEEM intensity (brighter line) during the pattern evolution; the time scale is the same as in the diagram.

been employed in order to systematically investigate this size dependence in experiments conducted at high pressure (>1 mbar) (Jacobs et al., 1997; Johanek et al., 2004; Somorjai et al., 2007; Zuburtikudis and Saltsburg, 1992). Since the diameter of the metal particles is still below the resolution of most spatially resolving techniques only integral measurements of the rates have been performed.


415

The concept of the group of experiments discussed here is different (Graham et al., 1994), With microstructured surfaces the use of spatially resolved methods should be possible in order to allow an in situ imaging of the dynamic processes. The aim is to study the influence of the boundary conditions on pattern formation and to observe effects like front nucleation at interfaces, ignition/extinction phenomena etc. which are controlled by design. With regard to catalysis the hope is to exploit these non-linear effects in order to improve yield and selectivity of chemical reactions. A kind of microchemical engineering has been proposed but although numerous promising effects exist, the practical problem of finding industrially interesting reaction systems with composite catalysts which are stable to survive at high pressure is still unsolved (Schütz et al., 1998). In the group of experiments discussed here microstructured composite surfaces are prepared by optical lithography in a negative photoresist process in which thin metal layers of typical 5–100 nm thickness are deposited onto a single crystal surface (Imbihl, 2002; Li et al., 2002). The macroscopic length scale of >1 µm of the resulting structures is convenient for the observation with PEEM and SPEM. Reaction systems of the automotive catalytic converter are studied using mostly combinations of the metals, Pt, Pd, Rh in composite surfaces which catalyze reactions like CO oxidation, NO reduction and the O2 + H2 reaction. Most of the experiments were accompanied by extensive mathematical modeling in the group of Kevrekidis (Li et al., 2002). Since the size of each of the microstructures is small vs. the single crystal surface area a large number of microstructures which may be also viewed as “microreactors” can be viewed simultaneously. Depending on the metal which is evaporated we can distinguish between experiments with unreactive boundaries and experiments with reactive boundaries. If a thin layer of titanium is evaporated onto the catalyst surface then the Ti under reaction conditions is rapidly converted to TiO2 which under the low pressure conditions applied here is practically inert. With titanium one thus essentially restricts the size of the active area. With reactive metals mobile adsorbates will diffuse across the domain boundaries resulting in dynamic coupling effects. This latter case is of course more interesting for catalysis. 9.7.3.2. Unreactive boundaries With lithographic techniques a large number of domains of different geometries and sizes can be generated on a single crystal surface with the active metal surface being surrounded by an inactive TiO2 layer (Graham et al., 1994). With catalytic CO oxidation pattern formation in Pt domains surrounded by an inert TiO2 layer has been studied on Pt(110) (Bär et al., 1996, 1995a; Bangia et al., 1996; Graham et al., 1995a, 1994) and Pt(100) (Haas et al., 1995). In the NO + CO reaction on Pt(100) pattern formation in squares, circles, rings, and dumbbell geometries has been studied (Christoph et al., 1999; Hartmann et al., 1996, 2000; Shvartsman et al., 2000). With circular domains the dependence of the rotational period of spiral waves on the domain diameter was investigated in the NO + CO reaction on Pt(100) (Hartmann et al., 1996, 2000). By restricting the active area one can isolate certain aspects of the propagation of pulses or fronts and study the effect of anisotropy or the curvature dependence on the propagation. In a dumbbell geometry the transition from a quasi 1D-propagation along narrow channels to a 2D-propagation into the circular domain can be observed as shown in

416

R. Imbihl

Fig. 9.36. PEEM images showing selected experiments with chemical wave patterns on microstructured surfaces. (a) Reaction fronts in catalytic CO oxidation propagating in dumbbell-shaped Pt(100) domains of varying size. The arrows indicate the directions of propagation. Experimental conditions: T = 360 K, 10−4 mbar range. (From Haas et al., 1995.) (b) Pinning of spiral waves by a grid of inert Ti squares during catalytic CO oxidation on Pt(110). Experimental conditions: T = 451 K, 10−4 mbar range. (From Bär et al., 1996.) (c) Interaction of pulses in the NO+CO reaction on a ring-shaped Pt(100) domain with a diameter of 320 µm. Shown are space–time plots of two data sets displaying the angular position of the pulses vs. the time. Experimental conditions: T = 427 K, 10−6 mbar range. For details see text. (From Hartmann et al., 2000.) (d) Oxygen pulses in a 22 mm Pt(110) channels surrounded by Rh in catalytic CO oxidation. Experimental conditions: T = 440 K, 10−4 mbar range. (From Pollmann et al., 2001.)

Fig. 9.36a (Haas et al., 1995). From the curvature dependence of the front velocity during this transition the diffusion constant of CO can be determined as demonstrated for CO oxidation and the NO + CO reaction on Pt(100) (Haas et al., 1995; Hartmann et al., 2000). Figure 9.36b shows how single and multi-armed spiral waves are pinned by a grid of inert TiO2 inclusions on a Pt(110) surface. In ring-shaped channels


417


on Pt(100) the effect of anomalous dispersion leading to the attraction of pulses in the NO + CO reaction was demonstrated (Christoph et al., 1999). As depicted in Fig. 9.36c with such a geometry various types of interactions between pulses can be seen: #1 pulse reflection, #2 pulse splitting, #3 partial annihilation, and #4 soliton-like behavior (Hartmann et al., 2000). 9.7.3.3. Reactive boundaries With microstructured composite surfaces of the metals Pt, Pd, and Rh various reactions of the automotive catalytic converter were studied: catalytic CO oxidation on Pt(110)/Rh (Pollmann et al., 2001) and Pt(110)/Pd (Lauterbach et al., 1998), the NO reducing reactions NO + CO and NO + H2 on Pt(100)/Rh (Esch et al., 1999, 1998; Schütz et al., 1998, 1996; Shvartsman et al., 1999, 2001), and the O2 + H2 reaction on Rh(110)/Pt (Günther et al., 2002b; Schütz et al., 1999) and Pt(100)/Rh (Schütz et al., 1998). Due to segregation and alloying effects the layer thickness was an important parameter in these studies (Esch et al., 1999). In addition, contaminants can decisively influence pattern formation (Günther et al., 2002b). Reactive boundaries can act as a drain or source for the reacting adsorbates and this property may lead to a significant size dependence of the reactivity of a domain. Pd or Rh surrounding Pt(110) domains can act as CO supplier in catalytic CO oxidation (Lauterbach et al., 1998; Pollmann et al., 2001). They may trigger the nucleation of CO pulses or modify the shape of propagating oxygen pulses as demonstrated by Fig. 9.36d. It was shown that even the geometry of a domain can facilitate or suppress the nucleation of chemical waves (Li et al., 2002). With the NO + CO reaction on Pt(100) it was demonstrated that the ignition temperature of a layer of molecularly coadsorbed NO and CO on Pt(100)/Rh decreases with shrinking diameter of the Pt(100) domains (Shvartsman et al., 1999). This happens because the surrounding Rh layer is more active in NO dissociation than Pt thus creating vacant sites whose autocatalytic proliferation ignites the “surface explosion” of the CO/NO-adlayer inside the Pt(100) domain (see Section 9.4.3.2). For a similar reaction

418

R. Imbihl

Fig. 9.37. PEEM images showing different stages in the titration of an oxygen saturated Pt(100)Rh microstructure with hydrogen at pH2 = 1.5 × 10−7 mbar and T = 580 K. Inside the circles the substrate is a Pt(100) surface and the surrounding area is covered with a 500 Å thick Rh film. The bright rings surrounding the Pt domains (frame b) are reaction fronts propagating away from the interface. Images were taken at t = 0 s, 120 s, 210 s, 270 s. (From Schütz et al., 1998.)

system, the NO + H2 reaction on Pt(100)/Rh, multistability and front ignition at the Pt/Rh interface was investigated both, theoretically and experimentally (Shvartsman et al., 1999). The PEEM images in Fig. 9.37 show the titration of an oxygen covered microstructured Pt(100)/Rh surface with hydrogen (Schütz et al., 1998). Two reaction fronts nucleate at the perimeter of two differently sized Pt(100) domains and spread out into the surrounding Rh surface while the Pt circles themselves remain still oxygen covered for some time. First the small circle becomes bright (oxygen free) while the larger one follows with a delay of ≈1 min. The difference in reactivity can be attributed to the stronger diffusional coupling of the smaller circle because the amount of hydrogen, which diffuses back from the Rh surface into the Pt circle, will be proportional to the perimeter, i.e. proportional to r, whereas the amount of oxygen to be reacted away is proportional to r 2 . A surprising variety of stationary and dynamic patterns was discovered in the O2 + H2 reaction on Rh(110)/Pt originally conceived as two bistable reaction systems coupled via


419

hydrogen diffusion (Schütz et al., 1999). It was soon discovered that this system is not simple and that the stationary patterns are created by additional processes. Due to the thin Pt layer of 30 nm thickness partial alloying with the underlying Rh substrate took place. Since adsorbed oxygen pulls Rh from the Pt layer towards the surface the oxygen spots in the stationary patterns on the Pt domains are connected with the local enrichment of Rh. When the presence of K contamination on the microstructured sample was discovered it became soon clear that the segregation effect is not the actual driving force of the patterns. It turned out that the primary pattern forming process is the condensation of K with oxygen into large islands through the mechanism of reactive phase separation discussed above (see Section 9.5.5.2) (Günther et al., 2002b). The distribution pattern of Rh on the Pt domain is a secondary process driven by the already existing stationary K + O concentration pattern. The latter example demonstrates how even seemingly simple systems can become very complex through alloying effects and contaminations. These effects and the potential lack of stability of microstructured surfaces at high pressure will certainly limit the practical applicability of the concept. For studying elementary dynamic effects on composite surfaces the concept of microstructured surfaces, however, turned out to be highly fruitful. 9.7.4. Manipulation by local laser heating For controlling patterns the ideal solution would be a method which addresses the different points of a medium individually with a corrective signal which varies in space and time and which is computed in real time from the momentarily observed pattern. The feasibility of such an approach has been demonstrated with a light sensitive variant of the BZ reaction using a digital video projector for generating a spatio-temporal perturbation (Lin et al., 2000; Petrov et al., 1977). On a catalytic surface a local perturbation can be achieved through local heating with a laser as was demonstrated recently with a very versatile tool (Wolff et al., 2001, 2003a, 2003b, 2003c; Wolff and Rotermund, 2003). By focusing a laser beam into a narrow spot of ≈80 µm diameter and the use of computer controlled galvanometer mirrors the laser spot can be moved within 1 ms to any place on the imaged area (1 × 2 mm2 ) with a precision of 5 µm. Through local heating of the surface the surface kinetics are modified locally. How a pulse with such a tool can be dragged across the surface is demonstrated in Fig. 9.38 where, after initiation of a target pattern, the laser spot was moved with constant velocity across the target pattern creating a V-shaped pulse reminiscent of a Mach cone. In combination with real time computation of the response this method allows a rather sophisticated control of patterns on catalytic surfaces.

9.8. Conclusions and outlook Despite the fact that the study of oscillatory effects and spatio-temporal patterns in catalytic reactions has reached a certain level of matureness – as judged from the degree of theoretical understanding and from the sophistication of the experiments – probably large areas in the field of non-linear dynamics on surfaces are still waiting to be explored. This is certainly the case for phenomena on a nanoscale where suitable methods for the exploration are just being developed. An even more important field where the indispensable role

420

R. Imbihl

Fig. 9.38. Dragging an oxygen pulse across the surface with a laser spot, in the system Pt(110)/CO + O2 . After establishing excitable reaction conditions a target pattern was first initiated by local heating with the pulse. The images show an area of 1.5 × 1.1 mm and 1.0 × 0.6 mm, respectively. The arrows indicate the position of the laser spot. Experimental conditions: T = 515 K, pressure in the 10−4 mbar range. (From Wolff et al., 2001.) Reproduced with permission of the authors.

of non-linear dynamics for a fundamental understanding still waits to be demonstrated is “real catalysis”. Structure and composition of a catalyst under working conditions will be determined largely by dynamic processes and deviations from the ideal structure which are stabilized by non-equilibrium conditions like defects or strain may generate a high catalytic activity. Furthermore, a catalytic reaction will never proceed spatially uniformly but self-organization processes will always lead to a certain spatio-temporal organization of the reaction which may just involve microscopically small adsorbate islands or even reach macroscopic dimensions in the case of chemical waves. It is the lack of suitable in situ methods which prevents the study of such phenomena in detail. Ironically, the phenomenon with which the whole development started, namely the rate oscillations observed at atmospheric pressure with CO oxidation over Pt catalysts is still mechanistically unsolved. This demonstrates, first of all, that the pressure & material gap is a severe scientific challenge and, secondly, that despite all the detailed insights from UHV studies the use of in situ methods which are applicable at high pressure is indispensable in order to resolve the mechanisms behind the oscillations and wave patterns at high pressure.

Acknowledgements The author is indebted to N. Hartmann and F. Lovis for carefully reading the manuscript.

References Adlhoch, W., Lintz, H.-G., Weisker, T., 1981. Surf. Sci. 103, 576. Andrade, R.F.S., Dewel, G., Borckmans, P., 1989. J. Chem. Phys. 91, 2675. Andrade, R.F.S., Lima, D., Dewel, G., et al., 1994. J. Chem. Phys. 100, 9192. Aranson, I.S., Kramer, L., 2002. Rev. Mod. Phys. 74, 99. Bär, M., Eiswirth, M., Rotermund, H.H., et al., 1992a. Phys. Rev. Lett. 69, 945. Bär, M., Zuelicke, C., Eiswirth, M., et al., 1992b. J. Chem. Phys. 96, 8595. Bär, M., Eiswirth, M., 1993. Phys. Rev. E 48, R1635. Bär, M., Gottschalk, N., Eiswirth, M., et al., 1994. J. Chem. Phys. 100, 1202. Bär, M., Kevrekidis, I.G., Rotermund, H.H., et al., 1995a. Phys. Rev. E 52, R5739. Bär, M., Nettesheim, S., Rotermund, H.H., et al., 1995b. Phys. Rev. Lett. 74, 1246. Bär, M., Bangia, A.K., Kevrekidis, I.G., et al., 1996. J. Phys. Chem. 100, 19106.


421

Bär, M., Meron, E., Utzny, C., 2002. Chaos 12, 204. Bailey, J.E., Horn, F., 1970. Ber. Bunsenges. Phys. Chem. 74, 611. Bangia, A.K., Bär, M., Kevrekidis, I.G., et al., 1996. Chem. Eng. Sci. 51, 1757. Banholzer, W.F., Park, Y.O., Mak, K.M., et al., 1983. Surf. Sci. 128, 176. Bare, S.R., Hofmann, P., King, D.A., 1984. Surf. Sci. 144, 347. Bassett, M.R., Imbihl, R., 1990. J. Chem. Phys. 93, 811. Battogtokh, D., Mikhailov, A., 1996. Physica D 90, 84. Bauer, E., 1994. Rep. Prog. Phys. 57, 895. Bauer, E.G., Allen, C.E., 1995. Prog. Surf. Sci. 49, 265–330. Berdau, M., Karpowicz, A., Yelenin, G.G., et al., 1997. J. Chem. Phys. 106, 4291. Berdau, M., Yelenin, G.G., Karpowicz, A., et al., 1999. J. Chem. Phys. 110, 11551. Bertram, M., 2002. Controlling turbulence and pattern formation in chemical reactions. Thesis, TU Berlin, Berlin. Bertram, M., Beta, C., Pollmann, M., et al., 2003. Phys. Rev. E 67, 036208. Beta, C., Moula, M.G., Mikhailov, A.S., et al., 2004. Phys. Rev. Lett. 93, 188302. Beusch, H., Fieguth, P., Wicke, E., 1972. CIT 44, 445. Bondzie, V.A., Kleban, P.H., Dwyer, D.J., 2000. Surf. Sci. 465, 266. Bonzel, H.P., Bradshaw, A.M., Ertl, G. (Eds.), 1989. Physics and Chemistry of Alkali Metal Adsorption. Elsevier, Amsterdam. Brown, W.A., Kose, R., King, D.A., 1998. Chem. Rev. 98, 797. Carabineiro, S.A.C., van Noort, W.D., Nieuwenhuys, B.E., 2002. Catal. Lett. 84, 135. Castets, V., Dulos, E., Boisonnade, J., et al., 1990. Phys. Rev. Lett. 64, 2953. Chavez, F., Vicente, L., Perera, A., et al., 1998. J. Chem. Phys. 109, 8617. Chavez, F., Vicente, L., Perera, A., 2000a. J. Chem. Phys. 113, 10353. Chavez, F., Vicente, L., Perera, A., et al., 2000b. J. Chem. Phys. 112, 8672. Christoph, J., Eiswirth, M., Hartmann, N., et al., 1999. Phys. Rev. Lett. 82, 1586. Clarke, B., 1980. Adv. Chem. Phys. 43, 1. Cobden, P.D., Siera, J., Nieuwenhuys, B.E., 1992. J. Vac. Sci. Technol. A10, 2487. Cobden, P.D., Janssen, N.M.H., van Breugel, Y., et al., 1996. Surf. Sci. 366, 432. Cobden, P.D., Nieuwenhuys, B.E., Esch, F., et al., 1998. Surf. Sci. 416, 264. Colen, R.E.R., Christoph, J., Pena, F., et al., 1998. Surf. Sci. 408, 310. Comelli, G., Lizzit, S., Hofmann, P., et al., 1992. Surf. Sci. 277, 31. Comelli, G., Dhanak, V.R., Kiskinova, M., et al., 1998. Surf. Sci. Rep. 32, 165. Coullet, P., Gil, L., Lega, J., et al., 1989. Phys. Rev. Lett. 62, 1619. Cox, M.P., Ertl, G., Imbihl, R., et al., 1983. Surf. Sci. 134, L517. Cross, M.C., Hohenberg, P.C., 1993. Rev. Mod. Phys. 65, 851. Dath, J.P., Fink, T., Imbihl, R., et al., 1992. J. Chem. Phys. 96, 1582. De Decker, Y., Marbach, H., Hinz, M., et al., 2004. Phys. Rev. Lett. 92, 198305. De Decker, Y., Mikhailov, A.S., 2004. J. Phys. Chem. B 108, 14759. de Wolf, C.A., Nieuwenhuys, B.E., Sasahara, A., et al., 1998. Surf. Sci. 411, L904. de Wolf, C.A., Nieuwenhuys, B., et al., 2001. J. Phys. Chem. B 105, 4254. de Wolf, C.A., Nieuwenhuys, B.E., 2001. Catal. Today 70, 287. Dicke, J., Erichsen, P., Wolff, J., et al., 2000. Surf. Sci. 462, 90. Doedel, E.J., 1986. AUTO – Software for Continuation and Bifurcation Problems in Ordinary Differential Equations. CalTech, Pasadena. Dwyer, D.J., Hoffmann, F.M., 1992. Surface Science of Catalysis, vol. 482. American Chemical Society, Washington. Egelhoff, W.F., 1982. Nitric oxide reduction. In: King, D.A., Woodruff, D.P. (Eds.), The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis, vol. 4. Elsevier, Amsterdam, pp. 401–426. Ehsasi, M., Seidel, C., Ruppender, H., et al., 1989. Surf. Sci. 210, L198. Ehsasi, M., Berdau, M., Rebitzki, T., et al., 1993. J. Chem. Phys. 98, 9177. Eiswirth, M., Schwankner, R., Ertl, G., 1985. Z. Phys. Chem. 144, 59. Eiswirth, M., Ertl, G., 1986. Surf. Sci. 177, 90. Eiswirth, M., Ertl, G., 1988. Phys. Rev. Lett. 60, 1526.

422

R. Imbihl

Eiswirth, M., Ertl, G., 1994. In: Kapral, R., Showalter, K., Kapral, K.S.R. (Eds.), Chemical Waves and Patterns. Kluwer, Dordrecht. Eiswirth, M., Krischer, K., Ertl, G., 1988. Surf. Sci. 202, 565. Eiswirth, M., Moeller, P., Wetzl, K., et al., 1989. J. Chem. Phys. 90, 510. Eiswirth, M., Freund, A., Ross, J., 1991a. J. Phys. Chem. 95, 1294. Eiswirth, M., Freund, A., Ross, J., 1991b. Adv. Chem. Phys. 80, 127. Eiswirth, M., Kruel, T.-M., Ertl, G., et al., 1992. Chem. Phys. Lett. 193, 305. Eiswirth, M., 1993. In: Fields, R.J., Györgyi, L. (Eds.), Chaos in Chemistry and Biochemistry. World Scientific, Singapore. Eiswirth, M., Bürger, J., Strasser, P., et al., 1996. J. Phys. Chem. 100, 19118. Emig, G., Liauw, M., 2002. Top. Catal. 21, 11. Engel, T., Ertl, G., 1979. Adv. Catal. 28, 1. Engel, W., Kordesch, M.E., Rotermund, H.H., et al., 1991. Ultramicroscopy 36, 148. Ernst, N., Bozdech, G., Gorodetskii, V., et al., 1994. Surf. Sci. 318, L211. Ertl, G., Norton, P.R., Ruestig, J., 1982. Phys. Rev. Lett. 49, 177. Ertl, G., 1990. Adv. Catal. 37, 213. Esch, F., Gunther, S., Schütz, E., et al., 1998. Catal. Lett. 52, 85. Esch, F., Gunther, S., Schütz, E., et al., 1999. Surf. Sci. 443, 245. Evans, J.W., 1991. Langmuir 7, 2514. Evans, J.W., Miesch, M.S., 1991. Phys. Rev. Lett. 66, 833. Evans, J.W., 1992. J. Chem. Phys. 97, 572. Evans, J.W., Madden, H.H., Imbihl, R., 1992. J. Chem. Phys. 96, 4805. Evans, J.W., Liu, D.-J., Tammaro, M., 2002. Chaos 12, 131. Falcke, M., Bär, M., Engel, H., et al., 1992. J. Chem. Phys. 97, 4555. Falcke, M., Engel, H., 1994a. Phys. Rev. E 50, 1353. Falcke, M., Engel, H., 1994b. J. Chem. Phys. 101, 6255. Falcke, M., Engel, H., Neufeld, M., 1995. Phys. Rev. E 52, 763. Falta, J., Imbihl, R., Henzler, M., 1990. Phys. Rev. Lett. 64, 1409. Falta, J., Imbihl, R., Sander, M., et al., 1992. Phys. Rev. B: Condens. Matter 45, 6858. Fechner, G.T., 1828. Schweiggers Journal für Chemie und Physik 53, 61. Fery, P., Moritz, W., Wolf, D., 1988. Phys. Rev. B 38, 7275. Fichthorn, K., Gulari, K.E., Ziff, R.M., 1989. Phys. Rev. Lett. 63, 1527. Field, R.J., Schneider, F.W., 1989. J. Chem. Educ. 66, 3. Fink, T., Dath, J.P., Bassett, M.R., et al., 1991a. Surf. Sci. 245, 96. Fink, T., Dath, J.P., Imbihl, R., et al., 1991b. J. Chem. Phys. 95, 2109. FitzHugh, R., 1961. Biophys. J. 1, 445. Flytzani-Stephanopoulos, M., Schmidt, L.D., Caretta, R., 1980. J. Catal. 64, 346. Freyer, N., Kiskinova, M., Pirug, G., et al., 1986. Surf. Sci. 166, 206. Ganduglia-Pirovano, M.V., Reuter, K., Scheffler, M., 2002. Phys. Rev B 65, 245426. Gardiner, C.W., 1983. Handbook of Stochastic Methods for Physics. Chemistry and the Natural Sciences Springer, Berlin. Gardner, P., Martin, R., Tüshaus, M., et al., 1990a. J. Electron Spec. Relat. Phenom. 54/55, 619. Gardner, P., Tüshaus, M., Martin, R., et al., 1990b. Surf. Sci. 240, 112. Gelten, R.J., Jansen, A.P.J., van Santen, R.A., et al., 1998. J. Chem. Phys. 108, 5921. Gohndrone, J.M., Masel, R.I., 1989. Surf. Sci. 209, 44. Gong, X.-Q., Raval, R., Hu, P., 2004. Phys. Rev. Lett. 93, 106104. Gorodetskii, V.V., Block, J.H., Drachsel, W., et al., 1993a. Appl. Surf. Sci. 67, 198. Gorodetskii, V.V., Drachsel, W., Block, J.H., 1993b. Catal. Lett. 19, 223. Gorodetskii, V.V., Block, J.H., Drachsel, W., 1994a. Appl. Surf. Sci. 76/77, 129. Gorodetskii, V.V., Drachsel, W., Block, J.H., 1994b. Appl. Surf. Sci. 76/77, 122. Gorodetskii, V.V., Lauterbach, J., Rotermund, H.H., et al., 1994c. Nature 370, 277. Gorodetskii, V.V., Elokhin, V.I., Bakker, J.W., et al., 2005. Catal. Today 105, 183. Gorte, R.J., Schmidt, L.D., Gland, J.L., 1981. Surf. Sci. 109, 367. Gottschalk, N., Mertens, F., Bär, M., et al., 1994. Phys. Rev. Lett. 73, 3483.


423

Graham, M.D., Kevrekidis, I.G., Asakura, K., et al., 1994. Science 264, 80. Graham, M.D., Bär, M., Kevrekidis, I.G., et al., 1995a. Phys. Rev. E 52, 76. Graham, M.D., Kevrekidis, I.G., Hudson, J.L., et al., 1995b. Chaos, Solitons, Fractals 5, 1817. Griffiths, K., Jackman, T.E., Davies, J.A., et al., 1984. Surf. Sci. 138, 113. Gritsch, T., Coulman, D., Behm, R.J., et al., 1989. Phys. Rev. Lett. 63, 1086. Gruyters, M., Ali, T., King, D.A., 1995. Chem. Phys. Lett. 232, 1. Gruyters, M., Pasteur, A.T., King, D.A., 1996. J. Chem. Soc. Faraday Trans. 92, 2941. Guckenheimer, J., Holmes, P., 1983. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields. Springer, Berlin. Guo, X.-C., Bradley, J.M., Hopkinson, A., et al., 1994. Surf. Sci. 310, 163. Günther, S., Kaulich, B., Gregoratti, L., et al., 2002a. Prog. Surf. Sci. 70, 187. Günther, S., Marbach, H., Hoyer, R., et al., 2002b. J. Chem. Phys. 117, 2923. Günther, S., Marbach, H., Lurssen, B., et al., 2002c. Surf. Rev. Lett. 9, 751. Günther, S., Marbach, H., Hoyer, R., et al., 2003. J. Chem. Phys. 119. Günther, S., Hoyer, R., Marbach, H., et al., 2006. J. Chem. Phys. 124, 014706. Haas, G., Bär, M., Kevrekidis, I.G., et al., 1995. Phys. Rev. Lett. 75, 3560. Haken, H., 1983. Synergetics. An Introduction. Springer, Berlin. Hartmann, N., Imbihl, R., Vogel, W., 1994a. Catal. Lett. 28, 373. Hartmann, N., Krischer, K., Imbihl, R., 1994b. J. Chem. Phys. 101, 6717. Hartmann, N., Bär, M., Kevrekidis, I.G., et al., 1996. Phys. Rev. Lett. 76, 1384. Hartmann, N., Kevrekidis, Y., Imbihl, R., 2000. J. Chem. Phys. 112, 6795. Hartmann, N., Schaak, A., Shaikhutdinov, S., et al., 2004. Surf. Sci. 548, 163. Heilmann, P., Heinz, K., Müller, K., 1979. Surf. Sci. 83, 487. Heinze, S., Schmatloch, V., Kruse, N., 1995. Surf. Sci. 341, 124. Hendriksen, B.L.M., Frenken, J.W.M., 2002. Phys. Rev. Lett. 89, 46101. Hendriksen, B.L.M., Bobaru, S.C., Frenken, J.W.M., 2005a. Catal. Today 234. Hendriksen, B.L.M., Bobaru, S.C., Frenken, J.W.M., 2005b. Top. Catal. 36, 43. Hildebrand, M., Mikhailov, A., 1996. J. Phys. Chem. 100, 19089. Hildebrand, M., Mikhailov, A.S., Ertl, G., 1998a. Phys. Rev. Lett. 81, 2602. Hildebrand, M., Mikhailov, A.S., Ertl, G., 1998b. Phys. Rev. E 58, 5483. Hildebrand, M., Kuperman, M., Wio, H., et al., 1999. Phys. Rev. Lett. 83, 1475. Hildebrand, M., 2002. Chaos 12, 144. Hildebrand, M., Ipsen, M., Mikhailov, A.S., et al., 2003. New J. Phys. 5, 1. Hinz, M., Guenther, S., Marbach, H., et al., 2004. J. Phys. Chem. B 108, 14620. Hopkinson, A., Bradley, J.M., Guo, X.-C., et al., 1993. Phys. Rev. Lett. 71, 1597. Hopkinson, A., King, D.A., 1993a. Chem. Phys. 177, 433. Hopkinson, A., King, D.A., 1993b. Faraday Discuss. 96, 255. Horsthemke, W., Lefever, R., 1984. Noise-induced Transitions. Springer, Berlin. Hoyle, R.B., Anghel, A.T., Proctor, M.R.E., King, D.A., 2007. Phys. Rev. Lett. 98, 226102. Hugo, P., 1970. Ber, Bunsen-Ges. Phys. Chem. 74, 121. Imbihl, R., Cox, M.P., Ertl, G., et al., 1985. J. Chem. Phys. 83, 1578. Imbihl, R., Cox, M.P., Ertl, G., 1986. J. Chem. Phys. 84, 3519. Imbihl, R., Ladas, S., Ertl, G., 1988a. Surf. Sci. 206, L903. Imbihl, R., Sander, M., Ertl, G., 1988b. Surf. Sci. 204, L701. Imbihl, R., 1989. Springer Ser. Synergetics 44, 26–64. Imbihl, R., Reynolds, A.E., Kaletta, D., 1991. Phys. Rev. Lett. 67, 275. Imbihl, R., 1992. Mod. Phys. Lett. B 6, 493–505. Imbihl, R., Fink, T., Krischer, K., 1992. J. Chem. Phys. 96, 6236. Imbihl, R., 1993. Prog. Surf. Sci. 44, 185. Imbihl, R., Veser, G., 1994. J. Vac. Sci. Technol., A 12, 2170. Imbihl, R., Ertl, G., 1995. Chem. Rev. 95, 697. Imbihl, R., 2002. Chaos 12, 182. Imbihl, R., 2003. New J. Phys. 5, 62. Irurzun, I.M., Hoyle, R.B., Proctor, M.R.E., et al., 2003. Chem. Phys. Lett. 377, 269.

424

R. Imbihl

Irurzun, I.M., Imbihl, R., Vicente, J.L., et al., 2004. Chem. Phys. Lett. 389, 212. Jackman, T.E., Davies, J.A., Jackson, D.P., et al., 1982. Surf. Sci. 120, 389. Jacobs, P.W., Wind, S.J., Ribeiro, F.H., et al., 1997. Surf. Sci. 372, L249. Jakubith, S., Rotermund, H.H., Engel, W., et al., 1990. Phys. Rev. Lett. 65, 3013. Janssen, N.M.H., Nieuwenhuys, B.E., Ikai, M., et al., 1994. Surf. Sci. 319, L29. Janssen, N.M.H., Nieuwenhuys, B.E., Cobden, P.D., et al., 1995. Catal. Lett. 35, 155. Janssen, N.M.H., Schaak, A., Nieuwenhuys, B.E., et al., 1996. Surf. Sci. 364, L555. Janssen, N.M.H., Cobden, P.D., Nieuwenhuys, B.E., 1997. J. Phys.: Condens. Matter 9, 1889. Johanek, V., Laurin, M., Grant, A.W., et al., 2004. Science 304, 1639. Kapral, R., Showalter, K., 1994. Chemical Waves and Patterns. Kluwer, Dordrecht. Katona, T., Somorjai, G.A., 1992. J. Phys. Chem. 96, 5465. Kellog, G.L., 1985. Phys. Rev. Lett. 55, 2168. Khrustova, N., Veser, G., Mikhailov, A., et al., 1995. Phys. Rev. Lett. 75, 3564. Khrustova, N., Mikhailov, A.S., Imbihl, R., 1997. J. Chem. Phys. 107, 2096. Kim, M., Bertram, M., Pollmann, M., et al., 2001. Science 292, 1357. Kiskinova, M., 1992. Poisoning and Promotion in Catalysis Based on Surface Science Concepts, vol. 70. Elsevier, Amsterdam. Kiskinova, M., 1996. Chem. Rev. 96, 1431. Kiskinova, M., Fabrizio, E.D., Gentili, M., et al., 1999. Surf. Rev. Lett. 6, 265. Kortluke, O., von Niessen, W., 1996. J. Chem. Phys. 105, 4761. Kortluke, O., Kuzovkov, V.N., von Niessen, W., 1998. Phys. Rev. Lett. 81, 2164. Kortluke, O., Kuzovkov, V.N., von Niessen, W., 1999. J. Chem. Phys. 110, 11523. Krischer, K., Eiswirth, M., Ertl, G., 1992a. J. Chem. Phys. 96, 9161. Krischer, K., Eiswirth, M., Ertl, G., 1992b. J. Chem. Phys. 97, 307. Kuramoto, Y., 1984. Chemical Oscillations, Waves and Turbulence. Springer, Berlin. Kuzovkov, V.N., Kortluke, O., von Niessen, W., 1998. J. Chem. Phys. 108, 5571. Ladas, S., Imbihl, R., Ertl, G., 1988a. Surf. Sci. 198, 42. Ladas, S., Imbihl, R., Ertl, G., 1988b. Surf. Sci. 197, 153. Ladas, S., Imbihl, R., Ertl, G., 1989. Surf. Sci. 219, 88. Ladas, S., Imbihl, R., Ertl, G., 1993. Surf. Sci. 280, 14. Langmuir, I., 1921. Trans. Faraday Soc. 17, 607. Langmuir, I., 1922. Trans. Faraday Soc. 17, 672. Lauterbach, J., Asakura, K., Rotermund, H.H., 1994. Surf. Sci. 313, 52. Lauterbach, J., Asakura, K., Rasmussen, P.B., et al., 1998. Physica D 123, 493. Lauterbach, J., Bonilla, G., Pletcher, T.D., 1999. Chem. Engrg. Sci. 54, 4501. Lele, T., Lauterbach, J., 2002. Chaos 12, 164. Lesley, M.W., Schmidt, L.D., 1985. Surf. Sci. 155, 215. Li, X.J., Kevrekidis, I.G., Pollmann, M., et al., 2002. Chaos 12, 190. Lim, Y.-S., Berdau, M., Naschitzki, M., et al., 1994. J. Catal. 140, 292. Lin, A.L., Bertram, M., Martinez, K., et al., 2000. Phys. Rev. Lett. 84, 4240. Liu, D.-J., Evans, J.W., 2002. J. Chem. Phys. 117, 7319. Liu, D.-J., Evans, J.W., 2005. Multiscale Model. Simul. 4, 424. Liu, H., Kevrekidis, I.G., Hinz, M., et al., 2008. Phys. Rev. E, submitted for publication. Locatelli, A., Mentes, T.O., Aballe, L., et al., 2006. J. Phys. Chem. B 110, 19108. Lombardo, S.J., Esch, F., Imbihl, R., 1992a. Surf. Sci. 271, L367. Lombardo, S.J., Slinko, M., Fink, T., et al., 1992b. Surf. Sci. 269/270, 481. Lombardo, S.J., Fink, T., Imbihl, R., 1993. J. Chem. Phys. 98, 5526. Lox, E.S.J., Engler, B.H., 1999. Environmental Catalysis. Wiley-VCH, Weinheim. Luss, D., 1992. Physica 188A, 68. Luss, D., Marhawa, B., 2002. Chaos 12, 172. Luss, D., Sheintuch, M., 2005. Catal. Today 105, 254. Luther, R., 1906. Z. Elektrochem. 32, 596. Müller, E.W., Tsong, T.T., 1969. Field Ion Microscopy. Elsevier, New York. Magtoto, N.P., Richardson, H.H., 1998. Surf. Sci. 417, 189.


425

Makeev, A., Imbihl, R., 2000. J. Chem. Phys. 113, 3854. Makeev, A., Hinz, M., Imbihl, R., 2001. J. Chem. Phys. 114, 9083. Makeev, A.G., 1996. Math. Modelling 2, 115 (in Russian). Makeev, A.G., Slinko, M.M., Janssen, N.M.H., et al., 1996. J. Chem. Phys. 105, 7210. Makeev, A.G., Nieuwenhuys, B.E., 1998a. Surf. Sci. 418, 432. Makeev, A.G., Nieuwenhuys, B.E., 1998b. J. Chem. Phys. 108, 3740. Makeev, A.G., Kevrekidis, I.G., 2004. Chem. Eng. Sci. 59, 1733. Malchow, H., Schimansky-Geier, L., 1985. Noise and Diffusion in Bistable Nonequilibrium Systems. Teubner, Berlin. Manrubia, S.C., Mikhailov, A.S., Zanette, D.H., 2004. Emergence of Dynamical Order, vol. 2. World Scientific, Singapore. Marbach, H., Gunther, S., Luerssen, B., et al., 2002a. Catal. Lett. 83, 161. Marbach, H., Hinz, M., Günther, S., et al., 2002b. Chem. Phys. Lett 364, 207. Marbach, H., Lilienkamp, G., Wei, H., et al., 2003. PCCP 5, 2730. Marbach, H., Günther, S., Neubrand, T., et al., 2004a. J. Phys. Chem. B 108, 15182. Marbach, H., Günther, S., Neubrand, T., et al., 2004b. Chem. Phys. Lett 395, 64. McMillan, N., Lele, T., Snively, C., et al., 2005. Catal. Today 105, 244. Medvedev, V.K., Suchorski, Y., 1996. Surf. Sci. 364, L540. Meissen, F., Patchett, A.J., Imbihl, R., et al., 2001. Chem. Phys. Lett. 336, 181. Meron, E., Bär, M., Hagberg, A., et al., 2001. Catal. Today 70, 331. Mertens, F., Imbihl, R., 1994. Nature 370, 124. Mertens, F., Imbihl, R., Mikhailov, A., 1994. J. Chem. Phys. 101, 9903. Mertens, F., Gottschalk, N., Bär, M., et al., 1995. Phys. Rev. E 51, R5193. Mertens, F., Imbihl, R., 1995. Chem. Phys. Lett. 242, 221. Mertens, F., Imbihl, R., 1996a. Surf. Sci. 347, 355. Mertens, F., Imbihl, R., 1996b. J. Chem. Phys. 105, 4317. Mertens, F., Schwegmann, S., Imbihl, R., 1997. J. Chem. Phys. 106, 4319. Mikhailov, A., 1994. Phys. Rev. E 49, 5875. Mikhailov, A.S., 1991. Foundations of Synergetics, vol. 1. Springer, Berlin. Mikhailov, A.S., Zykov, V.S., 1991. Physica 52D, 379. Mikhailov, A.S., Ertl, G., 1995. Chem. Phys. Lett. 238, 104. Mikhailov, A.S., Loskutov, A.Y., 1996. Foundations of Synergetics II, second ed. Springer, Berlin. Mikhailov, A.S., Showalter, K., 2006. Phys. Rep. 425, 79. Miners, J.H., Gardner, P., 2000. J. Phys. Chem. B 104, 10265. Moeller, P., Wetzl, K., Eiswirth, M., et al., 1986. J. Chem. Phys. 85, 5328. Mola, E.E., King, D.A., Irurzun, I.M., et al., 2004. Surf. Rev. Lett. 11, 57. Monine, M., Pismen, L.M., 2001. Catalysis Today 70, 311. Monine, M., Pismen, L., Bar, M., et al., 2002. J. Chem. Phys. 117, 4473. Monine, M.I., Schaak, A., Rubinstein, B.Y., et al., 2001. Catal. Today 70, 321. Monine, M.I., Pismen, L., Imbihl, R., 2004. J. Chem. Phys. 121, 11332. Mross, W.D., 1983. Catal. Rev.-Sci. Eng 25, 591. Murray, J., 1989. Mathematical Biology. Springer, Berlin. Nettesheim, S., von Oertzen, A., Rotermund, H.H., et al., 1993. J. Chem. Phys. 98, 9977. Nicolis, G., 1989. Physics of far-from-equilibrium systems and self-organisation. In: Davies, P.C.W. (Ed.), The New Physics. Cambridge University Press, Cambridge. Niehus, H., 1984. Surf. Sci. 145, 407. Over, H., Kim, Y.D., Seitsonen, A.P., et al., 2000. Science 287, 1474. Over, H., Muhler, M., 2003. Progr. Surf. Sci. 72, 3. Pasteur, A.T., Guo, X.-C., Ali, T., et al., 1996. Surf. Sci. 366, 564. Pavlenko, N., Evans, J.W., Liu, D.J., et al., 2002. Phys. Rev. E 65, 016121. Pavlenko, N., Imbihl, R., Evans, J.W., et al., 2003. Phys. Rev. E 63, 016212. Perera, A., Vicente, L., 2003. PCCP 5, 1888. Petrov, V., Quyang, Q., Swinney, H.L., 1977. Nature 388, 655. Pineda, M., Imbihl, R., Schimansky-Geier, L., et al., 2006. J. Chem. Phys. 124, 44701.

426

R. Imbihl

Pineda, M., Schimansky-Geier, L., Imbihl, R., 2007. Phys. Rev. E 75, 061107. Pismen, L.M., Imbihl, R., Rubinstein, B.Y., et al., 1998. Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 58, 2065. Pismen, L.M., Rubinstein, B.Y., 1999. Chaos 9, 55. Pismen, L.M., 2006. Patterns and Interfaces in Dissipative Dynamics. Springer, Berlin. Pollmann, M., Rotermund, H.H., Ertl, G., et al., 2001. Phys. Rev. Lett. 86, 6038. Punckt, C., Rotermund, H.H., 2007. Phys. Chem. Chem. Phys. 9, 3635. Rafti, M., Vicente, J.L., Uecker, H., et al., 2006. Chem. Phys. Lett. 421, 577. Rausenberger, B., Swiech, W., Rastomjee, C.S., et al., 1993. Chem. Phys. Lett. 215, 109. Rehmus, P., Ross, J., 1985. In: Field, R., Burger, M. (Eds.), Oscillations and Traveling Waves in Chemical Systems. Wiley, New York. Reichert, C., Starke, J., Eiswirth, M., 2001. J. Chem. Phys. 4829. Ressler, T., Hagelstein, M., Hatje, U., et al., 1997. J. Phys. Chem. B 101, 6680. Reuter, K., Scheffler, M., 2002. Phys. Rev. B 65, 034506. Reuter, K., Scheffler, M., 2003. Phys. Rev. Lett. 90, 046103. Reuter, K., Frenkel, D., Scheffler, M., 2004. Phys. Rev. Lett. 93, 116105. Reuter, K., Scheffler, M., 2004. Appl. Phys. A 78, 793. Rose, H., Hempel, H., Schimansky-Geier, L., 1994. Stat. Phys. A 206, 421. Rose, K.C., Battogtokh, D., Mikhailov, A., et al., 1996. Phys. Rev. Lett. 76, 3582. Rose, K.C., Berton, B., Imbihl, R., et al., 1997. Phys. Rev. Lett. 79, 3427. Rotermund, H.H., Jakubith, S., von Oertzen, A., et al., 1989. J. Chem. Phys. 91, 4942. Rotermund, H.H., Jakubith, S., Von Oertzen, A., et al., 1991. Phys. Rev. Lett. 66, 3083. Rotermund, H.H., Lauterbach, J., Haas, G., 1993. Appl. Phys. A 57, 507. Rotermund, H.H., Haas, G., Franz, R.U., et al., 1995. Science 270, 608. Rotermund, H.H., 1997. Surf. Sci. Rep. 29, 265. Rotermund, H.H., Pollmann, M., Kevrekidis, I.G., 2002. Chaos 12, 157. Sachs, C., Hildebrand, M., Volkening, S., et al., 2001. Science 293, 1635. Sales, B.C., Turner, J.E., Maple, M.B., 1982. Surf. Sci. 114, 381. Sander, M., Imbihl, R., 1991. Surf. Sci. 255, 61. Sander, M., Imbihl, R., Ertl, G., 1991. J. Chem. Phys. 95, 6162. Sander, M., Imbihl, R., Ertl, G., 1992a. J. Chem. Phys. 97, 5193. Sander, M., Imbihl, R., Schuster, R., et al., 1992b. Surf. Sci. 271, 159. Schüth, F., Henry, B.E., Schmidt, L.D., 1993. Adv. Catal. 39, 51. Schütz, E., Hartmann, N., Kevrekidis, Y., et al., 1998. Catal. Lett. 54, 181. Schütz, E., Esch, F., Gunther, S., et al., 1999. Catal. Lett. 63, 13. Schütz, E., Imbihl, R., 2002. Zeit. Phys. Chem. 216, 509. Schaak, A., Imbihl, R., 1997. J. Chem. Phys. 107, 4741. Schaak, A., Imbihl, R., 1998. Chem. Phys. Lett. 283, 386. Schaak, A., Gunther, S., Esch, F., et al., 1999a. Phys. Rev. Lett. 83, 1882. Schaak, A., Shaikhutdinov, S., Imbihl, R., 1999b. Surf. Sci. 421, 191. Schaak, A., Imbihl, R., 2000. J. Chem. Phys. 113, 9822. Schaak, A., Imbihl, R., 2002. J. Chem. Phys. 116, 9021. Schmidt, T., Heun, S., Slezak, J., et al., 1998. Surf. Rev. Lett. 5, 1287. Schmidt, T., Schaak, A., Gunther, S., et al., 2000. Chem. Phys. Lett 318, 549. Schüth, F., Wicke, E., 1989. Ber. Bunsen-Ges. Phys. Chem. 93, 191. Schuster, H.G., 1989. Deterministic Chaos. VCH Publishers, Weinheim. Schütz, E., Hartmann, N., Kevrekidis, Y., et al., 1996. Faraday Discussions 47. Schwankner, R.J., Eiswirth, M., Moeller, P., et al., 1987. J. Chem. Phys. 87, 742. Sheintuch, M., Schmitz, R., 1977. Catal. Rev. Sci. Eng. 15, 107. Sheintuch, M., 1981. Chem. Eng. Sci. 36, 893. Shvartsman, S.Y., Schütz, E., Imbihl, R., et al., 1999. Phys. Rev. Lett. 83, 2857. Shvartsman, S.Y., Schütz, E., Imbihl, R., et al., 2000. Phys. Rev. Lett. 85, 902. Shvartsman, S.Y., Schütz, E., Imbihl, R., et al., 2001. Catal. Today 70, 301. Sieben, B., Bozdech, G., Ernst, N., et al., 1996. Surf. Sci. 352–354, 167.


Siera, J., Cobden, P., Tanaka, K., et al., 1991. Catal. Lett. 10, 335. Silveston, P.L., Hanika, J., 2002. Chem. Eng. Sci. 57, 3373. Singh-Boparai, S.P., King, D.A., 1980. In: Proc. 4th Int. Congr. Surf. Sci., Cannes, p. 403. Slinko, M., Fink, T., Loeher, T., et al., 1992. Surf. Sci. 264, 157. Slinko, M.G., Slinko, M.M., 1978. Catal. Rev. Sci. Eng. 17, 119. Slinko, M.M., Jaeger, N., 1994. Oscillatory Heterogeneous Catalytic Systems. Elsevier, Amsterdam. Slinko, M.M., Ukharskii, A.A.V.P.N., et al., 2001. Catal. Today 70, 341. Somorjai, G.A., York, R.L., Butcher, D., et al., 2007. PCCP 9, 3500. Starke, J., Reichert, C., Eiswirth, M., et al., 2006. Europhys. Lett. 73, 820. Suchorski, Y., Beben, J., Imbihl, R., 1998a. Prog. Surf. Sci. 59, 343. Suchorski, Y., Beben, J., Imbihl, R., 1998b. Surf. Sci. 405, L477. Suchorski, Y., Imbihl, R., Medvedev, V.K., 1998c. Surf. Sci. 401, 392. Suchorski, Y., Beben, J., James, E.W., et al., 1999. Phys. Rev. Lett. 82, 1907. Suchorski, Y., Beben, J., Imbihl, R., 2000. Surf. Sci. 454, 331. Suchorski, Y., Beben, J., Imbihl, R., et al., 2001. Phys. Rev. B 63, 165417. Swiech, W., Rausenberger, B., Engel, W., et al., 1993. Surf. Sci. 294, 297. Swiech, W., Rastomjee, C.S., Imbihl, R., et al., 1994. Surf. Sci. 307–309, 138. Takoudis, C.G., Schmidt, L.D., 1983. J. Phys. Chem. 87, 964. Tammaro, M., Sabella, M., Evans, J.W., 1995. J. Chem. Phys. 103, 10277. Tammaro, M., Evans, J.W., 1998a. Surf. Sci. 395, L207. Tammaro, M., Evans, J.W., 1998b. J. Chem. Phys. 108, 762. Tammaro, M., Evans, J.W., 1998c. J. Chem. Phys. 108, 7795. Tammaro, M., Evans, J.W., Rastomjee, C.S., et al., 1998. Surf. Sci. 407, 162. Thiel, P.A., Behm, R.J., Norton, P.R., et al., 1983. J. Chem. Phys. 78, 7448. Thompson, J.M.T., Stewart, H.B., 1987. Nonlinear Dynamics and Chaos. J. Wiley, New York. Thullie, J., Renken, A., 1991. Chem. Eng. Sci. 46, 1083. Tsarouhas, G.E., Ross, J., 1989. J. Chem. Phys. 89, 715. Turing, A.M., 1952. Phil. Trans. R. Soc. B237, 37. Tyson, J.J., Keener, J.P., 1988. Physica 32D, 327. Uecker, H., Imbihl, R., Rafti, M., et al., 2003. Chem. Phys. Lett. 382, 232. Uecker, H., 2005. Phys. Rev. E 71, 016207. Völkening, S., Bedürftig, K., Jacobi, K., et al., 1999. Phys. Rev. Lett. 83, 2672. van Hove, M.A., Koestner, R.J., Stair, P.C., et al., 1981. Surf. Sci. 103, 189. van Kampen, N.G., 1987. Stochastic Processes in Physics and Chemistry. North Holland, Amsterdam. van Tol, M.F.H., Gielbert, A., Nieuwenhuys, B.E., 1992a. Catal. Lett. 16, 297. Van Tol, M.F.H., Siera, J., Cobden, P.D., et al., 1992b. Surf. Sci. 274, 63. van Tol, M.F.H., Wouda, P.T., Nieuwenhuys, B.E., 1994. J. Vac. Sci. Technol., A 12, 2176. van Tol, M.F.H., de Maaijer-Gielbert, J., Nieuwenhuys, B.E., 1993. Chem. Phys. Lett. 205, 207. Vanag, V.K., Epstein, I.R., 2004. Phys. Rev. Lett. 92, 128301. Vance, W., Tsarouhas, G.E., Ross, J., 1989. J. Progr. Theor. Phys. 99, 331. Vayenas, C.G., Georgakis, C., Michaels, J., et al., 1981. J. Catal. 67, 348. Vayenas, C.G., Michaels, J.N., 1982. Surf. Sci. 120, L405. Verdasca, J., Borckmans, P., Dewel, G., 2002. PCCP 4, 1355. Veser, G., Esch, F., Imbihl, R., 1992. Catal. Lett. 13, 371–382. Veser, G., Imbihl, R., 1992. J. Chem. Phys. 96, 7155. Veser, G., Mertens, F., Mikhailov, A., et al., 1993. Phys. Rev. Lett. 71, 935. Veser, G., Imbihl, R., 1994a. J. Chem. Phys. 100, 8492. Veser, G., Imbihl, R., 1994b. J. Chem. Phys. 100, 8483. Visart de Bocarme, T., Kruse, N., 2002. Chaos 12, 118. Vishnevskii, A.L., Latkin, E., Elokhin, V.I., 1995. Surf. Rev. Lett. 2, 459. von Oertzen, A., Mikhailov, A.S., Rotermund, H.H., et al., 1996. Surf. Sci. 350, 259. von Oertzen, A., Mikhailov, A.S., Rotermund, H.H., et al., 1998. J. Phys. Chem. B 102, 4966. von Oertzen, A., Rotermund, H.H., Mikhailov, A.S., et al., 2000. J. Phys. Chem. B 104, 3155. Voss, C., Kruse, N., 1995. Appl. Surf. Sci 87, 127.

427

428

Voss, C., Kruse, N., 1996. Appl. Surf. Sci 94, 186. Voss, C., Kruse, N., 1998. Ultramicroscopy 73, 211. Wang, J.G., Li, W.X., Borg, M., et al., 2005. Phys. Rev. Lett. 95, 256102. Wehner, S., Hoffmann, P., Schmeisser, D., et al., 2005. Phys. Rev. Lett. 95, 038301. Wei, H., Lilienkamp, G., Imbihl, R., 2004. Chem. Phys. Lett. 389, 284. Wei, H., Lilienkamp, G., Davidsen, J., et al., 2006. Phys. Rev. E 73, 016210. Wei, H., Lilienkamp, G., Imbihl, R., 2007. J. Chem. Phys. 127, 024703. Wei, T.C., Phillips, J.A.C., 1992. Adv. Catal. 41, 359. Werner, H., Herein, D., Schulz, G., et al., 1997. Catal. Lett. 49, 109. Wicke, E., Kummann, P., Keil, W., et al., 1980. Ber. Bunsenges. Phys. Chem. 84, 315. Wider, J., Greber, T., Wetli, E., et al., 1998. Surf. Sci. 417, 301. Winfree, A.T., 1991. Chaos 1, 303. Wintterlin, J., 2002. Chaos 12, 108. Wolff, J., Papathanasiou, A.G., Kevrekidis, I.G., et al., 2001. Science 294, 134. Wolff, J., Papathanasiou, A.G., Rotermund, H.H., et al., 2003a. Phys. Rev. Lett. 90, 148301. Wolff, J., Papathanasiou, A.G., Rotermund, H.H., et al., 2003b. J. Catal. 216, 246. Wolff, J., Papathanasiou, A.G., Rotermund, H.H., et al., 2003c. Phys. Rev. Lett. 90, 018302. Wolff, J., Rotermund, H.H., 2003. N.J. Phys. 5. Wu, X.-G., Kapral, R., 1992. Physica A 188, 284. Yamamoto, S.Y., Surko, C.M., Maple, M.B., et al., 1995. Phys. Rev. Lett. 74, 4071. Zhdanov, V.P., Kasemo, B., 1994a. Physica D 70, 383. Zhdanov, V.P., Kasemo, B., 1994b. Surf. Sci. Rep. 20, 111. Zhdanov, V.P., 1999a. J. Chem. Phys. 110, 8748. Zhdanov, V.P., 1999b. Phys. Rev. E 59, 6292. Zhdanov, V.P., 2004. Surf. Sci. Rep. 55, 1. Zhdanov, V.P., Kasemo, B., 2005a. Surf. Sci. Rep. 39, 25. Zhdanov, V.P., Kasemo, B., 2005b. Surf. Sci. 588, L220. Ziff, R.M., Gulari, E., Barshed, Y., 1986. Phys. Rev. Lett. 56, 2553. Zuburtikudis, I., Saltsburg, H., 1992. Science 258, 1337.

R. Imbihl

CHAPTER 10

Electron Transfer and Nonadiabaticity

Bengt I. LUNDQVIST Department of Applied Physics, Chalmers University of Technology SE-412 96, Göteborg, Sweden Center for Atomic-scale Materials Design, Department of Physics, Technical University of Denmark DK-2800 Kgs. Lyngby, Denmark

Anders HELLMAN Department of Applied Physics, Chalmers University of Technology SE-412 96, Göteborg, Sweden Competence Center for Catalysis, Chalmers University of Technology SE-412 96, Göteborg, Sweden

Igor ZORIC´ Department of Applied Physics, Chalmers University of Technology SE-412 96, Göteborg, Sweden



Contents 10.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Adiabaticity and nonadiabaticity . . . . . . . . . . . . . . . . . . . . . . . . 10.2.1. Mathematical preliminaries . . . . . . . . . . . . . . . . . . . . . 10.2.2. Adiabatic approximation . . . . . . . . . . . . . . . . . . . . . . . 10.2.3. Curve crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.4. One-state problem . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.5. Two-state problem . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.6. Nonadiabatic transitions between two states . . . . . . . . . . . . 10.2.7. Nonadiabatic transitions between multiple states – semiclassically 10.2.8. Nonadiabatic transitions between multiple states – a quantum case 10.2.9. Master equation for ET processes . . . . . . . . . . . . . . . . . . 10.2.10. In summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2.11. Time-dependent density-functional theory and nonadiabaticity . . 10.2.12. Friction treatment . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Electron transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.2. Image potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.3. Adsorbate-induced electron structure . . . . . . . . . . . . . . . . 10.3.4. Molecule–surface collisions . . . . . . . . . . . . . . . . . . . . . 10.3.5. Nonadiabaticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3.6. Dynamic electron transfer . . . . . . . . . . . . . . . . . . . . . . 10.3.6.1. Multitude of electron-transfer processes . . . . . . . . . 10.3.7. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4. Nonadiabatic processes at surfaces . . . . . . . . . . . . . . . . . . . . . . . 10.4.1. Electron and ion spectroscopy according to Hagstrum . . . . . . . 10.4.2. Surface chemiluminescence and the NNL model . . . . . . . . . . 10.4.3. Vibrational damping of adsorbates . . . . . . . . . . . . . . . . . . 10.4.4. Exoelectrons and chemicurrents . . . . . . . . . . . . . . . . . . . 10.4.5. Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4.6. Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5. Chemical reactions at surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.1. Sticking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2. Vibrational dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.3. Thermal chemical reactions at surfaces . . . . . . . . . . . . . . . 10.6. Conclusions and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

433 435 436 437 438 438 439 440 441 446 447 448 448 448 449 449 452 453 454 455 455 456 471 471 471 475 483 487 493 495 496 497 510 513 514 515

Abstract Electron processes at solid surfaces are numerous and important. Nonadiabaticity (NA) is a concept that is well documented but that seems to face some resistance towards general acceptance in surface dynamics. There is a great multitude of surface processes, and in many

430

subfields NA is just taken for granted, while in other areas some reluctance can be spotted. Even recently, gas–surface interactions could be reviewed cautiously by saying that there is “growing evidence” that “there exist cases where understanding means need to go beyond BOA” (the Born–Oppenheimer approximation), that is to apply NA. No doubt, surface dynamics does seldom provide spectacular manifestations of NA, but still there are numerous NA effects, as listed in this chapter. Electron transfer (ET) is part of such processes. The multitude of ET phenomena at surfaces is reviewed, with a focus on key concepts connected with electron-structure features in the surface region. Documentation for ET and NA in thermal surface processes is also provided. Finally, some attempts are made to find factors for surfacedynamical processes that can be described in adiabatic terms.

431


433

10.1. Introduction Wherever scanning the horizon, you see surfaces. Surfaces that provide scenes for many grand spectacles. Photosynthesis, catalysis, and growth perform for life. Paints, glow lamps, and lightworms bring light and colors. Radiators, suns, and furs give heat and warmth. Etcetera. Gas–surface interactions and surface dynamics encompass numerous important processes and phenomena. The limelight on the research area and its progress from awarding the 2007 Nobel Prize to Gerhard Ertl reflects this and is well deserved. Some technical terms are molecular scattering, energy accommodation, adsorption, desorption, molecular dissociation, surface diffusion, catalytic reactions, etching, deposition, surface chemiluminescence, and chemicurrents. Other chapters of this handbook provide accounts of many of them. This chapter picks up some general concepts and treats two relevant phenomena, electron transfer (ET) and nonadiabaticity (NA). Its aim is to define, illustrate, list, document, and discuss ET and NA, and to assess their relevance for many surface processes and phenomena. Understanding of NA of course calls for the concept of adiabaticity. The word “adiabatic” has a Greek origin and means “occurring without loss or gain of heat” (MerriamWebster, 2007). In quantum-mechanics a sharp and physically more proper definition is given, separating atomic and electronic motions in the adiabatic approximation, also called the Born–Oppenheimer approximation (BOA) (Born, 1927; Born and Oppenheimer, 1927). The BOA assumes electrons to follow the nuclear motion instantaneously, and the nuclear and electronic degrees of freedom are assumed to be uncoupled. This brings us to the potential-energy surface (PES) and adiabatic and diabatic descriptions (O’Malley, 1971), and to methods to calculate PES’s (Richardson and Holloway, 2000), like the density-functional theory (DFT) (Hohenberg and Kohn, 1964; Kohn and Sham, 1965; von Barth and Hedin, 1972; Gunnarsson et al., 1972; Gunnarsson and Lundqvist, 1976; Langreth and Mehl, 1981; Jones and Gunnarsson, 1989; Perdew et al., 1992, 1996, 1999). Theoretical approaches to the problem of NA electronic processes and phenomena that manifest NA and ET are at focus in this chapter. Although NA has been considered for at least half a century in many subfields of surface science, the evidences that the BOA might break down in some thermal surface reactions seem to have been exposed in leading media only recently (Metiu and Gadzuk, 1981; Wodtke, 2006), spurred, by, e.g., accounts for experimental results for state-selected highly vibrationally excited molecules (Hou et al., 1999a; Huang et al., 2000; White et al., 2005) and initial sticking (Nørskov and Lundqvist, 1979b; Zhdanov, 1997; Hellman et al., 2003; Behler et al., 2005). In many areas of physics, there are real highlights in the breakdown of the BOA. For instance, in (i) metals, where the continuum of easily excited electron–hole pairs (EHP’s) makes NA effects frequent, including the spectacular one of superconductivity (Schrieffer, 1974), the seminal one of electronic mass enhancement (Grimvall, 1986), and the also very spectacular Peierls (Peierls, 1929, 1955) and Jahn–Teller (Jahn, 1937, 1938) effects; (ii) dynamics of gas-phase reactions (Nikitin, 1974), like in the crossed molecular beam NA reaction of F (2 P3/2 , 2 P1/2 ) + HBr (DBr) → HF (DF) + Br (2 P3/2 , 2 P1/2 ), which involves low-lying excited spin-orbit states (Hepburn et al., 1981), with electron-energy transfer between nascent HF and Br, resulting in a transfer onto the upper PES; and (iii) reactions of

434

B.I. Lundqvist et al.

alkali metal surfaces, with halogens and oxygen, resulting in emission of exoelectrons and surface chemiluminescence (Kasemo, 1974; Kasemo and Walldén, 1975; Greber, 1997), i.e. electrons or photons, respectively, leaving the surface. Metallic surfaces, with their continua of low-energy EHP excitations give less prominent luminescence phenomena than insulators and semiconductors, which are not accounted for in this chapter. However, for very exothermic reactions on metal surfaces, there are cases with clear experimental evidences that the BOA breaks down and that energy is transferred from the reactants to electronic excitations in the substrate (e.g., EHP’s and, in nanoscale structures, localized surface plasmons), incl. point (iii) above. Moreover, excitation of hot electrons can be observed during adsorption of various gases on thin metal films by detection of a chemicurrent across a Schottky barrier on, e.g., an n-type silicon substrate (Nienhaus, 2002). Further examples of NA effects at metal surfaces include scattering of metastable atoms, ion scattering, and energy dissipation by scattering of highly vibrationally excited molecules, vibrational energy relaxation, dissociative adsorption, and associative desorption (Diekhöner et al., 2001a). Reactive processes at surfaces (Nørskov, 1990) are technologically important, as, e.g., heterogeneous catalysis and materials growth. They are often studied with metals as model substrates, as favorable energy landscapes may be provided here. Compared with the gas phase, reaction barriers might be reduced substantially. For the reaction dynamics, the proximity provided by the metal is also important. On the other side, NA is present or just around the corner at metals. Bond making and bond breaking involve ET from one electronic state to another. Surfaces are important by providing a scene for the large number of events constituting a gas–surface reaction. The slow progress in the analysis of surface chemical reactions might be due to the additional complexity caused by energy-dissipation channels provided by the surface that interacts with the reactants, particularly efficient on metals. For instance, initial sticking is the start of many important surface processes, such as epitaxial growth, heterogeneous catalysis, and oxidation. A seemingly simple elementary reaction event, the dissociation of an oxygen molecule on a simple metal surface like Al(111), has confronted the surface-science community with a number of puzzles over the years. Recently, an NA-based explanation appears to successfully account for the measured sticking behavior, however (Hellman et al., 2003). Mechanisms for ET are key elements of surface processes. However, the BOA provides a simple and tempting tool for description, with its concept of a nuclear motion that evolves on a PES. Many processes seem possible to describe in this way, within common experimental accuracy. To call into question the applicability of BOA demands extra efforts. The mere de facto use of the BOA and the transition state as concepts and approximations in surfaces processes, even on metal surfaces, is a fortunate circumstance. Inherent criteria say that NA should be the fundamental rule. The NA concept is well documented in many cases but seems to face some resistance against general acceptance in surface dynamics, varying from one subfield to another. Certainly, there is “growing evidence that understanding the dynamics of reactions at metal surfaces requires insights and approaches that go beyond the BOA” (Wodtke et al., 2004). The above comments set the scene for this chapter on NA and ET. On one hand, there are some very clear manifestations of these phenomena. And on the other hand, their effects


435

in a central and technically important subfield, chemical reaction dynamics, are subtle and have been waiting rather long for their disclosure. The remarkably successful “standard model of chemical reactivity” (Eyring and Polanyi, 1935) relies on the BOA. Its reactants evolve to the product state on one PES in an electronically adiabatic way. NA coupling effects are neglected. The approximation is valid, however, only if the involved PES’s are significantly separated. PES crossing happens often at, e.g., metal surfaces, with their continua of EHP’s. Then different electronic configurations couple to each other, i.e. an NA coupling is becoming relevant. Despite this fact, the BOA has frequent successes in descriptions of surface-reaction dynamics on the ground-state PES. Transition-state theory (Eyring, 1935; Wigner, 1938; Eyring, 1938) is used routinely in, e.g., such methods as kinetic Monte Carlo simulations (KMC) (Landau and Binder, 2000). “Future work will certainly focus on helping to better define under what conditions this standard model of reactivity can be applied to catalytically important reactions at metal surfaces” (Wodtke, 2006). “The next generation of chemical simulation packages . . . can take into account the role of excited electronic states in surface chemistry, going beyond the BOA” (Wodtke, 2006). There will be important contributions to our understanding of chemistry involving excited electrons in solids. “For example, our ability to learn how to power catalytic processes with light (photocatalysis), as opposed to heat (conventional thermal catalysis), will rely on new understanding of excited states in solids, an area of future technology that is essential to a world with diminishing cheap oil reserves” (Wodtke, 2006). The scope of this chapter is to (i) introduce BOA, NA, and ET carefully, (ii) give an extensive account of the multitude of NA processes at surfaces, including such with ET, (iii) describe chemical surface reactions adiabatically and, in particular, NA, with examples where that is needed, and (iv) sum up and assess the present situation. The significance of NA effects in dynamical processes at metal surfaces is today a topic of debate and frequent reviews, some of them with an aim to explain the many NA occurrences in a unifying view (Langreth and Suhl, 1984; Greber, 1997; Nienhaus, 2002; Wodtke et al., 2004; Hasselbrink, 2006; Luntz, 2007; Wodtke et al., 2008).

10.2. Adiabaticity and nonadiabaticity In molecular physics at large, huge efforts go into studies of adiabatic properties, experimentally by, e.g., various spectroscopies and theoretically by, e.g., solving the adiabatic equations to get static properties, including vibrations. Both adiabaticity and NA call for proper introductions. The following one closely follows that of O’Malley (1971). The adiabatic states are simply the eigenstates of the electronic Hamiltonian, He (see below). The adiabatic PES’s are the corresponding eigenvalues of He , defined for each nuclear configuration, R. They are, in fact, the potential energies that would exist for the atoms, if they were not able to move. Since the discovery of quantum mechanics, a key molecular-physics problem has been to understand the permanent bound electronic states, together with their low-lying vibrational and rotational levels, as revealed by the spectroscopic observation of transitions between these states.

436


In dynamics, a natural question is “What are the forces that two slowly moving atoms exert on each other?” (O’Malley, 1971). This question is meant to be answered by the so-called diabatic states of molecules, through their diabatic PES’s (O’Malley, 1971). Diabatic (i.e. not adiabatic) states thus provide corrections to the common misconception that “atoms move along the adiabatic PES’s, as long as they are much slower than the valence electrons”. Already early experiments have shown that the forces often differ from the adiabatic ones and caused a concern to define the actual forces (Smith, 1969), or more fundamentally the corresponding diabatic states. The search for such states follows a natural direction. Each step is designed to meet a specific experimental result (O’Malley, 1971). First traditional adiabatic states are presented as answers to questions about permanent or stationary states. Then some key definitions are introduced for the two- or many-state problem generated by atomic and electronmolecule collisions, albeit they have full generality. In, e.g., the Na + Cl system, the ionic and covalent states form the natural expansion basis for the molecular wavefunction. Another example is symmetric charge exchange, where due to experimental results the adiabatic states have been rejected in favor of single-configuration molecular-orbital wavefunctions. Finally, the dissociative attachment and recombination of electrons with molecules require an expansion of the wavefunction in a quasistationary representation of the electronic Hamiltonian, He , general enough to be able to embrace the earlier singleconfiguration and covalent-ionic representations. 10.2.1. Mathematical preliminaries A quantum-mechanical many-particle system is described by a many-body wavefunction Ψ (r, R, t), where the coordinates r and R give the positions and spins of all electrons and nuclei, respectively. The time-dependent Schrödinger equation describes the time evolution, ∂Ψ (r, R, t) = H Ψ (r, R, t), (10.1) ∂t where H is the total Hamiltonian of the system and h¯ Plancks constant. As most electronstate results here apply equally to di- and polyatomic systems, the explicit discussion can be limited to diatomics without loss of generality. The Hamiltonian i h¯

H = TR + He

(10.2)

separates the nuclear kinetic-energy operator TR = −(h¯ 2 /(2M))∇R2 , M being the reduced mass of the two nuclei, and R the internuclear radius vector, from the rest. The remaining electronic Hamiltonian, N N

e2 /rij He = − h¯ 2 /(2m) ∇i2 − ZA e2 /rAi − ZB e2 /rBi + i=1

+ ZA ZB /R,

j =i=1

(10.3)

where the sum is over all N electrons, rAi , rBi , and rij the vectors between electron i and nuclei A, B and electron j, respectively, m the electronic mass, with the small mass ratio m/M, and ZA and ZB the nuclear charges.


437

The solution is naturally expressed in some set of electronic basis functions, φi (r, R). The functions φi can be chosen “adiabatic” in one of the many senses of the term, namely with the internuclear coordinate R as a parameter R, rather than a dynamic variable. So far the φi ’s are otherwise undefined. In the stationary part of the Schrödinger equation Ψ (r, R) = (10.4) φi (r, R)χi (R) = φi χi , i

i

where χi (R) are eigenfunctions for the nuclear motion, is used. In the stationary part of the Schrödinger equation (10.1) this expansion is utilized. Multiplication from the left by φj∗ and integration over all coordinates give the formal set of coupled equations for the nuclear wavefunction χi (R),

Vij (R) + Tij + Tij χj (R), TR + Tii + Vii (R) − E χi (R) = − (10.5) j =i

where Vj k (R) = φj |He |φk are the electronic matrix elements of He , integrated over electronic coordinates, and T and T result from the action of the Laplacian operator on the product φi χi (∇ 2 φχ = φ∇ 2 χ + 2∇φ ∗ ∇χ + χ∇ 2 φ), Tij = −2(h¯ 2 /2M) φi |∇R |φj ∗ ∇R , (10.6) 2 2 Tij = −(h¯ /2M) φi |∇R |φj . (10.7) The odd parity of the operator ∇R in the Tij matrix element and the lack of rotational invariance make the diagonal matrix element Tii originally on the left side of Eq. (10.5) vanish. 10.2.2. Adiabatic approximation Equation (10.5) involves no approximations and is useless for most practical purposes. Approximations have to be made to get results for a realistic system. Because of their high mass, the atomic nuclei can be treated as slow classical objects described by a trajectory R(t). With this approximation, the electronic Schrödinger equation can be derived, where the electronic Hamiltonian He depends parametrically on R, i.e. on time t. Thus He is changing slowly with time, which implies that electrons that start out in an eigenstate of He (0) will adiabatically follow the time evolution and finally end up in an eigenstate of He (t). This is a reason why the BOA is also called adiabatic approximation. The huge ratio M/m between nuclear and electronic masses stands behind our understanding of small and big molecules. It is 2000 already for H, 30,000 for atmospheric gases and over 400,000 for the heaviest atoms. Its key consequence is the qualitative difference between fast electrons and utterly slow nuclei. To the fast electrons the nuclei appear static, and the electrons can immediately adjust to any change in nuclear positions, whereas the nuclei experience the distribution of electrons only through the PES, on which the nuclei move. This is the foundation for the adiabatic approximation or BOA (Born, 1927), according to which T and T are neglected. Equation (10.5) then becomes TR + Vii (R) − E χi (R) = (10.8) Vij (R)χj (R). j =i

438


In these simpler equations the potential energies for nuclear motion and the coupling terms are given by the electronic energies Vij . In the BOA the nuclear wavefunction χi (R; t) is a stationary state, i.e. a system once in the ground state will remain there. Thus the BOA solution is called the stationary adiabatic representation. Justification for the BOA can simply be assessed by the magnitudes of T and T operating on χ for the radial component ∂/∂R of the gradient operator ∇R (assuming that the rotational velocities are not much larger than the radial ones). Now |∇R χ| Kχ, where the wave number K is related to the nuclear kinetic energy E by K = (2ME/h¯ 2 )1/2 . As such a simple expression is lacking for the parametric variation of φ with R, one has to rely on the experience that normal, well-behaved electronic states do not vary greatly over a distance much smaller than the atomic unit of distance, a0 . This makes the assumption |∇R φj | a0−1 |φj | very reasonable, barring any anomalously impulsive jumps in φj . It also implies that the magnitudes of T and T operating on χi may be compared with T χi , T χ = Eχ,

(10.9)

|T χ| 2(Ry ∗ E)

1/2

1/2

(m/M)

|T χ| Ry ∗ m/M|χ|,

|χ|,

(10.10) (10.11)

where E is the local nuclear kinetic energy [E − Vii in Eq. (10.5)], Ry = h¯ 2 /(2ma02 ) the Rydberg unit of energy (13.6 eV), and m/M is the infinitesimal mass ratio. Further, Eq. (10.11) shows that the T operator is independent of energy and with a value in the range 10−2 to 10−4 eV, depending on M. It may be neglected at all energies. The energy-dependent T term is likewise seen to be very small, except in the limit of high kinetic energy, a limit that can be estimated from the fact that T attains the value of 2 for a nuclear velocity of one atomic unit (v0 = e2 /h¯ ). Thus nuclear velocities below e2 /h¯ (E M/m Ry) are referred to as the “adiabatic region”, where the BOA can normally be expected to be good. 10.2.3. Curve crossing In the adiabatic region the starting point is Eq. (10.8). However, there are situations with BOA failing even at low velocities. The assumption behind it can be invalid for “adiabatic” electronic states that cross or rather suffer an “avoided crossing” (see below). A closer look shows this problem to arise from a bad choice of the basis functions φi , however. There always exists some obvious and reasonable definition of φi that avoids this artificial difficulty and preserves the essential distinction between nuclei and electrons, which is lost when the BOA is abandoned (O’Malley, 1971). 10.2.4. One-state problem The understanding of the permanently bound electronic states, together with their lowlying vibrational and rotational levels, has been a focus in molecular physics since long. This is a part of “the one-state problem” in the present approach. The BOA reduces the exact coupled equations (10.5) to the form (10.8), where heretofore the electronic functions


439

φi have been arbitrary. Chosen as the stationary eigenfunctions of the electronic Hamiltonian He (Eq. (10.3)), He φiad = Viiad (R)φiad , φiad |He |φjad

(10.12)

= = they constitute a diagonal representaor equivalently tion of He in stationary states, the so-called adiabatic states. The superscripts “ad” means adiabatic (in the sense that R enters only parametrically). The stationary adiabatic representation should be a more proper name. There are countless other adiabatic but nonstationary representations. The terminology seems firmly established. These adiabatic states reduce Eq. (10.8) to the very simple BOA form TR + Viiad (R) − E χi (R) = 0, (10.13) Vijad

Vijad (R)δij ,

first derived by Born and Oppenheimer. The electronic energy eigenvalue Viiad (R) becomes the potential energy for the nuclear motion, in agreement with a qualitative picture of the electrons adjusting themselves rapidly to the slowly changing nuclear positions. An essential property of Eq. (10.13) is that the χi ’s are totally uncoupled, so that the states, i, are permanent. This makes the stationary adiabatic representation of He perfect for describing the ground and low-lying electronic states of molecules, as observed in traditional spectroscopies, which are in fact permanent (at least in between radiative transitions). In other words, Eq. (10.13) describes a one-state problem. The famous noncrossing rule of von Neumann and Wigner (1929) is another interesting property of the stationary adiabatic representation. It states that two PES’s, Viiad (R) and Vjjad (R), may not cross, if they have the same symmetry (spin, parity, angular momentum, . . . ). This rule can be derived rigorously and straightforwardly (von Neumann and Wigner, 1929) from the definition (10.12). However, the noncrossing rule holds true only for the diagonal representation of He , but not for nondiagonal ones. It is an artificial mathematical construct and not in any sense a law of nature, an often overlooked fact. 10.2.5. Two-state problem The Na + Cl collision system has two atoms with very different electronegativities. The nature of forces at wide separation is obvious (Zener, 1932). At finite separations, the answer to the force question brings in an excellent generalization of the one-state problem, The interacting Na + Cl electron system can be described as either covalent (Na + Cl) or ionic (Na+ + Cl− ) at large and moderate separations R, with wavefunctions φcov and φion , respectively. For large R, the covalent configuration is known to have a lower energy than the ionic one, while the opposite applies for small R values. Graphs of the expectation values Vcov = φcov |He |φcov and Vion = φion |He |φion as functions of R (Fig. 10.1) illustrate what happens to electronic states of the same symmetry (1 for both) which violate the noncrossing rule. The crossing occurs, because they are not (nor should be) the stationary eigenvalues which diagonalize He . Rather, in this two-state space the He matrix Vcov Vcov,ion Vij = φi |He |φj = (10.14) Vion,cov Vion

440


is nondiagonal. The off-diagonal elements are neither vanishing nor negligible, albeit small (they are much larger than T or T , so that T |Vcov,ion | 1 Ry). Collisions in this system are simple to describe (Zener, 1932). The wavefunctions φcov and φion of the two physical states are chosen as the electronic basis in the expansion (10.4) of the full molecular wavefunction, Φ(r, R) = φcov χcov (R) + φion χion (R).

(10.15)

The two relevant nuclear-motion wavefunctions χcov and χion satisfy the general equation (10.8) in the simple form TR + Vcov (R) − E χcov (R) = −Vcov,ion (R)χion (R), (10.16) TR + Vion (R) − E χion (R) = −Vion,cov (R)χcov (R). (10.17) Here the roles of elements in the electronic matrix He are clear. The diagonal V ’s are the potential energies for elastic motion in each particular state, while the nondiagonal elements, Vcov,ion and Vion,cov , provide the coupling between the two states. This nonvanishing coupling makes the present Eqs. (10.16) and (10.17) differ in an essential way from the corresponding Eq. (10.13) of the one-state problem, which explicitly rules out transitions. 10.2.6. Nonadiabatic transitions between two states Naturally, the adiabatic and diabatic representations give rise to the same physics, as long as the BOA is not used. However, the diabatic representation can be more convenient to use than to calculate the coupling between different adiabatic states. In it the coupling appears more naturally, as off-diagonal matrix elements of the electronic part of the Hamiltonian. The probability P of a transition from one state to another during the passage through the crossing point Re of Fig. 10.1 can be calculated from Eqs. (10.16) and (10.17) in a reasonable semiclassical approximation (Zener, 1932). It is found appreciable only in the

Fig. 10.1. Potential energy curves of the ionic and covalent 1 Σ states of NaCl, as functions of the internuclear separation R. In the adiabatic representation there is an avoided crossing of the adiabatic potential energy curves, whereas in the diabatic representation a crossing occurs at Re .


441

neighborhood of the crossing point Re , according to the explicit formula (Zener, 1932) P = 1 − e−2δ 2δ,

(10.18)

where δ=

(π)|Vcov,ion |2 / h¯ v d(Vcov − Vion )/dR .

(10.19)

The matrix elements Vij and the internuclear velocity v are evaluated at the crossing point Re . The approximate but very useful Eq. (10.18) is the famous Landau–Zener formula (Zener, 1932; Langmuir, 1932; Stückelberg, 1932). It predicts that the probability of leaving the covalent (ionic) state is small, when δ is so, i.e. at all but the very lowest velocities. Obviously, it is well worth to extend the application of the nondiagonal representation (10.14) of He and the resulting equations to describe elastic motion and transitions in other colliding systems. The picture with crossing PEC’s, as in Fig. 10.1, may be more than mathematically appropriate, even if the physics should not be a clean division into covalent and ionic. There are further classes of collision processes, in which the experimental facts also demand a nondiagonal representation of He (O’Malley, 1971). For them the motion of the system should be successfully described by the appropriate, and progressively more general, nondiagonal or diabatic representations, as suggested by theoretical considerations. These classes of collision processes include (i) charge exchange in helium scattering, (ii) dissociative recombination and attachment, and (iii) slow heavy-particle collisions (O’Malley, 1971). 10.2.7. Nonadiabatic transitions between multiple states – semiclassically The case with two electronic configurations with PES’s that cross can be generalized. Systems with several electronic configurations with close energies have several PES’s that cross and avoid crossing, respectively, dependent on the representation. The continuum of electronic states of metal surfaces, corresponding to the multitude of EHP’s, brings in the pictures of a multitude of PES’s (Fig. 10.2). For instance, a molecule close to a metal surface might have small spacings between electronic energies and a large Vmn , relatively speaking. This might happen, when the involved derivatives of the electronic wavefunction Ψn (r, R) with respect to the nuclear displacement and velocity get large, relatively speaking. This is the case, when the electronic state changes rapidly, when the spacings between the electronic levels are small, or when the velocities of the nuclei are large. Then parts neglected in the BOA get important and make it invalid. Each PES associated with a particular electronic configuration comes with a densely packed excitation spectrum (Fig. 10.2) due to the continuum of EHP’s. Hence, even small variations in the nuclear configuration can cause transitions in the electronic system, thus invalidating the BOA. Both adiabatic and NA aspects apply for an atom or molecule approaching a metal surface (Nørskov, 1981). The former give reaction paths, activation barriers, intermediates, adsorbate-induced electron and vibration structures, and much more. The latter give further complications in the theoretical description of the adsorption dynamics for gas-phase

442


Fig. 10.2. (Left side) Adapted from Nienhaus (2002). Schematic PES’s to explain an adiabatic transition and a NA transition. Here PES (1) may represent a molecule-surface potential and PES (2) the atom–surface interaction. The two diabatic PES’s cross at point Zcr . A precursor state with binding energy Ed is formed at ZΓ and an activation barrier Ea + Ed exists between the two potential minima. If the transition from (1) to (2) occurs adiabatically the interaction follows the dashed line (avoided crossing). However, in the case of an NA transition the system follows (1) beyond the crossing point and than makes a sudden transition to the dash-dotted PES (2) which describes (2) plus an e–h pair excitation. The continuum of EHP that can be created in the metal gives rise to a whole spectrum of excitation PES’s (2). (Right side) Adapted from Tully (1981). Schematic illustration of atom–surface PES’s, as functions of distance z from the surface and possible trajectories Tully (1981). (a) Barrier to adsorption. (b) No barrier to adsorption.

reactions often circumvented by invoking the BOA, but which with a continuum of lowlying EHP levels available may make electronic excitations easily occur for reactions at metal surfaces. Fortunately, the ground-state adiabatic PES is often a good starting point for the description of the motion of the adsorbate nuclei, and the electronic excitations can then often be considered as corrections to the adiabatic picture. In an adiabatic picture, the PES is in focus, and for, e.g., the sticking of an atom or molecule coming into the surface the entrance channel is particularly important. The required translational energy E might have to be high enough to surmount a possible activation energy Ea (Fig. 10.2). A model calculation in DFT for a typical adsorbate-substrate system gives a PES (Fig. 10.3) that illustrates this point and also how the adsorbate electron structure is changed upon approach. Adsorbate levels are shifted and broadened for an atom or molecule that is close to the surface. An initially unoccupied affinity level develops to a resonance, which in the BOA can get partly below the Fermi level of the substrate (Fig. 10.3), get (partially) filled up to the Fermi level. For an antibonding state this means a weakened intramolecular bond. Some stay there, like the spectroscopically identified O− 2 state for O2


443

Fig. 10.3. Adopted from Nørskov et al. (1981). Model-calculation results for the equipotential-energy curves for H2 parallel to the surface outside an Mg(0001) surface for a varying H–H separations and distances to the surface (left side). In addition, the development of the local density of states (LDOS) along the reaction path (right side), with intramolecular binding energy indicated. A conceptual picture of dissociative molecular adsorption (Hjelmberg et al., 1979; Lundqvist et al., 1979b; Nørskov et al., 1981; Lundqvist, 1990) develops by comparing the variation of the LDOS with the Fermi level and the corresponding filling of the antibonding molecular resonance, in the course of a possible dissociative molecular adsorption.

Fig. 10.4. Adapted from Kasemo and Lundqvist (1984). Competing mechanisms for filling of the adsorbate-induced hole state: resonance tunneling (left), Auger decay (middle), and radiative decay (right).

adsorption on Pt(111) (Fisher et al., 1980) and Ag(110) (Backx et al., 1981). Others get their molecular bonds broken, like H2 on Ni(100) (Andersson, 1978). The BOA might be viewed as having an infinite time to fill the resonance. In a NA approach, there is less time available, and the filling might be incomplete, thus forming a hole. The schematic Fig. 10.4 illustrates how during an adsorption process, electronic and optical excitations may appear in the deexcitation of the hole (Kasemo and Lundqvist, 1984). ET is part of several of the listed mechanisms, and the schematics for ET can be described semiclassically in the following way (Fig. 10.5) (Nørskov, 1981): An initially unoccupied affinity level a , like that for the antibonding state in H2 , is broadened (width ) and shifted below the Fermi level F ; Then it can be filled by ET (typically electronic tunneling) from the substrate, which gives the hole in the adsorbate level a finite lifetime

444


Fig. 10.5. Adapted from Nørskov (1981). Schematic picture of the excitation process. Both a one-electron (a) and a total-energy (b) picture are shown. The time development is indicated. Two sets of diabatic PEC’s are shown with the state |a filled and empty, respectively. The different curves within each set correspond to various electron–hole excitations in the substrate. The trajectory followed by an adsorbate of energy E0 is indicated by the full line. In this example the adsorbate is trapped. If the interaction between the two sets of diabatic curves is taken into account, the crossing disappears and the adiabatic solutions change smoothly from the |a-empty to |a-filled configuration results (dotted curves see inset).

τ = h/(2). The filling does not occur immediately after the level crossing but after a ¯ typical time τ has elapsed. Two characteristic times are involved, the ET time, τ , and a time, T , characterizing the passage of the molecule, typically chosen as the time for going forward and back in the adsorption well. The approach of the adsorbate to the surface makes the adsorbate level a thus be below F , when the ET takes place, and a hole is left behind in the substrate (Fig. 10.5(a)) (Nørskov and Lundqvist, 1979b). Similarly, an excitation process can take place on the way out (Nørskov and Lundqvist, 1979b). The net result is the creation of an EHP in the substrate during one round trip of the adsorbate in the potential well (Fig. 10.5(a)). The effects that an initially empty (filled) adsorbate state is broadened and shifted down (up) due to chemisorption are quite general (Gunnarsson et al., 1980). Further, for adsorption on metals, the delocalized metal electrons efficiently screen out the electrons transferred to the adsorbate. The excitation mechanism is quite general and well worth to study further, but already a simple semiclassical theory within a trajectory approximation is valuable to set the scene (Nørskov, 1981):


445

In the simple case of Fig. 10.5, the probability that an electron is excited from k < F to q > F during one round trip in the well is a product of several probabilities (see Fig. 10.5 for notation): (i) the probability that the hole survives in the adsorbate level |a from the time t0 , when a crosses F until the time tk it crosses k , exp[−2/h¯ (tk − t0 )]; (ii) the probability that the hole in |a is filled by a substrate electron in the energy range [k , k + dk ], (2/h¯ )[dk /˙a (tk )] = (2/h¯ ) dtk ; (iii) the probability that the electron survives in the adsorbate level |a from the time t1 , when a crosses F until the time tq it crosses q , exp[−2/h¯ (tq − t1 )]; (iv) the probability that the electron in |a is filling a substrateelectron state in the energy range [q , q + dq ], (2/h¯ )[dq /˙a (tq )] = (2/h¯ ) dtq . Thus P (k → q ) dk dq = e−(2/h¯ )(tk −t0 ) (2/h¯ ) dk /˙a (tk ) . (10.20) The exponential decay probabilities exp[−2(/h¯ )(tk − t0 )] and exp[−2(/h¯ )(tq − t1 )] are consequences of the assumption of a constant lifetime τ = h¯ /(2) of the hole in |a (Nørskov and Lundqvist, 1979a). If we also assume that the time derivative ˙a (t) = |da (tk )/dt| is constant in the narrow region, where most processes take place, (tq − t1 ) + (tk − t0 ) = (q − k )/˙a . This gives P (k → q ) = ω0−2 exp −(q − k )/ω0 , (10.21) where ω0 = h¯ ˙a /(2). The excitation probability P (ω) is F ∞ P (ω) = dk dq P (k → q )δ(q − k − ω) = ωω0−2 exp[−ω/ω0 ], −∞

F

(10.22)

which with account for spin degeneracy gives a total excitation probability ∞ ∞ Pt (ω) = dω1 dω2 P (ω1 )P (ω2 )δ(ω1 + ω2 − ω) 0

0

= (1/6)ω3 ω0−4 exp [−ω/ω0 ].

(10.23)

The results (10.22) and (10.23) are derived under the semiclassical assumption that only probabilities and not amplitudes are important. This is best justified if the adsorbate resonance is always either completely above or below F , that is, when the crossing of F is very fast. The probabilities P (ω) and Pt (ω) both peak at ω0 and fall off exponentially with increasing ω. The characteristic energy loss is thus roughly ω0 = (da (tk )/dt)(h¯ /(2)), that is how much the adsorbate level moves below the Fermi level during the lifetime h¯ /(2) of the hole. The EHP contribution to the sticking probability is ∞ σ ∼ P˜ (kT ) = Pt (ω) kT 1 1 = 1 + (kT /ω0 ) + (kT /ω0 )2 + (kT /ω0 )3 exp [−kT /ω0 ], (10.24) 2 6 where kT is the thermal heat of the incoming gas. For ω0 values close to kT or higher, the ET to EHP excitations is an efficient mechanism to trap the adsorbate (Nørskov and Lundqvist, 1979b).

446


10.2.8. Nonadiabatic transitions between multiple states – a quantum case The description of the coupled electronic and nuclear motions is of course basically a task for quantum mechanics (Brivio and Trioni, 1999). To treat the nuclei classically is much simpler, however, even more so in the trajectory approximation, with the nuclei given trajectories, R = R(t). This is not a fully consistent approach, as the trajectories typically consider neither the proper forces nor the energy dissipation (Schönhammer and Gunnarsson, 1980). The time-dependent Schrödinger equation then is He R(t) φ(t) = i h∂φ(t)/∂t. (10.25) ¯ The probability that an energy ω is lost in one round trip in the adsorption well can be written as

P (ω) = φ(∞)|δ ω − He (∞) − E0 (∞) |φ(∞), (10.26) where E0 (∞) is the adiabatic ground-state energy at time t = ∞ after the collision. An exact solution can be obtained with some simplifications, by using coherent particlehole excitations (rigorously obeying boson commutations) and by working in a subspace of states with no holes deep in the Fermi sea and no electrons high above the Fermi level, the relevant subspace for this problem (Schönhammer and Gunnarsson, 1980). The subspace can be chosen as small as wanted, simply by increasing T , as a perturbation varying on a time scale T produces excitations from the ground state of the instantaneous Hamiltonian only in a range ±1/T around the Fermi level. The exact P (ω) is then possible to get in terms of the instantaneous phase shifts at the Fermi level. With a predominance of low-energy EHP’s, assumed to behave as bosons, the system can be considered as a collection of perturbed (displaced) bosons (harmonic oscillators), and Eq. (10.26) results in an analytic form (Müller-Hartmann et al., 1971; Schönhammer and Gunnarsson, 1980), ∞

dt eiωt exp − |λα /ωα |2 1 − e−iωα t , P (ω) = 1/(2π) (10.27) −∞

α

where the sum runs over all sets α of EHP excitations of energy ωα . In the case of only one relevant adsorbate level the coupling constant becomes simply (Schönhammer and $∞ Gunnarsson, 1980) λα = (1/π) −∞ dt eiωα t δ˙F (t), where δ˙F is the time derivative of the instantaneous phase shift at the Fermi level, and thus δ˙F (t)/π can be interpreted as the number of states below F , signaling the rate at which new states are created below or above F , respectively. This fits the simple picture given earlier. It even brings further results, such as (Schönhammer and Gunnarsson, 1980)

P (ω) = P0 δ(ω) + (1 − P0 ) ω/ω02 exp [−ω/ω0 ], (10.28) where

P0 = 1/ 1 + [ω0 T /h¯ ]2 ,

(10.29)


447

obtained from Eq. (10.27) for the situation, when the number of states below F , δ˙F /π changes by more than 1/2. The simple semiclassical theory should apply in the limit ω0 T h, ¯ and it is gratifying that in this limit Eqs. (10.28) and (10.29) give Eq. (10.22). The quantum theory gives a new feature, the nonzero probability P0 of the no-loss, i.e. elastic, scattering. P0 reduces the role of electronic excitations as a trapping mechanism (Brako and Newns, 1980). Typically P0 is well below 100 per cent, however. With parameters for H2 at a metal surface, P0 is about 0.5. As two spin states contribute, the probability for elastic scattering is P02 ∼ 0.25, leaving quite some probability for excitations (Nørskov, 1981). 10.2.9. Master equation for ET processes In many dynamical phenomena at surfaces, charge is transferred from one part of the system to another, for instance, from a substrate surface to an approaching molecule. These important processes (Newns, 1989; Johnson et al., 1990; Nordlander and Tully, 1989) in most cases carried out by electrons, that is, through ET. Early calculations of ET probabilities were done in the time-dependent Anderson model (Anderson, 1961; Newns, 1969), with its hopping matrix element, Vak (t), between the impurity, |a, and a continuum state, |k, in the metal. Effects of the intra-atomic Coulomb correlations and spin effects are neglected in most treatments, like in Blandin et al. (1979a), Tully (1977), Nørskov and Lundqvist (1979b), Brako and Newns (1981), Lang (1983). Some attempts to describe such effects in special limits (Bloss and Hone, 1978; Grimley et al., 1983; Brako and Newns, 1985; Sebastian, 1985; Brako and Newns, 1991) are hampered by the fact that the correlations are not small and therefore cannot to be treated by, e.g., perturbation theory. For bulk problems, there has been some progress in the treatment of highly correlated systems, like the introduction of “slave bosons” (Coleman, 1984). This can be adapted to the time-dependent surface problem (Langreth and Nordlander, 1991). Intra-atomic correlation effects are shown to drastically alter the charge-transfer dynamics even for degenerate levels (Langreth and Nordlander, 1991). With powerful nonequilibrium methods (Kadanoff and Baym, 1962), a rate equation of motion can be derived for the tunneling electron (Langreth, 1976), and under certain circumstances the dynamics of the electronic processes are shown to be well described by a very simple master equation (Langreth and Nordlander, 1991),

dnσ (t) = −Γσ (t) f > σ (t) nσ (t) − f < σ (t) nB (t) , (10.30) dt where nB (t) = 1 − σ nσ (t). The intra-atomic correlation effects can combine to prevent neutralization, even though there are excited levels that are readily available for resonant neutralization. A channel for the survival of hyperthermal positive ions is thus provided in desorption and sputtering experiments (Langreth and Nordlander, 1991). In the so-called semiclassical approximation (SCA) and in the wide-band limit, important Kadanoff–Baym equations can be solved, and the electron self-energy takes a form, in

448

which the width function is given by 2 Γ (t) = 2π Vak (t) ρ,


(10.31)

where ρ is the density of free-electron states (Langreth and Nordlander, 1991). In the master equations of SCA, it is sufficient to take (Langreth and Nordlander, 1991) 2 Γσ (t) = 2π Veσ (t) ρ(eσ )(t). (10.32) Accurate calculations for these expressions have been proposed (Nordlander and Tully, 1988, 1990). In qualitative elaborations, the width of the adatom energy level is naturally assumed to vary exponentially, i.e. Γ (t) = exp[−aZ(t)] (Langreth and Nordlander, 1991), reflecting the exponential decays of the adatom and metal orbitals. 10.2.10. In summary While adiabatic and diabatic representations are equivalent, there are situations (see above), where the diabatic formulation is conceptually very simple. Quantitatively, it is best suited for strongly NA systems, where states cross in a narrow region (ω0 T h¯ ) (Nørskov, 1981). 10.2.11. Time-dependent density-functional theory and nonadiabaticity There are other theoretical tools, like time-dependent density-functional theory (TDDFT) that offer interesting possibilities for the study of NA. For instance, the strong electronically NA effects close to the spin transition during the chemisorption of a hydrogen atom on an Al(111) surface have been simulated, using TDDFT for the electrons in combination with Ehrenfest dynamics for the nuclei, and the dissipated energy as well as the EHP excitation spectra are calculated (Lindenblatt and Pehlke, 2006). The recent Newns–Andersonmodel approach by Mizielinski et al. (2005) is thus confirmed (Bird et al., 2008). The simulations illustrate the physical processes that contribute to internal exoelectron emission. 10.2.12. Friction treatment In other treatments of NA, the instantaneous adiabatic ground state and the adiabatic representation could be more convenient starting points, and the dissipation of kinetic energy into EHP excitations could be described in an average way through a friction coefficient (d’Angliano et al., 1975; Blandin et al., 1979b). The dissipation is assumed to occur by a large number of events, rather than by a single ET, as in the treatment above. For the energy dissipation during a single round trip of the adsorbate in the potential well such a friction description should not be valid (Schönhammer and Gunnarsson, 1980; Brako and Newns, 1980). Friction models based on classical mechanics have been used to describe the exchange of energy between molecular motion and EHP’s at metal surfaces (d’Angliano et al., 1975; Brako and Newns, 1980; Li and Wahnström, 1992; Head-Gordon and Tully, 1995). Most


449

treatments have used empirical friction parameters. However, some more rigorous treatments have been developed, including “molecular dynamics with electronic friction” (MDEF) (Head-Gordon and Tully, 1995). The MDEF involves the development of a classical-mechanical multidimensional Langevin equation (Head-Gordon and Tully, 1995; Tully, 2000), M R¨ = −∇V (R) − Ω− ∗ R˙ + F (t),

(10.33)

where the friction Ω− is an N × N matrix that accounts for the coupling of each mode to the EHP’s. There is a strong dependence of Ω−,i on the atomic coordinates, which here means the orientation of the admolecule and its distance from the surface, and explicit expressions for ab initio calculation of Ω−,i have been derived. The random force, F (t), is obtained from the fluctuation-dissipation theorem, FX (t)FX (t ) = 2kB Te ηXX δ(t − t ), (10.34) where ηXX is the friction (d’Angliano et al., 1975; Head-Gordon and Tully, 1995).

10.3. Electron transfer 10.3.1. Introduction At the heart of changes of matter, there are electron processes, where electrons change states in one way or another. This prominent role of ET is present in key processes in all areas of physics, chemistry, and biology. The light mass of an electron, compared to ionic masses, makes ET often synonymous with the concept of charge transfer (CT). However, in, e.g., ionic conduction, other forms of CT should of course also be considered. A process like ET has a before and an after. In quantum-mechanical terms, both initial and final states should ideally be completely characterized. Here dynamics, i.e. the course of events occurring between the initial and final states, is at focus, completely from the “crib” to the “grave” in the ideal case. Thus ET and NA encompass a great variety of phenomena. This section contains a quick reminder of the manifold of static ET phenomena, earlier described in many other articles in this volume and aired in many reviews (Lundqvist, 1990, 1991, 1994; Richardson and Holloway, 2000), and to which the dynamic ET processes, the main topic of this section, stand in sharp relief. Most of these static ET effects have key roles in the development of surface science and technology. The emphases have differed between different periods, let it be dipole layers, workfunction changes through cesiation, electron emitters in TV tubes, fields emitters, ion sources, photoemission equipment, or other manifestations. Thus for later reference, some textbook (Zangwill, 1988) examples of static ET results are listed and given pictorial presentations: the electronic density profile at the jellium surface (Fig. 10.6) (Lang, 1973; Kiejna and Wojciechowski, 1981); the effective electron potential at a jellium surface (Fig. 10.6) (Lang, 1973; Kiejna and Wojciechowski, 1981); the lowering of the workfunction by cesiation (Fig. 10.7); the face dependence of the workfunction (Fig. 10.7); adsorption mechanisms (Fig. 10.8) (Langmuir, 1932; Taylor and Langmuir, 1933; Lang and Williams, 1975; Wimmer et al., 1983; Ishida and

450


Fig. 10.6. Adapted from Kiejna and Wojciechowski (1981). Schematic representation of the step-potential model of a metal surface, where in (top) the region of the uniform positive-charge background and the surface electronic charge distribution n(z) are shown, and where in (bottom) all relevant energies are indicated. For further explanation of used symbols, see Kiejna and Wojciechowski (1981).

Teracuda, 1988; Ishida, 1989); energy-level shifts (Fig. 10.8); the resonance model of adsorbate electron structure (Fig. 10.9) (Gurney, 1935; Gadzuk, 1967, 1969; Grimley, 1971; Lang and Williams, 1975); dissociative molecular adsorption (Fig. 10.4) (Hjelmberg et al., 1979; Lundqvist et al., 1979b; Nørskov et al., 1981; Lundqvist, 1990) and development of the LDOS in the course of dissociative molecular adsorption on Mg (Fig. 10.3) (Nørskov et al., 1981); and the bonding-antibonding picture, as modeled on a Cu surface (Fig. 10.10) (Hammer et al., 1994). The static ET affects the work function, the nature of the chemisorption bond, and the frequencies of adsorbate vibrations, for instance, and it shifts and broadens (Gurney, 1935) adsorbate levels (Lang and Williams, 1975, 1978; Gunnarsson et al., 1976; Lundqvist, 1990). Over the years, adequate qualitative descriptions have been given by simple mod-


451

Fig. 10.7. Adapted from Lundqvist (1994). (Left) Lowering of workfunction by cesiating. (Right) Face dependence of the workfunction.

Fig. 10.8. Qualitative illustration of energy-level shifts for (a) a covalent system (H+ 2 ), (b) an ionic system (LiF) together with a schematic adsorption mechanisms for three models of alkali adsorption: (c) ionic, (d) resonance, and (e) polarized-atom models.

els, like the adsorbate-resonance model (Gadzuk, 1967; Grimley, 1971), the Newns– Anderson model (Anderson, 1961; Newns, 1969), and jellium-model calculations (Lang and Williams, 1975; Gunnarsson et al., 1976). The conceptual descriptions thus developed have successively been supplemented and reinforced by quantitative self-consistent electron-structure calculations (Richardson and Holloway, 2000). When the latter are properly based on the DFT, they give an impressive accuracy and agreement with experiment (Richardson and Holloway, 2000).

452


Fig. 10.9. Adapted from Gadzuk (1970). The canonized Gadzuk picture of the resonance model of adsorbate electron structure (Gurney, 1935; Gadzuk, 1967, 1969; Grimley, 1971; Lang and Williams, 1975). Here, the dashed curve represents the ion-core potential for the adsorbate at infinity, and the solid curve the combined adsorbate and metal potential at an adsorbate distance of s. Vi is the ionization energy of the adsorbate, E is the energy shift, and Γ is the broadening caused by the adsorbate–metal interaction.

Fig. 10.10. Adapted from Hammer et al. (1994). (Left) The PES for H2 dissociation over Cu(111). The inset shows the dissociation geometry. (Right) The molecular bonding (full line) and antibonding (dashed line) density of states along the optimum reaction path. The hatched region in (D) gives the position of the Cu d band.

10.3.2. Image potential The image force is an electrostatic interaction between an external charge and a surface, an important long-range force that should be included in any viable model for gas–surface interactions. At large separations z from the surface, the corresponding potential energy (Fig. 10.11) varies like (Hewson and Newns, 1974) Vim (z) = −

Ze2 , 4(z − zim )

(10.35)


453

Fig. 10.11. Adapted from Nørskov and Lundqvist (1979a). Variation of the affinity level a (z) of an atom outside a metal (Nørskov and Lundqvist, 1979a). The zero of the distance is at the image plane, which is 1.5–2 Å outside the first atomic layer. Below is shown the one-electron width , the survival probability P and their product.

where Ze is the charge of the adparticle and zim the location of the image plane, placed typically within one Å from the surface, i.e. in the chemisorption region (Lang, 1973). The image force has many important consequences, including its significant effects on molecule–surface processes. It can give substantial shifts in energy levels at the surface. For instance, due to the image-potential shift of the electron-affinity (ionization) level of a molecule in interaction with a metal surface, an incident molecule could find its affinity level degenerate with or lower than the substrate Fermi level somewhere outside the surface (Fig. 10.12). 10.3.3. Adsorbate-induced electron structure In addition to image shifts, there are significant adsorbate-induced effects in the electron structure caused by quantum-mechanical interference between adsorbate and substrate electron states. Adsorbate levels are shifted and broadened, and there are general variations in the adsorbate-induced local density and local density of states (LDOS), as shown both for jellium-like (Fig. 10.3) (Lang and Williams, 1975; Holmström, 1987) and more realistic surfaces (Fig. 10.10) (Hammer et al., 1994). The figure examples illustrate (i) how the induced LDOS for alkali adsorption is ionic well outside the surface and metallic at close distances d (Fig. 10.12) (Lang and Williams, 1975; Holmström, 1987), (ii) that the variation with d for the induced dipole moment, which clearly distinguishes between ionic (Li, Ca, Sr) and neutral adparticles (Mg, Be, B) (Holmström, 1987), and (iii) a typical variation with separation d for a molecular affinity level (Figs. 10.3b and 10.10b) (Nørskov et al., 1981; Hammer et al., 1994).

454


Fig. 10.12. (Left) Variation with separation z for a molecular affinity level. (Right) Adsorbate-induced local density and local density of states (LDOS). Induced LDOS for alkali adsorption, being ionic well outside the surface and metallic at close distances d (Lang and Williams, 1975; Holmström, 1987).

Fig. 10.13. Adapted from Lundqvist (1994). (Right) Examples of molecule–surface encounters. (Left) Schematic presentation of “particles” that might go into the surface and those going out from it.

10.3.4. Molecule–surface collisions Molecule–surface encounters are complex processes, with many examples of dynamic ET (Fig. 10.13). Within them, however, certain elementary process steps can be identified and be objects for characterization. Among these there are many direct NA processes, including such that involve ET (hopping or tunneling), Auger transitions, and radiative transitions (Fig. 10.4). No doubt, the adiabatic picture has been very successful in describing some surface processes. Numerous dynamical processes are classified as NA phenomena, however. In the following section an attempt is made to list some of the clear cases and to highlight some of the key concepts. The listing is made with reference to what is going into the surface and what is going out from it (Fig. 10.13).


455

10.3.5. Nonadiabaticity One role of NA processes is to introduce irreversibility into the course of events. For instance, only sufficiently strong dissipative forces can make an impinging particle lose enough kinetic energy to be trapped in the PES well at the surface. NA contributes to dissipation. Further, only NA transitions can create real electronic transition or intermediate states. The importance of NA processes was stressed early, long before the important colloquium on the physical basis of heterogeneous catalysis (Drauglis and Jaffee, 1975). Recently, there is an extra focus on the significance of NA for a thorough understanding of chemical reactions on metal surfaces (Wodtke, 2006; Nieto et al., 2006; Hasselbrink, 2006; Luntz, 2007). There is a quest for an accurate account of the role of electronic excitations in these processes. Our knowledge about energy dissipation into electronic degrees of freedom on a quantitatively accurate level is still limited, however. This might be due to lack of relevant experimental data that are capable of challenging the theoretical concepts (Hasselbrink, 2006). A few recent benchmark-type experiments (Österlund et al., 1997; Gergen et al., 2001) might change the situation quantitatively or at least semi-quantitatively, however. In great contrast, NA finds expression in quite a number of other well-established observations in surface science (Greber, 1997; Nienhaus, 2002). Early chemiluminescence and exoelectron emission are considered as established NA reaction events (Ertl and Hasselbrink, 1992). However, “the minute probability with which these events are observed has allowed practitioners in the field to behave such as these effects would be spurious occurrences” (Hasselbrink, 2006). 10.3.6. Dynamic electron transfer Dynamic ET is typically associated with dissipative processes. Most kinds of particlesurface collisions, fields that affect surfaces, dissociation dynamics of molecules, and NA electronic processes that accompany exothermic chemical reactions are associated with dissipation into EHP’s and of course phonons. At higher energies also possibly excitons and plasmons provide channels for the energy to flow from one part of the system to another. Most elementary processes coming after the ET sooner or later lead to excitation of phonons. In principle, of magnons and other elementary excitations can also result. For studies of initial and final states of a transfer and of static CT there are many experimental and theoretical methods available. Ways to study the pathways in between initial and final states in CT are more rare, however. Many different and alternative routes can be envisaged, and dynamics means that one follows such pathways to get additional information. Examples of ET appear in CT processes, in molecular dissociation within various gas–surface collisions and the associated dissociation dynamics (Rettner and Ashfold, 1991) and in many situations of scientific and technological importance, e.g., (i) spectroscopic studies of molecular assemblies – in particular femtosecond pump-and-probe experiments (Schoenlein et al., 1988, 1990), exploring dissociation and relaxation dynamics of molecules, (ii) almost all types of chemical reactions in molecular assemblies, e.g., in combustion and atmospheric chemistry, (iii) photochemistry, and (v) low-temperature plasmas. Dynamics of ET processes is an area very rich in physical phenomena.

456


10.3.6.1. Multitude of electron-transfer processes Typically, an ET process in the adparticle-surface system is initiated by exerting it to some perturbation, for instance, a collision, a temperature rise, an electric field, or a mechanical work. Then some effect of this perturbation is observed, for instance, a particle emitted from the surface, as in desorption. In principle, all these excitation mechanisms may also lead to deexcitation via electrons or ions, exoemission. Classifications, like that for relaxation in luminescence, may be adopted for exoemission phenomena, as well. It is done with respect to typical time constants, τ : spontaneous processes with τ < 10−8 s are called fluorescent, stimulated with τ > 10−8 s phosphorescent, and stimulated with 1010 y > τ > 102 s thermoluminescent (Greber, 1997). The manifold of dynamical manifestations of NA and ET can be illustrated by grouping them with respect to the external agents and to results of the surface event, e.g., emission of light, electrons, and ions, abstraction, and surface reactions (Fig. 10.13). It can be envisioned as the large number of combinations described by the statement “For i = 1, 2, for j = 1, 2, do F (i, j ), where both i and j = atom, molecule, ion, photon, electron, heat pulse, mechanical motion, electric field, . . . ” (Fig. 10.13). The matrix F (i, j ) is far from sparse, most matrix elements having one or several well-known processes. Such a listing in terms of particles in and out could look like atom-to-atom, atom-to-molecule, atomto-ion, atom-to-photon, atom-to-electron, atom-to-heat, molecule-to-atom, molecule-tomolecule, molecule-to-ion, molecule-to-photon, molecule-to-electron, molecule-to-heat, ion-to-atom, ion-to-ion, ion-to-heat, photon-to-molecule, photon-to-photon, photon-toelectron, electron-to-molecule, electron-to-electron, heat-to-molecule, mechanical motionto-heat, etc. Some of the common and typically well established surface processes are organized in the following way, hopefully showing the abundance of NA and ET processes (the richness of these phenomena (Lundqvist, 1994) is even greater, as they often appear under different names, and as there are many classes of surfaces active). Atom-to-atom: Atom-beam scattering (ABS) experiments (Celli, 1984) (Fig. 10.13(a)) are typically performed on well-characterized surfaces, using well-prepared beams of impinging atoms and well-characterized exiting atom beams (Celli, 1984; Manson, 2008). In an elastic-scattering mode, ABS informs about the atom-surface potential. In the inelastic-scattering mode, typically with He atoms, the loss or gain of energy quanta, informs about phonons of the substrate that result in heating of the surface or the gas phase, respectively (Manson, 2008). The process is usually simply described in terms of interaction potentials. With thermal atoms in their ground states, the odds for ET are slim. The probability for exciting EHP’s by impinging neutral He atoms is negligible (Gunnarsson and Schönhammer, 1982). Now the He electrons are very tightly bound, and no empty (filled) states available on the atom with an energy to match those of the valence (conduction) electrons of the solid. Other atoms are better in this respect, however. There are even some noble gases, such as Xe and Ar, that are able to excite EHP’s, observed by, e.g., the scanning-tunneling microscope (STM) (Eigler et al., 1991). With excited atoms, more energy can be fed into the system. The well-established experimental technique of metastable atom electron spectroscopy (MAES) (Harada et al., 1997) illustrates this (see below) (Sesselman et al., 1983). When excited atoms feed en-


457

Fig. 10.14. Adopted from Harada et al. (1997). Deexcitation mechanisms of a metastable atom: (a1 and a2 ) Resonance ionization (RI) followed by Auger neutralization (AN) on a metal surface, and (b) Auger deexcitation (AD) or Penning ionization (PI) on an insulator surface.

ergy into the system, there are several channels of energy dissipation ending up in EHP’s and emitted electrons. The dominant deexcitation mechanisms in MAES are (i) resonance ionization (RI) followed by Auger neutralization (AN), that is an electron tunneling from the excited electronic state on the impinging atom to the empty state above the Fermi energy in parallel with filling of the hole state on the metastable atom by ET from the valence band of the solid (Fig. 10.14(a)), and (ii) Auger deexcitation (AD) (Fig. 10.14(b)) that involves the hole on the metastable atom, an ET-providing state of the surface, the electron from the excited state of the metastable atom, and the electron ejected into the vacuum. Both processes release sufficient energy to enable a valence electron from the surface to be ejected, but they differ by leaving two and one holes, respectively, in the surface bands. In general, RI + AD

458


Fig. 10.15. Adapted from Gao et al. (1997). (a) Schematic picture of the atomic switch. (b) Double-well model for atom transfer based on truncated harmonic oscillators. In the vibrational heating mechanism, the atom transfer results from stepwise vibrational excitation of the adsorbate–substrate bond by inelastic electron tunneling (Gao et al., 1992, 1993, 1997). (c) Inelastic electron tunneling to an adsorbate-induced resonance with density of states ρa induces vibrational excitations, (1), while electronic excitations within the substrate induces vibrational relaxations, (2) (Stipe et al., 1997).

is the dominant decay channel for surfaces with a local workfunction value larger than the effective ionization potential of the impinging metastable atom, while AD dominates in cases with low workfunction and with energetic overlap between a metastable electronic state and an energy band of the substrate. In MAES the energy distribution of the emitted electrons is measured. One gets information about the density of empty and occupied surface states, respectively, for the two decay channels, with a more direct relation in the AD channel (Sesselman et al., 1983). The analysis of the MAES and INS experiments also gives valuable experiences for other surface processes. The atomic switch, where a Xe atom is manipulated to move between a Ni(110) surface and a W tip in an STM (Eigler et al., 1991), belongs to this category, in spite of the restricted motion of the atom. By switching the bias of the STM, the Xe atom is pushed between two positions: one on the Ni surface and one on the W tip (Fig. 10.15) (Eigler et al., 1991; Gao et al., 1992, 1993, 1997). The experiment informs about the position of the Xe atom, the direction of the motion in relation to the bias, and the conductance of the switch circuit. In particular, a delay time t between the switching of the bias and the build-up of the conductance to its new value is observed. The corresponding rate 1/t is found to relate to the magnitude of the bias V in a very explicit way, being proportional to V 5 (Eigler et al., 1991). A model (Gao et al., 1992; Walkup et al., 1993; Gao et al., 1997) for the mechanism of the atomic switch that benefits from ET’s between the Xe atom and the surface and tip, respectively, explains this behavior. Thanks to the shifting and broadening of the Xe affinity level upon adsorption (cf. Fig. 10.9), there is a tiny energetic overlap between the atom and the closest metal, sufficient for electron tunneling. Excess energy is given to EHP’s and then in turn to atomic vibrations in the Xe potential well. The atom climbs up the ladder of a few vibrational energy levels in its adsorption potential well on the Ni(110) surface or the W tip, depending on the sign of the bias. This is a very explicit experimental proof of ET, and the coupling between the displacement of the Xe atom, in this case vibrations, and the EHP’s (Gao et al., 1997). Atom-to-molecule: An example of a surface reaction is obtained, when an incoming atom causes the emission of a molecule (Fig. 10.13). This can occur as a simple pick-up process (Eley–Rideal mechanism) or as a result of a rather complex process, with several


459

intermediate states, finally ending up in the emission of a molecule (Rettner, 1992; Rettner and Auerbach, 1995). By necessity ET is involved. Atom-to-ion: When an atom hits a surface and gives away or picks up electrons it is called surface ionization. The conversion of a neutral atom A into a positive, A → A+ , or negative ion, A → A− , has been studied since long (Andersen, 1976). The presence of ET is obvious. Ejected negative particles were first discovered about a century ago, then from freshly prepared alkali metal surfaces (Thomson, 1905). Metastable deexcitation spectroscopy (MDS), including MAES, focuses on the detection of emitted electrons and their energy distributions. Detection of both exoelectrons and emitted ions gives additional information. In MDS of noble-gas atoms on a Cs surface (Woratschek et al., 1987a, 1987b), the maximal exoelectron-emission intensity coincides with the disappearance of MDS emission from Cs. This stage is interpreted as a transformation of outermost Cs layer into Cs+ species. This blocks the mechanism for MDS, as such do not offer any electrons below the Fermi level that by ET can fill the empty hole of the incoming metastable atom. This gives a great chance for the atom to come close to the surface before the hole is eventually filled and thus to release an energy sufficient to allow exoelectron emission (Hasselbrink, 2006). Change in ionization state of the particle might occur in processes like sputtering and secondary-ion emission, where typically one kind of energetic ion, say Ar+ , makes ionic fragments of the surface leave the surface. In such processes ET is a substantial ingredient. Outgoing negative ions can be produced in different ways, including by incoming atoms. On the way out, there is a chance for neutralization by ET, a truly NA process, while in an adiabatic process the yield of negative ions would have been zero. A yield of ions is found, however, and it relates to the workfunction Φ of the surface (Andersen, 1976). From the energetics it can be seen that a high probability α − for a sputtered particle to come out as a negative ion requires a high velocity v out from the surface (an analogous treatment applies for a positive ion). By using an irreversible-quantum-mechanical transport description (Blandin et al., 1979b) one can derive a formula (Nørskov and Lundqvist, 1979a), α − (v) ∝ exp −(Φ − A)/(cv) , (10.36) where A is the affinity of the free atom and c a constant. For an exiting atom/ion the electron-structure is such (Fig. 10.11) that an initially filled affinity level can get above the Fermi level of the substrate, thus forming an ionic state. Then it might get emptied by ET on the way out (Nørskov and Lundqvist, 1979a). To get the ion out with a high probability thus a high velocity v is required. By lowering the workfunction Φ, as is done by covering the surface with Cs, the probability is increased. This demonstrates one of the roles of Cs in ion sources (Yu, 1978). Atom-to-photon: An atom impinging on a surface can cause emission of electromagnetic radiation, surface chemiluminescence (Kasemo, 1974; Kasemo and Walldén, 1975; Poon et al., 2003). At approach, the system can undergo electronic rearrangements, and NA electronic excitations may be created. A simple case is that of an incoming atom with an affinity level that at large separations has an energy above the Fermi level of the substrate and close to the surface one below. Here NA is a prerequisite, and ET is a key step in the process. A radiative transition (Fig. 10.4) can occur as electron jumps from a higher

460


Fig. 10.16. Adopted from Nørskov et al. (1979c). Comparison of experimental and theoretical chemiluminescence spectra.

energy level to a lower one. In surface processes, such an excited electron (hole) state can be created by the motion of the atom, when it causes a Fermi level crossing during passage (cf. Fig. 10.5). One can envisage that for an affinity level the hole survival probability Ph (z) (Fig. 10.11) and its z dependence are important issues. Alkali-halogen reactions have long been known to exhibit strong chemiluminescence in the gas phase (Carrington and Polanyi, 1972; Oldenburg et al., 1974) and, albeit weaker, as surface chemiluminescence on, e.g., Na metal (Fig. 10.16) (Kasemo and Walldén, 1975). Typically this is for molecules, but also atoms should be able to exhibit photon emission, when an s valence electron of the alkali hops to the unoccupied p state of the halogen (Fig. 10.16) (Nørskov et al., 1979c), however, not necessarily in the visible. Atom-to-electron: Excited atoms, as well as ions, provide extra energy and open up truly NA channels that involve ET. For sufficiently excited atoms, there is energy enough to emit electrons. Well-established experimental techniques, like MDS and INS (Fig. 10.14) illustrate this (Hagstrum, 1978). Channels for energy dissipation ending up in EHP’s can be resonant ET and Auger transitions (Fig. 10.14). With, e.g., metastable He in the 1s2s singlet or triplet states, the available energy is sufficient to emit some of the electrons, to be detected and energy-analyzed in the spectroscopy. Similar results are obtained with incoming He+ ions (Hagstrum, 1978). Atom-to-phonon: An atom hitting a surface gives energy and momentum to the atoms in the surface lattice. For not too violent collisions, this just affects the vibrations of the lattice, that is, phonons are emitted and absorbed. For a cool surface, the phonon emission dominates. This means an increased occupation of phonon states, i.e. a heating of the surface.


461

Fig. 10.17. Adopted from Luntz (2007). Schematic of common bond making/breaking processes: (a) atomic and (b) molecular adsorption/desorption/scatterig, (c) direct and (d) precursor-mediated dissociation/associative, (e) Langmuir–Hinshelwood chemistry, (f) Eley–Rideal chemistry, (g) photochemistry/femtochemistry, and (h) single-molecule chemistry. Solid (dashed) figures generally represent typical initial (final) states of chemistry.

In inelastic atom scattering, an energy- and momentum-characterized beam of atoms is scattered against a well-characterized surface and the scattered beam is then energy and momentum analyzed. From energy and momentum conservation the energy and crystal momentum of the phonon created or annihilated is deduced, for instance, with the help of a so-called Celli diagram (Celli, 1984). The energy transfer to phonons is an energytransfer mechanism that competes with ET and typically dominates. For sticking of atoms, the energy and momentum transfer to phonons is of key importance (Persson and Andersson, 1994). Actually, for thermal He atoms, where electronic excitations lie outside the reach of the available energy, this is the only effective process (Schönhammer and Gunnarsson, 1980, 1981). Also for the atomic switch, there is energy exchange to phonons (Eigler et al., 1991; Gao et al., 1992, 1993, 1997). To detect and isolate the electronic mechanism described above, one has to go to low temperatures, in order to keep the occupation of phonon states low. The original atomic-switch experiment was performed at 4 K (Eigler et al., 1991). Molecule-to-atom: A molecule impinging on a surface that causes an atom to leave the surface is an example of molecular dissociation (Figs. 10.13 and 10.17). The surface may change the bond of the molecule to the extent that even a strongly bound molecule can break into pieces (Fig. 10.13f). For instance, the H2 molecule, with its free-molecule dissociation energy of about 4.5 eV, breaks into two H atoms on a Ni surface (Andersson, 1978). On the other hand, a Cu surface typically leaves the H2 molecule inert, unless extra energy is provided to the system. The odds for ET with thermal molecules are typically much better than with the corresponding atom, thanks to the delocalization of valence electrons within the molecule. This lowers the electronic excitation energies and makes the corresponding frontier orbitals (HOMO’s and LUMO’s) more accessible for interactions. In addition to the image-force effects (Fig. 10.11), the molecule–surface interactions are typically strong enough to affect

462


the LUMO’s of triple-bonded CO (2π ∗ MO) and even inert H2 O molecules sufficiently much to make them at least partially available for ET. Similarly for the HOMO’s of electropositive molecules, like Na2 . A channel that is open in some molecule–surface systems is abstraction. For instance, a dimer impinging on a surface can result in one atom that is adsorbed and one that stays in or returns to the gas phase. For instance, this is observed for F2 molecules impinging on a Si(111)-(7 × 7) surface (Jensen et al., 1995), and for an oxygen molecule interacting with the Al(111) surface (Binetti et al., 2000; Binetti and Hasselbrink, 2004). A very particular process under this heading is single-molecule dissociation (Stipe et al., 1997). The tunneling current from a STM is here used to image and dissociate single O2 molecules on the Pt(111) surface at low temperatures. After dissociation, the two oxygen atoms are found a couple of lattice constants apart. The dissociation rate has a characteristic current dependence, depending on the sample bias, which is explained in a general model for dissociation induced by intramolecular vibrational excitations via resonant inelastic electron tunneling (Fig. 10.15, where the PEC now should describe the energy variation with the O–O bondlength) (Stipe et al., 1997). So in this process NA and ET are key effects that enter in a very explicit way. Molecule-to-molecule: The class of molecular-beam scattering (MBS) experiments, with a molecule hitting a surface and then causing another molecule to leave the surface, contains the important cases of elastic scattering (same molecule in and out, having same energy), inelastic scattering (same molecule in and out, having different energies), and surface reactions, including heterogeneous catalysis (Fig. 10.13). There are several MBS experiments set up in the world, aiming at as complete characterizations of the incoming and outgoing beams as possible. These are important tools in revealing essential, critical, and practically useful information about the dynamics of the gas–surface interaction. Unfortunately, in the midst of a fruitful development, with several examples of experiment and theory in a quantitative dialog, the observation that “the experimental study of gas–surface dynamics has unfortunately dropped off precipitously in the last several years” (Luntz, 2007) seems to be true. Like He scattering, MBS experiments with H2 molecules have reached a stage with detailed mappings of the interactions. This is highlighted in an earlier chapter of this volume (Persson and Andersson, 2008) and by analyses and calculations of the sticking and scattering of H2 and D2 on Cu(100) (Andersson et al., 1988, 1989; Wilzén et al., 1991; Persson and Andersson, 1994). Detailed quantum-mechanical theory of elastic and inelastic scattering agrees well with minute experimental features. Specific information is obtained on, e.g., (1) a new H2 –Cu potential (Andersson et al., 1988), which is substantially deeper than the commonly accepted one, thereby provoking deepened studies of the physisorption interaction, (2) a working description of the sticking for H2 on Cu(100) in the quantum regime by a traditional model with the molecular motion coupling to the substrate phonons (Andersson et al., 1989), and (3) an accurately assessed, strongly reduced anisotropy of the physisorption interaction between H2 and Cu(100) (Wilzén et al., 1991), where inclusion of interactions between the metal-electron states and the antibonding 2σ ∗ resonance of H2 brings theory into agreement with experiment (Wilzén et al., 1991). The latter conclusion is particularly interesting, as it points towards reactive interactions, where surface modifica-


463

tion of antibonding molecular resonances is, indeed, a key concept (Lundqvist et al., 1979b; Nørskov et al., 1981). At smaller HOMO-LUMO gaps and more reactive surfaces, the phenomenon of making and breaking bonds at surfaces enters. The dynamics of these processes has key bearings on surface chemical reactions, as the latter represent the sum of several dynamical steps occurring on different time scales (Luntz, 2007). Usually such chemistry is only measured as a kinetic process at thermal equilibrium, ignoring details of the dynamics. However, reactants or products in the gas phase make experiments and theory that probe the detailed dynamics possible. Gas–surface reactions are more complicated than the gas-phase analogy. Surfaces introduce infinite dimensionality into the dynamics, and this provides a heat bath coupled to the reactive coordinates. In addition, real surfaces are not perfect, and there are steps, other defects, and impurities. Especially for activated dissociation of molecules at surfaces, e.g., where there is an energy barrier to dissociation, these minority sites may have reactivities many orders of magnitude larger than majority terrace sites. Thus sample heterogeneity is a significant issue in real-world dynamics, especially when trying to compare theory with experiment (Luntz, 2007). Also for chemical gas–surface reactions kinetic analyses are made, both experimental, where use of seeded molecular beam are particularly interesting, and theoretical ones. For adiabatic reactions the descriptions are particularly well developed, with important contributions, like the quantum-mechanical treatment in Refs. (Gross et al., 1994, 1995; Gross, 1996) that starts from 6D PES’s, takes vibrational zero-point energy into account, but neglects energy dissipation and classical molecular dynamics with electronics friction (Head-Gordon and Tully, 1995), MDEF, and developments from there. The multitude of bond making/breaking processes is illustrated in Fig. 10.17 (Luntz, 2007). An a priori prediction of the validity of the BOA is difficult. There is no doubt that there are chemical events on surfaces that dissipate energy by creating EHP’s in the metal. EHP’s are also created thermally or by other means (e.g., by absorption of photons or by STM currents) in a way that can induce surface chemistry (Amirav and Cardillo, 1986; Stipe et al., 1997; Huang et al., 2000), truly NA processes. Abrupt or strong changes in the electronic structure of the adiabatic states should also promote NA behavior, e.g., when a narrow adsorbate resonance crosses the Fermi level of the metal. This occurs when electronegative adsorbates, (e.g., NO, O2 , Cl2 ) adsorb on metals, especially lowworkfunction metals (Kasemo et al., 1979; Hellman et al., 2005a). More gradual but very strong changes in the electronic nature of the adiabatic state also occur at avoided crossings (Luntz, 2007), e.g., at the transition states in activated dissociative adsorption/associative desorption (Luntz and Persson, 2005). Some model systems have been well studied both experimentally and theoretically. Low-dimensional MDEF with ab initio results for the friction (Luntz and Persson, 2005) suggest that the associative desorption of H2 from Cu(111) is essentially adiabatic. As a contrast, there is a radical difference in the measured sticking S for N2 on Ru(0001) (Diekhöner et al., 2001a) and D2 on Cu(111). This causes confusion, and in Section 10.4 the ongoing attempts to solve the problem are indicated. Overall, there has been quite an evolution in our knowledge about the dynamics of bond making and breaking at surfaces. In the early years, detailed experiments and simple

464


Fig. 10.18. Adopted from Hellsing et al. (1983). The potential energy curves for H and H− as a function of the distance d from the jellium edge (density like for Ag). The curves are results from the effective medium theory (Nørskov, 1982) (dotted line) and the image potential (dash-dotted line) and the calculation in Hellsing et al. (1983) (solid line). The dashed line is the results from interpolation by hand between the separate limits.

models were the only ways to gain insight into it. Today the roles seem reversed. The firstprinciples dynamics studies now seem to provide insight significantly more easily (and more cheaply) than detailed dynamics experiments (Luntz, 2007). However, there are still major challenges in generalizing the theory beyond that of H2 + metals, e.g., how to include lattice coupling reasonably and to understand when the breakdown in the BOA is important. In the present situation, with its many interesting challenges, the threatening reduction in the number of experimental studies of gas–surface dynamics is very unfortunate, as it is absolutely essential to have access to the most detailed experiments to benchmark first-principles theory (Luntz, 2007). Molecule-to-ion: For a small energy gap between the workfunction of the metal and the electron affinity of the molecule, i.e. Φ − EA (I − Φ for an electropositive molecule), ET might convert a molecule impinging on a surface into an ion, B2 → B− 2 or B2 → B+ (Böttcher et al., 1993a). Ejection of negative ions can occur also from dissociation 2 fragments (Greber et al., 1993, 1994; Hellberg et al., 1995; Hasselbrink, 2006). For an ideal NA case, with a schematic energetics containing crossing M + H and M+ + H− PES’s, and a small energy difference Φ − EA at infinite separation, the crossing might occur far out. At the crossing point, a substrate electron could jump onto the molecule, that is, ET from M to H (Fig. 10.13h), in analogy with gas-phase harpooning (Polanyi, 1932; Magee, 1940; Struve et al., 1975; Gadzuk, 1985). Now electron and molecule times differ, and an electron jump from the M conduction band to the effective molecule affinity level might be delayed, that is, there is a hole on the affinity level (Figs. 10.5, 10.11). While the molecule moves, there is a hole on the affinity level, as long as it survives. Actually, the hole survival probability Ph (z) goes from unity to zero in a narrow region, the reaction zone (see below), located inside the crossing point (Fig. 10.11). This gives a transition closer to the surface (Nørskov et al., 1979c; Greber, 1994; Hasselbrink, 2006). Energetics


465

permitting, the molecule might then leave the surface on the M+ + H− PES, that is, as an ion. For O2 interacting with Cs, both exoelectron and exiting O− ions are observed with MBS (Böttcher et al., 1994a; Hasselbrink, 2005), however, with a yield of O− ions that is about a factor of 50 smaller than the exoelectron yield. Both scale as exp[α ∗ /v] (cf. Eq. (10.36)), where α ∗ is a constant and v the velocity of the incoming particle. Such a dependence is derived in the “chemical hole diving model” (Greber, 1994), however, with α ∗ values differing slightly from the model value (Greber, 1994). For the release of an O− ion, two ET’s are required, one after another, with negligible activation barrier (Greber et al., 1994). However, the number of detectable ions are reduced on the way out by recapture of escaping charge (Nørskov and Lundqvist, 1979a). Rather, a large abundance of neutral released O atoms is expected, an expectation that still waits for its experimental verification (Hasselbrink, 2006). Several cases of this kind of process have been demonstrated (Greber, 1997). Ejected O− ions have been observed (with very low probabilities) for O2 interacting with Ru(0001) covered with submonolayers of Cs (Böttcher et al., 1996). The yield of O− reaches a maximum at Cs coverages around 0.3–0.4 monolayer (1 ML is here equivalent to an absolute coverage ΘCs = 0.33), and is in this range much higher for low (similar to 260 K) than for high sample temperatures. These effects are traced back to the varying degree of “metallization” of the Cs overlayers as substantiated by MDS experiments. Also in experiments on halogen (chlorine, bromine) adsorption on group IIIA (Y, Sc) and group IA (Zr) surfaces, both clean and with Na predosing, ionic species have been observed (Bourdon and Prince, 1984; Prince et al., 1981). Molecule-to-photon: A molecule impinging on a surface can cause emission of electromagnetic radiation (Kasemo, 1974; Kasemo and Walldén, 1975). This is an early observed case of NA, and ET is a key step in the process (Kasemo, 1974; Kasemo and Walldén, 1975; Nørskov et al., 1979c; Andersson et al., 1985; Hellberg et al., 2002; Hasselbrink, 2006). However, observations are not that frequent, as emission of photons is both a low-probability process and taking place in the infrared for most molecule–surface reactions. It is not accidental that the few cases with emitted light in the visible range, what is normally called surface chemiluminescence (Fig. 10.13h) (Kasemo and Walldén, 1975), involve highly electronegative gases and low-workfunction surfaces. Such combinations can have an energetics that gives photons in the visible range. An energetic adsorbate hole might be created, and this hole is then filled by one of three competing mechanisms (Fig. 10.4) (Kasemo and Lundqvist, 1984): resonance tunneling, Auger decay, and radiative decay (Nørskov et al., 1979c). Fluorescence is the most improbable event, as it requires electronically excited states with long lifetimes (Roose and Offergeld, 1977; Nørskov et al., 1979c; Hasselbrink, 2006). Radiative transitions, which take place between a higher energy level and a lower one, are made possible in surface processes by energy-level shifts and delayed survivals of excitations. Typically, an excited electron (hole) state can be created by a filled ionization (empty affinity) level of a molecule being shifted up (down) in energy upon molecular approach to the surface. The affinity case (energy level a ) and its hole survival probability Ph (z) have been discussed above (Fig. 10.5), including the narrowness of the region where Ph (z) differs essentially from both zero and unity (Fig. 10.11).

466


Molecule-to-electron: A hundred years ago, observations of electron emission as a result of chemical reactions were first reported (Reboul, 1909; Haber and Just, 1909). Refined exoelectron-emission experiments are now performed, and the wide variety of processes can be classified (Oster et al., 1999). The molecule–surface reaction heat is also channeled to excite electrons and emit exoelectrons. One can measure the yield and energy distribution of exoelectrons. Low workfunction metals and electronegative atoms and molecules atoms or molecules, like oxygen (Böttcher et al., 1990) and halogens (Hellberg et al., 1995), form systems that are favorable for exoelectron emission. The magnitude of the electronic excitation energy is determined by the shift of the hole level of A, a (z) upon approach to M (separation z) and the hole survival probability, Ph (z) (Figs. 10.5 and 10.11). The system thus follows the diabatic PES’s, until a deexcitation process involving ET brings it back to the electronic ground state. The reaction enthalpy, H , plus possible kinetic and internal excitation energy of A, is available for the electronic excitation ∗ . To allow escape of the electron into vacuum, the latter energy has to be above the workfunction, ∗ > Φ. As for surface chemiluminescence, the NNL model (Nørskov et al., 1979c) provides a consistent picture of electron emission in the NA surface reactions of halogens and alkalis (Kasemo and Walldén, 1975; Hellberg et al., 1995; Poon et al., 2003) and of O2 with a Cs surface (Böttcher et al., 1990, 1991, 1993c). Only slight but significant extensions of the NNL model are needed to get a mechanism for exoelectron emission (Kasemo et al., 1979; Hellberg et al., 1995; Greber, 1997). In a recent experiment, important for the NA perspective, the conversion of large molecular vibrational excitation energies (9 v 18) of NO into exoelectron emission has been observed from a low workfunction surface (Cs on Au), as described below (White et al., 2005). Molecule-to-heat: Phonons are important also for the sticking of molecules (Persson and Andersson, 1994). As the Langmuir–Hinshelwood and Eley–Rideal mechanisms (Fig. 10.17) involve two and one adsorbed reactants, respectively, the conversion of molecular energy to heat is of key importance also for surface reactions. Ion-to-atom: When an ion is converted to an atom at a surface, one talks about surface neutralization. In ion-neutralization spectroscopy (INS) (Fig. 10.14) (Hagstrum, 1978), where energy is fed into the combined system by having excited or ionized atoms coming in to the surface, where there are several channels of energy dissipation ending up in EHP’s and emitted electrons. Ion-to-ion: Ion scattering can sometimes be performed in a mode that changes the ionization state of the particle. In secondary-ion emission and in sputtering, one kind of energetic ion, say Ar+ , makes ionic fragments of the surface leave the surface. In both types of processes, ET is a substantial ingredient. A particular kind of secondary-ion emission is used in ion sources, needed, for instance, for accelerators. In one example of such a source design (Andersen, 1976), negative Ta− ions are produced by having a plasma to send out keV Cs+ ions towards a Ta-metal surface. In the process, a monolayer of Cs is built up. The yield of Ta− ions relates to the workfunction Φ of the Cs overlayer (Andersen, 1976). In an adiabatic process the yield of negative ions would have been zero. The process is thus truly NA. The probability α − for a sputtered particle to come out as a negative ion


467

depends on its velocity v out from the surface, as in Eq. (10.36) (Nørskov and Lundqvist, 1979a), that is as exp(−V0 /v), where V0 depends on the workfunction Φ and the affinity A of the free atom (Eq. (10.36)). Ion sources are therefore constructed with hot plasmas and thus high velocities v and by covering the surface with Cs to have a low workfunction Φ (Yu, 1978). For an atom or other particle that moves sufficiently rapidly near a metal surface, the state of the electron gas may deviate from the adiabatic one, which applies, only if the particle moves infinitely slowly along its trajectory. In fast-ion scattering from surfaces the ion fraction in the scattered beam is given by NA. There is a simple semi-classical theory for this (Lang, 1983), like for surface chemiluminescence, occurring at thermal energies with NA emission of a photon. A more rigorous quantum-mechanical approach is necessary in, for example, sputtering (e.g., in SIMS) and in post ionization in field evaporation, and it can be applied to ionization of hyperthermal sodium atoms from a hot tungsten surface (Lang, 1983). Further, the possibility of energy loss to EHP’s providing a trapping mechanism for adsorbing particles can be approached by solving the Heisenberg equations of motion for a slowly moving particle (Lang, 1983). The formation of negative hydrogen ions by scattering protons from a metal surface can be described with two models both with the electron motion described quantum mechanically and the nuclear motion classically: a probability model, where the time evolution of the ionization probability is considered, and an amplitude model considering the time evolution of the corresponding wave function amplitude. The electron affinity level of an atom close to the metal is lowered by means of image forces and broadened due to resonant transition of an electron between the conduction band of the metal and the valence shell of the atom. The calculated position of the affinity level and the transition rate in both models give rise to maximum negative ionization efficiencies of 4% on W(110), 40% on cesiated tungsten and 15% on cesium (Rasser et al., 1982). Ion-to-electron: An example here is ion-neutralization spectroscopy (INS) (Hagstrum, 1978), totally dependent on NA and described below (Fig. 10.14). Ion-to-heat: There are many surface processes that involve ions, where ET plays a role. Here it suffices to mention the case of stopping power, where in short ionic translation energy is converted to heat in the receiving material. As intermediate steps, NA processes and ET occur. The traditional theory for stopping power expresses the energy loss per length unit −dE/dx in terms of a linear-response expression. For a beam of He ions moving with a steady velocity in aluminum, the relative contributions to the stopping power of the different charge states of He have been calculated from first principles (Arnau et al., 1990). In combination with dielectric results in the appropriate velocity range, good agreement with experiment is obtained. Results for He++ bombardment of Al (Fig. 10.19), considering the capture mechanisms (i) shell transitions, i.e. an electron bound to a lattice atom jumps over to He++ or He+ , (ii) resonant processes, i.e. an electron moving from the metal conduction band to a bound state of the ion, and (iii) Auger transitions involving conduction electrons (Arnau et al., 1990), show this. For the Auger rate, a scaling law discovered (Fondén and Zwartkruis, 1992, 1993a, 1993b) for He+ neutralization might have ramifications. Photon-to-molecule: Photons impinging on a surface can have many effects. Atoms and molecules adsorbed on the surface might desorb and photon-stimulated desorption

468


Fig. 10.19. Adopted from Arnau et al. (1990). Stopping power in atomic units for helium ions in AI as a function of ion speed. The thick solid line (TOTAL) is the result of Arnau et al. (1990) and the curve labeled LT is obtained from linear-response theory for a bare ion. The circles are the experimental data. The different contributions to the curve labeled TOTAL from the fractions of bare ions (He++ ), single ionized ions (He+ ), neutral atoms (He0 ), and capture and loss processes (C&L) are shown separately.

is a very active field (Ramsier and Yates, 1991; Ho, 1994; Hasselbrink, 2008). Dependent on conditions, the Mentzel–Gomer–Redhead (MGR) (Menzel and Gomer, 1964; Redhead, 1964), the Antoniewicz (Antoniewicz, 1980), the Knotek–Feibelman mechanisms (Knotek, 1984), or mechanisms involving hot electrons (Ho, 1983; Gadzuk et al., 1990; Gao, 1992, 2008) might be at work. NA and ET are obvious ingredients. An odd but early example of such a development is a mechanism (Holmberg and Lundqvist, 1985) proposed for the Honda electrolytic cell, where photons are producing hydrogen gas, intended for use as a fuel. The photon is absorbed in the outer submicron layer of a titanium-oxide electrode. This is a semiconductor with bent energy bands due to defects on the surface. The hole is created, when the photon lifts an electron over the energy gap. It there propagates to the surface. As indicated in Fig. 10.20, there is an OH admolecule on the surface with a 4σ ∗ -derived affinity resonance close to the top of the valence band. The OH is formed from water molecules in the electrolyte. When a hole is transferred to the 4σ ∗ resonance, the admolecule changes state, from “OH− ”, with a completely filled resonance, to “OHneutral ”. There is a corresponding transfer from one PES to another. On the “OHneutral ” PES the admolecule is expected to be quite mobile. This


469

Fig. 10.20. Adopted from Holmberg and Lundqvist (1985). Schematic representation of a possible mechanism for the desorption of water molecules in a Honda cell.

allows recombination of hydroxyl species to (OH)2 and even complexes of such peroxides. The latter ones disintegrate to produce O2 molecules. The hydrogen is transported through the electrolyte to form H2 molecules on the platinum electrode. The overall reaction is 2H2 O → O2 + 2H2 , and the photon provides the energy necessary to keep this endothermic reaction going. The process is NA, and ET is important. Photon-to-photon: Scattering of photons to photons at surfaces is listed, because infrared-reflection spectroscopy (IRS) is a valuable tool to study phonons and adsorbate vibrations very accurately (Ryberg, 1982). The detailed measurements of peak energies, widths, and shapes provide important information about, e.g., the decay of admolecular modes into phonons and EHP’s. Comparison of theory and experiment have documented that the latter mechanism is present at several instances (Ryberg, 1982; Chabal, 1985; Reuff et al., 1988). Actually, IRS has given the key information on EHP damping of adsorbate vibrations, verifying the Persson-and-Persson model (see below) for this phenomenon (Persson and Persson, 1980a). Raman scattering belongs also to this category, as a photon with one frequency comes in and one with another frequency goes out. The frequency shift gives information about excitations in the studied systems, typically vibrations. For some adsorbate-substrate systems there is a surface-enhanced Raman scattering (SERS). The enhancement can be many orders of magnitude. For quite some time, it has been known that both electric-field and adsorbate-chemical effects involving ET are behind the strong signal. However, how they are established has been debated. One model, developed on the basis of different types of experiments, shows that the long-range enhancement by resonances of the macroscopic laser and Stokes field can be separated quantitatively from the metal-electron-mediated resonance Raman effect (Otto et al., 1992). The latter mechanism proceeds by increased electron–photon coupling at an atomically rough surface and by temporary ET to orbits of the adsorbates. This model can account for the chemical specificity and vibrational selectivity of SERS and (partly) for the SERS specificity of the various metals. SERS can be modeled by the time-dependent evolution in the intermediate anionic state of the adsorbate, in analogy to Franck–Condon resonance Raman scattering (Otto, 2005). For adsorbates with a π ∗ state, the residence time of some femtoseconds in the anionic state leads to a separation of electron and hole, which quenches SERS at a smooth surface. At so-called SERS-active sites, the residence time of the hole is enhanced and therefore

470


there is no final EHP and the excitation of only a molecular vibration leads to SERS. In contrast, for molecules with only high-energy σ ∗ states, the residence time in the anionic state is less than one fs (analogous to the impulse mechanism in electron scattering), and the creation of EHP’s is less likely. This leads to first-layer electronic Raman scattering, especially by C–H stretch vibrations with an average enhancement of about 30–40fold. Since the mid-1990s, SERS has advanced greatly and gained wider application and a renewed interest. Among the new and creative developments, SERS of single molecules, nanostructures and transition metals, tip-enhanced Raman scattering (TERS), surfaceenhanced hyper-Raman scattering (SEHRS), ultraviolet-excited SERS (UV-SERS) and surface-enhanced resonance Raman scattering (SERRS) could be mentioned, together with their wide applications in biology, medicine, materials science, and electrochemistry. SERS is important not only for Raman spectroscopy and surface science but also for nanoscience (Tian, 2005). Photon-to-electron: In photoemission spectroscopy (PS) an incoming photon causes an ejected electron. PS is a major source for the characterization of the electronic structure of adsorbed species. From the dynamic-ET perspective, the application of fast and ultrafast lasers in two-photon electron spectroscopy (2PPE) (Schoenlein et al., 1988, 1990) might be even more relevant. In such pump-and-probe experiments the real-time decays of electronic and vibrational excitations at surfaces can be monitored. An interesting situation occurs in metallic nanoparticles, where an incident photon can resonantly excite an intermediate localized surface plasmon (a collective electronic excitation of the nanoparticle) (Calvayrac et al., 2000). The latter decays with typical lifetimes of a few tens fs into a cascade of EHP’s. The presence of the intermediate short-lived localized surface plasmon significantly enhances the probability for EHP creation due to the resonant character of this process. Recent resolved two-photon photoemission (2PPE) experiments (Watanabe et al., 2006) support the enhanced efficiency of EHP excitations in silver nanoparticles, when the incident photons are in resonance with the localized surface plasmon related to the nanoparticle. The single EHP created in a decay process further decays via an EHP cascade and eventually couples to the phonons, when the EHP’s energy gets in resonance with the phonon system. This novel channel opens a possibility for a “plasmon-mediated chemistry” on metallic nanoparticles (Langhammer et al., 2006). The electrons created in the process may tunnel to the adsorbates on the nanoparticle surface and induce a chemical reaction. Electron-to-molecule: As a supplement to photons, electrons can be used to stimulate desorption of atoms, molecules and ion – electron-stimulated desorption. In the very active field of DIET (desorption induced by electronic transitions) (Ho, 1983, 1994; Frischkorn and Wolf, 2006), the two phenomena are given a unified description. Either photons or electrons can be used to excite some hot electrons and/or some intermediate state of the combined adparticle-substrate system. Then one follows the development of these excitations, including the detection and characterization of the emitted particle. The mechanism illustrated in Fig. 10.20 seems to be prevalent in the “hot-electron” interpretation of DIET (Ho, 1994). Electron-to-electron: The important experiment of electron-energy-loss spectroscopy (EELS) allows studies of the vibrational structure of adsorbates on surfaces very im-


471

portant in the characterization of states before and after ET (Lindgren and Walldén, 2000). In particular, energy shifts in vibrational levels are sensitive to ET (Andersson, 1977). Heat-to-molecule: When a surface with adsorbed atoms or molecules is heated, desorption of the species occurs at a sufficiently high surface temperature. This thermal desorption is a standard tool in the surface laboratory. In this way different adsorption states are identified. The importance of NA and ET are issues for recent debate (Luntz and Persson, 2005; Diaz et al., 2006a; Lane et al., 2006; Ran et al., 2007; Cheng et al., 2007). Chemical reaction heat is covered under several headings above, which illustrate that chemically induced energy can give spontaneous and stimulated particle emissions, as well as many other NA effects. Examples of spontaneous exoemission can be found in the first stage of the oxidation of alkali metals and of stimulated exoemission in late stage of oxidation of alkali metals (Greber, 1997). Mechanical motion-to-heat: Another indirect evidence for ET and EHP excitation is obtained in atomic-scale experiments on friction. When an adsorbate layer is forced to move relative to its substrate, there is heat produced. This is common wisdom concerning friction and tribology. As such experiments get on the atomic scale (“nanotribology”), the similarities with adsorbate vibrational damping becomes clear. There is dissipation to EHP’s (Persson and Persson, 1980a; Persson, 1991), in addition to what is going into phonons. See Section 10.4.3. 10.3.7. Summary The above broad and sketchy particle-in-particle-out matrix overview gives an attempt to illustrate the multitude of ET and NA processes at surfaces. It appears that some types of NA processes are particularly frequent and worth a closer look. 10.4. Nonadiabatic processes at surfaces Dissociation dynamics, a key surface process, involves separation of nuclear coordinates and accompanying electronic rearrangements that evolve in time, after the initial perturbation that induces the dissociation. Questions about NA immediately appear. Are the dissipative forces such that the system smoothly and continuously adjusts to the new ground state of the system, by rapidly dissipating any excess energy in a “friction”-like manner? Such is the common picture of, e.g., growth and thermal chemical reactions. Or, in the other extreme, are intermediate excited states created that relax their energy by so large quantum jumps that they can be observed externally as individual events? Chemi- and bioluminescent reactions are examples of the latter. This section gives some further details on some examples of surface processes, where NA is well established. 10.4.1. Electron and ion spectroscopy according to Hagstrum Experimental results for band structures of solids, like Si(111), Si(100), Ge(111), Cu, and Ni were early obtained by INS, an electron-spectroscopical method tied to Hagstrum

472


Fig. 10.21. Adopted from Woratschek et al. (1987a). (Left) He∗ (21 S) and He∗ (23 S) MAES of a clean Cs surface. The peaks denoted by hω ¯ sp are due to surface plasmons. (Right) Energy diagrams showing two mechanisms of the S2T conversion of metastable He atoms: (a) two-step resonance process and (b) one-step Auger process.

(Hagstrum, 1961; Hagstrum and Becker, 1967). When an ion of sufficiently large ionization energy is neutralized at the atomically clean surface, two-electron, Auger-type transitions take place. An analysis of them gives total electron yield and kinetic energy distribution of ejected electrons in terms of a number of parameters (Hagstrum, 1961). Fit of theory to experiment yields results on (1) DOS of the valence bands, (2) energy dependence of the matrix element, (3) effective ionization energy near the solid surface, (4) energy broadening, and (5) electron escape over the surface barrier. Early results can be illustrated by over-all widths of valence bands (14–16 eV for both Si and Ge), widths of degenerate p bands (5.1 eV in Si and 4.3 eV in Ge), and effective ionization potentials (2.2 eV less than the free-space value for 10-eV He+ ions, decreasing linearly with ion velocity). The spectroscopic information comes via a deconvolution of the kinetic-energy distribution of electrons ejected in the radiationless, two-electron neutralization of slowly moving noble-gas ions at the solid surface. This leads to a transition-density function, which includes information about the density of states in the filled conduction band, transition probabilities of the Auger-type neutralization process, and possible many-body effects and final-state interactions. Hagstrum postulated that a rare-gas metastable atom incident on a metal surface would be deexcited to the ground state via resonance ionization (RI) followed by Auger neutralization (AN) (Hagstrum, 1954) (Fig. 10.14). It is now well-established that his prediction is correct (Harada et al., 1997). The AN process produces two holes in the valence bands (Fig. 10.21), which makes the analysis of data less direct. Here a metal electron with energy i is transferred to the lowest unoccupied orbital (ψa ) of the rare-gas ion and an another metal electron at j is emitted to an empty state (ψe ), simultaneously. For sufficiently large energy transfer, this electron may escape from the metal, with a kinetic energy Ek = Ei − i − j − 2Φ, where Ei is the effective ionization potential of the rare-gas atom in front of the surface, and Φ is the workfunction of the metal. The transition rate


473

$ P () = |Hn |2 N( − ζ )N ( + ζ ) dζ of the AN process, that is, a true ET, is given by integrating over all combinations of transitions. Here N () is the LDOS at which the AN process occurs, and Hn is the matrix element of the form

ψa∗ (r2 )ψe∗ (r1 )F |r1 − r2 | φi (r1 )φj (r2 ), Hij = (10.37) where φi and φj are the orbitals of the surface electrons involved, and F (|r1 − r2 |) is a screened Coulomb potential between the two surface electrons. The formalism and deconvolution procedure were discussed in detail by (Hagstrum and Becker, 1971; Boiziau et al., 1980; Sesselman et al., 1987). Equation (10.37) indicates that the relative transition probability from MO φi largely depends on the relative overlap between the relevant orbitals φi and ψa . The ET to the ion approaching the surface depends essentially on the overlap between the wavefunctions of the ion and surface, ψa |φj , at which the AN process takes place (typically at the distance of 3 Å in front of the first atomic layer) (Sesselman et al., 1987). Therefore an electronic wavefunction exposed further outside the surface gives more effective overlap than that distributed inside the surface. Taking the short-range nature of the screened Coulomb potential in Eq. (10.37) into account, the second electron is considered to be emitted also from the local area at the outermost layer. Consequently, the transition rate is determined largely by the electrons at and outside the surface. The transition probability of RN becomes appreciable at larger atom-surface distances than that of AN, as the wavefunction of the excited MO ψb is spatially more extended than that of the ground state ψa . Therefore, at a surface with a low workfunction an approaching ion converts to a metastable atom through RN before the AN process occurs. The metastable atom thus formed is finally deexcited through AD. In INS, where a rare-gas ion beam is incident on a metal surface, the mechanism is AN. The singlet-to-triplet (S2T) conversion, i.e. conversion of He∗ (21 S) to He∗ (23 S), has been the subject of intense experimental (Roussel, 1983; Lee et al., 1985; Woratschek et al., 1985, 1987a; Hemmen and Conrad, 1991, 1992; Bremten et al., 1992; Böttcher et al., 1994b) and theoretical (Makoshi and Newns, 1985; Kasai and Okiji, 1985; Nedeljkovic et al., 1989; Makoshi et al., 1990; Teiller-Billy and Gauyacq, 1990; Yoshimori and Makoshi, 1990; Makoshi, 1991; Yoshimori, 1993; Borisov et al., 1993, 1995; Makhmetov et al., 1995) studies, in particular on alkali metal surfaces. First it was verified (Roussel, 1983) by scattering of He∗ (21 S) and He∗ (23 S) as He+ ions from a Ni(111) surface and studying it as a function of the workfunction-lowering by K adsorption. On the clean surface, no ions are detected, as those formed by RI are neutralized by the AN process. With an increasing amount of K, K+ sites repel ions formed by the RI process before the AN takes place. The ion signal rises from zero for the clean surface and becomes zero again in the region, where the Fermi level is higher than the 2s levels of H∗ (the surface workfunction is lower than the ionization energies of the metastable atoms) and thus the RI process is suppressed. Almost all the singlet atoms are converted to the triplet before the AD process occurs (Roussel, 1983). This is demonstrated by, e.g., experimental spectra of Cs (10 or more monolayers) on a Cu(110) surface (Woratschek et al., 1987a). Compared to the He∗ (23 S) spectrum, that for He∗ (21 S) (Fig. 10.21) shows a very small peak β at higher kinetic energy of the emitted electrons that originates from the singlet species and the Cs-derived states

474


Fig. 10.22. Adopted from Böttcher et al. (1994b). (Left) Energy diagram for the singlet-triplet conversion of H∗ involving He− : (a) the formation of He− (1s2s2 ,2 S) due to the resonance ionization of He∗ (21 S), and (b) the formation of He∗ (23 S) by the autodetachment of an electron from He− (1s2s2 ,2 S). (Right) Intensities of the singlet and triplet peaks in the He∗ (21 S) spectrum of Na/Ru(0001) (top panel) compared with the S2T conversion ratio R and the workfunction Φ (bottom panel) as a function of Na coverage (Böttcher et al., 1994b).

(Harada et al., 1997). The much stronger α peaks in the He∗ (23 S) and He∗ (21 S) spectra demonstrate the almost complete conversion of the singlets. Several mechanisms have been proposed for the S2T conversion (Fig. 10.21), among which the following two, both involving ET, seem generally accepted: (I) In an Auger-type process, where a valence electron from the surface fills the hole in the 2s level of He∗ , and the 2s electron with opposite spin is raised to an energy above the Fermi level, where it can tunnel into the solid. This spin-flip process competes with the ordinary AD process involving the transfer of a valence electron of the surface to the 1s hole of He∗ (21 S). The former process, however, occurs with much higher probability, because the 2s state is much more diffuse and thus overlaps much more effectively with a surface orbital. Mechanism I is supported by several theoretical calculations (Kasai and Okiji, 1985; Nedeljkovic et al., 1989; Makoshi et al., 1990; Teiller-Billy and Gauyacq, 1990; Yoshimori and Makoshi, 1990; Makoshi, 1991; Borisov et al., 1993). (II) In this mechanism (Fig. 10.22), the energy positions of the occupied and affinity 2s levels of He∗ (21 S) and He∗ (23 S) vary with the atom–surface distance, the image force shifting the occupied levels up and the affinity levels down from their asymptotic positions (the electron affinities of He∗ (21 S) and He∗ (23 S) being 1.3 and 0.5 eV, respectively


475

(Schultz, 1973)). For low-workfunction surfaces the affinity level of He∗ (21 S) eventually crosses the Fermi level and can be resonantly ionized by a surface electron to form a negative ion He− (1s2s2 , 2S ) (Fig. 10.22a). This core-excited He− ion undergoes then rapid autodetachment to form a He∗ (23 S) atom, giving an electron to the solid (Fig. 10.22b) or a ground-state He(11 S). The S2T conversion occurs efficiently, when occupied and unoccupied states of the surface have high densities, because these states are directly involved in both mechanisms I and II (Fig. 10.22) (Böttcher et al., 1994b). The He∗ (21 S) results for a Ru(0001) surface covered with different concentrations of Na show the ratio R of S2T intensities, a measure for the S2T conversion efficiency, together with the workfunction Φ, as functions of the Na coverage. The variation of Φ, showing a minimum, is typical of alkali-metal overlayers (Fig. 10.7). The conversion efficiency R starts to increase around the minimum of Φ, where the “metallization” begins (Aruga and Murata, 1989) and reaches a maximum of 30 around 0.8 ML, where the overlayer is essentially metallic. This supports the idea that R is closely related to the DOS near the Fermi level, which increases in the process of metalization (Harada et al., 1997). For the Na layers, mechanism I is at work, as the workfunction is higher than 2.5 eV at all coverages. It should be clear that mechanism II corresponds to what is discussed in Section 10.4.2 below. The value of the hole survival probability Ph (z) for an electronegative atom or molecule there defined is on approach to the surface reduced from unity by ET, this time called RI. In summary, a conceptual framework has emanated from (Hagstrum, 1954), where ET is the mechanism responsible for the S2T conversion of metastable He. The mechanism bears on such general features that it should apply also to other similar surface collisions. For instance, for the collision of O2 with Al(111), discussed below, the conditions are such (that is, overlap between adparticle and metal orbitals act similar to those in the metastable-He case) that ET should be the mechanism here also, as suggested by (Nørskov and Lundqvist, 1979b; Strömquist et al., 1996; Zhdanov, 1997; Hellman et al., 2003). This should happen in the reaction zone described below. 10.4.2. Surface chemiluminescence and the NNL model Surface chemiluminescence has been studied in detail with, e.g., Na, K, and Al metals and halogen gases (Cl2 , Br2 , and I2 ) (Kasemo and Walldén, 1975; Hellberg et al., 1995; Poon et al., 2003). The most important findings are (i) emission distribution over rather broad emission bands, with peak intensities in the visible spectrum, (ii) spectral-distribution changes with surface coverage, (iii) emission-intensity proportionality to the gas pressure at low gas pressure, (iv) emission probabilities in the range 10−7 –10−5 per elementary reaction, (v) emission of electrons frequently accompanying the light emission, and (vi) absence of vibrational fine structure in high-resolution spectra (Kasemo and Lundqvist, 1984). Fig. 10.13h illustrates one case of relaxation, the reaction M +X → M + +X − , which in a NA way results in emission of a photon or in decay without radiation, by Auger deexcitation. With most molecule–surface combinations the emission of photons is in the infrared. In a few cases, including the model reaction of Cl2 impinging on a Na surface, the in-

476


volved energy is high enough to give light in the visible range, surface chemiluminescence (Fig. 10.16) (Kasemo and Walldén, 1975). The NNL model (Nørskov et al., 1979c) for surface chemiluminescence from reactive molecules impinging on metal surfaces involves ET at large molecule–surface separations, “harpooning” (Polanyi, 1932; Magee, 1940; Struve et al., 1975; Gadzuk, 1985), to form a negative ion, a dynamically created hole, i.e. an empty electronic level below the Fermi level, on the impinging molecule or its fragments (cf. Fig. 10.5), and a radiative decay of this hole close to the surface (Fig. 10.13h). The negative molecular ion is attracted to the surface by the image force until it is trapped, dissociated, or possibly reflected from the surface. It is conveniently described in a diabatic picture, where PES’s can be calculated self-consistently in simple models (Lang et al., 1986; Hellman et al., 2005a). An exothermic surface reaction gives most of its energy to heat, i.e. phonons of the surface. Only a small amount goes into NA, including electronic excitations and photons (Fig. 10.4) to a tiny fraction, the bases for exoelectron emission (Kasemo et al., 1979) and surface chemiluminescence. Each deexcitation channel has a decay rate, and to a first approximation the total decay rate of the hole may be written as = ET + Auger + Rad ,

(10.38)

where ET Auger Rad . Ideally, this occurs when electronegative adsorbates, e.g., Cl2 , O2 , and NO, adsorb on metals, especially low-workfunction metals, like Na and K (Kasemo et al., 1979; Hellman et al., 2004). The NNL description is made in terms of molecular orbitals, featuring the energy level a and survival probability Ph of the adatom hole, and their variations with the distance z to the surface. Assuming both a trajectory approximation and a constant velocity v of the incoming particle, the probability for the hole state, i.e. the 2p hole on the chlorine ion getting formed, to survive may be expressed as z (z ) dz , Ph (z) = exp −4π/(hv) (10.39) za

where z = za denotes the position, where a (z) crosses the Fermi level (Fig. 10.5). The rapid variation of Ph (Fig. 10.11) reflects that (z) typically decays exponentially with z. The very narrow region outside the surface, where Ph (z) varies from close to unity to close to zero, could be viewed as a reaction zone (Nørskov et al., 1979c). It lies typically at about twice the chemisorption equilibrium distance from the surface layer (Nørskov et al., 1979c), about 3 – 4 Å outside this layer, according to estimates for transition metals (Nørskov and Lundqvist, 1979a) and Na metal (Nørskov et al., 1979c). The initial NNL model was made for Cl2 molecules impinging on an Na surface. It gives a reasonable account for measured chemiluminescence spectra from halogens on sodium, in particular the fact that the emission peak follows the variation of the electron-affinity value of the incoming atom (Fig. 10.16) (Nørskov et al., 1979c). The NNL model has later been successfully applied to the Cl2 /K-metal chemiluminescence (Fig. 10.23) (Andersson et al., 1985; Andersson, 1989). The observed dramatic increase of photon-emission intensities for K (Andersson et al., 1985) can be explained within the model. The surface-modified Cl affinity level is shifted down in energy and eventually ends up below the bottom of the K valence band. Close to the surface there


477

Fig. 10.23. Adopted from Andersson et al. (1985). NNL model for Cl2 on Na and K.

is thus no resonance tunneling, the mechanism that so efficiently holds the chemiluminescence intensity down on Na. On K, the narrow conduction band allows resonance tunneling only rather distantly from the surface. This lets the hole survive, to allow Auger transitions occur close to the surface. The stronger overlaps there make the electron emission stronger on K. So, differences between the spectra from Na+Cl2 and from K+Cl2 can be explained by the different bandwidths, that is, by ultimately the difference in electronic densities in K and Na (Fig. 10.23) (Andersson et al., 1985). Experimental data for electron and photon emissions (kinetics, yields, and energy distributions) have also been obtained during the chemisorption of chlorine on pure and sodiumdoped zirconium (Prince et al., 1981) and of halogen (chlorine, bromine) adsorption on group IIIA (Y, Sc) and group IA (Zr) surfaces, both clean and with Na predosing (Bourdon and Prince, 1984). They can be explained in terms of a slightly extended NNL model. It is made quantitative (Bourdon and Prince, 1984) for an adequate description of the gross features of the electron-emission process (Prince et al., 1981). For species with high electron affinity, such as chlorine, the observed chemisorptive emission is strongly dependent on workfunction, varying from 10−6 to 10−2 electrons per incident atom, and following a cubic dependence on the excess energy available. Chemiluminescence is also observed, at levels of 10−8 photons per incident atom in accordance with previous estimates. Doubledosing experiments show that halogen dissociation is not a precursor step. Similarly, the NNL model with extensions provides a consistent picture of electron emission in the NA surface reactions of halogens and alkalis (Kasemo and Walldén, 1975; Hellberg et al., 1995; Poon et al., 2003) and of O2 with a Cs surface (Böttcher et al., 1990, 1991, 1993c). The exoelectron emission is described by extending it to account for the Auger transition involving conduction-band electrons, Cl 2p hole, and an exoelectron

478


Fig. 10.24. Adopted from Poon et al. (2003). Evolution of Cl2 and Cl− 2 PES as a function of the Cl2 /Al(111) reaction coordinate (z). The separation between the Cl2 and the surface (z) decreases from the left frame (a) to the right (d). (a) Adiabatic charge transfer is endothermic by 1.25 eV at infinite separation. (b) At z ∼ 2.9 Å the wave function overlap predicts small resonant adiabatic charge transfer probability. (c) At z ∼ 1.1 Å vertically − resonant charge transfer results in dissociation of Cl− 2 into Cl and Cl fragments with the excess translational energy shown between the arrows. (d) At z 1.1 Å vertically NA charge transfer results in the same products and excess energy as the vertically resonant process.

(Kasemo et al., 1979). For the reaction of O2 with a Cs surface (Böttcher et al., 1990, 1991, 1993c), the probability of an electronically excited intermediate state is calculated, estimating the molecular velocity and electronic structure with values in the region slightly outside the surface. In the deexcitation of this state, valence band Auger transitions are much more probable than radiative decay, but the photon emission probability is substantial enough to give experimental spectra in several cases. The Auger electrons in the tail above the workfunction barrier can be detected as exoelectrons. The photon and electron emission are direct evidences that chemisorption may take place via NA processes. The evolution of the Cl2 molecule approach to a metal surface can be demonstrated by the case of Al(111) in terms of a set of schematic PES’s (Fig. 10.24) (Poon et al., 2003): (a) For infinitely separated reactants, gas-phase PES’s for Cl2 and Cl− 2 (shifted up in energy by the Al(111) workfunction value, ΦA = 4.25 eV (Schochlin et al., 1995); (b) Closer to surface (2.9 Å), the Cl− 2 curve is shifted by the image potential, −3.6 eV/z(Å), where z is the separation; (c) At z = 1.1 Å, vertically resonant CT results in dissociation of Cl− 2 into Cl and Cl− fragments with the excess translational energy shown between the arrows; (d) At z < 1.1 Å, vertically NA electron transfer results in the same products and excess energy as the vertically resonant process. The NNL model ramifies to ion emission (Nørskov and Lundqvist, 1979a). Experimentally, the ionization probability of the ejected particles of a sputtered metal surface correlates with the substrate workfunction, the outward velocity v, and the ionization potential I or affinity A of the departing atom (Andersen, 1976). This can be accounted for by considering the probability that the initial occupation of the ionization or affinity level survives during the NA passage through the surface region (Fig. 10.11). With reasonable model assumptions for the variation of the position and width of the ionization or affinity level with distance from the surface, the ionization probability is shown for a large class of systems


479

to be roughly proportional to exp[−(I − Φ)/(cv)] (positive ions) or exp[−(Φ − A)/(cv)] (negative ions), where c is a constant (Nørskov and Lundqvist, 1979a). There is also a direct link between the NNL model and vibrational damping of adsorbates on metals (Persson and Persson, 1980a), as described in Section 10.4.2. The NNL model has also implications for sticking (Nørskov and Lundqvist, 1979b). The loss in kinetic energy of the impinging particle that is necessary to trap the impinging particle in the surface potential well can be provided by NA processes. The nonzero velocity of the impinging particle and the different scales of characteristic times for electrons and particles can cause an NNL-type excited electronic state, with a subsequent decay, as described Section 10.4.2. The initial support were some trends in the specificity of the sticking probability S (Nørskov and Lundqvist, 1979b), and Eq. (10.24) gives a typical model result. This adlevel-Fermi-level crossing mechanism (strong NA) differs from the friction-type sticking (weak NA) (d’Angliano et al., 1975). A focus of the NNL model is the variation of adparticle valence levels with the distance to the metal surface. Are there adlevels available to cross the Fermi level? Indeed, there are such levels or resonances, even such that shift almost a dozen eV within a couple of Å close to the surface (cf. Fig. 10.3). Once there is a crossing, there might be ET. Its efficiency is reflected in the hole survival probability Ph (z) and the location of the reactivity zone. It depends on orbital overlaps, and their typically exponential spatial decays make the reactivity zone very narrow. So, a strongly varying adlevel is important to provide a crossing with the Fermi level, but in expressions for the ET only adlevel-energy value(s) in the narrow zone are picked up (cf. Fig. 10.11). Detailed observation and analysis of exoelectron emission from velocity-selected Cl2 impinging on K (Hellberg et al., 1995) imply a NA reaction pathway, where ET (surface harpooning), surface-modified electron affinities, molecular dissociation, and Auger neutralization are key ingredients. The model used is consistent with the NNL chemiluminescence model (Nørskov et al., 1979c) but extends it in a significant way (Figs. 10.25 and 10.26) by trajectory calculations for Cl2 molecules approaching a K surface with different velocities and orientations, and by explicit use of gas-phase Morse potentials (Davidovits and McFadden, 1979) and ion–surface image forces (Fig. 10.25), clarA ), molecular adiabatic (E AA ), and vertical ifying the differences between atomic (ECl Cl2 AV (ECl2 ) electron-affinity levels, and a rather explicit picture of the dissociation dynamics (Fig. 10.26) (Hellberg et al., 1995). It accounts for the dependence of the exoelectron yield and the essentials of its energy dependence on the velocity of the incident Cl2 molecules. There are four key steps in the model (Fig. 10.26): (1) Around 4 Å from the surface (jellium edge) the harpooning electron is transferred from the K conduction band to the antibonding LUMO resonance of Cl2 . The Cl− 2 ion dissociates or forms a temporary negativeion state. As the Cl and Cl− fragments separate, the empty 3p adlevel resonance of the Cl atom is rapidly shifted downwards, producing an excited hole. This hole is either filled A is shifted below by resonant ET from the metal (2) or survives, until its affinity level ECl the bottom of the K conduction band (3). It then deexcites by an Auger transition (4) or radiative decay, producing measurable electrons or surface chemiluminescence. Electron A to end up clearly below the band bottom, since for K the workemission requires an ECl function Φ (2.3 eV) is larger than the bandwidth (2.1 eV). The highest measured exoelec-

480


Fig. 10.25. Adopted from Strömquist et al. (1996). Molecular PEC’s as functions of internuclear distance R with the electron applied from (a) the vacuum level and (b) the Fermi level of potassium. The dashed curve shows how the Cl− 2 PEC is image shifted im = 1.3 eV at 3.5 Å from the surface.

tron energy implies an affinity level between 2 and 2.5 eV below the band, corresponding to a hole-excitation energy between about 4.2 and 4.7 eV (Hellberg et al., 1995). The ET is the result of a complicated coordinated motion of nuclei and electrons, the latter in a correlated motion due to their Coulomb interaction. Nevertheless, as described above, a combination of realistic nuclear trajectory calculations and a simple master equation for the electrons is mostly used for the survival rate and CT processes (Langreth and Nordlander, 1993; Nordlander and Tully, 1990; Strömquist et al., 1996; Dutta and Nordlander, 2001). In (Strömquist et al., 1996; Hellman et al., 2003, 2005a) the ET, considered as a resonance tunneling of a substrate electron to the LUMO of the molecule, is assumed irreversible. That is, no back donation of electrons from the negative molecule, X − , is considered, where relevant X could be, for instance, O2 , NO, or N2 . In a semiclassical trajectory approximation, the time-dependence of the X − electron population, nX− , is described by the corresponding simplification of Eq. (10.30) (Langreth and Nordlander, 1993; Strömquist et al., 1996; Hellberg et al., 1995),

d −1 f E A z(t) 1 − nX− (t) , nX− (t) = τres dt

(10.40)

−1 is the resonance filling rate, f (E) the Fermi-distribution function, and where τres A E (z(t)) = VX− /Al + (z(t)) − VX/Al (z(t)), being the difference between the PES’s associated with the negative ion X− and the neutral molecule X, respectively.


481

Fig. 10.26. Adopted from Strömquist et al. (1996). Schematic picture of the model in the case of a molecule with A (R, z) an initial orientation of 20◦ . (Top) Events (1) – (4). (Bottom) The corresponding affinity-level positions ECl A (∞, z). Event (4) is the of the Cl atom are denoted. For comparison, the free-atom affinity level is shown as ECl Auger process of Cl producing the electron emission, and the Cl− ion is already screened out by the electron gas.

Studies of the lifetime have revealed an exponential dependence on the separation z (Borisov et al., 1998, 1996; Auth et al., 1998; Bahrim et al., 1994; Nordlander and Tully, 1990; Teillet-Billy et al., 1995). The resonance-filling rate can be approximated −1 = 2 exp(−α z), where as (Strömquist et al., 1996; Hellberg et al., 1995) hτ ¯ res 0 res 0 and αres are constants that have to be chosen appropriately. Equation (10.40) rests on assumptions about (i) a trajectory approximation (r(t) = z(t)), where r(t) is the position of the molecule X, (ii) tunneling rates with exponential spatial decay, (iii) occupation numbers nX− (t) representing expectation values for a macroscopic ensemble of the molecules 2 scattered against the surface. The latter means that the fractional occupancy of the MO resonance nX− (t) should be interpreted as an ensemble average of the occupation (Langreth and Nordlander, 1993). If the empty affinity level is within the range of the substrate electron band, process (a) is expected to dominate the decay (Nørskov et al., 1979c). In the mid-90s the NA situation could be summarized by addressing some fundamental processes in molecule–surface dynamics: ET from a metal surface to an incident or adsorbed particle, NA electronic processes, and bond breaking dynamics (Kasemo, 1996): (i) For Cl2 molecules impinging on a clean potassium surface (Hellberg et al., 1995) the mechanism is rather clear: At about 3.5 Å from the surface a “harpooning” electron is tunneling from the metal valence band to the diabatic electron affinity level of Cl2 , making a − Cl− 2 ion, an ET that can cause dissociation of Cl2 into Cl and Cl , the neutral Cl, thanks − to its escape from the Coulomb repulsion of the Cl ion, having its atomic affinity level rapidly ( zd , d (ω) − 1 . σd (ω) = d (ω) + 1

(11.17) (11.18)

The first term in Eq. (11.16) is the d-screened bare Coulomb interaction. The second term stems from polarisation charges at the boundary of the dielectric medium.

Dynamics of Electronic States at Metal Surfaces

533

11.2.4. Electron–phonon interaction The contribution of electron–phonon scattering to the total spectroscopic linewidth, Γ , can be written as (Hellsing et al., 2002) ωm α 2 Fki 1 + 2n(ω) + f (E + ω) − f (E − ω) dω, Γph = Γph (ki , E) = 2π 0

(11.19) where E and ki are the energy and parallel momentum of the surface state, f and n denote the Fermi–Dirac and the Bose–Einstein distribution function, respectively, and ωm the maximum phonon frequency. The so-called Eliashberg function α 2 F (ω), which is the phonon density of states weighted by the electron–phonon coupling function g, can be written in the quasielastic approximation as

g ν (q )2 δ ω − ων (q ) δ(f − k ), α 2 Fki (ω) = (11.20) i i,f ν,q ,f

where ων (q ) is the phonon dispersion relation for the νth branch. The last δ function indicates the quasielastic approximation (Grimvall, 1981). The electron–phonon coupling function includes the matrix elements between the initial (i) and final (f) electron states. For a translationally invariant system 1 μ ν (q ) = q ν (Rμ ) · ∇Rμ V˜q |i. f| gi,f (11.21) 2MNων (q ) μ In Eq. (11.21) the static screening of the electron-ion potential is used and thus one neglects the frequency dependence of the coupling function. The Rμ denote the positions of the ions, N is the number of ions in each atomic layer, M is the ion mass, μ is the layer μ index, is the phonon polarisation vector, and V˜q is the screened electron-ion potential. For T = 0 K Eq. (11.19) yields max(|E|,ωm ) Γph (ki , E) = 2π (11.22) α 2 Fki (ω) dω. 0

The electron–phonon coupling parameter, λ, can be calculated using the Eliashberg function: ωm 2 α Fki (ω) λ(ki ) = 2 (11.23) dω. ω 0 In the high temperature limit (kB T hω ¯ m , kB is Boltzmann’s constant) we have (Grimvall, 1981) Γph (ki , E) = 2πλ(ki )kB T .

(11.24)

Equation (11.24) may be used to extract the electron–phonon coupling parameter by measuring the linewidth at different temperatures. Given α 2 F many interesting quantities can be calculated. However, these calculations are demanding since α 2 F contains all the physics of electron–phonon coupling, like the phonon dispersion relation, the phonon polarisation vectors, the one-electron wave functions, and the gradient of the deformation

534

R. Berndt and J. Kröger

Table 11.1 Eliashberg coupling function and phonon contribution to decay as calculated within the Einstein and the twodimensional (2D) and three-dimensional (3D) Debye model with Einstein frequency ωE and Debye frequency ωD . E denotes the surface-state energy and λ the electron–phonon coupling parameter α 2 F (ω) 1 λω δ(ω − ω ) E E 2

λω2 , 2 ωD

0,

⎧ ⎨ λω ⎩

ω < ωD ω > ωD

2 −ω2 π ωD

0,

,

ω < ωD ω > ωD

Γph π λωE , E > ωE 0, E < ωE ⎧ 2π λω ⎨ 3 D , E > ωD

Reference Einstein model (Grimvall, 1981)

3

⎩ 2π λω E < ωD 2 , 3ωD ⎧ E > ωD ⎨ 2λωD , *

ω2 , E < ω 1 − 2λω 1 − D D ⎩ ω2 D

3D Debye model (Grimvall, 1981) 2D Debye mode (Kostur and Mitrović, 1993)

potential. These quantities can be obtained from first-principles calculations (Maksimov et al., 1997) for bulk metal electron states. For electron states at metal surfaces these evaluations are time consuming (Eiguren et al., 2003a). In particular, phonon dispersion relations and polarisation vectors can be calculated with reasonable accuracy using force-constant models (Kress and Wette, 1991) or the embedded-atom method (Nelson et al., 1988; Rahman et al., 2002; Sklyadneva et al., 1998). In recent calculations of Γph and λ for surface states, one-electron wave functions obtained by using the one-electron model potential (Chulkov et al., 1997, 1999b) have been used. The deformation potential can be described by using the bare pseudopotential and the static dielectric function, i.e.,

μ

μ V˜q (z) = ˜ −1 z, z , q V˜bare z , q dz , (11.25) μ where V˜bare is the two-dimensional Fourier transform of the bare electron-ion pseudopotential (Ashcroft, 1966). ν (q ) to be constant, the Eliashberg function is simply proportional to the Taking gi,f product of the phonon and electron densities of states. Assuming then a free electron gas model for the one-electron states one can derive simple analytical expressions for the Eliashberg function and for the phonon contribution to the linewidth broadening within the Debye and Einstein models for vibrational spectra (see Table 11.1). To estimate the phonon contribution to the decay of electronic surface states usually the three-dimensional Debye model was used (Grimvall, 1981). However, this approach is questionable since surface phonon modes are not taken into account. A more rigorous treatment of the electron– phonon contribution is needed especially for surface states close to the Fermi level, because for these states the electron–electron scattering is small and the electron–phonon interaction becomes dominant even at low temperatures. In the calculations by Eiguren et al. (2002, 2003b) the contribution from different phonon modes, in particular the Rayleigh surface mode, are taken into account. Figure 11.3 shows the calculated phonon dispersion (right panel) for Cu(111) and the corresponding Eliashberg coupling function at Γ (right panel) for Cu(111) (Eiguren et al., 2002, 2003b). The Rayleigh surface mode is split off


535

Fig. 11.3. (a) Phonon dispersion relation from a 31-layer slab calculation in the Γ M direction of the surface Brillouin zone of Cu(111). (b) Plot of the Eliashberg coupling function of the hole state at Γ for Cu(111) (solid line) and the contribution from the Rayleigh surface phonon to the Eliashberg function (dashed line). From Eiguren et al. (2002).

from the bulk phonon bands giving rise to a peak in the Eliashberg function of the hole state at Γ at ≈13 meV. The oscillations in the Eliashberg function reflect the finite number of layers of the model-potential calculation of the electron wave functions and thus have no physical meaning. Similar results have been obtained for the coupling of surface phonons to the n = 1 image-potential state on Cu(100) (Eiguren et al., 2003c). The use of the three-dimensional and two-dimensional Debye models in the evaluations of Γph for holes in surface states on the (111) surfaces of noble metals (Kliewer et al., 2000) and Al(100) normally lead to a fairly good agreement with the results of more sophisticated calculations (Hellsing et al., 2002; Eiguren et al., 2002, 2003b).

11.3. Photoelectron spectroscopy To study the dynamics of surface-localised electronic excitations by means of photoelectron spectroscopy an analysis of the measured lineshape is necessary. For this purpose the photoemission process must be described within a many-body picture. This section is organised as follows. The single-particle and many-particle approach to the photoemission process are briefly described in the first paragraph; in the second paragraph examples of the experimental study of electron dynamics at surfaces using angle-resolved photoelectron spectroscopy (ARPES) are presented. 11.3.1. Description of the photoemission process In a photoemission experiment irradiation of the sample with monochromatic light of energy hν can lead to an excitation of an occupied state below the Fermi energy, EF , to an

536


unoccupied state above EF . If the photon energy is high enough then the excited electron can overcome the work function, Φ, of the sample and escape into vacuum. The singleparticle picture of the photoemission process now assumes that the remaining electron system is not perturbed by removing the photoelectron, i.e., no modification of the individual wave functions takes place (“approximation of frozen orbitals”). Energy conservation of the photoelectric effect then reads (Einstein, 1905) Ekin = −En (k) + hν − Φ,

(11.26)

where Ekin is the kinetic energy of the photoemitted electron in vacuum and En (k) is the energy of the initial state (with band index n and wave vector k). For a thorough study of photoemission lineshapes the photoelectric effect is described within the more realistic many-body theory. The remaining photohole, rather than being an isolated state like in the single-particle approach, is coupled to the rest of the system by many-body interactions. The excited hole state relaxes when it is filled by occupied electronic states with higher energy. The average time the hole state exists as a quantum mechanical state is the so-called lifetime, τ , which can be related to the linewidth, Γh , in the excitation spectrum via τ = h¯ Γh−1 .

(11.27)

We notice that in the single-particle picture the hole state exhibits an infinitely long lifetime leading to a δ function in the photoemission spectrum. Let us consider the electronic system of a solid as an N electron system, which is described by the Hamiltonian H = H0 + H1 .

(11.28)

Here H0 is the Hamilton operator describing the motion of electrons in an effective singleparticle potential. The operator H1 contains many-body interactions, which can be written as a sum of the individual contributions H1 = Hee + Hph + Him + · · · .

(11.29)

Here, many-body interactions are restricted to electronelectron (ee), electron-phonon (ph), and electron-impurity (im) scattering. In order to arrive at a measurable quantity, namely the current of photoelectrons (I ), the interaction of the electron with the electromagnetic (em) field, A(r, t), of the light has to be considered. This interaction can be described by the Hamiltonian 1 2eh¯ eh¯ e2 Hem = (11.30) − A · ∇ − ∇ · A + 2 |A|2 . 2m ic ic c Using Fermi’s Golden Rule the photocurrent reads ψf |Hem |ψi 2 δ(hν + Ei − Ef ). I (hν) ∝

(11.31)

f

|ψi and |ψf are eigenfunctions of the many-body Hamiltonian H in Eq. (11.28). The initial state |ψi is the ground state of the system with energy Ei , while |ψf denotes a final state with energy Ef . The δ function in Eq. (11.31) ensures energy conservation.


537

Fig. 11.4. Ratio of the spectral function over the Fermi–Dirac distribution function, Ak /F (E), as a function of the binding energy. For the plot (k) = 40 meV, Re Σ(E) = 0, and Im Σ(E) ∝ E 2 were assumed.

With these ingredients one can show (Hedin and Lundquist, 1969; Almbladh and Hedin, 1983; Economou, 1990; Hedin et al., 1998) that the photocurrent measured in a photoemission experiment is proportional to the so-called spectral function Ak (E) =

1 Im Σ(E) f (E). π (E − (k) − Re Σ(E))2 + (Im Σ(E))2

(11.32)

Here, Σ(E) denotes the complex self-energy which contains the whole many-body physics as introduced by the Hamiltonian H1 in Eq. (11.29), and f (E) is the Fermi–Dirac distribution function; (k) are the eigenvalues of H0 . As a consequence, photoelectron spectroscopy offers, via the measured photocurrent, a direct access to many-body interactions. Figure 11.4 disolays the normalised spectral function Ak (E)/f (E) as a function of E assuming the Fermi energy at EF = 0 and further Im Σ ∝ E 2 . The resulting graph is referred to as the Breit–Wigner lineshape. The lineshape analysis using Eq. (11.32) can be simplified provided that the imaginary part of the self-energy varies weakly at E = (k) + Re Σ(E). For fixed k, Im Σ(E) may be considered as a constant: 1 Im Σ(E)|E=(k)+Re Σ(E) = Γh (k) + Re Σ(E) . (11.33) 2 With this assumption the spectral function becomes a Lorentzian with full width at half maximum (FWHM) 12 Γh [(k) + Re Σ(E)] multiplied with the Fermi–Dirac distribution function: 1 Γh Ak (E) = (11.34) f (E). π (E − (k) − Re Σ(E))2 + Γh2 For Σ → 0, i.e., for vanishing many-body interactions interaction-free particles are obtained:

1 1 f (E) = δ E − (k) f (E). lim Ak (E) = Im (11.35) Σ→0 π E − (k) − i0+

538


As expected, the spectral function then turns into a δ function at the single-particle binding energy (k) (multiplied by f (E)). Comparing the spectral function of non-interacting particles (see Eq. (11.35)) with the one of (weakly) interacting particles (see Fig. 11.4) in both cases a sharp line is found whose maximum depends on the actual value of k. In analogy to the interaction-free case, therefore the excitations in the presence of many-body interactions may be characterized as elementary excitations which are generally referred to as Landau’s quasiparticles. However, these excitations reveal a finite linewidth. Thus, by switching off the many-body interactions the quasiparticles are transformed into single-particle excitations of the noninteracting situation. 11.3.2. Influence of kinematic effects The excited photohole due to many-body interactions contributes a finite linewidth, Γh , to the total linewidth, Γ . Likewise the excited photoelectron gives a contribution, Γe , to the total linewidth. Smith et al. (1993) derived the total linewidth of a photoemission peak as a function of the linewidth contributions of the photohole and of the photoelectron. In particular, for photoemission from a surface state band any influence of Γe to Γ is absent. Owing to their two-dimensional character electronic surface states are thus an ideal test system for the analysis of quasiparticle lifetimes. Experimentally, kinematic effects have been investigated, for instance, by Hansen et al. (1998) and Matzdorf (1998). 11.3.3. Experimental results At the M point of the Cu(100) surface Brillouin zone a Tamm-type state is found. Theilmann et al. (1997) report a linear correlation between the photoemission linewidth of this state and the width of low-energy electron diffraction spots. This enables the authors to extrapolate Γh to the limiting case of perfectly ordered surfaces, i.e., to the case where the diffraction peaks exhibit a vanishing width. Fig. 11.5 displays results obtained at 300 K. The temperature dependence of the linewidth for photoemission from d-like surface states has been studied by Matzdorf et al. (1996). As a result they obtained for the electron– phonon coupling parameter λ = 0.09 ± 0.02 at M on Cu(100) and λ = 0.085 ± 0.015 at M on Cu(111). Extrapolating to T = 0 K upper inverse lifetime limits were determined as Γh (20 ± 3) meV at M on Cu(111) and Γh (13 ± 4) meV at M on Cu(100) (Theilmann et al., 1997). The M Tamm state on Cu(100) has also been investigated by Purdie et al. (1998). Figure 11.6 shows the photoemission signal of the Tamm state at M (the temperature of the sample was 10 K, the energy and angle resolution of the spectrometer were set to 5 meV and ±0.5◦ , respectively). The fit to the data (Lorentzian line shape convoluted with a Gaussian representing the instrumental energy resolution) leads to Γh = 7 meV corresponding to τh = 94 fs at an initial state energy Ei = −1.8 eV. The fit deviates from the data around −1.82 eV, which is most probably due to insufficient angle resolution and electron–defect scattering. A d-like quantum well state was reported for atomically uniform Ag films grown on Fe(100) at ≈4 eV below the Fermi level by Luh et al. (2000). The linewidth of the associated photoemission signal was as low as Γh = 13 meV. Values between 20 and 30 meV


539

Fig. 11.5. Left panel: Photoemission peak width (full width at half maximum) of the Tamm-type surface state at M on Cu(100) as a function of the width of low-energy electron diffraction peaks. Right panel: As left panel for Tamm surface state at M on Cu(111). From Theilmann et al. (1997).

Fig. 11.6. Tamm state at M measured with angle-resolved electron spectroscopy with an energy resolution of ΔE = 5 meV at 10 K. The full line is a fit to the data with a Lorentzian of linewidth 7 meV. From Purdie et al. (1998).

for Γh have been observed at the upper edges of the bulk d band structure at the X point of the Brillouin zone of Cu at Ei = −2 eV (Petek et al., 1999a, 1999b; Gerlach et al., 2001) and of Ag at Ei = −3.7 eV (Gerlach et al., 2002). The resulting hole lifetimes are large as compared to the Fermi-gas predictions (Gerlach et al., 2001, 2002; Campillo et al., 2000; Zhukov et al., 2001). This observation is explained by many-body quasiparticle GW calculations of electron–electron inelastic lifetimes of d holes (Campillo et al., 2000) and

540


Fig. 11.7. Photoelectron spectra of the Ag(111) L gap Shockley surface state measured by different groups. The data clearly demonstrate how the relevant resolution parameters (energy and angle) improved with time. From Reinert et al. (2001).

demonstrates the significant role that band structure effects play in the hole–decay mechanism. Let us now turn to experimental investigations of Shockley-type surface states. The experimental progress made during the last 25 years is illustrated in Fig. 11.7 which displays the photoemission signal of the Ag(111) L gap surface state as measured by Heimann et al. (1977), Kevan and Gaylord (1987), Paniago et al. (1995), and by Nicolay et al. (2000). It shows normal emission spectra of the Γ surface state on Ag(111). Obviously, the resolution parameters improve from top to bottom. The bottom spectrum which was acquired at 30 K gives a full width at half maximum of 9 meV. Correcting for the instrumental energy resolution the authors arrive at Γh = 6.2 meV (Nicolay et al., 2000). Presently, this is the sharpest photoemission peak observed on a metal surface. It is in excellent agreement with tunnelling spectroscopy results by Kliewer et al. (2000) and Pivetta et al. (2003). The Γ surface state of Cu(111) has attracted considerable interest. Slagsvold et al. (1983) measured a linewidth of 250 meV. Increasing crystalline disorder by ion bombard-


541

ment of the surface increased the width to 400 meV. This increase in width connected with increasing asymmetry into the direction of the k dispersion of the state was interpreted in terms of disorder-induced relaxation of the sharpness of k (Slagsvold et al., 1983). Applying an improved spectrometer energy resolution Kevan reported a width of 55 meV for the same Cu(111) surface state (Kevan, 1983). Interestingly, Kevan found that upon dispersing towards the Fermi level the width of the Cu(111) surface state increased. This observation clearly contradicts the expectation based on Fermi-liquid theory that Γh → 0 with Ei → EF due to vanishing phase space for scattering. Tersoff and Kevan interpreted this result in terms of weakened k conservation and attributed the effect to a finite mean free path for elastic electron scattering from a small number of defects or impurities on the surface. The most recent results for the Γ surface states on the noble metals have been obtained by Reinert et al. (2001); Nicolay et al. (2000, 2002). For instance, for the normal emission peak of the Cu(111) surface state Reinert et al. (2001) obtained a linewidth of 25.5 meV at 30 K which is roughly three times higher than the observed width for the Ag(111) surface state at the same temperature. This effect can be attributed to the different binding energies: For Ag(111) and Cu(111) the surface state binding energies are E0 = (63 ± 1) meV and E0 = (435 ± 1) meV, respectively. As a consequence, the intraband transitions filling the photohole at Γ are much more efficient on Cu(111) and shorten the lifetime τh than on Ag(111). The most interesting hole binding energy region is the one close to the Fermi level, in particular when the binding energy is less than the maximum phonon frequency ωm . In this region the linewidth broadening is mainly determined by

Fig. 11.8. (Top) Linewidth broadening of the Cu(111) surface hole state as a function of binding energy. Diamonds: Photoemission data, solid line: Γee + Γph , dotted line: Γph , dashed line: Contribution of Rayleigh wave to Γph . (Bottom) The same as in (top) for Ag(111). From Eiguren et al. (2002).

542


the electron–phonon coupling. The contribution from electron–electron interaction is very small. For Cu(111), Ag(111), and Au(111), Γee < 0.2 meV. In Fig. 11.8 calculated results together with experimental data for Cu(111) and Ag(111) are plotted (Eiguren et al., 2002). From a three-dimensional Debye model a cubic binding energy dependence would be expected in the region below ωm , which is obviously not the case in the experimental data. The saturation of Γph (dotted line) for binding energies exceeding ωm is clearly seen in the experiment. Adding the contribution from the electron–electron interaction, values close to the experimental results are obtained. The contribution from the Rayleigh wave gives about 38% of Γph beyond the maximum phonon frequencies, indicating that bulk phonons give most of the contributions in this range. But for binding energies below the maximum of the Rayleigh wave energy, this mode alone represents on an average of about 85% of the electron–phonon scattering rate. Before turning to other materials than the noble metals a Shockley surface state within the Γ L gap projected around Y on the (110) noble metal surfaces is addressed. Only for Cu(110) a linewidth analysis has been performed so far (Straube et al., 2000) resulting in an upper limit for the imaginary part of the self-energy Im Σ = Γh /2 (16 ± 3) meV. This value corresponds to a photohole lifetime of τh (21 ± 5)fs at Y for T → 0 K. This has been interpreted by inelastic electron–hole interactions with about 50% contributions each from intra-surface-band and from coupling to substrate decay channels. The effect of substrate phonons on the electronic system has been investigated on Be(0001) (Balasubramanian et al., 1998; Hengsberger et al., 1999a, 1999b). Detailed calculations of the electronic structure of Be(0001) (Chulkov et al., 1987) predict the existence

Fig. 11.9. Experimental surface state linewidth W at Γ on Be(0001). The solid line is a fit using the Debye model. Inset: Photoemission spectra of the surface state at the indicated temperatures. From Silkin et al. (2001).


543

of surface states in large gaps of the projected bulk density of states in good agreement with photoemission data (Karlsson et al., 1984; Bartynski et al., 1985). The Γ surface state of Be(0001), which is of interest here, is widely decoupled from bulk states and forms a nearly ideal two-dimensional electron gas on a poorly conducting substrate. In close analogy to Cu(111) at Γ the surface state of Be(0001) is located inside a wide energy gap. The contribution of intraband decay to the photohole inverse lifetime at Γ is therefore the dominant factor and gives about 80% of the total value Γee = 265 meV (Silkin et al., 2001). The temperature dependence of the experimental linewidth is shown in Fig. 11.9. The solid line is the Debye model fit and the inset shows experimental spectra exhibiting Lorentzian lineshapes at different temperatures. Extrapolation to T = 0 K gives Γee + Γim = (281 ± 7) meV. The difference of 16 meV with respect to the computed value Γee = 265 meV is attributed to Γim . For Cu(111) the surface state linewidth at Γ was calculated to be Γee = 21.7 meV (Kliewer et al., 2000) and observed with Γee = (23 ± 1) meV in photoemission (Reinert et al., 2001). From this one can conclude that the large increase in Γee for Be(0001) as compared to Cu(111) is almost exclusively determined by the increase in binding energy at Γ , which gives much more electrons between EF and Ei to fill the Γ hole by intraband transitions. The first experiment pointing at an unusually large electron–phonon coupling was the photoemission study of the Γ surface state of Be(0001) by Balasubramanian et al. (1998). The authors obtained λ = 1.15 ± 0.10, which in a subsequent experiment by the same group was corrected to λ = 0.7 ± 0.1 (LaShell et al., 2000). In Fig. 11.10 photoemission spectra of the Γ surface state of Be(0001) along the Γ M direction of the surface Brillouin zone (Hengsberger et al., 1999a). Raw data is depicted as dots while full lines represent calculated spectral functions. The spectra were taken at a sample temperature of 12 K and are labelled by the wave vector Δk = k − kF with respect to kF = k(EF ). Hengsberger et al. (1999a) determined the initial state energy Ei = −2.73 eV at Γ with an effective mass of m∗ = 1.19me along Γ M (and m∗ = 1.14me along Γ K). Figure 11.10 suggests that the surface state is considerably modified near EF . In particular, the lineshape near the Fermi level is no longer a Lorentzian: Upon approaching the Fermi level a second peak appears at ≈−70 meV. Its intensity increases strongly towards kF where it finally dominates the spectral function. This second quasiparticle peak is caused by electron–phonon coupling. Using conventional many-body theory Hengsberger et al. (1999a) describe precisely this exceptional double peak evolution of the experimental spectra (full lines in Fig. 11.10). They extract an electron–phonon coupling parameter of λ = 1.18 ± 0.07. Further, a contribution of Γim = 75 meV is found, which is translated into a mean free path of ≈15 Å in the surface plane. Using a slightly different analytic approach, LaShell et al. (2000) arrived at λ = 0.7 ± 0.1. Their experimental data, however, are in excellent agreement with those of Hengsberger et al. (1999a) where comparable. An overview on experiments dealing with electron–phonon coupling at metal surfaces has recently been given by Kröger (2006). Shockley-type surface states have also been resolved on the Be(1010) surface. Balasubramanian et al. (2001) determined the surface electronic band structure in a combined photoemission and density functional theory study. After subtraction of Γph the linewidth of a surface state at A (Ei = −0.42 eV) is determined as (51 ± 8) meV. This result is in good agreement with the calculated total contribution Γee = 53 meV to the total width and indicates that Γim is most probably much smaller than Γee . The calculated width

544


Fig. 11.10. Photoemission spectra of the Γ surface state on Be(0001) at 12 K along the Γ M direction. Solid lines are calculated spectral functions. The right column gives the wave vector distance to the Fermi vector, i.e., Δk = k − kF . Inset: Close-up view of spectrum 9 to reveal the sharp peak at EF and the second structure at h¯ ωm = −70 meV. From Hengsberger et al. (1999a).

is dominated by intraband decay rates indicating the necessity to consider the details of the surface band structure for lifetime calculations. Tang et al. (2002) obtained electron– phonon coupling parameters of λ1 = 0.677 and λ2 = 0.491 for two surface states at A on Be(1010) with binding energies E1 = −0.37 eV and E2 = −2.62 eV, respectively. Valla et al. (1999) studied a d-like surface state on Mo(110), which disperses rapidly with polar angle to EF . They observed that the associated photoemission peak sharpens upon approaching the Fermi level and that there is a small but clearly resolved change in band velocity near EF . In their experiment Valla et al. succeeded in separating the contributions from electron–electron scattering, electron–phonon scattering, and electron-impurity scattering to the total linewidth of the surface state. We conclude this paragraph by mentioning a recently published technique, which allows to extract directly the Eliashberg coupling function from the measured self-energy


545

(Junren et al., 2004). The key idea is to obtain the Eliashberg function from an integral equation (Grimvall, 1981; Junren et al., 2004). Applying this method Junren et al. (2004) arrive at an electron-coupling parameter for Be(1010) of 0.68 ± 0.08. One of the main advantages of this method is that no ad hoc assumptions on phonon models have to be made. Further, low-temperature measurements are not required. In principle, since ARPES allows measurements along different crystal directions, the Eliashberg coupling function can be determined on the whole Fermi surface.

11.4. Scanning tunnelling techniques In this section we discuss various approaches which are based on low-temperature scanning tunnelling microscopy (STM) and spectroscopy (STS) to determine surface-state and image state lifetimes. The wave functions of surface states have significant amplitude outside of a crystal, making it an ideal subject for STM which probes wave functions several Angstrom above a surface. The ability of the STM to detect surface topology and to identify minute amounts of contamination, well below the limits of conventional surface analytical techniques, ensures that effectively defect-free surfaces can be studied. Moreover, using the spatial resolution of the STM individual nanostructures on surfaces can be addressed enabling spatially resolved measurements of electronic lifetimes. 11.4.1. Surface states viewed by the scanning tunnelling microscope To large extent electrons in the Shockley surface states of noble metals behave like a two-dimensional electron gas. The Ag(111) surfaces state with its isotropic, parabolic dispersion (Fig. 11.11) is a typical example. The local density of states (LDOS) for a twodimensional electron gas can be modelled as (Economou, 1983) 1 E − E0 1 . ρ(E) ∝ + arctan (11.36) 2 π Γ /2 Its characteristics are a sharp increase at the bottom of the surface state band, E0 , and an energy-independent density of states at higher energies. The width of the band onset varies with the imaginary part of the self-energy, Im(Σ) = Γ /2 = h/(2τ ), where τ is the ¯ lifetime and Γ is the linewidth in photoelectron spectroscopy. STM being sensitive to the LDOS, this electron gas can be investigated through scanning tunnelling spectroscopy or by imaging its scattering at defects or in confining structures (Davis et al., 1991; Hasegawa and Avouris, 1993; Crommie et al., 1993). The most common mode of scanning tunnelling spectroscopy is to record the differential conductance dI /dV of the tunnelling gap vs. the sample bias voltage V . This resulting spectrum is closely related to the local density of states of the sample. Experimental data from noble metal (111) surfaces (Fig. 11.12), which exhibit surface states with parabolic dispersion, is consistent with the picture of a two-dimensional electron gas. Consequently, a line shape analysis using Eq. (11.36) provides access to the lifetime (Li et al., 1998). Scattering of surface states at defects causes standing wave patterns which can be imaged by recording, e.g., maps of dI /dV . Figure 11.13 shows typical cross-sectional profiles

546


Fig. 11.11. Surface band structure of Ag(111). The shaded area and solid line are the projected continuum and surface state dispersion from ab-initio calculations. Symbols are STM-derived dispersion data: circles from Li et al. (1997); squares from Vitali et al. (2003). The dashed lines are parabolic dispersions with effective masses m∗ = 0.42 (surface state band) and 0.24 (band edge). Adapted from Becker et al. (2006).

Fig. 11.12. dI /dV spectra for the surface states on Ag(111), Au(111) and Cu(111) from defect-free surface areas. The Au(111) spectrum constitutes an average over 17 single spectra taken across various positions across the surface reconstruction. Adapted from Kliewer et al. (2000).

of a dI /dV image of a monatomic step on Cu(111) (Bürgi et al., 1999). The observed oscillatory behaviour of dI /dV at a given sample voltage V represents an energy resolved (E = eV ) Friedel oscillation and can be used to extract the dispersion of the surface state. Its amplitude decreases with lateral distance x from the step for two reasons. First, the image represents a superposition of surface state wavefunctions with fixed energy but varying wavevector components perpendicular to the step which can be described by a zeroth order Bessel function J0 (Avouris et al., 1994). Second, scattering processes cause a limited phase coherence length LΦ of the surface state and modify the LDOS through an additional exponential damping (Bürgi et al., 1999). Recently Crampin et al. (2005b) calculated the surface state LDOS near a step including lifetime effects to be ρ(x) ∝ 1 − |r|e−x/LΦ J0 (2kx) , (11.37)


547

Fig. 11.13. dI /dV data perpendicular to a descending Cu(111) step. Solid lines indicate fits which take into account a phase coherence length LΦ . Neglecting inelastic processes by setting L → ∞ leads to a much slower decay rate than observed. Values of LΦ must be halved (see footnote 1). Adapted from Bürgi et al. (1999).

with r reflectivity of the step and k wavevector of the surface state at the energy E(k) = eV .1 11.4.2. Lifetimes from STS 11.4.2.1. Spectroscopy of “perfect” surfaces The first measurements of electronic lifetimes τ with STM were reported by Li et al. (1998) who analysed the width Δ of the surface-state-induced rise in dI /dV tunnelling spectra. A geometrical definition illustrated in Fig. 11.12 was adopted to quantify Δ, by extrapolating the slope at the midpoint of the rise to the continuation of the conductance above and below the onset. This definition of the width has the advantage of allowing an analytical treatment. In particular at 5 K, where kT Γ , Li et al. obtained a linear relation between the geometrical width of the onset and the self-energy: π Δ Γ 1 + O(T /Γ )2 . (11.38) 2 Provided that instrumental effects, which would broaden this rise, can be neglected or corrected for, the width can be used to estimate Γ or the lifetime τ = h¯ /Γ of (a hole in) the surface state at the very band minimum. The linear relation of Eq. (11.38) was confirmed by detailed calculations – within the framework of the many-body tunnelling theory and band structure calculations including lifetime effects via an imaginary part of a self-energy –, but with a slope π2 β, where β ≈ 1. 1 Note that in Bürgi et al. (1999) and much of the subsequent work a factor 2 has erroneously be included in the exponent. Consequently, published values of LΦ must be halved. See Crampin et al. (2005b) for details.

548


From an analysis of the data from Fig. 11.12, Kliewer et al. (2000) arrived at lifetimes of τ = 120, 35 and 27 fs for the states at the surface state band edge on Ag, Au and Cu, respectively. For Ag(111) and Au(111) STM measures lifetimes longer by a factor of 3 than previous PES data (Matzdorf, 1998; LaShell et al., 1996), illustrating the importance of defect scattering. For Cu(111), the PES value is an extrapolation to zero defect density (Theilmann et al., 1997). The good agreement found here with the STM value provides evidence to support the use of this procedure for removing defect-induced broadening in PES. More recent PES data of Reinert et al. (2001) are in excellent agreement with the lifetimes reported by Kliewer et al. (2000). Apparent remaining differences of the measured binding energies of the order of a few meV have been identified as being due to modifications of the electrostatic potential at the sample surface caused by the STM tip through a Stark effect, the contact potential and the modified image potential (Limot et al., 2003). Becker et al. (2006) verified via model calculations that the Stark shift in STM has no significant impact on the lifetimes of noble metal surface states. The lineshape analysis presented above has recently been extended to lanthanide films (Bauer et al., 2002; Wegner et al., 2006) and alkali-covered surfaces (Chulkov et al., 2003; Corriol et al., 2005). In the lanthanide case, the small dispersion of the surface state leads to modified line shape. Alkali layers on Cu(111) exhibit “tunable quantum well states” which are strongly localised to the adlayer and whose binding energy can be widely varied by changing surface coverage. Occupied states were investigated for Na on Cu(111) at coverages close to a saturated (3/2 × 3/2) monolayer (Chulkov et al., 2003). Despite their small binding energies the states exhibit fairly large linewidths owing to efficient intraband transitions (Table 11.4). As to unoccupied states, a comparison of p(2 × 2) Cs and Na layers revealed unexpectedly large and similar linewidths despite widely different binding energies for Cs and Na (Table 11.4). This striking result has been explained in terms of Brillouin zone backfolding which turns the quantum well state into a resonance and strongly enhances elastic scattering (Corriol et al., 2005). 11.4.2.2. Spectroscopy of nanostructures and adatoms The spectroscopic approach of measuring lifetimes presented above appears to be limited to electronic states at band minima (or maxima). However, electron confinement to natural or artificial nanoscale electron resonators, results in energy quantisation, inducing spectral structure in the form of a series of resonant levels at energies that can be controlled through hangs in the dimensions and geometry of the resonator. The quasiparticle lifetime is then reflected in the level widths (Crampin and Bryant, 1996; Kliewer et al., 2001), but additional contributions arise due to lossy boundary scattering that must be accounted for if lineshape analysis is to be used to determine the intrinsic quasiparticle lifetime (Jensen et al., 2005; Crampin et al., 2005a). Kliewer et al. (2001) constructed electron resonators (Fig. 11.14a) by laterally moving adatoms on the surface with the STM tip which exhibit electronic features over a range of energies, including the vicinity of EF (Fig. 11.14b). For analysis, the LDOS of the adatom arrays was calculated using a multiple-scattering technique introduced by Heller et al. (1994) extended to include an energy-dependent lifetime of the electronic states. Adatoms were modelled as perfectly absorbing scatterers – an approximation that presumably underestimates lossy scattering – and good agreement with the experimental spectra


549

Fig. 11.14. (a) Circular array of 35 Mn atoms on Ag(111). Diameter ∼22 nm. (b) dI /dV spectra from the centre of the array as measured (top) and calculated (bottom). Adapted from Kliewer et al. (2001).

was obtained (Fig. 11.14). The extracted linewidths are shown in Fig. 11.16 (squares). Jensen et al. (2005) investigated nanoscale hexagonal and triangular vacancy islands on Ag(111) and developed a model that includes the dependence of the lifetime on electron energy and reflection losses at the island edges to describe STS data of confined states. They showed that reflection losses are the predominant source of broadening except for very large island diameters (Fig. 11.15). Linewidth data corrected for this additional broadening are displayed in Fig. 11.16 (•). Localisation of surface states at adatoms has recently been reported by Limot et al. (2005) and Olsson et al. (2004). Limot et al. (2005) tentatively interpreted the resulting lineshapes in terms of lifetimes which are modified in an element specific manner by the adatoms. 11.4.3. Lifetimes from STM of scattering patterns Interference patterns near steps are affected by inelastic scattering processes as described by Eq. (11.37) via a phase relaxation length LΦ . Bürgi et al. (1999) analysed these patterns to extract LΦ for surface-state electrons on Ag(111) and Cu(111) over a range of energies above the Fermi level. Figure 11.13 displays characteristic data from a downward step on Cu(111) and fits (solid lines) using Eq. (11.37) (cf. footnote 1). To convert LΦ to a lifetime ∗ is the group velocity. τ , the authors used the relation τ = LΦ /v, where v = hk/m ¯ ∗ The effective mass m was assumed to be independent of the wavevector. Energy-resolved lifetimes thus determined at low temperatures are included in Fig. 11.16 ( ). Temperature causes the standing wave patterns to decay over a shorter range owing to broadening of the Fermi level of the tip and to increased electron–phonon scattering. Jeandupeux et al. (1999) used these effects to establish upper limits of the surface-state line width at EF . In the interesting energy range close to the Fermi level LΦ increases strongly rendering difficult the above analysis. An alternative approach demonstrated by Braun and Rieder (2002) relies on interference patterns in a triangular electron resonators which according to the authors are more sensitive to the phase coherence length than patterns at steps. For

550


Fig. 11.15. Linewidth contributions to states at the centre of circular Ag adatom corrals on Ag(111). Intrinsic width due to e–e and e–p scattering (solid line) compiled from theoretical values. Lossy boundary scattering: 60 atoms corral, radius 100 Å (squares); 120 atoms, radius 200 Å (circles); 240 atoms, radius 300 Å (lozenges). Inset: the LDOS normalised by ρC = m∗ /π h¯ 2 at the centre of the 120 atom corral.

Fig. 11.16. Lifetime τ vs. binding energy E–EF of Ag(111) surface state. Experimental data from STS of vacancy hexagons (•, Jensen et al. (2005)), circular and rectangular corrals of Mn adatoms (◦, Kliewer et al. (2001)); STM of triangular corrals of Ag adatoms ( , Braun and Rieder (2002)), scattering patterns near steps ( , Vitali et al. (2003)); Vacancy data has been corrected for lossy-boundary scattering. Phase coherence length data ( , ) has been divided by a factor of 2 (see footnote 1).

quantitative analysis, maps of dI /dV recorded from selected areas inside triangular resonators were compared to calculated LDOS patterns which take a phase coherence length LΦ into account. The extracted values the lifetime are included in Fig. 11.16. More recently, Wahl et al. (2003) analysed standing wave patterns at steps for imagepotential states. In STM, these states are significantly Stark-shifted to higher energies due


551

to the strong electric field of the STM tip (Binnig et al., 1985; Becker et al., 1985). Subsequent work by Crampin et al. (2005b) showed that the lifetimes reported are reduced from those measured with two-photon photoemission (Berthold et al., 2002b) by a factor of two. Crampin (2005) demonstrated that the Stark shift increases the number of inelastic scattering channels that are available for decay. Moreover, field-induced changes in the image state wave function increase the efficiency of the inelastic scattering through greater overlap with final state wave functions.

11.5. Two-photon photoemission Two-photon photoemission is ideally suited to study surface dynamics on the femtosecond timescale directly in the time domain (Schoenlein et al., 1988, 1990, 1991). To this end laser pulses of a few femtoseconds durations are generated with Ti:sapphire lasers (Huang et al., 1992; Baltuška et al., 1997; Xu et al., 1997) and are irradiated with a time delay which can be controlled with sub-femtosecond resolution on the sample surface (Ogawa et al., 1997). Two operation modes of 2PPE can be distinguished: (i) in time-resolved measurements the time delay between the two laser pulses is scanned while the energy of the detected electrons is kept fixed; (ii) for energy-resolved spectroscopy the kinetic energy of the emitted electrons is measured at fixed time delay. Both operation modes are performed at constant photon energy. An overview of the results obtained from both measuring modes of 2PPE for image-potential states on Cu(001) is shown in Fig. 11.17. This set of energy-resolved spectra taken at different time delays illustrates the wealth of information which can be obtained in 2PPE experiments. The experimental requirements to obtain this kind of data set are well documented in Refs. Shumay et al. (1998), Fauster (2002, 2003).

Fig. 11.17. Two-photon photoemission spectra of image-potential states on Cu(001) for various time delays between the pump and probe pulses. The n = 1 state is seen to decay rapidly. The spectra taken at large delays are dominated by the emission from states with n 2.

552


Fig. 11.18. Two-photon photoemission signal for the lowest image-potential state on Cu(001) as a function of pump-probe delay. From Boger et al. (2004a).

Fig. 11.19. Time-resolved spectra for the surface (open circles, n = 0) and image-potential (open squares, n = 1) state on Pd(111). The dots show the cross-correlation determined for the occupied surface state on Cu(111). From Schäfer et al. (2000).

11.5.1. Lifetimes of surface states on clean metal surface Figure 11.17 suggests that the energy position of the peaks is independent of the time delay. Consequently, it is possible to tune the energy analyser to a fixed kinetic energy and scan the time delay. Results from such time-resolved measurements for the lowest imagepotential state on a clean Cu(001) surface are shown in Fig. 11.18. The intensity is plotted


553

Fig. 11.20. Time-resolved 2PPE signal of the n = 1 state of Cu(100) for three different values of the parallel momentum k . The two basic k -dependent decay processes mediated by bulk electrons are depicted schematically in the upper right corner; left: Interband decay to the bulk; right: Intraband decay within the n = 1 band. From Berthold et al. (2002b).

on a logarithmic scale to reveal the exponential decay over several orders of magnitude for long time delays. Obviously, the lifetimes increase strongly with quantum number n as the probability density of the image-potential state wave function is concentrated further and further away from the metal (see Section 11.2). The occupied surface states of the (111) surfaces of noble metals (see the preceding section) can be seen as initial states in 2PPE (Giesen et al., 1985; Reuß et al., 1996) and linewidths comparable to regular photoelectron spectroscopy are observed (Fauster and Steinmann, 1995; Hertel et al., 1997; Petek and Ogawa, 1997). On Pd(111) the intrinsic surface state is unoccupied (Hulbert et al., 1986). Figure 11.19 shows 2PPE spectra for the Pd(111) surface state (n = 0) and the first image-potential state (n = 1) as a function of the time delay between pump and probe pulse. The exponential decay for the two states on Pd(111) is observed for different signs of the time delay between the two laser pulses due to an interchanged excitation sequence for these states (Schäfer et al., 2000; Fischer et al., 1993). The shorter lifetime of (13 ± 3) fs for the unoccupied surface state at 1.35 eV binding energy compared to (25±4) fs for the first image-potential state at 4.90 eV binding energy indicates a much stronger coupling to bulk bands. 11.5.2. Momentum dependence of lifetimes Like with conventional photoemission experiments the dispersion curves E(k ) can be measured with 2PPE by detecting electrons emitted into different polar directions. These

554


Fig. 11.21. Experimental (dots) and theoretical (solid lines) decay rates of the first two image-potential states n = 1, 2 on Cu(100) as a function of excitation energy E(k ) above the respective band bottom. Computed decay rates without the contribution of intraband relaxation are shown as dashed lines. Inset: Measured dispersion of the image-potential states parallel to the surface. From Berthold et al. (2002b).

measurements with 2PPE can now be extended to unoccupied states and owing to the pump-probe technique can be used to give insight into momentum-dependent relaxation and scattering processes at surfaces (Haight, 1995). Interestingly, although image-potential state electrons on Cu(001) or Ag(001) move almost freely parallel to the surface (their effective mass is close to the free-electron mass), their decay reveals a pronounced dependence on k (Berthold et al., 2002b). For instance, Figure 11.20 displays the 2PPE intensity of the first (n = 1) image-potential state of Cu(100) for three different crystal momenta (Berthold et al., 2002b). The lifetime of this state decreases from τ = 40fs at k = 0 to τ = 25 fs at k = 0.24 Å−1 . For this behaviour two factors are responsible. First, with increasing parallel wave vector the energy of the image-potential state increases, which leads to an increased decay to bulk electronic states. In Fig. 11.21 this interband decay channel is indicated as a dashed line. Second, intraband scattering plays an equally important role. In this process electrons stay within the image-potential state band and relax towards the band minimum. For delay times around 400 fs the 2PPE signal shows a weak shoulder that decays with a time constant of 120–130 fs. This phenomenon can be explained in terms of interband scattering between different image-potential state bands. It has been shown that electrons excited into the n = 2 state at k ≈ 0 are able to scatter roughly without energy loss into the n = 1 state at k ≈ 0.3 Å−1 (Berthold et al.,


555

Fig. 11.22. Linewidth of the n = 1 and n = 2 peaks (circles and squares) from spectra as shown in the inset as a function of pump-probe delay compared to the results of calculations (full lines). From Boger et al. (2002).

2001). This interband scattering can be mediated by defects (Boger et al., 2004b, 2004a), such as steps which owing to studies on vicinal surfaces have been shown to be a particularly efficient source for this resonant interband scattering (Roth et al., 2002a, 2004). Although some theoretical progress has been made for defect scattering (Borisov et al., 2002b, 2003a, 2003b) the quasielastic scattering from defects is not as well understood as, for instance, electron–electron scattering. Probably, the most important reason why the measured lifetimes of the n 2 image-potential states in earlier work (Höfer et al., 1997; Shumay et al., 1998) are systematically shorter than predicted theoretically is the scattering of electrons from defects. 11.5.3. Linewidths of image-potential states Figure 11.22 shows that the linewidths of image-potential states depends on the pumpprobe delay: The linewidths of the n = 1 and n = 2 image-potential states on Cu(001) decrease with increasing pump-probe delay. This phenomenon can be explained by modelling the 2PPE process in the Liouville–van Neumann formalism (Blum, 1983; Loudon, 1983; Boger et al., 2002; Wolf et al., 1999). 2PPE probes a time-dependent population of the intermediate state (Boger et al., 2002): If the laser pulses are longer than the lifetime of the involved states the linewidth is given approximately by the sum of decay and twice the dephasing rate (Wolf et al., 1999). For comparable time scales the time dependence includes the temporal shape of the pump pulse and the decay of the excited electron, which is assumed to be exponential after the pump pulse is over. The linewidth for long delays is given by the dephasing rate and spectral width of the laser pulses only. The decay rate does not contribute for Gaussian-shaped laser pulse envelopes in 2PPE (Boger et al., 2002). This is in contrast to one-photon photoemission which proceeds from a constant initial-state population and where the spectral lineshape is interpreted in terms of the hole lifetime. In Fig. 11.23 the decay rates of Cu(111), Cu(117), and Cu(001) image-potential

556


Fig. 11.23. Decay rates for the image-potential states as a function of binding energy for Cu(111) (squares), √ 3 Cu(117) (circles), and Cu(001) (diamonds). The dashed lines indicate a E dependence.

√ 3 states are plotted as a function of their binding energies. The dashed lines indicate a E dependence, which can be explained in terms of a classical particle bouncing with a rate √ 3 ∝ E at the surface (Fauster et al., 2000). As can be inferred from Fig. 11.23 the ex√ 3 perimental data is well described by the E behaviour for low binding energies, i.e., for high quantum numbers n. For n 2, however, the decay rates are below the expectation from the simple picture. Cu(111) and Cu(001) represent extreme cases of high and low penetration of the wave function into the bulk because the states are located just outside and in the centre of the band gap, respectively (Fauster and Steinmann, 1995). As a consequence, the decay rates differ by approximately one order of magnitude. For the vicinal surface Cu(117) (and Cu(119), not shown here) the decay rates of the higher imagepotential states contain a significant contribution from the interband scattering into lower image-potential bands at the same energy but at a different momentum (Roth et al., 2002a; Weinelt, 2002). The decay due to electron–electron scattering with bulk or surface electrons would therefore be close to the Cu(001) line. 11.5.4. Influence of defects The preceding paragraph showed that the decay rates of the Cu(001) image-potential states are significantly higher than the dephasing rates. Inelastic scattering with bulk or surface electrons is therefore more frequent than quasielastic scattering from phonons or defects. A negligible temperature dependence of linewidths corroborates this picture (Weinelt, 2002). This contrasts the situation for Cu(111) with a large penetration of the image-potential state wave function into the bulk and pronounced temperature effects (Knoesel et al., 1998). Figure 11.24 shows the 2PPE signal of the coherently excited n = 3 and n = 4 image-potential states of Cu(001) upon covering the surface with small amounts of Cu and CO (Boger et al., 2004a; Reuß et al., 1999; Boger et al., 2004b; Fauster et al., 2000; Weinelt et al., 1999). CO has almost no influence on the lifetime


557

Fig. 11.24. Influence of adsorbates on quantum beats on Cu(001). From Fauster (2002).

Fig. 11.25. Ratio of decay to dephasing rate as a function of binding energy. For adsorbate-covered surfaces the change of the rates with coverage is evaluated.

but destroys the quantum-beat oscillations effectively. In contrast, small concentrations of Cu adatoms decrease the lifetime substantially. The same behaviour is found for the n = 1 and n = 2 image-potential states (Reuß et al., 1999; Fauster et al., 2000; Weinelt et al., 1999). Although the reason for the different behaviour of CO and Cu is not clear at present a good starting point for a theoretical modelling might be the distinction between attractive and repulsive scattering potential presented by electronegative and electropositive adsorbates to the electrons in the image-potential states (Borisov et al., 2003a; Fauster et al., 2000). Figure 11.25 displays an evaluation of the ratio between the decay and dephasing rates for clean and adsorbate-covered surfaces. All adsorbate systems lead to a decrease of the ratio compared to the clean substrates. This is an indication that disorder on the surface favours quasielastic over inelastic scattering. For Cu on Cu(001) decay

558


is comparable or larger than dephasing, while for CO on Cu(001) dephasing dominates. Interestingly, for CO on Cu(001) the ratio increases with binding energy while the opposite trend is observed for the other systems. Assuming that decay and dephasing occurs by scattering events close to the surface, then the ratio is expected to be independent of the quantum number of the image-potential state. Further studies have to be performed to understand this behaviour. 11.5.5. Adsorbate states The majority of adsorbate states is located between the Fermi level and the vacuum level and delocalises on a sub-femtosecond timescale due to charge transfer into the hosting substrate (Gauyacq et al., 2001; Feulner and Menzel, 1995; Wurth and Menzel, 2000). Consequently, with the time resolution of present 2PPE experiments it is difficult to study the dynamics of such states directly in the time domain. Mainly the substrates Cu(111) and Ag(111) with their large sp gap at Γ have been investigated. This gap can effectively hinder resonant charge transfer between the adsorbate-induced states and the metal. However, for the decay of the 2π ∗ resonance in CO/Cu(111) only an upper limit of 5 fs could be given (Bartels et al., 1998). Larger lifetimes have been observed for the antibonding sp state of alkali metal atoms (Ogawa et al., 1999; Bauer et al., 1997, 1999; Petek et al., 2000a; Petek and Ogawa, 2002). For instance, lifetimes between 10 and 50 fs for Cs/Cu(111) have been reported for different temperatures (Ogawa et al., 1999; Bauer et al., 1999) (see Fig. 11.26). Upon excitation of the antibonding state the Cs atom is repelled from the surface. Due to the long time it takes for the excitation to delocalise (Borisov et al., 2001) the beginning of the desorption could be monitored. It manifests itself in a time-dependent shift of the energy of the 6s-derived state and a non-exponential decay (Petek et al., 2000b, 2001). A molecular resonance with finite lifetime has been observed for C6 F6 /Cu(111) (Gahl et al., 2000; Vondrak and Zhu, 1999). The lifetime of the lowest unoccupied molecular orbital (≈3 eV above EF (Vondrak and Zhu, 1999)) increases as a function of coverage and reaches a value of 30 fs for 4–5 ML (Gahl et al., 2000) (see Fig. 11.27). In contrast, transient electronic states observed for benzene adsorbed on Cu(111) (Velic et al., 1998; Munakata, 2000) and Ag(111) (Gaffney et al., 2000) are most likely modified image-potential states (Zhu, 2002). 11.6. Discussion In this section, the available experimental and theoretical results for metal surfaces will be compared and a fairly consistent picture of surface and image state lifetimes will be presented. 11.6.1. Surface states The most thoroughly investigated surface states are those of the noble metal (111) surfaces at the Γ point. Agreement between STM and PES data has been achieved in these cases and instrumental issues as well as broadening due to defect scattering appear by now to be well under control. Table 11.2 summarises the experimental data for the linewidth at the


559

Fig. 11.26. Interferometric cross-correlation measurements for Cs/Cu(111): (a) Temperature-dependent 2PPE spectra; (b) inelastic decay times; (c) dephasing times. From Ogawa et al. (1999).

Fig. 11.27. Time-resolved 2PPE spectroscopy of the lowest unoccupied molecular orbital of C6 F6 on Cu(111). From Gahl et al. (2000).

560


Table 11.2 Binding energies E0 and linewidths (both in meV) for surface states at low temperatures. Calculated linewidths (Γcalc ) are decomposed in electron–electron (Γee ) and electron–phonon (Γep ) contributions. Experimental values Γexp in parentheses were measured at room temperature. Apparent differences between PES and STM concerning E0 are due a Stark effect in STM (Limot et al., 2003). a: Kliewer et al. (2000); b: Reinert et al. (2001); c: Keyling et al. (2002); d: Straube et al. (2000); e: Pivetta et al. (2003); f: Reinert et al. (2001); g: Silkin et al. (2001); h: Bartynski et al. (1985); i: Balasubramanian et al. (2001); j: Tang et al. (2002); k: Chulkov et al. (2000); l: Bartynski et al. (1986); m: Karlsson et al. (1984); n: Eiguren et al. (2003b); o: Levinson et al. (1983); p: Kevan et al. (1985); q: Schäfer et al. (2000). Adapted from Echenique et al. (2004) E0

Γee

Γep

Γcalc

Γexp

calc.

exp.

14

8

22

a

2

4

6

14

4

18

8 265 72 53

80 80 80

345 152 133

83

25

108

131 67 336 37

18

149

36

372

24 23 ± 1 6, 5 6 ± 0.5 18 21 ± 1 32 350 (380) 130 185 (∼500) (∼200) (500) (450) (∼1500) (54)

a b a, e f a f d g h i j l m o p p q

Cu(111)

Γ

Ag(111)

Γ

Au(111)

Γ

Cu(110) Be(0001) Be(0001) Be(1010)

Y Γ M1 A

Mg(0001)

Γ

Al(100)

Γ

−445 −435 −67 −63 −505 −484 −510 −2730 −1800 −420 −390 −1600 −1700 −2750

Al(111) Pd(111)

Γ Γ

−4560 +1350

a a c g g i k n c k q

minimum E0 of the surface state from various metal surfaces along with calculated values. Satisfactory consistency of experiment and theory is found in most cases. In discussing trends, it is useful to decompose the surface state linewidth into contributions from electron–phonon and electron–electron scattering. For binding energies beyond the Debye energy, the electron–phonon part is fairly energy independent. Under this condition, which is fulfilled at the binding energy E0 of the states listed in Table 11.2, the importance of intraband scattering within the surface-state band is found to depend on the size of the energy gap the surface state is located in. For surface states in a wide energy gap – Γ state on Cu(111), Ag(111), Au(111), Be(0001) (Kliewer et al., 2000; Chulkov et al., 2000, 2001; Silkin et al., 2001; Silkin and Chulkov, 2000), A state of Be(1010) (Balasubramanian et al., 2001) – the intraband contribution to the hole linewidth dominates the electronic decay. These transitions are more efficient in filling the hole than those arising from bulk states 3D → 2D (interband) transitions because of the greater overlap of the initial- and final-state wave functions which exists in the region where the imaginary part of the screened interaction is larger than in the bulk (see Fig. 11.28). For Ag(111), intraband transitions within the surface state band itself (2D → 2D transitions) contribute ∼80% of the total electron–electron decay rate. Estimates on the basis of a 3D electron gas take into account only 3D transitions and neglect both band structure and surface effects.


561

Fig. 11.28. There is only a gradual evolution in the wave functions φn (z) of states through the occupied portion of the Ag(111) surface state band, leading to large overlap with the hole wave function φ0 at the band minimum. In contrast, wave functions associated with levels within the continuum of bulk states exhibit a different spatial distribution near the surface, leading to smaller overlap and making interband transitions less important than intraband transitions in filling the hole. The screened interaction is significantly enhanced near the surface (in the low part of the figure Im W (z, z) is shown, in atomic units), both for 2D intra- and 3D interband transitions, coinciding with where the hole state is concentrated. This results in a hole decay rate that is much larger than that of bulk states at a similar energy. From Kliewer et al. (2000).

As a consequence, electron–electron scattering is significantly underestimated. In contrast, 3D → 2D interband transitions are the dominant process for surface states in a narrow gap, e.g. the Γ states Al(100), Al(111), Mg(0001), M state of Mg(0001) (Chulkov et al., 2000; Silkin and Chulkov, 2000; Eiguren et al., 2003b). While on noble metals the d-bands are not directly involved in lifetime limiting transitions owing to there large binding energies they do modify the electron–electron interaction via screening. (Kliewer et al., 2000; Reinert et al., 2001). The amount of available high-resolution experimental data is much smaller when states at k = 0 are considered. Some detailed PES results are available for the occupied parts of the surface states of Ag(111) and Cu(111) which will be discussed below in the context of electron phonon scattering. Information on unoccupied states comes from surface state scattering at steps and in confining structures. Data for Ag(111) is compiled in Fig. 11.29 along with calculated linewidths. STS data from electron confining islands (•) and STM of scattering patterns at steps agree fairly. The results from adatom corrals (◦) is not corrected for scattering losses at the confining adatoms and thus represents a lower limit of the lifetimes. The data of Braun and Rieder (2002) ( ) appears to exhibit scatter which the authors tentatively interpreted in terms of details of the electronic structure of the confining arrays. The overall trends are well reproduced by the calculations (lines) in which screening by the d-band was neglected. Below energies of ≈0.1 eV, electron–phonon scattering is the dominant decay process. In an intermediate range up to energies of ≈0.5 eV, intraband transitions are the most efficient decay channel. Their contribution, however, increases less rapidly than interband electron–electron scattering, which exhibits the typical (E − EF )2 behaviour of a free electron gas and thus determines the decay for energies above ≈0.5 eV. Vitali et al. (2003) suggested that the relative importance of intra- and interband transitions may be understood by considering the surface state wave functions. The surface-state band

562


Fig. 11.29. Comparison between the experimental and calculated surface state inverse lifetime Γ for Ag(111). Calculated linewidth displayed as solid line: total linewidth, dashed: intraband contribution to Γee , dotted: interband contribution to Γee , dash-dotted: Γep . The values for Γep at energies below 40 meV have been calculated for occupied states. Experimental data indicated by dots: Jensen et al. (2005), circles: Kliewer et al. (2001), squares: Braun and Rieder (2002), lozenges: Vitali et al. (2003). The latter two data set have been corrected by a factor 2 according to Crampin et al. (2005b).

is well separated from bulk states at low energies, but it approaches the bulk band as its energy increases edge. In an effective mass approximation, it enters the bulk continuum at ∼0.45 eV for Ag(111) where it loses its surface weight and the intraband contribution decreases. Echenique et al. (2001) calculated a similar scenario for Cu(111). More recent band structure calculations (Becker et al., 2006), however, showed that the surface state enters the bulk continuum at significantly larger energies (Fig. 11.11). Hence the calculated rapid rise of γ at ≈0.5 eV appears to be a peculiarity of the effective mass band structure. Much of the previous discussion focused on electron–electron interaction. However, electron–phonon interaction is significant as indicated by Table 11.2. Since the temperature dependence of electron–electron scattering can be neglected for binding energies exceeding kB T , the electron–phonon contribution to the linewidth may be enhanced by performing experiments at elevated temperatures (Fig. 11.9) and good agreement has been achieved for the hole state at Γ of Cu(111), Ag(111) and Au(111) (Eiguren et al., 2002, 2003b, 2003c). Close to the Fermi level, at energies below the maximum phonon energy h¯ ωm , the electron–phonon exhibits interesting structure and simultaneously Γee is very small, Γee < 0.2 meV for Cu(111), Ag(111) and Au(111). Figure 11.8 shows calculated Γep , at T = 30 K, for Cu(111) and Ag(111) along with the experimental linewidth (Eiguren et al., 2002). Deviations from the cubic binding-energy dependence in the region below ωm expected in a simple 3D Debye model are obvious in the calculation. The experimental data which has been corrected for instrumental broadening shows exhibits comparable structure. A detailed analysis shows that the Rayleigh surface phonon contributes most of the linewidth


563

for hole binding energies below the energy of the Rayleigh phonon. In the case of a surface state which is less localised to the surface, namely the Γ state on Al(100), the influence of the Rayleigh mode is significantly reduced Eiguren et al. (2003b). 11.6.2. Image-potential states The coupling of image-potential states to bulk bands being weak the phonon contribution to the linewidth of the image-potential states is expected to be small (Weinelt, 2002) although electron–phonon coupling may be enhanced when an image-potential state is close to the bulk band edge (Knoesel et al., 1998). For Cu(100), Eiguren et al. (2003c) calculated Γep < 1 meV for the first image-potential state on Cu(100) and Ag(100). The relevant decay channels therefore are purely electronic. From the bottom of the n = 1 image-potential state electrons may decay through interband transitions into bulk and surface states. Higher n image-potential states may also undergo transitions into lower lying image-potential states. Table 11.3 shows recent measured and calculated lifetimes at single-crystal surfaces for k = 0. In most cases, good agreement between first-principles theory and experiment has been achieved when realistic effective masses of the involved states are used. The main factor limiting lifetimes turns out to be the penetration of the image state wave function into the metal (Echenique et al., 2004) although in the case of Ag with its particularly low plasmon energies an influence of collective excitations has been reported as well (Shumay et al., 1998; García-Lekue et al., 2002). On d-metals short lifetimes are found owing to a large density of states around the Fermi level (Rhie et al., 2003). While an electron excited to an image-potential state at Γ may only decay via electron– electron interaction electrons with a non-vanishing momentum may also undergo transitions within the image-potential-state band itself. Berthold et al. (2002b) investigated these processes for the n = 1, 2 image-potential states on Cu(100) and found that intraband transitions to contribute ≈50% to the increase of Γ compared to the k = 0 value. Owing to the good spatial overlap of the initial- and final-state wave functions the importance of the transitions is comparable to interband decay to the bulk metal despite a much smaller phase space. 11.6.3. Overlayer states Rare-gas layers have been used as model systems to manipulate the dynamics imagepotential states (Xe/Ag(111): (Wong et al., 1999; McNeill et al., 1997); Xe/Cu(111): (Wolf et al., 1996; Hotzel et al., 1999, 2000); Ar, Kr and Xe on Cu(100): (Berthold et al., 2002a; Marinica et al., 2002; Berthold et al., 2004); Xe/Ru(0001): (Berthold et al., 1998, 2000)). Model calculations agree quite well with the experimental data and indicate that the lifetimes can be estimated by scaling the measured values of the clean surface with the inverse penetration of the image state wave function although in a few cases discrepancies occur which remain to be understood (Echenique et al., 2004). Alkali layers on Cu(111) are among the most extensively systems in surface science and thus are ideal to extend the experimental and theoretical methods present above to a more complex situation. STS measurements and calculations of the hole linewidth of a

564


Table 11.3 Lifetimes in fs for image-potential states on clean metal surfaces measured by time-resolved two-photon photoemission and calculated using the GW approximation. a: Lehmann et al. (1999); b: Link et al. (2001b); c: Shumay et al. (1998); d: Höfer et al. (1997); e:; f: Sarría et al. (1999); g: Roth et al. (2002b); h: Knoesel et al. (1998); i: Weinelt (2002); j: Shen et al. (2002); k: Shumay et al. (1998); l: García-Lekue et al. (2003); m: Lingle et al. (1996); n: Schäfer et al. (2000); o: Link et al. (2001a); p: Berthold et al. (2000); q: Gahl (2004); r: Echenique et al. (2001); t: Chulkov et al. (1999a) τ1 C(0001) Ni(111) ⎧ ⎨ Cu(001) ⎩

exp. exp. exp. exp. theo. Cu(119) exp. Cu(117) exp. exp. Cu(111) theo. Cu(775) exp. exp. Ag(001) theo. exp. Ag(111) theo. exp. Pd(111) theo. exp. Pt(111) theo. exp. Ru(0001) theo. Li(110) theo.

τ2

40 ± 6 7±3 40 ± 6 41.3 38 15 ± 5 15 ± 5 18 ± 5 29 18 ± 2 55 ± 5 55 32 ± 10 18 25 ± 4 22 26 ± 7 29 11 14 18

τ3

120 ± 15 150 168 39 ± 5 39 ± 5 14 ± 3

300 ± 20 406 480 105 ± 15 95 ± 15 40 ± 6

160 ± 10 219 20

360 ± 15 658

τ4

τ5

Refs.

630

1200

200 ± 20 190 ± 20

350 ± 40 350 ± 40

89 62 ± 7 73

44

a b c, d e f g g h, i f j k l m l n n o o p, q r t

Table 11.4 Comparison of experimental ΓSTM (all energies in meV) and calculated decay rates for Na and Cs superstructures on Cu(111). E0 is the experimental binding energy. Γelastic elastic, Γe–e electron–electron, and Γe–ph electron– phonon scattering. Γtotal is the resulting total width. 95% indicates a coverage some 5% below the (3/2 × 3/2) saturated overlayer. Data from Chulkov et al. (2003); Corriol et al. (2005)

Na (3/2 × 3/2) Na 95% (3/2 × 3/2) Na p(2 × 2) Cs p(2 × 2)

E0

ΓSTM

Γee

Γep

−127 −42 410 40

21.0 ± 2.0 14.5 ± 1.5 16 18

13 4 0.4 100 eV). For these processes the termini electron stimulated desorption (ESD) and photon stimulated desorption (PSD) had been coined. Many of these processes were attributed to core excitations of the adsorbate. Surface photochemistry adapted two models from this earlier work for the interpretation of the experimental observations. Independently, Menzel and Gomer (1964) as well as Redhead (1964) suggested the so called MGR-model. Fig. 13.1 illustrates such a scenario in the most simplified manner. As a result of an electronic excitation the system undergoes a transition from a bonding to an anti-bonding state. In the excited state the interaction is characterised by a repulsive potential energy curve on which the particle is accelerated away from the surface. However, due to the ultrashort lifetime in the vicinity of a metal or semi-conductor substrate ( 3.5 eV) depletion of the oxygen molecules and photodissociation into adsorbed oxygen atoms are observed (Fig. 13.2). Complementary measurements using the time-offlight method to characterise the photodesorption process (see below) show that the oxygen molecules desorb with a mean translational energy, Etrans /2k, of 750 K. This energy is expressed as a temperature to allow a comparison with the sample temperature, TS . The translational energy exceeds the sample temperature significantly, as the experiments have been carried out at TS = 100 K. More precisely the latter is the sample temperature between laser pulses, which rises by approx. 15 K as consequence of absorption of the laser pulse (10 mJ/cm2 ) for a period of ca. 100 ns. The finding that the translational energy is larger than the sample temperature suggests that the photodesorption process is of nonthermal nature. The same translational energy expressed in eV is Etrans = 65 meV. This energy is small in comparison to the photon energy of several eV. Even if one takes into account that for the desorption process a binding energy of ca. 0.5 eV has to be spent, one has to conclude that only a small fraction of the excess energy is transferred to the desorbing molecule. This conclusion is further corroborated by the observation that Etrans changes only little when the photon energy is varied (Weik et al., 1993). A closer inspection of the molecular orbital structure of O2 sheds some light on the electronic processes causing molecular desorption and fragmentation into surface oxygen. The HOMO of gas phase O2 is the 1πg orbital which is two-fold degenerate. Following Hund’s rule, each orbital carries one electron resulting in a triplet ground state. As hinted at above, O2 adsorbs in a side-on geometry. When the molecules interacts with the surface, the degeneracy is lifted and the 1πg orbital, which is bonding to the surface since the

626

E. Hasselbrink

Fig. 13.2. HREEL spectra of molecular oxygen adsorbed on Pd(111) at TS = 100 K. The spectra were recorded after various exposures to UV–laser light (hν = 6.4 eV). With increasing exposure the coverage in the molecular states (α1 - to α3 -O2 ) decreases and atomic oxygen is formed. The primary electron beam energy was 3.15 eV. The photon fluence is noted on the right (Wolf et al., 1991a).

lobes point towards the surface, is lower in energy than the 1πg⊥ orbital which is nonbonding (Fig. 13.3). (As reference plane for the nomenclature we used a plane containing the surface normal and the molecular axis.) Thus two electrons will populate the 1πg orbital to facilitate bonding. Charge transfer from the surface to the adsorbed molecules will result in partial occupation of the πg⊥ orbital, which is hence located at the Fermi level. As a consequence the intramolecular bond is weakened. At a higher energy the 3σu orbital is located. This orbital is anti-bonding with respect to the intramolecular bond. It is this orbital which needs to be populated by an electron in order to dissociate the molecule. This interpretation is consistent with the observation that it takes a minimal photon energy of approx. 3.5 eV to induce the photochemical processes, which in turn can be identified with the position of the orbital with respect to the Fermi level. Varying the photon energy, one observes a quasi-exponential increase of the relevant cross sections. In the case of the O2 /Pd(111) system, the increase is 38-fold for hν from 3.9 to 6.9 eV (Weik et al., 1993). For a direct excitation involving two molecular orbitals, one would expect a width of the resonance of approx. 1.5–2 eV arising from the convolution of the widths of the two levels involved. The experimental observation of a monotonic increase of the cross section is at variance with such an interpretation.

Photon Driven Chemistry at Surfaces

627

Fig. 13.3. Molecular orbitals of O2 in the side-on geometry interacting with a metal surface. The 1πg orbital is bonding to the surface, the 1πg⊥ orbital is non-bonding. Initially both are filled with one electron. Upon approaching the surface, the degeneracy is lifted and for bonding both electrons should fill the 1πg orbital. Upon adsorption the 1πg⊥ orbital will end partially filled at the Fermi level. Surface photochemistry using ns-pulses is attributed to a temporary electron transfer into the 3σu orbital.

This finding is easily reconciled if one considers that it is not an electronic excitation which promotes one of the molecular electrons to the 3σu orbital, but rather an indirect process is at work. The overwhelming amount of light is absorbed in a ca. 20 nm thick layer of the substrate. From these absorption processes excited electrons result. They lose their energy in scattering processes with other electrons on the length scale of the mean electron free path. For electron energies of 3–5 eV the latter is on the order of 10 nm. Hence, the electrons excited in the top region of the substrate will scatter with a large probability at the surface with the adsorbed molecules. Provided the molecule exhibits an affinity level in which the electron can be captured for some time, a transient negatively charged adsorbate–substrate complex results. The phrase negative ion resonance has been used for this phenomenon, which has first been observed in conventional low energy electron scattering with an adsorbate layer. This scenario contains all the ingredients to explain the experimental observations discussed so far. The negatively charged molecule will experience some additional attraction towards the surface due to the image charge. This may be described by a potential of the form V (r) = −

1 e2 , 4π0 4(z − zim )

(13.1)

where z is the distance from the surface and zim the location of the image plane. Unfortunately, a precise evaluation is hampered since it is unclear where the image plane is located with respect to the plane of the nuclei of the top layer of the substrate. Nevertheless, from the image charge attraction a scenario results similar to the one discussed in the context

628

E. Hasselbrink

Fig. 13.4. Schematic illustration of the steps involved in the surface photochemistry of O2 on metals. Light absorption results in creation of excited electrons. With a large probability these scatter with the adsorbate layer. On the way to the surface, secondary electrons may also be generated. In a scattering process with an adsorbed molecule sufficient energy may be transferred to result in dissociation or desorption of the molecule.

of the Antoniewicz model in Section 13.1. At the same time the occupation of the antibonding σu orbital with an electron causes a repulsive force in the intramolecular bond, ultimately leading to dissociation (Fig. 13.4). The latter scenario maps well onto the MGR model. Since the residence time of the electron in the resonance is short, only a fraction of the scattering processes will lead to observable chemical processes. For too short residence times vibrationally excited adsorbates may result, however the energy transferred to the molecule is too small to break either the intramolecular or the molecule–surface bond. We will discuss later the optimal lifetime of the negative ion resonance, to allow for desorption or dissociation. Light absorption in a metallic substrate does not result in excited electrons of a specific energy. Rather all energies between the Fermi niveau and a level located by the photon energy above may be populated. The exact distribution of the excited electrons depends on the band structure of the material and the matrix elements for the electronic excitations, which are unfortunately often not known well enough to allow a meaningful calculation. Those electrons out of the distribution which have an energy located in the width of the adsorbate affinity level, may directly cause the chemical processes in the adsorbate layer. Moreover, electrons resulting from inelastic scattering processes may contribute in case the photon energy is larger than the energetic separation between affinity level and Fermi niveau. The first may be called primary and the latter secondary electrons. Due to the broad energetic distribution of the electrons excited in the substrate and the additional contribution from secondary electrons the resonance structure is washed out. With higher photon energies generally a larger number of excited electrons at the level of the affinity niveau results, which is consistent with the monotonic increase of the cross section with photon energy observed in this particular system and many other cases. 13.2.2. Methyl nitrite The methyl nitrite (CH3 ONO)/Ag(111) system studied by Pressley et al. (1996) and the tert-butyl nitrite ((CH3 )3 CONO)/Ag(111) system studied by Kleyn and coworkers (Jenniskens et al., 1997) shows a drastically different behaviour from the oxygen systems


629

Fig. 13.5. Time-of-flight spectra for NO ejected from methylnitrite photolysis on Ag(111) at hν = 3.5 and 5 eV, corresponding to λ = 355 and 248 nm, respectively. The translational energies are determined from the fits to the data. Reprinted with permission from Pressley et al. (1996).

discussed above. Dissociation into NO and an alkoxy proceeds as in the gas phase. The first has been detected in these experiments using a mass spectrometer. The NO time-of-flight spectrum from adsorbed methylnitrite exhibits two channels (Fig. 13.5), one of which indicates NO molecules with large translational energies and the other molecules which have accommodated with the sample temperature. For tert-butyl nitrite the hyperthermal channel is observed at coverages as large as 50 ML without much of a change in the translational energy. Since the hyperthermal NO must originate from the outermost layers, it is concluded that a direct excitation mechanism must be operative at all coverages. The accommodated NO is interpreted to result from NO temporarily trapped at the surface, which is consistent with the observed coverage dependent yield in both channels. Between 3 and 4 eV in photon energy alkyl nitrite dissociates in the gas phase via a long lived (125 fs) predissociative S1 state. This transition is localised on the O–N=O group and out of the O–N=O plane. At photon energies larger than 4 eV, dissociation via the dissociative S2 state sets in and results in a cross section which is an order of magnitude larger, however, yielding the same reaction products. The photodissociation cross sections determined by Jenniskens et al. are identical to the ones determined for the gas phase if one considers the large uncertainty in determining absolute cross sections. Moreover they show the identical dependence on the photon energy, namely a sharp increase of the cross section for photon energies larger than 2.5 eV (Jenniskens et al., 1998a). Moreover, the translational energy of the released NO increases significantly with increasing photon energy. The time-of-flight spectrum obtained when tert-butyl nitrite is adsorbed on top of a 20 ML thick methanol spacer layer resembles the one obtained for the molecule adsorbed on Ag(111) corroborating the interpretation that the photodissociation is due to a local excitation, since it is not conceivable that substrate electrons will penetrate through 20 ML of methanol (Jenniskens et al., 1998b).

630

E. Hasselbrink

In summary methyl nitrite similar molecules are an example for cases where surface photochemistry, or more specifically phono induced fragmentation, is not mediated by substrate electrons most likely for the simple reason that the molecules does not provide an affinity level to which an electron could attach and induce fragmentation. Similar processes are also not known to occur in the gas phase. The molecules is also too heavy that it would desorb intact. 13.2.3. CCl4 on Ag The idea of an indirect excitation mechanism mediated by substrate electron put forward when discussing the oxygen photochemistry became directly evident when John Polanyi and coworkers studied carbon tetrachloride on Ag(111) (Dixon-Warren et al., 1991). Dissociative electron attachment (DEA) in low-energy gas phase collisions of e− with CCl4 leading to CCl3 + Cl− has been well studied and found to have a large cross section (Olthoff et al., 1986). Upon irradiation (hν 5 eV) of a CCl4 covered Ag(111) surface the desorption of Cl− ions was observed. No complementary CCl+ 3 ions could be detected. Bipolar dissociation in the gas phase appears also to be impossible from energetic reasons at these photon energies. The yield of Cl− ions showed an interesting dependence on the photon energy of the laser radiation used (Fig. 13.6). Cl− was observed using 6.4, 5.4 and 5.0 eV, but not with 4.0 and 3.5 eV photon energies. From energetic and kinematic considerations the maximum energy of the ion is given by Emax = (1 − β)(hν − Φ + Vim − Ebind + EA ) − Vim ,

(13.2)

where β is the ratio of the ion to the molecular mass, Vim is the image-potential energy at the position of the adsorbed molecule, Ebind is the bond-dissociation energy and EA is the electron affinity of the desorbing ion. The work function Φ of Ag(111) is 4.77 eV. However it was observed that some CCl4 dissociated thermally, resulting in surface chlorination which is know to increase the work function. Dixon-Warren et al. assumed that Φ = 5 eV,

Fig. 13.6. Relative cross sections for the release of Cl− as a function of the incident photon energy for CCl4 /Ag(111). Adapted from Dixon-Warren et al. (1991).


631

and further Vim ≈ 1 eV and β ≈ 0.23 and the gas phase values for Ebind and EA (3.06 and 3.61 eV, respectively), and found for hν = 6.4 eV light Emax = 1.3 eV. The lowest energy at which a Cl− ion could desorb is then hν0 = 4.75 eV, which is in good agreement with the data depicted in Fig. 13.6. This result is in so far especially interesting as electrons with energies smaller than the work function cause negative ion desorption. It unambiguously proves the role of excited substrate electrons in surface photochemistry and the significant role of electron attachment processes. The cross section for Cl− emission was small with a value of 10−25 cm2 for hν = 6.4 eV. In comparable studies of Cl− release from other chloromethanes such as CH3 Cl it is observed that the ion desorption cross section is even smaller and also much smaller than the cross section for surface photodissociation into neutral products (Dixon-Warren et al., 1993). This finding results from the high energetic barrier as the work function has to be spent and only the electron affinity is gained. 13.2.4. Oxides The photochemical response of adsorbates on oxides differs in terms of cross sections from the one observed on metals. They are consistently one to two orders of magnitude larger (Budde et al., 1988). However, many of the features observed in the dynamics are rather similar. The electronic excitation mechanism varies from system to system studied. Charge transfer excitations and molecular excitations to higher lying states are the alternatives, since the common mechanism observed in metals, electron transfer from the substrate, is ruled out because of the large band gap in oxides. In the excited state, the forces on the adsorbate molecule can result from electrostatic interactions between the molecule and the ions forming the oxide substrate or alternatively from the covalent bond between the molecule to the surface atoms. From these alternative scenarios rather complex dynamics result. The electronic structure of the adsorbate–substrate complex has been studied by Klüner and Staemmler for a series of systems, such as NO/NiO(100), CO/NiO(100) and CO/Cr2 O3 (0001), using cluster methods (Mehdaoui and Klüner, 2007). In these cases the substrate is represented by a cluster of M+ and O− atoms with an adsorbate molecule bound to one of the surface atoms (Fig. 13.7). This approach allows straight forward the calculation of electronically excited states (Klüner et al., 1996). A large number of electronically excited states is found which however is composed from sets of parallel potential energy curves. The latter are in reality continua, which are represented by a number of discrete states because of the finite size of the cluster model. The width arises from coupling to a continuum of substrate states. From the topology of these curves and an analysis of the corresponding charge distributions the character of the states can be inferred. In the case of NO/NiO(100) it is found that the lowest lying electronically excited state is a charge transfer state, with one electron transferred from the cluster to the 2π ∗ orbitals of the NO molecule resulting in an NO− (NiO+ )-like intermediate. Whereas the ground state has a minimum energy geometry in which the molecule is tilted from the surface normal, the charge transfer state favours an upright and with a smaller binding energy a parallel geometry. From this feature arises a bifurcation in the dynamics.

632

E. Hasselbrink

Fig. 13.7. Minimum energy geometry of NO adsorbed on a nickeloxide cluster. For computational reasons the outer Ni atoms have been replaced by Mg atoms. The cluster is embedded in 2906 point charges which are used to represent the Madelung field. The NO molecule is bound in a titled configuration. Figure courtesy of T. Klüner.

However, Pykavy et al. (2002) have argued that an intermolecular excitation to the CO a 3 Π state, which corresponds to a one-electron transfer from the 5σ to the lowest unoccupied 2π ∗ orbital gives rise to the observed photodesorption in the case of CO on Cr2 O3 (0001). Charge transfer is in this system unlikely since CO− is unstable and the formation of CO+ requires very large energies. Sophisticated calculations addressing the dynamics of the desorption process have been carried out where the potential topologies have been inferred from the results of the cluster calculations (Klüner et al., 1998; Mehdaoui et al., 2006). These calculation excel in many aspects their counterparts which have been performed in the context of adsorbate/metal systems, in particular with respect to the number of degrees of freedom considered. However, it has not yet been possible to extract the lifetime of the electronic excitations from these calculation. Hence, the latter is used as an empirical fitting parameter. In about every case it is found that these lifetimes are on the order of ten fs, which is consistent with the rather large cross sections. 13.2.5. Weakly bound systems Physisorbed molecules have been demonstrated to exhibit surface photochemistry with features distinct from what has been discussed so far. Since the orbital overlap between the adsorbate and the substrate is much smaller in the case of physisorption, which is inherent to the nature of physisorption, electron transfer to a physisorbed molecule is less likely when compared to the chemisorbed case. We expect the typical equilibrium distance for physisorbed molecules to be in the range of 3 to 5 Å, whereas 1 to 2 Å are typical for chemisorption. Matsumoto and coworkers have reported that methane adsorbed on Pt(111) photodissociated to produce methyl and hydrogen adsorbates upon irradiation with 193 nm light


633

(6.4 eV) (Gruzdkov et al., 1994; Matsumoto et al., 1996). The binding energy of methane on Pt(111) is only 0.23 eV (Ukraintsev and Harrison, 1993). CH4 in the gas phase is transparent at this wavelength and shows appreciable absorption cross sections only for photon energies larger than 8.5 eV. One would expect strong interactions as prerequisite for a red shift of the excitation energy by 2 eV. However, Watanabe et al. (1996) have interpreted the dependence of the photo cross section on the polarisation and angle of incidence to indicate that direct electronic excitation of the methane adsorbate plays an important role in the photochemistry of methane. Hence, they concluded that the excited state of methane mixes significantly with substrate electronic states which the ground state does not. Similarly Kr and Xe adsorbed on Si(100) – the binding energy of Xe on Si(100) is expected to be 0.15 to 0.2 eV – was desorbed by the irradiation with photons in the energy range from 1.2 to 6.4 eV (Watanabe et al., 2000). The kinetic energy distributions of the desorbed rare gas atoms did not depend on the photon energy and the laser fluence. The distributions were well represented by the Maxwell–Boltzmann distribution with a mean energy of 380 ± 20 K which is higher than expected for surface heating with the laser intensity employed. Matsumoto and co-workers proposed that desorption is a result of direct energy flow to the desorption channel from hot surface phonons generated by charge recombination via surface states before they decay into the bulk. Wright and Hasselbrink (2001) have observed a marked difference in the photochemistry of chemisorbed and physisorbed disilane (Si2 H6 ) on hydrogen terminated Si(100). Chemisorbed Si2 H6 fragments when irradiated with 6.4 eV photons, leaving fragments on the substrate and releasing some into the gas phase. In contrast physisorbed Si2 H6 solely photodesorbs intact which is observed for photon energies larger than 3.6 eV. This difference is in particular noteworthy since the binding energy of both species only differs by about 0.1 eV. The cross of the physisorbed molecules sections are substantially larger than those for the chemisorbed state and do not show the same strong photon energy dependence. Clearly, the surface gap states are the key to understanding the mechanism which is responsible for photodesorption in the physisorbed cases. In Si hot electrons loose energy by scattering with the relative large number of electrons that accumulate at the bottom of the conduction band and with the holes that accumulate at the top of the valence band. Recombination of electrons and holes occurs at defects and at the surface through surface states. Matsumoto and co-workers have suggested that the energy released when electrons and holes recombine, which is mediated by surface states, causes desorption. In fact photodesorption of Xe and Kr is quenched when surface gap states are removed by oxygen adsorption. The authors speculated that each recombination event excites low energy surface phonons. Desorption occurs if a significant number of quanta accumulate in the appropriate coordinate before the energy dissipates into the bulk. Howe and Dai (1998) also studied the surface photochemistry of a number of physisorbed molecules such as HFCO, H2 CO, CH2 CO and CH3 Cl on Ag(111). The translational energy distribution of the desorbing molecules is characterised by a Maxwell– Boltzmann distribution with a mean energy in the range 110–150 K. No threshold energy could be established when studying photon energies between 1.2 and 4.7 eV. Howe and Dai suggested that photodesorption of these physisorbed molecules involves excited substrate electrons but that these do not attach to form a molecular ion. Rather these electrons in-

634

E. Hasselbrink

teract with the adsorbate through a dipole mechanism. Such a mechanism does not require electron wave function overlap between the adsorbate and the substrate electron. Rather the substrate electron which is internally reflected at the surface excites through a dipole mechanism molecular vibrations. Such a mechanism may be operative over the distance of about 3 Å at which a physisorbed molecule is located from the surface as the hot electron is not screened at the turning point. For a dipole transition the selection rule v = ±1 holds. Hence, the model requires that an intramolecular bond is excited, the energy of which is larger than the molecule– surface bond. Subsequently, this energy is transferred to the latter bond which thereupon breaks. The energy difference would be available as the translational energy of the desorbed molecule, which is consistent with the rather small energies observed. In summary, it may be said in all fairness that these photochemical processes of physisorbed species are much less understood than the chemisorption systems discussed previously. 13.2.6. Multiple electronic excitations It was discovered by several groups for a variety of systems that using short-pulse ( 0; (b) hole attachment, VS < 0; (c) electronic transition, VS > 0; (d) electronic transition, VS < 0; (e) electron–hole pair attachment, VS > 0; (f) electron–hole pair attachment, VS < 0. Reproduced with permission from Mayne et al. © 2006 American Chemical Society.

or unoccupied states or electronic excitation (Dujardin et al., 1992; Bartels et al., 1998; Avouris et al., 1996a). The general concept of atomic or molecular dynamics following electronic excitation has been formulated by Tolk et al. (1983) and Avouris and Walkup (1989) in the framework of the DIET (Desorption Induced by Electronic Transitions) process. The DIET concept relies on the following sequence of events based on the Menzel–Gomer–Redhead (MGR) model (Menzel and Gomer, 1964; Redhead, 1964); (i) a vertical electronic transition from the electronic ground state to the excited electronic state, (ii) the dynamical evolution of the system in its excited electronic state, (iii) the electronic relaxation to the electronic ground state, and (iv) the subsequent dynamics in the electronic ground state with vibrational excitation. The role of DIET in surface chemistry over the last 40 years has recently been reviewed by Menzel (2006). Electronic excitations can be grouped into three categories; electron attachment, electron transition and electron–hole pair attachment which are schematically shown in Fig. 14.10 (Mayne et al., 2006a).

700

A.J. Mayne and G. Dujardin

14.3.3.2. Electron (hole) attachment Electrons from the STM tip can be attached temporally on an unoccupied orbital of an adsorbed atom or molecule, producing a negatively charged species (Fig. 14.10a). Such a process is the surface analogue of a negative ion resonance in the gas phase as explained by Palmer and Rous (1992). Using the STM, this manipulation requires a positive voltage on the surface with respect to the tip, VS > 0. Consider the electronic structure of an atom or a molecule adsorbed on a surface. The atom or molecule will have unoccupied states that lie above the Fermi level of the surface. Some of these unoccupied states will be above the vacuum level (of the surface) while others will be below. The states lying above the vacuum level are accessible with far field techniques such as electron impact scattering (Huels et al., 1994). The states lying below the vacuum level can be investigated by inverse photoemission experiments as described in the review by Smith (1988). The STM tip can be used to induce electronic excitation by injecting electrons into these antibonding orbitals. In fact, this enables access to all the antibonding states, even those that lie below the vacuum level of the surface. Similarly, holes can be attached temporarily to an occupied orbital of an adsorbed atom or molecule. This requires a negative surface voltage, VS < 0, for the electrons to tunnel from this occupied orbital to unoccupied states on the tip (see Fig. 14.10b). Electron (hole) attachment results in the formation of a transient ionic state of the adsorbed atom and molecule which will induce its dynamics as discussed in Section 14.3.3.1. Note that such ionic states on surfaces can be very short-lived on metal surface and to a lesser extent on semiconductor surfaces, whereas they can be very long lived on insulating layers (Repp et al. 2004a, 2004b). 14.3.3.3. Electronic transition Inelastic tunnelling of electrons (Fig. 14.10c) or holes (Fig. 14.10d) can also induce an electronic transition, i.e. the transition of an electron from an occupied orbital to an unoccupied orbital, in the adsorbed atom or molecule. This process should occur at a higher surface (tip) voltage compared to the electron (hole) attachment, such that the electrons may no longer be in the tunnel regime but rather in the field emission regime, depending on the HOMO–LUMO energy gap of the adsorbed atom or molecule (Highest occupied molecular orbital – HOMO; lowest unoccupied molecular orbital – LUMO). 14.3.3.4. Electron–hole pair attachment The third type of electronic excitation of a molecule is the simultaneous attachment of an electron into an unoccupied π ∗ orbital and a hole into an occupied π orbital. Such an electronic scheme can only occur if the molecule interacts weakly with both the STM tip and the surface. In such a case, the orbital energies of the molecule may be shifted by the electric field between the tip and the surface as shown in Figs. 14.10e and 14.10f. 14.3.3.5. Advantages of electronic excitation If we compare electronic excitation with vibrational excitation or direct contact (between the STM tip and the molecule), electronic excitation has several advantages if one wants to induce dynamical processes. Firstly, electronic excitation should provide a relatively large amount of energy (1–4 eV), thus enabling the molecule to be excited into

STM Manipulation and Dynamics

701

far-from equilibrium conformations, resulting in very rapid, efficient and more easily controllable molecular dynamic processes. Secondly, the transfer of electronic energy inside a molecule should be more rapid. Thirdly, electronic excitation can be used to activate a broader range of molecular functions. These can be different in nature, for example, an electronic function might involve a change in transport properties of the molecule. A mechanical function might involve a change in configuration and an optical function might be inducing fluorescence from the molecule. Lastly, quantum control of isolated molecules has been demonstrated in the gas phase using the laser to induce electronic excitation (Charron et al., 1995). However, it remains to be seen if this is possible on a surface using tunnel electrons from the STM tip to induce electronic excitation. 14.3.3.6. Experimental manipulation methods in electronic excitation To illustrate the experimental manipulation methods by electronic transition using the STM, we will take the example of the desorption of individual hydrogen atoms from the fully hydrogenated Si(100)-2 × 1 surface and also the partially hydrogenated Ge(111)c(2 × 8) surface. The hydrogenated silicon (100) surface has been studied extensively over the last 20 years (Lyding et al., 1994; Shen et al., 1995; Avouris et al., 1996a, 1996b; Sakurai et al., 1997; Foley et al., 1998; Stokbro et al., 1998a, 1998b; Thirstrup et al., 1999; Soukiassian et al., 2003a, 2003b) while the partially hydrogenated Ge(111)-c(2 × 8) surface has been studied more recently (Mayne et al., 2000; Dujardin et al., 2001a, 2001b, 2002). This is due to the importance of silicon in microelectronics, the relative ease of preparation of the hydrogenated surface and the simplicity of the Si–H bond. Indeed, the Si–H bond comprises a σ bond. Thus it is easier to compare the different mechanisms, electron attachment to the unoccupied σ ∗ orbital or direct excitation of the σ –σ ∗ transition. There are two modes in which the STM tip can be operated: the first is called the “constant current” mode and the second is called “constant height” mode. For each of these modes, there are two experimental methods for desorbing the hydrogen atom with electrons from the STM tip: the stationary mode and the scanning mode. 14.3.3.6.1. Stationary mode In the constant current mode (Binnig et al., 1982b), the STM tip is placed above the chosen hydrogen atom whereupon both the set-point voltage and current are rapidly changed to a different value from the initial setting (usually larger, for example, −2 V and 1 nA to +3 V and 10 nA). Since the feedback loop is on, the tip-surface distance changes. The desorption event is detected by the appearance of a sharp peak (or depression) in the tunnel current trace as shown in Fig. 14.11 (Soukiassian et al., 2003a). At this moment, the abrupt change in current induces the feedback loop to adjust the tip–surface distance to re-establish the fixed current value. After a fixed time (several hundred ms), the tip is returned to the initial voltage and current settings. The surface is then imaged to verify that the extraction of the hydrogen atom has taken place and the process repeated on another H atom. The constant current method works well if there is a significant change in the local density of states (LDOS) between the adsorbed atom site and the clean surface. This is the case for hydrogen on silicon. In the constant height mode (Bryant et al., 1986), as used in the experiments on Ge(111) surface (Mayne et al., 2000; Dujardin et al., 2001a, 2001b, 2002), the STM tip is placed above the hydrogen atom under computer control, the feedback loop is opened (switched

702


Fig. 14.11. Stationary mode desorption. (a) An empty state topography of 40 × 60 Å was recorded at VS = +1.5 V and I = 0.4 nA and an H atom (marked by the arrow) was selected and a voltage pulse was applied. (b) After the pulse the same region of the surface is scanned. A Si dangling bond has appeared as a bright feature replacing the H atom (the dark defect serves as a landmark). (c) A recording of the tunnel current during a VS = +2.5 V and Ides = 6 nA pulse. The first peak marks the beginning of the pulse (change of voltage) and the second peak is due to desorption of a hydrogen atom. The duration of the pulse is 160 ms. (d) Distribution of the exposure time before desorption for VS = +2.5 V and Ides = 3 nA pulses. The bin width is t = 10 ms. The data were fitted with an exponential decay. Reproduced with permission from Soukiassian et al. (http://link.aps.org/abstract/PRB/v68/p035303). © 2003 American Physical Society.

off), the tip is displaced a fixed vertical distance from the surface (either retracting as much as 20 Å or approaching up to 5 Å) and a positive sample bias (2–10 V) applied for a short time period (200 ms) during which the tunnel current is measured. After this, the applied bias is removed, the tip is returned to its initial z position, the initial bias reapplied and the feedback restored. Again, the surface is imaged afterwards to verify that the extraction of the hydrogen atom has taken place and the process repeated on another H atom. The constant height mode is not appropriate on the Si(100):H surface as the tunnel current increases greatly when the hydrogen is removed and since neighbouring hydrogen atoms are under the tip, multiple desorption events can occur during the same pulse. This makes the interpretation of the results particularly difficult. On the other hand, the removal of hydrogen from the Ge surface gives rise to a small change in the current during


703

Fig. 14.12. An STM topography (20 × 20 nm) of the Si(100)-2 × 1:H surface showing the pattern of an OR gate formed from Si DBs created by extracting the hydrogen atoms with the STM tip. The image was taken at −1.5 V and 0.5 nA. Reproduced with permission from Soukiassian et al. © 2003 Elsevier.

the pulse which was not possible to detect using the constant current mode and difficult using the constant height mode. So a modified constant height mode had to be employed (Mayne et al., 2000). A pulse composed of a train of about 100 short “write” and “read” pulses was created. Between each “write” and “read” pulse, the feedback loop was not restored, so that after the ”write” pulse, a ”read” pulse was made by applying the initial voltage bias and the current measured permitting the removal of the H atom to be clearly detected. By applying a sequence of a hundred or so pulses (duration 0.1 to 2 ms), each followed by a measure of the current at the image bias, before re-establishing the feedback loop, it was possible to deduce the current required and the time at which the change occurred. 14.3.3.6.2. Scanning mode The constant current and constant height modes, as described above, involve positioning the STM tip above a single hydrogen atom during the pulse. That is the tip is stationary with respect to the x, y plane of the surface. Both these modes can be operated by scanning the tip across the surface while applying the modified voltage and current settings (with respect to the imaging conditions). Thus hydrogen atoms can be removed thereby tracing lines of newly created silicon dangling bonds on the surface. This scanning method has been widely used ever since the initial experiment by Lyding and co-workers (1994). The advantage of this method is that it is quicker and so patterns can be created easily on the surface as illustrated in Fig. 14.12 (Soukiassian et al., 2003b). However, in this scanning mode, measuring the reaction time of each desorption event is not possible, so the desorption yield has to be estimated from the following relation: Y = N e/I t, where N is the number of desorption events, I is the tunnel current and, t is the time needed for the tip to write a line of given length at a given speed and e is the

704


Fig. 14.13. STM tip-induced desorption of hydrogen atoms from the Si(100)-2 × 1 surface. H-atom desorption yield as a function of the sample bias voltage. The tunnel current was 0.01 nA. The sample was As-doped, 5 × 10−3 -cm. Reproduced with permission from Avouris et al. © 1996 Elsevier.

electron charge. This relation is valid provided that the number of dangling bonds created during the scan is much less than the number of irradiated H atoms. At high scan speeds (>100 nm/s), the approximation is valid since the yield is independent of the scan speed (Soukiassian et al., 2003b). 14.3.3.7. An example of experiments using electronic excitation: Desorption of hydrogen from semiconductor surfaces The results from the early studies on the desorption of hydrogen from the fully hydrogenated Si(100)-2 × 1 surface showed the presence of two desorption regimes. In the first regime at low voltage, the desorption yield was observed to vary dramatically with the current. In this regime, the STM tip injects electrons into the σ ∗ (Si–H) antibonding orbitals. The process of electron attachment provides energy to the system creating an excited state which has the form of a temporary negative ion resonance, in this case, Si–H− (Palmer and Rous, 1992). When the system relaxes, the electron leaves to the surface and the energy dissipated causes the hydrogen atom to desorb. The second regime was found to be independent of the current and the desorption yield showed an increase of several orders of magnitude from 10−10 to 10−6 H atoms per electron where the threshold was around 6.5 eV as shown in Fig. 14.13 (Avouris et al., 1996a). This higher energy regime corresponds to a direct electronic excitation of the σ –σ ∗ transition of the Si–H bond. In the low voltage or tunnel regime, that is below 5 V, the desorption yield was found to be in the range 10−10 to 10−8 H atoms per electron. Moreover, the desorption yield showed a distinct power law behaviour as a function of the current for a given applied voltage. For each voltage between 2 V and 4 V, the yield increased by more than a factor of 10 as the current varied from 1 to 3 nA (Shen et al., 1995; Avouris et al., 1996a). The desorption


705

Fig. 14.14. Desorption yield of hydrogen atoms from the hydrogenated Si(100)-2 × 1:H surface as a function of the tunnelling current in stationary mode for p-type samples (up-triangles) and in scanning mode for p-type samples (down-triangles) and n-type samples (squares). The solid lines are the corresponding least-squares fit to a power law, I n . The exponents are n = 0.3 ± 0.1, 1.3 ± 0.3, and 0.8 ± 0.3, respectively. The values of the yield from previous studies as a function of the tunnelling current (Stokbro et al., 1998a, 1998b) (circles) and Shen et al., 1995 (diamonds)) and the respective least-squares fit to a power law, I n . The exponents are n = 15 and n = 10, respectively. Reproduced with permission from reference Soukiassian et al. (http://link.aps.org/abstract/PRB/v68/p035303). © 2003 American Physical Society.

yield depends on the current according to the formula Y = e/I n τ , where Y is the yield, e is the electric charge, I is the tunnel current, n is the power factor which depends on the number of electrons involved and τ is the mean value of the extraction time. The best fit of the data gave n = 10 for a surface voltage of 2.5 V, i.e. a strong dependence of the yield on the current. From this, it was proposed that the desorption mechanism involved the electron attachment in the σ ∗ (Si–H) orbital via vibrational heating of the Si–H bond where 10 or more electrons were needed – each electron giving 1 quanta of energy to the Si–H bond. This is only a small fraction of the energy of the Si–H bond (Lyding et al., 1994; Shen et al., 1995, Avouris et al. 1996a, 1996b; Sakurai et al., 1997; Foley et al., 1998; Stokbro et al., 1998a, 1998b). Later, more detailed studies on Si(100):H were carried out over a larger current range (1–10 nA). In these studies, the desorption yield varied only a weakly with the current with and n = 1.3 for n-type samples, and n = 0.8 for p-type samples using the scanning mode. For comparison, the desorption yield obtained using the stationary mode gave n = 0.3 (Fig. 14.14). This implies that at most only 2 electrons are required to break the Si–H bond (Soukiassian et al., 2003b). Indeed, it should be stressed that the desorption yield had the same absolute value as that found in previous studies but only a very small dependence on the current. Further investigation showed that the structure of the tip plays an important role in the experiments which is hard to quantify (Soukiassian et al., 2003b). It was observed that the silicon dangling bond lines created by the tip were segmented such that from time to time no hydrogen was removed. This suggests that the tip had “on” and “off”

706


modes. This could be due to the fact that a large number of hydrogen atoms are removed in a short space of time so the tip is easily passivated which could change the efficiency of the desorption process. These more recent results (Soukiassian et al., 2003a, 2003b) are more compatible with the coherent excitation of the Si–H bond as proposed by Salam, Persson and Palmer (1994). In this model, the desorption process still involves the electron attachment to the σ ∗ (Si–H) orbital forming a negative ion resonance. However, as the electron leaves (to the surface) it transfers a much larger part of its energy to the Si–H bond (several quanta) thereby climbing the vibrational ladder several levels at a time. Consequently, only two electrons are needed to induce hydrogen desorption. This casts doubt over the validity of the vibrational heating mechanism as the description of the hydrogen extraction process. There are a number of practical complications related to the early results (Lyding et al., 1994; Shen et al., 1995; Avouris et al., 1996a, 1996b; Sakurai et al., 1997; Foley et al., 1998; Stokbro et al., 1998a, 1998b). The first and most critical was the lack of precision inherent in the line scan method. It was difficult to determine the exact number of electrons involved since the tip is scanned over the surface. The whole line receives a certain dose of electrons so depending on the speed, several electrons can interact with a single hydrogen atom or between hydrogen atoms, which means that the inelastic coupling will vary from site to site. This renders an understanding of the physical process of desorption more difficult. As a consequence, it seems hard to justify the vibrational heating mechanism given the experimental and physical uncertainties and especially the lack of experimental data points. 14.3.4. Manipulation by vibrational excitation 14.3.4.1. Vibrational inelastic electron tunnelling spectroscopy In recent years, a new and important technique has developed, that of vibrational inelastic electron tunnelling spectroscopy (IETS). This uses the sub-angstrom spatial resolution capability of the STM to probe and measure the vibrations of individual chemical bonds within a molecule adsorbed on a surface. It is a logical extension of the tunnelling spectroscopy obtained in STM, which has taken a while to develop due to the difficulty in observing the phenomenon. Tunnelling spectroscopy using the STM is based on the fact that the tunnel current corresponds approximately to the local density of states (LDOS) at the Fermi level of the tip (or more exactly the differential conductance, dI /dV ) as defined by the theory of Tersoff and Hamann (1983, 1985). In practice, one can measure the variation of the current as function of energy by scanning the applied voltage between two values at a fixed tipsample distance. This gives an I –V curve. For a more detailed mathematical treatment of tunnelling spectroscopy, see the reviews by Stroscio and Feenstra (1993), and also Kubby and Boland (1996). It is important to realise that the spectroscopy curves obtained with the STM are a complex combination of the electronic states of the tunnel junction, that is, the metal tip, the substrate and any adsorbed atom or molecule that might be between the two. This renders any interpretation difficult and is further complicated by the fact that, for an adsorbed species, the electronic levels are broadened and often shifted in energy due to interactions with the surface.


707

Fig. 14.15. Schematic showing the emergence of inelastic tunnelling at the threshold for vibrational excitation. The change in the tunnelling current due to vibrational excitation is too small to be measured from the I –V curve. While a change in the differential conductance, dI /dV , can be seen for strong modes, more often vibrational features needs to be extracted from d2 I /dV 2 . An important characteristic of vibrational inelastic electron tunnelling spectroscopy (IETS) is the occurrence of a peak of the opposite sign on the negative bias side. Lacking an isotope shift analysis, the assignment of a feature to vibrational excitation needs to be confirmed by a corresponding feature with the opposite polarity at the opposite bias. This schematic depicts an increase in the conductance, associated with a positive (negative) peak for positive (negative) sample bias. In contrast, electronic spectra arise from elastic tunnelling; peaks are positive and occur on either positive (unoccupied states) or negative (occupied states) sample bias. Reproduced with permission from Ho. © 2002 American Institute of Physics.

The I –V curves contain all the spectroscopic information from which the properties of the single atom or molecule can be deduced. Taking the derivative (dI /dV ) gives the differential conductance (or ac conductance) and the second derivative (d2 I /dV 2 ) gives the differential change in the conductance. Tunnelling is a predominately elastic process; however, inelastic processes can occur if the electrons couple with a vibrational mode of an adsorbate. If during the scanning of the sample bias, it crosses the threshold for exciting a vibrational mode, an inelastic channel in the tunnel current will be opened. Provided that the opening of this channel causes a significant change in the tunnel current, then the slope in the current will change as the voltage continues to change. In the conductance (dI /dV ) curve, this will be seen as a small step (Fig. 14.15). It is easier to detect this change by recording the differential of the conductance (d2 I /dV 2 ) where peaks (or troughs) appear (Fig. 14.16). See Fig. 14.29 for the STM images corresponding to these IETS curves. Several review papers by Ho (2002) and Komeda (2005) have been published giving more detailed explanations of IETS.

708


Fig. 14.16. Single-molecule vibrational spectra obtained by STM-IETS, showing the C–O stretch of Fe(CO) and Fe(CO)2 . The differential change of the ac conductance (d2 I /dV 2 ) as a function of the sample bias is displayed. For each scan, the dc sample bias was ramped from 180 to 280 mV and back down in 2.5-mV steps with a 300-ms dwell time per step (1 meV = 8.065 cm−1 ). Each spectrum has been signal averaged with repeated scans. The root-mean square ac modulation at 200 Hz was 7 mV. Peak positions have an uncertainty of ±1 meV. (Line A) Spectrum taken over clean Ag(110) surface and signal averaged with 100 scans. (Line B) Spectrum averaged with 210 scans over the edge of the lobe in the image of single-molecule Fe(12 C16 O). (Line C) Spectrum averaged with 210 scans over the edge of the lobe in the image of Fe(13 C18 O). (Line B–C) Difference between spectra B and C. (Line D) Spectrum averaged with 100 scans over the left protrusion in the STM image of the single-molecule Fe(12 C16 O)2 . (Line E) Corresponding spectrum averaged with 100 scans for Fe(13 C18 O)2 . (Line D–E) Difference between spectra D and E. (Line F–G) Difference between spectra recorded separately over the left (12 C16 O, 697 scans) and right (13 C18 O, 280 scans) protrusions in the STM image of the single-molecule Fe(12 C16 O)(13 C18 O). Dashed lines denote zero level in difference spectra. Reproduced with permission from Science (http://www.aaas.org), Lee and Ho. © 1999 American Association for the Advancement of Science.

The vibrational modes of small molecules are best detected in molecules adsorbed on metal surfaces. There are several factors that contribute to this. Such vibrational modes are usually found at low voltage where metal surfaces have the advantage that their electron


709

Fig. 14.17. STM image (a) and single-molecule vibrational spectra (b) of three acetylene isotopes on Cu(100) at 8 K. The vibrational spectra on Ni(100) are shown in (c). The imaged area in (a), 56 Å × 56 Å, was scanned at 50 mV sample bias and 1 nA tunnelling current. Inelastic tunnelling spectra in (b) are the average of 16 scans of 2 min each while the vibrational spectra in (c) are the average of 25, 50, and 222 scans for C2 H2 , C2 D2 , and C2 HD, respectively. Background spectra over the bare surfaces have been subtracted. A root-mean square (rms) modulation voltage of 5 mV at 200 Hz was used. The tip was positioned over the centre of the molecule in the recording of the spectra and was fixed in position vertically to give a dc tunnelling current of 1 nA at 249 mV sample bias. Reproduced with permission from Stipe et al. (http://link.aps.org/abstract/PRL/v82/p1724). © 1999 American Physical Society.

density is uniform so that a vibration will stand out from the background. In addition, for the molecules to remain still, the metal surfaces have to be cold (0.2 V to 3 at 0.1 V. Rotation was believed to be due to inelastic electron tunnelling through an adsorbate-induced resonance. The energy is transferred to the hindered rotational mode at a current-dependent rate. When the energy of the electrons is less than the energy barrier for rotation, more than one electron is required. Recent theoretical calculations allowed a more detailed understanding (Teillet-Billy et al., 2000). Rotation occurred through reso-


723

nant electron scattering where a transient negative ion is formed as the molecule captures an electron before being scattered to the substrate. Indeed, it was necessary to take into account angular momentum conservation. The transfer of a finite angular momentum leads to efficient rotational excitation. Within this model (Teillet-Billy et al., 2000), the efficiency of the inelastic electron process is independent of the electron resonance lifetime contrary to the electron (hole) attachment processes discussed in Section 14.3.3.2. 14.4.4. Desorption Desorption implies bond breaking between an adsorbed species and the surface or between a surface atom itself and the surface. Desorption has been studied using a number of excitation methods including photon impact, electron impact, ion impact or metastable atom impact (Raseev and Dujardin, 2003; Tanimura and Ueba, 2005; Comtet and Dujardin, 2006a, 2006b). In most cases, the desorption reactions are monitored through the detection of the desorbed species by mass spectrometry methods. This explains why desorption reactions have been so widely studied since other surface reactions producing no detectable species are more difficult to monitor. Consequently, a number of powerful concepts have been developed over the years based on the DIET (Desorption Induced by Electronic Transitions) model introduced by Menzel, Gomer, and Redhead (Menzel and Gomer, 1964; Redhead, 1964) as Menzel’s recent review highlights (Menzel, 2006). With the STM, instead of detecting desorbing species, one monitors the disappearance of species from the surface after excitation. The variety of processes producing desorption reactions which can be induced using tunnel electrons from the tip are illustrated by the two examples below. Both vibrational (NH3 on Cu(100)) and electronic (halobenzene on Si(100)) excitation are able to produce desorption reactions. Energy barriers are usually larger for desorption than for other molecular reactions such as surface diffusion or dissociation. Therefore, desorption is often in competition with other reactions. The two examples below show that desorption can be selectively activated either through the control of the tunnel conditions (current and surface voltage) or the functionalization of the STM tip. 14.4.4.1. NH3 on Cu(100) NH3 molecules adsorbed on the Cu(100) surface at 5 K present an interesting case (Pascual et al., 2003). Careful choice of the tunnel conditions can be used to select different reaction pathways; either translation of the NH3 across the surface or desorption of the molecule. As shown below, this is due to the relative balance of the different excitations involved as well as the number of electrons. Translation of NH3 is observed for tunnel currents below 0.5 nA, with a threshold voltage at 400 mV. Desorption occurs preferentially at higher tunnel currents >1 nA and at a lower threshold at 270 mV. An isotopic shift is observed for ND3 where the thresholds are 300 and 200 mV for translation and desorption (Fig. 14.26). These energies correspond to N–H (N–D) stretch at the translation threshold and the first overtone of the N–H3 (N–D3 ) umbrella mode for the desorption threshold. Thus translation occurs via the excitation of the stretch mode and desorption via excitation of the umbrella mode. Measuring the reaction yield as a function of the current enables a power law to be extracted and hence deduce the number of electrons involved in each process. Translation was found to be a one electron process involving intermode coupling

724


Fig. 14.26. Ammonia adsorbed on Cu(100) at 5 K. Thresholds of electron energy needed to induce molecular motion. The thresholds are statistically probed by fixing the STM tip above a molecule with VS = ±100 mV, and slowly increasing the bias magnitude. A sudden current drop reveals the occurrence of a reaction. The panels show the distribution of electron energy (eVS ) reached when the reaction occurs. At low tunnelling currents (It < 0.5 nA) NH3 shows a threshold at ∼400 mV (a), which shifts down to ∼300 mV in ND3 (b), matching correspondingly the energy of N-H and N-D stretch modes (dashed lines). In this reaction mechanism (M1) 60% of the events produced molecular translation. (c) For It > 1 nA, an additional threshold appears gradually at ∼270 mV in NH3 , consistent with the energy of two umbrella-mode quanta (dotted lines). This reveals a second bond-cleavage mechanism (M2), which produced molecular desorption in 75% of the events. In ND3 , the corresponding onset at ∼200 mV is observed (d) only after reaching tunnelling currents higher than 10 nA. Reproduced with permission from Pascual et al., Nature (http://www.nature.com). © 2003 Nature Publishing Group.

to overcome the 300 meV barrier as had been suggested in earlier experiments either on ammonia (Bartels et al., 1999) or other molecules (Komeda et al., 2002). Desorption has a higher energy barrier of 600 meV and so is a 3-electron process involving vibrational heating. In this example, two different manipulation results are observed by the excitation of two different vibrational modes of the molecule requiring a different number of electrons (Lorente et al., 2005). This illustrates that selectivity can be achieved between different reactions in the same molecule. 14.4.4.2. Halobenzene on Si(111) Chlorobenzene (PhCl) is a source of chlorine used to chemical attack silicon surfaces. However, desorption is a competing process with dissociation as described in Section 14.4.2.3. Desorption of PhCl from the Si(111)-7 × 7 surface can be induced by STM manipulation (Sloan and Palmer, 2005b). By choosing the appropriate tip condition, desorption can become the dominant channel of STM manipulation. If the mole-


725

Fig. 14.27. Experimental desorption yield per electron of C6 H5 Cl molecules from the Si(111)-7 × 7 surface as a function of the sample bias voltage in the STM (black squares), compared with (solid lines) calculated partial density of p states at (a) the carbon atom in the ring and (b) the chlorine atom. Reproduced with permission from Sloan et al. (http://link.aps.org/abstract/PRL/v91/p118301). © 2003 American Physical Society.

cule was seen as a protrusion at +2 V then the tip induced desorption, whereas if the molecule was observed as a depression at sample bias voltages of both +1 V and +2 V, then dissociation could be induced. The difference between the two tips is probably that one type of tip has a foreign atom or molecule attached to it, while the other is metallic (Bartels et al., 1997b). The desorption rate (s−1 ) showed a linear dependence on the current implying a one electron process. In addition, the desorption yield (events per electron) was independent of the tip–sample distance. These findings rule out a vibrational heating mechanism, an electric field effect and a mechanical interaction. However, the desorption yield showed clear thresholds as a function of the voltage at −1.5 V and +2.5 V (Fig. 14.27). This indicated that desorption is driven by tunnelling electrons (or holes) populating a negative (or positive) ion resonance of the chemisorbed PhCl molecule. Calculations were carried out using density functional theory (DFT). The measured, asymmetric thresholds for desorption correlate nicely with the density of states for the ring orbital, but not at all well with the Cl orbital. This suggests that the resonant states are associated with the occupied π and unoccupied π ∗

726


orbitals of the benzene ring, in agreement with other experiments (Becker et al., 1990; Patitsas et al., 2000). 14.4.5. Molecular molds The idea of creating reactive sites by STM desorption of hydrogen atoms on the Si(100):H surface for reacting with other atoms or molecules was first put forward by Lyding et al. (1994) and Abeln et al. (1998) when the initial desorption experiments on Si(100):H were performed. Hydrogenated semiconductor surfaces are particularly adapted to the study of molecular manipulation because the presence of hydrogen passivates the reactive semiconductor surfaces. As a result there have been numerous studies on these surfaces (Mayne et al., 2006b). Indeed, isolated Si dangling bonds have been used as reactive sites to adsorb a variety of complex organic molecules, for example, Norbornadiene, copper phthalocyanine (CuPc) and C60 where one molecule is adsorbed per site (Hersam et al., 2000). However, the molecules do not always attach themselves in the same way to the dangling bond. For instance, the CuPc can attach via the central metal atom or through a π interaction with one of the pyrrole groups. However, individual dangling bonds are not selective in the sense that they are all identical. The controlled creation of groups of silicon dangling bonds is required to not only select the position of molecular adsorption but can also select the molecule by its size and orient the molecule on the surface with respect to the underlying surface. These groups of dangling bonds behave as molecular moulds as shown in Fig. 14.28 (Mayne et al., 2004b). At room temperature, biphenyl molecules could selectively adsorbed into the dangling bond moulds on the hydrogenated silicon surface provided that the mould was at least the size of the molecule i.e. 4 or more adjacent DBs on 2 or 3 neighbouring dimers. The biphenyl was not observed to adsorb on 1, 2 or 3 DBs. Also the dynamics of the adsorbed molecule were modified by the presence of the hydrogen atoms surrounding the mould. The pivoting molecule in the mould was seen to fix itself spontaneously unlike on the clean surface where electrons from the tip are needed to fix the molecule in its stable site. Adsorption of a larger poly-phenyl molecule called Trima (1,4 -paraterphenyldimethylacetone) in such moulds was much more complicated due to the relatively more reactive molecule (Mayne et al., 2004b; Soukiassian et al., 2005). The Trima molecule contains two ketone groups, one at each end of a triphenyl chain, which are very reactive towards the surface. The Trima molecule reacted immediately with the hydrogenated surface before attaching to or in a mould since new DB sites appeared several tens of Angstroms away from the mould. The biphenyl molecule just diffuses across the hydrogenated silicon surface into the mould without any apparent modification of the nearby surface. 14.4.6. Chemical reaction Manipulation of atoms and molecules on a surface with the STM can lead to a chemical reaction between different adsorbed species. This is the result of bond formation creating new species which can be distinguished from the reactants. This often requires using a combination of different manipulation procedures – dissociation, translation, desorption


727

Fig. 14.28. A chronological sequence of 4 filled-state STM topographies (each 50 × 50 Å, −1.5 V and 0.5 nA) of the same area of the hydrogenated Si(100)-2 × 1 surface. The left-hand column shows, (a) the clean hydrogenated surface, (c) the surface after fabricating the molecular mold, (e) a moving biphenyl molecule adsorbed in the mold, and (g) the biphenyl molecule is fixed in the mold. These four stages are shown schematically in the right-hand column (b), (d), (f), and (h). Note in (f), only one of the bistable states of the molecule is shown. The structure of the molecule is shown at the bottom. Reproduced with permission from Mayne et al. © 2004 American Institute of Physics.

728


and deposition. Thus, chemical reaction is the logical extension of the elementary manipulation steps and marks the beginning of our ability to use the STM as a tool to build nanostructures from individual elements. 14.4.6.1. Metal surfaces Electronic excitation with the STM can be used to induce bond making between two molecules or between a molecule and an atom adsorbed on a surface. For example, a possible method consists of transferring a molecule from the surface to the STM tip, and then bringing the STM tip with this molecule close to the targeted atom or molecule. The desired bond is then created between the two species by an increased surface voltage and tunnel current (Lee and Ho, 1999). Starting from an Fe atom and two separate CO molecules adsorbed on a Ag(110) surface at 13 K, Fe(CO) and Fe(CO)2 molecules can be formed (Fig. 14.29). The fabricated molecules were characterised by vibrational spectroscopy using vibrational inelastic electron tunnelling spectroscopy (Lee and Ho, 2000). Another example is the bonding of a CO molecule with an O atom to produce a CO2 molecule (Hahn and Ho, 2001). Lateral manipulation of the CO molecule with the STM tip brings it close to the O atom on the Ag(110) surface at 13 K. Bonding between the CO molecule and the O atom was then induced by tunnelling electrons. Reaction producing the CO2 molecule is evidenced by the desorption of CO2 from the surface. A third example used vibrational excitation of a H2 S molecule to induce it to diffuse until it was next to a dicarbon (CC) species (Lauhon and Ho, 2000a, 2000b). Facile hydrogen transfer occurs between the two adjacent species forming SH and CCH which have different intensities in the STM images (Figure 14.30) and can be verified by vibrational spectroscopy. We emphasise that using tunnel electrons for inducing bond making through electronic excitation has been studied much less than bond breaking (see the sections on dissociation and desorption). It would be very interesting in the future to elucidate the electronic excitation processes involved. One recent example has made steps in this direction. When adsorbed on a gold surface, a cobalt phthalocyanine molecule shows no Kondo effect. However, it was possible to induce a Kondo resonance by creating molecular bonds between the molecular ring and the surface (Zhao et al., 2005). This was achieved by extracting individual hydrogen atoms from the molecular ring with the STM tip. 14.4.6.2. Semiconductor surfaces Another example of the use of STM manipulation to enable a chemical reaction to occur has been performed on the Ge(111)-c(2 × 8) surface at room temperature. Vertical manipulation with the direct contact method was used to create single atomic vacancies by the controlled removal of individual Ge adatoms (Dujardin et al., 1998). It was observed that at room temperature these vacancies diffuse across the surface due to the thermally activated hopping of the adjacent Ge atoms into the neighbouring vacancy (Molinàs-Mata et al., 1998; Mayne et al., 2001; Brihuega et al., 2004). A number of studies had shown that the perfect reconstructed surface was not reactive towards oxygen (Klitsner et al., 1991; Hirshorn et al., 1991). Indeed, an atomic scale analysis of the reactivity of individual atomic vacancies on the surface allowed the sticking coefficient to be estimated at around 10−5 (Dujardin et al., 1999). On the contrary, it was found that the STM-created atomic vacancies were highly reactive to oxygen due to the vacancies having an electron density


729

Fig. 14.29. Formation of Fe(CO) and Fe(CO)2 molecules starting from an Fe atom and two separate CO molecules adsorbed on an Ag(110) surface at 13 K. 25 Å by 25 Å STM topographic images recorded at 70-mV bias and 0.1 nA without a CO molecules attached to the tip for (A) Fe, (B) CO, (C) Fe(CO), and (D) Fe(CO)2 . (E) Atomically resolved STM topographic image recorded at 22-mV bias and 2.5 nA tunnelling current with a CO molecule attached to the tip. All species, including CO, image as protrusions. The Fe(CO) image appears similar to that of Fe(CO)2 because of frequent 180◦ flips during the scan with these tunnelling parameters. In this image, it is not the tip height (z) that is displayed but its derivative (dz/dy), where y is the scan direction (from top to bottom). This has the effect of illuminating the scan area from the top of the image and accentuating small corrugations. Therefore, each protrusion shows a bright illuminated side facing the top and a dark shadow facing the bottom. A grid is drawn through the Ag(110) surface atoms to guide the determination of the adsorption sites. (F) The side view and (G) the top view of Fe(CO) show the CO to be tilted by an angle τ and bent by an angle β as suggested by the asymmetry in the image (C). The four-fold adsorption site is determined from (E). (H) The side view and (I) the top view of Fe(CO)2 show a similar tilt and bent geometry with angles τ and β as implied by the images (D and E). Reproduced with permission from Lee and Ho, Science (http://www.aaas.org). © 1999 American Association for the Advancement of Science.

close to the Fermi level (Gadzuk, 1991). Furthermore, the process of oxidation of the vacancy gave rise to a very interesting behaviour (Dujardin et al., 1999). It was observed that while the vacancy became dark due to the adsorption of oxygen, another vacancy was observed next to the oxidised site. This new vacancy could diffuse and be oxidised subsequently (Figure 14.31). In other words, the oxidation of a vacancy resulted in the spontaneous creation of another vacancy, thus a chain reaction could be produced by using

730


Fig. 14.30. Single H atom abstraction from H2 S by CC on Cu(001) at 9 K. All images were taken at 0.1 nA and 2 100 mV sample bias. (a) 100×100 Å STM image of H2 S and CC. By exciting the H2S with tunnelling electrons at 0.1 nA and 250 mV sample bias, we made the molecule to hop between lattice sites in a random manner. The path of the tip following the adsorbate motion is indicated by the zig-zag line, starting near the middle of the image and ending toward the right-hand side next to the CC. (b) STM image taken after the induced diffusion. The H2 S 2 has reacted with a CC. The product SH(CCH) is less bright (darker) than the reactant H2 S(CC). (c) 30 × 30 Å image of the area in the red box of panel b showing the newly formed SH and CCH reaction products (for colours see the web version of this book). (d–f) The SH was separated from the CCH by induced diffusion (at 0.1 nA and 250 mV sample bias) to show the products more clearly. The CCH rotates rapidly under the imaging conditions. This rotation is responsible for the streaky halo around the depression. Reproduced with permission from Lauhon and Ho. © 2000 American Chemical Society.

the controlled manipulation of a surface atom alternately with an exposure to a small dose of oxygen. 14.4.7. Charging Electron (hole) attachment using tunnel electrons, as described in Section 14.3.3.2, involves the formation of negatively (positively) charged species having lifetimes which are usually very short (about 1 ps) due to rapid neutralisation by the substrate. Having a charge with a long lifetime and ultimately permanent would be especially valuable for controlling atomic-scale chemical reactions. Charged atomic vacancies are known to exist on ultrathin MgO films on an Ag substrate (Sterrer et al., 2006). However, being able to charge up and discharge a single atom or molecule in a controlled manner is a key step towards using individual atoms or molecules in some form of memory device (Piva et al., 2005).


731

Fig. 14.31. A sequence of STM topographs (100 × 100 Å) of the same area of the Ge(111)-c(2 × 8) surface taken at positive sample bias (+1 V, 1 nA tunnel current) showing: (a) the clean surface (the arrow indicates the adatom which was selectively extracted); (b) the created vacancy; (c) after exposure to 1 L O2 , the oxidised vacancy (dark), and the new induced vacancy (light); (d) the induced vacancy is seen to move; (e) after a second exposure to 1 L of oxygen, the second vacancy has been oxidised as well as the neighbouring site; (f ) no further motion occurs. The time sequence is (a) 0, (b) 100 s, (c) 450 s, (d) 500 s, (e) 850 s, and (f) 1000 s. Reproduced with permission from Dujardin et al. (http://link.aps.org/abstract/PRL/v82/p3448). © 1999 American Physical Society.

A recent STM manipulation experiment has provided the first example of reversible permanent charging of a single atom (Repp et al., 2004a). Au atoms were adsorbed on an ultra thin insulating NaCl film supported on a Cu(111) surface. The Au adatoms show up as bright protrusions on the NaCl layer. The adsorption state of an individual Au adatom could be changed by applying a voltage pulse. Indeed, this process was perfectly reversible; a positive sample voltage (> +0.6 V) caused the appearance of the atom to change (into a sombrero form) and a negative sample voltage (−1 V) switched the Au adatom back to its original state (Figure 14.32). Two tests showed that the sombrero form was a negative charge state. First, the 2D electron gas at the NaCl/Cu interface showed long-range scattering around the modified Au atom. Secondly, approaching a negatively biased tip towards the modified Au atom pushed it away. DFT calculations found two stable configurations; one nearly neutral (Au0 ) and the other a negatively charged (Au− ). However, the positions

732


Fig. 14.32. Manipulation of the adatom state of Au atoms on a NaCl thin layer. After recording the image (A), the STM tip was positioned above one of the Au adatoms (arrow) and a positive voltage pulse was applied to the sample. After a time t, a sharp decrease in the tunnelling current can be observed (B). A subsequent STM image (C) shows that the manipulated Au adatom has a different appearance but did not change its position. By applying a negative voltage pulse, one can switch the manipulated adatom back to its initial state (D). The Au adatoms in the initial state (E) do not scatter interface-state electrons of NaCl/Cu(111), whereas the manipulated adatom (F) acts as a scatterer. Reproduced with permission from Repp et al., Science (http://www.aaas.org). © 2004 American Association for the Advancement of Science.

of the atoms are very different in the second configuration. The Cl− underneath the Au− is forced down and the surrounding Na+ atoms move up. This relaxation of the NaCl film creates an attractive potential for the Au− state which is further stabilised by the screening charge in the metal substrate. Indeed, this shifts the Au(6s) state of the Au0 down by 1 eV and so becomes fully occupied for the Au− adatom. Simulated STM images fitted very well the observed images. While this illustrates reversible permanent charging of a single atom, it requires that the gold atom is isolated from the Cu surface by an insulating layer. Indeed, the experiments by Bennewitz et al. (1999) and Repp et al. (2004b) on thin insulating layers on metal surfaces have allowed the STM imaging of individual molecular orbitals such as those of pentacene (Repp et al., 2005). Consequently, by combining lateral manipulation techniques, a gold atom could be brought into contact with a pentacene molecule, modifying the molecular orbitals (Repp et al., 2006). A recent follow-up experiment has studied the adsorption of silver (Ag) on the same surface of NaCl on Cu (Olsson et al., 2007). Silver atoms adsorb in a neutral state and can be charged negatively by electron tunnelling to form Ag− as for gold and positively by electron extraction to form Ag+ by applying an appropriate voltage to the surface (+1.3 V


733

or −1.3 V, respectively). Confirmation of the presence of a positively charged adatom is provided by the appearance of a localised interface state in the dV /dI spectrum and the scattering of the interface state electrons (Olsson et al., 2004; Limot et al., 2005). These charge states can be reversed by applying an opposite bias. However, the stability of the Ag− is low because it can easily loose an electron to the substrate and requires only a low bias of −0.2 V and Ag+ can be desorbed as well as returned to the Ag0 state. 14.4.8. Conformational changes Conformational change is probably the most important basic process which can be operated at the level of a single molecule. Indeed, conformational changes are often reversible. By controlling molecular conformational changes, applications can be found in molecular electronics (see Section 14.5.1) and molecular nanomachines (see Section 14.5.2). Five different examples given below illustrate the variety of conformational changes that can be induced by STM manipulation. 14.4.8.1. Porphyrin on NiAl(110) Reversible conformational change has been demonstrated in the case of electronic excitation with the STM tip of Zn(II) Etioporphyrin I (ZnEtioI) molecules adsorbed on a NiAl(110) surface at 13 K (Qiu et al., 2004). STM images show the presence of two molecular conformations; a type I where two of the four lobes are distinctly brighter than the other two and a type II conformation where all four lobes have roughly the same intensity. Resolution of the surface atoms reveals that the centre of the molecule is over the bridge site along the Ni trough. Thus a pair of opposite pyrrole rings is over the neighbouring Al atoms and the other two rings over the Ni trough. As a consequence the four pyrrole rings experience different interactions with the surface leading to a non-planar geometry. A similar distortion is found in ligand-coordinated Zn porphyrin molecules (Shelnutt et al., 1998). Molecular manipulation can be induced by taking I(V) spectra and show up as abrupt jumps in the current. A conformation change is observed when subsequently imaged. The threshold at +1.0 V was independent of the tip–sample separation whereas at −1.3 V the threshold shifted linearly towards lower voltages (more negative) as the tip– sample separation was increased. A pronounced peak at +1.25 V only could be seen in the derivative (dI /dV ) and was attributed to the π ∗ LUMO of the molecule. Thus the conformation change requires two different mechanisms depending on the surface polarity. At positive surface voltage, an electron attachment occurs forming a negative ion resonance, while at negative voltage, the electric field overcomes the surface dipole created by the Zn2+ –Ni interaction, reducing the energy barrier to conformational change. This shows that while conformational change can be reversible, different mechanisms can be involved. It is worth mentioning at this point that porphyrin and phthalocyanine molecules are good candidates for molecular electronics for a number of reasons. These molecules are fluorescent with applications in opto-electronics and their properties can be changed by changing the metal atom at the centre. A number of these molecules have been studied at the atomic scale and on a wide variety of surfaces (Papageorgiou et al., 2003, 2004). Recent STM experiments have shown that the molecules can be charged as in the case of copper

734


Fig. 14.33. STM images of (a) trans-2-butene and (b) 1,3-butadiene molecules (area = 45 × 45 A2 , VS = −300 mV, It = 1.00 nA). Schematic representation of the adsorption site of (c) trans-2-butene and (d) 1,3-butadiene molecule on Pd(110), respectively. STM images of the molecules adsorbed on the Pd surface (area = 20 × 20 A2 , VS = −200 mV, It = 0.86 nA) (e) before and (f) after dosing tunnelling electrons on a target molecule of trans-2-butene marked with an arrow in (e). The trans-2-butene, reaction product, and 1,3-butadiene molecules are labelled T , P , and B, respectively, in (f). Reproduced with permission from Kim et al. (http://link.aps.org/abstract/PRL/v89/p126104). © 2005 American Physical Society.

phthalocyanine (CuPc) on the Al2 O3 /NiAl(110) surface. In the experiment by Mikaelian et al. (2006), the CuPc can be charged negatively by resonant electron tunnelling but only at a certain position within the molecule. This is an example of molecular charging in comparison with atom charging discussed in Section 14.4.7. Naphthalocyanine molecules containing two hydrogen atoms in the centre (instead of a metal atom) can be manipulated such that the hydrogen atoms undergo a reversible switching of position. This tautomerization shows up beautifully in the change of symmetry of the LUMO (Liljeroth et al., 2007). These molecules can also be adsorbed on semiconductor surfaces such as the hexagonal 6H-SiC(0001) surface (Baffou et al., 2007). In this case hydrogen phthalocyanine molecules are observed to “walk” across the surface when under the STM tip. 14.4.8.2. Trans-2-butene on Pd(110) The transformation of trans-2-butene into 1,3-butadiene by electron induced dehydrogenation demonstrates that conformational changes can take the form of an isomerization that modifies the chemical nature of the molecule as shown in Fig. 14.33 (Kim et al., 2002). The trans-2-butene molecules were adsorbed on Pd(110) surface at 5 K. Transformation occurred above a voltage of 365 mV corresponding to the threshold of the vibrational excitation of the C–H stretching mode. Again, the reaction rate varies as a power law in I n with n = 2 for voltages less than 800 mV and n = 1 above 800 mV. This suggests that the dehydrogenation reaction occurs via vibrational heating unless the electron energy is higher than the barrier. One of the consequences of this reaction is visible in the STM


735

images; namely, the product 1,3-butadiene appears rotated with respect to the initial trans2-butene. This rotation is thought to occur during the transition state or during relaxation. Rotation allows the C=C bonds of the 1,3-butadiene adsorb on-top sites which is the most stable configuration for C=C double bonds on the Pd(110) surface (Katano et al., 2002). In this case, however, the conformational change is irreversible due to dehydrogenation of the molecule. Nevertheless, the product molecule is rotated so this could be used in a prototype memory device. 14.4.8.3. Pyrrolidine on Cu(100) Some high-speed semi-conductor devices can be operated using the negative differential resistance (NDR) as described by Sze (1981) and had been first observed in the p-n junction of a Ge diode (Esaki, 1958). NDR manifests itself in a system which can be flipped between two states; one conducting and the other not. Thus such systems act as gates in semiconductor devices. NDR has been reportedly observed in a single molecule (Pyrrolidine C4 H8 NH) adsorbed on the Cu(100) surface at 9 K (Gaudioso et al., 2000). When the STM tip is placed above the molecule and the feedback loop switched off, the current trace shows jumps indicating that the molecule is switching between two states. The molecule undergoes a conformational change which modifies the conductance of the molecule. In this case, the NDR effect occurs at a low voltage threshold of 375 mV suggesting that it is vibrationally mediated. Previous examples of NDR on the atomic scale were observed in the tunnel junction (Lyo and Avouris, 1989; Bedrossian et al., 1989; Xue et al., 1999) or in self-assembled monolayers (Chen et al., 1999). 14.4.8.4. Benzene on Si(100) Initially, the study of the adsorption of unsaturated hydrocarbons on silicon was motivated by their use as precursors in SiC film growth for example in the experiments by Bozso et al. (1985), Cheng et al. (1993), and Widdra et al. (1996). A number of the early experiments focused on the adsorption of small hydrocarbon molecules such as ethylene and acetylene on semiconductor surfaces using surface sensitive techniques (Huang et al., 1994; Matsui et al., 1998) including STM experiments to determine the molecular adsorption sites (Mayne et al., 1993; Kim et al., 2001; Mezhenny et al., 2001) and the distribution of the molecules across the surface (Mayne et al., 1996). These were complemented by a number of theoretical studies (Yoshinobu et al., 1987; Fisher et al., 1997; Briggs and Fisher, 1999; Silvestrelli et al., 2001). Studying the adsorption of benzene on semiconductor surfaces at the atomic scale was a logical continuation. At room temperature, STM images of benzene adsorbed on the Si(100)-2 × 1 surface showed the presence of two different bonding configurations (Self et al., 1998; Lopinski et al., 1998a, 1998b). A metastable site is produced by the adsorption of the benzene on top of a single Si dimer through two C–Si bonds forming a butterfly structure and a more stable site where the benzene bridges two dimers in a tight-bridge structure. These adsorption configurations were confirmed by angle-resolved ultraviolet photoemission spectra (Gokhale et al., 1998) and NEXAFS studies (Kong et al., 1998) and they had been suggested by earlier calculations (Craig, 1993; Jeong et al., 1995). It was found that the metastable benzene site could be converted to the more stable site by thermal energy even at room temperature. Furthermore, the STM tip could induce conversion from

736


the stable site back to the metastable site. By scanning at −3 V and 40 pA, while some molecules are desorbed, 75% of the remaining molecules are converted back to the on-top single dimer site. 14.4.8.5. Polyphenyls on Si(100) STM manipulation of poly-aromatic molecules is potentially interesting for several reasons. First, by changing the molecular conformation it may be possible to modulate the conductivity of the π system of the molecule. See Section 14.5 for a discussion of molecular conductivity. Second, it may be possible to transport energy by exciting one part of the molecule thereby inducing a modification in another part. A recent STM study by Soukiassian et al. (2005) has investigated the adsorption and manipulation of a functionalized tri-phenyl 1,4 -paratriphenyldimethylacetone (called Trima for short). On the Si(100)2 × 1 surface at room temperature, molecule adsorbs with its long axis parallel and on top of the silicon dimer row. In support of this, NEXAFS spectra showed that the C=O bond in the ketone groups reacted with the silicon surface to form C–O–Si bonds and valance-band photoemission spectra indicated that the benzene rings had only a very weak interaction with the surface. This suggests that the Trima molecule is chemisorbed on the surface through the ketone groups. Individual molecules were manipulated by injecting electrons into the central benzene ring with the STM tip. Different parts of the Trima molecule could be selectively modified by choosing the appropriate applied bias (Fig. 14.34). With 4.0 eV electrons, the end of the molecule changed, with 4.5 eV electrons the middle of the molecule was modified and with 5.0 eV electrons, dissociation of the molecule was observed. Furthermore, the synchrotron studies suggested a π–π ∗ transition at 4.5 eV, corresponding closely to the threshold observed in the STM manipulation experiments, thus indicating a direct electronic excitation of the π–π ∗ transition. All the observed changes of configuration were irreversible. This is most probably due to the fact that at room temperature at which the experiments were done, only very stable molecular configurations could be observed with the STM. There have been several recent experiments demonstrating other conformational changes of individual molecules using STM manipulation. The STM studies on individual molecules adsorbed on a Au(111) surface indicated that controlled isomerization could be induced by electron excitation. A reversible bistable character was found for a simple azobenzene (Choi et al., 2006; Henzl et al., 2006) but irreversible for a more complicated azobenzene derivative (Henzl et al., 2007). A monolayer of azobenzene molecules can be adsorbed on a Au(111) surface at low temperature (Alemani et al., 2006). In this experiment, the conformation of individual molecules could be modified by increasing the applied electric field between the STM tip and the surface. In another experiment, single chlorophyll-a molecules were adsorbed on a Au(111) surface. These contain a C=C double bond in a phytyl chain and could be folded into four different configurations when manipulated with the STM tip (Iancu and Hla, 2006). Another class of molecules that show isomerization are the family of diarylethenes (Matsuda and Irie, 2006). These molecules undergo a photon induced reversible cyclization which changes the molecular structure (Kobatake et al., 2007). The different forms of the molecule can be observed at the atomic scale (Bellec et al., 2007) but it remains to be seen whether the molecules can be manipulated with the STM. Nevertheless, molecules such as azobenzene and diarylethenes have


737

Fig. 14.34. STM manipulation of the Trima molecule on the Si(100)-2 × 1 surface. Three pairs of STM images show different molecules before manipulation (a), (c) and (e), and after manipulation (b), (d) and (f). In each case the electrons were injected by the STM tip into the central bright part of the molecule. For a voltage pulse at a sample bias of 4.0 V (a), the end of the molecule is modified (b). For a voltage pulse of 4.5 eV (c), the centre of the molecule is modified (d). For a voltage pulse of 5.0 eV (e), the molecule is dissociated (f). Note that in images (e) and (f), the poorer resolution is due to the tip. The structure of the molecule is shown at the bottom. Reproduced with permission from Soukiassian et al. © 2005 American Institute of Physics.

potentially interesting applications in the field of molecular nanomachines. Quinone derivatives show potential as memory devices since Bandyopadhyay et al. (2006) showed that the molecule could be switched reversibly between different charge states using the STM.

738


14.5. Applications of STM manipulation and dynamics The ability to manipulate individual atoms and molecules in an ever increasingly controlled manner has opened up new perspectives to fabricate atomic-scale devices and to investigate their mechanical, electronic, magnetic and chemical properties. This has applications in several areas including molecular electronics and molecular nanomachines. 14.5.1. Molecular electronics The idea of using molecules as individual components in an electronic device was first put forward by Aviram and Ratner (1974). However, no real advance was made until the advent of the STM followed by the pioneering experiments of Eigler and Schweizer (1990) and Becker et al. (1987). Being able to see and manipulate individual atoms and molecules gave a huge impetus to this idea of molecular electronics. Over the last decade or so, research in this field has concentrated on “hybrid molecular electronics” by finding the molecular equivalents of transistors, wires, switches, diodes and so on (Joachim et al., 2000). In “hybrid molecular electronics”, the circuit architectures are much smaller but otherwise identical to those in standard microelectronics, except that the electronic components are replaced by molecules. Implicitly, this requires that the molecules are conducting and hence there have been a number of theoretical studies on various molecules (Lang and Avouris, 2001; Heurich et al., 2002; Emberly and Kirczenow, 2003). Some of these have been in response to experiments on the conductivity of such molecules attached to two metal electrodes (Reed et al., 1997; Reichert et al., 2004). Indeed, one key issue facing the use molecular electronics is the electron transport, both inside molecules and through the molecule-wire junction (Joachim and Ratner, 2005). Two principal conduction mechanisms have been identified; either the molecule has states in resonance with the metal electrode states near the Fermi level or super-exchange tunnelling enables the electron to be transmitted because of the presence of high lying molecular states (Nitzan, 2001). More recently, a novel approach has been tested theoretically by Joachim et al. (2000), namely, the “mono-molecular electronics”. Within this approach, it has been suggested that a single molecule could replace a whole electronic circuit by integrating a set of electronic functions inside a single molecule leading to the fabrication of a single molecule logic gate (Joachim et al., 2000; Joachim, 2002). Several architectures of “mono-molecular electronics” have been proposed by Ami et al. (2003a). In the semi-classical mono-molecular electronics approach, the architecture is designed such that several components and wires are integrated into a single molecule (Ami et al., 2003b). In the quantum Hamiltonian architecture, there is no longer any need to distinguish components and wires. Here, logic gate functions are obtained through the control of the molecular Hamiltonian (Fiurášek et al., 2004; Duchemin and Joachim, 2005). Since mono-molecular electronics has been limited so far to theoretical investigations, we will limit our review to the applications of STM manipulations to hybrid molecular electronics. 14.5.1.1. Molecular transistor The first example of a transistor effect in a molecule is that of C60 adsorbed on the Au(110) surface (Joachim and Gimzewski, 1995; Joachim et al., 1995). The idea was to


739

use the STM tip to deform the adsorbed C60 molecule in such a way that the transport of electrons through the molecule was modified. The experiments showed that mechanical deformation of the C60 molecule did give rise to transistor-like behaviour; I(V) spectroscopy curves taken at different tip–sample separations showed a linear variation of the current with voltage and the slope varied as a function of tip–sample distance (Joachim and Gimzewski, 1997; Joachim et al., 1998). Indeed, at electrical contact, it was deduced that the electron transfer was more efficient through the molecule than if the molecule was absent. This is due to the suppression of Coulomb blockade effects due to the small HOMO-LUMO gap of the molecule. Based on these results, a theoretical treatment of a hybrid molecular electronic circuit was performed by incorporating the C60 molecules into an electronic circuit. In principle, a hybrid molecular architecture in the form of a half-adder logic circuit could be used to perform simple calculations. A planar geometry was assumed with AFM-like cantilevers to activate each C60 molecule (Ami and Joachim, 2001; Stadler et al., 2001). These highlighted a number of potential problems and serious constraints. The circuit must be twodimensional so that all C60 molecules can be activated by cantilevers. This severely limits the types of logic circuits that could be constructed. The operational frequency is limited by the cantilever frequency ( 0◦ ). Wetting corresponds to layer-by-layer growth, while nonwetting behaviour leads to the formation of three-dimensional islands. For a microscopic understanding of wetting behaviour we have to consider the balance of intermolecular forces between the substrate B and the overlayer A. Take ΦAA , ΦAB and ΦBB to be the energies of bond formation for A–A, A–B and B–B, respectively. If ΦAB is greater in magnitude than ΦAA , then the system prefers to form A–B bonds over the maintenance of A–A bonds and wetting occurs. When the opposite condition prevails, the system attempts to minimize the contact area between A and B and droplet formation occurs. By defining the energy difference according to = ΦAA − ΦAB ,

(16.5.2)

we can restate the wetting condition as < 0. Nonwetting prevails for > 0. Eq. (16.5.1) is valid only at equilibrium. Growth is often performed in a vacuum chamber in which the pressure p of the gas phase differs from the equilibrium vapour pressure p ∗ of the material being deposited. Thus we must account for the accompanying pressure dependence in the Gibbs energy of the system, G = nkB T ln ζ,

(16.5.3)

where n is the number of moles and ζ = p/p ∗

(16.5.4)

is the degree of supersaturation. Control over ζ and thereby the Gibbs energy change in the system, as we shall see below, potentially gives us a control variable (in addition to T ) that can be used to switch the growth mode of the system at hand. There are three limiting cases of growth under equilibrium control. These three cases are illustrated in Fig. 16.5. Below we define the conditions under which each of these is observed. If the overlayer A wets the substrate B, then the contact angle ψ = 0◦ , that is, flat layers are expected. Once we identify that the surface Gibbs energy Gs is equal to the surface energy γ , we can use Eq. (16.5.3) and the Young equation to determine the condition for wetting γBg γAB + γAg + CkB T ln(ζ ).

(16.5.5)

802

K.W. Kolasinski

Fig. 16.5. Three modes of thermodynamically controlled growth of an overlayer A on substrate B in the presence of a gas (or more generally a fluid or vacuum) g. (a) Layer-by-layer growth (Frank–van der Merwe, FM) of two lattice matched (ε0 = 0) materials. (b) Layer-plus-island growth (Stranski–Krastanov, SK). The strained wetting layer does not exhibit dislocations at either of its interfaces; however, the islands continuously relax with a lattice distortion in the growth direction. (c) Island growth (Volmer–Weber, VW) of lattice mismatched (ε0 = 0) materials with dislocations at the interface.

If strain can be neglected (ε ≈ 0), Eq. (16.5.5) represents a simple energetic statement of when layer-by-layer growth occurs. Layer-by-layer growth is also known as Frank–van der Merwe (FM) growth (Frank and van der Merwe, 1949a). At equilibrium ζ = 1 and the third term vanishes. For finely balanced systems ( ≈ 0), this third term acts as a switch that can be used under nonequilibrium conditions to push the system in an out of the Frank–van der Merwe regime even though the system is still under thermodynamic control. The growth window represents the range of temperatures and compositions over which layer-by-layer growth is followed. The MBE growth window is particularly large for the Inx Ga1−x As/GaAs system. Substrate temperatures of 670–870 K are used. The V:III flux ratio can be varied between 2.5:1 and 25:1. The In content can extend from 0 x 0.25 (Dunstan, 1997). At the other extreme in which ψ > 0◦ , γBg < γAB + γAg + CkB T ln(ζ ),

(16.5.6)

and the overlayer does not wet the substrate. Adsorbate-adsorbate interactions are stronger than adsorbate-substrate interactions and the system attempts to minimize the contact area between the deposit and the substrate by balling up into three-dimensional (3D) islands.

Growth and Etching of Semiconductors

803

Three-dimensional island growth is also known as Volmer–Weber growth (Volmer and Weber, 1926). A large lattice mismatch can also lead to Volmer–Weber growth. The most commonly observed growth mode corresponds to neither of these limiting cases and cannot be explained in the absence of strain. In Stranski–Krastanov growth (Stranski and Kr’stanov, 1938), a wetting layer of about 1–4 ML first coats the substrate. However, the lattice misfit is too substantial to be relaxed (exceeding about 2%) and the overlayer cannot continue to grow psuedomorphically in the strained structure. At this point the growth switches from layer-by-layer to the formation of three-dimensional islands. This leads to a natural definition of what is a small as opposed to a large lattice misfit. A small lattice misfit, ε0 < 1.5%, favours layer-by-layer growth. A large lattice misfit, ε0 > 2%, does not accommodate layer-by-layer growth. The best known examples of Stranski–Krastanov growth are InAs/GaAs and Ge/Si for which the critical coverages for island formation are 1.6 ML and 3.5 ML, respectively (Priester and Lannoo, 1997). In these cases, the islands are dislocation free and known as coherent islands. The energetics of the transition to SK growth have been reviewed by Müller and Saúl (2004). Under favourable conditions, the islands form with a narrow size distribution and may even form ordered arrays (Shchukin and Bimberg, 1999). The self-assembly process that leads to the spontaneous formation of islands has attracted great interest because of the unique optical and electronic properties of semiconductor quantum dots (Alivisatos, 1996; Yoffe, 1993). Apart from temperature and pressure, the one other variable at the experimentalist’s control to change the growth mode is the presence of a surfactant (Horn-von Hoegen et al., 1991; Copel et al., 1989). As mentioned in the discussion of strain relief, a surfactant can lower the surface energy and thereby change the strain profile of an interface. The change in strain can also lead to a change in growth mode. Si grows on Ge(100) in a Volmer– Weber mode, whereas Ge grows on Si(100) in a Stranski–Krastanov mode. This inversion of growth modes is a general rule and is a direct result of Ge possessing a lower surface energy than Si and, therefore, that Eq. (16.5.2) changes sign when the roles of substrate and overlayer are switched. Both As and Sb can be used to change the growth mode of Ge on Si. They do so by acting as a surfactant that segregates to the top surface. At the surface, they lower the energy of a flat film and inhibit the formation of islands. In this way, the surfactant promotes layer-by-layer growth at the expense of Stranski–Krastanov growth. 16.5.2. Ostwald ripening The chemical potential μi of an unstrained island of radius ri can be written (Liu et al., 2001) νσ , μi (ri ) = (16.5.7) ri where ν is the surface area per atom in the island and σ is the island edge (step) energy. In the absence of strain, Eq. (16.5.7) shows that the chemical potential of an island is inversely proportional to its radius. Therefore, a thin film that does not wet a surface will break up into a distribution of 3D islands. If the system evolves without constrain and in the absence of strain, coarsening of the initial layer/island distribution will tend to form one large island rather than any number of smaller islands.

804

K.W. Kolasinski

In any real finite system, the communication distance between islands is limited. Once the distance between the contracting islands exceeds the diffusion length, the islands become independent of one another and their accretion is arrested. During accretion, small islands feed material to larger islands. Since the position of islands is random, the number and size of initial neighbouring islands is random and the amount of material that can ultimately be accreted into the largest island will vary randomly across the surface. There is no optimum island size and the size distribution of islands will be broad. This process is known as Ostwald ripening, which was first observed and described for the growth of grains in solution (Ostwald, 1900). STM observation of the growth and decay of two-dimensional islands, e.g. for Ag on Ag(111) (Morgenstern et al., 1998, 1996), allows for the determination of the energetic parameters that control the evolution of the size distribution. As will be shown below, even at equilibrium and with no constraint on diffusion, ripening can be observed under appropriate conditions in the presence of strain. The chemical potential of a strained island can be written (Liu et al., 2001) σ −α α ri − ln , μi (ri ) = ν (16.5.8) ri ri a0 where α=

4πF 2 (1 − ν 2 ) , μ

(16.5.9)

F is the misfit strain induced elastic force monopole along the island edge, ν the Poisson ratio, μ the Young modulus and a0 is a cutoff length on the order of the surface lattice constant. The strained system exhibits a thermodynamically stable island size that is resistant to further coarsening and a system of many islands evolves until they reach a radius given by r0 = a0 exp(σ/α).

(16.5.10)

At finite temperatures, entropic effects broaden the size distribution into a Gaussian distribution (Priester and Lannoo, 1995) about a mean value that is somewhat different than r0 (Shchukin, 2001). Lagally and co-workers (Liu et al., 2001) have shown that further considering the influence of island-island interactions, strain leads not only to the establishment of a preferred size, but also that it can result in self-organization and a narrowing of the size distribution. 16.5.3. Equilibrium overlayer structure and growth mode All three of the growth modes outlined above can be observed at equilibrium. The development of a theory to describe the formation of different equilibrium layer structures remains an active pursuit (Priester and Lannoo, 1997, 1995; Shchukin and Bimberg, 1999; Shchukin et al., 2004; Wang et al., 1999; Daruka et al., 1999; Tersoff et al., 1996). A theory to describe the role of strain in growth at equilibrium has been derived by Daruka and Barabási (1997) from the theoretical framework of Shchukin and Bimberg (1999). They consider the deposition of H monolayers of atom A on substrate B. The system is allowed to equilibrate forming n1 monolayers of A in a wetting layer with the remaining H − n1


805

monolayers distributed into 3D islands. The islands are assumed to take on pyramidal shapes with a constant aspect ratio. Neglecting entropic contributions, the internal energy of the deposit can be written u(H, n1 , n2 , ε) = Eml (n1 ) + n2 Eisl + (H − n1 − n2 )Erip .

(16.5.11)

The first term is the contribution of the wetting layer (the monolayer ml), the second from the islands and the third from the ripened islands. The wetting layer contribution is given by an integral over the binding and elastic energy densities in the layer. The effects of strain must be properly accounted for in the calculation of both energy densities. The uniformly strained wetting layer has an energy density, G, given by the sum of the strain energy density and the binding energy of the layer G = Cε2 − ΦAA ,

(16.5.12)

where C is a material dependent constant and ε and −ΦAA are, as before, the lattice mismatch and the strength of an A–A bond, respectively. In a strained layer of finite thickness, the binding energy density of A atoms increases from −ΦAB in the first monolayer to −ΦAA far from the interface. Accounting for this, the total energy stored in the wetting layer is written n1 n−1 Eml (n1 ) = (16.5.13) dn G + Θ(1 − n) + Θ(1 − n) exp − , a 0 where Θ(1 − n) = 0 if (1 − n) < 0 and Θ(1 − n) = 1 if (1 − n) > 0. is the energy difference defined in Eq. (16.5.2). The second term in Eq. (16.5.11) describes the free energy per atom of the islands and the island-island interaction. Islands assemble if the strain energy density of an island is lower than that of the wetting layer and is accounted for by a form factor g that assumes values from 0 to 1. The binding energy is as before. The elastic energy of an edge of length L is proportional to −L ln L. The edge energy density is, therefore, proportional to −(ln L)L2 . In addition, account must be taken of the facet energy and for the interaction of the homo- and hetero-epitaxial stress fields. Writing x = L/L0 (the reduced island size) and the island-island interaction as Eii yields α 2

Eisl = gCε2 − ΦAA + E0 − 2 ln e1/2 x + (16.5.14) + Eii , x x where E0 is the characteristic energy and α = p(γ − ε). p and γ are constants that describe the coupling between the stress fields and the extra surface energy introduced by the islands, respectively. The lateral interaction term, Eii , depends on the island spacing, shape and size as well as ε. Eii is dominated by dipole-dipole interactions, thus at low coverages of islands, it can be neglected. Erip , the total elastic energy per atom of the ripened islands, is obtained from Eq. (16.5.11) by taking the limit x → ∞. This yields Erip = gCε2 − ΦAA .

(16.5.15)

The phase diagram calculated by Daruka and Barabási using Eq. (16.5.11) appears in Fig. 16.6. It demonstrates that FM, SK and VW can each represent the equilibrium growth mode for the appropriate combinations of deposited material and lattice mismatch. Two

806

K.W. Kolasinski

Fig. 16.6. Phase diagram calculated by Daruka and Barabási as a function of coverage H and misfit ε. The small panels at the top and bottom schematically represent the growth modes observed. The small empty islands indicate the presence of stable islands, while the large shaded ones refer to ripened islands. Reprinted with permission from I. Daruka and A.-L. Barabási, Physical Review Letters 79 (1997) 3708. © 1997 American Physical Society.

SK phases are found, differing as to whether the wetting layer forms before (SK1 ) or after (SK2 ) the islands form. Also found in Fig. 16.6 are three distinct ripening phases. R1 corresponds to classic Ostwald ripening in the presence of a wetting layer. R2 is a modified ripening phase with a wetting layer and stable small islands. The islands are formed during the SK stage of growth. Subsequently, their growth is arrested but not all of them are lost to the ripening islands. The R3 phase is similar to R2 but lacks the wetting layer. Just as for atomic and molecular adsorbates, lateral interactions become progressively more important as the coverage increases (Liu et al., 2001; Shchukin et al., 2004), in which case the Eii term cannot be neglected. If the strain fields that engender the lateral interac-


807

tions, represented by Eii , are anisotropic, they have the potential to lead to ordered arrays of islands, just as appropriate lateral interactions can lead to ordered adsorbate structures. In the construction of Eq. (16.5.11), the entropic contribution has been neglected. Consequently, the phase diagram of Fig. 16.6 is valid only at T = 0 K. Some temperature effects are included implicitly. One effect arises when the substrate and overlayer have different coefficients of thermal expansion. This results in a temperature dependence to ε. This effect, assuming all other material dependent parameters are constant, is accounted for within the theory outlined above. The extension of the theory to include entropic effects remains an outstanding challenge. This is required to describe the temperature dependence of growth systems. The temperature plays a vital role because surface energies are temperature dependent. The most obvious consequence of this is that the most stable reconstruction of a surface is temperature dependent. Furthermore, some surfaces are known to undergo reversible roughening transitions at a specific temperature. Whether islands are truly equilibrium structures has been questioned by Scheffler and co-workers (Wang et al., 2000b, 2000a, 1999). They showed that any equilibrium theory that includes energetic contributions only from island surface energy and elastic relaxation would not predict a finite equilibrium size distribution. In this case, strain relaxation dominates and at high coverage, small islands are unstable with respect to large islands and Ostwald ripening occurs. They offer a theory of so-called constrained equilibrium in which island size is determined by island density and coverage. Nuclei grow to a size determined by material transport between the wetting layer and the islands, which is governed by energy balance. They calculate the elastic energy of islands and substrate within continuum elasticity theory and use density functional theory to calculate the surface energies of the wetting layer as well as the islands facets. The result is a three phase process: (i) a nucleation phase that determines the island density, (ii) island growth at the expense of the wetting layer, and (iii) Ostwald ripening. However, the theory of Wang et al. fails to predict the correct form of the dependence of critical thickness on growth conditions (Joyce and Vvedensky, 2004). Critical thickness increases with increasing surface temperature Ts . The theory of Wang et al. predicts a decrease. Critical thickness is independent of number density at constant Ts . Contrarily, the theory predicts an increase in critical thickness with increasing number density. Further developments in theory are awaited with great anticipation. 16.5.4. Catalytic growth: Vapour–liquid–solid (VLS) etcetera mechanisms Catalytic growth has been used to create a range of high aspect ratio features including nanowires (solid core structures with diameters below ∼100 nm), nanotubes (single or multi-walled hollow core structures with diameters below ∼100 nm) and whiskers (larger solid core structures). Several recent reviews have appeared including ones on the formation of semiconductor nanowires from solutions and supercritical fluids (Wang et al., 2006b) as well as from vapours (Fan et al., 2006; Lu and Lieber, 2006), in addition to carbon nanofibres (De Jong and Geus, 2000) and carbon nanotubes (Ajayan, 1999; Dai, 2002). Increasingly a mechanistic understanding is developing of the processes involved in and which link the seemingly disparate techniques known variously as vapour–liquid– solid (VLS), vapour–solid–solid (VSS), supercritical fluid–liquid–solid (SFLS), solution–

808

K.W. Kolasinski

liquid–solid (SLS) and solid–liquid–solid growth (unfortunately also SLS) (Kolasinski, 2006). The prototypical growth mechanism involving reactions catalyzed by metal particles is the VLS mechanism (Wagner and Ellis, 1964), which leads to structures of the type shown in Fig. 16.7. A metal particle is placed on a solid surface. Classically it is thought that the metal should form an alloy (eutectic) with the material of the substrate with a relatively low melting point. The liquid alloy acts as a preferred deposition site for material from the vapour phase. For instance, if a CVD type process is used, dissociative adsorption might preferentially occur on the liquid metal/alloy particle rather than on the substrate. Alternatively, especially if an MBE process is used, initial adsorption may occur elsewhere but in the absence of random nucleation surface diffusion delivers material to the catalytic particles such that atoms are only incorporated into the growing structure at this position (Blakely and Jackson, 1962). Because of alloy formation and the feeding of material into the liquid particle by dissociative adsorption/diffusion, the particle becomes supersaturated and material begins to precipitate through the bottom of the particle. This allows the particle to detach from the substrate and sit atop the whisker that grows beneath it. Diffusion, whether on the substrate, the sides of the growing whisker, on the surface of the particle and through the bulk of the particle, obviously plays an important role in the growth dynamics (Wang and Fischman, 1994). In addition, as we shall see from the work of Tromp and co-workers (Hannon et al., 2006), diffusion of the metal atoms that make up the catalyst particle itself may also play a role in the growth dynamics. A specific example is the deposition of Au particles on Si, which are then exposed to a mixture of SiCl4 and H2 at 1220 K. Using these simple conditions, Wagner and Ellis were able to demonstrate the growth of crystalline Si columns grown along the 111 direction with diameters from 100 nm to 0.2 mm. Ni, Pd and Pt (Givargizov, 1975) as well as Ti (Liu et al., 1996) have also been used as catalysts for Si. Size-selected gold colloids offer the opportunity to control the diameter of the Si columns since the diameter of the column closely matches the diameter of the Au particle. The three-step process of alloying, nucleation and axial growth has been observed by in situ TEM for Au catalyzed Ge nanowire (NW) growth. Yang and co-workers (Hochbaum et al., 2005) have used this technique to grow Si columns as small as 39 nm in diameter with spatial control over where the columns form. Lieber’s group has demonstrated diameter controlled growth of a wide range of nanowires, including group IV (Morales and Lieber, 1998; Cui et al., 2001), III–V (Duan and Lieber, 2000; Duan et al., 2002), and II–VI (Duan and Lieber, 2000; Barrelet et al., 2003) materials via the VLS mechanism. A CdS nanowire laser has been created in this fashion (Duan et al., 2003) among other devices (Duan et al., 2002; Cui et al., 2003; McAlpine et al., 2003). Also branched Si and GaN nanowires can be grown (Wang et al., 2004). A Ni catalyst is used for GaN nanowire growth. Branched structures are formed by interrupting VLS growth after a defined growth period, depositing a secondary set of catalyst particles (of similar or different sizes) that attach to the nanowires grown in the first growth cycle and then proceeding to a second growth stage. The cycle can be repeated to create an arbitrary number of branches on branches. The use of an Fe or FeAu catalyst to grow a SiNW (Morales and Lieber, 1998) allows one to attach a carbon nanotube to the SiNW simply by switching the process gas from


809

Fig. 16.7. Scanning electron micrograph of GaP nanowires grown via the VLS mechanism. The inset shows a hemispherical Au particle atop the GaP nanowire. Reprinted in part with permission from J. Johansson, C.P.T. Svensson, T. Mårtensson, L. Samuelson and W. Seifert, Journal of Physical Chemistry B 109 (2005) 13567. © 2005 American Chemical Society.

SiH4 to C2 H4 and using the same catalyst particle for growth (Hu et al., 1999). Similarly, the composition, in terms of either doping or change of stoichiometry, can be controlled by changing the composition of the gas phase. In this manner, junctions of n-type and p-type Si or GaP with GaAs have been produced (Gudiksen et al., 2002). In addition to using a gas source, the vapour phase can also be provided by laser ablation of a solid target (Morales and Lieber, 1998; Duan and Lieber, 2000; Gudiksen et al., 2002). The source for SiNW growth can also be a standard MBE source, as employed by Dujardin et al. (2006). The use of an MBE source is quite significant because of a basic difference in the dynamics of the interaction of an atomic vapour of Si as compared to SiH4 . Whereas the sticking coefficient of SiH4 is low on a Si and practically zero on a H-terminated Si surface, it is much higher on the surface of a metal catalyst. In contrast, the sticking coefficient of Si atoms is unity on a clean or H-terminated Si surface as well as on a metal catalyst particle. The success of MBE in semiconductor processing relies largely on the fact that the sticking coefficient of numerous evaporated semiconductor materials is virtually unity on a solid substrate regardless of its composition. VLS growth in a MBE configuration has, for example, been demonstrated for nanowires composed of Si, SiGe and III–V compounds. A sticking coefficient difference alone cannot account for the formation of nanowires. Other factors must also be considered that allow the nanowire/catalytic particle interface to act as a sink for the incorporation of new material into the nanowire at a greater rate than the growth of the sidewalls or the thickness of the substrate in between catalyst sites. For instance, the catalytic particle can lower the barrier that is present for the incorporation of new material at the growth interface as compared to the nucleation of an island on a sidewall or the substrate. This fact puts the lie to a rather widely reported misunderstanding (Tan et al., 2003; Wagner and Ellis, 1964; Givargizov, 1975) that VLS growth occurs because the sticking coefficient on a liquid is unity and must be higher than the sticking coefficient on the solid. For growth of SiNWs from silanes, which have a much higher dissociative sticking coefficient on the particle than on the substrate or sidewalls, the reason for the greater rate of

810

K.W. Kolasinski

dissociation is because of the catalytic action of the metal in the particle not the fact that it is liquid. There is no general evidence for the assertion that the sticking coefficient must be larger on a liquid than a metal, and the assertion does not even apply generally to VLS growth. The dependence of SiNW diameter on the diameter of the catalyst particle has been studied, for example by Lieber and co-workers (Cui et al., 2001). Nanoclusters were deposited from colloidal Au suspensions. TEM studies of the materials grown from 5 nm (4.9 ± 1.0 nm), 10 nm (9.7 ± 1.5 nm), 20 nm (19.8 ± 2.0 nm), and 30 nm (30.0 ± 3.0 nm) Au nanoclusters revealed that the average NW diameters were 6.4±1.2, 12.3±2.5, 20±2.3, and 31.1±2.7 nm, respectively. TEM also demonstrated that the nanowires have crystalline Si cores, even for cores as small as 2 nm in diameter, sheathed with 1–3 nm of amorphous oxide. The smooth structure of the SiNWs indicates that NWs grown this way have fewer defects than those grown by SiO2 -mediated or Si thermal evaporation methods (Lee et al., 2000). The breadth of the SiNW diameter distribution mirrors that of the Au colloid distribution, suggesting that the dispersion in SiNW diameter is limited only by the dispersion in catalyst particle diameter. The catalytic nanoparticles need not be deposited directly; rather, they can be formed under growth conditions. After deposition of 2 ML of Au on Si(111), heating to 873 K leads to the formation of Au droplets resting on a 1 ML thick wetting layer (Hannon et al., 2006). Further annealing leads to coarsening (a change in the island size distribution) and Ostwald ripening (growth of large particles at the expense of small ones), which clearly indicates that Au atoms are mobile on the growth surfaced at this temperature. Kikkawa et al. (2005) deposited a 0.5 nm Au film on Si(111) then exposed it to SiH4 (1% in Ar, total pressure of 98 kPa) at 640 K Ts 770 K in a flow tube. Most of the resulting Si nanowires were 3–12 nm in diameter with lengths often in excess of several hundred nm. Theoretical approaches to explain nanowire growth can be separated into at least three different categories: molecular dynamics, thermodynamics and kinetics. Molecular dynamics has been used, for instance, by Rosén and co-workers Ding et al., 2006, 2004b, 2004a to examine the growth of CNTs. Kinetics (mass transport) based models have been considered by Verheijen et al. (2006), Dubrovskii et al. (2006), Tersoff and co-workers (Kodambaka et al., 2006; Ross et al., 2005), Persson et al. (2004) and Johansson et al. (2006, 2005). Thermodynamic approaches trace back to Blakely and Jackson (1962) as well as Givargizov (1975) and more recently have been treated by Kwon and Park (2006), Wang et al. (2006a), Chandrasekaran et al. (2006), Chen and Cao (2006), Mohammad (2006), and Tan, Li and Gösele (Tan et al., 2004, 2003; Li et al., 2007). The Gibbs–Thomson effect expresses how a curved interface affects the chemical potential of a body, an effect that causes vapour pressures and solubilities to become dependent on the size of a catalyst particle. Thermodynamic treatments are able to show how the Gibbs–Thomson effect leads to nanowire growth rates that depend not only on the growth parameters (pressure and temperature) but also the diameter of the catalyst particle. The small critical diameter measured in the study of Kikkawa et al. (1.7 nm) is of note because it is considerably smaller the smallest critical diameter found in an earlier study by Givargizov (1975), who also presented the first thermodynamic theory to describe the growth rate and critical diameter in terms of the chemical potential difference in the vapour phase and the whisker.


811

It is often found that, the growth rate should decrease with decreasing diameter (Givargizov, 1975; Chen and Cao, 2006). However, as pointed out by Johansson et al. (2005), the extent of supersaturation within the catalyst depends on the temperature and gas-phase composition. A transition from smaller diameters having lower growth rates to smaller diameter having higher growth rates can be expected as temperature and gas-phase composition are changed. This may explain why Si NWs grown under MBE conditions (Schubert et al., 2004) exhibit higher growth rates for smaller diameter rather than the opposite conclusion found by Givargizov under CVD conditions. Gösele and co-workers (Tan et al., 2004, 2003; Li et al., 2007) have examined whether there is a thermodynamically determined minimum size and the parameters that affect not only the nanowire size but also the size of the catalytic particle. Their results indicate that during SiNW growth in the presence of a Au catalyst, the catalyst minimum size is determined by the vapour pressures of Si and the metal. The SiNW minimum size is determined by the catalyst composition and its size. They conclude that there is no thermodynamically determined minimum size, rather, that the minimum size is determined by kinetic limitations. In the case of SiNWs, they (Tan et al., 2004) predict that the catalyst should be larger than the nanowire, but this is clearly not universally true for all materials. The aforementioned mobility of Au atoms can play an important role in the growth dynamics. First, as we have seen above, mobility can be important for determining the Au particle size and therefore the nanowire size distribution. Second, under some growth conditions, the Au particles may continue to coarsen during growth. This will lead to tapering in the NW diameter, eventual cessation of growth when the particle is exhausted of Au, and the expectation that thinner NWs will have a shorter ultimate length. The importance of Au atom mobility has been observed by Tromp and co-workers (Hannon et al., 2006) for growth at 873 K on Si(111) in the presence of 20% Si2 H6 in 80% He at 5 × 10−5 Torr. The substrate surface is found to always be covered by a Au monolayer and their results also strongly suggest that Au must at least partially wet the sidewalls of the nanowires. Since tapering is not observed for several other systems, particularly when high pressure CVD is used, they suggest that the presence of impurities such as O2 may play a role in hindering the diffusion of Au atoms for some growth conditions. Clearly transport dynamics are extremely important in VLS growth and the presence of co-adsorbates as well as the temperature; chemical identity of the metal and substrate; and the sticking coefficient of the gas on the particle, sidewall and substrate all have important roles to play. In a real system, the interface between the liquid and the growing nanotube is not sharp. Instead a transition region exists (Kuo and Clancy, 2004). One consequence of this as found by Gösele and co-workers (Schmidt et al., 2005) is that at diameters below 20 nm, most of the nanotubes grow along the 110 direction, whereas above 40 nm they grow along the 111 direction as found originally by Wagner and Ellis. How versatile is the VLS mechanism and must the catalytic particle be liquid? Quite general is the answer to the first question, as it may play a role in a number of systems including (Kolasinski, 2006) the growth of SiOx (a substoichiometric silicon oxide); SiO2 ; Si1−x Gex ; Ge; AlN; γ -Al2 O3 ; oxide-coated B; CNx ; CdO; CdS; CdSe; CdTe; α-Fe2 O3 (hematite), ε-Fe2 O3 and Fe3 O4 (magnetite); GaAs; GaN; Ga2 O3 ; GaP; InAs; InN (hexangular structures); InP; In2 O3 ; In2 Set3 ; LiF; SnO2 ; ZnO nanowires and nanoplates; ZnS; ZnSe; Mn doped Zn2 SO4 ; and ZnTe. To address the second question we look at a model

812

K.W. Kolasinski

developed in Lund by Seifert and co-workers (Johansson et al., 2005, 2006; Persson et al., 2004; Jensen et al., 2004). In their model a nanowire grows from a metal particle under the following assumptions: (i) the metal particle is hemispherical, (ii) there is steady-state adatom diffusion on the substrate and nanowire sides toward the metal particle, (iii) the processes within the metal particle (diffusion) as well as at the metal-semiconductor interface (nucleation) need not be considered in detail, and (iv) the interwire separation is fairly large. Assumption (iii) limits the generality of the model because the authors note that there are clear indications of these processes being rate limiting under certain circumstances. Nonetheless, for the specific case of III/V nanowire growth, experimental growth conditions can easily be set to validate these assumptions. Their model describes a mass-transport-limited system in which deposition occurs on the substrate as well as on the nanowire walls and the metal particle. Growth is favoured at the metal/semiconductor interface, which acts as a sink, and is kinetically hindered on the substrate and nanowire walls. Transport via diffusion across the substrate and up the walls plays a vital role. The microscopic details of this model are sketchy and the explanation for why incorporation does not occur on the substrate and walls is vague, nonetheless its phenomenological description of length versus radius for various temperatures is quite good. The model also demonstrates that the catalyst particle need not necessarily be liquid. Experimental results (Persson et al., 2004) for GaAsNWs grown from Au particles at 540 ◦ C, as well as InAs and InP NWs which only grow below the eutectic melting point (Johansson et al., 2006), are consistent with growth from solid catalyst particles, or what we might call a vapour-solid-solid (VSS) process. The success of the model of Johansson et al. should not be interpreted to mean that the catalytic particle is never liquid. There is clear evidence (Harmand et al., 2005) from TEM and energy dispersive X-ray spectroscopy (EDX) that the metal particle used to grow Au-catalyzed GaAsNWs under Ga and As2 fluxes is liquid at the growth temperature of 590 ◦ C. Each of the metallic phases found in the catalytic particles at the end of the NWs corresponds to a well-defined composition in the Au–Ga phase diagram. During the growth of carbon nanotubes or fibres both liquid (Harutyunyan et al., 2005) and solid (Helveg et al., 2004) catalyst particles have been reported. These reports demonstrate that there are circumstances under which the catalyst can be liquid and others in which it can be solid and the nanowires that result do not appear to be materially affected. In other words, while it is important for characterizing the growth mechanism, as far as the dynamics of nanowire and nanotube formation are concerned, whether the catalytic particle is liquid or solid is not essential for determining whether growth occurs. The nature of the growth mechanism involved in carbon nanotube growth – VLS or VSS – remains controversial. The transport of C and Ni atoms along the C/Ni catalyst interface rather than through the bulk has been reported by Helveg et al. (2004) during the growth of carbon fibres. However, the most important metals known to catalyze the growth of carbon nanotubes are Fe, Co, Ni (pure or as alloys) as well as Cr, V and Mo. All of these metals dissolve C and/or form metal carbides and most reports suggest that some form of VLS/VSS mechanism is involved (De Jong and Geus, 2000). The typical sources of C are CH4 , CO, syngas (H2 + CO), C2 H2 or C2 H4 and the reaction is run at 700–1200 K. The metal particles can either be in the form of powders or be supported on a substrate such as an oxide or Si wafer. It may well be the case that rather than corresponding to any limiting


813

case, transport of C atoms by surface diffusion, subsurface diffusion and bulk diffusion all play a role. Takagi et al. (2006) have shown that pyrolysis of ethanol on the coinage metals (Cu, Ag and Au) can also be used to catalyze CNT growth. For these three metals to work not only do they have to be clean to start with (as opposed to the metals mentioned above, which tend to self-clean under reaction conditions), they must also be smaller than 5 nm in diameter for growth to be efficient. C atoms are believed to precipitate onto the surface of the nanoparticles, resulting in the formation of a hemispherical cap with a graphitic structure as the precursor of SW-CNT growth. An important conclusion of the molecular dynamics studies of Ding et al. (2006, 2004a, 2004b) is that a thermal gradient is not required for the growth of single-walled carbon nanotubes. Their study shows that a concentration gradient is more important than a thermal gradient for the growth of SW-CNTs on small metal particles. Furthermore, SW-CNTs growth can occur in the presence of an opposing thermal gradient, i.e., the SW-CNTs grow from the hot region of the catalyst particle. While the growth of one nanowire per catalyst particle, with the particle sitting atop the growing nanowire is the best understood system, catalytic growth does not occur exclusively in this manner. Four major distinctions in the growth process must be made: root vs float growth and multiprong vs single-prong growth. The particle may either end up at the bottom (root growth) or top (float growth) of the nanowire. In multiprong growth, more than one nanowire grows from a single particle. In this case, the radius of the nanowire rw must be less than the radius of the catalytic particle rp . In single-prong growth there is a one-to-one correspondence between particles and nanowires. The usual means to exercise control over the nanowire diameter in single-prong growth is accomplished when the nanowire radius determines the nanowire diameter and rw ≈ rp . In single-prong growth, rw ≈ rp , as has been demonstrated for SiNW growth, is usually taken for granted but for other materials the catalyst particle sometimes is significantly larger and occasionally is somewhat smaller than the nanowire radius. In multiprong growth rw is not determined directly by rp but must be related to other structural factors such as the curvature of the growth interface and lattice matching between the catalytic particle and the nanowire. Chandrasekaran et al. (2006) have investigated the nucleation of uniformly sized Ge nanowires in multipronged root growth from Ga particles. They found that rw depends on the molar volume, the interfacial energy, and the ratio of solute concentration at the point of instability to the corresponding equilibrium solubility at a given temperature, T . The growth of nanowires demands that all of the growth occurs at the NW/catalyst interface. The walls of the NW must be passivated or else act as diffusion conduits upon which nucleation is suppressed and material is not incorporated. As will be discussed more thoroughly in Section 16.8.1, adsorbed H atoms passivate Si surfaces against growth from silanes. Thus one way to ensure that growth does not occur on the walls in this case is to maintain H passivation. Alternatively, it may be interesting to make core/shell structures in which one material such as Si coats nanowires of a different composition, e.g. Ge (Lauhon et al., 2002). Hence catalytic growth contains aspects of both thermodynamic control (precipitation out of the catalyst particle, nucleation and growth of the solid phase) as well as kinetic control (greater rate of dissociation on the catalyst rather than the nanowire walls, preferred incorporation into the growing lattice at the site of the particle rather than elsewhere), which lead us into the next topic.

814

K.W. Kolasinski

16.6. Growth under kinetic control 16.6.1. Thermodynamics versus dynamics Pressure and temperature can be controlled by the experimentalist to drive a system away from equilibrium and into a regime in which dynamical and kinetic parameters determine the layer morphology. Temperature exerts control over the rates of desorption and diffusion. In addition it can change sticking coefficients and surface energies. Pressure also affects the rate of adsorption and through this, changes in partial pressure can be used to control the composition of the surface. If diffusion is anisotropic and the temperature is low enough for islands to nucleate, highly nonequilibrium structures can arise. Under these conditions and, in particular, if the amount of material deposited is large, the size and shape of the islands formed is determined by the interplay between the kinetics of adsorption and diffusion. Pyramidal islands are frequently the predominant feature of a generally rough surface profile. Phenomenological models and scaling relations have been applied to describe the morphologies produced (Tong and Williams, 1994). 16.6.2. Nucleation Nucleation of a new phase is naturally resisted by a pure system. A condensing vapour generally does not form droplets unless the vapour is supersaturated. A liquid does not solidify unless supercooled in the absence of some heterogeneous catalyst for the process. The resistance to nucleation is associated with the variation in surface energy with the size of small particles. The surface energy, or change in the Gibbs energy, is unfavourable for addition of matter to small clusters up to a critical size. After attaining the critical nucleus size, further addition of material to the cluster is energetically favourable. The formulation of this size dependence on the change in Gibbs energy is the basis of classical nucleation theory. The change in Gibbs energy upon forming a spherical cluster or droplet of radius r from N atoms (or molecules) is G = −Nμ + 4πr 2 γ .

(16.6.1)

The first term is the chemical potential change brought about by the phase transformation of N atoms into a sphere with a surface energy given by the second term. The number of atoms and the radius are related by 4π r 3 NA ρ 4π 3 (16.6.2) ¯ = r ρ, 3 M 3 where NA is the Avogadro constant, ρ the density, M the molar mass and ρ¯ the number density. Combining Eqs. (16.6.1) and (16.6.2) and differentiating with respect to either N or r yields values for the size of the critical classical nucleus N=

Nc =

32πγ 3 , 3ρ¯ 2 μ3

(16.6.3)


rc =

2γ . ρμ ¯

815

(16.6.4)

This is the smallest structure for which the probability of growth is greater than that of decay. Substitution of the critical radius into Eq. (16.6.1) gives the energy barrier to nucleation. 4πrc2 γ . (16.6.5) 3 The rate of nucleation depends exponentially on the nucleation barrier. Nonspherical shapes, such as the equilibrium crystal shape, will lead to slight modifications of the energy terms but the same principles are followed. The presence of a substrate, rather than nucleation of droplets out of the gas phase, leads to further complications. For instance, on a semiconductor surface, the need for the substrate to reconstruct under a growing island will lead to modification of the energetic requirements for nucleation, namely, the energy required to reconstruct the substrate must be added (Pala and Liu, 2005) to Eq. (16.6.1), and this may lead to a change in the size of smallest stable entity. The analogy to droplets in contact with a vapour phase of adatoms has been used to estimate the conditions under which islands will nucleate and form stable entities. This is the basis of the Burton–Cabrera–Frank (BCF) theory of surface nucleation (Burton et al., 1951), which considers a stepped surface with terraces of width l on which atoms diffuse with a mean diffusion length λ. If the atoms, which are deposited with flux J , ”reevaporate” from the step with a lifetime τs , then the equilibrium concentration of terrace and step atoms can be evaluated (Nishinaga, 2004). This analysis shows that for a given set of flux and diffusion parameters, a critical temperature exists below which nucleation will occur. Above this temperature, the atoms accumulate at the steps. This is the basis of step-flow growth, which is discussed in more detail in Section 16.6.3. The difficulty of nucleating the new phase is expressed by the positive value of max G. It is unfavourable for small clusters to grow until the critical size is surpassed. The accretion and release of atoms from small particles will lead to some distribution of small particles. Random fluctuations will take these small clusters past the critical size at which point they will then rapidly grow larger. The rate of formation of clusters with the critical size determines the rate of formation of the new phase. On a surface, this rate will be related to the rate at which atoms are deposited and the rate of surface diffusion to and from steps and between the islands. In addition to energetic factors, nucleation can be affected by structural factors. Steps and/or elastic strain interactions can lead to preferential sites for nucleation, which can have dramatic implications for structure formation (Röder et al., 1993). Numerous metal-on-metal growth systems show evidence for nucleation at preferred sites (Brune, 1998). Below we will examine in more detail how preferential nucleation can be used to influence the structure of multilayer structures in the III–V family. Transport processes as well as the critical nucleus size affect the evolution of layer morphology as material is added to the system (Brune, 1998), in particular, the evolution of the island size distribution if the system is not following step flow growth. In the nucleation regime, addition of material to the substrate leads to the formation of islands, which share the same mean size. The number of islands changes but not their size distribution. In the growth regime, the island density no longer changes but the size of the islands increases. max G =

816

K.W. Kolasinski

The mobility and coverage of atoms added to the substrate must be such that they are more likely to encounter and add to an island than to find another adatom and form a nucleus. The extent of the nucleation phase will, therefore, determine the density of islands. Because of the importance of transport processes – the interplay between deposition rate, diffusion rate and the distance between islands – the critical nucleus size can be determined not only by thermodynamics as outlined above, but by kinetic phenomena. Venables (Venables, 1973, 1987, 1994; Venables et al., 1984) developed a kinetic framework to describe critical nucleus size, cluster size and the density of clusters during growth. The single atom population density, n1 , (assuming atomic deposition) determines the critical nucleus density, ni . In turn, n1 is determined by the rate of atom impingement J and the characteristic times for evaporation, τa , nucleation, τn , and diffusion capture, τc , by stable clusters. At high coverage direct incorporation of impinging atoms into the stable clusters can also become significant. Venables has shown (Venables, 1987) that when reevaporation is negligible and taking σ0 to be the substrate atom density and β = (kB Ts )−1 , the relative coverage of stable clusters nx can be written i+2 i J nx = f (Z0 , i) 2 exp(βEi ), (16.6.6) σ0 σ0 D where f (Z0 , i) is nominally a constant that depends on the maximum cluster density Z0 and the number of atoms in the critical nucleus i. The diffusion coefficient D is given by D = D0 exp(−βEdif ).

(16.6.7)

Ei is the critical cluster binding energy and Edif the diffusion activation energy. Equation (16.6.7) demonstrates explicitly how the cluster coverage depends on kinetic parameters such as the impingement rate, which is directly proportional to the growth rate, and the temperature dependence of diffusion and detachment from the critical cluster. One consequence of the importance of transport dynamics is that the critical cluster size as well as the island density can depend on the flux of incident atoms. Furthermore, anisotropic diffusion, strain fields or anisotropic accommodation of adatoms to islands can lead to the formation of anisotropic islands such as elongated chains on Si(100). On Si(100), the critical cluster size is simply a dimer (i = 2). Lagally and co-workers have shown that elongated chains results from anisotropic accommodation (Hamers et al., 1990; Mo et al., 1989, 1990a, 1990b; Liu et al., 2001). Of course, any distribution of islands that is created using kinetic control will be subject to Ostwald ripening once the temperature is raised sufficiently to facilitate diffusion in all directions. 16.6.3. Nonequilibrium growth modes When kinetic rate equation are used to describe nucleation and growth behaviour, the expressions derived are highly useful for describing experimental data such as the size of the critical nucleus and its dependence on temperature and incident flux because most experiments are not performed at true thermodynamic equilibrium (Venables et al., 1984; Venables, 1994). When growth occurs away from equilibrium, the relative rates of the processes delineated in Fig. 16.8 control the formation of surface structures. Note that the processes


817

Fig. 16.8. The surface processes involved in film growth. Reprinted with permission from K.W. Kolasinski, Surface Science: Foundations of Catalysis and Nanoscience. © 2002 John Wiley & Sons, Chichester.

defined in Fig. 16.8 also are prevalent during growth at equilibrium; however, in that case the temperature is sufficiently high that all energetic barriers may be overcome at a substantial rate. The kinetics of five processes must be considered (Rosenfeld et al., 1997): deposition from the gas phase, terrace diffusion, accommodation to the steps, stepdown diffusion and nucleation. Diffusion across a terrace is generally more facile than transport across a step. In addition, atoms are generally bound more tightly at the bottom of a step. As a result the barrier for step-up diffusion is substantially greater than that for step-down diffusion and only step-down diffusion is kinetically significant unless the temperature is very high. The added barrier for step-down diffusion compared to terrace diffusion is know as the Ehrlich–Schwöbel barrier (Ehrlich and Hudda, 1966; Schwöbel and Shipsey, 1966) The absence of strain in homoepitaxy means that the only growth mode possible at equilibrium is layer-by-layer growth. Nonetheless, the three growth modes depicted in Fig. 16.9 have been observed, which indicates the presence of kinetic control. The first of these modes resembles layer-by-layer growth. Near equilibrium terrace diffusion is so rapid that nucleation of islands is suppressed. Instead, atoms make their way to steps and aggregate there. There is no interlayer transport and step-flow growth is observed. Step-flow growth can be used in heteroepitaxy to grow nanowires (Himpsel et al., 1999). This has been shown, for instance, in the case of GaAs/GaAlAs and can be used for any combination of a high surface energy substrate, e.g. W or Mo, with a low surface energy non-alloying adsorbate (Cu). The difference between step-flow growth and layer growth is best illustrated by their different behaviours when observed by reflection high energy electron diffraction (RHEED). In step-flow growth, all atoms are added at the step edge and the surface roughness (i.e. step density) does not change during growth. In layer growth, in contrast, the density of islands rises until they begin to coalesce and eventually form one uniform layer. Therefore, the roughness goes through a maximum when the island density is highest, decreases at the onset of coalescence, and reaches a minimum when the layer is complete. As a consequence, step-flow growth leads to no variation in the intensity of RHEED reflexes, whereas distinct oscillations in reflex intensity are observed during layer growth. The transition from 2D nucleation (layer growth) to step-flow growth was observed experimentally using RHEED intensity oscillations by Joyce and co-workers (Neave et al., 1985). They studied the homoepitaxy of GaAs by MBE as a function of Ts and the fluxes of Ga and As, JGa and JAs , respectively. The evaporator conditions can be chosen to preferentially dose either As2 or As4 . The GaAs(001) crystal was misoriented slightly

818

K.W. Kolasinski

Fig. 16.9. Three classes of nonequilibrium growth: (a) step-flow growth, (b) layer growth by island coalescence also known as 2D nucleation growth and (c) multilayer growth.

to give either a mean terrace width of l = 72 Å in the (2 × 4) reconstruction or l = 400 Å in the (1 × 3) reconstruction (the difference in reconstruction being determined by the higher Ts used in experiments on the latter crystal). The terrace diffusion length λ increases with increasing Ts , decreases with increasing JGa and is shorter with As2 than with As4 for a constant flux of atoms. Step-flow growth occurs when λ l, whereas 2D nucleation occurs for λ < l. The transition from layer growth to step-flow growth allows for the determination of λ according to these conditions. For particular fluxes of Ga and As2 , the transition temperature was found to be 853 K and 923 K for the l = 72 Å and l = 400 Å terraces, respectively. The wider terrace, of course, requires a higher Ts to induce diffusion over greater lengths. These results also allow determination of the diffusion rate parameters. Since the activation barrier for diffusion Edif of Ga is 1.31 eV (Neave et al., 1985) as compared to 0.25 eV for As (Foxon and Joyce, 1975), we can assume that diffusion of Ga atoms is determining the growth mode. Thus, the mean displacement distance for random isotropic diffusion in the one dimension across the terrace is x 2 = 2Dτ,

(16.6.6)

where D is the diffusion constant and τ is the average time of arrival of Ga atoms at a specific site σ0 , τ= (16.6.7) JGa where σ0 is the areal density of surface atoms. By varying JGa and observing the temperature at which RHEED oscillations disappear (where step-flow growth occurs and l = λ), D and Edif can be determined from D=

l 2 σ0 2JGa

(16.6.8)


and

−Edif . D = D0 exp k B Ts

819

(16.6.9)

Far from equilibrium rapid nucleation of islands characterizes growth. If interlayer transport is rapid, then islands nucleate and coalesce completing a layer before the next begins to form. This mode can be accompanied by a proliferation of domain boundaries but the growth is exclusively two-dimensional. In the absence of interlayer transport, islands nucleate atop islands. Three-dimensional multilayer growth results. Multilayer growth is associated with the limit of the activation barrier for step-down diffusion tending toward infinity. However, in the opposite limit, when this barrier tends to zero, ideal layer growth does not occur as long as there is a finite probability for island nucleation. Since this probability never vanishes, ideal layer growth is never observed. Some systems approach this behaviour initially; however, after the deposition of many layers, all nonequilibrium systems roughen and, as noted above, tend toward the formation of pyramidal structures. Comsa and co-workers (Rosenfeld et al., 1997) have introduced the following notation to quantify the distinction between layer and multilayer growth. Let θc be the critical coverage at which nucleation occurs on top of the growing islands, and let θcoal be the coverage at which islands coalesce to form a connected layer. Layer growth is then defined by θc > θcoal ⇒ layer growth

(16.6.10)

θc < θcoal ⇒ multilayer growth.

(16.6.11)

and

The critical coverage is not constant. Typical values range from 0.5 to 0.8. θc increases with increasing temperature and decreasing deposition rate, which are the general conditions required to approach equilibrium. As the system transitions from nonequilibrium to equilibrium growth, the mode will switch to step-flow before crossing into the equilibrium mode. Whether the transition is directly from multilayer growth to step-flow or whether a transitory layer growth regime intervenes depends on the value of the Ehrlich–Schwöbel barrier. In the absence of a barrier, all three growth modes will be observed as the temperature is decreased. However, if the barrier is finite, a direct multilayer to step-flow transition is observed.

16.7. Etching of semiconductors The removal of material from surfaces is just as important as the addition of material for creating structures. Etchants are generally categorized either as isotropic or anisotropic, that is, they are categorized based on whether the etch rate does not or does depend on the crystallography of the etching surface. Etching is also classified as dry, in which surface atoms are volatilized by chemical reaction, or wet, in which reaction products are released into a solution phase above the surface. Etching is not the reverse of growth. While growth is often observed and carried out at equilibrium (or at least at pressures for which the chemical potential is not

820

K.W. Kolasinski

strongly perturbed from equilibrium), the reverse of equilibrium homoepitaxial growth is simple evaporation (or dissolution in solution). Evaporation is not an etching reaction. Most commonly etching is carried out under kinetic control, sometimes in steady state but rarely at equilibrium. This distinguishes etching from dissolution of a mineral phase in contact with a saturated (or nearly saturated) solution phase, for which the reaction dynamics can be addressed in terms of growth models (Dove et al., 2005; Lasaga and Luttge, 2001). Etching modes can resemble nonequilibrium growth modes in reverse. Step flow modes are observed for both. Pits play an analogous role in etching to that played by islands in growth and, therefore, some of the same terminology can be used to describe the two processes. Adsorption of etchants, diffusion of reactants and substrate atoms and desorption of products all have roles to play in etching reactions. It is often claimed that the wet etching of semiconductors proceeds by the oxidation of the semiconductor followed by the dissolution of the oxide. This is not always the case and its widespread dissemination is derived partly from the ambiguity of the term oxidation. Take Si etching as an example. When Si is anodically etched by aqueous HF in the electropolishing regime, a surface oxide forms and is then chemically etched by the fluoride solution (Smith and Collins, 1992). However, at less positive bias, Si is still etched to form porous silicon as will be discussed in detail below. Si is oxidized in the electrochemical sense, that is its oxidation state changes from zero to a more positive value, but a surface oxide need not play a role in the etching mechanism. There are several types of etching processes. Etching can be either electrochemical (involving free charge transfer) or chemical, dry or wet. Dry etching is chemical but wet etching can be either chemical or electrochemical. Electrochemical etching can be further distinguished as either anodic, in which the sample and a counter electrode are connected to a power supply and a bias is applied to control current flow; electroless, in which no electrodes or power supply is used but instead a redox couple is formed between the sample and a species in solution; or photoelectrochemical, in which photons act to provide charge carriers. Photoelectrochemical etching can be performed either with or without the presence of a counter electrode and power supply. In the latter case, it is sometimes called contactless etching. For charge injection to occur, the solution species must have an electronic level that overlaps energetically either with the conduction band (for electron injection) or with the valence band (for hole injection). Photoelectrochemical etching in the absence of surface states is possible only for irradiation with above band gap radiation to create an electron–hole pair. In almost all cases, semiconductors exhibit band bending in the selvage and this will cause charge separation under illumination. For instance, in n-type Si, the bands are bent upward such that holes are forced to the surface, whereas for p-type Si, the bands bend downward and electrons are driven to the surface. Thus control of the doping type allows for unequivocal identification of the charge species that is responsible for initiating etching or other reactions at the surface of a semiconductor. 16.7.1. Etch morphologies Etching leads to one of three general classes of substrate morphology: flat surfaces, rough surfaces or porous films. If one starts with a rough surface and material is preferentially removed from asperities compared to flat regions, the surface becomes progressive flatter


821

as it is etched. This is the basic mechanism of electropolishing in which, for example, electrical field enhancement at asperities increases the rate of an electrochemical reaction and leads to smoother surfaces. To achieve an atomically flat surface, we similarly need to consider the relative rates of etching at different sites on the surface and how these affect the morphology. Consider a system consisting of a real single-crystal surface, that is, one with terraces separated by one-atom-high steps that are randomly spaced across the surface and which have kinks sites randomly distributed along the step. The steps are randomly shaped and there may also be a random array of pits on the terraces. Now consider the relative rate of removal of a step atom compared to a terrace atom. The removal of a terrace atom corresponds to the nucleation of an etch pit. The formation of a pit also transforms the atoms that ring the pit from terrace atoms to step atoms. If the removal rate is different for step and terrace atoms, then etch pit formation also changes the etch rate for these atoms. If the rate of terrace atom etching (etch pit nucleation) is small compared to the step atom etch rate, the surface etches by zipping off the steps and the morphology of the terraces is unaffected. Atomically flat terraces separated by steps result from a mechanism that is analogous to step-flow growth. The terrace atom etch rate does not need to be zero, simply low enough that the rate of pit nucleation is low compared to the rate at which an entire terrace is removed by step flow. Any pits that initially populated the surface will be removed as they expand until they encounter and merge with receding steps. Importantly, since terrace atom removal is slow, it is extremely unlikely that a pit is nucleated within a pit. Therefore, no new steps are created, which would lead to a rougher surface. If the kink atom etch rate is greater than that of normal step atoms, kink sites are preferentially removed and the step becomes straight. If the kink atoms are removed more slowly than step atoms, the steps are roughened. If by some mechanism the steps repel one another – for instance, by a dipole interaction resulting from Smoluchowski smoothing – the etch rate will slow as the steps approach each other and the steps will tend toward regular spacing. On the other hand, if they tend to attract each other – for instance, if a strain field near the step accelerates the etching – then the steps will tend to bunch. Step bunching instabilities driven by elasticity have been reviewed by Müller and Saúl (2004). Step bunching can lead to an increase in the step height, a reduction in the number of steps and an increase in the mean terrace width. If the steps do not merge, then step bunching leads to packs of closely spaced steps (short terraces) separated by broad atomically flat regions. It is also important to note that the etchant, particularly for etching in solution, provides a means of communication and coupling between different regions of the crystal. If inhomogeneities in concentration or temperature arise in the etchant, they can couple to the surface reactions and lead to step bunching, as shown by Hines and co-workers (Garcia et al., 2004) for KOH etching of Si(100). Couplings in reaction-diffusion systems can lead to autocatalytic phenomena that affect the rate and nature of pitting corrosion (Punckt et al., 2004). Such couplings have also been shown to lead to spatiotemporal pattern formation and rate oscillations in numerous heterogeneous reactions (Rotermund et al., 1990; Imbihl and Ertl, 1995; Flätgen et al., 1995; Ertl, 1998; Rotermund, 1997; Rotermund et al., 1995; Wolff et al., 2001). In the opposite extreme, terrace atoms etch more rapidly than step atoms. Now pits form readily inside of pits. The terraces become pocked with pits and the pits increase in

822

K.W. Kolasinski

depth until the surface is composed of nothing but steps. The surface roughens and forms morphologies much like what might be observed from multilayer growth. If the steps sites have zero etch rate, perhaps because they are passivated by selective adsorption, and only the terrace atoms etch, a rough surface profile composed of pyramidal structures is the ultimate morphology of the surface. A middling case is difficult to concoct but is observed for the important and useful case of short-term oxygen etching of graphite (Chang and Bard, 1991). Graphite presents a surface with wide flat terraces populated by a few random vacancy defects. The defects etch significantly faster than terrace atoms. Thus, short duration etching in O2 or air leads to widening of the defects into circular etch pits. Since the rate of nucleation of new pits is low, the pits tend to have mostly single atom high walls and the pit density is determined by the initial defect density. Such pits, which can range from 50 to 5000 Å in diameter, have been used by Beebe and co-workers (Patrick et al., 1994) as molecular corrals to study, for instance, the dynamics of ordering in the formation of self-assembled monolayers. 16.7.2. Porous solid formation Let us now investigate the transition from a rough surface to the formation of a porous solid. Electrochemical etching has been used to form porous semiconductors from, e.g., Si, Ge, SiC, GaAs, InP, GaP and GaN (Langa et al., 2005) as well as Ta2 O5 (Sieber and Schmuki, 2005), TiO2 (Beranek et al., 2003), WO3 (Tsuchiya et al., 2005a) and ZrO2 (Tsuchiya et al., 2005b). We leave behind for the moment an atomistic description and consider a more coarse-grained approach. Three rates become important: pit nucleation, pit wall etching and pit bottom etching. If pit nucleation and pit wall etching are both rapid and faster than pit bottom etching, pits merge frequently and more rapidly than they grow in depth. If pit wall etching is much faster than pit nucleation, flat surfaces result (step flow etching). If pit bottom etching is much more rapid than wall etching, a rough surface results if new pits are continually nucleated at the tops of the pit walls. However, if the pits do not move, that is, if nucleation can be arrested after the first wave of nucleation events, then stable pore growth results. An extreme example of this is shown in Fig. 16.10. Lehmann and co-workers (Müller et al., 2000; Leonard et al., 2000, 1999; Lehmann and Rönnebeck, 1999; Lehmann and Grünning, 1997; Ottow et al., 1996; Lehmann, 1993) used lithography and anisotropic wet etching of Si in KOH to define nucleation sites to initiate etching. The etching was performed photoelectrochemically in acidic fluoride solution. Both etching processes are highly anisotropic for much different reasons. The etch rate of Si in aqueous KOH, as detailed in Section 16.8.5, is highly dependent on the exposed crystallographic plane and is used to create an inverted pyramidal tip. Etching in fluoride, viz Section 16.8.4, is enhanced at the pyramidal tips because electric field lines preferentially direct holes generated by photon absorption at the back of the Si wafer to the tips. The result is that the pit bottom etches rapidly while the pit walls are inert. In this fashion a regular array of rectangular macropores is created that can extend hundreds of µm deep, indeed, through the entire depth of the wafer. The lithographically produced array of nucleation sites is not required for the formation of straight walled macropores; however, it does lead to the most regular arrays.


823

Fig. 16.10. Surface, cross section, and a 45◦ bevel of an n-type silicon sample showing a predetermined pattern of macropores. Pore growth was induced by a regular pattern of etch pits produced by standard photolithography as shown in the inset. Reprinted with permission from S. Ottow, V. Lehmann and H. Föll, Journal of the Electrochemical Society 143 (1996) 385. © 1996 Electrochemical Society.

The above case is an extreme example of pore formation. The formation of a porous solid usually results from a combination of nucleation, growth, branching and coalescence of pores (Smith and Collins, 1992; John and Singh, 1995; Smith et al., 1988; Cullis et al., 1997; Parkhutik, 1999). An in depth review of, in particular, macropore formation in silicon has been given by (Föll et al., 2002) and the reader is referred to this article for a more thorough discussion of pore formation models. Passivation, either chemical or as we have discussed above and in more detail in Section 16.8.4 electrical passivation, must play an important role because if pore wall etching is not somehow arrested, the pores will disappear. The relative rates determine not only whether a porous solid forms, but also the pore morphology and pore size distribution. Porous solids are classified according to their mean pore size. The International Union of Pure and Applied Chemistry (IUPAC) recommendations define samples with a mean pore size 100 nm as macroporous. The term nanoporous is currently in vogue but undefined.

824

K.W. Kolasinski

16.7.3. Anisotropic etching Pore formation obviously results from a form of anisotropic etching. Isotropic etching can only lead to flat or rough surfaces. In anisotropic etching, the initial shape of the surface being etched determines whether fast or slow etching planes are survivors. A convex object, e.g. a sphere, etches to reveal fast etching planes. Conversely, a concave object such as a hollow, etches to reveal the slow etching planes (Wind and Hines, 2000). Any factor that affects the rate of reaction can affect the degree of anisotropy of an etchant, including temperature, concentration, solvent, surfactant addition, addition of an electrochemically active species or applying a bias to the surface. There are two ways in which anisotropy appears in etching. One is that when comparing one crystallographic plane to another, there are differences in etch rates. The second is that within a single plane, different sites can etch at different rates. Thus, the surface morphology that results from etching depends not only on the crystallographic orientation of the initial surface but also on the relative etch rates of kinks versus steps versus terraces. Masking or patterning of the substrate prior to etching will also play a role. While it is impossible to predict a priori what morphologies will result from an arbitrary combination of rate parameters, simple rule-based simulations can be run, for instance, based on a kinetic Monte Carlo scheme to predict expected steady-state morphologies (Hines, 2003) and to extract relative rates. 16.7.4. Ablation Ablation is a general approach to removing material by photon or particle irradiation. Here I will discuss this topic in terms of laser ablation, primarily of silicon, since laser ablation in general will be treated elsewhere in this volume. For ablation to occur the laser fluence (pulse energy per unit area) must be high enough for the absorbed energy to exceed the binding energy of the solid. Sufficiently high laser powers melt the solid, heating it to the boiling point and beyond to the critical temperature until an inhomogeneous phase of vapour bubbles and liquid is formed. In this high fluence regime, the material is superheated but remains solid until regions of liquid or gas nucleate. The liquid and/or gas phase expands into the material. Explosive evaporation ensues in a process known as ablation. Just above threshold, disordering takes tens of ps consistent with a thermal melting process and ablation is well characterized by this quasi-thermal process. Nonetheless, ablation will always encompass some degree of both photolytic and photothermal processes. Particularly for molecular solids and polymers irradiated in the deep UV, photochemistry may play a role in ablation (Anisimov and Luk’yanchuk, 2002). When ultrafast lasers are used with pulse widths of 1 ps or less, a large fraction of electrons can be excited to the conduction band, which is antibonding, and if a sufficient fraction of the electrons are excited (exceeding ∼10%), a highly repulsive state is achieved. The resulting forces cause non-thermal mechanisms of melting and ablation (Shank et al., 1983a, 1983b; Downer and Shank, 1986; Stampfli and Bennemann, 1995, 1994; Sokolowski-Tinten et al., 1995, 1998, 1999, 2001; Rethfeld et al., 2002; Sokolowski-Tinten et al., 2003; Sundaram and Mazur, 2002). In Si and GaAs non-thermal disordering has been observed by optical measurements at excitation fluences >1.5 times the damage threshold.


825

Ablation can occur in a single laser shot. However, it can also occur via multiple shots. Repeated laser shots lead to rapid heating, melting and resolidification, which can incorporate a growing number of defects into the lattice with each successive shot. The lattice defects correspond to the build up of energy in the lattice and make it progressively easier to ablate (Emel’yanov and Babak, 2002). The pulse width of the laser is very important in determining the processes that occur during laser ablation and the comparison of femto- to nano-second lasers is particularly illustrative (Kolasinski, 2004a). Consider the absorption of near IR, visible or UV light by a solid. The electrons absorb the radiation. They equilibrate among themselves within a couple hundred femtoseconds. The hot electrons then equilibrate with the lattice over the course of the next several picoseconds. Thermal diffusion is important during ns irradiation but does not play a role in removing energy from the irradiated region when using fs pulses. Because of these timescales, the interaction of fs pulses in the ablation regime is much different than that of ns pulses. To obtain an idea of some typical parameters, consider the following. Pedraza et al. (2001) have used the data of de Unamuno and Fogarassy (1989) to estimate the melting time tmelt and melting depth hmelt of Si exposed to 20 ns, 248 nm laser pulses as a function of the laser fluence and found tmelt (ns) = −11.79φ02 + 134.18φ0 − 53.64, hmelt (nm) =

−33.57φ02

+ 509.4φ0 − 273.3,

(16.7.1) (16.7.2)

where φ0 is the laser fluence expressed in J cm−2 . A fluence of 3 mJ cm−2 , a typical value for pillar formation, leads to a melt depth of ∼950 nm and a melt time of ∼240 ns. For ultrafast irradiation, many of the magnitudes are much different (Anisimov and Luk’yanchuk, 2002). For the same energy deposited, the peak temperature is higher for fs versus ns pulses, it is achieved sooner in time and the melt depth is slightly shorter. For a 1 ps pulse incident on a metal at 0.15 J cm−2 , the electron temperature peaks after ∼1.8 ps, the lattice temperature at 27.2 ps and ablation is completed by 1 ns. The lifetime of the melt, on the other hand, is still the same as in ns irradiation since resolidification is determined by the same materials properties in both cases. Femtosecond pulses are absorbed before the lattice has reached its peak temperature and the onset of ablation; hence, they do not interact with ablated material (the plume), which begins to form in the first 10 ps. Before 1 ns has elapsed, ablation ceases. Some molten material may still exist at the bottom of the ablation pit but most of the melted material is removed. Nanosecond pulses heat the sample comparatively slowly (even though heating rates can exceed 1010 K s−1 ). The onset of steady-state ablation is at roughly 5 ns and continues for many ns after the pulse is over; therefore, considerable irradiation of the plume occurs. This significantly heats the plume, induces photochemistry and increases the plume pressure (Claeyssens et al., 2003). The plasma formed in the plume also shadows the surface and decreases the laser power that can be coupled into the substrate. Ablation with ultrashort (1 ps) pulses compared to ns or longer pulses reveals major differences because of changes in laser/plume interactions and the lack of heat diffusion during the laser pulse in the former case. A decrease in the ablation threshold of two orders of magnitude was observed by Preuss et al. (1994) for thin films of Ni when the pulse

826

K.W. Kolasinski

length was decreased from 14 ns to 500 fs. Stuke and co-workers (Preuss et al., 1995) have developed a general formula to describe the ablation depth per pulse d under the following assumptions (i) absorption is described by the Beer–Lambert law; (ii) reflectivity is constant; (iii) ablation occurs after the laser pulse is over, hence, shadowing can be neglected; (iv) thermal conduction is negligible; (v) the laser fluence exceeds the ablation threshold; and (vi) redeposition is negligible. For a Gaussian laser beam the ablation depth is given by 1 φ0 d = ln (16.7.3) , α φth where α is the absorption coefficient, φ0 is the laser fluence at beam centre and φth is the ablation threshold fluence. The diameter of the ablated spot, D, is given by (Jandeleit et al., 1996) φ0 D 2 = 2w02 ln (16.7.4) , φth where w0 is the beam radius. The ablation threshold fluence, φth , is (Emel’yanov and Babak, 2002) φth =

U Cv , kB α(1 − R)

(16.7.5)

where U is the binding energy of the solid, Cv is the specific heat capacity, kB is the Boltzmann constant and R the optical reflection coefficient. The formation dynamics of an ablation plume is quite complex (Geohegan, 1994). Whereas etched material is often forgotten about once it has left the surface, ablation is used for transferring material from the target to a substrate as well as for texturing the target. Hence, plume interactions (plume/laser, plume/surface and plume/ambient gas) are very important in laser ablation. Once the desorption rate exceeds approximately 0.1 ML per laser shot, collisions among the ejected particles become significant. For ejection into vacuum, the collisions lead to an expansion phenomenon much like that experienced in a supersonic molecular beam expansion. For ejection into an ambient gas, a shock wave forms. Pressures in excess of 100 atm can be exerted on the surface. Furthermore, free expansion into vacuum means that redeposition out of the plume is unimportant, whereas redeposition is extremely important when an ambient gas is present to impede plume expansion. Redeposition has a measurable effect on ablation rates (Preuss et al., 1995) and can decrease them by a factor of 3 or more. The presence of gas above the surface prior to the onset of ablation influences more than just the formation of the plume (Zhigilei, 2003). Collisions can lead to the formation of clusters (Anisimov and Luk’yanchuk, 2002). Indeed, laser ablation into a buffer gas creates clusters from the ejecta very efficiently (Makimura et al., 2000). The clusters are often observed to form aerosol dispersions when ablation is carried out at pressures on the order of 100 mbar or more. These aerosol particles can scatter a significant amount of laser light. Considerable chemistry can also result from the interactions of the ambient gas, plume and surface. The presence of a reactive gas significantly increases the ablation rate. For instance, when a Si surface is irradiate in air with roughly 3 mJ cm−2 of 248 nm light


827

from an KrF excimer laser, the ablation rate is ∼24 greater than when the air is replaced by Ar (Pedraza et al., 1999a). Si removal is faster yet in the presence of SF6 . SF6 does not react with the surface but F, released either by photodissociation or the interaction of SF6 with the plume plasma, does as discussed in Section 16.8.3. The enhanced ablation can either occur because of direct etching by the fluorine to produce SiF4 or because adsorbed species such as SiF, SiF2 and SiF3 are more volatile and have a higher desorption rate than Si. Plume chemistry can also influence redeposition. Redeposition can occur by way of clusters that precipitate on the surface, by the condensation of atoms previously ejected from the surface or by the adsorption of compounds that have formed in the plume and then subsequently encountered the surface. The topology of the surface remaining after ablation depends sensitively on the power density, the number of shots, the laser pulse duration and the composition and pressure of the ambient gas (Riedel et al., 2004; Mills and Kolasinski, 2004; Kolasinski, 2004a; Crouch et al., 2004). Below the ablation threshold and particularly for exposure to multiple shots, the formation of laser induced periodic surface structures (LIPSS) is observed (Young et al., 1983, 1984; Sipe et al., 1983). These form by a variety of mechanisms and under a wide range of illumination conditions (Siegman and Fauchet, 1986). At power densities at and just below the single shot ablation threshold, columnar or conical structures often form, especially when ablation is performed in the presence of a reactive gas (Foltyn, 1994; Her et al., 1998; Pedraza et al., 1999b). This is how the pillars in Fig. 16.2(b) were formed. Ordered arrays of pillars can be produced either by the use of masks or diffractioninduced modulation of the laser intensity profile (Shen et al., 2003; Mills and Kolasinski, 2005, 2004; Riedel et al., 2004). At intensities significantly above the single shot ablation threshold and particularly for fs irradiation, ablation leads to the formation of pits with vertical walls and flat bottoms. Irradiation in this regime can be used for direct writing of pits, trenches and patterns as small as a few hundred nm across. 16.7.5. Ion irradiation effects Ion irradiation can itself remove material from the surface or catalyze the removal of material via chemical processes. The former process, in which the energy and momentum of the projectile are used to physically remove material, is known as sputtering (Smentkowski, 2000). Sputtering can be enhanced by chemical processes that make surface species more volatile or that cause ejected material to become inactive for redeposition. The introduction of defects into the surface and selvage can enhance the rate of chemical reactions that lead to etching. Sputtering, ion enhanced chemistry and chemical etching make important contributions during plasma etching and must be included in any attempt to model kinetics and structure formation (Blauw et al., 2003; Belen et al., 2005). Another important issue is that of sidewall passivation and surface chemistry (Oehrlein and Kurogi, 1998). These issues will not be dealt with here but can be found elsewhere (Chuang and Coburn, 1990; Winters et al., 1983; Winters and Coburn, 1985; Chang and Coburn, 2003; Flamm, 1990). The directionality of ion irradiation plays an important role in the formation of etched structures. If ion irradiation is performed at normal incidence through a mask then only the areas directly under the openings in the mask will be irradiated. If only this material is sputtered and removed through ion-enhanced chemical etching, then we should observe

828

K.W. Kolasinski

features with vertical walls. Two processes conspire against this ideal case. One is the angular distribution of sputtered material, which is not collimated along the surface normal. As the etched trench deepens, the sputtered material has an increasing probability of encountering the trench walls at which point it may deposit. In some cases, this may be beneficial because it can lead to the formation of a passivating layer. However, if the sputtered material has a lower sputter yield than the substrate because it has, e.g., reacted with the process gas to form a polymer or other hard coating, then the top of the trench may begin to close up. The second process is chemical etching. Chemical etching, often referred to as spontaneous etching, is usually not directional and can etch sidewalls just as easily as the bottom of the trench. This can lead to undercutting, in other words, the substrate etches in regions below closed areas of the mask. Sidewall passivation can arrest spontaneous etching and ameliorate undercut.

16.8. Case studies 16.8.1. Silicon epitaxy: Silane/Si The epitaxial growth of Si from silane (SiH4 ) and disilane (Si2 H6 ) is closely related to the surface chemistry of hydrogen on Si (Jasinski et al., 1987; Jasinski and Gates, 1991; Ji and Shen, 2004; Dippel et al., 1997; Rauscher, 2001). There is also a close correspondence between the chemistry of silane and germane on Si surfaces (Turner et al., 1995; Zhang et al., 1994; Wintterlin and Avouris, 1994, 1993; Suemitsu et al., 1994; Mahajan et al., 1994; Dutartre et al., 1994; Klug et al., 1993; Isobe et al., 1993; Eres and Sharp, 1993b, 1993a), which is of interest for the formation of strained Si and SiGe superlattice structures via CVD (Mooney, 1996; Paul, 2004; Rosei, 2004). The growth of hydrogenated amorphous silicon (a-Si:H) and hydrogenated microcrystalline silicon (μc-Si:H) thin films from SiH4 or SiH4 /H2 plasmas (Matsuda, 2004) also share much surface chemistry in common with epitaxial Si growth. SiH2 Cl2 is also often used in Si CVD and doping is accomplished with addition of, e.g., AsH3 , PH3 or B2 H6 . An understanding of the surface chemistry of H/Si will also help us to understand better the chemical processes involved in the etching of Si by halogens. Deposition of thin films with a column/void structure has been achieved with a microwave plasma struck in SiH4 diluted in H2 (Kalkan et al., 2005). The porosity can be controlled by adding SiF4 to the gas mix. These films can be deposited at temperatures as low as 375 K on virtually any type of substrate and are being investigated for use in biomedical, bioanalytical and sensor applications. The chemisorption of atomic hydrogen on and desorption of molecular hydrogen from Si surfaces have been studied extensively using numerous surface science techniques (Chabal et al., 1993; Boland, 1993; Kolasinski, 1995; Doren, 1996; Höfer, 1996; Waltenburg and Yates, 1995; Kolasinski, 2004b; Raschke and Höfer, 2001; Dürr et al., 2002b; Dürr and Höfer, 2004; Yilmaz et al., 2004; Namiki, 2006; Dürr and Höfer, 2006). For Si(100)-(2 × 1), adsorption of a single H atom onto a Si dimer breaks the interaction of one set of dangling bonds that leads to dimer formation and creates a lone dangling bond on the Si atom not involved in the Si–H bond (Boland, 1991a). There are two alternatives to describe the bonding in the clean dimer (Kolasinski, 1995). A π-bonding


829

interaction would lead to a symmetric dimer. A Peierls distortion leads to an asymmetric dimer and a transfer of charge from the down to the up atom in the dimer (Ancilotto et al., 1990). Since the preferred structure of the dimer is buckled (Wolkow, 1992), the Peierls distortion rather than the π bond is the major source of stabilization in the clean dimer. However, the buckling motion of the dimer is a low frequency vibration that rapidly converts the dimer from a tilt in one direction through the symmetric dimer to a tilt in the other direction (Dabrowski ˛ et al., 1994; Dabrowski ˛ and Scheffler, 1992). This makes the time-averaged dimer appear symmetric at room temperature and leads to a natural coupling between vibrations and electronic structure. Saturation of the monohydride phase, that is, adsorption of two H atoms per dimer, stabilizes the symmetric dimer and relieves some of the subsurface strain associated with the reconstruction (Craig and Smith, 1990; Ciraci et al., 1984). The chemisorption of molecular hydrogen continues to be a source of great controversy (Kolasinski et al., 1992, 1994a, 1994b; Shane et al., 1992b; Shane et al., 1992a; Brenig et al., 2003; Brenig and Hilf, 2001; Zimmermann and Pan, 2000; Yilmaz et al., 2004; Shi et al., 2005; Matsuno et al., 2005; Sagara et al., 2002; Namiki, 2006; Dürr and Höfer, 2006). The sticking coefficient at room temperature is below 10−9 (Bratu and Höfer, 1995) and depends strongly on the temperature of the surface (Kolasinski et al., 1994a; Bratu and Höfer, 1995; Bratu et al., 1996; Dürr et al., 1999). Vibrational and translational excitation of the incident molecule facilitate adsorption (Dürr and Höfer, 2004), though rotational excitation hinders it (Kolasinski et al., 1992; Shane et al., 1992b, 1992a). The strong dependence of the sticking coefficient on Ts reveals an important aspect of the potential energy surface (PES) that governs the interaction of an adsorbate with the surface of a covalent solid: the PES depends strongly on displacements of the surface atoms. Relaxations and vibrations of surface atoms must be taken into account to describe correctly the energetics of the interaction of hydrogen with Si. This means that not only the barrier to adsorption is strongly influenced by surface atom motion and relaxation but also that the diffusion barrier for the movement of H atoms on the surface is strongly influenced by surface atom motion and relaxation (Vittadini et al., 1993; Nachtigall et al., 1994). This is a general feature of semiconductor surfaces and we shall see in Section 16.8.2 that the response of In atom diffusion to a strain field changes the growth dynamics of InAs islands on a GaAs substrate. More important for Si CVD from silanes is the kinetics of hydrogen desorption. Frequently it is stated that desorption from the monohydride phase on the Si(100)-(2 × 1) surface follows a first-order rate law (Sinniah et al., 1989), while desorption from Si(111)-(7× 7) follows a second-order rate law (Koehler et al., 1988). However in both cases, because of the ability of H2 to desorb through multiple pathways distinguished by the configuration of the H and Si atoms prior to desorption, the order of desorption is coverage dependent. Desorption from Si(111)-(7 × 7) at low coverage is described by kinetics that are intermediate between first- and second-order, which has been explained by a model involving multiple binding and desorption sites (Reider et al., 1991). The deviations from pure first- or second-order kinetics for desorption from Si(100)-(2 × 1) arise from second-order desorption via at least two separate channels as well as the influence of lateral interactions in the chemisorbed layer that lead to adatom pairing and clustering (Zimmermann and Pan, 2000; Brenig et al., 2003).

830

K.W. Kolasinski

Fig. 16.11. Surface atom configurations associated with the desorption pathways of H2 from Si(100)-(2 × 1). Filled circles (") correspond to adsorbed H atoms. Open circles (!) correspond to bare Si surface atoms within dimer units.

The near first-order desorption kinetics on the Si(100)-(2 × 1) surface is related to the interplay of the desorption dynamics with surface structure. Wise et al. (1991) suggested a pairing model in which recombination occurs across a Si–Si dimer pair (Wise et al., 1991). D’Evelyn et al. (1992) reported a kinetic model that incorporated the energetic stabilization of the π-bonding interaction. They showed that π-bonding (or equivalently the Peierls distortion) leads to pairing prior to desorption and that first-order desorption kinetics ensue. Pairing in the adsorbed phase was observed by Boland using STM (Boland, 1991a). The kinetics of desorption at high surface coverage are complicated by the presence of at least two distinct pathways to desorption (multiple inter- and intra-dimer mechanisms), which are also sensitive to the configuration of neighbouring H atoms (Zimmermann and Pan, 2000; Yilmaz et al., 2004; Brenig et al., 2003; Dürr and Höfer, 2004; Dürr et al., 2002b; Dürr et al., 2002a). It now appears to be the case that desorption of molecular hydrogen from Si(100)-(2×1) is dominated by interdimer desorption via the H2 and H4 pathways, cf. Fig. 16.11, whereas the intradimer H2* pathway envisioned in the original pairing models is of minor importance. The formation of mono-, di-, and tri-hydride silicon species follows the adsorption of either silanes or atomic hydrogen. Each of these species has clearly identifiable spectral features in the IR (Chabal et al., 1993). Although the source of adsorbed hydrogen has little effect upon the properties of these hydrides (Jansson and Uram, 1989; Uram and Jansson, 1989, 1991; Bozso and Avouris, 1988; Wu and Nix, 1994), the stabilities of the three surface hydrides depend on Ts and coverage. At an elevated Ts of ∼400 K during adsorption, formation of higher hydrides is suppressed and ensures that the Si surface is predominantly covered with the monohydride (Uram and Jansson, 1991). Such elevated temperatures and suppression of higher hydride formation is also important for the suppression of etching during H atom exposure (Olander et al., 1987; Cheng and Yates, 1991; Gates et al., 1989b). At saturation coverage all of the surface dangling bonds are capped with adsorbed hydrogen, H(a), which is important for CVD in that dissociation of silanes occurs with an appreciable rate only in the presence of dangling bonds. Therefore, the rate of dissociation can be limited by the rate at which empty sites are created by H2 desorption. This self-limiting nature of silanes adsorption can be used as the basis of ALE schemes (Lubben et al., 1991; Suda et al., 1989; Tanaka et al., 1990; Hirva and Pakkanen, 1989; Sakuraba et al., 1993; Akazawa and Utsumi, 1995). Homoepitaxy of Si requires some mobility of adsorbing Si subsequent to dissociative adsorption so that the deposited material may reach the proper lattice sites. On the (111)


831

surface, epitaxy has been reported (Kulkarni et al., 1990) at Ts = 773 K using disilane as the source gas. Because the surface was continually dosed with disilane from a molecular beam, the temperature required for epitaxy was determined not by the mobility of Si but by the requirement that the desorption rate of H2 exceeded the delivery rate of adsorbing H atoms. Growth of an ordered Si:H(1 × 1) phase at Ts as low as 673 K indicates that either H(a) or SiHx units must be mobile at this temperature. On the (100) surface (Gossmann and Feldman, 1985), MBE at Ts as low as 570 K has resulted in epitaxial growth and STM has confirmed diffusion of adsorbed Si at Ts = 520 K (Mo et al., 1990a; Mo et al., 1990b) and 580 K (Hamers et al., 1990). Mobility of Si on Si(100) has been demonstrated even at room temperature (Mo et al., 1990a) though this is most likely the result of mass transport through a precursor state. A decrease in the inhomogeneous linewidth of the Si–H stretching mode upon annealing or adsorption of H atoms at elevated temperatures lead Chabal (Chabal et al., 1983) to conclude rearrangement of monohydride units into more ordered structures can take place below the hydrogen desorption temperature. Gas source MBE using Si2 H6 proceeds by step flow growth as expected (Mokler et al., 1992a, 1992b). However, for Ts < 870 K, a transition to three-dimensional growth was observed, presumably because the deposition rate exceeded the rate at which step-down diffusion could smooth the resulting layers. Step flow growth following SiH4 dissociation on Si(111) has been observed at 730 K (Masson and Thibaudau, 2005). (1×1) islands form in the first monolayer above the originally (7 × 7) reconstructed substrate; however, once the first monolayer fills in, growth proceeds by step flow on the (1 × 1) reconstructed surface. STM studies (Boland, 1991c; Lin et al., 1992) confirmed the expectation that either H(a), Si(a), SiH(a) or SiH2 (a) units are mobile following Si2 H6 adsorption well below the monohydride desorption temperature. Boland found ordered, monohydride-capped Si(100)-(2 × 1) surfaces after briefly annealing a Si2 H6 saturated Si(100)-(2 × 1) surfaces to 670 K. Hamers and co-workers (Wang et al., 1994) dispute his assignment. Instead, they find that STM images after annealing to 570 K are consistent with islands of bare Si adatoms and phase segregation of adsorbed H atoms onto other regions of the substrate. The formation of epitaxial islands of Si results from diffusion and nucleation of Si atoms on a partially H-covered surface. The diffusion of H atoms is responsible for the formation of extended areas covered with monohydrides. These temperatures are one hundred degrees or more below the monohydride desorption temperature. Therefore, because of the reordering of the monohydride-covered surface and the tendency of the Si(100) surface to form dimers even in the presence of defects (Hamers et al., 1986), the majority of the adsorption sites during growth should correspond to the (2 × 1) dimer reconstruction, even if the surface has some degree of disorder and H atoms adsorbed in sites other than ideal Si dimer units. Theoretical studies (Rockett and Barnett, 1988; Srivastava et al., 1991) also suggest that the (2 × 1) reconstruction is maintained at the growing vacuum–solid interface. We now turn to the dynamics of the decomposition of silanes on Si surfaces. Two cautionary notes on such studies must be made. The first is that since Si2 H6 is roughly 104 more reactive when room temperature gas is used, impurities of Si2 H6 in SiH4 can have a dramatic impact on the apparent reactivity of SiH4 . The second is that SiH3 and SiH2 are unstable in the presence of dangling bonds and, therefore, the initial products of decompo-

832

K.W. Kolasinski

sition can only be isolated at low temperature, furthermore, the hydride distribution on the surface is sensitive both to Ts and the hydrogen coverage. There is a major difference in the adsorption and, therefore, growth characteristics of SiH4 compared to Si2 H6 because of the need to activate an Si–H bond in the former rather than the weaker Si–Si bond in the latter. When silane is adsorbed at cryogenic temperatures, the physisorbed molecule interacts very weakly with the surface and SiH4 merely desorbs intact. The dissociative chemisorption of SiH4 is a direct activated process with a moderate activation barrier. Gates (Gates et al., 1989a, 1990b) reported a value of 13.8 kJ mol−1 obtained from an Arrhenius analysis of s0 vs Ts data; however, this is not a true activation energy since the 300 K gas was not in equilibrium with the surface. Kavulak et al. (2005) reported a reaction threshold energy of 19 kJ mol−1 (0.20 eV), which according to a microcanonical unimolecular rate theory analysis of various sticking data should correspond to the true zero Kelvin barrier height on the potential energy surface. This moderate activation barrier is consistent with the observed weak surface temperature dependence of s0 when adsorption proceeds on the clean surface. s0 is a strong function of Ts in the region dominated by H2 desorption but at high Ts , i.e. on clean surfaces, s0 only increases weakly with Ts . This is true for both Si(100) and Si(111). On Si(111), however, there is a large increase in s0 above 800 ◦ C associated with the (7 × 7) to (1 × 1) reconstruction (Jones et al., 1994). Kavulak et al. favour the reaction of SiH4 with two dangling bond sites located on the same dimer on the Si(100)-(2 × 1) surface (intradimer mechanism). Attempts by ab initio computational methods have delivered scattered estimates of the barrier height on the (100) surface: 59.4–74.9 kJ mol−1 with density functional theory (DFT) methods at the BLYP level of theory for intradimer dissociation on a Si9 H12 cluster (Brown and Doren, 1999); 37.7 kJ mol−1 for interdimer dissociation across dimers in adjacent rows with configuration interaction (CI) calculations on a Si19 H21 cluster (Jing and Whitten, 1991); 69.5 kJ mol−1 using DFT with a generalized gradient approximation (GGA) on a periodic slab for an interdimer mechanism (Lin and Kuo, 2000; Lin et al., 2000); and 31 kJ mol−1 for intradimer adsorption or 59.8 kJ mol−1 for intradimer adsorption using DFT at the B3LYP level but only 18.8 kJ mol−1 with a hybrid functional and clusters as large as Si37 H36 (Kang and Musgrave, 2001a, 2001b). Smardon and Srivastava (2005) applied DFT within the GGA using norm-conserving pseudopotential to treat the electron–ion interactions. They used a periodic slab geometry to represent a Si(100)-(2×2) surface. A paired configuration of the initial dissociation products, that is, adsorbing SiH3 and H on the same dimer, is found to be 0.18 eV more stable than the adsorption of SiH3 and H on the same ends of two adjacent dimers. This compares to a pairing energy of 0.40 eV found by Brown and Doren (1999). The lowest barrier pathway to this configuration also has the lowest overall barrier height of 77.2 kJ mol−1 . This pathway begins in between two dimers, rather than an approach that begins directly above the dimer as found by others (Jing and Whitten, 1991; Brown and Doren, 1999; Kang and Musgrave, 2001a, 2001b). The continuing significant disagreement between different computational methods demonstrates that much still needs to be learned before we will be able to definitively identify the lowest energy dissociation pathway and its characteristics from ab initio methods.


833

The sticking coefficient of SiH4 depends nearly exponentially on the incident translational energy Ei (Jones et al., 1994; Xia et al., 1995b; Mullins et al., 1997). However, the angular dependence of s0 is complicated by the highly corrugated nature of reconstructed Si surfaces. The sticking coefficient is related to normal energy scaling to the local surface normal. The analysis of Engstrom and co-workers (Jones et al., 1994; Xia and Engstrom, 1994) demonstrates that in the absence of shadowing, a single “universal” scaling function exists s0 ∝ Ei Θ(ϑi ), which is given by the expression Θ(ϑi ) = (1 − ) cos2 ϑi + 3 sin2 ϑi ,

(16.8.1)

where ϑi is the angle of incidence, is a corrugation parameter (0 1) and only inplane corrugation has been considered. The analysis neglects mechanical energy transfer to the lattice (recoil) to the lattice and the effects of the attractive well. It also assumes that the corrugation is incident energy independent and that reactivity is uniform across the unit cell, i.e. no active sites. For SiH4 , they found that = 0.20 ± 0.04 on Si(100) at Ts = 900 ◦ C; while for Si(111) at Ts = 700 ◦ C and 900 ◦ C, = 0.20 ± 0.04 and 0.16 ± 0.05, respectively. In light of the strong role played by corrugation in the dissociative adsorption of silanes, it would be interesting to apply this analysis to the angular dependence of H2 dissociation on Si surfaces to determine whether corrugation is similarly important. Engstrom and co-workers (Jones et al., 1994) measured only a small kinetic isotope effect on Si(100). At Ts = 900 ◦ C and 0.5 EK 0.75 eV, s0 for SiH4 is 1.1–1.6 times greater than s0 for SiD4 . They conclude that this is not consistent with a strong influence of tunneling; however, it should also be noted that under these conditions of translational energies far in excess of the barrier height (E0 ≈ 0.2 eV) and high Ts , it would seem unusual for tunneling to play a strong role in the dissociation dynamics. Even though the primary mechanism for dissociation of SiH4 is direct activated dissociative chemisorption involving the breaking of a Si–H bond, nonetheless, the mechanism of dissociation appears to be complex. Dissociation on clean Si(100)-(2 × 1) follows the expected course SiH4 (g) + 2db → SiH3 (a) + H(a),

(16.8.2)

which requires two empty dangling bond (db) sites. All of the above mentioned ab initio studies agree with this reaction, although they disagree on whether adsorption occurs via an intradimer or interdimer mechanism. The agreement between a recent combination of experiment (Kavulak et al., 2005) and theory (Kang and Musgrave, 2001a, 2001b) favours the intradimer mechanism in which the SiH3 (a) and H(a) are adsorbed on the same dimer unit (analogous to H2* in Fig. 16.11). On clean Si(111)-(7 × 7), the lack of significant SiH3 coverage lead to the proposal (Farnaam and Olander, 1984; Gates et al., 1990b) of the following dissociative pathway SiH4 (g) + 2db → 2SiH2 (a).

(16.8.3)

Surprisingly, the dissociative sticking coefficient of SiH4 on Si(100)-(2 × 1) and Si(111)(7 × 7) at Ts = 673 K is 3 × 10−5 for both surfaces (Gates et al., 1990b). The value for Si(111)-(7 × 7) is supported by direct STM measurements (Albertini et al., 1996). This equality of magnitudes as well as SiH being the largest peak in the SIMS data (Gates et

834

K.W. Kolasinski

al., 1990b) for adsorption on Si(111)-(7 × 7), leads this author to suggest that the initial step in the reaction in the dissociation of SiH4 on both Si(100)-(2 × 1) and Si(111)-(7 × 7) is the breaking of a Si–H bond and the formation of SiH3 (a) and H(a) as in Rxn. (16.8.2). However, on Si(111)-(7 × 7), the SiH3 (a) is not stable at the site at which dissociation occurs and immediately fragments into lower hydrides (SiH2 (a) and SiH(a)). On both Si(100)-(2×1) and Si(111)-(7×7), the sticking coefficient of SiH4 is a complex function of H coverage (Gates et al., 1990b) and does not follow simple Langmuirian kinetics. Adsorbed H atoms block sites and suppress SiH4 adsorption. Gates reported that there appears to be a small number of sites on both surfaces with significantly higher reactivity, which is consistent with the observation that steps and domain boundaries may provide the required defects on Si(111)-(7×7) (Memmert et al., 1995). Different sites within the (7×7) unit cell appear to have different reactivities with either the corner adatom (Memmert et al., 1995) or the centre adatom (Albertini et al., 1996) exhibiting the highest reactivity. In order to increase the dissociation probability of SiH4 to useful levels, the surface and/or gas temperature must be raised to considerable levels. Because high temperatures above Ts = 770 K are generally used to dissociate SiH4 , H2 desorption is rapid, db sites are available, and unless the SiH4 flux is extraordinarily high, the deposition rate is proportional to the SiH4 sticking coefficient on the clean surface. Below this temperature, H2 desorption, i.e. the creation of empty sites, represents the rate limiting event (Gates and Kulkarni, 1991; Liehr et al., 1990). High temperatures during growth are not only important for the removal of H(a) to free up adsorption sites. The maintenance of a low H atom coverage is also important to ensure high crystallinity in the homoepitaxial layer (Horn-von Hoegen and Golla, 1995). The presence of adsorbed H atoms leads to differences in the nucleation behaviour between MBE and CVD growth of Si homoepitaxial layers (Andersohn et al., 1996; Köhler et al., 1997). Whereas classical nucleation theory with a critical nucleus of 5–7 describes the island density versus growth rate data for MBE growth on Si(111) at 773 K, similar disilane CVD data is not consistent with the predictions of classical nucleation theory. Adsorbed H atoms hinder the diffusion of Si adatoms and stabilize (1 × 1) islands, which have a different nucleation behaviour than (7 × 7) regions. During SiH4 CVD at 730 K, the nucleation rate is 2.5 times higher on (1 × 1) islands than on (7 × 7) regions. Adatoms are more easily incorporated into the (1 × 1) islands because there is no need to unreconstruct the island (Masson and Thibaudau, 2005). Si2 H6 interacts more strongly with a clean Si surface than SiH4 and readily forms an undissociated physisorbed molecule. The dissociation dynamics of Si2 H6 are dependent upon how the Si2 H6 is dosed. Whereas direct dissociative adsorption is observed at translational energies in excess of 1 eV (Engstrom et al., 1993a), under normal growth conditions, disilane is dosed from a thermal (room temperature) source and dissociates via precursor mediated dynamics (Gates, 1988; Wu and Nix, 1994). The physisorbed molecule dissociates upon heating to 170 K or above as long as free sites are available (Imbihl et al., 1989). Consistent with precursor mediated adsorption dynamics, the dissociation probability decreases with increasing Ts for moderate Ts . However, the small negative apparent activation energy (Gates, 1988; Imbihl et al., 1989) (−2.6 kcal mol−1 ) associated with precursor mediated dynamics appears vida infra to have little effect on the growth kinetics. STM measurements (Wang et al., 1994) indicate that SiH3 is the primary initial decomposition product and that steps do not play an important role in dissociation of Si2 H6 .


835

The lack of a kinetic isotope effect when Si2 D6 dissociative adsorption is compared to Si2 H6 also suggests that the initial dissociation step involves the Si–Si bond rather than a Si–H bond as observed for SiH4 (Gates, 1988). The primary route to dissociative adsorption for thermal Si2 H6 dosed onto surfaces at Ts below the hydrogen desorption temperature is via Si–Si bond scission (Uram and Jansson, 1989; Suda et al., 1990; Lin et al., 1992; Wu et al., 1993; Bronikowski et al., 1993; Bramblett et al., 1994; Wang et al., 1994), k1

Si2 H6 (g) → Si2 H6 (p) + 2db −→ 2SiH3 (a).

(16.8.4)

This is followed by successive hydride decomposition, Si incorporation and H2 desorption as long as Ts is sufficiently high, k2

SiH3 (a) + db −→ SiH2 (a) + H(a), k3a

2SiH2 (a) −→ 2SiH(a) + H2 (g), k3b

SiH2 (a) + db −→ 2SiH(a), k4

2SiH(a) −→ 2Si(a) + H2 (g).

(16.8.5) (16.8.6) (16.8.7) (16.8.8)

Even at room temperature, as long as there are available dangling bonds, SiH3 (a) is unstable toward dissociation (Gates et al., 1990a; Lubben et al., 1991; Wang et al., 1994). Rxn. (16.8.6) corresponds to H2 desorption through the β2 TPD feature with a peak temperature of Tpeak ≈ 680 K and Rxn (16.8.8) to desorption through the β1 feature (Tpeak ≈ 800 K). The SiH2 moieties become reactive and mobile above Ts = 500 K (Lin et al., 1992, 1993). If dangling bonds are available, diffusion of SiH2 competes with decomposition to form two monohydrides. An unimportant pathway for Si2 H6 decomposition from thermal dosing (Gates and Kulkarni, 1991; Bronikowski et al., 1993; Wang et al., 1994) involves direct SiH4 liberation Si2 H6 (g) → SiH4 (g) + SiH2 (a).

(16.8.9)

Supersonic molecular beam investigations (Xia et al., 1995a) failed to observed evidence of Rxn. (16.8.9) on the clean Si(100)-(2×1) and Si(111)-(1×1) surfaces. A silane production channel during supersonic dosing was observed on Si(111)-(7 × 7) and Si(100)-(2 × 1) in the presence of a finite coverage of either adsorbed H or P atoms. Under the condition most relevant to Si epitaxy, therefore, it is Rxn. (16.8.8) in combination with the second order nature of Si2 H6 decomposition, Rxn (16.8.4), that determines the rate of Si growth. Using this reaction scheme, Greene and co-workers (Bramblett et al., 1994) found that the growth rate as a function of Ts and disilane flux (supplied by a room temperature tubular doser) is well fitted with a Ts independent initial dissociative sticking coefficient for disilane of s0 = 0.036. The weak Ts dependence of s0 found by (Engstrom et al., 1993b) was not found to make a significant difference to the growth kinetics. Werner et al. (1994b, 1994a), using reflection high energy electron diffraction (RHEED) oscillations and a similar kinetic scheme to that above, measured the sticking coefficient of disilane, injected at Tgas = 338 K, on the Si(001) surface to be 0.16 ± 0.06 independent of Ts in the range 737–853 K. Gates found (Gates, 1988) that the reactive sticking coefficient for a

836

K.W. Kolasinski

room temperature Si2 H6 incident on Si(111)–(7×7) at Ts = 300 K is 0.47±0.1 and Si2 H6 is, therefore, approximately 104 times more reactive than SiH4 . As a result of diametrically opposed Ts dependence for SiH4 (s0 increases with increasing Ts ) and Si2 H6 (s0 decreases weakly with increasing Ts ), the difference is much less as the temperatures commonly used during growth. As mentioned above, Engstrom and co-workers (Engstrom et al., 1993a, 1993b; Xia et al., 1995a, 1995b) have observed direct dissociative adsorption dynamics for Si2 H6 incident on Si surfaces at high translational energies, which they attribute to Si–H rather than Si–Si bond activation. Direct dissociation at high incident kinetic energy, which is enhanced by increasing Ts , was also found by Mullins and co-workers (Pacheco et al., 1995) who also observed that the change in adsorption dynamics from precursor mediated to direct dissociation does not affect the quality of the film grown (Pacheco et al., 1996). Direct dissociation is observed for both CH4 and C2 H6 , for instance, on Pt(111) (DeWitt et al., 2006b, 2006a) as well as some other metal surfaces (Weaver et al., 2003); thus, it may not be so surprising that analogous direct dissociation dynamics are found (at least at high incident energy) for the low silanes and alkanes. Consistent with a change in dissociation dynamics, s0 for disilane is 0.47 for low (room temperature thermal distribution) Ei , drops precipitously with increasing Ei to reach 3 ×10−2 at 0.6 eV, then increases above 1 eV with increasing Ei . Engstrom and co-workers also suggest that SiH4 production during Si2 H6 adsorption is related to dissociation via Si–H bond activation (Xia et al., 1995a). In this scheme, Si2 H6 dissociation is not a direct process; rather, it proceeds via a chemisorbed intermediate according to Si2 H6 (g) + 2db → Si2 H5 (a) + H(a),

(16.8.10)

Si2 H5 (a) + db → SiH2 (a) + SiH3 (a),

(16.8.11)

Si2 H5 (a) + 2db → Si2 H4 (a) + H(a),

(16.8.12)

Si2 H5 (a) → SiH4 (g) + SiH(a).

(16.8.13)

Within this scheme, it then makes sense that SiH4 (g) production is favoured by high H atom coverage as this will suppress Rxns. (16.8.11) and (16.8.12) but not (16.8.13). DFT calculations (Smardon and Srivastava, 2005) do not support dissociation via Si–H bond activation on clean Si(100). Using the same formalism applied to SiH4 dissociation, Smardon and Srivastava found that such a pathway has a significantly higher barrier than the lowest barrier pathway to SiH3 formation via Si–Si bond scission. The preferred pathway followed a trajectory much like that found for SiH4 , beginning between two dimers and ending with the two SiH3 moieties adsorbed on either end of the same dimer unit. The barrier height was determined to be 1.45 eV (140 kJ mol−1 ). Note, however, that this barrier height far exceeds the value found for SiH4 by this group (0.8 eV). In the direct regime (Ei 1 eV), the sticking coefficients of SiH4 and Si2 H6 at the same translation energy are nearly equal (slightly higher for Si2 H6 ) and this is one of the bits of evidence that lead Xia et al. to conclude that Si–H bond activation might be occurring. The calculation of Smardon and Srivastava also found an activation barrier to the formation of physisorbed Si2 H6 , which is at variance with experimental results.


837

Just as for SiH4 , surface corrugation and structure strongly influence the sticking coefficient of Si2 H6 . On Si(111) and abrupt increase in s0 above ∼ 825 K seems to coincide with the loss of the (7 × 7) reconstruction and the formation of the (1 × 1) structure (Engstrom et al., 1993b). For Si2 H6 on Si(111) at Ts = 1173 K, the angular dependence of s scales as cosn ϑ with n = 0.87 ± 0.06. Again a model was proposed in which normal energy scaling involving the local surface normal is important. The two parameters of the local normal model are found to be Ts dependent on Si(111) but not on Si(100). For Si CVD from Si2 H6 on Si(100), both the growth mode and the reactive sticking coefficient are influenced by the presence of B on the surface (Wang and Hamers, 1995). Boron atoms deposited by dissociative adsorption of B2 H6 coalesce into c(4 × 4) islands. Si2 H6 does not dissociate on these islands but its dissociation on the clean portions of the Si(100)-(2 × 1) surface is little affected by the presence of B atoms. The presence of B islands also reduces the mobility of Si atoms on the surface. Thereby the growth mode of Si layers from Si2 H6 at Ts = 815 K changes from layer-by-layer on the clean surface to island growth on the B covered surface. Furthermore, at this temperature the B is able to segregate to the top of the growing layer so that they can continue to enhance the nucleation of islands as growth proceeds. When dosing is performed with a supersonic molecular beam and direct dissociative adsorption dynamics are operative, the sticking coefficients of silanes are suppressed by the presence of pre-adsorbed P atoms (Maity et al., 1995). The probability of dissociative adsorption of both SiH4 and Si2 H6 is found to be proportional to the quantity 1−θP2 , where θP is the fractional coverage of adsorbed P atoms. These results are inconsistent with a long-range electronic effect of adsorbed P on s0 . The dependence of s0 on θP for both SiH4 and Si2 H6 is consistent with a simple site blocking model in which doubly occupied Si dimers covered with P on both ends are unreactive. 16.8.2. Compound semiconductor epitaxy: III–V growth Growth of III–V compounds has been the subject of several reviews (Nishinaga, 2004; Franchi et al., 2003; Joyce and Vvedensky, 2004). In the early days of MBE growth of layers and superlattices, which built upon the pioneering work of Arthur (1968) and Foxon and Joyce (1975, 1977), the formation of 3D islands was usually seen as something to be avoided. Then it was realized that the islands act as zero-dimensional (quasi-atomic) structures in which the electronic states are confined in all three spatial dimensions. These quantum dots (QD) have electronic properties that depend on their size due to quantum confinement (Yoffe, 1993). The onset of quantum confinement occurs when both the mean free path and the de Broglie wavelength of free carriers exceed the size of the dot. The size dependence of the electronic properties translates into size-dependent optical properties, in particular, the luminescence wavelength can be tuned. The emission wavelength depends both on the characteristics of the quantum dot (shape, size, strain and composition) as well as the composition and residual strain of the layer in which it is embedded (Franchi et al., 2003). III–V materials are especially prized in this respect because they emit efficiently from the IR, which is of interest e.g. for communications, to the visible. Nitrides extend this range to the near UV. Thus they are suitable for a range of photonic devices, including

838

K.W. Kolasinski

lasers, photodetectors, amplifiers, solid state lighting and potentially in quantum computing. MBE is the bedrock technique for the formation of III–V superlattices. Solid-source MBE uses evaporation of, for example, In, Ga and As from high purity ingots held within crucibles in a so-called Knudsen cell. The molecular beam that emanates from a Knudsen cell is a thermal rather than supersonic molecular beam. Gas source MBE uses volatile reagents such as AsH3 and PH3 to deliver the Group V element. Metallorganic MBE (also called metal-organic vapour phase epitaxy MOVPE) delivers the Group III element with molecules such as Ga(CH3 )3 (trimethylgallium, TMGa), triethylgallium (TEGa) or trimethyleindium (TMIn). MBE allows for exquisite control of the composition and thickness of the deposited layers, which makes possible the growth of binary, ternary and quaternary compounds in superlattice structures of defined thickness. As we have seen above, GaAs grows by either step-flow growth or layer growth as is expected for such a strain-free homoepitaxial system. InAs has a lattice mismatch of 7% with respect to GaAs. This level of strain is too large to support layer-by-layer growth indefinitely and it follows Stranski–Krastanov growth. A strained wetting layer grows until a critical thickness of 1.6 ML is surpassed at which point 3D islands form on top of the wetting layer (Franchi et al., 2003). However, from the Vegard law relationships in Eqs. (16.2.8) and (16.2.9), we see that the misfit strain can be adjusted by controlling the composition of the overlayer. Thus, Inx Ga1−x As grown on GaAs will exhibit a layer-bylayer mode for smallxand transitions to Stranski–Krastanov growth as x increases. Similarly, other mixtures of Al, Ga and In with P, As and Sb can be grown in either lattice matched or lattice mismatched combinations depending on the stoichiometry of the layers. As2 adsorbs through a highly mobile precursor (Foxon and Joyce, 1977). This is an essential part of the growth kinetics. It allows the As2 to be held in what amounts to a physisorbed reservoir from which it can be funneled into strongly bound sites. However, as pointed out in the model of Joyce and Vvedensky (2004) described below, the As2 is not to be thought of as an opportunistic bystander. Rather, it plays an active role in the incorporation of Ga into the surface lattice. Growth of epitaxial layers on III–V materials, and GaAs in particular, is complicated by the need to incorporate at least two (and perhaps three or even four) chemically dissimilar atoms into the lattice. Furthermore, the surfaces exhibit numerous reconstructions that depend not only on Ts but also on the relative composition (Joyce and Vvedensky, 2004). The GaAs(001)-(2 × 4) surface is most commonly used in growth studies. It consists of alternate rows of As dimer pairs and missing dimer trenches. Growth on this surface follows a layer-by-layer mode and strain is relaxed by the formation of misfit dislocations. For GaAs substrates, quantum dots are only formed on the GaAs(001)-c(4 × 4) surface (Joyce and Vvedensky, 2004). The formation of the stable growth nucleus is proposed to conform to the following mechanism (Joyce and Vvedensky, 2004), which is depicted in Fig. 16.12. Ga atoms do not readily break and insert into the As dimer bond, whereas there is no barrier to adding another As dimer (the addimer) on top of a dimer pair. A pair of Ga atoms then binds to the addimer. The addimer decomposes, releasing As2 back into a mobile physisorbed state while the Ga atoms incorporate into the surface lattice, each bridging one of the original dimers. Another Ga atom (or two) then binds into the long bridge site next to and in the


839

Fig. 16.12. GaAs growth mechanism. Reprinted with permission from B.A. Joyce and D.D. Vvedensky, Self-organized growth on GaAs surfaces, Materials Science & Engineering R-Reports 46 (2004) 127. © 2004 with permission from Elsevier.

same row as the dimer that have already accommodated the first two Ga atoms. The stable nucleus is completed when As2 forms a dimer atop the four Ga atoms. In a device structure, the quantum dots are capped with an overgrown layer. For instance, InAs QDs grown on GaAs might be capped with GaAs. The capping layer must also relax and this sets up a strain field in the layer unless its stoichiometry is chosen to lattice match the QDs and their wetting layer. The strain field in the capping layer influences the growth characteristics of subsequent layers as shown in Figs. 16.12 and 16.13. Fig. 16.13 demonstrates that the density, size and size distribution of dots grown in subsequent layers are influenced by the strain field. The effect is dependent on the thickness h of the capping layer (Howe et al., 2004). Complete strain relief – and loss of memory of the underlying QDs – was found for capping layers 50 nm in bilayer structures and 60 nm in trilayer structures. The parameters σd and σH (full width at half maximum divided by the mean value for the diameter and height, respectively) is used to characterize the breadth of the distributions. For h = 50 nm, almost all QDs have d < 35 nm, σd = 0.119 and σH = 0.247, in close accord with the values found for the growth on the bare substrate. For thinner layers, only QDs with a d > 35 nm are observed and the breadth of the size distribution is changed, with σd = 0.064 and σH = 0.146. The extremely narrow size distribution translates into a narrow photoluminescence linewidth (Le Ru et al., 2003) from the quantum dots. Reduced linewidth is an important property because laser gain is inversely proportional to the linewidth. Intermediate values of h lead to the formation of a bimodal distribution of small (d < 35 nm) and large (d > 35 nm) dots. The large d QDs, therefore, appear to form as a consequence of the strain field. Beyond a critical thickness, the strain field abates and the influence of the underlying islands is no longer felt. Fig. 16.14 demonstrates that also the location of the subsequently grown QDs is affected by the strain field (Tersoff et al., 1996). Vertical stacking, confirmed by TEM, is observed in many cases, not only for III–V materials (Xie et al., 1995; Mi and Bhattacharya, 2005) but also for SiGe superlattices (Tersoff et al., 1996), whereas antistacking has also be observed

840

K.W. Kolasinski

Fig. 16.13. Contact-mode AFM images (1 × 1 µm2 ) of uncapped InAs/GaAs QD samples; (a) a single QD layer, (b)–(d) the second QD layer with h = 10, 20 and 50 nm. Histograms of the measured QD heights and diameters are also shown. The dashed vertical line in the first two diameter histograms denotes the distinction between small and large dot. In all cases, 2.5 ML of InAs were deposited at 475 ◦ C. Reprinted with permission from P. Howe, B. Abbey, E.C. Le Ru, R. Murray and T.S. Jones, Strain-interactions between InAs/GaAs quantum dot layers, Thin Solid Films 464–465 (2004) 225. © 2005 with permission from Elsevier.

(Franchi et al., 2003). Hence, this can be considered a method of strain patterning to align as well as to make the QDs more uniform (Tersoff et al., 1996). Not only are the QDs aligned and larger with a narrower size distribution, they also nucleate at a reduced critical coverage compared to the unstrained substrate (Schmidt et al., 1999). Due to the strain fields produced by the dots in the first layer (stressor dots), the lattice constant of the GaAs layer is expanded directly above the buried dots, that is, it exhibits tensile strain above the dots. However, the island and the substrate around the island are under compressive strain.


841

(a)

(b) Fig. 16.14. Mechanism of quantum dot stacking. (a) Schematic of a bi-layer quantum dot heterostructure that shows perfect vertical coupling of the active (second) quantum dot layer due to the strain field generated by the stressor quantum dots. The dashed lines indicate the lateral extent of the tensile strain field in the GaAs barrier; (b) Schematic layout of a quantum dot heterostructure grown by molecular beam epitaxy. Reused with permission from Zetian Mi and Pallab Bhattacharya, Journal of Applied Physics 98 (2005) 023510. © 2005 American Institute of Physics.

Since we have seen above that the diffusion and binding of H atoms respond strongly to Si surface atom relaxations, we should not be surprised that the lattice deformations associated with the strain field in the capping layer changes the energetics of In atoms on a strained In(Ga)As layer. The excess strain δε induced at a lateral position x by a spherical island at a distance L below a surface is given by (Priester and Lannoo, 1997) −3/2

2 3L2 δε(x) = C 1 − 2 (16.8.14) . x + L2 2 x +L The coefficient C depends on the volume of the buried island, the misfit strain and the elastic constants of the spacer layers (Maradudin and Wallis, 1980). The excess strain is negative and exhibits a deep minimum √ at x = 0, that is, directly above the inclusion. A slight maximum occurs at x = ± 2L, at which point δε is positive. Scheffler and co-workers (Penev et al., 2001; Kratzer et al., 2002) have investigated the effects of strain on the diffusion and binding of In on GaAs(001). Changes from the unstrained lattice alter both the prefactor and the activation energy for diffusion, the latter being perhaps the more important factor. They (Penev et al., 2001) have shown that the binding energy of In increases with tensile strain. Therefore the equilibrium concentration of In at steady state will be higher above the underlying stressor dots and this will enhance nucleation at these sites. They have also shown that Edif displays a maximum at compressive strain and is a decreasing function for tensile strain. This has implications for the growth kinetics of InAs islands and can lead to a self-limiting nature of island growth because at a certain size a repulsive interaction between the quantum dot and a diffusing In atom occurs. This result is consistent with the data of Jones and co-workers (Howe et al., 2004;

842

K.W. Kolasinski

Le Ru et al., 2003), who reported a narrower size distribution for second layer QDs grown on a 10 nm capping layer. One other fascinating aspect of the growth of structures on III–V materials is the consequences of growth on patterned substrates, in particular the consequences of growth involving diffusion between different crystal facets (intersurface diffusion). Nishinaga and co-workers (Shen et al., 1994) demonstrated that the direction of intersurface diffusion can be reversed by varying the As pressure in GaAs MBE. Growth at equilibrium leads to the ECS according to the Wulff construction (Wulff, 1901), subject to some modification introduced by strain. However, under kinetic control, it is intersurface diffusion that will determine the profile of facets. When the substrate on which growth begins is convex, the slowest growth rate facet dominates the final structure, whereas if the initial structure is concave, the fastest growth rate structure dominates (Nishinaga, 2004). (As an aside, recall that this is the opposite pattern to that followed by etching, i.e., a convex object etches to reveal fast etching planes; while a concave object etches to reveal the slow etching planes.) By controlling the As pressure, either pyramids or flat-topped pyramids can be grown. Furthermore, control of diffusion by temperature, pressure and composition can be used to control the profile of structures grown on mesas (Williams et al., 2005, 2004). 16.8.3. Silicon dry etching: F/Si etching The chemistry of H atoms with Si surfaces shares much in common with the halogens. In all cases, the atoms have a tendency to cap the dangling bonds of Si and all of them exhibit stable SiX4 (X = F, Cl, Br or I) compounds that are more volatile than crystalline Si. Because the most facile reaction is with dangling bonds, the presence of adsorbed H or X inhibits further dissociative adsorption of H2 or X2 . Etching can occur for both sets of systems. The Si–X bond strength for adding one atom to the Si(100)–(2×1) surface is much stronger for F (5.8 eV) or Cl (4.0 eV) (Walch, 2002) compared to H (3.2 eV on Si(100)(2×1) and 3.1 eV on Si(111)-(7×7) (Raschke and Höfer, 2001)) and, therefore, the former two species have a much greater tendency to etch Si rather than to desorb recombinatively. The size of the atom also influences the surface chemistry. The small H and F atoms have a tendency to diffuse into the Si lattice. H atoms diffuse into interstitial sites (Gorostiza et al., 2003). F atoms can also diffuse into the bulk to take up interstitial or substitutional sites (Watanabe and Shigeno, 1992; Makino et al., 2000). All of the species can form adsorbed SiXn , n = 1–3; however, increasing atomic size leads to strain (Boland, 1991b; Chen and Boland, 2003) playing an important role in understanding the difference in reactivity between H and the halogens. Whereas the Si–H bond is virtually nonpolar, the Si–Cl and, even more exaggeratedly, the Si–F bonds are highly polar and this plays an important role in determining the relative chemical differences (Gerischer et al., 1993). In particular, the halogens polarize the backbonds of surface Si atoms and make them more susceptible to reaction via insertion. The partial negative charges on adsorbed halogens lead to repulsive lateral interactions whereas the nearly neutral adsorbed H atoms will experience much less of this. The chemistry of H2 molecules is much different than that of molecular halogens. As mentioned above, the sticking coefficient of room temperature H2 on Si is vanishingly small and is enhanced by increasing Ts as well as molecular translational and vibrational


843

energy. In contrast, the sticking coefficient of F2 on clean Si is ∼0.5 and does not depend strongly on Ts or translational energy (Engstrom et al., 1989). Thermal etching of Si can be achieved with F2 , XeF2 , and halogen fluorides such as ClF3 , BrF3 and IF5 . Even with its high clean surface sticking coefficient, the etch rate of F2 is 103 –104 times less than that of XeF2 (Winters and Coburn, 1979; Holt et al., 2002) because F2 does not stick effectively on a F-covered surface whereas XeF2 does. F atoms are much more efficient at engendering etching than F2 . The initial sticking coefficient for atoms is near unity (Carter and Carter, 1996b, 1996a) but drops with increasing coverage. Reported (Humbird and Graves, 2004) etch rates (Si atoms etched per incident F atom) depend on doping and the F atom source (as well as other parameters when plasmas are involved) and range from (0.04–3) × 10−2 . The most general etching scheme for H and the halogens involves the following Langmuir-Hinshelwood mechanism in which adsorbed atoms are sequentially added to SiXn (a) to form SiX4 , which then desorbs. This scheme is written explicitly for H as Si(a) + H(a) → SiH(a),

(16.8.15)

SiH(a) + H(a) SiH2 (a),

(16.8.16)

SiH2 (a) + H(a) SiH3 (a),

(16.8.17)

SiH3 (a) + H(a) → SiH4 (g).

(16.8.18)

The reversibility of steps (16.8.16) and (16.8.17) indicates that higher hydrides tend not be to stable in the presence of neighbouring dangling bonds. SiH4 is bound very weakly on the surface and immediately desorbs. The binding of SiX4 is somewhat stronger and the condensation of these etch products will be increasingly likely as the temperature drops significantly below room temperature. This highly simplified scheme is complicated by noting that H(a) exists as SiH(a), SiH2 (a) or SiH3 (a) and that diffusion of either H(a) or a SiHx (a) unit can occur. Transport as SiHx (a) rather than by H atom hopping has implications for the surface structure because of the concomitant substrate atom motion. Furthermore, Eley–Rideal reaction steps, in which an H atom from the gas phase reacts with an adsorbed species to form the next higher hydride, may also play a role under some circumstances (Dinger et al., 2001; Jo et al., 1997; Buntin, 1997, 1996; Koleske et al., 1994, 1993; Rahman et al., 2004; Kutana et al., 2003). The final step in the etch mechanism is perhaps the most likely to include a contribution from an Eley–Rideal reaction. As mentioned above, this etching sequence can be suppressed by high Ts , which inhibits the formation of higher hydrides. We must consider the effects of bond strength and atomic size on the reaction mechanism when the discussion is expanded to the halogens. Room temperature steady-state etching of Si by F atoms appears primarily to follow the above etch mechanism. SiF4 is the main product (Winters and Coburn, 1992) but there are complications. A molecular dynamics simulation found (Humbird and Graves, 2004) that for steady-state etching at 300 K, the etch products are composed of SiF4 (71%), Si2 F6 (27%) and Si3 F8 (1%). Higher hydrides are also found among the etch products for H atom etching (Gates et al., 1989b). While the percentages should be interpreted loosely, they do indicate that pathways to higher silanes exist for low temperature etching.

844

K.W. Kolasinski

Surface temperature strongly influences the product distribution. When XeF2 is dosed to the surface, there is a sharp increase in the etch rate around Ts = 500 K because surface SiFx groups begin to rearrange, dissociate and desorb (Winters and Coburn, 1992). When F atoms are dosed onto Si(100) at 120 K and then heated, the formation of SiF4 (g), as expected according to the mechanism outlined above, is observed (Engstrom et al., 1989). However, not only the amount but also the proportion of SiF4 produced is coverage and Ts dependent because SiF2 (g) is also produced via the desorption reaction SiF2 (a) → SiF2 (g).

(16.8.19)

When Si(100) is exposed to F2 in steady state at 650 K Ts 1200 K, SiF2 is the predominant etch product. Si2 F6 production was found to peak at 423 K and then become negligible at high temperature (Winters and Coburn, 1992). SiCl2 is the primary etch product resulting from chlorine exposure for Ts > 600 K, but SiCl4 is the primary product at Ts < 500 K (Szabò and Engel, 1994). The other halogens (Br and I) etch to produce SiX2 exclusively (Aldao and Weaver, 2001) at the elevated temperatures where the etch rate is substantial. First-principles calculations (Aizawa et al., 1999) confirm that both strain and the weakening of backbonds by bond polarization are important in the formation of SiX2 etch products. Neither of these effects are strong for H atoms; hence, SiH2 has much too high a desorption energy to be a significant product. A reaction akin to Rxn. (16.8.19) does not occur appreciably during H atom etching but is the primary means of Si atom removal for halogen atom etching at high Ts . Considerable second layer etching and surface roughening occur during the thermal etching of Si exposed to F atoms and then heated. The effects of roughening led many to believe that a relatively thick fluorinated layer is formed during etching. The high coverage of F during etching, however, is due to the combined effects of higher fluoride (SiF2 and SiF3 ) formation along with the surface area increase resulting from roughening (Humbird and Graves, 2004). In contrast, this is not observed when Cl or Br are used (Nakayama and Weaver, 1999), which again relates to a steric effect. Bridge bonding of F atoms between Si atoms is unlikely because of the short Si–F bond. Cl bridge bonding is unfavourable on the defect-free Si(100) surface, but the large Si–Cl bond length enables it at defect sites. The stability of this structure makes second layer etching unlikely for halogens other than F. In plasma etching of Si (Winters and Coburn, 1992; Chang and Coburn, 2003), F is rarely added as pure F2 . Instead various combinations of fluorinated compounds (NF3 , SF6 , CF4 , CFCl3 , C2 F6 ) are added to Cl2 , O2 , H2 and/or inert gases (He, Ar, Kr). Each mixture has its own properties regarding etch rate and anisotropy (Flamm, 1990). SiF6 /O2 is particularly prized for its anisotropic etching and ability to etch deep trenches with vertical sidewalls (Demic et al., 1992). SiF4 can be used alone to etch Si selectively over SiO2 or Si3 N4 or, when combined with O2 or H2 , to deposit fluorinated a-SiO2 or a-Si:H, F layers, respectively (Williams et al., 2002). The etch products formed during plasma etching are more difficult to ascertain than during thermal etching because of the large variety of species in the plasma and the complicated nature of the plasma/surface interaction. The etch product distribution undoubtedly is dependent upon the composition of the plasma, the pressure and the conditions used to produce the plasma. Under many conditions, particularly for Ts < 600 K, it is believed (Winters and Coburn, 1992) that SiF4 comprises a major portion of the desorption flux.


845

Above 600 K, SiF2 makes the leading contribution. SiF may also be released from the surface (Winters et al., 1983; Flamm, 1990). Flamm (1990) suggested that the production of SiF2 was signaled by chemiluminescence from the reaction of SiF2 with F or F2 to form SiF3 *. The formation of Si clusters (Anisimov and Luk’yanchuk, 2002) in plasmas created by spark ablation (Saunders et al., 1993), laser ablation (Makimura et al., 2000), laser pyrolysis (Fojtik et al., 1993; Ehbrecht and Huisken, 1999; Botti et al., 2000), microwaves (Kravets et al., 2005), and magnetron sputtering (Bera et al., 2001) points out one of the difficulties in identifying the products of plasma etching: there is significant chemistry that occurs in the plasma. The species that are formed in the plasma can either enhance etching or, alternatively, they can redeposit material on the substrate. Winters and Coburn (1992) have questioned whether the SiF2 detected by Flamm and others is the primary etch product. They believe that Si2 F6 represents as much as 40% of the etch product flux and that SiF2 results from cracking of Si2 F6 . Low SF2 production compared to Si2 F6 production is consistent with molecular dynamics simulations performed at Ts = 300 K (Humbird and Graves, 2004). Ion-molecule reactions in the plasma and ion-surface reactions play an important part in plasma processing, especially for the formation of polymer layers (Williams et al., 2002). 16.8.4. Silicon wet etching: HF/Si How does HF react with a Si surface in aqueous solution? The short answer is that it does not. A Si surface exposed to an acidic fluoride solution is rapidly stripped of any native oxide layer and then terminated with H atoms. With all of the dangling bonds terminated, this surface is almost perfectly impervious to attack from HF. The sticking coefficient of F− (aq) on the H-terminated Si surface (H/Si), which we will see below is involved in the first step of the etching mechanism, is estimated (Kolasinski, 2003) to be at most 5 × 10−11 . However, to say that HF does not react with Si is almost heresy since every Si wafer is cleaned in HF and fluoride etching of Si is so highly versatile that it can either produce nearly perfect atomically flat surfaces or else it can create nanostructured porous Si (Kolasinski, 2003) in which quantum confinement dramatically changes the optical, electrical and thermal properties of Si (Canham, 1990; Cullis et al., 1997; Pavesi, 1997; Pavesi et al., 2000; Bisi et al., 2000; Collins et al., 1997; Fiory and Ravindra, 2003). To understand this paradox and how to control the etching of Si in fluoride solutions, we need to understand what initiates the (electro)chemical interaction of HF(aq) with the surface. First, note that the reaction of HF with Si to form SiF4 is thermodynamically favourable, and it is even more favourable to form H2 SiF6 (aq), which is the end product of the hydrolysis of SiF4 in aqueous solution. Etching is stopped by kinetic constraints, that is, there is a significant barrier to adsorption caused by the H-termination of the Si surface. No significant barrier exists for the reaction of HF(aq) with SiO2 ; hence, any oxide present on the surface is rapidly stripped from the surface. This is essential for the occurrence of Si electropolishing. At high enough anodic bias (+0.7 V with respect to the normal hydrogen electrode (Rappich and Lewerenz, 1996b)), oxide grows on the Si surface and this is then removed by HF via chemical etching (Gerischer and Lübke, 1988; Peiner and Schlachetzki, 1992). Growth and dissolution of the oxide can even lead to sustained current oscillations (Cattarin et al., 1998). Electropolishing is not observed (Propst and Kohl, 1994) in an

846

K.W. Kolasinski

environment that is both anhydrous and oxygen free because of the lack of oxide growth. In short, one way to make Si react with HF is to form an oxide layer, which is readily removed chemically. It is the initial step of the etching reaction of HF with Si that presents the kinetic barrier; therefore, it is control of this initiation step (Kolasinski, 2003) that allows us to turn on the reactivity. Once the first step of the reaction occurs, the following steps are rapid because the presence of F on the surface polarizes and weakens the Si back bonds and makes them susceptible to attack (Gerischer et al., 1993; Trucks et al., 1990, 1991). One way to turn on the initiation step is to inject a hole in the valence band (Gerischer and Mindt, 1968; Gerischer et al., 1993; Gerischer, 1997, 1990). The presence of a hole in a bulk electronic state localized near the surface increases the sticking coefficient of F− (aq) by roughly 10 orders of magnitude (Kolasinski, 2003). Three methods of hole injection are commonly encountered that can lead to efficient etching of Si (i) apply a sufficient voltage to induce electrochemistry (Lehmann, 2002; Campbell and Lewerenz, 1998), (ii) illuminate with a laser or lamp using above band gap radiation to form electron-hole pairs (Noguchi and Suemune, 1993; Koker and Kolasinski, 2000, 2001), or (iii) add an oxidant to the solution that can induce charge transfer because it possesses an acceptor level at the right energy (Turner, 1960; Nahidi and Kolasinski, 2006; Kolasinski et al., 2006). These three methods – electrochemistry, photoelectrochemistry and stain etching, respectively – can also be used in conjunction with one another. All three of these methods are used to produce porous Si (Canham, 1997; Cullis et al., 1997). The third method of inducing etching is to raise the pH and thereby the concentration of OH− . The hydroxide ion has a much lower barrier to reacting with the H/Si surface. Its chemistry is reviewed in the next section. Once a hydroxide has substituted for a H atom on the surface, fluoride is able to etch the surface effectively. The onset of etching induced by hydroxide initiation is correlated with the onset of step-flow etching in fluoride solutions and the production of atomically flat, H-terminated Si(111)-(1 × 1) surfaces, as proposed by Jakob et al. (1992) and Allongue et al. (Allongue et al., 1995, 2000; Munford et al., 2001). Note that a 40% NH4 F solution, as is commonly used to produce atomically flat Si(111)-(1 × 1) surfaces, has a pH of ∼8. The mechanism of Si etching, known as the Gerischer mechanism, is illustrated in Fig. 16.15. This model was refined from the models of Gerischer, Allongue and Costa Kieling (Gerischer et al., 1993; Allongue et al., 1995); Kooij and Vanmaekelbergh (1997); Kang and Jorné (1998); and Lehmann and Gösele (1991) in conjunction with a meta-analysis of experimental data to formulate a comprehensive mechanism (Kolasinski, 2003). Comparatively few theoretical studies have been carried out on the HF/Si etching system (Trucks et al., 1990; Sacher and Yelon, 1991; Trucks et al., 1991). To begin to understand this mechanism we must first understand the composition of fluoride solutions (Kolasinski, 2005). The structure of HF in solution is highly controversial. It exists as some combination of the undissociated molecule and a contact ion pair − H3 O+ •F− . It also forms H+ , F− , HF− 2 , H2 F3 and higher polymeric ions at high concentration. Dimers are only formed at very high concentrations. Equilibrium constants are fairly well understood and well behaved such that reliable compositions and activity coefficients can be calculate for solutions in the range of about 1–6 mol dm−3 in HF as well as for solutions created with other sources of fluoride. Kinetics studies (Koker


847

and Kolasinski, 2001) performed in this range show that the species F− , HF and HF− 2 have to be considered in the reaction mechanism. These species lead to a surface that is predominantly H-covered throughout etching (Higashi et al., 1990; Burrows et al., 1988; Rappich and Lewerenz, 1996a, 1995; Jungblut et al., 2002; Rao et al., 1991; Chazalviel and Ozanam, 1997; Dubin et al., 1995, 1994; Cattarin et al., 1998; Ozanam et al., 1996; Belaïdi et al., 1999; Dubin et al., 1993; Chazalviel et al., 2000; Safi et al., 2002; Peter et al., 1989; Bjorkman et al., 1995). However, the observation (Lewerenz et al., 2003) of a small oxygen coverage indicates that H2 O and OH− may also play a competing role in some of the steps under some conditions. The roles of H2 O and OH− will, of course, be enhanced by low fluoride concentration and high pH. Very little if any oxygen or fluorine is present on the surface after etching. There have been reports of adsorbed F on the surface (Canham et al., 1991; Ogata et al., 1995; Sato and Maeda, 1994; Takahagi et al., 1991; Matsumura and Fukidome, 1996); however, these may be due to physisorbed etch products rather than chemisorbed reaction intermediates since they are removed by a water rinse (Watanabe et al., 1991; Takahagi et al., 1991; Vasquez et al., 1992; Chabal et al., 1993; Dumas and Chabal, 1991; Benner et al., 1989; Zazzera and Evans, 1993; Watanabe and Shigeno, 1992). A fluorine signal may also be attributable to F absorbed in interstitial sites (Watanabe and Shigeno, 1992), which ab initio electronic structure calculations suggest is favourable (Makino et al., 2000). While the reactivity of step sites is undoubtedly greater than that of terrace sites (Koker and Kolasinski, 2000; Gerischer et al., 1993; Gerischer and Lübke, 1987), reaction cannot occur exclusive at these sites as that would lead to step-flow etching and preclude the formation of por-Si. One other consideration to delineate before discussing the etching mechanism is the nature of the products. H2 is not evolved when etching occurs in anhydrous CH3 CN (Propst and Kohl, 1994). H2 is, therefore, not a direct product of the etching mechanism. Instead, it is connected to hydrolysis of the primary etch product(s). A variety of primary etch products are possible with the general formula SiHx Fy (OH)z , with x + y + z = 4 (Kooij and Vanmaekelbergh, 1997; Gerischer et al., 1993; Kang and Jorné, 1998; Lehmann and Gösele, 1991), none of which are stable in aqueous HF, thus SiF2− 6 (H2 SiF6 ) is the final product as evinced by the precipitation of hexafluorosilicates onto the etching Si surface (Seo et al., 1993; Hassan et al., 1995; Koker et al., 2002) in some circumstances. SiO2 is etched by H2 SiF6 . Therefore, the final etch product of SiO2 , relevant for Si etching in the electropolishing regime, is a higher fluorosilicate (Somashekhar and O’Brien, 1996). The mechanism shown in Fig. 16.15 is somewhat abbreviated from the full version (Kolasinski, 2003) in that it does not show the pathway involved in the current quadrupling regime. Electron injection accompanies etching, which leads to photocurrent multiplication (Matsumura and Morrison, 1983; Gerischer and Lübke, 1987; Peter et al., 1990b, 1990a; Stumper and Peter, 1991). Under the most common conditions current doubling is observed and that is what we concentrate upon here. Step (1) is the creation of a hole in the bulk as a result of photon absorption or an applied voltage. The transport of a bulk hole to a bulk state at the surface comprises Step (2). In the case of hole injection from an oxidant, these steps merge into one, though there may still be hole transport along the surface. Step (3) is the first chemical step – the adsorption of F− at a Si–H2 site activated by the hole. Alternatively, OH− might participate in this step; however, as suggested by Propst and Kohl (Propst and Kohl) subsequent steps would be

848

K.W. Kolasinski

Fig. 16.15. The Gerischer mechanism of Si etching in acidic fluoride solutions. Adapted from Kolasinski (2003).

retarded due to a reduced inductive effect. Step (4), which is the second slowest step after Step (1), is branched to allow for the kinetic competition between HF− 2 and HF observed by Koker and Kolasinski (2001). At Step (5) a branching between current doubling and current quadrupling paths is possible, though only the current doubling path is shown. Step (6) represents the final addition of HF− 2 , H2 O or HF to a SiX3 (a) moiety and the release of the primary etch product into solution.


849

Step (3) occurs either by (i) formation of a pentavalent transition state by addition of F− to SiH2 –h+ (SiH2 –h+ represents a surface site at which a bulk hole has been localized due to electrostatic interactions with an F− ion) followed by loss of H+ to form SiHF or (ii) abstraction of H+ from SiH2 –h+ by F− followed by addition of F− . In either case an electron is donated to the conduction band and current doubling is the result. It would be of considerable interest to see the results of ab initio calculations addressing the relative energetics of these two different reaction paths and transition states. Why does por-Si form under some conditions rather than, as we shall see for etching with OH− , simply etching away the entire surface? For the answer to this we need to investigate the self-limiting nature of the etching reaction imposed by quantum confinement as detailed by Lehmann and Gösele (1991) and Frohnhoff et al. (1995). Because Si etching in HF is essentially an electrochemical process, it responds to the electronic structure of the Si. As Si nanostructures become comparable to and smaller than the size of the exciton radius (∼5 nm) in Si, quantum confinement widens the band gap by pushing the conduction band up in energy while decreasing the energy of the valence band minimum (van Buuren et al., 1998). A depletion of holes, required to initiate etching, within the confined structures results. Instead, holes are directed to the bottoms of pores, which are connected to unconfined bulk Si and etching proceeds there. The etching of OH− is not similarly constrained by quantum confinement effects. Porous Si is not stable in OH− and is quite efficiently removed by it. 16.8.5. Silicon wet etching: KOH/Si Here we are concerned with the etching of Si in alkaline solution at open circuit potential (OCP) rather than anodic etching (Allongue et al., 1993b, 1993a; Cattarin and Musiani, 1999; Bressers et al., 1996; Xia et al., 2001; Xia and Kelly, 2001). In the reactions of any aqueous system with Si, there are three species that must always be considered: H2 O itself, OH− added intentionally or produced through autoionization and dissolved O2 . Whereas H2 O dissociates readily on clean Si surfaces (Queeney et al., 2003), the H/Si surface is extraordinarily unreactive not only to HF but also to O2 , OH− and H2 O. The sticking coefficient of O2 on H/Si(111), based on the data of Ye et al. (2001) is 1.6×10−14 . Dissolved O2 , reacts much differently and increases the reactivity of water (Watanabe and Sugita, 1995; Kanaya et al., 1995; Ogawa et al., 1995; Wade and Chidsey, 1997), which is otherwise quite low. From the data of Boonekamp et al. (1994) an upper limit for the sticking coefficient of H2 O on H/Si of 7 × 10−17 is estimated. However, even this low rate of dissociation may only be due to the presence of OH− . Hydroxide is thought to play a catalytic role (Allongue et al., 1993a, 1993b) in the reaction of water with H/Si, which may be considered as an SN-type reaction (Baum and Schiffrin, 1998), in which OH− first attacks the Si–H bond and water reacts with the departing “hydride” (Xia and Kelly, 2001). For basic pH, the reactivity of water with H/Si is dominated by OH− . This is also important for neutral water solutions at high temperatures since the autoionization of water is strongly temperature dependent and the solubility of gases is reduced. However, as the concentration of OH− declines, the effects of dissolved oxygen (Watanabe and Sugita, 1995; Kanaya et al., 1995; Ogata et al., 2000; Wade and Chidsey, 1997; Garcia et al., 2003), which etches less anisotropically than OH− (Hines, 2003), will become more pronounced.

850

K.W. Kolasinski

Fig. 16.16. Initiation steps in the etching of Si by OH− (aq).

Fig. 16.16 depicts the first two steps of etching by OH− (aq) (Allongue et al., 1993a, 1993b; Xia and Kelly, 2001; Xia et al., 2001). Subsequent steps are analogous to those depicted in Fig. 16.15 for fluoride etching, which results in a H-terminated surface (Rappich et al., 1993). What these two steps reveal is that the reaction is chemical (Smith et al., 1987; Glembocki et al., 1985), the action of the OH− is catalytic (Allongue et al., 1993a, 1993b) and that water is directly involved (Allongue et al., 1993b; Campbell et al., 1993). At appropriate anodic bias or in the presence of a sufficient concentration of a strong oxidizing agent such as ferricyanate, OH(a) forms on the surface at a high enough coverage that neighbouring groups can condense to form bridge bonding Si–O–Si species and a surface oxide (Bressers et al., 1995). This oxide passivates the surface, in contradistinction to what can occur in acidic fluoride solutions in the absence of an applied bias. The first step in Fig. 16.16 involves the breaking of a Si–H bond, which is consistent with the kinetic isotope effect found when deuterated solutions are used (Baum and Schiffrin, 1997). The etching of Si in hot concentrated KOH is one of the staples of Si processing because the etch rate is highly anisotropic with the (111) surface being etched as much as ∼100 times slower than the (100) and other planes (Williams and Muller, 1996). The etch rate and anisotropy are affected by the concentration of KOH, addition of i-propanol and the temperature (Wind and Hines, 2000). The basis for the anisotropy is believed (Baum and Schiffrin, 1998) to relate to the geometry of the pentavalent transition state that is formed in the first step. A pentavalent transition state was first proposed by Hines et al. (1994). Steric hindrance is much greater on Si(111) than other planes and this leads to a significantly higher barrier to reaction on the (111). The formation of pyramidal hillocks is often observed and can be controlled by changes in concentration, addition of an oxidizing agent such as ferricyanate or dissolved oxygen, or by the application of an applied potential (Bressers et al., 1996).


851

KOH etches Si with a step-flow mechanism (Flidr et al., 1998). Steric factors in the transition state can be used to explain not only the relative differences of etch rate with crystallographic plane, but also in the etch rates of step, kink and terrace sites. This in turn can explain the observation of step-flow etching and the shape of the steps (Hines et al., 1994; Wind and Hines, 2000). The etch rates follow the series kink > dihydride step monohydride step terrace. There are a great many similarities in the etching of Si by F− and OH− ; however, there are two important distinctions. Fluoride etching is exclusively electrochemical in nature and is subject to the influence of quantum confinement. Thus it can form por-Si. Hydroxide etching has a chemical pathway and does not form por-Si. Hydroxide etching can lead to a passivating oxide layer if the concentration of adsorbed hydroxide is sufficiently high to allow condensation into an oxide. There is no spontaneous formation of an oxide layer in an acidic fluoride solution.

16.9. Future perspectives So long as devices are being made out of semiconductors, we will continue to ask the question of how particular structures can be fabricated from various combinations of growth and etching. The more demanding our specifications – whether it be by forcing devices into the quantum regime by virtue of their small size or linking macroscopic arrays of structures with extreme tolerances – the more necessary it becomes to understand these processes on the atomic scale and to translate this atomistic understanding into models that can be used to describe the evolution of structures over macroscopic scales (Esashi and Ono, 2005). The theoretical description of these processes is extremely challenging in that processes need often be followed over long time and distance scales. Truly ab initio atomistic dynamics calculations of relevantly large systems are only just beginning to be possible. Connecting these calculations to continuum models and simulations as well as to thermodynamics will greatly enhance our understanding of the processes that shape nanoscale features across macroscopic samples. The intersection of top–down engineering with bottom–up chemical assembly contains some of the most exciting aspects of this field (Mendes and Preece, 2004). The two are almost antithetical approaches to structure formation. The former, best exemplified by the lithographic methods used to fabricate integrated circuits, attempts to make vast numbers of structures with maximum invasiveness (Watts, 1988; Helbert, 2001). The strategy is to control every step of the process with ever increasing precision but to win overall processing gains by using repetitiveness and economies of scale to great advantage. The latter, often considered to be a mime of biological processes, attempts to fashion structures with a minimally invasive scheme: let the system form itself by using inherent error correction and self-assembly. Chemists have been performing atom-resolved assembly since the foundation of this branch of science but the paradigm shifting difference is that now chemical architectures include not just the individual rooms but the whole building. Formerly chemists and materials scientists synthesized molecules with the appropriate composition in the appropriate stereoconfiguration. Now they synthesize not

852

K.W. Kolasinski

only the desired molecule but also, they assemble these into the appropriate superstructure.

Acknowledgements I gratefully acknowledge the comments provided by David Dunstan, Nils Hartmann and David Mills, which helped to shape this manuscript. The manuscript also benefited greatly from a critical reading by Frank Meyer zu Heringdorf. This work was supported by the National Science Foundation IGERT Program under Grant #9972790 and the NSF CREST Project HRD-9805059.

References Aizawa, H., Tsuneyuki, S., Ogitsu, T., 1999. Population analysis study of etching processes at Si(100) surfaces with adsorbed halogens and hydrogens. Surf. Sci. 438, 18–25. Ajayan, P.M., 1999. Nanotubes from carbon. Chem. Rev. 99, 1787–1799. Akazawa, H., Utsumi, Y., 1995. Reaction kinetics in synchrotron-radiation-excited Si epitaxy with disilane. 1. Atomic layer epitaxy. J. Appl. Phys. 78, 2725–2739. Albertini, D., Thibaudau, F., Masson, L., Salvan, F., 1996. STM study of low pressure adsorption of silane on Si(111)7 × 7. Surf. Sci. 350, L216–L220. Aldao, C.M., Weaver, J.H., 2001. Halogen etching of Si via atomic-scale processes. Prog. Surf. Sci. 68, 189–230. Alivisatos, A.P., 1996. Semiconductor clusters, nanocrystals and quantum dots. Science 271, 933–937. Allongue, P., Costa-Kieling, V., Gerischer, H., 1993a. Etching of silicon in NaOH solutions I. J. Electrochem. Soc. 140, 1009–1018. Allongue, P., Costa-Kieling, V., Gerischer, H., 1993b. Etching of silicon in NaOH solutions II. J. Electrochem. Soc. 140, 1018–1026. Allongue, P., Henry de Villeneuve, C., Morin, S., Boukherroub, R., Wayner, D.D.M., 2000. The preparation of flat H–Si(111) surfaces in 40% NH4 F revisited. Electrochim. Acta 45, 4591. Allongue, P., Kieling, V., Gerischer, H., 1995. Etching mechanism and atomic structure of H–Si(111) surfaces prepared in NH4 F. Electrochim. Acta 40, 1353. Ancilotto, F., Andreoni, W., Selloni, A., Car, R., Parrinello, M., 1990. Structural, electronic, and vibrational properties of Si(111)-2 × 1 from ab initio molecular dynamics. Phys. Rev. Lett. 65, 3148–3151. Andersohn, L., Berke, T., Kohler, U., Voigtlander, B., 1996. Nucleation behavior in molecular beam and chemical vapor deposition of silicon on Si(111)-(7 × 7). J. Vac. Sci. Technol. A 14, 312–318. Andreev, A.F., Kosevich, Y.A., 1981. JETP 54, 761. Zh. Eksp. Teor. Fiz. 81 (1981) 1435. Anisimov, S.I., Luk’yanchuk, B.S., 2002. Selected problems in laser ablation theory. Phys. Usp. 45, 293–324. Arthur Jr., J.R., 1968. Interaction of Ga and As2 molecular beams with GaAs surfaces. J. Appl. Phys. 39, 4032– 4043. Bahr, D., Press, W., Jebasinski, R., Mantl, S., 1993. X-ray reflectivity and diffuse-scattering study of CoSi2 layers in Si produced by ion-beam synthesis. Phys. Rev. B 47, 4385–4393. Barrelet, C.J., Wu, Y., Bell, D.C., Lieber, C.M., 2003. Synthesis of CdS and ZnS nanowires using single-source molecular precursors. J. Am. Chem. Soc. 125, 11498–11499. Baum, T., Schiffrin, D.J., 1997. Kinetic isotopic effects in the anisotropic etching of p-Si100 in alkaline solutions. J. Electroanal. Chem. 436, 239–244. Baum, T., Schiffrin, D.J., 1998. Mechanistic aspects of anisotropic dissolution of materials etching of singlecrystal silicon in alkaline solutions. J. Chem. Soc., Faraday Trans. 94, 691. Belaïdi, A., Safi, M., Ozanam, F., Chazalviel, J.-N., Gorochov, O., 1999. Surface chemistry during porous silicon formation in dilute fluoride electrolytes. J. Electrochem. Soc. 146, 2659–2664.


853

Belen, R.J., Gomez, S., Kiehlbauch, M., Cooperberg, D., Aydil, E.S., 2005. Feature-scale model of Si etching in SF6 plasma and comparison with experiments. J. Vac. Sci. Technol., A 23, 99–113. Benner, D.B., Biegelsen, D.K., Bringans, R.D., 1989. Silicon surface passivation by hydrogen termination: A comparative study of preparation methods. J. Appl. Phys. 66, 419–424. Bera, S.K., Chaudhuri, S., Bandyopadhyay, A.K., Chakraborty, B.R., Pal, A.K., 2001. Quantum size effect in silicon nanocrystals prepared by dc magnetron sputtering. J. Phys. D: Appl. Phys. 34, 273–278. Beranek, R., Hildebrand, H., Schmuki, P., 2003. Self-organized porous titanium oxide prepared in H2 SO4 /HF electrolytes. Electrochem. Solid State Lett. 6, B12–B14. Bisi, O., Ossicini, S., Pavesi, L., 2000. Porous silicon: A quantum sponge structure for silicon based optoelectronics. Surf. Sci. Rep. 38, 1–126. Bjorkman, C.H., Fukuda, M., Yamazaki, T., Miyazaki, S., Hirose, M., 1995. Atomic scale morphology of hydrogen-terminated Si(100) surfaces studied by Fourier-transform infrared attenuated total reflection spectroscopy and scanning probe microscopies. Jpn. J. Appl. Phys. 34, 722. Blakely, J.M., Eizenberg, M., 1981. Morphology and composition of crystal surfaces. In: King, D.A., Woodruff, D.P. (Eds.), The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis. Elsevier, New York, pp. 1– 80. Blakely, J.M., Jackson, K.A., 1962. Growth of crystal whiskers. J. Chem. Phys. 37, 428–430. Blauw, M.A., van der Drift, E., Marcos, G., Rhallabi, A., 2003. Modeling of fluorine-based high-density plasma etching of anisotropic silicon trenches with oxygen sidewall passivation. J. Appl. Phys. 94, 6311–6318. Boland, J.J., 1991a. Evidence of pairing and it role in the recombinative desorption of hydrogen from the Si(100)2 × 1 surface. Phys. Rev. Lett. 67, 1539–1542. Boland, J.J., 1991b. Role of bond-strain in the chemistry of hydrogen on the Si(100) surface. Surf. Sci. 261, 17–28. Boland, J.J., 1991c. Role of hydrogen desorption in the chemical-vapor deposition of Si(100) epitaxial films using disilane. Phys. Rev. B 44, 1383–1386. Boland, J.J., 1993. Scanning tunneling microscopy of the interaction of hydrogen with silicon surfaces. Adv. Phys. 42, 129–171. Boonekamp, E.P., Kelly, J.J., van de Ven, J., Sondag, A.H.M., 1994. The chemical oxidation of hydrogenterminated silicon (111) surfaces in water studied in situ with Fourier transform infrared spectroscopy. J. Appl. Phys. 75, 8121. Botti, S., Coppola, R., Gourbilleau, F., Rizk, R., 2000. Photoluminescence from silicon nano-particles synthesized by laser-induced decomposition of silane. J. Appl. Phys. 88, 3396–3401. Bozso, F., Avouris, P., 1988. The thermal and electron-beam-induced reaction of disilane on Si(100)-(2×1). Phys. Rev. B 38, 3943–3947. Bramblett, T.R., Lu, Q., Karasawa, T., Hasan, M.-A., Jo, S.K., Greene, J.E., 1994. Si(001)2 × 1 gas-source molecular-beam epitaxy from Si2 H6 : Growth kinetics and boron doping. J. Appl. Phys. 76, 1884–1888. Bratu, P., Höfer, U., 1995. Phonon-assisted sticking of molecular hydrogen on Si(111)-7 × 7. Phys. Rev. Lett. 74, 1625–1628. Bratu, P., Kompa, K.L., Höfer, U., 1996. Optical second-harmonic investigations of H2 and D2 adsorption on Si(100)2 × 1: The surface temperature dependence of the sticking coefficient. Chem. Phys. Lett. 251, 1–7. Brenig, W., Hilf, M.F., 2001. Reaction dynamics of H2 and D2 on Si(100) and Si(111). J. Phys.: Cond. Matter 13, R61. Brenig, W., Kreuzer, H.J., Payne, S.H., 2003. Reactions of hydrogen on silicon: Kinetics and dynamics. Phys. Rev. B 67, 205419. Bressers, P.M.M.C., Kelly, J.J., Gardeniers, J.G.E., Elwenspoek, M., 1996. Surface morphology of p-type (100) silicon etched in aqueous alkaline solution. J. Electrochem. Soc. 143, 1744–1750. Bressers, P.M.M.C., Pagano, S.A.S.P., Kelly, J.J., 1995. Ferricyanide reduction as a probe for the surfacechemistry of silicon in aqueous alkaline-solutions. J. Electroanal. Chem. 391, 159–168. Brillouin, L., 1938. Nécessité de distingjer deux definitions différentes de la tension superficielle. J. Phys. Radium 9, 462–464. Bronikowski, M.J., Wang, Y., McEllistrem, M.T., Chen, D., Hamers, R.J., 1993. Adsorption and dissociation of disilane on Si(001) studied by STM. Surf. Sci. 298, 50–62. Brown, A.R., Doren, D.J., 1999. Dissociative adsorption of silane on the Si(100)-(2 × 1) surface. J. Chem. Phys. 110, 2643–2651.

854

K.W. Kolasinski

Brune, H., 1998. Microscopic view of epitaxial metal growth: Nucleation and aggregation. Surf. Sci. Rep. 31, 125–229. Bublik, V.T., Gorelik, S.S., Zaitsev, A.A., Polyakov, A.Y., 1974. Calculation of the binding energy of GeSi solid solution. Physica Status Solidi B-Basic Research 65, K79–K84. Buntin, S.A., 1996. Hyperthermal H atom interactions with D/Si(100): Effects of incident H atom kinetic energy on the removal of adsorbed D. J. Chem. Phys. 105, 2066–2075. Buntin, S.A., 1997. H atom abstraction of D adsorbed on Si(100): Dynamical evidence for an Eley–Rideal mechanism. Chem. Phys. Lett. 278, 71–76. Burrows, V.A., Chabal, Y.J., Higashi, G.S., Raghavachari, K., Christman, S.B., 1988. Infrared spectroscopy of Si(111) surfaces after HF treatment: Hydrogen termination and surface morphology. Appl. Phys. Lett. 53, 998–1000. Burton, W.K., Cabrera, N., Frank, F.C., 1951. The growth of crystals and the equilibrium structure of their surfaces. Philos. Trans. R. Soc. London, A 243, 299–358. Campbell, S.A., Lewerenz, H.J. (Eds.), 1998. Semiconductor Micromachining: Fundamental Electrochemistry and Physics. John Wiley & Sons, Chichester. Campbell, S.A., Schiffrin, D.J., Tufton, P.J., 1993. Chemical and electrochemical anisotropic dissolution of silicon in ethylenediamine plus pyrocatechol plus water media. J. Electroanal. Chem. 344, 211–233. Canham, L. (Ed.), 1997. Properties of Porous Silicon. Institution of Electrical Engineers, London. Canham, L.T., 1990. Silicon quantum wire array fabrication by electrochemical and chemical dissolution of wafers. Appl. Phys. Lett. 57, 1046–1048. Canham, L.T., Houlton, M.R., Leong, W.Y., Pickering, C., Keen, J.M., 1991. Atmospheric impregnation of porous silicon at room temperature. J. Appl. Phys. 70, 422–431. Carter, L.E., Carter, E.A., 1996a. Ab initio-derived dynamics for F2 reactions with partially fluorinated Si(100) surfaces: Translational activation as a possible etching tool. J. Phys. Chem. 100, 873–887. Carter, L.E., Carter, E.A., 1996b. Simulated reaction dynamics of F atoms on partially fluorinated Si(100) surfaces. Surf. Sci. 360, 200–212. Cattarin, S., Chazalviel, J.-N., Da Fonseca, C., Ozanam, F., Peter, L.M., Schlichthörl, G., Stumper, J., 1998. In situ characterization of the p-Si/NH4 F interface during dissolution in the current oscillations regime. J. Electrochem. Soc. 145, 498–502. Cattarin, S., Musiani, M.M., 1999. Electrodissolution and passivation of silicon in aqueous alkaline media: A volammetric and impedance investigation. J. Phys. Chem. B 103, 3162–3169. Chabal, Y.J., Chaban, E.E., Christman, S.B., 1983. High resolution infrared study of hydrogen chemisorbed on Si(100). J. Electron Spectrosc. Relat. Phenom. 29, 35. Chabal, Y.J., Harris, A.L., Raghavachari, K., Tully, J.C., 1993. Infrared spectroscopy of H-terminated silicon surfaces. Internat. J. Mod. Phys. B 7, 1031. Chandrasekaran, H., Sumanasekara, G.U., Sunkara, M.K., 2006. Rationalization of nanowire synthesis using low-melting point metals. J. Phys. Chem. B 110, 18351–18357. Chang, H., Bard, A.J., 1991. Scanning tunneling microscopy studies of carbon–oxygen reactions on highly oriented pyrolytic graphite. J. Am. Chem. Soc. 113, 5588–5596. Chang, J.P., Coburn, J.W., 2003. Plasma–surface interactions. J. Vac. Sci. Technol., A 21, S145–S151. Chazalviel, J.-N., Belaïdi, A., Safi, M., Maroun, F., Erné, B.H., Ozanam, F., 2000. In situ semiconductor surface characterisation: A comparative infrared study of Si, Ge and GaAs. Electrochim. Acta 45, 3205. Chazalviel, J.-N., Ozanam, F., 1997. In situ infrared characterization of the silicon surface in hydrofluoric acid. J. Appl. Phys. 81, 7684–7686. Chen, D.X., Boland, J.J., 2003. Steric interaction model of roughening and vacancy reorganization on halogenterminated Si(100)-2 × 1 surfaces. Phys. Rev. B 67, 195328. Chen, Z., Cao, C.B., 2006. Effect of size in nanowires grown by the vapor–liquid–solid mechanism. Appl. Phys. Lett. 88, 143118. Cheng, C.C., Yates Jr., J.T., 1991. H-induced surface restructuring on Si(100): Formation of higher hydrides. Phys. Rev. B 43, 4041–4045. Chuang, M.-C., Coburn, J.W., 1990. Molecular beam study of gas-surface chemistry in the ion-assisted etching of silicon with atomic and molecular hydrogen and chlorine. J. Vac. Sci. Technol., A 8, 1969. Ciraci, S., Butz, R., Oellig, E.M., Wagner, H., 1984. Chemisorption of hydrogen on the Si(100) surface: Monohydride and dihydride phases. Phys. Rev. B 30, 711–720.


855

Claeyssens, F., Henley, S.J., Ashfold, M.N.R., 2003. Comparison of the ablation plumes arising from ArF laser ablation of graphite, silicon, copper, and aluminum in vacuum. J. Appl. Phys. 94, 2203–2211. Clausen, B.S., Schiøtz, J., Gråkæk, L., Ouesen, C.V., Jacobsen, K.W., Nørskov, J.K., Topsøe, H., 1994. Wetting/non-wetting phenomena during catalysis: Evidence from in situ on-line EXAFS studies of Cu-based catalysts. Topics in Catalysis 1, 367–376. Collins, R.T., Fauchet, P.M., Tischler, M.A., 1997. Porous silicon: From luminescence to LEDs. Phys. Today 50 (1), 24. Copel, M., Reuter, M.C., Kaxiras, E., Tromp, R.M., 1989. Surfactants in epitaxial growth. Phys. Rev. Lett. 63, 632–635. Craig, B.I., Smith, P.V., 1990. The structure of the Si(100)2 × 1:H surface. Surf. Sci. 226, L55–L58. Crouch, C.H., Carey, J.E., Warrender, J.M., Aziz, M.J., Mazur, E., Génin, F.Y., 2004. Comparison of structure and properties of femtosecond and nanosecond laser-structured silicon. Appl. Phys. Lett. 84, 1850–1852. Cui, Y., Lauhon, L.J., Gudiksen, M.S., Wang, J.F., Lieber, C.M., 2001. Diameter-controlled synthesis of singlecrystal silicon nanowires. Appl. Phys. Lett. 78, 2214–2216. Cui, Y., Zhong, Z.H., Wang, D.L., Wang, W.U., Lieber, C.M., 2003. High performance silicon nanowire field effect transistors. Nano Lett. 3, 149–152. Cullis, A.G., Canham, L.T., Calcott, P.D.J., 1997. The structural and luminescence properties of porous silicon. J. Appl. Phys. 82, 909–965. D’Evelyn, M.P., Yang, Y.L., Sutcu, L.F., 1992. π -bonded dimers, preferential pairing and first-order desorption kinetics of hydrogen on Si(100)-(2 × 1). J. Chem. Phys. 96, 852–855. Dabrowski, ˛ J., Müssig, H.-J., 2000. Silicon Surfaces and Formation of Interfaces: Basic Science in the Industrial World. World Scientific, Singapore. Dabrowski, ˛ J., Pehlke, E., Scheffler, M., 1994. Calculation of the surface stress anisotropy for the buckled Si(001)(1 × 2) and p(2 × 2) surfaces. Phys. Rev. B 49, 4790–4793. Dabrowski, ˛ J., Scheffler, M., 1992. Self-consistent study of the electronic and structural properties of the clean Si(001)(2 × 1) surface. Appl. Surf. Sci. 56–58, 15–19. Dai, H., 2002. Carbon nanotubes: Opportunities and challenges. Surf. Sci. 500, 218–241. Daruka, I., Barabási, A.-L., 1997. Dislocation-free island formation in heteroepitaxial growth: A study at equilibrium. Phys. Rev. Lett. 79, 3708–3711. Daruka, I., Tersoff, J., Barabási, A.L., 1999. Shape transition in growth of strained islands. Phys. Rev. Lett. 83, 2753–2756. De Jong, K.P., Geus, J.W., 2000. Carbon nanofibers: Catalytic synthesis and applications. Catal. Rev. 42, 481– 510. de Unamuno, S., Fogarassy, E., 1989. A thermal description of the melting of c- and a-silicon under pulsed excimer lasers. Appl. Surf. Sci. 36, 1–11. Demic, C.P., Chan, K.K., Blum, J., 1992. Deep trench plasma-etching of single-crystal silicon using SF6 /O2 gas-mixtures. J. Vac. Sci. Technol. B 10, 1105–1110. Depero, L.E., Curri, M.L., 2004. Inorganic self-assembly. Curr. Opin. Solid State Mater. Sci. 8, 103–109. Desjonquères, M.C., Spanjaard, D., 1996. Concepts in Surface Physics. Springer-Verlag, Berlin. DeWitt, K., Valadez, L., Abbott, H., Kolasinski, K.W., Harrison, I., 2006a. Effusive molecular beam study of C2 H6 dissociation on Pt(111). J. Phys. Chem. B 110, 6714–6720. DeWitt, K., Valadez, L., Abbott, H., Kolasinski, K.W., Harrison, I., 2006b. Using effusive molecular beams and microcanonical unimolecular rate theory to characterize CH4 adsorption on Pt(111). J. Phys. Chem. B 110, 6705–6713. Ding, F., Bolton, K., Rosen, A., 2004a. Nucleation and growth of single-walled carbon nanotubes: A molecular dynamics study. J. Phys. Chem. B 108, 17369–17377. Ding, F., Rosen, A., Bolton, K., 2004b. Molecular dynamics study of the catalyst particle size dependence on carbon nanotube growth. J. Chem. Phys. 121, 2775–2779. Ding, F., Bolton, K., Rosén, A., 2006. Molecular dynamics study of SWNT growth on catalyst particles without temperature gradients. Comput. Mater. Sci. 35, 243–246. Dinger, A., Lutterloh, C., Küppers, J., 2001. Interaction of hydrogen atoms with Si(111) surfaces: Adsorption, abstraction, and etching. J. Chem. Phys. 114, 5304–5312. Dippel, O., Wright, S., Hasselbrink, E., 1997. Photochemical routes to silicon epitaxy. J. Vac. Sci. Technol., A 15, 1135.

856

K.W. Kolasinski

Doren, D.J., 1996. Kinetics and dynamics of hydrogen adsorption and desorption on silicon surfaces. In: Prigogine, I., Rice, S.A. (Eds.), Adv. Chem. Phys., vol. 95. John Wiley & Sons, New York, pp. 1–60. Dove, P.M., Han, N., De Yoreo, J.J., 2005. Mechanisms of classical crystal growth theory explain quartz and silicate dissolution behavior. Proc. Natl. Acad. Sci. USA 102, 15357–15362. Downer, M.C., Shank, C.V., 1986. Ultrafast heating of silicon on sapphire by femtosecond optical pulses. Phys. Rev. Lett. 56, 761–764. Duan, X., Lieber, C.M., 2000. General synthesis of compound semiconductor nanowires. Adv. Mater. 12, 298. Duan, X.F., Huang, Y., Agarwal, R., Lieber, C.M., 2003. Single-nanowire electrically driven lasers. Nature (London) 691, 241–245. Duan, X.F., Huang, Y., Lieber, C.M., 2002. Nonvolatile memory and programmable logic from molecule-gated nanowires. Nano Lett. 2, 487–490. Dubin, V.M., Chazalviel, J.-N., Ozanam, F., 1993. In situ photoluminescence and photomodulated infrared study of porous silicon during etching and in ambient. J. Lumin. 57, 61. Dubin, V.M., Ozanam, F., Chazalviel, J.-N., 1994. Electronic states of photocarriers in porous silicon studied by photomodulated infrared spectroscopy. Phys. Rev. B 50, 14867–14880. Dubin, V.M., Ozanam, F., Chazalviel, J.-N., 1995. In situ luminescence and IR study of porous silicon during and after anodic oxidation. Thin Solid Films 255, 87–91. Dubrovskii, V.G., Sibirev, N.V., Cirlin, G.E., Harmand, J.C., Ustinov, V.M., 2006. Theoretical analysis of the vapor–liquid–solid mechanism of nanowire growth during molecular beam epitaxy. Phys. Rev. E 73, 021603. Dujardin, R., Poydenot, V., Devillers, T., Favre-Nicolin, V., Gentile, P., Barski, A., 2006. Growth mechanism of Si nanowhiskers and SiGe heterostructures in Si nanowhiskers: X-ray scattering and electron microscopy investigations. Appl. Phys. Lett. 89, 153129. Dumas, P., Chabal, Y.J., 1991. Electron-energy-loss characterization of H-terminated Si(111) and Si(100) surfaces obtained by etching in NH4 F. Chem. Phys. Lett. 181, 537–543. Dunstan, D.J., 1997. Strain and strain relaxation in semiconductors. J. Mater. Sci. 8, 337. Dunstan, D.J., Young, S., Dixon, R.H., 1991. Geometrical theory of critical thickness and relaxation in strainedlayer growth. J. Appl. Phys. 70, 3038–3045. Dürr, M., Biedermann, A., Hu, Z., Höfer, U., Heinz, T.F., 2002a. Probing high-barrier pathways of surface reactions by scanning tunneling microscopy. Science 296, 1838–1841. Dürr, M., Höfer, U., 2004. Molecular beam investigation of hydrogen dissociation on Si(001) and Si(111) surfaces. J. Chem. Phys. 121, 8058–8067. Dürr, M., Höfer, U., 2006. Dissociative adsorption of molecular hydrogen on silicon surfaces. Surf. Sci. Rep. 61, 465–526. Dürr, M., Hu, Z., Biedermann, A., Höfer, U., Heinz, T.F., 2002b. Real-space study of the pathway for dissociative adsorption of H2 on Si(001). Phys. Rev. Lett. 88, 046104. Dürr, M., Raschke, M.B., Höfer, U., 1999. Effect of beam energy and surface temperature on the dissociative adsorption of H2 on Si(001). J. Chem. Phys. 111, 10411–10414. Dutartre, D., Warren, P., Provenier, F., Chollet, F., Pério, A., 1994. Fabrication of relaxed Si1−x Gex layers on Si substrates by rapid thermal chemical vapor deposition. J. Vac. Sci. Technol., A 12, 1009. Ehbrecht, M., Huisken, F., 1999. Gas-phase characterization of silicon nanoclusters produced by laser pyrolysis of silane. Phys. Rev. B 59, 2975–2985. Ehrlich, G., Hudda, F.G., 1966. Atomic view of self-diffusion: Tungsten on tungsten. J. Chem. Phys. 44, 1039– 1049. Emel’yanov, V.I., Babak, D.V., 2002. Defect capture under rapid solidification of the melt induced by the action of femtosecond laser pulses and formation of periodic surface structures on a semiconductor surface. Appl. Phys. A 74, 797–805. Engstrom, J.R., Hansen, D.A., Furjanic, M.J., Xia, L.Q., 1993a. Dynamics of the dissociative adsorption of disilane on Si(100): Energy scaling and the effect of corrugation. J. Chem. Phys. 99, 4051–4054. Engstrom, J.R., Nelson, M.M., Engel, T., 1989. The adsorption and reaction of fluorine on the Si(100) surface. Surf. Sci. 215, 437–500. Engstrom, J.R., Xia, L.-Q., Furjanic, M.J., Hansen, D.A., 1993b. Dissociative adsorption of Si2 H6 on silicon at hyperthermal energies: The influence of surface structure. Appl. Phys. Lett. 63, 1821–1823. Eres, G., Sharp, J.W., 1993a. Investigation of the kinetics of digermane chemisorption and reaction product desorption in thin film growth of germanium. J. Vac. Sci. Technol., A 11, 2463.


857

Eres, G., Sharp, J.W., 1993b. The role of hydride coverage in surface-limited thin-film growth of epitaxial silicon and germanium. J. Appl. Phys. 74, 7241–7250. Eriksson, J.C., 1969. Thermodynamics of surface phase systems: V. Contribution to the thermodynamics of the solid–gas interface. Surf. Sci. 14, 221–246. Ertl, G., 1998. Pattern formation at electrode surfaces. Electrochim. Acta 43, 2743. Esashi, M., Ono, T., 2005. From MEMS to nanomachine. J. Phys. D: Appl. Phys. 38, R223–R230. Fan, H.J., Werner, P., Zacharias, M., 2006. Semiconductor nanowires: From self-organization to patterned growth. Small 2, 700–717. Farnaam, M.K., Olander, D.R., 1984. The surface chemistry of the thermal cracking of silane on silicon(111). Surf. Sci. 145, 390–406. Fiory, A.T., Ravindra, N.M., 2003. Light emission from silicon: Some perspectives and applications. J. Electron. Mater. 32, 1043–1051. Fitzgerald, E.A., 1991. Dislocations in strained-layer epitaxy: Theory, experiment, and applications. Mater. Sci. Reports 7, 87–140. Flamm, D.L., 1990. Mechanisms of silicon etching in fluorine-containing and chlorine-containing plasmas. Pure Appl. Chem. 62, 1709–1720. Flätgen, G., Krischer, K., Pettinger, B., Doblhofer, K., Junkes, H., Ertl, G., 1995. Two-dimensional imaging of potential waves in electrochemical systems by surface plasmon microscopy. Science 269, 668–671. Flidr, J., Huang, Y.C., Newton, T.A., Hines, M.A., 1998. Extracting site-specific reaction rates from steady surface morphologies: Kinetic Monte Carlo simulations of aqueous Si(111) etching. J. Chem. Phys. 108, 5542–5553. Fojtik, A., Giersig, M., Henglein, A., 1993. Formation of nanometer-size silicon particles in a laser induced plasma in SiH4 . Ber. Bunsen-Ges. Phys. Chem. 97, 1493. Föll, H., Christophersen, M., Carstensen, J., Hasse, G., 2002. Formation and application of porous silicon. Mater. Sci. Eng., R 39, 93–141. Foltyn, S.R., 1994. Surface modification of materials by cumulative laser irradiation. In: Chrisey, D.B., Hubler, G.K. (Eds.), Pulsed Laser Deposition of Thin Films. Wiley, New York, pp. 89–113. Foxon, C.T., Joyce, B.A., 1975. Interaction kinetics of As4 and Ga on {100} GaAs surfaces using a modulated molecular beam technique. Surf. Sci. 50, 434–450. Foxon, C.T., Joyce, B.A., 1977. Interaction kinetics of As2 and Ga on {100} GaAs surfaces. Surf. Sci. 64, 293– 304. Franchi, S., Trevisi, G., Seravalli, L., Frigeri, P., 2003. Quantum dot nanostructures and molecular beam epitaxy. Prog. Cryst. Growth Characterization Mater. 47, 166–195. Frank, F.C., van der Merwe, J.H., 1949a. One-dimensional dislocations. I. Static theory. Proc. R. Soc. London, A 198, 205–216. Frank, F.C., van der Merwe, J.H., 1949b. One-dimensional dislocations. II. Misfitting monolayers and oriented overgrowth. Proc. R. Soc. London, A 198, 216–225. Frank, M.M., Chabal, Y.J., Green, M.L., Delabie, A., Brijs, B., Wilk, G.D., Ho, M.Y., da Rosa, E.B.O., Baumvol, I.J.R., Stedile, F.C., 2003a. Enhanced initial growth of atomic-layer-deposited metal oxides on hydrogenterminated silicon. Appl. Phys. Lett. 83, 740–742. Frank, M.M., Chabal, Y.J., Wilk, G.D., 2003b. Nucleation and interface formation mechanisms in atomic layer deposition of gate oxides. Appl. Phys. Lett. 82, 4758–4760. Frank, M.M., Wilk, G.D., Starodub, D., Gustafsson, T., Garfunkel, E., Chabal, Y.J., Grazul, J., Muller, D.A., 2005. HfO2 and Al2 O3 gate dielectrics on GaAs grown by atomic layer deposition. Appl. Phys. Lett. 86, 152904. Frohnhoff, S., Marso, M., Berger, M.G., Thönissen, M., Lüth, H., Münder, H., 1995. An extended quantum model for porous silicon formation. J. Electrochem. Soc. 142, 615–620. Garcia, S.P., Bao, H., Hines, M.A., 2003. Understanding the pH dependence of silicon etching: The importance of dissolved oxygen in buffered HF etchants. Surf. Sci. 541, 252–261. Garcia, S.P., Bao, H., Hines, M.A., 2004. Etchant anisotropy controls the step bunching instability in koh etching of silicon. Phys. Rev. Lett. 93, 166102. Gates, S.M., 1988. Adsorption kinetics of SiH4 , Si2 H6 , and Si3 H8 on the Si(111)-(7 × 7) surface. Surf. Sci. 195, 307–329. Gates, S.M., Greenlief, C.M., Beach, D.B., 1990a. Decomposition mechanisms of SiHx species on Si(1000-(2×1) for x = 2, 3, and 4. J. Chem. Phys. 93, 7493–7503.

858

K.W. Kolasinski

Gates, S.M., Greenlief, C.M., Beach, D.B., Holbert, P.A., 1990b. Decomposition of silane on Si(111)-(7 × 7) and Si(100)-(2 × 1) surfaces below 500 ◦ C. J. Chem. Phys. 92, 3144–3153. Gates, S.M., Greenlief, C.M., Beach, D.B., Kunz, R.R., 1989a. Reactive sticking coefficient of silane on the Si(111)-(7 × 7) surface. Chem. Phys. Lett. 154, 505–510. Gates, S.M., Kulkarni, S.K., 1991. Kinetics of surface reactions in very low-pressure chemical vapor deposition of Si from SiH4 . Appl. Phys. Lett. 58, 2963–2965. Gates, S.M., Kunz, R., Greenlief, C.M., 1989b. Silicon hydride etch products from the reaction of atomic hydrogen with Si(100). Surf. Sci. 207, 364–384. Geohegan, D.B., 1994. Diagnostics and characteristics of pulsed laser deposition laser plasmas. In: Chrisey, D.B., Hubler, G.K. (Eds.), Pulsed Laser Deposition of Thin Films. Wiley, New York, pp. 115–165. Gerischer, H., 1990. The impact of semiconductors on the concepts of electrochemistry. Electrochim. Acta 35, 1677. Gerischer, H., 1997. Principles of Electrochemistry. In: Gellings, P., Bouwmeester, H. (Eds.), The CRC Handbook of Solid State Electrochemistry. CRC Press, Boca Raton, pp. 9–73. Gerischer, H., Allongue, P., Costa Kieling, V., 1993. The mechanism of the anodic oxidation of silicon in acidic fluoride solutions revisited. Ber. Bunsen-Ges. Phys. Chem. 97, 753. Gerischer, H., Lübke, M., 1987. The electrochemical behaviour of n-type silicon(111)-surfaces in fluoride containing aqueous electrolytes. Ber. Bunsen-Ges. Phys. Chem. 91, 394. Gerischer, H., Lübke, M., 1988. Electrolytic growth and dissolution of oxide layers on silicon in aqueous solutions of fluorides. Ber. Bunsen-Ges. Phys. Chem. 92, 573. Gerischer, H., Mindt, W., 1968. The mechanisms of the decomposition of semiconductors by electrochemical oxidation and reduction. Electrochim. Acta 13, 1329. Gibbs, J.W., 1928. Collected Works. Thermodynamics, vol. 1. Longmans, London. Givargizov, E.I., 1975. Fundamental aspects of VLS growth. J. Cryst. Growth 31, 20–30. Glembocki, O.J., Stahlbush, R.E., Tomkiewicz, M., 1985. Bias-dependent etching of silicon in aqueous KOH. J. Electrochem. Soc. 132, 145–151. Gorostiza, P., Allongue, P., Díaz, R., Morante, J.R., Sanz, F., 2003. Electrochemical characterization of the opencircuit deposition of platinum on silicon from fluoride solutions. J. Phys. Chem. B 107, 6454–6461. Gossmann, H.-J., Feldman, L.C., 1985. Initial stages of silicon molecular-beam epitaxy: Effects of surface reconstruction. Phys. Rev. B 32, 6–11. Gudiksen, M.S., Lauhon, L.J., Wang, J., Smith, D.C., Lieber, C.M., 2002. Growth of nanowire superlattice structures for nanoscale photonics and electronics. Nature (London) 415, 617–620. Halsey, G.D., 1978. Surface tension and spreading pressure of mobile and registered monolayers. Surf. Sci. 72, 1–7. Hamers, R.J., Köhler, U.K., Demuth, J.E., 1990. Epitaxial growth of silicon on Si(001) by scanning tunneling microscopy. J. Vac. Sci. Technol., A 8, 195. Hamers, R.J., Tromp, R.M., Demuth, J.E., 1986. Scanning tunneling microscopy of Si(001). Phys. Rev. B 34, 5343–5357. Hannon, J.B., Kodambaka, S., Ross, F.M., Tromp, R.M., 2006. The influence of the surface migration of gold on the growth of silicon nanowires. Nature (London) 440, 69–71. Harmand, J.C., Patriarche, G., Péré-Laperne, N., Mérat-Combes, M.-N., Travers, L., Glas, F., 2005. Analysis of vapor–liquid–solid mechanism in Au-assisted GaAs nanowire growth. Appl. Phys. Lett. 87, 203101. Harutyunyan, A.R., Tokune, T., Mora, E., 2005. Liquid as a required catalyst phase for carbon single-walled nanotube growth. Appl. Phys. Lett. 87, 051919. Hassan, H.H., Chazalviel, J.-N., Neumann-Spallart, M., Ozanam, F., Etman, M., 1995. Chemical limitations to the anodic dissolution of p-Si in fluoride media in the presence of alkali metal cations. J. Electroanal. Chem. 381, 211. Helbert, J.N., 2001. In: Helbert, J.N. (Ed.), Handbook of VLSI Microlithography. Noyes Publications, Norwich, NY. Helveg, S., López-Cartes, C., Sehested, J., Hansen, P.L., Clausen, B.S., Rostrup-Nielsen, J.R., Abild-Pedersen, F., Nørskov, J.K., 2004. Atomic-scale imaging of carbon nanofibre growth. Nature (London) 427, 426–429. Her, T.-H., Finlay, R.J., Wu, C., Deliwala, S., Mazur, E., 1998. Microstructuring of silicon with femtosecond laser pulses. Appl. Phys. Lett. 73, 1673–1675.


859

Higashi, G.S., Chabal, Y.J., Trucks, G.W., Raghavachari, K., 1990. Ideal hydrogen termination of the Si(111) surface. Appl. Phys. Lett. 56, 656–658. Himpsel, F.J., Jung, T., Kirakosian, A., Lin, J.-L., Petrovykh, D.Y., Rauscher, H., Viernow, J., 1999. Nanowires by step decoration. MRS Bulletin 24 (8), 20. Hines, M.A., 2003. In search of perfection: Understanding the highly defect-selective chemistry of anisotropic etching. Annu. Rev. Phys. Chem. 54, 29–56. Hines, M.A., Chabal, Y.J., Harris, T.D., Harris, A.L., 1994. Measuring the structure of etched silicon surfaces with Raman spectroscopy. J. Chem. Phys. 101, 8055–8072. Hirva, P., Pakkanen, T.A., 1989. Theoretical studies on the growth mechanisms of silicon thin films by atomic layer epitaxy. Surf. Sci. 220, 137–151. Hochbaum, A.I., Fan, R., He, R., Yang, P., 2005. Controlled growth of Si nanowire arrays for device integration. Nano Lett. 5, 457–460. Höfer, U., 1996. Nonlinear optical investigations of the dynamics of hydrogen interactions with silicon surfaces. Appl. Phys. A 63, 533. Holt, J.R., Hefty, R.C., Tate, M.R., Ceyer, S.T., 2002. Comparison of the interactions of XeF2 and F2 with Si(100)(2 × 1). J. Phys. Chem. B 106, 8399–8406. Horn-von Hoegen, M., Golla, A., 1995. Influence of H on low temperature Si(111) homoepitaxy. Surf. Sci. 337, L777–L782. Horn-von Hoegen, M., LeGoues, F.K., Copel, M., Reuter, M.C., Tromp, R.M., 1991. Defect self-annihilation in surfactant-mediated epitaxial growth. Phys. Rev. Lett. 67, 1130–1133. Howe, P., Abbey, B., Le Ru, E.C., Murray, R., Jones, T.S., 2004. Strain-interactions between InAs/GaAs quantum dot layers. Thin Solid Films 464–465, 225–228. Hu, J.T., Min, O.Y., Yang, P.D., Lieber, C.M., 1999. Controlled growth and electrical properties of heterojunctions of carbon nanotubes and silicon nanowires. Nature (London) 399, 48–51. Hull, R., Bean, J.C., Bahnck, D., Peticolas, L.J., Short, K.T., Unterwald, F.C., 1991. Interpretation of dislocation propagation velocities in strained Gex Si1−x /Si(100) heterostructures by the diffusive kink pair model. J. Appl. Phys. 70, 2052–2065. Hull, R., Stach, E.A., 1996. Equilibrium and metastable strained layer semiconductor heterostructures. Curr. Opin. Solid State Mater. Sci. 1, 21–28. Humbird, D., Graves, D.B., 2004. Atomistic simulations of spontaneous etching of silicon by fluorine and chlorine. J. Appl. Phys. 96, 791–798. Ibach, H., 1997. The role of surface stress in reconstruction, epitaxial growth and stabilization of mesoscopic structures. Surf. Sci. Rep. 29, 193–263. Ibach, H., 1999. Erratum to: “The role of surface stress in reconstruction, epitaxial growth and stabilization of mesoscopic structures” [Surf. Sci. Rep. 29 (1997) 193]. Surf. Sci. Rep. 35, 71–73. Imbihl, R., Demuth, J.E., Gates, S.M., Scott, B.A., 1989. Interaction of Si2 H6 with a Si(111)-7 × 7 Surface. Phys. Rev. B 39, 5222–5233. Imbihl, R., Ertl, G., 1995. Oscillatory kinetics in heterogeneous catalysis. Chem. Rev. 95, 697–733. Isobe, C., Cho, H.-C., Crowell, J.E., 1993. The photo-induced reaction of digermane with the Si(100)(2 × 1):D surface. Surf. Sci. 295, 117–132. Jain, S.C., Maes, H.E., Van Overstraeten, R., 1997. Semiconductor strained layers. Curr. Opin. Solid State Mater. Sci. 2, 722–727. Jakob, P., Chabal, Y.J., Raghavachari, K., Becker, R.S., Becker, A.J., 1992. Kinetic model of the chemical etching of Si(111) surfaces by buffered HF solutions. Surf. Sci. 275, 407–413. Jandeleit, J., Urbasch, G., Hoffmann, H.D., Treusch, H.G., Kreutz, E.W., 1996. Picosecond laser ablation of thin copper films. Appl. Phys. A 63, 117–121. Jansson, U., Uram, K., 1989. The adsorption of hydrogen on Si(111)-7 × 7 as studied by multiple internal reflection spectroscopy. J. Chem. Phys. 91, 7978–7987. Jasinski, J.M., Gates, S.M., 1991. Silicon chemical vapor deposition one step at a time: Fundamental studies of silicon hydride chemistry. Acc. Chem. Res. 24, 9–15. Jasinski, J.M., Meyerson, B.S., Scott, B.A., 1987. Mechanistic studies of chemical vapor deposition. Annu. Rev. Phys. Chem. 38, 109. Jensen, L.E., Björk, M.T., Jeppesen, S., Persson, A.I., Ohlsson, B.J., Samuelson, L., 2004. Role of surface diffusion in chemical beam epitaxy of InAs nanowires. Nano Lett. 4, 1961–1964.

860

K.W. Kolasinski

Ji, J.-Y., Shen, T.-C., 2004. Low-temperature silicon epitaxy on hydrogen-terminated Si(001) surfaces. Phys. Rev. B 70, 115309. Jing, Z., Whitten, J.L., 1991. Ab initio studies of silane decomposition on Si(100). Phys. Rev. B 44, 1741–1746. Jo, S.K., Gong, B., Hess, G., White, J.M., Ekerdt, J.G., 1997. Low-temperature Si(100) etching: Facile abstraction of SiH3 (a) by thermal hydrogen atoms. Surf. Sci. 394, L162. Johansson, J., Svensson, C.P.T., Mårtensson, T., Samuelson, L., Seifert, W., 2005. Mass transport model for semiconductor nanowire growth. J. Phys. Chem. B 109, 13567–13571. Johansson, J., Wacaser, B.A., Dick, K.A., Seifert, W., 2006. Growth related aspects of epitaxial nanowires. Nanotechnology 17, S355–S361. John, G.C., Singh, V.A., 1995. Porous silicon: Theoretical studies. Phys. Rep. 263, 93. Jones, M.E., Xia, L.Q., Maity, N., Engstrom, J.R., 1994. Translationally activated dissociative chemisorption of SiH4 on the Si(100) and Si(111) surfaces. Chem. Phys. Lett. 229, 401–407. Joyce, B.A., Vvedensky, D.D., 2004. Self-organized growth on GaAs surfaces. Mater. Sci. Eng., R 46, 127–176. Jungblut, H., Jakubowicz, J., Schweizer, S., Lewerenz, H.J., 2002. Mechanism of initial structure formation on highly doped n-Si (111). J. Electroanal. Chem. 527, 41. Kalkan, A.K., Henry, M.R., Li, H., Cuiffi, J.D., Hayes, D.J., Palmer, C., Fonash, S.J., 2005. Biomedical/analytical applications of deposited nanostructured Si films. Nanotechnology 16, 1383–1391. Kanaya, H., Usuda, K., Yamada, K., 1995. Examination of Si(100) surfaces treated by ultrapure water with 5 ppb dissolved oxygen concentration. Appl. Phys. Lett. 67, 682. Kang, J.K., Musgrave, C.B., 2001a. Prediction of transition state barriers and enthalpies of reaction by a new hybrid density-functional approximation. J. Chem. Phys. 115, 11040–11051. Kang, J.K., Musgrave, C.B., 2001b. Theoretical study of the chemical vapor deposition of (100) silicon from silane. Phys. Rev. B 64, 245330. Kang, Y., Jorné, J., 1998. Photoelectrochemical dissolution of n-type silicon. Electrochim. Acta 43, 2398. Kavulak, D.F., Abbott, H.L., Harrison, I., 2005. Nonequilibrium activated dissociative chemisorption: SiH4 on Si(100). J. Phys. Chem. B 109, 685–688. Kikkawa, J., Ohno, Y., Takeda, S., 2005. Growth rate of silicon nanowires. Appl. Phys. Lett. 86, 123109. Klug, D.-A., Du, W., Greenlief, C.M., 1993. Adsorption and decomposition of Ge2 H6 on Si(100). J. Vac. Sci. Technol., A 11, 2067. Koch, R., 1997. Intrinsic stress of epitaxial thin films and surface layers. In: King, D.A., Woodruff, D.P. (Eds.), The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis: Growth and Properties of Ultrathin Epitaxial Layers, vol. 8. Elsevier, Amsterdam, p. 448. Kodambaka, S., Tersoff, J., Reuter, M.C., Ross, F.M., 2006. Diameter-independent kinetics in the vapor–liquid– solid growth of Si nanowires. Phys. Rev. Lett. 96, 096105. Koehler, B.G., Mak, C.H., Arthur, D.A., Coon, P.A., George, S.M., 1988. Desorption kinetics of hydrogen and deuterium from Si(111)-7 × 7 studied using laser-induced thermal desorption. J. Chem. Phys. 89, 1709. Köhler, U., Andersohn, L., Bethge, H., 1997. The influence of hydrogen on CVD-growth on Si(111) surfaces. Phys. Status Solidi A 159, 39. Koker, L., Kolasinski, K.W., 2000. Investigations of the photoelectrochemical etching of Si and porous Si in aqueous HF. Phys. Chem. Chem. Phys. 2, 277–281. Koker, L., Kolasinski, K.W., 2001. Laser-assisted formation of porous silicon in diverse fluoride solutions: Reactions kinetics and mechanistic implications. J. Phys. Chem. B 105, 3864–3871. Koker, L., Wellner, A., Sherratt, P.A.J., Neuendorf, R., Kolasinski, K.W., 2002. Laser–assisted formation of porous silicon in diverse fluoride solutions: Hexafluorosilicate deposition. J. Phys. Chem. B 106, 4424–4431. Kolasinski, K.W., 1995. Dynamics of hydrogen interactions with Si(100) and Si(111) surfaces. Internat. J. Mod. Phys. B 9, 2753–2809. Kolasinski, K.W., 2008. Surface Science: Foundations of Catalysis and Nanoscience, second ed. John Wiley & Sons, Chichester. Kolasinski, K.W., 2003. The mechanism of Si etching in fluoride solutions. Phys. Chem. Chem. Phys. 5, 1270– 1278. Kolasinski, K.W., 2004a. Laser-assisted restructuring of silicon over nano-, meso- and macro-scales. In: Pandalai, S.G. (Ed.), Recent Research Advances in Applied Physics. Transworld Research Network, Kerala, India, pp. 267–292.


861

Kolasinski, K.W., 2004b. Nonadiabatic and ultrafast dynamics of hydrogen adsorbed on silicon. Curr. Opin. Solid State Mater. Sci. 8, 353–366. Kolasinski, K.W., 2005. The composition of fluoride solutions. J. Electrochem. Soc. 152, J99–J104; J. Electrochem. Soc. 153 (2006) L28–L29. Kolasinski, K.W., 2006. Catalytic growth of nanowires: Vapor–liquid–solid, vapor–solid–solid, solution–liquid– solid and solid–liquid–solid growth. Curr. Opin. Solid State Mater. Sci. 10, 182–191. Kolasinski, K.W., Mills, D., Nahidi, M., 2006. Laser assisted and wet chemical etching of silicon nanostructures. J. Vac. Sci. Technol. A 24, 1474–1479. Kolasinski, K.W., Nessler, W., Bornscheuer, K.-H., Hasselbrink, E., 1994a. Beam investigations of D2 adsorption on Si(100): On the importance of lattice excitations in the reaction dynamics. J. Chem. Phys. 101, 7082–7094. Kolasinski, K.W., Nessler, W., de Meijere, A., Hasselbrink, E., 1994b. Hydrogen adsorption on and desorption from Si: Considerations on the applicability of detailed balance. Phys. Rev. Lett. 72, 1356–1359. Kolasinski, K.W., Shane, S.F., Zare, R.N., 1992. Internal-state distributions of recombinative hydrogen desorption from Si(100). J. Chem. Phys. 96, 3995–4006. Koleske, D.D., Gates, S.M., Jackson, B., 1994. Atomic H abstraction of surface H on Si: An Eley–Rideal mechanism? J. Chem. Phys. 101, 3301–3309. Koleske, D.D., Gates, S.M., Schultz, J.A., 1993. Facile abstraction of chemisorbed D on Si(100) by atomic H. J. Chem. Phys. 99, 5619–5622. Kooij, E.S., Vanmaekelbergh, D., 1997. Catalysis and pore initiation in the anodic dissolution of silicon in HF. J. Electrochem. Soc. 144, 1296–1301. Kratzer, P., Penev, E., Scheffler, M., 2002. First-principles studies of kinetics in epitaxial growth of III–V semiconductors. Appl. Phys. A 75, 79–88. Kravets, V.G., Meier, C., Konjhodzic, D., Lorke, A., Wiggers, H., 2005. Infrared properties of silicon nanoparticles. J. Appl. Phys. 97, 084306. Kulkarni, S.K., Gates, S.M., Greenlief, C.M., Sawin, H.H., 1990. Mechanisms of disilane decomposition on Si(111)-7 × 7. Surf. Sci. 239, 13. Kuo, C.L., Clancy, P., 2004. MEAM molecular dynamics study of a gold thin film on a silicon substrate. Surf. Sci. 551, 39–58. Kutana, A., Makarenko, B., Rabalais, J.W., 2003. Kinetics of H atom adsorption on Si(100) at 500–650 K. J. Chem. Phys. 119, 11906–11911. Kwon, S.J., Park, J.-G., 2006. Theoretical analysis of the radius of semiconductor nanowires grown by the catalytic vapour–liquid–solid mechanism. J. Phys.: Cond. Matter 18, 3875–3885. Langa, S., Carstensen, J., Christophersen, M., Steen, K., Frey, S., Tiginyanu, I.M., Föll, H., 2005. Uniform and nonuniform nucleation of pores during the anodization of Si, Ge, and III–V semiconductors. J. Electrochem. Soc. 152, C525–C531. Lasaga, A.C., Luttge, A., 2001. Variation of crystal dissolution rate based on a dissolution stepwave model. Science 291, 2400–2405. Lauhon, L.J., Gudiksen, M.S., Wang, C.L., Lieber, C.M., 2002. Epitaxial core–shell and core–multishell nanowire heterostructures. Nature (London) 420, 57–61. Le Ru, E.C., Howe, P., Jones, T.S., Murray, R., 2003. Strain-engineered InAs/GaAs quantum dots for longwavelength emission. Phys. Rev. B 67, 165303. Lee, S.T., Wang, N., Lee, C.S., 2000. Semiconductor nanowires: Synthesis, structure and properties. Mater. Sci. Eng., A 286, 16–23. Lehmann, V., 1993. The physics of macropore formation in low doped n-type silicon. J. Electrochem. Soc. 140, 2836–2843. Lehmann, V., 2002. Electrochemistry of Silicon: Instrumentation, Science, Materials and Applications. WileyVCH, Weinheim. Lehmann, V., Gösele, U., 1991. Porous silicon formation: A quantum wire effect. Appl. Phys. Lett. 58, 856–858. Lehmann, V., Grünning, U., 1997. The limits of macropore array fabrication. Thin Solid Films 297, 13–17. Lehmann, V., Rönnebeck, S., 1999. The physics of macropore formation in low-doped p-type silicon. J. Electrochem. Soc. 146, 2968–2975. Leonard, S.W., Mondia, J.P., van Driel, H.M., Toader, O., John, S., Busch, K., Birner, A., Gösele, U., Lehmann, V., 2000. Tunable two-dimensional photonic crystals using liquid-crystal infiltration. Phys. Rev. B 61, 2389.

862

K.W. Kolasinski

Leonard, S.W., van Driel, H.M., Busch, K., John, S., Birner, A., Li, A.-P., Müller, F., Gösele, U., Lehmann, V., 1999. Attenuation of optical transmission within the band gap of thin two-dimensional macroporous silicon photonic crystals. Appl. Phys. Lett. 75, 3063. Lewerenz, H.J., Aggour, M., Murrell, C., Kanis, M., Jungblut, H., Jakubowicz, J., Cox, P.A., Campbell, S.A., Hoffmann, P., Schmeisser, D., 2003. Initial stages of structure formation on silicon electrodes investigated by photoelectron spectroscopy using synchrotron radiation and in situ atomic force microscopy. J. Electrochem. Soc. 150, E185–E189. Li, L., 2000. Ferroelectric/superconductor heterostructures. Mater. Sci. Eng., R 29, 153–181. Li, N., Tan, T.Y., Gösele, U., 2007. Chemical tension and global equilibrium in VLS nanostructure growth process: From nanohillocks to nanowires. Appl. Phys. A 86, 433–440. Liehr, M., Greenlief, C.M., Kasi, S.R., Offenberg, M., 1990. Kinetics of silicon epitaxy using SiH4 in a rapid thermal chemical vapor deposition reactor. Appl. Phys. Lett. 56, 629. Lin, D.-S., Hirschorn, E.S., Chiang, T.-C., Tsu, R., Lubben, D., Greene, J.E., 1992. Scanning-tunnelingmicroscopy studies of disilane adsorption and pyrolytic growth on Si(100)-(2 × 1). Phys. Rev. B 45, 3494. Lin, D.-S., Miller, T., Chiang, T.-C., Tsu, R., Greene, J.E., 1993. Thermal reactions of disilane on Si(100) studied by synchrotron-radiation photoemission. Phys. Rev. B 48, 11846. Lin, J.S., Kuo, Y.T., 2000. Density functional study of silane adsorption onto Si(100) surface. Thin Solid Films 370, 192–198. Lin, J.S., Kuo, Y.T., Lee, M.H., Lee, K.H., Chen, J.C., 2000. Density functional study of structural and electronic properties of silane adsorbed Si(100) surface. J. Molec. Struc.-Theochem 496, 163–173. Liu, F., Li, A.H., Lagally, M.G., 2001. Self-assembly of two-dimensional islands via strain-mediated coarsening. Phys. Rev. Lett. 87, 126103. Liu, J.L., Shi, Y., Wang, F., Yu, Y., Zhang, R., Han, P., Gu, S.L., Zheng, Y.D., 1996. A method for fabricating silicon quantum wires based on SiGe/Si heterostructure. Appl. Phys. Lett. 68, 352–354. Lu, W., Lieber, C.M., 2006. Semiconductor nanowires. J. Phys. D: Appl. Phys. 39, R387–R406. Lubben, D., Tsu, R., Bramblett, T.R., Greene, J.E., 1991. Mechanisms and kinetics of Si atomic-layer epitaxy on Si(100)(2 × 1) from Si2 H6 . J. Vac. Sci. Technol., A 9, 3003–3011. Mahajan, A., Kellerman, B.K., Russell, N.M., Banerjee, S., Campion, A., Ekerdt, J.G., Tasch, A., White, J.M., Bonser, D.J., 1994. Surface chemistry of diethylsilane and diethylgermane on Si(100): An atomic layer epitaxy approach. J. Vac. Sci. Technol., A 12, 2265–2270. Maity, N., Xia, L.Q., Engstrom, J.R., 1995. Effect of PH3 on the dissociative chemisorption of SiH4 and Si2 H6 on Si(100) – Implications on the growth of in-situ doped Si thin-films. Appl. Phys. Lett. 66, 1909–1911. Makimura, T., Mizuta, T., Murakami, K., 2000. Formation dynamics of silicon nanoparticles after laser ablation studied using plasma emission caused by second-laser decomposition. Appl. Phys. Lett. 76, 1401–1403. Makino, O., Sakata, K., Yamazaki, H., Iguchi, K., Tachibana, A., 2000. Theoretical study of penetration reaction of fluorine atoms and ions into hydrogen-terminated Si(111) thin film. Thin Solid Films 374, 143–149. Maradudin, A.A., Wallis, R.F., 1980. Elastic interactions of point defects in a semi-infinite medium. Surf. Sci. 91, 423–439. Masson, L., Thibaudau, F., 2005. Role of steps in deposition rate in silane chemical vapor deposition on Si(111). Phys. Rev. B 71, 085314. Matsuda, A., 2004. Thin-film silicon – Growth process and solar cell application. Jpn. J. Appl. Phys., Part 1 43, 7909–7920. Matsumura, M., Fukidome, H., 1996. Enhanced etching rate of silicon in fluoride containing solutions at pH 6.4. J. Electrochem. Soc. 143, 2683–2686. Matsumura, M., Morrison, S.R., 1983. Anodic properties of n-Si and n-Ge electrodes in HF solution under illumination and in the dark. J. Electroanal. Chem. 147, 157. Matsuno, T., Niida, T., Tsurumaki, H., Namiki, A., 2005. Coverage dependent desorption dynamics of deuterium on Si(100) surfaces: Interpretation with a diffusion-promoted desorption model. J. Chem. Phys. 122, 024702. Matthews, J.W., Crawford, J.L., 1970. Accommodation of misfit between single-crystal films of nickel and copper. Thin Solid Films 5, 187–198. McAlpine, M.C., Friedman, R.S., Jin, S., Lin, K.H., Wang, W.U., Lieber, C.M., 2003. High-performance nanowire electronics and photonics on glass and plastic substrates. Nano Lett. 3, 1531–1535. Meade, R.D., Vanderbilt, D., 1989. Origins of stress on elemental and chemisorbed semiconductor surfaces. Phys. Rev. Lett. 63, 1404–1407.


863

Memmert, U., Berkó, A., Behm, R.J., 1995. Site selective SiH4 adsorption on Si(111)(7 × 7) surfaces. Surf. Sci. 325, L441–L447. Mendes, P.M., Preece, J.A., 2004. Precision chemical engineering: Integrating nanolithography and nanoassembly. Curr. Opin. Colloids Interface Sci. 9, 236–248. Mi, Z., Bhattacharya, P., 2005. Molecular-beam epitaxial growth and characteristics of highly uniform InAs/GaAs quantum dot layers. J. Appl. Phys. 98, 023510. Mijatovic, D., Eijkel, J.C.T., van den Berg, A., 2005. Technologies for nanofluidic systems: Top-down vs. bottomup – a review. Lab on a Chip 5, 492–500. Mills, D., Kolasinski, K.W., 2004. Laser-etched silicon pillars and their porosification. J. Vac. Sci. Technol., A 22, 1647–1651. Mills, D., Kolasinski, K.W., 2005. A non-lithographic method to form ordered arrays of silicon pillars and macropores. J. Phys. D: Appl. Phys. 38, 632–636. Mo, Y.-W., Kariotis, R., Savage, D.E., Lagally, M.G., 1989. Anisotropic growth and “layer-by-layer” epitaxy. Surf. Sci. 219, L551–L559. Mo, Y.-W., Kariotis, R., Swartzentruber, B.S., Webb, M.B., Lagally, M.G., 1990a. Scanning tunneling microscopy study of diffusion, growth, and coarsening of Si on Si(001). J. Vac. Sci. Technol., A 8, 201–206. Mo, Y.W., Kariotis, R., Swartzentruber, B.S., Webb, M.B., Lagally, M.G., 1990b. Growth of Si on flat and vicinal Si(001) surfaces: A scanning tunneling microscopy study. J. Vac. Sci. Technol. B 8, 232–236. Mohammad, S.N., 2006. Self-catalysis: A contamination-free, substrate-free growth mechanism for single-crystal nanowire and nanotube growth by chemical vapor deposition. J. Chem. Phys. 125, 094705. Mokler, S.M., Liu, W.K., Ohtani, N., Joyce, B.A., 1992a. A reflection high-energy electron diffraction study of Si surfaces during gas source MBE growth from disilane. Appl. Surf. Sci. 60–61, 92–98. Mokler, S.M., Ohtani, N., Zhang, J., Joyce, B.A., 1992b. Growth anisotropy observed on Si(001) surface during Si-GSMBE using disilane. Surf. Sci. 275, 401–406. Mooney, P.M., 1996. Strain relaxation and dislocations in SiGe/Si structures. Mater. Sci. Eng., R 17, 105–146. Morales, A.M., Lieber, C.M., 1998. A laser ablation method for the synthesis of crystalline semiconductor nanowires. Science 279, 208–211. Morgenstern, K., Rosenfeld, G., Comsa, G., 1996. Decay of two-dimensional Ag islands on Ag(111). Phys. Rev. Lett. 76, 2113–2116. Morgenstern, K., Rosenfeld, G., Lœgsgaard, E., Besenbacher, F., Comsa, G., 1998. Measurement of energies controlling ripening and annealing on metal surfaces. Phys. Rev. Lett. 80, 556–559. Müller, F., Birner, A., Gösele, U., Lehmann, V., Ottow, S., Föll, H., 2000. Structuring of macroporous silicon for applications as photonic crystals. J. Porous Mater. 7, 201–204. Müller, P., Saúl, A., 2004. Elastic effects on surface physics. Surf. Sci. Rep. 54, 157–258. Mullins, C.B., Pacheco, K.A., Banerjee, S., 1997. Growth and characterization of silicon thin films employing supersonic jets of SiH4 on polysilicon and Si(100). J. Appl. Phys. 82, 6281–6288. Munford, M.L., Cortes, R., Allongue, P., 2001. The preparation of ideally ordered flat H–Si(111) surfaces. Sens. Mater. 13, 259. Munt, T.P., Jesson, D.E., Shchukin, V.A., Bimberg, D., 2004. Manipulating the size distributions of quantum dots associated with strain-renormalized surface energy. Appl. Phys. Lett. 85, 1874–1886. Nachtigall, P., Jordan, K.D., Sosa, C., 1994. Theoretical study of the mechanism of recombinative hydrogen desorption from the monohydride phase of Si(100): The role of defect migration. J. Chem. Phys. 101, 8073– 8081. Nahidi, M., Kolasinski, K.W., 2006. The effects of stain etchant composition on the photoluminescence and morphology of porous silicon. J. Electrochem. Soc. 153, C19–C26. Nakayama, K.S., Weaver, J.H., 1999. Si(100)-(2 × 1) etching with fluorine: Planar removal versus three dimensional fitting. Phys. Rev. Lett. 83, 3210–3213. Namiki, A., 2006. Desorption related to adsorption of hydrogen via detailed balance on the Si(100) surfaces. Prog. Surf. Sci. 81, 337–366. Neave, J.H., Dobson, P.J., Joyce, B.A., Zhang, J., 1985. Reflection high-energy electron-diffraction oscillations from vicinal surfaces – A new approach to surface–diffusion measurements. Appl. Phys. Lett. 47, 100–102. Niinistö, L., 1998. Atomic layer epitaxy. Curr. Opin. Solid State Mater. Sci. 3, 147–152. Nishinaga, T., 2004. Atomistic aspects of molecular beam epitaxy. Prog. Cryst. Growth Characterization Mater. 48/49, 104–122.

864

K.W. Kolasinski

Noguchi, N., Suemune, I., 1993. Luminescent porous silicon synthesized by visible light irradiation. Appl. Phys. Lett. 62, 1429–1431. Nozières, P., Wolf, D.E., 1988. Interfacial properties of elastically strained materials I. Thermodynamics of a planar interface. Z. Phys. B: Condens. Matter 70, 399–407. Oehrlein, G.S., Kurogi, Y., 1998. Sidewall surface chemistry in directional etching processes. Mater. Sci. Eng., R 24, 153–183. Ogata, Y., Niki, H., Sakka, T., Iwasaki, M., 1995. Hydrogen in porous silicon: Vibrational analysis of SiHx species. J. Electrochem. Soc. 142, 195–201. Ogata, Y.H., Ikeda, T., Sakka, T., Kobayashi, T., 2000. Influence of dissolved oxygen on intensity modulated photocurrent spectroscopy (IMPS) at a silicon-hydrofluoric acid interface. Electrochim. Acta 45, 2219. Ogawa, H., Ishikawa, K., Suzuki, M.T., Hayami, Y., Fujimura, S., 1995. Effects of dissolved oxygen in HF solution on silicon surface morphology. Jpn. J. Appl. Phys. 34, 732. Olander, D.R., Balooch, M., Abrefah, J., Siekhaus, W.J., 1987. Modulated molecular-beam studies of the surface chemistry of silicon. J. Vac. Sci. Technol. B 5, 1404–1409. Ostwald, W., 1900. Über die vermeintliche Isomerie des roten und gelben Quecksilberoxyds und die Oberflächenspannung fester Körper. Z. Phys. Chem. 34, 495–503. Ottow, S., Lehmann, V., Föll, H., 1996. Processing of three-dimensional microstructures using macroporous ntype silicon. J. Electrochem. Soc. 143, 385–390. Ozanam, F., Djebri, A., Chazalviel, J.-N., 1996. The hydrogenated silicon surface in organic electrolytes probed through in situ IR spectroscopy in the ATR geometry. Electrochim. Acta 41, 687. Pacheco, K.A., Ferguson, B.A., Banerjee, S., Mullins, C.B., 1996. Surface morphology of homoepitaxial silicon thin films grown using energetic supersonic jets of disilane. Appl. Phys. Lett. 69, 1110–1112. Pacheco, K.A., Ferguson, B.A., Li, C., John, S., Banerjee, S., Mullins, C.B., 1995. Epitaxial silicon growth using supersonic jets of disilane – A model study of energetic jet deposition. Appl. Phys. Lett. 67, 2951–2953. Pala, R.G.S., Liu, F., 2005. Critical epinucleation on reconstructured surfaces and first-principle calculation of homonucleation on Si(100). Phys. Rev. Lett. 95, 136106. Parkhutik, V., 1999. Porous silicon-mechanisms of growth and applications. Solid-State Electron. 43, 1121. Patrick, D.L., Cee, V.J., Beebe Jr., T.P., 1994. “Molecular corrals” for studies of monolayer organic films. Science 265, 231–234. Paul, D.J., 2004. Si/SiGe heterostructures: From material and physics to devices and circuits. Semicond. Sci. Technol. 19, R75–R108. Pavesi, L., 1997. Porous silicon dielectric multilayers and microcavities. Riv. Nuovo Cimento 20, 1. Pavesi, L., Dal Negro, L., Mazzoleni, C., Franzò, G., Priolo, F., 2000. Optical gain in silicon nanocrystals. Nature (London) 408, 440. Payne, M.C., Roberts, N., Needs, R.J., Needels, M., Joannopoulos, J.D., 1989. Total energy and stress of metal and semiconductor surfaces. Surf. Sci. 211/212, 1–20. Pedraza, A.J., Fowlkes, J.D., Lowndes, D.H., 1999a. Self-organized silicon microcolumn arrays generated by pulsed laser irradiation. Appl. Phys. A 69, S731. Pedraza, A.J., Fowlkes, J.D., Lowndes, D.H., 1999b. Silicon microcolumn arrays grown by nanosecond pulsedexcimer laser irradiation. Appl. Phys. Lett. 74, 2322–2324. Pedraza, A.J., Jesse, S., Guan, Y.F., Fowlkes, J.D., 2001. Laser-induced surface perturbations in silicon. J. Mater. Res. 16, 3599. Peiner, E., Schlachetzki, A., 1992. Anodic dissolution during electrochemical carrier-concentration profiling of silicon. J. Electrochem. Soc. 139, 552–557. Penev, E., Kratzer, P., Scheffler, M., 2001. Effect of strain on surface diffusion in semiconductor heteroepitaxy. Phys. Rev. B 64, 085401. Persson, A.I., Larsson, M.W., Stenström, S., Ohlsson, B.J., Samuelson, L., Wallenberg, L.R., 2004. Solid-phase diffusion mechanism for GaAs nanowire growth. Nature Mater. 3, 677–681. Peter, L.M., Blackwood, D.J., Pons, S., 1989. In situ characterization of the illuminated silicon-electrolyte interface by Fourier-transform infrared spectroscopy. Phys. Rev. Lett. 62, 308–311. Peter, L.M., Blackwood, D.J., Pons, S., 1990a. Photocorrosion of n-Si in ammonium fluoride solutions: An investigation by in-situ Fourier transform infrared spectroscopy. J. Electroanal. Chem. 294, 111. Peter, L.M., Borazio, A.M., Lewerenz, H.J., Stumper, J., 1990b. Photocurrent multiplication during photodissolution of n-Si in NH4 F. J. Electroanal. Chem. 290, 229.


865

Preuss, S., Demchuk, A., Stuke, M., 1995. Sub-picosecond UV laser ablation of metals. Appl. Phys. A 61, 33–37. Preuss, S., Matthias, E., Stuke, M., 1994. Subpicosecond UV-laser ablation of Ni-films – strong fluence reduction and thickness-independent removal. Appl. Phys. A 59, 79–82. Priester, C., Lannoo, M., 1995. Origin of self-assembled quantum dots in highly mismatched heteroepitaxy. Phys. Rev. Lett. 75, 93–96. Priester, C., Lannoo, M., 1997. Growth aspects of quantum dots. Curr. Opin. Solid State Mater. Sci. 2, 716–721. Propst, E.K., Kohl, P.A., 1994. The electrochemical oxidation of silicon and formation of porous silicon in acetonitrile. J. Electrochem. Soc. 141, 1006–1013. Punckt, C., Bölscher, M., Rotermund, H.H., Mikhailov, A.S., Organ, L., Budiansky, N., Scully, J.R., Hudson, J.L., 2004. Sudden onset of pitting corrosion on stainless steel as a critical phenomenon. Science 305, 1133–1136. Queeney, K.T., Weldon, M.K., Chabal, Y.J., Raghavachari, K., 2003. The microscopic origin of optical phonon evolution during water oxidation of Si(100). J. Chem. Phys. 119, 2307–2313. Rahman, F., Kuroda, M., Kiyonaga, T., Khanom, F., Tsurumaki, H., Inanaga, S., Namiki, A., 2004. Modulated hydrogen beam study of adsorption-induced desorption of deuterium from Si(100)-3 × 1:D surfaces. J. Chem. Phys. 121, 3221–3227. Rao, A.V., Ozanam, F., Chazalviel, J.-N., 1991. In situ Fourier-transform electromodulated infrared study of porous silicon formation: Evidence for solvent effects on the vibrational linewidths. J. Electrochem. Soc. 138, 153–159. Rappich, J., Lewerenz, H.J., 1995. In situ Fourier transform infrared investigation of the electrolytic hydrogenation of n-silicon(111). J. Electrochem. Soc. 142, 1233–1237. Rappich, J., Lewerenz, H.J., 1996a. In situ FTIR investigation of the electrochemical microstructuring of nSi(111). Electrochim. Acta 41, 675. Rappich, J., Lewerenz, H.J., 1996b. Photo- and potential-controlled nanoporous silicon formation on n-Si(111): An in situ FTIR investigation. Thin Solid Films 276, 25. Rappich, J., Lewerenz, H.J., Gerischer, H., 1993. The surface of Si(111) during etching in NaOH studied by FTIR spectroscopy in the ATR technique. J. Electrochem. Soc. 140, L187–L189. Raschke, M.B., Höfer, U., 2001. Chemisorption energy of hydrogen on silicon surfaces. Phys. Rev. B 63, 201303. Rauscher, H., 2001. The interaction of silanes with silicon single crystal surfaces: Microscopic processes and structures. Surf. Sci. Rep. 42, 207–328. Reider, G.A., Höfer, U., Heinz, T.F., 1991. Desorption kinetics of hydrogen from the Si(111)7 × 7 surface. J. Chem. Phys. 94, 4080–4083. Rethfeld, B., Sokolowski-Tinten, K., von der Linde, D., Anisimov, S.I., 2002. Ultrafast thermal melting of laserexcited solids by homogeneous nucleation. Phys. Rev. B 65, 092103. Riedel, D., Hernández-Pozos, J.L., Kolasinski, K.W., Palmer, R.E., 2004. Arranged silicon conical spike structures from optical diffraction and ultrafast laser etching in halogen gas. Appl. Phys. A 78, 381–385. Rockett, A., Barnett, S.A., 1988. Monte Carlo simulations of the role of (2 × 1) reconstruction dimers and surface diffusion on the growth of Si(001). J. Vac. Sci. Technol., A 6, 2039. Röder, H., Hahn, E., Brune, H., Bucher, J.-P., Kern, K., 1993. Building one and two dimensional nanostructures by diffusion controlled aggregation at surfaces. Nature (London) 366, 141–143. Rosei, F., 2004. Nanostructured surfaces: Challenges and frontiers in nanotechnology. J. Phys.: Cond. Matter 16, S1373–S1436. Rosenfeld, G., Poelsema, B., Comsa, G., 1997. Epitaxial growth modes far from equilibrium. In: King, D.A., Woodruff, D.P. (Eds.), The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis: Growth and Properties of Ultrathin Epitaxial Layers, vol. 8. Elsevier, Amsterdam, p. 66. Ross, F.M., Tersoff, J., Reuter, M.C., 2005. Sawtooth faceting in silicon nanowires. Phys. Rev. Lett. 95, 146104. Rotermund, H.H., 1997. Imaging of dynamic processes on surfaces by light. Surf. Sci. Rep. 29, 265–364. Rotermund, H.H., Engel, W., Kordesch, M., Ertl, G., 1990. Imaging of spatio-temporal pattern evolution during carbon monoxide oxidation on platinum. Nature (London) 343, 355. Rotermund, H.H., Haas, G., Franz, R.U., Tromp, R.M., Ertl, G., 1995. Imaging pattern formation in surface reactions from ultrahigh vacuum up to atmospheric pressures. Science 270, 608–610. Rottman, C., Wortis, M., 1984. Statistical mechanics of equilibrium crystal shapes: Interfacial phase diagrams and phase transitions. Phys. Rep. 103, 59–79. Sacher, E., Yelon, A., 1991. Comment on “Mechanism of HF etching of silicon surfaces: A theoretical understanding of hydrogen passivation”. Phys. Rev. Lett. 66, 1647.

866

K.W. Kolasinski

Safi, M., Chazalviel, J.-N., Cherkaoui, M., Belaïdi, A., Gorochov, O., 2002. Etching of n-type silicon in (HF + oxidant) solutions: In situ characterisation of surface chemistry. Electrochim. Acta 47, 2573. Sagara, T., Kuga, T., Tanaka, K., Shibataka, T., Fujimoto, T., Namiki, A., 2002. Translational heating of D2 molecules thermally desorbed from Si(100) and Ge(100) surfaces. Phys. Rev. Lett. 89, 086101. Sakuraba, M., Murota, J., Ono, S., 1993. Silicon atomic layer growth using flash heating in CVD. J. Phys. IV C3, 449. Sander, D., 2003. Surface stress: Implications and measurements. Curr. Opin. Solid State Mater. Sci. 7, 51–57. Sato, Y., Maeda, M., 1994. Study of HF-treated heavily-doped Si surface using contact angle measurements. Jpn. J. Appl. Phys. 33, 6508. Saunders, W.A., Sercel, P.C., Lee, R.B., Atwater, H.A., Vahala, K.J., Flagan, R.C., Escorcia-Aparcio, E.J., 1993. Synthesis of luminescent silicon clusters by spark ablation. Appl. Phys. Lett. 63, 1549–1551. Schlom, D.G., Haeni, J.H., Lettieri, J., Theis, C.D., Tian, W., Jiang, J.C., Pan, X.Q., 2001. Oxide nano-engineering using MBE. Mater. Sci. Eng., B 87, 282–291. Schluter, M., 1988. Structural and electronic properties of elemental semiconductors and surfaces. In: King, D.A., Woodruff, D.P. (Eds.), In: The Chemical Physics of Solid Surfaces and Heterogeneous Catalysis: Surface Properties of Electronic Materials, vol. 5. Elsevier, Amsterdam, p. 37. Schmidt, O.G., Kienzle, O., Hao, Y., Eberl, K., Ernst, F., 1999. Modified Stranski–Krastanov growth in stacked layers of self-assembled islands. Appl. Phys. Lett. 74, 1272–1274. Schmidt, V., Senz, S., Gösele, U., 2005. Diameter-dependent growth direction of epitaxial silicon nanowires. Nano Lett. 5, 931–935. Schubert, L., Werner, P., Zakharov, N.D., Gerth, G., Kolb, F.M., Long, L., Gosele, U., Tan, T.Y., 2004. Silicon nanowhiskers grown on 111Si substrates by molecular-beam epitaxy. Appl. Phys. Lett. 84, 4968–4970. Schwöbel, R.L., Shipsey, E.J., 1966. Step motion on crystal surfaces. J. Appl. Phys. 37, 3682–3686. Seo, Y.H., Nahm, K.S., Lee, K.B., 1993. Mechanistic study of silicon etching in HF-KBrO3 -H2 O solution. J. Electrochem. Soc. 140, 1453–1458. Shane, S.F., Kolasinski, K.W., Zare, R.N., 1992a. Internal-state distributions of H2 desorbed from mono- and dihydride species on Si(100). J. Chem. Phys. 97, 3704–3709. Shane, S.F., Kolasinski, K.W., Zare, R.N., 1992b. Recombinative desorption of H2 on Si(100)-(2×1) and Si(111)(7×7): Comparison of internal state distributions. J. Chem. Phys. 97, 1520–1530. Shank, C.V., Yen, R., Hirlimann, C., 1983a. Femtosecond-time-resolved surface structural dynamics of optically excited silicon. Phys. Rev. Lett. 51, 900–902. Shank, C.V., Yen, R., Hirlimann, C., 1983b. Time-resolved reflectivity measurements of femtosecond-opticalpulse-induced phase transitions in silicon. Phys. Rev. Lett. 50, 454–457. Shchukin, V.A., 2001. Entropy-driven effects in self-organized formation of quantum dots. Phys. Status Solidi B 224, 503–508. Shchukin, V.A., Bimberg, D., 1999. Spontaneous ordering of nanostructures on crystal surfaces. Rev. Mod. Phys. 71, 1125–1171. Shchukin, V.A., Bimberg, D., Munt, T.P., Jesson, D.E., 2004. Elastic interaction and self-relaxation energies of coherently strained conical islands. Phys. Rev. B 70, 085416. Shen, M.Y., Crouch, C.H., Carey, J.E., Younkin, R., Mazur, E., Sheehy, M., Friend, C.M., 2003. Formation of regular arrays of silicon microspikes by femtosecond laser irradiation through a mask. Appl. Phys. Lett. 82, 1715–1717. Shen, X.Q., Kishimoto, D., Nishinaga, T., 1994. Arsenic pressure-dependence of surface-diffusion of Ga on nonplanar Gaas substrates. Jpn. J. Appl. Phys., Part 1 33, 11–17. Shi, J., Kang, H.C., Tok, E.S., Zhang, J., 2005. Evidence for hydrogen desorption through both interdimer and intradimer paths from Si(100)-(2 × 1). J. Chem. Phys. 123, 034701. Shuttleworth, R., 1950. The surface tension of solids. Proc. Phys. Soc. A 63, 444–457. Sieber, I.V., Schmuki, P., 2005. Porous tantalum oxide prepared by electrochemical anodic oxidation. J. Electrochem. Soc. 152, C639–C644. Siegman, A.E., Fauchet, P.M., 1986. Stimulated Wood’s anomalies on laser-illuminated surfaces. IEEE J. Quantum Electron. 22, 1384–1403. Sinniah, K., Sherman, M.G., Lewis, L.B., Weinberg, W.H., Yates Jr., J.T., Janda, K.C., 1989. A new mechanism for hydrogen desorption from covalent surfaces: The monohydride phase on Si(100). Phys. Rev. Lett. 62, 567–570.


867

Sipe, J.E., Young, J.F., Preston, J.S., Van Driel, H.M., 1983. Laser-induced periodic surface-structure. I. Theory. Phys. Rev. B 27, 1141–1154. Smardon, R.D., Srivastava, G.P., 2005. Ab initio surface reaction energetics of SiH4 and Si2 H6 on Si(001)-2 × 2. J. Chem. Phys. 123, 174703. Smentkowski, V.S., 2000. Trends in sputtering. Prog. Surf. Sci. 64, 1. Smith, R.L., Chuang, S.-F., Collins, S.D., 1988. A theoretical model of the formation morphologies of porous silicon. J. Electron. Mater. 17, 533. Smith, R.L., Collins, S.D., 1992. Porous silicon formation mechanisms. J. Appl. Phys. 71, R1–R22. Smith, R.L., Kloeck, B., Derooij, N., Collins, S.D., 1987. The potential dependence of silicon anisotropic etching in KOH at 60 ◦ C. J. Electroanal. Chem. 238, 103–113. Sokolowski-Tinten, K., Bialkowski, J., Cavalleri, A., von der Linde, D., Oparin, A., Meyer-ter-Vehn, J., Anisimov, S.I., 1998. Transient states of matter during short pulse laser ablation. Phys. Rev. Lett. 81, 224–227. Sokolowski-Tinten, K., Bialkowski, J., von der Linde, D., 1995. Ultrafast laser-induced order-disorder transitions in semiconductors. Phys. Rev. B 51, 14186–14198. Sokolowski-Tinten, K., Blome, C., Blums, J., Cavalleri, A., Dietrich, C., Tarasevitch, A., Uschmann, I., Förster, E., Kammler, M., Horn-von-Hoegen, M., von der Linde, D., 2003. Femtosecond X-ray measurement of coherent lattice vibrations near the Lindemann stability limit. Nature (London) 422, 287–289. Sokolowski-Tinten, K., Blome, C., Dietrich, C., Tarasevitch, A., Horn von Hoegen, M., von der Linde, D., Cavalleri, A., Squier, J., Kammler, M., 2001. Femtosecond X-ray measurement of ultrafast melting and large acoustic transients. Phys. Rev. Lett. 87, 225701. Sokolowski-Tinten, K., Cavalleri, A., von der Linde, D., 1999. Single-pulse time- and fluence-resolved optical measurements at femtosecond excited surfaces. Appl. Phys. A 69, 577. Somashekhar, A., O’Brien, S., 1996. Etching SiO2 films in aqueous 0.49% HF. J. Electrochem. Soc. 143, 2885– 2891. Srivastava, D., Garrison, B.J., Brenner, D.W., 1991. Modeling the growth of semiconductor epitaxial films via nanosecond time scale molecular dynamics simulations. Langmuir 7, 683–692. Stampfli, P., Bennemann, K.H., 1994. Time dependence of the laser-induced femtosecond lattice instability of Si and GaAs: Role of longitudinal optical distortions. Phys. Rev. B 49, 7299–7305. Stampfli, P., Bennemann, K.H., 1995. Theory for the laser-induced femtosecond phase transition of silicon and GaAs. Appl. Phys. A 60, 191. Stranski, I.N., Kr’stanov, L., 1938. Sitzungsber. Akad. Wiss. Wien 146, 797. Stumper, J., Peter, L.M., 1991. A rotating ring-disc study of the photodissolution of n-Si in ammonium fluoride solutions. J. Electroanal. Chem. 309, 325. Suda, Y., Lubben, D., Motooka, T., Greene, J.E., 1989. Thermal and photostimulated reactions on Si2 H6 -adsorbed Si(100)2×1 surfaces: Mechanisms of Si film growth by atomic-layer epitaxy. J. Vac. Sci. Technol. B 7, 1171– 1175. Suda, Y., Lubben, D., Motooka, T., Greene, J.E., 1990. Adsorption and thermal dissociation of disilane (Si2 H6 ) on Si(100)2 × 1. J. Vac. Sci. Technol., A 8, 61–67. Suemitsu, M., Kim, K.-J., Miyamoto, N., 1994. Role of Ge in SiGe epitaxial growth using silane/germane gassource molecular beam epitaxy. J. Vac. Sci. Technol., A 12, 2271–2275. Sundaram, S.K., Mazur, E., 2002. Inducing and probing non-thermal transitions in semiconductors using femtosecond laser pulses. Nature Mater. 1, 217–224. Szabò, A., Engel, T., 1994. Etching of Si surfaces with hot chlorine beams: Translational and vibrational excitation of the incident chlorine particles. J. Vac. Sci. Technol. A 12, 648–657. Takagi, D., Homma, Y., Hibino, H., Suzuki, S., Kobayashi, Y., 2006. Single-walled carbon nanotube growth from highly activated metal nanoparticles. Nano Lett. 6, 2642–2645. Takahagi, T., Ishitani, A., Kuroda, H., Nagasawa, Y., 1991. Fluorine-containing species on the hydrofluoric acid etched silicon single-crystal surface. J. Appl. Phys. 69, 803–807. Tan, T.Y., Li, N., Gösele, U., 2003. Is there a thermodynamic size limit of nanowires grown by the vapor–liquid– solid process? Appl. Phys. Lett. 83, 1199–1201. Tan, T.Y., Li, N., Gösele, U., 2004. On the thermodynamic size limit of nanowires grown by the vapor–liquid– solid process. Appl. Phys. A 78, 519–526. Tanaka, T., Fukuda, T., Nagasawa, Y., Miyazaki, S., Hirose, M., 1990. Atomic layer growth of silicon by excimer laser induced cryogenic chemical vapor deposition. Appl. Phys. Lett. 56, 1445–1447.

868

K.W. Kolasinski

Tersoff, J., Teichert, C., Lagally, M.G., 1996. Self-organization in growth of quantum dot superlattices. Phys. Rev. Lett. 76, 1675–1678. Tong, W.M., Williams, R.S., 1994. Kinetics of surface growth: Phenomenology, scaling and mechanisms of smoothening and roughening. Annu. Rev. Phys. Chem. 45, 401. Trucks, G.W., Raghavachari, K., Higashi, G.S., Chabal, Y.J., 1990. Mechanism of HF etching of silicon surfaces: A theoretical understanding of hydrogen passivation. Phys. Rev. Lett. 65, 504–507. Trucks, G.W., Raghavachari, K., Higashi, G.S., Chabal, Y.J., 1991. Trucks et al. reply. Phys. Rev. Lett. 66, 1648. Tsuchiya, H., Macak, J.M., Sieber, I., Taveira, L., Ghicov, A., Sirotna, K., Schmuki, P., 2005a. Self-organized porous WO3 formed in NaF electrolytes. Electrochem. Commun. 7, 295–298. Tsuchiya, H., Macak, J.M., Taveira, L., Schmuki, P., 2005b. Fabrication and characterization of smooth high aspect ratio zirconia nanotubes. Chem. Phys. Lett. 410, 188–191. Turner, A.R., Pemble, M.E., Fernández, J.M., Joyce, B.A., Zhang, J., Taylor, A.G., 1995. Real time observation of reflectance anisotropy and reflection high-energy electron diffraction intensity oscillations during gas-source molecular-beam-epitaxy growth of Si and SiGe on Si(001). Phys. Rev. Lett. 74, 3213–3216. Turner, D.R., 1960. On the mechanism of chemically etching germanium and silicon. J. Electrochem. Soc. 107, 810–816. Uram, K.J., Jansson, U., 1989. Comparison of disilane and hydrogen adsorption on Si(111)-7 × 7. J. Vac. Sci. Technol. B 7, 1176–1181. Uram, K.J., Jansson, U., 1991. The adsorption of disilane on Si(111)-7×7 as studied by multiple internal reflection spectroscopy. Surf. Sci. 249, 105–116. van Buuren, T., Dinh, L.N., Chase, L.L., Siekhaus, W.J., Terminello, L.J., 1998. Changes in the electronic properties of Si nanocrystals as a function of particle size. Phys. Rev. Lett. 80, 3803–3806. Vanderbilt, D., 1987. Absence of large compressive stress on Si(111). Phys. Rev. Lett. 59, 1456–1459. Vasquez, R.P., Fathauer, R.W., George, T., Ksendzov, A., Lin, T.L., 1992. Electronic structure of light-emitting porous Si. Appl. Phys. Lett. 60, 1004–1006. Vegard, L., 1921. Die Konstitution der Mischkristalle und die Raumfüllung der Atome. Z. Phys. 5, 17. Venables, J.A., 1973. Kinetic basis of nucleation theory. Philos. Mag. 27, 693. Venables, J.A., 1987. Nucleation calculations in a pair-binding model. Phys. Rev. B 36, 4153–4162. Venables, J.A., 1994. Atomic processes in crystal growth. Surf. Sci. 299/300, 798–817. Venables, J.A., Spiller, G.D.T., Hanbucken, M., 1984. Nucleation and growth of thin-films. Rep. Prog. Phys. 47, 399–459. Verheijen, M.A., Immink, G., de Smet, T., Borgström, M.T., Bakkers, E.P.A.M., 2006. Growth kinetics of heterostructured GaP-GaAs nanowires. J. Am. Chem. Soc. 128, 1353–1359. Vittadini, A., Selloni, A., Casarin, M., 1993. Energetics of atomic hydrogen diffusion on Si(100). Surf. Sci. 289, L625–L630. Volmer, M., Weber, A., 1926. Keimbildung in übersättigten Gebilden. Z. Phys. Chem. 119, 277–301. Wade, C.P., Chidsey, C.E.D., 1997. Etch-pit initiation by dissolved oxygen on terraces of H-Si(111). Appl. Phys. Lett. 71, 1679–1681. Wagner, R.S., Ellis, W.C., 1964. Vapor–liquid–solid mechanism of single crystal growth. Appl. Phys. Lett. 4, 89–90. Walch, S.P., 2002. Computed energetics for etching of the Si(100) surface by F and Cl atoms. Surf. Sci. 496, 271–286. Waltenburg, H.N., Yates Jr., J.T., 1995. Surface chemistry of silicon. Chem. Rev. 95, 1589–1673. Wang, C.X., Hirano, M., Hosono, H., 2006a. Origin of diameter-dependent growth direction of silicon nanowires. Nano Lett. 6, 1552–1555. Wang, D., Qian, F., Yang, C., Zhong, Z., Lieber, C.M., 2004. Rational growth of branched and hyperbranched nanowire structures. Nano Lett. 4, 871–874. Wang, F.D., Dong, A.G., Sun, J.W., Tang, R., Yu, H., Buhro, W.E., 2006b. Solution–liquid–solid growth of semiconductor nanowires. Inorg. Chem. 45, 7511–7521. Wang, H., Fischman, G.S., 1994. Role of liquid droplet surface diffusion in the vapor–liquid–solid whisker growth mechanism. J. Appl. Phys. 76, 1557–1562. Wang, L.G., Kratzer, P., Moll, N., Scheffler, M., 2000a. Size, shape, and stability of InAs quantum dots on the GaAs(001) substrate. Phys. Rev. B 62, 1897–1904.


869

Wang, L.G., Kratzer, P., Scheffler, M., 2000b. Energetics of InAs thin films and islands on the GaAs(001) substrate. Jpn. J. Appl. Phys. 39, 4298–4301. Wang, L.G., Kratzer, P., Scheffler, M., Moll, N., 1999. Formation and stability of self-assembled coherent islands in highly mismatched heteroepitaxy. Phys. Rev. Lett. 82, 4042–4045. Wang, Y., Bronikowski, M.J., Hamers, R.J., 1994. An atomically resolved scanning tunneling microscopy study of the thermal decomposition of disilane on Si(001). Surf. Sci. 311, 64–100. Wang, Y., Hamers, R.J., 1995. Boron-induced morphology changes in silicon chemical vapor deposition: A scanning tunneling microscopy study. Appl. Phys. Lett. 66, 2057–2059. Watanabe, S., Nakayama, N., Ito, T., 1991. Homogeneous hydrogen-terminated Si(111) surface formed using aqueous HF solution and water. Appl. Phys. Lett. 59, 1458–1460. Watanabe, S., Shigeno, M., 1992. Fluorine adsorption and etching on Si(111): SiH surface during immersion in HF solution. Jpn. J. Appl. Phys. 31, 1702. Watanabe, S., Sugita, Y., 1995. The role of dissolved oxygen in hot water during dissolving oxides and terminating silicon surfaces with hydrogen. Surf. Sci. 327, 1–8. Watts, R.K., 1988. Lithography. In: Sze, S.M. (Ed.), VLSI Technology. McGraw Hill, Boston. Weaver, J.F., Carlsson, A.F., Madix, R.J., 2003. The adsorption and reaction of low molecular weight alkanes on metallic single crystal surfaces. Surf. Sci. Rep. 50, 107–199. Werner, K., Butzke, S., Radelaar, S., Balk, P., 1994a. Determination of the sticking coefficient of disilane on Si(001) using the first reflection high energy electron diffraction oscillation period. J. Cryst. Growth 136, 322–327. Werner, K., Butzke, S., Radelaar, S., Balk, P., 1994b. Evidence for non-hydrogen desorption limited growth of Si from disilane at very low temperatures in gas source molecular beam epitaxy? J. Cryst. Growth 136, 338–343. Wilk, G.D., Wallace, R.M., 1999. Electrical properties of hafnium silicate gate dielectrics deposited directly on silicon. Appl. Phys. Lett. 74, 2854–2856. Wilk, G.D., Wallace, R.M., 2000. Stable zirconium silicate gates deposited directly on silicon. Appl. Phys. Lett. 76, 112–114. Wilk, G.D., Wallace, R.M., Anthony, J.M., 2000. Hafnium and zirconium silicates for advanced gate dielectrics. J. Appl. Phys. 87, 484–492. Wilk, G.D., Wallace, R.M., Anthony, J.M., 2001. High-k gate dielectrics: Current status and materials properties considerations. J. Appl. Phys. 89, 5243–5275. Williams, K.L., Martin, I.T., Fisher, E.R., 2002. On the importance of ions and ion–molecule reactions to plasmasurface interface reactions. J. Am. Soc. Mass Spectrom. 13, 518–529. Williams, K.R., Muller, R.S., 1996. Etch rates for micromachining processing. J. Microelectromechanical Systems 5, 256. Williams, R.S., Ashwin, M.J., Jones, T.S., Neave, J.H., 2004. Influence of the growth conditions on the ridge morphology during GaAs deposition on GaAs (001) patterned substrates. J. Appl. Phys. 95, 6112–6118. Williams, R.S., Ashwin, M.J., Jones, T.S., Neave, J.H., 2005. Ridge structure transformation by group-III species modification during the growth of (Al,Ga)As on patterned substrates. J. Appl. Phys. 97, 044905. Wind, R.A., Hines, M.A., 2000. Macroscopic etch anisotropies and microscopic reaction mechanisms: A micromachined structure for the rapid assay of etchant anisotropy. Surf. Sci. 460, 21–38. Winters, H.F., Coburn, J.W., 1979. The etching of silicon with XeF2 vapor. Appl. Phys. Lett. 34, 70–73. Winters, H.F., Coburn, J.W., 1985. Plasma-assisted etching mechanisms – The implications of reaction probability and halogen coverage. J. Vac. Sci. Technol. B 3, 1376–1383. Winters, H.F., Coburn, J.W., 1992. Surface science aspects of etching reactions. Surf. Sci. Rep. 14, 161–269. Winters, H.F., Coburn, J.W., Chuang, T.J., 1983. Surface processes in plasma-assisted etching environments. J. Vac. Sci. Technol. B 1, 469–480. Wintterlin, J., Avouris, P., 1993. Hydrogen effects in CVD: Selective etching and metastable phases. Surf. Sci. 286, L529–L534. Wintterlin, J., Avouris, P., 1994. Scanning tunneling microscopy (STM) studies of the chemical vapor deposition of Ge on Si(111) from Ge hydrides and a comparison with molecular beam epitaxy. J. Chem. Phys. 100, 687–704. Wise, M.L., Koehler, B.G., Gupta, P., Coon, P.A., George, S.M., 1991. Comparison of hydrogen desorption from Si(111)7 × 7 and Si(100)2 × 1. Surf. Sci. 258, 166–176.

870

K.W. Kolasinski

Wolf, D.E., Nozières, P., 1988. Interfacial properties of elastically strained materials II. Mechanical and melting equilibrium of a curved interface. Z. Phys. B: Condens. Matter 70, 507–513. Wolff, J., Papathanasiou, A.G., Kevrekidis, I.G., Rotermund, H.H., Ertl, G., 2001. Spatiotemporal addressing of surface activity. Science 294, 134–137. Wolkow, R.A., 1992. Direct observation of an increase in buckled dimers on Si(001) at low temperatures. Phys. Rev. Lett. 68, 2636–2639. Wu, Y.M., Baker, J., Hamilton, P., Nix, R.M., 1993. MIRIRS studies of the growth of Si and Si–Ge alloys from molecular precursors. Surf. Sci. 295, 133–142. Wu, Y.M., Nix, R.M., 1994. Adsorption of hydrogen and disilane on Si(100) and Si–Ge surfaces. Surf. Sci. 306, 59–68. Wulff, G., 1901. Zur Frage der Geschwindigkeit des Wachsthums und der Auflösung der Krystallflächen. Z. Kristallogr. Mineral. 34, 449–530. Xia, L.-Q., Jones, M.E., Maity, N., Engstrom, J.R., 1995a. Dissociation and pyrolysis of Si2 H6 on Si surfaces: The influence of surface structure and adlayer composition. J. Chem. Phys. 103, 1691–1701. Xia, L.Q., Engstrom, J.R., 1994. The role of surface corrugation in direct translationally activated dissociative adsorption. J. Chem. Phys. 101, 5329–5342. Xia, L.Q., Jones, M.E., Maity, N., Engstrom, J.R., 1995b. Supersonic molecular-beam scattering as a probe of thin-film deposition processes. J. Vac. Sci. Technol., A 13, 2651–2664. Xia, X., Ashruf, C.M.A., French, P.J., Rappich, J., Kelly, J.J., 2001. Etching and passivation of silicon in alkaline solution: A coupled chemical/electrochemical system. J. Phys. Chem. B 105, 5722–5729. Xia, X.H., Kelly, J.J., 2001. Chemical etching and anodic oxidation of (100) silicon in alkaline solution: The role of applied potential. Phys. Chem. Chem. Phys. 3, 5304–5310. Xie, Q., Madhukar, A., Chen, P., Kobayashi, N.P., 1995. Vertically self-organized InAs quantum box islands on GaAs(100). Phys. Rev. Lett. 75, 2542–2545. Ye, S., Saito, T., Nihonyanagi, S., Uosaki, K., Miranda, P.B., Kim, D., Shen, Y.-R., 2001. Stability of the Si–H bond on the hydrogen-terminated Si(111) surface studied by sum frequency generation. Surf. Sci. 476, 121– 128. Yilmaz, M.B., Rajagopal, A., Zimmermann, F.M., 2004. Quenching of optical second harmonic generation at the Si(001) surface by hydrogen adsorption. Phys. Rev. B 69, 125413. Yoffe, A.D., 1993. Low-dimensional systems: Quantum size effects and electronic properties of semiconductor microcrystallites (zero-dimensional systems) and some quasi-two-dimensional systems. Adv. Phys. 42, 173– 262. Young, J.F., Preston, J.S., van Driel, H.M., Sipe, J.E., 1983. Laser-induced periodic surface structure. II. Experiments on Ge, Si, Al, and brass. Phys. Rev. B 27, 1155–1172. Young, J.F., Sipe, J.E., Van Driel, H.M., 1984. Laser-induced periodic surface-structure. III. Fluence regimes, the role of feedback, and details of the induced topography in germanium. Phys. Rev. B 30, 2001–2015. Zazzera, L., Evans, J.F., 1993. In situ internal reflection infrared study of aqueous hydrofluoric acid and ultraviolet/ozone treated silicon (100) surfaces. J. Vac. Sci. Technol., A 11, 934–939. Zhang, J., Marinopoulou, A., Hartung, J., Lightowlers, E.C., Anwar, N., Parry, G., Zie, M.H., Mokler, S.M., Wu, X.D., Joyce, B.A., 1994. Growth and characterization of Si1−x Gex /Si multilayers on patterned Si(001) substrates using gas source molecular beam epitaxy. J. Vac. Sci. Technol., A 12, 1139–1141. Zhigilei, L.V., 2003. Dynamics of the plume formation and parameters of the ejected clusters in short-pulse laser ablation. Appl. Phys. A 76, 339–350. Zimmermann, F.M., Pan, X., 2000. Interaction of H2 with Si(001)-(2 × 1): Solution of the barrier problem. Phys. Rev. Lett. 85, 618–621.

CHAPTER 17

Sputtering and Laser Ablation

Herbert M. URBASSEK Fachbereich Physik Universität Kaiserslautern Erwin-Schrödinger-Straße D-67663 Kaiserslautern, Germany E-mail: [email protected] url: http://www.physik.uni-kl.de/urbassek/



Contents 17.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2. Sputtering by ion bombardment . . . . . . . . . . . . . . . . . . . . . 17.2.1. Stopping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.1.1. Potentials . . . . . . . . . . . . . . . . . . . . . . . 17.2.1.2. Cross sections . . . . . . . . . . . . . . . . . . . . . 17.2.1.3. Stopping power . . . . . . . . . . . . . . . . . . . . 17.2.1.4. Electronic stopping . . . . . . . . . . . . . . . . . . 17.2.2. Implantation of the ion and damage in the target . . . . . . . . 17.2.2.1. Ranges . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.2.2. Recoil generation: The linear cascade . . . . . . . . 17.2.2.3. Deposited energy . . . . . . . . . . . . . . . . . . . 17.2.2.4. Recoil spectrum and damage . . . . . . . . . . . . . 17.2.3. Sputtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.2.3.1. Sputter yields . . . . . . . . . . . . . . . . . . . . . 17.2.3.2. Threshold sputtering . . . . . . . . . . . . . . . . . 17.2.3.3. Surface binding energy . . . . . . . . . . . . . . . . 17.2.3.4. Sputtered atom energy distribution . . . . . . . . . . 17.2.3.5. Angular distribution . . . . . . . . . . . . . . . . . . 17.2.3.6. Depth of origin . . . . . . . . . . . . . . . . . . . . 17.2.3.7. Fluctuations in sputtering . . . . . . . . . . . . . . . 17.2.3.8. Surface topography changes . . . . . . . . . . . . . 17.2.3.9. Cluster emission . . . . . . . . . . . . . . . . . . . . 17.3. High energy densities: Sputtering from spikes . . . . . . . . . . . . . 17.3.1. Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.2. Cluster impact . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.3. Crater formation . . . . . . . . . . . . . . . . . . . . . . . . . 17.3.4. Linear vs nonlinear sputtering . . . . . . . . . . . . . . . . . . 17.4. Ablation of metals by ultrafast laser pulses . . . . . . . . . . . . . . . 17.4.1. From energy absorption to the two-temperature model . . . . 17.4.1.1. Light absorption . . . . . . . . . . . . . . . . . . . . 17.4.1.2. Two-temperature model . . . . . . . . . . . . . . . 17.4.1.3. Electronic heat conduction . . . . . . . . . . . . . . 17.4.1.4. Example: Instantaneous homogeneous excitation . . 17.4.2. Materials effects . . . . . . . . . . . . . . . . . . . . . . . . . 17.4.2.1. Overview . . . . . . . . . . . . . . . . . . . . . . . 17.4.2.2. Thermomechanical spallation . . . . . . . . . . . . 17.4.2.3. Pressure wave . . . . . . . . . . . . . . . . . . . . . 17.4.2.4. Ablation yield and threshold . . . . . . . . . . . . . 17.4.3. Further effects: Longer laser pulses and post-emission effects 17.5. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

872

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

875 875 875 875 878 879 880 881 881 882 883 883 885 885 886 886 887 888 889 889 890 890 891 893 895 896 896 898 898 898 899 900 900 901 901 904 905 906 907 909 910 910

Abstract The physical mechanisms and processes underlying the erosion of a surface induced by ion bombardment or short-pulse laser irradiation are presented. The stopping of the ion, its energy deposition in the electronic and atomic system and the build-up of a linear collision cascade are reviewed. Linear sputtering occurs when recoiling target atoms can surpass the surface barrier. When the average energy delivered per atom in the vicinity of the surface becomes comparable to the cohesive energy of the solid, sputtering from a so-called spike may result. This scenario is particularly pronounced in the case of cluster impact. The materials phenomena occurring after ultra-fast laser irradiation of a metal in the ps- or fs-regime are highlighted, and the thermomechanical spallation process is characterized in detail.

873


875

17.1. Introduction In this chapter, the physics underlying the erosion of solids by ion impact or laser irradiation will be presented. While differing in many aspects, these two forms of irradiation deliver such a high amount of energy to the surface of the irradiated material that it is sufficient for bond breaking and hence induces surface erosion. Furthermore, in contrast to temperature-induced evaporation, the processes occur out of equilibrium since, as a consequence of the sudden perturbation induced by the irradiation, erosion occurs fast. Both these aspects, and in particular their non-equilibrium nature, make the processes occurring under energetic ion bombardment or intense laser irradiation a fascinating subject also for fundamental interest. In this presentation, I will restrict attention to the case of elemental targets, thus avoiding the complications arising as a consequence of the bombardment-induced concentration changes in compound targets. Furthermore, in the case of laser irradiation, I will concentrate on (ultra-) short pulse irradiation, i.e., pulse durations in the regime of several ps and below. It is exactly in this range of short pulses, where analogies between the erosion behavior due to ion and laser irradiation are most pronounced, cf. Fig. 17.1.

17.2. Sputtering by ion bombardment 17.2.1. Stopping The projectile ion delivers its energy in a series of collisions to the target atoms, and also to the target electronic system. These two channels of energy loss have been termed elastic and inelastic, respectively. Lindhard et al. (1968) introduced the approximation that these two energy loss channels can be treated independently, such that the total energy loss is given by the sum of these two contributions. In the following, we shall first discuss the physics underlying elastic collisions. 17.2.1.1. Potentials The interaction potential between a projectile and a target atom, and also between two target atoms, has the following generic form: (i) At small distances, the interaction is purely repulsive, due to the electrostatic repulsion of the nuclei and of the inner electron shells, but also – towards lower interaction energies – due to the so-called Pauli repulsion. The latter denotes an increase of the potential energy of two atoms with overlapping electron shells, where the Pauli principle promotes electrons to higher energy states. (ii) At larger separation, the interaction potential is attractive due a covalent, ionic or metallic chemical bonding or due to van-der-Waals interactions. In this attractive regime, the potential energy is generally of a many-body nature, i.e., the attraction between two atoms depends on the number and positions of the surrounding neighbor atoms. This attractive force is considered not to be of importance for particle stopping. It however influences sputtering both directly, since it is responsible

876

H.M. Urbassek

Fig. 17.1. Schematical survey of the sputtering and ablation regimes covered in this review: (a) collision-cascade or linear sputtering induced by ion impact; (b) sputtering from a so-called spike, which might be induced by cluster impact; (c) laser ablation. The shaded areas denote regions of high energy density.

for the surface binding energy, which sputtered particles have to overcome (Section 17.2.3), and indirectly, since it governs phase changes (melting, phase explosion, gasification), which are the mechanisms for spike sputtering (Section 17.3). The repulsive part of the potential is usually well described as a pair potential between the energetic projectile and a low-energy target atom, since the surrounding neighbors are too distant to influence this close collision. Due to their relevance for particle stopping, the two-atom repulsive potentials have been calculated repeatedly in the past. This has been done using either methods from quantum chemistry or from density functional theory. In principle, and also in practice, stopping calculations can be based on the individual


877

projectile-target-atom system of interest. However, often it has been found useful to invoke scaling properties, which are based on the Thomas–Fermi theory of the atom. These scaling relations, while being accurate only up to several percent for individual systems, allow to analyze the stopping behavior in more general terms. Consider the collision of two atoms with nuclear charges Z1 and Z2 and masses M1 and M2 . In the repulsive regime, the binary interaction potential V (r) may be written as Z1 Z2 e2 r V (r) = (17.1) Φ , r a where Gaussian units have been adopted, i.e., e2 = 14.40 eV Å. The screening function Φ introduces a length scale, the so-called screening radius a. In the original formulation of Lindhard et al. (1968), which was based on similarity considerations within the Thomas– Fermi theory of the atom, the screening length is given by

2/3 2/3 −1/2 , aTF = 0.885a0 Z1 + Z2 (17.2) where a0 is the Bohr radius. In the often employed so-called ZBL potential developed by Ziegler et al. (1985), a different screening length has been adopted on purely empirical grounds:

−1 aZBL = 0.8853a0 Z10.23 + Z20.23 . (17.3) The ZBL potential is employed in the widespread TRIM (Eckstein and Biersack, 1984) and SRIM (Ziegler, 2000, 2004) program codes, which calculate particle stopping, energy deposition, ranges, and sputtering on the basis of a Monte Carlo algorithm. The screening function Φ is parameterized as a sum of exponentials Φ(x) =

n

ci e−di x ,

(17.4)

i=1

where ci and di are constants and n assumes values up to n = 3 or 4. For the ZBL potential, it is n = 4 and ΦZBL (x) = 0.1818e−3.2x + 0.5099e−0.9423x + 0.2802e−0.4029x + 0.02817e−0.2016x .

(17.5)

We plot in Fig. 17.2 the potential between an Ar and a Au atom as calculated using density functional theory and compare to the ZBL potential. Figure 17.2 exemplifies that the repulsive binary interaction is at higher energies reasonably well described by using available parameterizations, such as the ZBL potential, while the latter becomes too repulsive at smaller energies (Karolewski, 2006). Since a single length scale, a, governs the potential equation (17.1), the so-called Thomas–Fermi (or Lindhard) energy EL = Z1 Z2 e2 /aTF

(17.6)

gives a convenient energy scale. For example, the Lindhard energy amounts to EL = 220 keV for Ar → Au and 1.16 MeV for Au → Au collisions. It allows to introduce

878

H.M. Urbassek

Fig. 17.2. Potential V (r) describing the interaction between an Ar and a Au atom at short distances, r < 1.5 Å. Ab-initio data calculated by density functional theory (Karolewski, 2006). The agreement with the often employed so-called ZBL potential is reasonably good, in particular for higher energies.

a reduced energy as =

Erel aTF Erel = , EL Z1 Z2 e2

(17.7)

where Erel is the relative interaction energy between the two collision partners. It is Erel = M2 E/(M1 + M2 ) for an atom of mass M1 and energy E hitting an atom of mass M2 at rest. For qualitative orientation it proves often useful to approximate the potential by a power law V (r) = Kr −1/m ,

(17.8)

where the exponent m may have a value between 0 and 1. For a fixed value of m this is obviously possible only over a restricted range of energies. For very high energies 1, m = 1 is valid. For 0.1 2, m = 1/2 has been advocated. For 10−3 , a value of m with 0 m 1/4 should be characteristic. 17.2.1.2. Cross sections When describing the collisions of two atoms, one is often not interested in the details of the trajectory, but rather in the statistics of energy transfer. This is conveniently given by the cross section. It may be defined as follows: If a particle of energy E moves a small distance R in a random medium of atomic number density N consisting of particles at rest, the probability dP of undergoing a collision with energy transfer between T and T + dT is dP = NRσ (E, T ) dT .

(17.9)

The cross section σ can be calculated from the potential V by well known rules (Landau and Lifshits, 1960). For power potentials, Eq. (17.8), it is straightforward to show by a


879

scaling argument (Landau and Lifshits, 1960; Lindhard et al., 1968) that T dT , σ (E, T ) dT = Cm E −2m g E E

(17.10)

with a constant Cm and a dimensionless function g. Lindhard et al. (1968) recommend the approximation g(t) = t −1−m , such that σ (E, T ) dT = Cm E −m T −1−m dT .

(17.11)

The cross section constant Cm is given in terms of the masses and the atomic charges of the colliding atoms as m π 2Z1 Z2 e2 2m 2 M1 Cm = λm a , 2 M2 a

(17.12)

where λm is a dimensionless number depending on the cross section exponent m, tabulated by Sigmund (1981); cf. also Vicanek et al. (1989). 17.2.1.3. Stopping power An important information to be obtained from the cross section is the average energy loss E a particle of energy E suffers when travelling a path length R. With the probability dP of Eq. (17.9) it is E = T dP = N R T σ (E, T ) dT . (17.13) For infinitesimal R, this leads to the concept of the (nuclear) stopping power dE = −NSn (E), dR

(17.14)

with the stopping cross section Sn (E) = T σ (E, T ) dT .

(17.15)

For power law cross sections, Eq. (17.11), it is Sn (E) =

1 Cm E 1−2m 1−m

(17.16)

for M1 = M2 . Hence, for a hard interaction m = 0, the stopping power increases linearly with E, whereas for m = 1/2, i.e. around the maximum of the stopping power, it is constant. For potentials (17.1) a dimensionless stopping power sn () can be introduced which is related to Sn (E) by Sn (E) = 4πaZ1 Z2 e2

M1 sn (). M1 + M2

(17.17)

880

H.M. Urbassek

Fig. 17.3. Reduced nuclear and electronic stopping cross sections, sn () and se (), as a function of the reduced energy . The nuclear stopping has been calculated for the ZBL potential, Eq. (17.18), and the electronic stopping using the Lindhard–Scharff formula, Eq. (17.21), for the cases Ar → Au and Au → Au. The reduced energies for 100 keV projectiles have been indicated.

For the ZBL potential, the reduced stopping power can be approximated by (Ziegler et al., 1985) ln(1+1.1383) , 30, sn () = 2(+0.01321 0.21226 +0.19593 0.5 ) (17.18) (ln )/(2), > 30. Figure 17.3 illustrates this function; for convenience, the reduced energies for a 100 keV Ar → Au and a 100 keV Au → Au collision have also been indicated. From Fig. 17.3 we observe three features: (i) The stopping has a broad maximum for 0.1 < < 1. Here the maximum stopping power is sn = 0.35 at = 0.33. This amounts to dE/dx = 90 eV/Å for Ar → Au collisions at 80 keV and 1.0 keV/Å for Au → Au collisions at 700 keV. (ii) Towards smaller energies, the stopping power decays towards 0 like 1−2m , cf. Eq. (17.16). The ZBL stopping power, Eq. (17.18), corresponds to m = 0.11. Note that a linear stopping power, Sn ∝ E, is predicted for m = 0 and for a steep repulsive potential, i.e., hard spheres. (iii) Towards high energies 1, the stopping power falls off like (ln )/. This is characteristic of Coulomb scattering, where cross sections decay ∝ 1/E 2 for large energies. 17.2.1.4. Electronic stopping The energy transfer of an energetic ion to the target electrons, or to the solid target as a whole, is still a subject of intense research. In particular for slow ions, with energies 25 eV) of a 10 keV Ar → Au cascade. Two properties of the cascade are of particular interest for sputtering. These are the spatial structure of the cascade (Section 17.2.2.3), and the energetic structure of the cascade, in particular the question with which energies the recoils are created (Section 17.2.2.4).


883

Fig. 17.5. Structure of a collision cascade as calculated by a molecular-dynamics simulation. The lateral size of the picture is about 20 nm. The projectile Ar atom enters a Au (111) crystal with 10 keV energy and an angle of 7◦ towards the surface normal. Only atoms which receive a kinetic energy above 25 eV (see scale at the upper right) are displayed. After a time of about 0.5 ps, the ion and all target atoms have an energy below the threshold, and the trajectories appear to end. Two atoms are sputtered as a dimer.

17.2.2.3. Deposited energy The so-called deposited-energy distribution FD (z) dz is defined as the amount of energy deposited in low-energy recoil motion in the target between depth z and z + dz. This distribution gives the gross shape of the defect distribution in a cascade and is thus of considerable interest to the study of ion-induced damage and in particular point defects like vacancies and interstitials. The energy density at the surface, FD (z = 0), is decisive for calculating the sputter yield. The center position of the deposited-energy distribution follows a law similar to Eq. (17.25). The variance of the distribution in the direction perpendicular to the target surface and lateral to it can be taken to indicate the damage zone induced by a single projectile. Note, however, that this is an average over many individual collision-cascade shapes. Figure 17.6 displays the deposited-energy distribution for the case of 10 keV Ar → Au, using the same SRIM calculation, with which the range distribution, Fig. 17.4, has been obtained. 17.2.2.4. Recoil spectrum and damage Consider a cascade started by an ion of energy E0 . Let us denote by F (E0 , E) dE the average number of recoils with energy in the interval (E, E + dE). It can be shown that F (E0 , E) = Γm

E0 E2

(17.26)

884

H.M. Urbassek

Fig. 17.6. Deposited-energy distribution FD (z) for the case of a 10 keV Ar atom bombarding a Au target. Data obtained by a SRIM (Ziegler, 2000) calculation.

with a number Γm ∼ = 0.6 only slightly depending on the potential exponent m. The total number of defects created in a collision cascade can be calculated from Eq. (17.26): Assume a displacement threshold Ed such that each recoil created with an energy above Ed leads to a stable Frenkel pair, while recoils generated with energies below Ed return to their initial lattice position, and no defect is formed. Then the number of defects (Frenkel pairs or vacancies) is given by Ndefects = 0.4

E0 . Ed

(17.27)

More refined estimates are available (Averback and Diaz de la Rubia, 1998). It may be noted that the actual number of defects found after a single ion impact may lead to amorphization in semiconductors, whereas in metals the formation of a heat spike and the consecutive melting of the cascade core may anneal out part of the damage formed. An up-to-date report on ion-induced damage has been presented by Averback and Diaz de la Rubia (1998); for low-energy ion-induced damage, in particular in semiconductors, see also Gnaser (1999). It is often of interest to determine the depth distribution of the damage formed. Thus, for studies of sputtering and sputter-induced surface changes, it may be relevant to know the amount of damage formed close to the surface. The deposited-energy distribution gives the gross shape of a cascade. However, in a cascade of course the interstitial and vacancy distributions will not coincide; the vacancies will lie rather more in the center of the cascade, while the interstitials will have been pushed out towards the periphery. In particular, the interstitial distribution will also be strongly determined by crystal effects, like replacement collision sequences.


885

17.2.3. Sputtering A consequence of the energetic ion impact is the sputtering of the target, i.e., the emission of atoms, molecules or clusters from the target surface. In this section, we will consider the scenario of linear sputtering, or sputtering from a linear collision cascade, where particles are emitted from the surface due to momentum transfer from the projectile ion or a recoil in the collision cascade developing in the target. A different scenario, sputtering from dense cascades, will be considered in Section 17.3. 17.2.3.1. Sputter yields In a sputter experiment, the quantity that is easiest to measure is the sputter yield Y . For an elementary target, it is defined as the average number of sputtered atoms per incident projectile. The yield depends on quite a few parameters of the projectile-target-system under consideration: on mass and atomic number of the projectile, and the target atom; on bombarding energy and direction; and on target material properties, in particular the surface binding that hinders atoms from leaving the surface. Nevertheless there exists a remarkably simple formula due to Sigmund (1969, 1972, 1981), which describes the sputter yield from collision cascades in a quantitative way. The formula reads 1 Γm FD (z = 0)x (17.28) . 8 U Here it is assumed that recoil atoms can only leave the surface, if their velocity component perpendicular to the surface is high enough to overcome a surface barrier of height U . x is a length characterizing the depth out of which recoils may be ejected. It is known by the name of depth of origin of sputtered atoms. With the exponent m describing the interaction potential of low-energy recoils, it reads Y =

x =

1 1 U 2m . 1 − 2m N Cm

(17.29)

This is proportional to the range of a recoil of energy U , cf. Eq. (17.25). Equation (17.28) has the following interpretation (Sigmund, 1969): – FD (z = 0)x is the amount of energy deposited near the surface that is available for sputtering. – The number of recoils created with energy above U is Γm FD x/U . – The remaining factor 1/8 accounts for the fraction of recoils that have the right direction to surpass the planar surface barrier. The sputter yield formula (17.28) has been shown to describe the sputtering of elemental targets with a sufficient degree of accuracy when the assumptions on which it is based are fulfilled: Sputtering proceeds via a collision cascade mechanism, and the bombarding energy E0 is high enough to guarantee the establishment of an isotropic, well developed cascade (Sigmund, 1987). For dimensional reasons, the energy deposited near the surface must be proportional to the projectile stopping power FD (E0 , z = 0) = αNSn (E0 ),

(17.30)

886

H.M. Urbassek

where the proportionality factor α depends mainly on the ratio of target atom to projectile mass and the bombarding angle. For normal incidence and self-sputtering α ∼ = 0.25, but it increases substantially for light projectiles, since these may easily be reflected back to the surface to deposit there more energy. 17.2.3.2. Threshold sputtering Sputtering ceases when the energy of the impinging ion has dropped below a certain threshold energy, Eth . From a large body of experimental and simulational data, Eckstein et al. (1993) found that the threshold energy depends on the surface binding energy U and on the mass ratio μ = M2 /M1 of the target atom mass M2 to the projectile ion mass M1 as Eth /U = 7.0μ−0.54 + 0.15μ1.12 .

(17.31)

At this threshold, sputtering sets in rather sharply. An empirical expression for the rise of sputtering at near-threshold energies has been given by Bohdansky (1984) as 1 Γm FD (z = 0)x Eth ·η , Yth (E0 ) = (17.32) 8 U E0 where the first term in brackets on the right is the sputter yield of Eq. (17.28) and η(x) denotes the threshold function

η(x) = 1 − x 2/3 (1 − x)2 (0 < x < 1). (17.33) In the near-threshold regime, the collision cascade is not yet fully established. Since only few collisions occur and the number of recoils is small, this regime has been called the single-knock-on regime (Sigmund, 1981). Recently, Wittmaack (2003) has compiled the available experimental data base of sputtering of a Si target; the data are displayed in Fig. 17.7a. Impact energies vary over more than three decades, and both light and heavy projectiles have been included. Wittmaack showed that this data set can be well described by the theory presented here, cf. Fig. 17.7b: To this end, observe that the energy dependence of the sputter yield Y , Eq. (17.28) is only given by the stopping power, Y = C · sn (), where C contains all remaining (energyindependent) factors of Eq. (17.28). Introducing therefore a reduced sputter yield as Y (17.34) , C the above analysis shows that all experimental sputter data should align on a single line y = sn (). (17.35) η y=

As Fig. 17.7b shows, the agreement between experiment and theory is more than satisfactory. It should be noted that Wittmaack (2003) employed the KrC stopping power as given by Garcia-Rosales et al. (1994) rather than the ZBL stopping power, Eq. (17.18). 17.2.3.3. Surface binding energy The surface binding energy U is generally set equal to the cohesive energy (sublimation energy) of the solid. This is justified for an atomically rough surface, as it is created after


887

Fig. 17.7. (a) Compilation of experimentally determined sputter yields of Si with normally incident ions. Data taken from Wittmaack (2003). Lines are to guide the eye. (b) Reduced sputter yields, Eq. (17.34), align well with the reduced nuclear stopping cross section. Analysis due to Wittmaack (2003).

prolonged sputtering: then, each surface atom will be bound on average with half the number of bonds that it has in the bulk. In fact, when several layers have been sputtered away, the cohesive energy has to be invested per atom for the atomization of these layers. This argument also shows that the surface binding energy of an atom in a dense crystal surface plane should be higher than the cohesive energy, since every atom will have a relatively high number of neighbors. For the case that bonding is better described by a many-body interaction, such as is adequate for metals, this argument was extended to give a quantitative description of the crystallinity dependence of the surface binding energy (Gades and Urbassek, 1992, 1994b). 17.2.3.4. Sputtered atom energy distribution In a sputter experiment the flux f (E) of atoms out of the target surface is measured. This must be related to the recoil density at the target surface z = 0 (Falcone and Sigmund, 1981). To this purpose, the space dependence can be included in the recoil density by

888

H.M. Urbassek

Fig. 17.8. Kinetic energy distribution of neutral Ag atoms sputtered from a polycrystalline Ag sample by bombardment with 5 keV Ar ions at 45◦ impact angle. Fit to Eq. (17.38) with surface binding energy U = 2.94 eV and power exponent m = 0.15. Data taken from Wahl and Wucher (1994).

replacing E0 in Eq. (17.26) by FD : Γm FD (z) . E2 Hence the sputter flux can be obtained as F (E, z) =

f (E) dE ∝ F (E, z = 0) dEx ∝

(17.36)

FD (z = 0) 2m E dE. E2

(17.37)

Obviously, for a hard interaction (m = 0), the sputtered flux follows an E −2 -law. For an interaction characterized by m > 0 it changes to a softer decay, due to the fact that higher-energy particles may originate from larger depths to contribute to the flux. In order to escape from the surface, particles have to overcome a surface barrier. Usually a planar barrier is assumed, which acts only on the velocity component of the particle perpendicular to the surface, leaving the parallel velocity component unchanged. When one assumes the flux to be cosine-distributed in angle, a planar barrier of height U changes the flux (17.37) to f (E) ∝

E . (E + U )3−2m

(17.38)

For m = 0 this equation is known as Thompson’s formula (1968). For m ∼ = 0, it well describes measured energy distributions of sputtered atoms. Figure 17.8 gives an example of the excellent quality of the Thompson formula for a specific case. 17.2.3.5. Angular distribution When an energetic projectile produces a recoil cascade in an infinite medium (i.e. one without a surface), high-generation recoils will have forgotten the initial direction of the projectile and the cascade becomes isotropic. Hence the particle flux through an imaginary surface is cosine-distributed.


889

For sufficiently high energy bombardment, E0 10 keV, say, measured sputtered particle angular distributions however tend to become overcosine, i.e. there is an excess of particles sputtered in the direction perpendicular to the surface (Andersen et al., 1985). This may be rationalized as the influence of the target surface: Since the surface acts as a drain on target recoils, near to the surface recoils move preferentially towards the surface, rather than into the target. This establishes a cascade anisotropy that is present even for high-generation recoils and agrees with the experimentally measured overcosine distributions (Waldeer and Urbassek, 1987). The angular distribution is however quite sensitive to the influence of target surface roughness and topography (Littmark and Hofer, 1978) and to target crystallinity. It changes furthermore if the target does not have a homogeneous composition in the first surface layers (Sigmund et al., 1982). This may be the case in nonelemental targets, after high implantation doses, or when adsorption layers form on the target surface. 17.2.3.6. Depth of origin The depth from which sputtered atoms originate is relevant for assessing the depth resolution of sputtering-based depth-profiling methods. Since for low-energy recoils, m = 0, Eq. (17.29) gives the average depth of origin as x = 1/NC0 .

(17.39)

With the original value of the low-energy stopping cross section C0 , this resulted in 5 Å (Sigmund, 1969). However, this value has been recalculated by Vicanek et al. (1989) who found that C0 should be increased by a factor of 2, thus resulting in a depth of origin of x = 2.5 Å. Recent computer simulations (Glazov et al., 1998; Shulga and Eckstein, 1998) demonstrated that the escape depth is a factor of 4 smaller than the original estimate by Sigmund (1969). This is in reasonable agreement with experimental data by Wittmaack (1997, 2003), who showed particles to be mainly sputtered from the topmost surface layer. We note that the computer simulations of Shulga and Eckstein (1998) predict a dependence on the atomic number density proportional to N −0.86 instead of N −1 as in Eq. (17.39). The depth of origin of a sputtered particle can be expected to be slightly dependent on its energy E. In analogy to Eq. (17.29), we obtain x =

E 2m 1 . 1 − 2m N Cm

(17.40)

This means that it can be expected that higher-energy sputtered particles originate from greater depths. This expectation is corroborated by computer simulation (Biersack and Eckstein, 1984; Shulga and Eckstein, 1998). Molecular-dynamics simulations typically show that for low impact energies, all sputtered atoms originate from the first target layer (Betz et al., 1994). With increasing energy, also deeper-lying atoms start to be emitted. 17.2.3.7. Fluctuations in sputtering Even for a well defined target surface and well defined ion bombardment energy and angle, the collision cascade and the sputtering induced may fluctuate strongly between individual ion impacts. This is a consequence of the stochastic nature of the projectile slowing down and recoil generation. In fact, consider the extreme cases of the ion hitting

890

H.M. Urbassek

head-on centrally on a surface atom, or on the other hand hitting the surface in the middle of an open channel. In the first case, the collision cascade will develop close to the surface and induce strong sputtering; in the latter case, it may lead to only moderate recoil generation in the vicinity of the surface and hence little sputtering. In many experiments, only the average over a large number of individual ion impacts is measured, and hence fluctuations are of minor relevance. However, the surface topographical features evolving around each individual impact point will also fluctuate along with the energy deposition close to the surface and the sputter yield, and hence all sputter experiments based on direct inspection of the surface changes induced by single ion impacts measure these fluctuations. Mainly from computer simulation data, it has been found that the sputter yield distribution is broad. While details depend on the bombardment condition (Eckstein, 1988), as a general rule of thumb it is found that the standard deviation of the sputter yield distribution is of the same order of magnitude as the average sputter yield (Conrad and Urbassek, 1990). An issue of particular interest is the fact that – albeit with a small frequency – highyield events may occur, in which the individual sputter yield is considerably larger than the average. This issue is important for explaining cluster sputtering data. 17.2.3.8. Surface topography changes As the result of ion impact on the target, damage will be created not only in the bulk, but also at the surface. Atomic defects may be categorized as surface vacancies (missing atoms in the topmost target layer) and adatoms. Among extended defects one finds adatom and vacancy islands, and even craters. Surface topographical features are accessible to measurements by scanning tunnelling microscopy (Michely and Comsa, 1991; Michely and Krug, 2004). These yield valuable information on the surface-near features of the cascade and on the sputter process. Since the consequences of individual ion impact become visible, also information on the sputter statistics becomes accessible. Linear cascade theory can predict the adatom yield Ya , i.e., the average number of adatoms due to ion impact. Let us denote by Ua the energy necessary to form an adatom. For example, for the (111) surface of an fcc metal, it is Ua ∼ = 0.4U . The recoil energy spectrum, Eq. (17.26), can be used to calculate the adatom yield Ya , with the result Ya /Y ∼ = 4. This is in agreement with experimental (Michely and Teichert, 1994) and simulational (Gades and Urbassek, 1994a) results of rare-gas bombardment of a Pt (111) surface for energies above 100 eV. 17.2.3.9. Cluster emission It is a common experience that besides monatomic particles, clusters are found in the sputtered particle flux. In fact, surprisingly large clusters can be emitted. As an example, in 15 keV Xe bombardment of a Ag sample, (neutral) Agn clusters up to n ∼ = 60 have been found (Staudt et al., 2000). In fact, emission of clusters containing more than 500 atoms has been observed by Rehn et al. (2001) for 500 keV rare-gas ion impact on Au (cf. also the molecular-dynamics simulation displayed in Fig. 17.9). Since clusters must originate from a single ion impact event, the fact that clusters larger than the average cluster yield are sputtered, points at the importance of sputter yield fluctuations. Observed mass distributions follow a polynomial decay with cluster size Yn ∝ n−δ ,

(17.41)


891

where Yn denotes the yield of clusters containing n atoms; the exponent δ has been empirically found to be inversely correlated to the average sputter yield Y (Wahl and Wucher, 1994). Up to now, sputter theory has not come up with an intuitive quantitative argument to explain such a polynomial decay, even though a number of theoretical investigations into the nature of large-cluster emission have been performed, cf. Reimann (1993). However, from molecular-dynamics simulations, it appears that large-cluster emission requires the correlated ejection of a group of neighboring atoms from the surface (Betz and Husinsky, 1995). The simulation of particularly energetic events, where a high amount of energy is deposited close to the surface, shows that even quasi-hydrodynamic droplet emission (clusters containing more than 100 atoms) is possible in specific cases, cf. the discussion in Section 17.3.2 and Fig. 17.9. Sputtered clusters are internally hot after emission (Urbassek and Hofer, 1993). As a rule, the larger clusters are found to be metastable and to evaporate off atoms to cool. This experimental finding is corroborated by molecular-dynamics simulations which find sputtered clusters to possess high internal energy (Wucher and Garrison, 1992a, 1992b). Dimers are the most abundant cluster species sputtered. Originally it was thought that dimers form upon sputtering from an elemental target when two atoms are sputtered more or less independently and combine to form a cluster in the vicinity of the surface (so-called recombination or double-collision model) (Gerhard and Oechsner, 1975; Können et al., 1974, 1975). Such a model led to the prediction of a translational energy spectrum which decays at high energies like E −5 ; an even higher fall-off was predicted for larger clusters. This prediction is not confirmed by experiment, where in particular the larger clusters show comparable high-energy decay laws, cf. e.g. Wahl and Wucher (1994). This must be taken as evidence that clusters do not originate from individual (independent) emission events.

17.3. High energy densities: Sputtering from spikes The linear-cascade picture, where an energetic atom collides with target atoms at rest, which in their term recoil and collide with other target atoms at rest, etc., breaks down, when the majority of atoms in a certain volume have been set in motion. This is invariably the fate of every linear collision cascade at large times. However, often, this late stage is of little experimental consequence since the energy of the moving atoms has dropped too low. However, in cases of high energy density, a number of physical effects may be induced in this so-called spike stage. As Sigmund (1974) notes, “Spike effects may be important when the spike life time is larger than the duration of the initiating cascade. Spikes have been considered as the origin of a variety of experimental results over the years. The more compelling evidence seems to come from sputtering experiments”. The notion of a spike has been invoked in particular for the discussion of ion-induced damage since the work by Brinkman (1954) and Seitz and Köhler (1956). Various epithet have been introduced to qualify a spike either as a displacement spike, a thermal spike, or an elastic collision spike. We shall not be concerned with these differences here. In a spike, thermal equilibrium can be established locally quite fast among the moving atoms. From gas-phase physics it is known that such an equilibrium will be established

892

H.M. Urbassek

Fig. 17.9. Perspective view of a Au target bombarded by a Au8 cluster with an energy of 16 keV/atom. The snapshot taken at 32 ps after cluster impact has been obtained by a molecular-dynamics simulation. The gray-shade (color online) illustrates the local temperature (i.e., the local kinetic energy averaged over a sphere of 6.2 Å radius in the center-of-mass frame). Color online: Blue: 0 K. Green: Molten (1338 K). Purple: Hotter than 2667 K.

when each atom has undergone a few (of the order of 3–10) collisions (Weller and Weller, 1982; Waldeer and Urbassek, 1991). In a solid, the time may accordingly be estimated to be ttherm ∼ = (3–10)/νD , where νD is the Debye frequency. For instance, in the case of Au, which has a small Debye frequency, it is νD = 3.4 THz, and thus after around 1–3 ps, thermal equilibrium will have been established locally. After this time it is possible to describe the state of matter using hydrodynamic and thermodynamic concepts like the density n(r, t) the temperature T (r, t) and the pressure p(r, t). These concepts allow to describe the behavior of the material after irradiation, and in particular the sputtering process, in a way which is different from the collision-cascade physics described above. Figure 17.9 presents an atomistic view of the processes occurring in a Au target after impact of a Au8 cluster with 128 keV total energy. It is clearly seen that the violent materials effects occurring cannot be described using the model of a linear-collision cascade. Rather, macroscopic concepts like temperature and boiling are appropriate for describing this process. The spike concept has been used to describe the phenomena of defect formation and in particular defect annealing in the aftermath of a collision cascade, and it has been found that in metals the spike systematically tends to reduce the number of point defects formed due to the thermally activated annihilation processes occurring (Nastasi et al., 1996; Averback and Diaz de la Rubia, 1998). The concept of a spike has also been extensively used to describe ion-induced mixing and it has been found that mixing processes can be enhanced or reduced in the spike phase after the collision cascade (Cheng, 1990;


893

Nastasi et al., 1996; Süle et al., 2003). For ion-induced sputtering, spikes have been invoked to describe particularly large sputter yields and also strong surface-topographical features, in particular crater formation under ion impact (Andersen, 1993; Reimann, 1993). Quite generally, spike effects on sputtering can be assumed to be strong, when in a limited volume close to the target surface, the energy per atom E0 is sizeable compared to the cohesive energy Ecoh of the target; or, in other terms, when the local temperature T is sizeable when compared to the critical temperature Tc of the liquid-vapor phase transition. 17.3.1. Models A number of models have been developed to describe sputtering from spikes. None of these appear to be able to describe quantitatively the variety of experimental results available. But each of them features a characteristic element important for the sputtering phenomena: Surface boiling, phase explosion, the importance of high pressures, etc. So several of the models available will be described briefly: (i) Surface evaporation: Assume the energized region to be of cylindric form with the axis aligned with the ion impact direction; this will be in the case considered here directed perpendicular into the solid. When the initial lateral distribution of the deposited energy or ‘temperature’ is of Gaussian form with maximum T0 , sputtering can be described from the interplay of surface evaporation over the surface barrier U and lateral heat conduction. An evaluation of this idea gives (Johnson and Evatt, 1980; Sigmund and Claussen, 1981) kT0 dE 2 Y =g (17.42) . · U dx The function g depends on details of the model. In any case, g(kT0 /U ) approaches a constant for kT0 U . Thus, for a hot narrow spike the yield is quadratic in the stopping power dE/dx – in contrast to the linear dependence of Eq. (17.28). The cylindrical surface geometry appears particularly apt to describe sputtering in the electronic energy deposition regime, and for heavy projectiles, which fly on a straight line into the target. In fact, the quadratic dependence of Eq. (17.42) has been observed in several experiments, such as the sputtering of molecular solids under MeV light projectiles (Johnson, 1990; Johnson and Sundqvist, 1992; Johnson and Schou, 1993). (ii) Phase-explosion or gas flow models: The idea that a high energy-density region may reach temperatures beyond the critical temperature Tc of the liquid-gas phase transition appears to have occurred first in describing the sputtering of condensedgas targets by keV ions (Michl, 1983). The sputter yield can then be described by the competition of heat conduction, which freezes the volume of high energy density, and the thermal flow velocity which drives the high-pressure, hightemperature gas out of the high-energy-density zone into the vacuum and leads to sputtering (Kelly, 1990; Reimann, 1993; Miotello and Kelly, 1997). This idea has been used to quantitatively describe the sputtering of weakly bound systems in a variety of cases (Urbassek and Michl, 1987; Balaji et al., 1990, 1995).

894

H.M. Urbassek

Fig. 17.10. Schematics of the phase-explosion model of spike sputtering. The temperature-density phase diagram of a model material is outlined. Triple-point temperature Tt and critical temperature Tc are indicated. The spinodal line, separating the metastable part of the liquid-gas coexistence region from the unstable part is indicated (color online: red). Schematic temperature-density trajectories of the energized spike volume are displayed (color online: blue). Trajectories 1–3 correspond to increased deposited energy densities. Trajectory 1 does not contribute to sputtering, while 2 enters the liquid-gas coexistence region and starts (non-equilibrium) surface evaporation. Trajectory 3 even passes the spinodal: The unstable material will spontaneously start boiling (phase explosion).

As Fig. 17.10 shows, the phase-space trajectory of a suddenly (isochorically) heated volume element may pass after (adiabatic) expansion deep into the liquidgas coexistence region. If the spinodal line (defined such that the material inside the spinodal is unstable and immediately starts to undergo phase separation) is approached, the system will start bulk boiling; this process has been termed phase explosion. This model has also been invoked to explain the high abundance of large-mass clusters sputtered under spike conditions (Urbassek, 1988): If the trajectory passes close to the critical point, the material will decompose into a mixture of clusters of all sizes following a power law in cluster size n, Eq. (17.41), with δ given by a critical exponent, δ = 7/3. (iii) Shock-wave and pressure-pulse models: These models emphasize that the high energy density in the spike region leads not only to a high temperature, but also to a large pressure in this volume. The pressure gradient to the surrounding material, and in particular to the vacuum above the surface, will induce a pressure pulse or shock wave, which may lead to sputtering when it unloads at the surface. Depending on the details of the modelling, a dependence of the yield Y ∝ (dE/dx)n with n = 32 or 3 is obtained (Kitazoe et al., 1981; Bitensky and Parilis, 1987; Johnson et al., 1989; Reimann, 1995). Energy distributions of sputtered particles typically exhibit a broad low-energy maximum at energies E U , see Fig. 17.11. This maximum is attributed to thermal processes occurring at the spike temperature. Both surface-evaporation and phase-explosion models predict the maximum to be at energies in the range of E = kT , where T is a temperature between the boiling temperature Tb and the critical temperature Tc of the target, and


895

Fig. 17.11. Kinetic energy distributions of neutral In atoms sputtered from a polycrystalline In sample under impact of a Au atom (triangles) and a Au2 dimer (circles). The impact energy is in both cases 5 keV/atom. The fit to a Thompson distribution of linear-cascade theory, Eq. (17.38), describes well the case of atom bombardment, but not the excess of low-energy atoms which are sputtered by dimer impact. Data taken from Samartsev et al. (2005).

k is Boltzmann’s constant. Thus, an experimental energy spectrum like that of 10 keV Au2 impact into In (Fig. 17.11) can be interpreted to be due to two different mechanisms: a linear-cascade contribution operative at energies E > 1 eV, which is well described by the Thompson distribution, Eq. (17.38), and an additional spike distribution, which is responsible for the excess of low-energy atoms (E < 1 eV). 17.3.2. Cluster impact When a cluster impinges on a surface, it will break up into its constituent atoms; each of these will deliver its energy in the solid. At sufficiently small energies the total cluster energy will hence be delivered in a roughly hemispherical volume, and the deposited energy density can reach high values; this constitutes an ideal setting for spike sputtering. At higher bombarding energies, the stochastics of energy deposition and scattering may lead to an only incomplete overlap of the individual collision cascades. Bouneau et al. (2002) and Bouneau et al. (2002) illustrated the huge sputtering yields which may be achieved as a consequence of cluster impact in a series of beautiful experiments performed with Aun clusters (n = 1–13) bombarding a Au target, see Fig. 17.13. At smaller projectile energies, molecular-dynamics simulation is an adequate tool to analyze the effects of cluster impact (Colla et al., 2000). Figure 17.9 displays the results of a Au8 impact on a Au surface at 16 keV/atom impact energy, i.e., around the lowest energies where experiments have been performed. The simulation results show that the sputtering process corresponds to a phase explosion, in which sputtering occurs by the gasification of the high-energy-density zone, as long as this is situated sufficiently close to the surface (cf. Section 17.3.1, item (ii)). In the bombardment of metallic targets under spike conditions, late ejection of large clusters (droplets, containing some 1000 atoms) is observed (Colla et al., 2000), cf. Fig. 17.9. These were emitted as compact ‘fingers’

896

H.M. Urbassek

(Nordlund et al., 2003), which remained as protrusions left over the violent disruption of the surface; their fate is decided by the interplay of surface tension pulling the fingers back towards the surface and the drift velocity which the fingers obtained during the ‘explosion’ of the highly pressurized spike region. 17.3.3. Crater formation The formation of a crater under heavy-ion or cluster impact has been observed repeatedly in computer simulation. For Cu cluster bombardment of a Cu surface, and analogously also for Au, with energies of the order of 10 keV per cluster, large craters were observed in molecular-dynamics simulations (Aderjan and Urbassek, 2000; Colla and Urbassek, 2000; Bringa et al., 2001). In these cases it was found that about half of the atoms that left the crater were sputtered; the other half was deposited on the surface in the form of a crater rim. Nordlund et al. (2001) investigated the dependence of the size of ion-induced craters on the materials properties of the target. They could show that the crater size scales inversely proportional to the cohesive energy and to the melting temperature of the material. Nordlund et al. (2003) identified various macroscopic features connected to crater production by 2 show a scaling like Y = n2 f (E/n), Eq. (17.45). Data taken from Bouneau et al. (2002).

f (x) = α

x 1+b . (xc + x)b

(17.44)

Thus, sputtering has been found to become linear with projectile energy per atom in this system above a threshold, E xc nU . Cluster-induced sputter yields Y of Au sputtered by Aun clusters of energy E have been measured over a wide range of energies and cluster sizes by Bouneau et al. (2002), cf. Fig. 17.13. The data exhibit a scaling like

898

H.M. Urbassek

Yn = n2 f (E/n),

(17.45)

i.e., sputtering is not additive but pronouncedly nonlinear. Note that monomer and – to a lesser degree – dimer impact does not yet fulfill this scaling completely, which is however astonishingly well fulfilled for clusters with sizes n = 3–13. A convincing argument for Eq. (17.45) is still missing.

17.4. Ablation of metals by ultrafast laser pulses The physical processes occurring in a material irradiated by an intense laser beam strongly depend on the type of material (insulator or metal), the laser frequency and intensity, and the laser pulse duration. In this paper, I shall focus on ultrashort laser pulses in the picosecond or femtosecond regime since here the analogy to ion-beam sputtering is most evident. Several of the processes complicating the physics of nanosecond laser pulse ablation – plasma formation, laser interaction with the ablation cloud, expansion physics of the ablation cloud, cluster aggregation and fragmentation in the cloud – will be sketched in Section 17.4.3. We shall focus on metallic targets irradiated by visible or UV lasers where the laser energy is absorbed in the electronic system and then transferred to the atomic system. In the case of insulators, where the band gap is wider than the photon energy, the material is transparent to the radiation. The light is hence only absorbed within the volume of the material, and ablation starts only after a first few incubation pulses which induce defect formation and the ensuing decomposition of the material (Bäuerle, 2000). Ultrafast laser ablation shows advantages in applications over ns-pulse ablation, since the “heat affected zone” which shows signs of damage due to melting is reduced or avoided, and wide-band-gap materials can be processed due to multi-photon processes. 17.4.1. From energy absorption to the two-temperature model 17.4.1.1. Light absorption The typical setup considered in this section is the excitation of a metal by a femtosecond or picosecond laser pulse in the visible or UV regime. When laser light with intensity I0 shines on a metal surface, a fraction R is reflected, while the remainder is absorbed inside the material. The absorbed intensity I follows an exponential attenuation law I (z, t) = (1 − R)I0 (t)e−z/-

(17.46)

with the attenuation length -. Here z measures the depth into the solid. Both the reflection coefficient R and the absorption length - are a function of the laser frequency ω. In particular, for frequencies above the plasma frequency, ω > ωp , the metal becomes transparent, R → 0 and - → ∞. For metals, the plasma frequency is in the UV; in this contribution we shall only consider the case ω < ωp . The attenuation length is given by the skin depth 2 δ= (17.47) μσ ω


899

where σ = σ (ω) is the AC conductivity of the metal, and μ its permeability. For example, Au has R = 0.47 and - = 22 nm for 500 nm incident light (Bäuerle, 2000). This energy deposited in the conduction electron system will thermalize internally on a time scale of a few femtoseconds; the details of this process have been analyzed by Rethfeld et al. (1999, 2002a). Electrons may be excited in the conduction band up to energies h¯ ω above the Fermi energy. From Fermi liquid theory (Pines and Nozières, 1966) and two-photon photoemission experiments (Aeschlimann et al., 1996) it is known that the lifetime of an excited electron is limited by electron–electron collisions and decays like 1/(E)2 with the height of the electron energy E = h¯ ω above the Fermi level. Thus, high-energy non-thermalized electrons can penetrate far into the solid from the laser absorption regime; since these electrons run on straight trajectories, unlike diffusion, they are termed ballistic electrons. 17.4.1.2. Two-temperature model After thermalization, the electron energy is dissipated by diffusion deeper into the metal, and by electron–atom collisions to the atomic system of the metal. These processes are conveniently summarized in the so-called two-temperature model, which assumes that the electronic system and the atomic system have two separate temperatures Te and Ta , respectively, which equilibrate in the course of time. This model has originally been developed by Anisimov et al. (1974) to describe the processes occurring in a metal after swift-ion impact in the electronic-stopping regime, cf. Section 17.2.1.4 above. However, it has now been applied with great success to describe the ultra-short laser irradiation of metals. The model is described by the following equations: ∂Te ∂Te ∂ = Λe − g(Te − Ta ) + Q(z, t), ∂t ∂z ∂z ∂Ta Ca = g(Te − Ta ). ∂t Here, the source term Ce

(17.48) (17.49)

1 I (z, t) (17.50) has been used. The various terms and coefficients appearing in Eqs. (17.48) and (17.49) will now be discussed. The atomic (volumetric) specific heat Ca is taken as 3kn according to the Dulong–Petit law, where n is the atomic number density and k is Boltzmann’s constant; in numerical treatments, such as the molecular-dynamics study of Ivanov and Zhigilei (2003), better estimates are available. The electronic specific heat is proportional to Te as Q(z, t) =

Ce (Te ) = γ Te .

(17.51)

The constant γ is in many cases well described by Sommerfeld’s result π 2 ne k/(2TF ), where ne is the electron density and TF is the Fermi temperature. For Au, it is γ = 67.6 J/m3 K2 and TF = 6.4 × 104 K. We note that Eq. (17.51) holds true for temperatures Te TF only.

900

H.M. Urbassek

The coefficient g has been termed the electron–phonon coupling coefficient, even though it is applied also in cases where the atomic system has already molten. Typical values range from 21–360 GW/K cm3 ; these values represent Au and Ni, respectively, as examples of a weakly and a strongly coupled metal. The dependence of g on Te and Ta is not known, and even its value at room temperature is not well measured. For Cu, for example, a recent compilation (Bonn et al., 2000) quotes literature values between 10 and 200 W/K cm3 . 17.4.1.3. Electronic heat conduction The lattice heat conduction can be ignored in good approximation, as it is small compared to the electronic heat conduction. The electronic heat conduction, however, is important. It has been discussed intensely by Anisimov and Rethfeld (1997). They propose a functional dependence Λe = α

(θe 2 + 0.16)5/4 (θe 2 + 0.44)θe , (θe 2 + 0.092)1/2 (θe 2 + βθa )

(17.52)

where θa = Ta /TF and θe = Te /TF . For Au, the parameters in Eq. (17.52) assume values α = 353 W/K m and β = 0.16. Expression (17.52) is valid over a wide range of temperatures, in particular also when Te approaches (or exceeds) the Fermi temperature (Anisimov and Rethfeld, 1997). For smaller Te , Eq. (17.52) can be simplified to (Wang et al., 1994) Λe =

ATe aTe 2 + bTa

.

(17.53)

This expression models the gas-kinetic heat conduction, which is proportional to Ce (Te )/ν. The collision frequency ν has contributions from electron–electron and electron–atom scattering, which are described by the terms aTe2 and bTa , resp. For Au, it is A = 4.35 × 1013 W/K s m, a = 1.2 × 107 /s K2 , and b = 1.23 × 1011 /s K. Figure 17.14 illustrates this dependence. Above the Fermi temperature of Au, Λe strongly increases, since Coulomb collisions become inefficient in a high-temperature plasma. The maximum observed in Fig. 17.14 reflects the competition of the increase of specific heat with temperature and the hindrance of transport by electron–electron collisions. Note that the high electronic heat conductivity smears the laser attenuation profile Eq. (17.46) to considerable higher depths and acts as a heat source for atoms. Since electronic heat conduction is quick, and electron–phonon coupling is slow, the processes of electronic energy dissipation and heat transfer to the atoms may be considered decoupled to a good approximation. 17.4.1.4. Example: Instantaneous homogeneous excitation We will illustrate the characteristic features of the two-temperature model with the help of a simple example. Consider a free-standing thin film; when its electronic system is instantaneously energized by a femtosecond laser pulse, ballistic electrons will homogenize the electron energy quickly throughout the film. Thus assume that at time t = 0 an energy E0 has been given to the electrons such that they acquire a temperature T0 , which is large


901

Fig. 17.14. Electronic heat conductivity Λe in Au as a function of the electron temperature Te for fixed atom temperature Ta = 300 K. The complete expression (17.52) (full line) is compared to the approximation (17.53) (dashed) valid below the Fermi temperature TF = 6.4 × 104 K.

compared to the lattice temperature. Then, as long as Te Ta , the solution of Eqs. (17.48) and (17.49) reads 2 t t t , Ta = T∞ 2 − , Te = T0 1 − (17.54) τ τ τ and hence

t 2 , E e = E0 1 − τ

where the relaxation time γ 2γ τ = T0 = E0 g g

E a = E0 − Ee ,

(17.55)

(17.56)

and the final atom temperature E0 (17.57) Ca have been introduced. The results hold for t τ . The temperature evolution and the resulting energy evolution in the electronic and atomic system are displayed in Fig. 17.15. It is seen that the electron temperature initially reaches very high values due to the small electronic heat capacity. After thermalization, however, for the same reason, almost all the energy has been transferred to the atomic system. T∞ =

17.4.2. Materials effects 17.4.2.1. Overview Figure 17.16 illustrates the materials effects to be expected in a metallic target irradiated with increasing laser fluence. These results from a molecular dynamics simulation

902

H.M. Urbassek

Fig. 17.15. Temporal evolution of temperatures (top) and energies (bottom) in the atomic and electronic subsystem of an ultrafast laser-excited metal. The simplified analysis, Eqs. (17.54) and (17.55), holds as long as Te Ta , and hence for times t τ . Here τ is the relaxation time of the system, Eq. (17.56), determined by the materials parameters and the initial energy input E0 . This figure demonstrates that in spite of the extremely high electronic temperatures reached, the laser energy efficiently dissipates into the atomic system.

have been calculated for computational convenience for a thin-slab geometry, in which the atoms can be assumed to be instantaneously and homogeneously energized. The energy, which each atom receives from the laser, is denoted the so-called ‘energization’ E0 . We see that with increasing E0 the material starts melting, then voids nucleate, which finally lead to the thermomechanical spallation of the metal; at the highest energies shown the material gasifies into a vapor cloud, containing monomers and clusters of various sizes. Note that with increasing laser fluence the probability of electron emission from the metal and ionization of emitted atoms and clusters will increase, and hence actually a plasma may form; this process had not been included in the simulation underlying Fig. 17.16. The phenomenon of non-thermal or cold melting has been discussed for insulating and semi-conducting systems for a while (Stampfli and Bennemann, 1990; Graves and Allen, 1998; Jeschke et al., 2001). In these systems, for sufficiently high electronic excitation density, electrons are excited out of their covalent bonding states to non-bonding or antibonding orbitals. This change in the potential energy of the solid induces forces on the atoms, and hence disordering of the lattice, before the electron–phonon coupling mechanism of the two-temperature model energizes the atoms. However, thermal melting can


903

Fig. 17.16. Series of snap shots of a short-pulse laser-irradiated Al film featuring various materials processes. All figures taken at 5 ps after irradiation. The energy E0 which each atom receives from the laser is denoted at the bottom. The expansion of the system due to the increased energy input is clearly observable. Gray-scale (color online) denotes the local temperature. (a) Material is heated, but remains crystalline. (b) Material melts. (c) A void forms temporarily. (d) The film breaks (thermomechanical spallation). (e) Cluster formation.

also occur ultrafast, i.e., within a few ps, if a sufficiently large superheating (of the order of T ∼ = 1.5Tm is achieved (Rethfeld et al., 2002b). The laser ablation of molecular substrates with increasing energization occurs in close analogy to the processes in metals mentioned above. Zhigilei et al. (2003) enumerate the following ablation mechanisms, which they analyzed in detail from the moleculardynamics simulation of the laser ablation of molecular substrates: surface desorption, overheating and phase explosion, photomechanical spallation, photochemical ablation. Figure 17.17 shows the materials processes occurring after laser irradiation in a temperature-density phase diagram. In analogy to previous simulational work by Zhakhovskii et al. (2000), Perez and Lewis (2002, 2003) and others, this presentation allows to obtain an overview of how increasing laser fluences lead to different materials effects, in analogy to the real-space atomistic snap shots shown for Al in Fig. 17.16. The following sequence of processes is visible: (i) heating in the crystalline state, (ii) melting,

904

H.M. Urbassek

Fig. 17.17. Laser-irradiation-induced processes in a van-der-Waals solid. The phase diagram of a Lennard-Jones material is outlined. Trajectories 1–4: Temperature-density trajectories of a Lennard-Jones solid, suddenly excited to an energy/atom of E0 , measured relative to the cohesive energy Ecoh . The trajectories are calculated by a molecular-dynamics simulation for a slab of 30 monolayers thickness and present averages over the central third part of the system. The sudden excitation leads to a rapid isochoric temperature increase, followed by a rapid expansion and cooling. At small energies this process ends with target melting, at higher energies the solid- (or liquid-) vapor coexistence region is entered and the material fails by spallation or complete vaporization. The analogous processes for a laser-irradiated metal are visualized atomistically in Fig. 17.16.

(iii) void formation and collapse, (iv) complete target disintegration and cluster formation. We note that for larger simulation crystallites, the process of target melting does not occur (Anisimov et al., 2003; Inogamov, 2006); this is a consequence of the small liquid regime in van-der-Waals solids. Note the analogy of the processes occurring under laser ablation to those of sputtering in a spike scenario, cf. Fig. 17.10. 17.4.2.2. Thermomechanical spallation The thermoelastic pressure induced by the deposited energy density leads to thermomechanical effects in the sample. Paltauf and Dyer (2003) These result in particular from the tensile pressure which originates due to the interaction of the initially compressive pressure with the free sample surface. The process occurring at the ablation threshold has been termed thermomechanical spallation. Spallation may be defined as the rupture of a sample due to a strong tensile pressure, such as it occurs due to pressure and shock waves induced by projectile impacts, explosions, or laser pulses (Vidal et al., 2004). Alternatively, the process of vapor-bubble formation in liquids is termed cavitation, or the generic term fragmentation may be employed (Vogel and Venugopalan, 2003). The spallation mechanism can be illustrated by a molecular-dynamics simulation, Fig. 17.18. This simulation was performed close to the ablation threshold in Cu, at a laser fluence of 170 mJ/cm2 and a laser pulse width of 0.5 ps. Simultaneously with the atomic motion simulated by molecular dynamics, the electron heat transport and energy transfer between the electronic the and atomic system was taken into account by a finite-difference solution of the electron temperature equation, Eq. (17.48). Figure 17.18 shows that at the time of spallation a strongly negative tensile pressure has developed with peak values of


905

Fig. 17.18. Atomistic view of the spallation process in a Cu crystal irradiated at time t = 0 by a 0.5 ps laser pulse of a fluence of 170 mJ/cm2 . Results of a molecular-dynamics simulation (Schäfer, 2001; Schäfer et al., 2002). The series of snapshots shows only the relevant part of the crystallite, where fracture occurs at times around 32 ps after the irradiation. The target surface is not visible. The cross sections shown have a height of 60 Å and a width of 21 Å; they are 10 Å thick. Atoms are gray-shaded (color online) according to their local pressures in GPa. Fracture occurs at a depth of 20 nm, which is beyond the laser attenuation length of 14 nm.

the order of −6 GPa. Since the system has already molten by this time, the liquid cannot sustain the large tensile pressure peak and tears. Thus the main result of this and similar simulations (Leveugle et al., 2004; Vidal et al., 2004) has been to show that it is not only the absorbed energy density which defines the ablation threshold; also pressure is an important variable, and it is in particular the tensile pressure which develops in the molten target, which leads to thermomechanical spallation, and hence ablation. 17.4.2.3. Pressure wave The origin of the tensile pressures developing in the solid can be clarified using the analysis of Bushnell and McCloskey (1968) and Dingus and Scammon (1991). An exponential energy deposition profile, analogous to Eq. (17.46), will establish a similar compressive pressure profile Φ0 −z/(17.58) , e where Φ0 is the surface value of the absorbed fluence and Γ is the Grüneisen coefficient. The latter is a materials constant which relates the induced thermoelastic pressure to the absorbed energy density; its value is around 2, but shows a slight temperature dependence (Slater, 1940; Zel’dovich and Raizer, 1966). The inhomogeneous pressure distribution will send a compressive wave into the target inner. Note, however, that the initial thermoelastic pressure is highest at the surface: this is an unstable situation, since mechanical equilibrium requires p = 0 at the surface. The material responds by unloading at the surface; this means that the surface expands into vacuum and a second, so-called rarefaction wave is p(z) = Γ

906

H.M. Urbassek

Fig. 17.19. Laser-irradiation-induced pressure wave in a Cu crystallite: Results of a molecular-dynamics simulation (Schäfer, 2001). The crystallite has been irradiated with a laser pulse of duration τL = 5 ps with a fluence of 20 mJ/cm2 , and an exponential absorption profile, Eq. (17.46), with absorption length - = 14 nm. The figure shows the resulting pressure profile during (top) and after (bottom) the pulse. The data are compared to an analytical expression of the pressure profile given by Bushnell and McCloskey (1968) and Dingus and Scammon (1991), where the sound velocity has been assumed to c = 45 Å/ps and the Grüneisen constant to Γ = 1.2.

sent into the target inner. The resulting wave thus has a bipolar struction, where the first compressive peak is followed by a tensile pressure peak, cf. Fig. 17.19. It is the second tensile peak which has catastrophic materials consequences, in particular when the target is molten and thus has a small yield strength. 17.4.2.4. Ablation yield and threshold Figure 17.20 shows the ablation rate measured under short-pulse laser irradiation of Cu (Preuss et al., 1995; Nolte et al., 1997). An ablation threshold at a fluence of Φth = 140–170 mJ/cm2 is observed. This figure also compares to molecular-dynamics computer simulation results, which observe the threshold at a similar value, Φth = 170 mJ/cm2 . The threshold in the simulation is sharp; above Φth , a sizeable amount of material, between


907

Fig. 17.20. Ablation rate versus fluence for short-pulse laser irradiation of Cu. : Experiment by Nolte et al. (1997), at a wavelength of 780 nm and a pulse duration τL = 0.15 ps. : Experiment by Preuss et al. (1995) at a wavelength of 248 nm and τL = 0.5 ps. : Simulation results for τL = 0.5 ps by Schäfer et al. (2002).

20 and 30 nm, is ablated. This is due to the spallation mechanism described above, Section 17.4.2.2. Experiment, however, shows only small ablation rates above the threshold, of the order of a few nm/pulse. It is believed that this is due to the measurement technique, which involves of the order of 1000 shots to measure the ablation rate; hence, irradiationinduced surface modification may influence the results (Schäfer et al., 2002). The existence and size of an ablation threshold Φth as well as the fluence dependence of the yield above the threshold can be derived using the following argument, which will be presented in its simplest form valid for insulators (Brannon et al., 1985; Johnson, 1994). Here it is assumed that the energy density deposited by the laser Φ(z) = Φ0 e−z/-

(17.59)

determines the ablation process. Φ0 is the absorbed fluence at the surface. Assuming that ablation occurs if the local energy density, (z) = Φ(z)/-, increases above a critical value, c , Eq. (17.59) leads to an ablated depth of zabl = - · ln

Φ , Φth

(17.60)

where the ablation threshold is given by Φth = Ec -.

(17.61)

Note that in metals the electronic heat conduction will spread the profile, hence effectively increasing -. Furthermore, the two-temperature equations (17.48) and (17.49) have to be taken into account, cf. Bäuerle (2000), p. 271f; these consideration do not seriously change the result (17.61). 17.4.3. Further effects: Longer laser pulses and post-emission effects In this contribution I concentrated on those aspects of laser ablation which show the most similarities to ion sputtering in the spike regime. However, due to the wide variability of

908

H.M. Urbassek

laser frequencies, pulse length, and the materials involved, fascinating physical processes are involved, which I briefly mention below. For a fuller coverage of this subject the reader should consult reviews like Phipps and Dreyfus (1993), Bäuerle (2000), Anisimov and Luk’yanchuk (2002), Georgiou and Hillenkamp (2003), Schou (2006). (i) Ablation mechanisms: Beyond the essentially thermal ablation mechanism discussed in detail above, photons may directly break bonds in the target and thus induce photochemical ablation. Also, defects induced by the irradiation may lead to stresses or even to electrical fields inside the sample and thus induce material failure, i.e., ablation. (ii) Post-emission processes in the ablation cloud: Immediately after ablation, the ablated material may still possess a rather high density, in particular for short-pulse ablation. Then, collisions in the gas cloud will be frequent. The consequences of these collisions are manifold: In the course of the adiabatic expansion process, internal energy will be converted into translational center-of-mass energy of the cloud. In other words, the cloud will internally cool, while its drift velocity off the surface increases. As a consequence, the energy distribution of ablated particles will change. This is of major importance for pulsed-laser deposition applications, since the properties of thin films formed from these particles may depend crucially on the particle energies involved (Chrisey and Hubler, 1994). Also the angular distribution will become more focussed, in the direction towards the target surface normal. Of course, these collision processes will be strongly changed, if the expansion is not into a vacuum, but into a background gas. Detailed hydrodynamic and also kinetic expansion models are available, which capture this physics quantitatively (Sibold and Urbassek, 1993; Anisimov et al., 1996; Anisimov and Luk’yanchuk, 2002). (iii) Clusters: The flux of ablated particles consists not only of monomers, but also of clusters. Often, also droplets, i.e., large liquid clusters, are ablated. This cluster distribution will change due to the collision processes mentioned above, which induce both cluster agglomeration and fragmentation processes, and also due to cluster evaporation, which cools the clusters internally until a metastable equilibrium has been reached. (iv) A plasma may form: Due to the interaction of the intense laser beam with the target, which may lead to the emission of both electrons and ions from the target, also the interaction of the laser beam with the ablation cloud will ionize it. Once a plasma has formed above the target, two main consequences arise: (i) The plasma may shield the target from the laser and thus reduce ablation. (ii) Due to internal electrical fields which are set up in particular at the plasma front, electrons and ions are strongly accelerated. While the plasma cloud in total is quasineutral, it obtains a pronounced internal structure, in which leading highly-charged ions are followed by singly-charged ions. Three-body recombination collisions may partly neutralize the plasma and thereby deliver potential energy to heat up and accelerate the plasma even more. As a consequence, the ions – and also the (re-neutralized) atoms – may reach kinetic energies, which are far above the thermal energies available from thermal processes. Thus, kinetic energies of above 1 keV have been measured for, e.g., the ablation of Ti by 80-fs pulses (Ye and Grigoropoulos, 2001).


909

17.5. Outlook Research in the areas of solid erosion by ion or laser irradiation is an active field which is driven both by the development of new applications and by the analysis of fundamental aspects of the underlying mechanisms. In the following, I highlight some of the issues around which current research concentrates. (i) While the phenomena of linear sputtering have been clarified, many aspects of nonlinear sputtering have still not been settled. What is the explanation of the surprisingly simple scaling in the measured data of nonlinear sputtering by cluster impact, Eq. (17.45)? Under which conditions is sputtering additive, i.e., is the sputter yield under molecule or ion impact equal to the sum of the sputter yields of the atomic constituents of the projectile? (ii) What is the role of energy loss to the electronic system, both during the cascade, and in the thermal spike developing later? In a metal, electrons can quench the cascade quite rapidly, and hence freeze mixing processes in a compound and sputtering processes occurring from spikes. However, it is not clear how large the electron mean free path – and hence the heat conductivity – is in the strongly distorted underdense liquid spike region. This question has also immediate consequences for the yield of electronically excited and ionized sputtered particles. (iii) A basic question of interest both in ion bombardment and laser irradiation is the validity of the interatomic empirical potentials used in the regime of high energy densities. While two-body potentials have been measured over the years with a high accuracy by spectroscopic and gas-phase-collision methods, the validity of many-body interaction potentials at high densities and temperatures is still yet poorly known. This applies even more, when simultaneously the electronic system has been highly excited. (iv) Both in ion bombarded and laser irradiation, clusters are formed and even large chunks of matter (droplets) may be emitted from the target. Often, the abundance distribution of emitted clusters can be characterized by a power-law decay, Eq. (17.41). A quantitative understanding is still lacking. (v) In the case of cluster impact into solids, the collision-cascade contribution to the sputter yield may be negligible compared to the contribution of the thermal spike involved. Here the analogy to the case of ultrashort laser irradiation appears to be most pronounced. Besides the different involvement of the electronic system, the lateral width of the energized zone, which is considerably larger for laser irradiation, forms the strongest difference. (vi) For laser irradiation, the two-temperature model presents a sound basis to understand many aspects of ablation. Here the question of what happens before an electronic and an atomic temperature can be defined – i.e., before the corresponding subsystems have internally thermalized – is still open. Also, the processes occurring while the two subsystems are still in strong mutual non-equilibrium (Te Ta ) strongly depends on the materials parameters adopted; their justification still awaits assessment.

910

H.M. Urbassek

(vii) The role of ballistic electrons transporting energy away from the irradiation zone needs to be implemented into a consistent picture of energy dissipation and ablation. (viii) The thermodynamics and kinetics of the (non-equilibrium) phase transitions occurring under intense laser irradiation are fascinating. This applies in particular to the processes occurring around the ablation threshold (thermomechanical spallation in the liquid phase), but also to the regimes of phase explosion and plasma formation.

Acknowledgements Thanks are due to Christian Anders, Cemal Engin, Yudi Rosandi, Christian Schäfer, Arun Upadhyay, and Steffen Zimmermann for preparing calculations and figures for this review, and to Michael Wahl, Klaus Wittmaack, and Andreas Wucher for providing original data. Discussions with Bärbel Rethfeld and Nail Inogamov on various issues connected to Section 17.4 are gratefully acknowledged.

References Aderjan, R., Urbassek, H.M., 2000. Nucl. Instrum. Meth. B 164–165, 697. Aeschlimann, M., Bauer, M., Pawlik, S., 1996. Chem. Phys. 205, 127. Anders, C., Urbassek, H.M., Johnson, R.E., 2004. Phys. Rev. B 70, 155404. Andersen, H.H., 1993. Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 43, 127. Andersen, H.H., Stenum, B., Sørensen, T., Whitlow, H.J., 1985. Nucl. Instrum. Meth. B 6, 459. Anisimov, S.I., Kapeliovich, B.L., Perel’man, T.L., 1974. Sov. Phys. JETP 39, 375. Anisimov, S.I., Luk’yanchuk, B.S., 2002. Phys. Usp. 45, 293. Anisimov, S.I., Luk’yanchuk, B.S., Luches, A., 1996. Appl. Surf. Sci. 96–98, 24. Anisimov, S.I., Rethfeld, B., 1997. Izvestia Akad. Nauk (Fiz.) 61, 1642. Anisimov, S.I., Zhakhovski, V.V., Inogamov, N.A., Nishihara, K., Oparin, A.M., Petrov, Y.V., 2003. JETP Lett. 77, 606. Averback, R.S., Diaz de la Rubia, T., 1998. In: Ehrenreich, H., Spaepen, F. (Eds.), Solid State Physics, vol. 51. Academic Press, Boston, p. 281. Balaji, V., David, D.E., Magnera, T.F., Michl, J., Urbassek, H.M., 1990. Nucl. Instrum. Meth. B 46, 435. Balaji, V., David, D.E., Tian, R., Michl, J., Urbassek, H.M., 1995. J. Phys. Chem. 99, 15565. Bäuerle, D., 2000. Laser Processing and Chemistry, third ed. Springer, Berlin. Betz, G., Husinsky, W., 1995. Nucl. Instrum. Meth. B 102, 281. Betz, G., Kirchner, R., Husinsky, W., Rüdenauer, F., Urbassek, H.M., 1994. Radiat. Eff. Defects Solids 130–131, 251. Biersack, J.P., Eckstein, W., 1984. Appl. Phys. A 34, 73. Birtcher, R.C., Donnelly, S.E., 1996. Phys. Rev. Lett. 77, 4374. Bitensky, I.S., Parilis, E.S., 1987. Nucl. Instrum. Meth. B 21, 26. Bohdansky, J., 1984. Nucl. Instrum. Meth. B 2, 587. Bonn, M., Denzler, D.N., Funk, S., Wolf, M., Wellershoff, S.-S., Hohlfeld, J., 2000. Phys. Rev. B 61, 1101. Bouneau, S., Brunelle, A., Della-Negra, S., Depauw, J., Jacquet, D., LeBeyec, Y., Pautrat, M., Fallavier, M., Poizat, J.C., Andersen, H.H., 2002. Phys. Rev. B 65, 144106. Brannon, J.H., Lankard, J.R., Baise, A.I., Burns, F., Kaufman, J., 1985. J. Appl. Phys. 58, 2036. Bringa, E.M., Nordlund, K., Keinonen, J., 2001. Phys. Rev. B 64, 235426. Brinkman, J.A., 1954. J. Appl. Phys. 25, 961.


911

Bushnell, J.C., McCloskey, D.J., 1968. J. Appl. Phys. 39, 5541. Cheng, Y.-T., 1990. Mat. Sci. Rep. 5, 45. Chrisey, D.B., Hubler, G.K. (Eds.), 1994. Pulsed Laser Deposition of Thin Films. Wiley, New York. Colla, T.J., Aderjan, R., Kissel, R., Urbassek, H.M., 2000. Phys. Rev. B 62, 8487. Colla, T.J., Urbassek, H.M., 2000. Nucl. Instrum. Meth. B 164–165, 687. Conrad, U., Urbassek, H.M., 1990. Nucl. Instrum. Meth. B 48, 399. Dingus, R.S., Scammon, R.J., 1991. SPIE Proc. Series 1427, 45. Donnelly, S.E., Birtcher, R.C., 1997. Phys. Rev. B 56, 13599. Eckstein, W., 1988. Nucl. Instrum. Meth. B 33, 489. Eckstein, W., Biersack, J., 1984. Nucl. Instrum. Meth. B 2, 550. Eckstein, W., Garcia-Rosales, C., Roth, J., Laszlo, J., 1993. Nucl. Instrum. Meth. B 83, 95. Falcone, G., Sigmund, P., 1981. Appl. Phys. 25, 307. Gades, H., Urbassek, H.M., 1992. Nucl. Instrum. Meth. B 69, 232. Gades, H., Urbassek, H.M., 1994a. Phys. Rev. B 50, 11167. Gades, H., Urbassek, H.M., 1994b. Nucl. Instrum. Meth. B 88, 218. Garcia-Rosales, C., Eckstein, W., Roth, J., 1994. J. Nucl. Mater. 218, 8. Georgiou, S., Hillenkamp, F. (Eds.), 2003. Thematic Issue: Laser Ablation of Molecular Substrates. Chem. Rev., vol. 103. Gerhard, W., Oechsner, H., 1975. Z. Phys. B 22, 41. Glazov, L.G., Shulga, V.I., Sigmund, P., 1998. Surf. Interface Anal. 26, 512. Gnaser, H., 1999. Low-Energy Ion Irradiation of Solid Surfaces. Springer Tracts in Modern Physics, vol. 146. Springer, Berlin. Graves, J.S., Allen, R.E., 1998. Phys. Rev. B 58, 13627. Inogamov, N., 2006. Private communication. Ivanov, D.S., Zhigilei, L.V., 2003. Phys. Rev. B 68, 064114. Jeschke, H.O., Garcia, M.E., Bennemann, K.H., 2001. Phys. Rev. Lett. 87, 015002. Johnson, R.E., 1990. Energetic Charged-Particle Interactions with Atmospheres and Surfaces. Springer, Berlin. Johnson, R.E., 1994. Int. J. Mass Spectrom. Ion Proc. 139, 25. Johnson, R.E., Evatt, R., 1980. Radiat. Eff. 52, 187. Johnson, R.E., Schou, J., 1993. Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 43, 403. Johnson, R.E., Sundqvist, B.U.R., 1992. Phys. Today 45 (5), 28. Johnson, R.E., Sundqvist, B.U.R., Hedin, A., Fenyö, D., 1989. Phys. Rev. B 40, 49. Karolewski, M.A., 2006. Nucl. Instrum. Meth. B 243, 43. Kelly, R., 1990. Nucl. Instrum. Meth. B 46, 441. Kitazoe, Y., Hiraoka, N., Yamamura, Y., 1981. Surf. Sci. 111, 381. Können, G.P., Tip, A., de Vries, A.E., 1974. Radiat. Eff. 21, 269. Können, G.P., Tip, A., de Vries, A.E., 1975. Radiat. Eff. 26, 23. Landau, L.D., Lifshits, E.M., 1960. Mechanics. Pergamon, New York. Leveugle, E., Ivanov, D.S., Zhigilei, L.V., 2004. Appl. Phys. A 79, 1643. Lindhard, J., Nielsen, V., Scharff, M., 1968. Mat. Fys. Medd. Dan. Vid. Selsk. 36 (10). Lindhard, J., Scharff, M., 1961. Phys. Rev. 124, 128. Littmark, U., Hofer, W.O., 1978. J. Mater. Sci. 13, 2577. Michely, T., Comsa, G., 1991. Phys. Rev. B 44, 8411. Michely, T., Krug, J., 2004. Islands, Mounds, and Atoms. Springer Series in Surface Science, vol. 42. Springer, Berlin. Michely, T., Teichert, C., 1994. Phys. Rev. B 50, 11156. Michl, J., 1983. Int. J. Mass Spectrom. Ion Phys. 53, 255. Miotello, A., Kelly, R., 1997. Nucl. Instrum. Meth. B 122, 458. Nastasi, M., Mayer, J.W., Hirvonen, J.K., 1996. Ion–Solid Interactions: Fundamentals and Applications. Cambridge University Press, Cambridge. Nolte, S., Momma, C., Jacobs, H., Tünnermann, A., Chichkov, B.N., Wellegehausen, B., Welling, H., 1997. J. Opt. Soc. Am. B 14, 2716. Nordlund, K., Henriksson, K.O.E., Keinonen, J., 2001. Appl. Phys. Lett. 79, 3624. Nordlund, K., Tarus, J., Keinonen, J., Donnelly, S.E., Birtcher, R.C., 2003. Nucl. Instrum. Meth. B 206, 189.

912

H.M. Urbassek

Paltauf, G., Dyer, P.E., 2003. Chem. Rev. 103, 487. Perez, D., Lewis, L.J., 2002. Phys. Rev. Lett. 89, 255504. Perez, D., Lewis, L.J., 2003. Phys. Rev. B 67, 184102. Phipps, C.R., Dreyfus, R.W., 1993. In: Vertes, A., Gijbels, R., Adams, F. (Eds.), Laser Ionization Mass Analysis. In: Chemical Analysis, vol. 124. Wiley, New York, p. 369. Pines, D., Nozières, P., 1966. The Theory of Quantum Liquids, vol. 1. Normal Fermi Liquids. Benjamin, New York. Preuss, S., Demchuk, A., Stuke, M., 1995. Appl. Phys. A 61, 33. Rehn, L.E., Birtcher, R.C., Donnelly, S.E., Baldo, P.M., Funk, L., 2001. Phys. Rev. Lett. 87, 207601. Reimann, C.T., 1993. Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 43, 351. Reimann, C.T., 1995. Nucl. Instrum. Meth. B 95, 181. Rethfeld, B., Kaiser, A., Vicanek, M., Simon, G., 1999. Appl. Phys. A [Suppl.] 69, S109. Rethfeld, B., Kaiser, A., Vicanek, M., Simon, G., 2002a. Phys. Rev. B 65, 214303. Rethfeld, B., Sokolowski-Tinten, K., von der Linde, D., Anisimov, S.I., 2002b. Phys. Rev. B 65, 092103. Samartsev, A.V., Duvenbeck, A., Wucher, A., 2005. Phys. Rev. B 72, 115417. Schäfer, C., 2001. Molecular-dynamics of laser ablation of metals. Diploma thesis, University Kaiserslautern. Schäfer, C., Urbassek, H.M., Zhigilei, L.V., 2002. Phys. Rev. B 66, 115404. Schou, J., 2006. In: Pauleau, Y. (Ed.), Materials Surface Processing by Directed Energy Techniques. Elsevier. Chapter 2. Seitz, F., Köhler, J.S., 1956. In: Seitz, F., Turnbull, D. (Eds.), Solid State Physics, vol. 2. Academic Press, Boston, p. 305. Shulga, V.I., Eckstein, W., 1998. Nucl. Instrum. Meth. B 145, 492. Sibold, D., Urbassek, H.M., 1993. J. Appl. Phys. 73, 8544. Sigmund, P., 1969. Phys. Rev. 184, 383. Sigmund, P., 1972. In: Kurepa, M.V. (Ed.), Physics of Ionized Gases. Institute of Physics, Beograd, Yugoslavia, p. 137. Sigmund, P., 1974. Appl. Phys. Lett. 25, 169, 27 (1975) 52. Sigmund, P., 1981. In: Behrisch, R. (Ed.), Sputtering by Particle Bombardment I. Springer, Berlin, p. 9. Sigmund, P., 1987. Nucl. Instrum. Meth. B 27, 1. Sigmund, P., Claussen, C., 1981. J. Appl. Phys. 52, 990. Sigmund, P., Oliva, A., Falcone, G., 1982. Nucl. Instrum. Meth. 194, 541. Slater, J.C., 1940. Phys. Rev. 57, 744. Stampfli, P., Bennemann, K.H., 1990. Phys. Rev. B 42, 7163. Staudt, C., Heinrich, R., Wucher, A., 2000. Nucl. Instrum. Meth. B 164–165, 677. Süle, P., Menyhard, M., Nordlund, K., 2003. Nucl. Instrum. Meth. B 211, 524. Thompson, M.W., 1968. Philos. Mag. 18, 377. Urbassek, H.M., 1988. Nucl. Instrum. Meth. B 31, 541. Urbassek, H.M., Hofer, W.O., 1993. Mat. Fys. Medd. K. Dan. Vidensk. Selsk. 43, 97. Urbassek, H.M., Michl, J., 1987. Nucl. Instrum. Meth. B 22, 480. Vicanek, M., Jimenez Rodriguez, J.J., Sigmund, P., 1989. Nucl. Instrum. Meth. B 36, 124. Vidal, F., Johnston, T.W., Kieffer, J.-C., Martin, F., 2004. Phys. Rev. B 70, 184125. Vogel, A., Venugopalan, V., 2003. Chem. Rev. 103, 577. Wahl, M., Wucher, A., 1994. Nucl. Instrum. Meth. B 94, 36. Waldeer, K.T., Urbassek, H.M., 1987. Nucl. Instrum. Meth. B 18, 518. Waldeer, K.T., Urbassek, H.M., 1991. Physica A 176, 325. Wang, X.Y., Riffe, D.M., Lee, Y.-S., Downer, M.C., 1994. Phys. Rev. B 50, 8016. Weller, R.A., Weller, M.R., 1982. Nucl. Instrum. Meth. 194, 573. Winterbon, K.B., Urbassek, H.M., Sigmund, P., Gras-Martí, A., 1987. Phys. Scripta 36, 689. Wittmaack, K., 1997. Phys. Rev. B 56, R5701. Wittmaack, K., 2003. Phys. Rev. B 68, 235211. Wucher, A., Garrison, B.J., 1992a. Surf. Sci. 260, 257. Wucher, A., Garrison, B.J., 1992b. Phys. Rev. B 46, 4855. Ye, M., Grigoropoulos, C.P., 2001. J. Appl. Phys. 89, 5183.


913

Zel’dovich, Y.B., Raizer, Y.P., 1966. Physics of Shock Waves and High-Temperature Hydrodynamic Phenomena, vol. 1. Academic Press, New York. Zhakhovskii, V.V., Nishihara, K., Anisimov, S.I., Inogamov, N.A., 2000. JETP Lett. 71, 167. Zhigilei, L.V., Leveugle, E., Garrison, B.J., Yingling, Y.G., Zeifman, M.I., 2003. Chem. Rev. 103, 321. Ziegler, J.F., 2000. Srim: http://www.srim.org/. Ziegler, J.F., 2004. Nucl. Instrum. Meth. B 219–220, 1027. Ziegler, J.F., Biersack, J.P., Littmark, U., 1985. The Stopping and Range of Ions in Solids. Pergamon, New York.


Author index Aballe, L., see Locatelli, A. 399, 400 Abbey, B., see Howe, P. 839, 841 Abbott, H., see DeWitt, K. 836 Abbott, H.L., see Kavulak, D.F. 832, 833 Abeln, G.C. 726 Abeln, G.C., see Avouris, Ph. 602, 616, 699, 701, 704–706 Abeln, G.C., see Lyding, J.W. 701, 703, 705, 706, 726 Abeln, G.C., see Shen, T.-C. 701, 704–706 Abild-Pedersen, F. 295, 297, 299 Abild-Pedersen, F., see Helveg, S. 812 Abrahams, J.P. 744 Abrams, B.L., see Chianelli, R.R. 330, 331, 333 Abrefah, J., see Olander, D.R. 830 Adelman, S.A. 6, 19 Aderjan, R. 896 Aderjan, R., see Colla, T.J. 895 Adler, P.M. 256 Adlhoch, W. 371 Adzic, R.R., see Zhang, J.L. 291 Aeschlimann, M. 899 Aeschlimann, M., see Bauer, M. 558, 565 Agarwal, R., see Duan, X.F. 808 Aggour, M., see Lewerenz, H.J. 847 Agrait, N. 591 Aharoni, C., see Rudzinski, W. 249 Aizawa, H. 844 Ajayan, P.M. 807 Akai, H., see Dederichs, P.H. 500 Akati, H.C., see Avouris, Ph. 699, 701, 704–706 Akazawa, H. 830 Akerman, W., see Otto, A. 469 Akpati, H.C. 746 Al-Rawi, A., see Rahman, T.S. 534 Al-Sarraf, N., see Borroni-Bird, C.E. 763 Ala-Nissila, T. 254

Ala-Nissila, T., see Masin, M. 256 Alavi, A., see Michaelides, A. 304 Alavi, S. 616 Albada, S.B.v., see Gastel, R.v. 777 Albertini, D. 833, 834 Alcalá, R. 302 Aldao, C.M. 844 Alemani, M. 736 Alemani, M., see Gross, L. 748 Alemani, M., see Moresco, F. 740, 751 Ali, T., see Gruyters, M. 362 Ali, T., see Pasteur, A.T. 357 Alivisatos, A.P. 803 Alivisatos, A.P., see Klein, D.L. 747 Alla-Nissila, T., see Chvoj, Z. 256 Allan, M., see Skalicky, T. 720 Allen, C.E., see Bauer, E.G. 382 Allen, R.E., see Graves, J.S. 902 Allenspach, R., see Weber, W. 781 Allongue, P. 846, 849, 850 Allongue, P., see Gerischer, H. 842, 846, 847 Allongue, P., see Gorostiza, P. 842 Allongue, P., see Munford, M.L. 846 Almbladh, C.-O. 537 Alnasrallah, W., see Dayo, A. 487 Althoff, F. 111 Althoff, F., see Andersson, T. 46, 104, 111 Althoff, F., see Wilzén, L. 103, 163, 462 Amar, J.G. 779 Ambaye, H. 44, 74, 75 Amemiya, K., see Nakai, I. 257 Ami, S. 738, 739 Ami, S., see Stadler, R. 739 Amirav, A. 463 Ancillotto, F., see Silvestrelli, P.L. 735 Ancilotto, F. 829 Anders, C. 896, 897

915

916

Andersen, H. 459, 466, 478 Andersen, H.H. 889, 893 Andersen, H.H., see Bouneau, S. 895, 897 Andersohn, L. 834 Andersohn, L., see Köhler, U. 834 Anderson, A.B. 694 Anderson, D.R., see Marston, J.B. 7, 12 Anderson, P.W. 4, 14, 16, 284, 447, 451 Andersson, B., see Olsson, L. 323 Andersson, D. 465, 476, 477, 482 Andersson, M.P. 294, 316–318 Andersson, M.P., see Sehested, J. 317, 318 Andersson, P.U., see Nagard, M.B. 44 Andersson, P.U., see Tomsic, A. 44 Andersson, S. 100, 102–105, 113, 114, 120, 124, 135, 150, 157, 163, 274, 443, 461, 462, 471 Andersson, S., see Althoff, F. 111 Andersson, S., see Andersson, T. 46, 101, 102, 104, 111 Andersson, S., see Borroni-Bird, C.E. 763 Andersson, S., see Gustafsson, K. 98 Andersson, S., see Harris, J. 150 Andersson, S., see Hassel, M. 130, 135 Andersson, S., see Linde, P. 46, 101, 102 Andersson, S., see Persson, M. 105, 117, 461, 462, 466 Andersson, S., see Svensson, K. 101, 128 Andersson, S., see Wilzén, L. 103, 163, 462 Andersson, T. 46, 101, 102, 104, 111 Andersson, T., see Althoff, F. 111 Andersson, Y., see Hult, E. 500 Andrade, R.F.S. 362 Andreev, A.F. 792 Andreev, S.A., see Maragakis, P. 306 Andreoni, W., see Ancilotto, F. 829 Andresen, P., see Budde, F. 631, 644 Andresen, P., see Mull, T. 646 Anger, G., see Rendulic, K.D. 183 Anghel, A.T., see Hoyle, R.B. 361 Angher, A.T., see Hoyle, R.B. 259 Angot, T., see Papageorgiou, N. 733 Anisimov, S.I. 824–826, 845, 899, 900, 904, 908 Anisimov, S.I., see Rethfeld, B. 824, 903 Anisimov, S.I., see Sokolowski-Tinten, K. 824 Anisimov, S.I., see Zhakhovskii, V.V. 903 Antczak, G. 254 Anthony, J.M., see Wilk, G.D. 797

Author index

Antoniewicz, P. 468 Antoniewicz, P.R. 127, 623 Antoniewicz, P.R., see Koel, B.E. 623 Anwar, N., see Zhang, J. 828 Aono, A., see Sakurai, M. 701, 705, 706 Aparicio, L.M., see Dumesic, J.A. 233, 305, 310 Apell, P. 486 Arakawa, E.T., see Gesell, T.F. 487 Aranson, I.S. 382, 410 Arias, J., see Lee, J. 473 Armand, G. 66 Arnaldsson, A., see Olsen, R.A. 306 Arnau, A. 467, 468 Arnau, A., see Corriol, C. 548, 564 Arnau, A., see Monturet, S. 588 Arnold, D.W., see Korolik, M. 220 Arnold, D.W., see Manolopoulos, D.E. 510, 512 Arnolds, H., see Lane, I.M. 471 Arthur, D.A., see Koehler, B.G. 829 Arthur Jr., J.R. 837 Artsyukhovich, A.N. 672 Aruga, T. 475 Arumainayagam, C.R. 31, 173 Aryasetiawan, F., see Zhukov, V.P. 539 Asada, H. 205 Asaki, M.T., see Huang, C.-P. 551 Asakura, K., see Graham, M.D. 345, 415 Asakura, K., see Lauterbach, J. 390, 417 Ashcroft, N.W. 534, 582 Ashfold, M.N.R., see Claeyssens, F. 825 Ashfold, M.N.R., see Rettner, C.T. 56, 455 Ashruf, C.M.A., see Xia, X. 849, 850 Ashwin, M.J., see Williams, R.S. 842 Asscher, M. 3, 50, 513, 644 Asscher, M., see Romm, L. 237, 513 Atkins, P.W. 307 Atwater, H.A., see Saunders, W.A. 845 Auerbach, D.J., see Barker, J.A. 31, 56, 199 Auerbach, D.J., see Becker, C.A. 671 Auerbach, D.J., see Berenbak, B. 45, 209 Auerbach, D.J., see Gulding, S.J. 177 Auerbach, D.J., see Head-Gordon, M. 46, 122, 125–127 Auerbach, D.J., see Hou, H. 178, 222, 433, 512 Auerbach, D.J., see Huang, Y. 148, 189, 190, 221, 222 Auerbach, D.J., see Huang, Y.H. 433, 463, 511, 512

Author index

Auerbach, D.J., see Janda, K.C. 82, 199 Auerbach, D.J., see Kimman, J. 85, 97, 212 Auerbach, D.J., see Kleyn, A.W. 83–85, 206–208, 215, 216 Auerbach, D.J., see Luntz, A.C. 215 Auerbach, D.J., see Michelsen, H.A. 172, 175, 179 Auerbach, D.J., see Mullins, C.B. 48, 49, 122 Auerbach, D.J., see Ran, Q. 471, 513 Auerbach, D.J., see Rettner, C.T. 6, 31, 48, 49, 82–84, 87, 88, 148, 164, 167, 172, 175, 177, 183, 189, 190, 212, 220, 459 Auerbach, D.J., see White, J.D. 189–191, 433, 466, 496, 511, 512 Auerbach, D.J., see Wodtke, A.M. 3, 189, 240, 434, 435, 496, 497, 512, 515 Auerbach, D.J., see Wodtke, J. 4, 6, 19 Auerback, D.J., see Hou, H. 334 Auerback, D.J., see Huang, Y.H. 334 Auerback, D.J., see White, J.D. 334 Aumiller, G.D., see Prybyla, J.A. 634 Auth, C. 481, 502 Averback, R.S. 884, 892 Avery, A.R., see Mayne, A.J. 735 Avery, N.R. 718 Aviram, A. 738 Aviram, A., see Joachim, C. 738 Avouris, Ph. 546, 602, 616, 685, 694, 696, 697, 699, 701, 704–706 Avouris, Ph., see Akpati, H.C. 746 Avouris, Ph., see Bozso, F. 830 Avouris, Ph., see Dujardin, G. 601, 687, 699, 718 Avouris, Ph., see Foley, E.T. 701, 705, 706 Avouris, Ph., see Hasegawa, Y. 545, 771 Avouris, Ph., see Lang, N.D. 738 Avouris, Ph., see Lyo, I.-W. 696, 735 Avouris, Ph., see Persson, B.N.J. 746 Avouris, Ph., see Shen, T.-C. 701, 704–706 Avouris, Ph., see Walkup, R.E. 458, 616, 694, 696 Avouris, Ph., see Wintterlin, J. 828 Aydil, E.S., see Belen, R.J. 827 Ayissi, S., see Dobrin, S. 696 Ayissi, S., see Harikumar, K.R. 696 Aziz, M.J., see Crouch, C.H. 827 Babak, D.V., see Emel’yanov, V.I. 825, 826 Bach, C. 19, 23

917

Back, C.H., see Weber, W. 781 Backus, S., see Huang, C.-P. 551 Backx, C. 443 Baddorf, A., see Heskett, D. 300 Badjic, J.D. 744 Baer, R., see Citri, O. 148 Baer, Y., see Hengsberger, M. 542–544 Baer, Y., see Purdie, D. 538, 539 Baerends, E.J., see Kirchner, E.J.J. 32, 33 Baerends, E.J., see Kroes, G.J. 3, 32, 146, 168, 179 Baerends, E.J., see McCormack, D.A. 146, 159, 169, 175 Baerends, E.J., see Nieto, P. 31 Baerends, E.J., see Pijper, E. 146 Baerends, E.J., see Somers, M.F. 161, 162, 169, 184, 189 Baffou, G. 734 Baffou, G., see Mayne, A.J. 696, 746 Bag, B.C., see Banerjee, D. 237 Bahn, S., see Jacobsen, C.J.H. 294 Bahn, S., see Nørskov, J.K. 302, 303 Bahnck, D., see Hull, R. 798 Bahr, D. 797 Bahrim, B. 481, 502 Bahrim, B., see Auth, C. 481, 502 Bahrim, B., see Teillet-Billy, D. 481, 502 Bailey, J.E. 411 Baise, A.I., see Brannon, J.H. 907 Baker, J., see Wu, Y.M. 835 Bakker, J.W., see Gorodetskii, V.V. 402 Bakkers, E.P.A.M., see Verheijen, M.A. 810 Balaji, V. 893 Balandin, A.A. 313 Balasubramanian, T. 542, 543, 560 Balasubramanian, T., see LaShell, S. 543 Balasubramanian, T., see Silkin, V.M. 542, 543, 560 Balasubramanian, V., see Chakrabarti, N. 658 Bald, D.J. 240, 246 Bald, D.J., see Stuckless, J.T. 763, 765 Baldo, P.M., see Rehn, L.E. 890 Bales, G.S., see Brune, H. 767, 769, 778, 782 Balk, P., see Werner, K. 835 Ballone, P., see Blandin, P. 778 Balooch, M., see Olander, D.R. 830 Baltuška, A. 551 Balzani, V., see Badjic, J.D. 744 Balzer, F., see Mathews, C.M. 513

918

Bammerlin, M., see Bennewitz, R. 732 Bandyopadhyay, A. 737 Bandyopadhyay, A.K., see Bera, S.K. 845 Banerjee, D. 237 Banerjee, S., see Mahajan, A. 828 Banerjee, S., see Mullins, C.B. 833 Banerjee, S., see Pacheco, K.A. 836 Bangia, A.K. 415 Bangia, A.K., see Bär, M. 415, 416 Banholzer, W.F. 371 Banik, S.K., see Banerjee, D. 237 Bansmann, J., see Klar, F. 488 Bao, H., see Garcia, S.P. 821, 849 Bär, M. 356, 383–385, 389, 390, 392, 410, 415, 416 Bär, M., see Bangia, A.K. 415 Bär, M., see Falcke, M. 385 Bär, M., see Gottschalk, N. 378, 385, 389 Bär, M., see Graham, M.D. 415 Bär, M., see Haas, G. 415, 416 Bär, M., see Hartmann, N. 415 Bär, M., see Meron, E. 389 Bär, M., see Mertens, F. 389, 390 Bär, M., see Monine, M. 393 Barabási, A.L., see Daruka, I. 799, 804 Baranov, A.N., see Stepanyuk, V.S. 772, 774 Baratoff, A., see Bennewitz, R. 732 Bard, A.J., see Chang, H. 822 Bardeen, J. 585 Bare, S.R. 357 Bare, S.R., see Strongin, D.R. 326 Barends, E.J., see Hellman, A. 304, 325, 327, 328 Barends, E.J., see McCormack, D.A. 179, 180 Barends, E.J., see Olsen, R.A. 147 Barends, E.J., see Watts, E. 161, 167, 169, 183, 184, 189 Barker, J.A. 31, 56, 199 Barker, J.A., see Kimman, J. 85, 97, 212 Barker, J.A., see Rettner, C.T. 212 Barnett, S.A., see Rockett, A. 831 Barredo, D., see Nieto, P. 31, 455 Barrelet, C.J. 808 Barshed, Y., see Ziff, R.M. 355, 357, 407 Barski, A., see Dujardin, R. 809 Barteau, M.A., see Kitchin, J.R. 292 Barteau, M.A., see Linic, S. 308 Barteau, M.A., see Linik, S. 325

Author index

Bartels, L. 4, 558, 638, 686–691, 699, 715, 724, 725 Bartels, L., see Hla, S.-W. 686, 692 Bartelt, M.C., see Evans, J.W. 763 Bartelt, N.C., see Hannon, J.B. 776, 777 Bartelt, N.C., see Theis, W. 775, 776 Barth, J.V. 4, 254, 775, 780 Barth, J.V., see Fischer, B. 775 Barth, J.V., see Zambelli, T. 718 Bartos, I., see van Hove, M.A. 290 Bartram, M.E. 320 Bartynski, R.A. 543, 560 Barwich, V., see Bennewitz, R. 732 Bassett, M.R. 368 Bassett, M.R., see Fink, T. 373 Bassi, D., see Scoles, G. 31 Battogtokh, D. 382 Battogtokh, D., see Rose, K.C. 388 Batzill, M., see Jerdev, D.I. 323 Bauer, A. 548 Bauer, A., see Wegner, D. 548 Bauer, E. 11, 380, 764 Bauer, E., see Kolaczkiewicz, J. 764 Bauer, E.G. 382 Bauer, M. 558, 565 Bauer, M., see Aeschlimann, M. 899 Bauer, M., see Gauyacq, J.P. 7 Bäuerle, D. 898, 899, 907, 908 Baule, B. 58, 105, 185 Baum, T. 849, 850 Baumeister, B., see Mull, T. 646 Bäumer, M., see Mavrikakis, M. 302 Baumvol, I.J.R., see Frank, M.M. 791 Baurichter, A., see Diekhöner, L. 434, 463, 511–514 Baurichter, A., see Mortensen, H. 82, 513, 514 Bawendi, M.G., see Murray, C.B. 747 Baym, G., see Kadanoff, L.P. 447 Beach, D.B., see Gates, S.M. 832–835 Bean, J.C., see Hull, R. 798 Bearden, R., see Chianelli, R.R. 330, 331, 333 Beauregard, J.N. 175 Beben, J., see Suchorski, Y. 404–407 Beck, R.D. 181 Beck, R.D., see Maroni, P. 181 Beck, R.D., see Schmid, M.P. 181 Becker, A.J., see Becker, R.S. 685, 687, 726 Becker, A.J., see Jakob, P. 846 Becker, C.A. 671

Author index

Becker, C.A., see Janda, K.C. 199 Becker, G.E., see Hagstrum, H.D. 472, 473 Becker, M. 546, 548, 562 Becker, O., see Asscher, M. 513 Becker, R.S. 551, 685, 687, 726, 738 Becker, R.S., see Jakob, P. 846 Becker, R.S., see Klitsner, T. 728 Beckerle, J.D. 98 Beckmann, D., see Reichert, J. 738 Bedrossian, P. 735 Bedürftig, K., see Völkening, S. 380, 401 Beebe Jr., T.P., see Patrick, D.L. 822 Behler, J. 149, 245, 334, 433, 496, 499–501, 507, 508, 510 Behm, R., see Wintterlin, J. 766 Behm, R.J. 293 Behm, R.J., see Barth, J.V. 780 Behm, R.J., see Brune, H. 73, 493, 494, 765, 766 Behm, R.J., see Gritsch, T. 362 Behm, R.J., see Günther, C. 780 Behm, R.J., see Günther, S. 767 Behm, R.J., see Hitzke, A. 767 Behm, R.J., see Memmert, U. 834 Behm, R.J., see Thiel, P.A. 357, 358 Behm, R.J., see Wintterlin, J. 685 Behringer, E.R., see Marston, J.B. 7, 12 Belaïdi, A. 847 Belaïdi, A., see Chazalviel, J.-N. 847 Belaïdi, A., see Safi, M. 847 Belen, R.J. 827 Belić, D.S. 647 Bell, D.C., see Barrelet, C.J. 808 Bellec, A. 736 Bellman, J., see Hassel, M. 130, 135 Benderskii, V.A. 4, 5 Benedek, G. 56 Benetti, M., see Komrowski, A.J. 494, 495, 500 Bengaard, H., see Nørskov, J.K. 302, 303 Bennemann, K.H., see Jeschke, H.O. 902 Bennemann, K.H., see Stampfli, P. 824, 902 Benner, D.B. 847 Bennett, C.H. 306 Bennewitz, R. 732 Bent, B.E., see Somorjai, G.A. 290 Bent, S.F., see Kong, M.J. 735 Benvenuti, C. 130 Bera, S.K. 845 Beranek, R. 822

919

Berdau, M. 356, 366, 383 Berdau, M., see Ehsasi, M. 368, 369 Berdau, M., see Lim, Y.-S. 403 Berenbak, B. 42, 43, 45, 209 Berenbak, B., see Butler, D.A. 45, 209 Berenbak, B., see Komrowski, A.J. 506, 507 Berenbak, B., see Riedmüller, B. 45 Berenbak, B., see Ternow, H. 507 Berge, K., see Gerlach, A. 539 Berge, K., see Straube, P. 542, 560 Berge, S., see Gartland, P.O. 528 Berger, M.G., see Frohnhoff, S. 849 Bergh, H.S., see Nienhaus, H. 148, 240, 489–491, 493 Bergmann, K. 498 Berhault, G., see Chianelli, R.R. 330, 331, 333 Berke, T., see Andersohn, L. 834 Berkó, A., see Memmert, U. 834 Bernard, R. 748 Bernasek, S.L. 671 Bernasek, S.L., see Bald, D.J. 240, 246 Berndt, R., see Becker, M. 546, 548, 562 Berndt, R., see Chulkov, E.V. 529, 548, 564, 565 Berndt, R., see Corriol, C. 548, 564 Berndt, R., see Crampin, S. 546–548, 551, 562 Berndt, R., see Echenique, P.M. 527, 531, 560, 563 Berndt, R., see Jensen, H. 548–550, 562 Berndt, R., see Kliewer, J. 527, 529, 531, 535, 540, 543, 546, 548–550, 560–562, 686 Berndt, R., see Kuntze, J. 739 Berndt, R., see Li, J. 545–547, 608, 609 Berndt, R., see Limot, L. 548, 549, 560, 733 Bernstein, R.B. 209 Bernstein, R.B., see Levine, R.D. 664 Bernstein, R.B., see Mackay, R.S. 210 Bertel, E. 527 Berthold, W. 551, 553, 554, 563, 564 Berthold, W., see Marinica, D.C. 563 Bertino, M.F., see Tate, M.R. 482, 483 Berton, B., see Rose, K.C. 396, 397 Bertram, M. 411, 413 Bertram, M., see Kim, M. 413, 414 Bertram, M., see Lin, A.L. 412, 419 Besenbacher, F. 296, 298, 717 Besenbacher, F., see Bollinger, M.V. 330 Besenbacher, F., see Christensen, A. 764 Besenbacher, F., see Helveg, S. 330, 331

920

Besenbacher, F., see Kibsgaard, J. 330 Besenbacher, F., see Lauritsen, J.V. 331 Besenbacher, F., see Morgenstern, K. 777, 804 Besenbacher, F., see Österlund, L. 246 Besenbacher, F., see Otero, R. 4 Besenbacher, F., see Rosei, F. 739, 748 Besenbacher, F., see Schunack, M. 739 Besenbacher, F., see Vang, R.T. 298, 300, 781 Beta, C. 413 Beta, C., see Bertram, M. 413 Bethge, H., see Köhler, U. 834 Bethune, D., see Kulginov, D. 122 Bethune, D.S., see Rettner, C.T. 48, 49 Bethune, D.S., see Winters, H.F. 49, 50 Beton, P.H. 691 Beton, P.H., see Keeling, D.L. 691, 692, 751 Beton, P.H., see Moriarty, P. 691 Betz, G. 889, 891 Beusch, H. 345 Beyvers, S., see Vazhappilly, T. 675 Bhasu, V.C.J., see Grimley, T.B. 447 Bhattacharya, P., see Mi, Z. 839 Bialkowski, J., see Sokolowski-Tinten, K. 824 Biberian, J.P., see van Hove, M.A. 290 Biczysko, M., see Hellman, A. 304, 325, 327, 328 Biedermann, A., see Dürr, M. 715, 828, 830 Biedermann, A., see Schmid, M. 73 Biegelsen, D.K., see Benner, D.B. 847 Biersack, J., see Eckstein, W. 877 Biersack, J.P. 889 Biersack, J.P., see Ziegler, J.F. 877, 880 Bihlmayer, G., see Link, S. 564 Bihlmayer, G., see Wegner, D. 548 Bilic, A., see Braun, J. 101 Billas, I.M.L. 781 Billing, G.D. 3, 9 Billing, G.D., see Henriksen, N.E. 237 Biloen, P., see Backx, C. 443 Bimberg, D., see Munt, T.P. 797 Bimberg, D., see Shchukin, V.A. 797, 803, 804, 806 Binder, K., see Landau, D.P. 304, 435 Binder, K., see Patrykiejew, A. 246 Binetti, M. 73, 149, 462, 493–495, 499, 500, 508 Binetti, M., see Ambaye, H. 74, 75 Binetti, M., see Bornscheuer, K.-H. 663 Binetti, M., see Komrowski, A.J. 73

Author index

Binetti, M., see Weiße, O. 72, 73, 175, 183 Binnig, G. 551, 579, 593, 685, 701 Bird, D.M. 148, 189, 448 Bird, D.M., see Mizielinski, M.S. 148, 189, 448, 493, 611 Bird, D.M., see Trail, J.R. 8, 15, 19, 148, 189, 493 Bird, D.M., see White, J.A. 154, 168, 498 Birkenheuer, U., see Gokhale, S. 735 Birner, A., see Leonard, S.W. 822 Birner, A., see Müller, F. 822 Birtcher, R.C. 896 Birtcher, R.C., see Donnelly, S.E. 896 Birtcher, R.C., see Nordlund, K. 896 Birtcher, R.C., see Rehn, L.E. 890 Birtwistle, D.T. 19 Bischof, A., see Weber, W. 781 Bisi, O. 845 Bitensky, I.S. 894 Björk, M.T., see Jensen, L.E. 812 Bjorkman, C.H. 847 Björkman, G. 484 Black, J.E. 107 Blackwood, D.J., see Peter, L.M. 847 Blaha, P., see Nicolay, G. 541 Blaha, P., see Perdew, J.P. 433 Blakely, J.M. 792, 808, 810 Blanc, M., see Gambardella, P. 302 Blandin, A. 8, 19, 447, 448, 459 Blandin, P. 778 Blatt, J.M. 3 Blauw, M.A. 827 Bligaard, T. 277, 278, 286, 289, 294, 307, 310–312, 314, 328, 330 Bligaard, T., see Andersson, M.P. 294, 316–318 Bligaard, T., see Hellman, A. 304, 325, 327, 328 Bligaard, T., see Nilsson, A. 287 Bligaard, T., see Nørskov, J.K. 302, 303, 331–333 Bligaard, T., see Sehested, J. 317, 318 Bligaard, T., see Skúlason, E. 333 Blöchl, P.E., see Fisher, A.J. 735 Block, J.H., see Gorodetskii, V.V. 403, 404 Blome, C., see Sokolowski-Tinten, K. 824 Bloss, H. 447 Bludau, H., see Grobecker, R. 482, 488 Blügel, S., see Dederichs, P.H. 500 Blügel, S., see Link, S. 564

Author index

Blum, J., see Demic, C.P. 844 Blum, K. 555 ´ Blüm, M.-C., see Cavar, E. 687, 688, 746 Blums, J., see Sokolowski-Tinten, K. 824 Bobaru, S.C., see Hendriksen, B.L.M. 366, 367, 369, 370 Bockris, J.O.M. 330, 331 Bocquet, M.-L. 597, 599, 600 Bocquet, M.-L., see Lesnard, H. 599 Boendgen, G., see Saalfrank, P. 650, 658 Boger, K. 552, 555, 556 Bogicevic, A. 319–322, 775 Bogicevic, A., see Ovesson, S. 304, 319–325, 771, 775 Bohdansky, J. 886 Böheim, J. 120, 123 Bohnen, K.P., see Schochlin, J. 478 Boisen, A., see Jacobsen, C.J.H. 312 Boisen, A., see Schumacher, N. 314, 315 Boisonnade, J., see Castets, V. 378 Boiziau, C. 473 Boland, J.J. 828, 830, 831, 842 Boland, J.J., see Chen, D.X. 842 Boland, J.J., see Kubby, J.A. 706 Bolis, C., see Mayne, A.J. 694, 728 Bollinger, M., see Nørskov, J.K. 302, 303 Bollinger, M.V. 330 Bölscher, M., see Punckt, C. 821 Bolton, K., see Ding, F. 810, 813 Bonczek, F., see Bauer, E. 764 Bonde, J., see Hinnemann, B. 330, 331, 333 Bonde, J., see Jaramillo, T.F. 277, 330–333 Bondzie, V.A. 368 Bönig, L. 19 Bonilla, G., see Lauterbach, J. 368 Bonn, M. 35, 483, 674, 900 Bonser, D.J., see Mahajan, A. 828 Bonzel, H.P. 398 Boonekamp, E.P. 849 Booth, M.F., see Busch, D.G. 635 Boragno, C., see Brune, H. 767, 769, 778, 780, 782 Boragno, C., see Vattuone, L. 172 Borazio, A.M., see Peter, L.M. 847 Borckmans, P., see Andrade, R.F.S. 362 Borckmans, P., see Verdasca, J. 385 Borg, M., see Wang, J.G. 367 Borgström, M.T., see Verheijen, M.A. 810

921

Borisov, A.G. 7, 473, 474, 481, 502, 555, 557, 558, 565 Borisov, A.G., see Auth, C. 481, 502 Borisov, A.G., see Chulkov, E.V. 7, 20 Borisov, A.G., see Gauyacq, J.P. 7, 558, 565 Borisov, A.G., see Makhmetov, G.E. 473 Borisov, A.G., see Marinica, D.C. 563 Borisov, A.G., see Olsson, F.E. 549, 733 Borkovec, M., see Hänggi, P. 8, 20 Borkovec, M., see Hanggi, P. 237 Born, M. 433, 437 Bornscheuer, K.-H. 663 Bornscheuer, K.-H., see Kolasinski, K.W. 829 Bornscheuer, K.-H., see Nessler, W. 660, 661 Borowecki, T., see Rudzinski, W. 249 Borroni-Bird, C.E. 763 Bortolani, V. 56, 62–64 Bos, A.N.R. 233 Bosch, C. 325 Boszo, F. 325 Bott, M. 780, 782 Bott, M., see Michely, T. 782, 783 Böttcher, A. 148, 464–466, 473–475, 477, 478, 488, 489, 512 Böttcher, A., see Greber, T. 464, 465, 494 Böttcher, A., see Grobecker, R. 482, 488, 512 Böttcher, A., see Hermann, K. 488 Botti, S. 845 Boudart, M. 261, 299, 304, 325, 326 Bouju, X., see Grill, L. 751 Boukherroub, R., see Allongue, P. 846 Bouneau, S. 895, 897 Bourdon, E.B.D. 465, 477 Bowen, K.H., see Burgert, R. 508 Bowering, N., see Brandt, M. 482, 488, 489, 512 Bowman, J.M., see Gazdy, B. 8 Box, F.M.A., see Raukerma, A. 245 Boyer, C.B. 38 Boyer, P.D. 744 Bozco, F., see Lee, J. 473 Bozdech, G., see Ernst, N. 404 Bozdech, G., see Sieben, B. 404 Bozso, F. 299, 735, 830 Bradforth, S.E., see Manolopoulos, D.E. 510, 512 Bradley, J.M., see Guo, X.-C. 357 Bradley, J.M., see Hopkinson, A. 373 Bradshaw, A.M., see Bonzel, H.P. 398

922

Brako, R. 7, 8, 10, 12–14, 59, 63, 65, 73, 106, 148, 447, 448, 483 Bramblett, T.R. 835 Bramblett, T.R., see Lubben, D. 830, 835 Brandbyge, M. 19, 585, 637 Brandbyge, M., see Frederiksen, T. 592, 595, 615 Brandbyge, M., see Newns, D.M. 19 Brandbyge, M., see Paulsson, M. 585, 597 Brandt, M. 482, 488, 489, 512 Brannon, J.H. 907 Bratlie, K.M., see Somorjai, G.A. 492 Brattain, W.H. 528 Bratu, P. 829 Braun, J. 101 Braun, K.-F. 549, 550, 561, 562 Braun, K.F., see Hla, S.-W. 686 Braun, O.M. 254 Bredow, T., see Henzl, J. 736 Breitschwerdt, A., see Herrero, C.P. 715 Bremten, H. 473 Brenig, W. 98, 106, 113, 120, 166, 829, 830 Brenig, W., see Gross, A. 148, 189, 221, 463, 498 Brenig, W., see Schlichting, H. 46, 111, 125, 126 Brenig, W., see Sedlmeir, R. 106 Brenig, W., see Stutzki, J. 113 Brenner, D.W. 32 Brenner, D.W., see Srivastava, D. 831 Bressers, P.M.M.C. 849, 850 Briggs, G.A.D. 735 Briggs, G.A.D., see Fisher, A.J. 735 Briggs, G.A.D., see Mayne, A.J. 735 Brihuega, I. 694, 728 Brihuega, I., see Paz, O. 590 Brijs, B., see Frank, M.M. 791 Brillouin, L. 792 Bringa, E.M. 896 Bringans, R.D., see Benner, D.B. 847 Brinkman, J.A. 891 Brivio, G. 120, 125, 446, 483, 484 Brivio, G.P. 3 Broadbelt, L.J., see Dooling, D.J. 261 Bromann, K. 778, 780, 781, 783 Bromann, K., see Brune, H. 775, 779 Bromann, K., see Hamilton, J.C. 781 Bronikowski, M.J. 835 Bronikowski, M.J., see Wang, Y. 831, 834, 835

Author index

Bronnikov, D.K. 204 Brønsted, N. 299 Brooks, B.R., see Chu, J.W. 306 Brooks, C., see Yaccato, K. 317 Brouet, V., see Junren, S. 545 Brouwer, A.M. 744 Brown, A.R. 832 Brown, J.K. 48 Brown, W.A. 357, 764 Bruch, L.W. 487 Brumer, Y., see Maragakis, P. 306 Brunauer, S., see Emmett, P.H. 325 Brune, H. 73, 493, 494, 763, 765–767, 769, 775, 778–780, 782, 815 Brune, H., see Barth, J.V. 775, 780 Brune, H., see Bogicevic, A. 320, 775 Brune, H., see Bromann, K. 778, 780, 781, 783 Brune, H., see Bürgi, L. 546, 547, 549 Brune, H., see Cren, T. 780 Brune, H., see Fischer, B. 775 Brune, H., see Gambardella, P. 778, 781 Brune, H., see Hamilton, J.C. 781 Brune, H., see Hofer, W.A. 765, 766 Brune, H., see Jeandupeux, O. 549, 771 Brune, H., see Knorr, N. 609, 771–774 Brune, H., see Müller, B. 778 Brune, H., see Röder, H. 775, 778, 779, 815 Brune, H., see Rusponi, S. 779, 781 Brune, H., see Venables, J.A. 771, 775 Brune, H., see Wintterlin, J. 685, 766 Brunel, F., see Corkum, P.B. 636 Brunelle, A., see Bouneau, S. 895, 897 Brunner, T., see Brenig, W. 166 Brunner, T., see Schlichting, H. 46, 111, 125, 126 Bruno, P., see Stepanyuk, V.S. 772, 774, 775 Brusdeylins, G. 55, 61 Bryant, A. 701 Bryant, O.R., see Crampin, S. 548 Bryant, O.R., see Li, J. 545, 547 Bublik, V.T. 795 Bucher, J.-P., see Röder, H. 815 Bucher, J.P., see Röder, H. 775, 778, 779 Buck, U., see Scoles, G. 31 Budde, F. 82, 631, 634, 635, 644 Budde, F., see Modl, A. 210 Budiansky, N., see Punckt, C. 821 Buhro, W.E., see Wang, F.D. 807 Buluschek, P., see Rusponi, S. 779, 781

Author index

Buntin, S.A. 641, 646, 669, 843 Buntin, S.A., see Gadzuk, J.W. 468, 641, 642, 650, 659 Buntin, S.A., see Richter, L.J. 641 Buntin, S.A., see Struck, L.M. 483 Bürger, J., see Eiswirth, M. 352 Burgert, R. 508 Bürgi, L. 546, 547, 549 Bürgi, L., see Jeandupeux, O. 549, 771 Burgin, T.P., see Reed, M.A. 738 Burke, K. 8, 58, 64, 111 Burke, K., see Andersson, T. 46, 111 Burke, K., see DiRubio, C.A. 58 Burke, K., see Gumhalter, B. 64, 113 Burke, K., see Marques, M.A.L. 580 Burke, K., see Perdew, J.P. 277, 433 Burns, A.R. 644, 660 Burns, A.R., see Orlando, T.M. 718 Burns, F., see Brannon, J.H. 907 Burrows, V.A. 847 Burton, W.K. 815 Busch, D.G. 635, 636, 638 Busch, D.G., see Kao, F.-J. 634 Busch, K., see Leonard, S.W. 822 Bushnell, J.C. 905, 906 Busnengo, C., see Crespos, C. 172, 186 Busnengo, H.F. 146, 147, 151, 172, 182, 185, 186 Busnengo, H.F., see Díaz, C. 151, 155, 156, 186 Busnengo, H.F., see Farías, D. 146, 154, 155, 191 Busnengo, H.F., see Olsen, R.A. 147 Busnengo, H.F., see Somers, M.F. 161, 162, 169, 184, 189 Butcher, D., see Somorjai, G.A. 414 Butler, D.A. 45, 209 Butler, D.A., see Berenbak, B. 45 Butler, D.A., see Raukerma, A. 245 Buttet, J., see Vandoni, G. 769 Büttiker, M. 584 Butz, R., see Ciraci, S. 829 Butzke, S., see Werner, K. 835 Cabrera, N. 275, 497 Cabrera, N., see Burton, W.K. 815 Cabria, I., see Gambardella, P. 778, 781 Calcott, P.D.J., see Cullis, A.G. 823, 845, 846 Calder, R.S., see Benvenuti, C. 130 Callcott, T.A., see Gesell, T.F. 487

923

Calvayrac, F. 470 Camillone III, N., see Francisco, T.W. 204, 223, 224 Campargue, R. 31, 56 Campbell, C.T. 330, 513, 671 Campbell, C.T., see Frenkel, F. 207 Campbell, C.T., see Ovesen, C.V. 314 Campbell, C.T.J., see Stuckless, J.T. 763, 765 Campbell, J., see Hellberg, L. 465, 483 Campbell, S.A. 846, 850 Campbell, S.A., see Lewerenz, H.J. 847 Campillo, I. 539 Campillo, I., see Gerlach, A. 539 Campion, A., see Hatch, S.R. 672 Campion, A., see Mahajan, A. 828 Canham, L. 846 Canham, L.T. 845, 847 Canham, L.T., see Cullis, A.G. 823, 845, 846 Cant, N.W. 319 Cao, C.B., see Chen, Z. 810, 811 Cao, G.Y., see Moula, M.G. 240 Cao, Y. 720 Cao, Y., see Li, Z.-H. 720 Capehart, T.W., see Schoenlein, R.W. 455, 470, 483, 551 Car, R. 499 Car, R., see Ancilotto, F. 829 Carabineiro, S.A.C. 377 Carabineiro, S.A.C., see Peskov, N.V. 259 Carbone, C., see Gambardella, P. 778, 781 Carbone, M., see Soukiassian, L. 701, 703–706 Cardillo, M.J., see Amirav, A. 463 Cardillo, M.J., see Grimmelmann, E.K. 36, 46 Caretta, R., see Flytzani-Stephanopoulos, M. 363 Carey, J.E., see Crouch, C.H. 827 Carey, J.E., see Shen, M.Y. 827 Carhart, R., see Yaccato, K. 317 Carlsson, A., see Hellman, A. 261, 274, 296, 304, 309, 325, 327, 328 Carlsson, A., see Honkala, K. 263, 274, 277, 296, 301, 304, 308, 325–330 Carlsson, A., see Kao, C.L. 49 Carlsson, A.F., see Weaver, J.F. 31, 836 Carpene, E. 637 Carrazza, J., see Strongin, D.R. 326 Carrington, T. 460 Carstensen, J., see Föll, H. 823 Carstensen, J., see Langa, S. 822

924

Carter, E.A. 306 Carter, E.A., see Carter, L.E. 843 Carter, L.E. 843 Casarin, M., see Vittadini, A. 829 Cassuto, A., see Hasselbrink, E. 644, 646 Castets, V. 378 Cattarin, S. 845, 847, 849 Cavalleri, A., see Sokolowski-Tinten, K. 824 Cavanagh, R.R., see Buntin, S.A. 641, 646, 669 Cavanagh, R.R., see Gadzuk, J.W. 468, 641, 642, 650, 659 Cavanagh, R.R., see Richter, L.J. 641 Cavanagh, R.R., see Struck, L.M. 483 ´ Cavar, E. 687, 688, 746 Cederbaum, L.S., see Domcke, W. 19, 20 Cederbaum, L.S., see Moiseyev, N. 6 Cee, V.J., see Patrick, D.L. 822 Celli, V. 56, 64, 98, 107, 157, 456, 461 Celli, V., see Manson, J.R. 62, 70 Cellota, R., see Whitman, L.J. 695 Celotta, R.J., see Stroscio, J.A. 765, 767 Cercla, J., see Mendez, J. 717 Cerda, J. 585 Cerf, N.J., see Fiurášek, J. 738 Ceyer, S.T., see Beckerle, J.D. 98 Ceyer, S.T., see Guthrie, W.L. 641 Ceyer, S.T., see Holt, J.R. 843 Ceyer, S.T., see Li, Y.L. 482, 483, 493 Ceyer, S.T., see Tate, M.R. 482, 483 Chab, V., see Chvoj, Z. 256 Chabal, Y.J. 469, 483, 485, 828, 830, 831, 847 Chabal, Y.J., see Becker, R.S. 685, 687, 726 Chabal, Y.J., see Burrows, V.A. 847 Chabal, Y.J., see Dumas, P. 847 Chabal, Y.J., see Frank, M.M. 791 Chabal, Y.J., see Higashi, G.S. 847 Chabal, Y.J., see Hines, M.A. 850, 851 Chabal, Y.J., see Jakob, P. 846 Chabal, Y.J., see Queeney, K.T. 849 Chabal, Y.J., see Reuff, J.E. 469 Chabal, Y.J., see Trucks, G.W. 846 Chaban, E.E., see Chabal, Y.J. 831 Chakarov, D. 482 Chakarova, R., see Oner, D.E. 498 Chakarova-Käck, S.D. 608 Chakrabarti, N. 658 Chakraborty, B.R., see Bera, S.K. 845 Chakraverty, B.K. 775 Chalopin, Y., see Bellec, A. 736

Author index

Chan, C.-M., see van Hove, M.A. 246 Chan, K.K., see Demic, C.P. 844 Chandler, D. 306 Chandrasekaran, H. 810, 813 Chang, H. 822 Chang, J.P. 827, 844 Chang, L.D., see Jensen, J.H. 113 Chapon, C., see Henry, C.R. 290 Charron, E. 701 Chase, L.L., see van Buuren, T. 849 Châtelain, A., see Billas, I.M.L. 781 Chatfield, D.C. 174 Chaudhuri, S., see Bera, S.K. 845 Chavez, F. 362 Chavy, C., see Girard, C. 695 Chavy, C., see Joachim, C. 738 Chazalviel, J.-N. 847 Chazalviel, J.-N., see Belaïdi, A. 847 Chazalviel, J.-N., see Cattarin, S. 845, 847 Chazalviel, J.-N., see Dubin, V.M. 847 Chazalviel, J.-N., see Hassan, H.H. 847 Chazalviel, J.-N., see Ozanam, F. 847 Chazalviel, J.-N., see Rao, A.V. 847 Chazalviel, J.-N., see Safi, M. 847 Chem, J., see Yamamoto, T. 306 Chen, C., see Wang, J.-T. 695 Chen, D., see Bronikowski, M.J. 835 Chen, D.M., see Bedrossian, P. 735 Chen, D.X. 842 Chen, F., see Guo, H. 659 Chen, J. 511, 512, 735 Chen, J., see White, J.D. 189–191, 334, 433, 466, 496, 511, 512 Chen, J.C., see Kitchin, J.R. 292 Chen, J.C., see Lin, J.S. 832 Chen, J.G., see Nørskov, J.K. 331–333 Chen, L., see Zhao, A. 728 Chen, P., see Xie, Q. 839 Chen, P.J., see Cheng, C.C. 735 Chen, T.C., see Avouris, Ph. 701, 705, 706 Chen, W., see Madhavan, V. 608, 609 Chen, X., see Wu, S.W. 747 Chen, Z. 810, 811 Cheng, C.C. 735, 830 Cheng, H. 6, 10, 12 Cheng, H., see Shenvi, N. 6 Cheng, H.Z. 471 Cheng, Y.-T. 892 ´ Chergui, M., see Cavar, E. 687, 688, 746

Author index

Cherkaoui, M., see Safi, M. 847 Chevary, J., see Perdew, J. 433 Chianelli, R.R. 330, 331, 333 Chiang, T.-C., see Hansen, E.D. 538 Chiang, T.-C., see Lin, D.-S. 831, 835 Chiang, T.-C., see Luh, D.-A. 538 Chiang, T.C., see Hirshorn, E.S. 728 Chiarello, R., see Krim, J. 486 Chiarotti, G. 528 Chiavaralloti, F. 748 Chiba, Y., see Gross, A. 148, 189 Chichkov, B.N., see Nolte, S. 906, 907 Chidsey, C.E.D., see Wade, C.P. 849 Chiesa, M. 102, 103 Chizmeshya, A. 99–102 Cho, H.-C., see Isobe, C. 828 Choi, B.-Y. 736 Choi, H., see Abeln, G.C. 726 Chollet, C., see Skalicky, T. 720 Chollet, F., see Dutartre, D. 828 Chorkendorff, I. 277, 308 Chorkendorff, I., see Abild-Pedersen, F. 299 Chorkendorff, I., see Besenbacher, F. 296, 298 Chorkendorff, I., see Dahl, S. 296–298, 300, 311, 326, 327, 330, 781 Chorkendorff, I., see Hinnemann, B. 330, 331, 333 Chorkendorff, I., see Jaramillo, T.F. 277, 330–333 Chorkendorff, I., see Schumacher, N. 314, 315 Choyke, W.J., see Bozso, F. 735 Choyke, W.J., see Cheng, C.C. 735 Choyke, W.J., see Mezhenny, S. 735 Chrisey, D.B. 908 Christensen, A. 764 Christensen, C.H. 234, 261 Christensen, C.H., see Andersson, M.P. 294, 316–318 Christensen, C.H., see Bligaard, T. 294, 311, 312, 314 Christensen, C.H., see Hellman, A. 261, 274, 296, 304, 309, 325, 327, 328 Christensen, C.H., see Honkala, K. 263, 274, 277, 296, 301, 304, 308, 325–330 Christensen, C.H., see Nilsson, A. 287 Christensen, C.H., see Sehested, J. 317, 318 Christman, S.B., see Burrows, V.A. 847 Christman, S.B., see Chabal, Y.J. 831 Christman, S.B., see Reuff, J.E. 469

925

Christoph, J. 415, 417 Christoph, J., see Colen, R.E.R. 368 Christophersen, M., see Föll, H. 823 Christophersen, M., see Langa, S. 822 Chu, J.W. 306 Chuang, M.-C. 827 Chuang, S.-F., see Smith, R.L. 823 Chuang, T.J., see Fukutani, K. 641, 643 Chuang, T.J., see Hussla, I. 660 Chuang, T.J., see Modl, A. 210 Chuang, T.J., see Winters, H.F. 827, 845 Chubb, J.N. 130 Chulkov, E., see Echenique, P.M. 527, 531, 560, 563 Chulkov, E.V. 7, 20, 529, 534, 542, 548, 560, 561, 564, 565 Chulkov, E.V., see Balasubramanian, T. 543, 560 Chulkov, E.V., see Berthold, W. 551, 553, 554, 563 Chulkov, E.V., see Borisov, A.G. 558, 565 Chulkov, E.V., see Corriol, C. 548, 564 Chulkov, E.V., see Echenique, P.M. 529, 530, 562, 564 Chulkov, E.V., see Eiguren, A. 534, 535, 541, 542, 560–563 Chulkov, E.V., see García-Lekue, A. 531, 532, 563, 564 Chulkov, E.V., see Hellsing, B. 533, 535 Chulkov, E.V., see Kliewer, J. 527, 529, 531, 535, 540, 543, 546, 548, 560, 561 Chulkov, E.V., see Link, S. 564 Chulkov, E.V., see Sarría, I. 564 Chulkov, E.V., see Schäfer, A. 552, 553, 560, 564 Chulkov, E.V., see Silkin, V.M. 529, 542, 543, 560, 561 Chulkov, E.V., see Sklyadneva, I.Yu. 534 Chulkov, E.V., see Vitali, L. 546, 550, 561, 562 Chulkov, E.V., see Wegner, D. 548 Chulkov, E.V., see Zhukov, V.P. 539 Chvoj, Z. 256 Chvoj, Z., see Masin, M. 256 Ciacchi, L.C. 498, 499, 503 Ciccaci, F., see Crampin, S. 529 Ciccotti, G., see Carter, E.A. 306 Ciccotti, G., see Sprik, M. 306 Cini, M. 8 Ciobîcã, I.M. 302

926

Ciobîcã, I.M., see Riedmüller, B. 45 Ciraci, S. 829 Cirlin, G.E., see Dubrovskii, V.G. 810 Ciszek, J.W., see Long, D.P. 592 Citri, O. 148 Citri, O., see Kosloff, R. 148 Claeyssens, F. 825 Clancy, P., see Kuo, C.L. 811 Clarke, B. 352 Clary, D.C., see Hellman, A. 304, 325, 327, 328 Claude, L., see Rusponi, S. 781 Clausen, B.S. 799 Clausen, B.S., see Besenbacher, F. 296, 298 Clausen, B.S., see Helveg, S. 330, 331, 812 Clausen, B.S., see Jacobsen, C.J.H. 294, 312 Clausen, B.S., see Kibsgaard, J. 330 Clausen, B.S., see Lauritsen, J.V. 331 Clausen, B.S., see Topsøe, H. 330, 331 Clausen, B.S., see Vang, R.T. 298, 300, 781 Claussen, C., see Sigmund, P. 893 Clemen, L., see Cheng, C.C. 735 Cobden, P., see Siera, J. 373, 374 Cobden, P.D. 374, 410 Cobden, P.D., see Janssen, N.M.H. 370, 374 Cobden, P.D., see van Tol, M.F.H. 370, 374 Coburn, J.W., see Chang, J.P. 827, 844 Coburn, J.W., see Chuang, M.-C. 827 Coburn, J.W., see Winters, H.F. 827, 843–845 Cococcioni, M., see Sit, P.H.L. 500 Cohen, D., see Kao, F.-J. 634 Cohen, M.H., see Wilke, S. 498 Cohen-Tannoudji, C. 586, 587 Colaianni, M.L., see Cheng, C.C. 735 Cole, M.W., see Vidali, G. 99 Coleman, C. 447 Colen, R.E.R. 368 Colla, T.J. 895, 896 Collins, R.T. 845 Collins, S.D., see Smith, R.L. 820, 823, 850 Coltrin, M.E., see Kay, B.D. 225, 662 Comelli, G. 374, 377, 390 Commaux, N., see Mayne, A.J. 726 Comsa, G. 125 Comsa, G., see Bott, M. 780, 782 Comsa, G., see Kern, K. 101 Comsa, G., see Michely, T. 782, 783, 890 Comsa, G., see Morgenstern, K. 776, 777, 804 Comsa, G., see Rosenfeld, G. 817, 819 Comtet, G. 688, 723, 743

Author index

Comtet, G., see Baffou, G. 734 Comtet, G., see Bellec, A. 736 Comtet, G., see Bernard, R. 748 Comtet, G., see Cranney, M. 696, 745, 746 Comtet, G., see Lastapis, M. 602, 688, 744 Comtet, G., see Mayne, A.J. 4, 22, 688, 696, 699, 726, 746 Comtet, G., see Soukiassian, L. 726, 736 Cong, P.J. 313 Conrad, H., see Chvoj, Z. 256 Conrad, H., see Hemmen, R. 473 Conrad, H., see Pascual, J.I. 592, 615, 616, 688, 723 Conrad, H., see Sesselman, W. 456, 458 Conrad, U. 890 Contreras, A.M., see Park, J.Y. 492 Conway, B.E. 330, 331, 333 Coon, P.A., see Koehler, B.G. 829 Coon, P.A., see Wise, M.L. 715, 830 Cooper, B.H., see DiRubio, C.A. 58 Cooper, B.H., see Marston, J.B. 7, 12 Cooper, B.H., see Powers, J. 57, 59 Cooper, V.R., see Thonhauser, T. 500 Cooperberg, D., see Belen, R.J. 827 Copel, M. 803 Copel, M., see Horn-von Hoegen, M. 798, 803 Coppola, R., see Botti, S. 845 Corkum, P.B. 636 Corriol, C. 172, 173, 175, 183, 548, 564 Corriol, C., see Saalfrank, P. 650 Corriol, C., see Savio, L. 158, 159 Corriol, C., see Weiße, O. 72, 73, 175, 183 Cortes, R., see Munford, M.L. 846 Cortright, R.D. 256, 302 Costa Kieling, V., see Gerischer, H. 842, 846, 847 Costa-Kieling, V., see Allongue, P. 849, 850 Costantini, G.C., see Barth, J.V. 4 Costello, S.A. 623 Cottrell, C., see Hodgson, A. 169 Coufal, H., see Winters, H.F. 49, 50 Coullet, P. 411 Coulman, D., see Gritsch, T. 362 Cowin, J.P. 164, 639 Cowin, J.P., see Becker, C.A. 671 Cowin, J.P., see Janda, K.C. 82, 199 Cowin, J.P., see Rettner, C.T. 164 Cox, M.P. 357, 359, 360 Cox, M.P., see Imbihl, R. 359–362, 379, 381

Author index

Cox, P.A., see Lewerenz, H.J. 847 Craig, B.I. 735, 829 Crampin, S. 529, 546–548, 551, 562 Crampin, S., see Becker, M. 546, 548, 562 Crampin, S., see Chulkov, E.V. 529, 548, 564, 565 Crampin, S., see Jensen, H. 548–550, 562 Crampin, S., see Kliewer, J. 527, 529, 531, 535, 540, 543, 546, 548–550, 560–562, 686 Crampin, S., see Li, J. 545, 547 Cranney, M. 696, 745, 746 Cranney, M., see Bellec, A. 736 Crawford, J.L., see Matthews, J.W. 799 Credi, A., see Badjic, J.D. 744 Cren, T. 780 Cren, T., see Rusponi, S. 779, 781 Crespos, C. 172, 186 Crespos, C., see Busnengo, H.F. 172, 182, 186 Crommie, M.F. 545, 686, 688, 694, 771, 775 Crommie, M.F., see Heller, E.J. 548, 686 Crommie, M.F., see Lu, X. 610 Crommie, M.F., see Madhavan, V. 608, 609 Crommie, M.F., see Yayon, Y. 765, 766 Cross, M.C. 382 Cross, P.C., see Wilson, E.B. 76 Crouch, C.H. 827 Crouch, C.H., see Shen, M.Y. 827 Crowell, J.E., see Isobe, C. 828 Cuenya, B.R. 491 Cuevas, J.C., see Heurich, J. 738 Cui, J. 130 Cui, Y. 808, 810 Cuiffi, J.D., see Kalkan, A.K. 828 Cullis, A.G. 823, 845, 846 Cunningham, R.E., see Gwathmey, A.T. 298 Curri, M.L., see Depero, L.E. 791 Curtiss, T.J., see Mackay, R.S. 210 Custance, O., see Brihuega, I. 694, 728 Custance, O., see Oyabu, N. 694 da Costa, D.G., see Kao, F.-J. 634 Da Fonseca, C., see Cattarin, S. 845, 847 da Rosa, E.B.O., see Frank, M.M. 791 Dabrowski, ˛ J. 792, 829 d’Agliano, E.G. 3, 14, 19 Dahan, M. 747 Dahl, S. 296–298, 300, 311, 326, 327, 330, 781 Dahl, S., see Bligaard, T. 294, 307, 311, 312, 314, 328, 330

927

Dahl, S., see Hellman, A. 261, 274, 296, 304, 309, 325, 327, 328 Dahl, S., see Honkala, K. 263, 274, 277, 296, 301, 304, 308, 325–330 Dahl, S., see Jacobsen, C.J.H. 294, 312 Dahl, S., see Logadottir, A. 299, 301, 326, 330 Dahl, S., see Nørskov, J.K. 302, 303 Dahl, S., see Schumacher, N. 314, 315 Dahl, S., see Vang, R.T. 298, 300, 781 Dai, H. 807 Dai, H.-L. 19 Dai, H.-L., see Howe, P.-T. 633, 656 Dai, J. 44, 59, 66 Dai, J.Q. 175 Dal Negro, L., see Pavesi, L. 845 Dallmeyer, A., see Gambardella, P. 778, 781 Daly, C. 486 D’Andrea, A., see Cini, M. 8 Dang, T.T., see Beck, R.D. 181 Dang, T.T., see Maroni, P. 181 d’Angliano, E.G. 448, 449, 479, 483, 484 Darling, G.R. 145, 151–153, 157, 169, 172, 174–176, 179, 183, 187, 188, 213, 498, 499 Darling, G.R., see Corriol, C. 172, 173, 175, 183 Darling, G.R., see Harris, S. 650, 666–668 Darling, G.R., see Kay, M. 151, 159, 183 Darling, G.R., see Kinnersley, A.D. 168, 169, 174, 176 Darling, G.R., see Saalfrank, P. 663 Darling, G.R., see Savio, L. 158, 159 Darling, G.R., see Wang, Z.S. 146, 159–161, 170, 171, 186, 187 Darling, G.R., see Weiße, O. 72, 73, 175, 183 Daruka, I. 799, 804 Dath, J.P. 371, 412 Dath, J.P., see Fink, T. 371–373 Datta, S. 592 Datta, S., see Xue, Y. 735 Davda, R.R., see Cortright, R.D. 302 David, D.E., see Balaji, V. 893 David, R., see Comsa, G. 125 David, R., see Kern, K. 101 David, R., see Wetzig, D. 178 Davidovits, P. 479 Davidsen, J., see Wei, H. 384 Davies, J.A., see Griffiths, K. 357 Davies, J.A., see Jackman, T.E. 357 Davies, J.C. 293

928

Davis, L.C. 545 Davison, S.G. 529 Daw, M.S., see Nelson, J.S. 534 Dayo, A. 487 de Beer, V.H.J., see Prins, R. 331 de Decker, Y. 259, 398 de Dominicis, C.T., see Nozières, P. 15 de Donder, T. 310, 313 de Gironcoli, S., see Eiguren, A. 534 de Groot, C., see Backx, C. 443 de Gurtubay, I.G., see Zhukov, V.P. 539 de Jong, K.P. 807, 812 de la Rosa, M.P., see Chianelli, R.R. 330, 331, 333 de Maaijer-Gielbert, J., see van Tol, M.F.H. 403 de Meijere, A., see Kolasinski, K.W. 829 de Meijere, A., see Weik, F. 625, 626, 669 de Rougemont, F., see Budde, F. 634, 635 de Smet, T., see Verheijen, M.A. 810 de Unamuno, S. 825 de Vries, A.E., see Können, G.P. 891 de Wette, F.W., see Kress, W. 534 de Wolf, C.A. 370, 377 de Yoreo, J.J., see Dove, P.M. 820 Decius, J.C., see Wilson, E.B. 76 Dederichs, P.H. 500 Dederichs, P.H., see Gambardella, P. 778, 781 Delabie, A., see Frank, M.M. 791 Delchar, T.A., see Woodruff, D.P. 293 Delerue, C., see Dubois, M. 745 Deliwala, S. 634 Deliwala, S., see Her, T.-H. 827 Della-Negra, S., see Bouneau, S. 895, 897 Delley, B., see Behler, J. 149, 245, 334, 433, 496, 499–501, 507, 508, 510 Delley, B., see Li, J. 608, 609 Delmon, B., see Parvulescu, V.I. 319 DeLouise, L.A., see Rettner, C.T. 164 DelSesto, D.F., see Smith, R.R. 181 Demchuk, A., see Preuss, S. 826, 906, 907 Demic, C.P. 844 Demuth, J.E., see Hamers, R.J. 816, 831 Demuth, J.E., see Imbihl, R. 834 Deng, J.F., see Cao, Y. 720 Denzler, D.N. 675 Denzler, D.N., see Bonn, M. 674, 900 Depauw, J., see Bouneau, S. 895, 897 Depero, L.E. 791 DePristo, A.E. 8, 13

Author index

DePristo, A.E., see Evans, J.W. 767 DePristo, A.E., see Sanders, D.E. 767 Dereux, A., see Girard, C. 714 Dereux, A., see Martin, O.J.F. 714 Derooij, N., see Smith, R.L. 850 Desjonquères, M.-C. 605, 606, 799 D’Evelyn, M.P. 174, 830 Devenney, M., see Strasser, P. 293 Devillers, T., see Dujardin, R. 809 Devonshire, A.F., see Lennard-Jones, J.E. 55, 113 Dewel, G., see Andrade, R.F.S. 362 Dewel, G., see Verdasca, J. 385 DeWitt, K. 836 Dhanak, V.R., see Comelli, G. 377, 390 Dhesi, S.S., see Gambardella, P. 778, 781 Dianat, A. 146 Díaz, C. 151, 155, 156, 186, 471, 513, 514 Díaz, C., see Farías, D. 146, 154, 155, 191 Díaz, R., see Gorostiza, P. 842 Diaz de la Rubia, T., see Averback, R.S. 884, 892 Dick, K.A., see Johansson, J. 810, 812 Dicke, J. 381 Diekhöner, L. 434, 463, 511–514 Diekhöner, L., see Mortensen, H. 82, 513, 514 Diekhöner, L., see Wahl, P. 550, 609 Dierkes, G., see Borisov, A.G. 565 Dietrich, C., see Sokolowski-Tinten, K. 824 Dietrich, H. 298, 300 DiLabio, G.A., see Piva, P.G. 696, 730 Ding, F. 810, 813 Dinger, A. 843 Dingus, R.S. 905, 906 Dinh, L.N., see van Buuren, T. 849 Diño, W.A. 145 Diño, W.A., see Miura, Y. 165 Dion, M. 500, 608 Dion, M., see Rydberg, H. 500 Dippel, O. 828 Dirksen, R.J., see Raukema, A. 38 DiRubio, C.A. 58 DiRubio, C.A., see Marston, J.B. 7, 12 Diu, B., see Cohen-Tannoudji, C. 586, 587 Dixon, R.H., see Dunstan, D.J. 799 Dixon-Warren, St.J. 630, 631, 720 Djebri, A., see Ozanam, F. 847 Djéga-Mariadassou, G., see Boudart, M. 304 Doak, R.B., see Brusdeylins, G. 55, 61

Author index

Doblhofer, K., see Flätgen, G. 821 Dobrin, S. 696 Dobson, P.J., see Neave, J.H. 817, 818 Doedel, E.J. 352 Dogel, S., see Piva, P.G. 696, 730 Dohnálek, Z. 715 Doll, J.D., see Adelman, S.A. 6, 19 Domcke, W. 19, 20 Domen, K., see Katano, S. 735 Dominko, A., see Rudzinski, W. 249 Donath, M., see Bertel, E. 527 Dondi, M.G. 102 Dong, A.G., see Wang, F.D. 807 Dong, W., see Busnengo, H.F. 146, 147, 172, 182, 185, 186 Dong, W., see Crespos, C. 172, 186 Donnelly, S.E. 896 Donnelly, S.E., see Birtcher, R.C. 896 Donnelly, S.E., see Nordlund, K. 896 Donnelly, S.E., see Rehn, L.E. 890 Doolen, R.D., see Cong, P.J. 313 Dooling, D.J. 261 Dopheide, R., see Wetzig, D. 178 Doren, D.J. 828 Doren, D.J., see Brown, A.R. 832 Dove, P.M. 820 Downer, M.C. 824 Downer, M.C., see Wang, X.Y. 900 Drachsel, W., see Gorodetskii, V.V. 403, 404 Dragoset, R.A., see Whitman, L.J. 695 Drauglis, E. 273, 277, 455 Dreyfus, R.W., see Phipps, C.R. 908 Dröge, H., see Gokhale, S. 735 Du, W., see Klug, D.-A. 828 Duan, X. 808, 809 Duan, X.F. 808 Dubin, V.M. 847 Dubois, M. 745 Dubrovskii, V.G. 810 Duchemin, I. 738 Duchemin, I., see Fiurášek, J. 738 Dujardin, G. 601, 602, 686, 687, 693, 699, 701, 718, 728, 729 Dujardin, G., see Baffou, G. 734 Dujardin, G., see Bellec, A. 736 Dujardin, G., see Bernard, R. 748 Dujardin, G., see Comtet, G. 688, 723, 743 Dujardin, G., see Cranney, M. 696, 745, 746 Dujardin, G., see Lastapis, M. 602, 688, 744

929

Dujardin, G., see Martín, M. 746 Dujardin, G., see Mayne, A.J. 4, 22, 602, 688, 694, 696, 699, 701, 703, 726, 728, 746 Dujardin, G., see Molinàs-Mata, P. 694, 728 Dujardin, G., see Papageorgiou, N. 733 Dujardin, G., see Ra¸seev, G. 723 Dujardin, G., see Riedel, D. 715 Dujardin, G., see Soukiassian, L. 701, 703–706, 726, 736 Dujardin, R. 809 Duke, C.B. 233 Dulos, E., see Castets, V. 378 Dumas, P. 847 Dumesic, J.A. 233, 234, 304, 305, 310, 326 Dumesic, J.A., see Alcalá, R. 302 Dumesic, J.A., see Cortright, R.D. 256, 302 Dumesic, J.A., see Kandoi, S. 308, 309 Dumesic, J.A., see Schumacher, N. 314, 315 Dunn, A.W., see Beton, P.H. 691 Dunstan, D.J. 794, 795, 799, 802 Dürig, U. 689 Dürr, H.A., see Link, S. 564 Dürr, H.A., see Rhie, H.-S. 563 Dürr, M. 183, 715, 828–830 Dutartre, D. 828 Dutta, C.M. 480 Duvenbeck, A., see Samartsev, A.V. 895 Dwyer, D.J. 346 Dwyer, D.J., see Bondzie, V.A. 368 Dyer, P.E., see Paltauf, G. 904 Eberhardt, W., see Link, S. 564 Eberhardt, W., see Rhie, H.-S. 563 Eberl, K., see Schmidt, O.G. 840 Echenique, P.M. 484, 527, 529–531, 560, 562–564 Echenique, P.M., see Arnau, A. 467, 468 Echenique, P.M., see Balasubramanian, T. 543, 560 Echenique, P.M., see Berthold, W. 551, 553, 554, 563 Echenique, P.M., see Borisov, A.G. 558, 565 Echenique, P.M., see Campillo, I. 539 Echenique, P.M., see Chulkov, E.V. 7, 20, 529, 534, 548, 560, 561, 564, 565 Echenique, P.M., see Corriol, C. 548, 564 Echenique, P.M., see Eiguren, A. 534, 535, 541, 542, 560–563 Echenique, P.M., see Föhlisch, A. 566

930

Echenique, P.M., see García-Lekue, A. 531, 532, 563, 564 Echenique, P.M., see Gerlach, A. 539 Echenique, P.M., see Kliewer, J. 527, 529, 531, 535, 540, 543, 546, 548, 560, 561 Echenique, P.M., see Link, S. 564 Echenique, P.M., see Sarría, I. 564 Echenique, P.M., see Schäfer, A. 552, 553, 560, 564 Echenique, P.M., see Silkin, V.M. 542, 543, 560 Echenique, P.M., see Vitali, L. 546, 550, 561, 562 Echenique, P.M., see Wegner, D. 548 Echenique, P.M., see Zhukov, V.P. 539 Eckman, S.C., see Tate, M.R. 482, 483 Eckstein, W. 877, 886, 890 Eckstein, W., see Biersack, J.P. 889 Eckstein, W., see Garcia-Rosales, C. 886 Eckstein, W., see Shulga, V.I. 889 Economou, E.N. 537, 545 Edgell, R.G., see Fishlock, T.W. 691 Eesley, G.L., see Schoenlein, R.W. 455, 470, 483, 551 Egeberg, R.C., see Dahl, S. 296–298, 300, 326, 327, 330, 781 Egelhoff, W.F. 370, 767 Egelhoff, W.F., see Nyberg, G.L. 767 Eguiluz, A.G. 486 Ehbrecht, M. 845 Ehm, D., see Nicolay, G. 540, 541 Ehm, D., see Reinert, F. 540, 541, 543, 548, 560, 561 Ehrlich, G. 765, 771, 817 Ehrlich, G., see Antczak, G. 254 Ehrlich, G., see Gölzhäuser, A. 765 Ehrlich, G., see Koh, S.J. 773, 774 Ehrlich, G., see Kyuno, K. 782 Ehrlich, G., see Wang, S.C. 765, 768 Ehsasi, M. 368, 369 Eichler, A. 182, 183, 311, 497 Eichler, A., see Gajdos, M. 288 Eichler, A., see Mittendorfer, F. 497 Eigler, D.M. 456, 458, 461, 601, 685, 688, 694, 717, 738 Eigler, D.M., see Crommie, M.F. 545, 686, 688, 694, 771, 775 Eigler, D.M., see Fiete, G.A. 686 Eigler, D.M., see Heinrich, A.J. 686, 709, 742 Eigler, D.M., see Heller, E.J. 548, 686

Author index

Eigler, D.M., see Manoharan, H.C. 609, 686 Eigler, D.M., see Stroscio, J.A. 686, 688 Eigler, D.M., see Weiss, P.S. 765 Eigler, D.M., see Zeppenfeld, P. 689, 690 Eiguren, A. 534, 535, 541, 542, 560–563 Eiguren, A., see Hellsing, B. 533, 535 Eijkel, J.C.T., see Mijatovic, D. 791 Einstein, E. 536 Einstein, T.L. 247 Eiswirth, M. 345–348, 352, 357, 359, 361, 364, 378, 381, 410–412 Eiswirth, M., see Bär, M. 356, 383–385, 390, 392, 410 Eiswirth, M., see Christoph, J. 415, 417 Eiswirth, M., see Krischer, K. 361, 411 Eiswirth, M., see Moeller, P. 362 Eiswirth, M., see Reichert, C. 384, 408, 409 Eiswirth, M., see Schwankner, R.J. 412 Eiswirth, M., see Starke, J. 409 Eizenberg, M., see Blakely, J.M. 792 Ekardt, W., see Keyling, R. 560 Ekerdt, J.G., see Jo, S.K. 843 Ekerdt, J.G., see Mahajan, A. 828 Ekvall, I., see Wahlström, E. 772 El-Aziz, A.M., see Kibler, L.A. 292, 293 Ellis, W.C., see Wagner, R.S. 808, 809 Ellison, M.D., see Mathews, C.M. 513 Elokhin, V.I., see Gorodetskii, V.V. 402 Elokhin, V.I., see Vishnevskii, A.L. 362 Elwenspoek, M., see Bressers, P.M.M.C. 849, 850 Emberly, E.G. 738 Emel’yanov, V.I. 825, 826 Emig, G. 411 Emmett, P.H. 325 Engbæk, J., see Abild-Pedersen, F. 299 Engdahl, C. 494, 719 Engel, H., see Falcke, M. 385, 388, 393 Engel, T. 356 Engel, T., see Engstrom, J.R. 843, 844 Engel, T., see Szabò, A. 844 Engel, W. 379 Engel, W., see Jakubith, S. 381, 385, 387 Engel, W., see Rotermund, H.H. 821 Engel, W., see Swiech, W. 380 Engelhardt, R., see Karlsson, U.O. 543, 560 Engler, B.H., see Lox, E.S.J. 370 Engstrom, J.R. 834–837, 843, 844 Engstrom, J.R., see Jones, M.E. 832, 833

Author index

Engstrom, J.R., see Maity, N. 837 Engstrom, J.R., see Xia, L.Q. 833, 835, 836 Epple, M., see Cren, T. 780 Epple, M., see Knorr, N. 609, 771–774 Epple, M., see Rusponi, S. 781 Epstein, I.R., see Vanag, V.K. 378 Eres, G. 828 Erichsen, P., see Dicke, J. 381 Eriksson, J.C. 792 Erné, B.H., see Chazalviel, J.-N. 847 Ernst, F., see Schmidt, O.G. 840 Ernst, N. 404 Ernst, N., see Sieben, B. 404 Ernzerhof, M., see Perdew, J.P. 277, 433 Erskine, J.L., see Koel, B.E. 623 Ertl, G. 3, 233, 325, 345, 346, 357, 455, 489, 496, 781, 821 Ertl, G., see Bartels, L. 4, 558, 687, 699 Ertl, G., see Barth, J.V. 780 Ertl, G., see Bonn, M. 483, 674 Ertl, G., see Bonzel, H.P. 398 Ertl, G., see Boszo, F. 325 Ertl, G., see Böttcher, A. 148, 464–466, 473–475, 477, 478, 488, 489, 512 Ertl, G., see Bozso, F. 299 Ertl, G., see Brune, H. 73, 493, 494, 765, 766 Ertl, G., see Budde, F. 82, 631, 644 Ertl, G., see Campbell, C.T. 671 Ertl, G., see Cox, M.P. 357, 359, 360 Ertl, G., see Denzler, D.N. 675 Ertl, G., see Dietrich, H. 298, 300 Ertl, G., see Eiswirth, M. 345, 346, 357, 359, 361, 364, 381, 410–412 Ertl, G., see Engel, T. 356 Ertl, G., see Flätgen, G. 821 Ertl, G., see Frenkel, F. 207 Ertl, G., see Greber, T. 464, 465, 482, 494 Ertl, G., see Grobecker, R. 482, 488, 512 Ertl, G., see Hasselbrink, E. 644, 646 Ertl, G., see Hermann, K. 488 Ertl, G., see Hertel, T. 553, 660, 663 Ertl, G., see Hildebrand, M. 398 Ertl, G., see Hotzel, A. 563 Ertl, G., see Imbihl, R. 259, 345, 346, 359–363, 370, 371, 379, 381, 821 Ertl, G., see Krischer, K. 361, 411 Ertl, G., see Ladas, S. 363, 364, 368, 369, 395, 396 Ertl, G., see Mendez, J. 717

931

Ertl, G., see Mikhailov, A.S. 397 Ertl, G., see Modl, A. 210 Ertl, G., see Paal, Z. 325 Ertl, G., see Pollmann, M. 416, 417 Ertl, G., see Rotermund, H.H. 821 Ertl, G., see Sander, M. 359, 361, 363, 365 Ertl, G., see Segner, J. 208, 209 Ertl, G., see Sesselman, W. 456, 458, 473 Ertl, G., see Trost, J. 771 Ertl, G., see Velic, D. 558 Ertl, G., see Wintterlin, J. 321, 671, 719 Ertl, G., see Wolf, M. 625, 626, 647 Ertl, G., see Wolff, J. 821 Ertl, G., see Woratschek, B. 459, 472, 473 Ertl, G., see Zambelli, T. 298, 330, 718 Ertl, G., see Zhu, X.-Y. 647 Esaki, L. 735 Esashi, M. 851 Esch, F. 417 Esch, F., see Cobden, P.D. 374 Esch, F., see Lombardo, S.J. 374 Esch, F., see Schaak, A. 374, 376, 390 Esch, F., see Schütz, E. 417, 419 Esch, F., see Veser, G. 374, 375, 385, 388 Escorcia-Aparcio, E.J., see Saunders, W.A. 845 Estermann, I. 55 Etman, M., see Hassan, H.H. 847 Evans, D., see Celli, V. 98 Evans, J.F., see Zazzera, L. 847 Evans, J.W. 355–357, 382, 383, 385, 763, 767 Evans, J.W., see Liu, D.-J. 382, 407 Evans, J.W., see Liu, D.J. 234 Evans, J.W., see Pavlenko, N. 407 Evans, J.W., see Tammaro, M. 355, 373, 382, 383, 385, 386, 407 Evans, M.G. 299, 308, 311 Evans, S.T., see Kandoi, S. 308, 309 Evatt, R., see Johnson, R.E. 893 Everson, M.P., see Davis, L.C. 545 Eyring, H. 304, 305, 435, 511 Fabre, F., see Rettner, C.T. 87, 88, 148, 183, 189, 190, 212, 220 Fabrizio, E.D., see Kiskinova, M. 379 Fain, S.C., see Cui, J. 130 Falcke, M. 385, 388, 393 Falcone, G. 887 Falcone, G., see Sigmund, P. 889 Fallavier, M., see Bouneau, S. 895, 897

932

Falta, J. 363, 378, 396 Family, F., see Amar, J.G. 779 Fan, H.J. 807 Fan, K.-N., see Li, Z.-H. 720 Fan, Q., see Cong, P.J. 313 Fan, Q., see Strasser, P. 293 Fan, R., see Hochbaum, A.I. 808 Fan, X.L. 508 Fang, G., see Ward, C.A. 249 Fano, U. 16, 22 Farías, D. 146, 154, 155, 191 Farías, D., see Nieto, P. 31 Farnaam, M.K. 833 Fassaert, D.J.M. 279 Fathauer, R.W., see Vasquez, R.P. 847 Fauchet, P.M., see Collins, R.T. 845 Fauchet, P.M., see Siegman, A.E. 827 Fauster, Th. 529, 551, 553, 556, 557 Fauster, Th., see Boger, K. 552, 555, 556 Fauster, Th., see Echenique, P.M. 527, 531, 560, 563 Fauster, Th., see Fischer, R. 553 Fauster, Th., see Höfer, U. 555, 564 Fauster, Th., see Reuß, Ch. 553, 556, 557 Fauster, Th., see Roth, M. 555, 556, 564 Fauster, Th., see Schäfer, A. 552, 553, 560, 564 Fauster, Th., see Shumay, I.L. 551, 555, 563, 564 Fauster, Th., see Weinelt, M. 556, 557 Favre-Nicolin, V., see Dujardin, R. 809 Fawcett, R.H.J., see Keeling, D.L. 692 Fechner, G.T. 345 Fedorov, A.V., see Valla, T. 544 Feenstra, R.M. 696, 746 Feenstra, R.M., see Stroscio, J.A. 706 Feibelman, P.J. 498, 765 Feibelman, P.J., see Harris, J. 103 Feldman, L.C., see Gossmann, H.-J. 831 Félix, C., see Vandoni, G. 769 Fenn, J.B., see Mantell, D.A. 672 Fenyö, D., see Johnson, R.E. 894 Ferencz, K., see Xu, L. 551 Ferguson, B.A., see Pacheco, K.A. 836 Ferm, P.M. 130 Ferm, P.M., see Budde, F. 82, 631, 644 Fermi, E. 500 Fernández, J.M., see Turner, A.R. 828 Ferrando, R., see Ala-Nissila, T. 254 Ferrando, R., see Braun, O.M. 254

Author index

Ferrante, J. 487 Ferre, D.C., see van Bavel, A.P. 251 Ferrell, R.A., see Quinn, J.J. 530, 531 Ferreni, S., see Dondi, M.G. 102 Ferro, Y., see Papageorgiou, N. 733 Fery, P. 357 Fetter, A.L. 612 Feulner, P. 558 Feulner, P., see Berthold, W. 551, 553, 554, 563, 564 Feulner, P., see Föhlisch, A. 566 Feulner, P., see Marinica, D.C. 563 Feynman, R.P. 685 Fichthorn, K.A. 237, 323, 407, 774, 775 Fichtnerendruschat, S., see Greber, T. 482 Fick, D., see Hermann, K. 488 Fick, D., see Zubkov, T. 302 Fieguth, P., see Beusch, H. 345 Field, R.J. 345 Field, R.W., see Silva, M. 511, 512 Fiete, G.A. 686 Filhol, J.S. 292 Filimonov, Y.G., see Bronnikov, D.K. 204 Findlay, R.D., see Ward, C.A. 249 Finger, K. 23, 653, 657, 659 Finger, K., see Saalfrank, P. 658 Fink, A., see Föhlisch, A. 566 Fink, T. 371–373 Fink, T., see Dath, J.P. 371, 412 Fink, T., see Imbihl, R. 373 Fink, T., see Lombardo, S.J. 374 Fink, T., see Slinko, M. 374 Finlay, R.J., see Deliwala, S. 634 Finlay, R.J., see Her, T.-H. 827 Fiolhais, C., see Perdew, J. 433 Fiory, A.T. 845 Fischer, B. 775 Fischer, B., see Barth, J.V. 775 Fischer, B., see Müller, B. 778 Fischer, C., see Mull, T. 646 Fischer, N., see Fischer, R. 553 Fischer, R. 553 Fischman, G.S., see Wang, H. 808 Fisher, A.J. 735 Fisher, A.J., see Briggs, G.A.D. 735 Fisher, E.R., see Bauer, E. 11 Fisher, E.R., see Williams, K.L. 844, 845 Fisher, G. 443 Fisher, G.B., see Gland, J.L. 718

Author index

Fishlock, T.W. 691 Fitting, H.J., see Klar, F. 488 Fitzgerald, E.A. 799 FitzHugh, R. 353 Fiurášek, J. 738 Flagan, R.C., see Saunders, W.A. 845 Flamm, D.L. 827, 844, 845 Flätgen, G. 821 Flidr, J. 851 Flödstrom, S.A., see Karlsson, U.O. 543, 560 Flores, F., see Arnau, A. 467, 468 Flytzani-Stephanopoulos, M. 363 Fogarassy, E., see de Unamuno, S. 825 Föhlisch, A. 566 Fojtik, A. 845 Foley, E.T. 701, 705, 706 Föll, H. 823 Föll, H., see Langa, S. 822 Föll, H., see Müller, F. 822 Föll, H., see Ottow, S. 822 Fölsch, S., see Olsson, F.E. 549, 733 Foltyn, S.R. 827 Fonash, S.J., see Kalkan, A.K. 828 Fondén, T. 467 Foord, J.S., see Prince, R.H. 465, 477 Ford, G.W., see Li, X.L. 6, 8, 19 Forsblom, M. 486 Forshaw, M., see Stadler, R. 739 Förster, E., see Sokolowski-Tinten, K. 824 Fortier, T.M.G., see Lopinski, G.P. 735 Foster, A., see Hofer, W.A. 589 Fowler, D.E., see Luntz, A.C. 98 Fowlkes, J.D., see Pedraza, A.J. 825, 827 Foxon, C.T. 818, 837, 838 Franchi, S. 837, 838, 840 Francisco, T.W. 204, 223, 224 Frank, F.C. 799, 802 Frank, F.C., see Burton, W.K. 815 Frank, K.H., see Binnig, G. 551 Frank, M.M. 791 Franz, R.U., see Rotermund, H.H. 349, 381, 409, 821 Franzò, G., see Pavesi, L. 845 Frechard, F., see Hermse, C.G.M. 254 Frederiksen, T. 592, 595, 615 Frederiksen, T., see Paulsson, M. 597 Freed, K.F. 711 Freeman, A.J., see Wimmer, E. 449 Freihube, K., see Hermann, K. 488

933

Freihube, T., see Greber, T. 464, 465, 494 Freitag, J., see Klüner, T. 631 French, P.J., see Xia, X. 849, 850 Frenkel, D., see Reuter, K. 304, 308, 319, 324, 325, 370 Frenkel, F. 207 Frenkel, F., see Segner, J. 208, 209 Frenken, J.W.M. 717 Frenken, J.W.M., see Gastel, R.v. 777 Frenken, J.W.M., see Hendriksen, B.L.M. 366, 367, 369, 370, 380 Frésard, R., see Hengsberger, M. 542–544 Freund, A., see Eiswirth, M. 352 Freund, H.-J., see Budde, F. 631, 644 Freund, H.-J., see Ertl, G. 781 Freund, H.-J., see Klüner, T. 631, 632 Freund, H.-J., see Libuda, J. 263 Freund, H.-J., see Mavrikakis, M. 302 Freund, H.-J., see Mehdaoui, I. 632 Freund, H.-J., see Mull, T. 646 Freund, H.-J., see Sterrer, M. 730 Freund, H.-J., see Watanabe, K. 470 Freund, H.-J., see Yulikov, M. 275 Frey, A.M., see Sehested, J. 317, 318 Frey, S., see Langa, S. 822 Freyer, N. 357 Frick, D., see Greber, T. 464, 465, 494 Fricke, A., see Fischer, B. 775 Fridell, E. 319 Fridell, E., see Olsson, L. 323 Friedel, J. 770 Friedman, R.S., see Atkins, P.W. 307 Friedman, R.S., see Chatfield, D.C. 174 Friedman, R.S., see McAlpine, M.C. 808 Friend, C.M., see Shen, M.Y. 827 Frigeri, P., see Franchi, S. 837, 838, 840 Frisch, M.J., see Li, X. 6 Frisch, O.R. 55 Frisch, R., see Estermann, I. 55 Frischkorn, C. 19, 22, 470 Frischkorn, C., see Denzler, D.N. 675 Frischkorn, C., see Luntz, A.C. 675 Frischkorn, C., see Wagner, S. 675 Frochot, C., see Brouwer, A.M. 744 Frohnhoff, S. 849 Fromm, 497 Fromm, E. 275 Fuchs, H., see Binnig, G. 551 Fuhrmann, D. 496

934

Fuhrmann, D., see Braun, J. 101 Fujimoto, J.G., see Schoenlein, R.W. 455, 470, 483, 551 Fujimoto, T., see Sagara, T. 829 Fujimura, S., see Ogawa, H. 849 Fujino, T., see Igarashi, H. 293 Fukidome, H., see Matsumura, M. 847 Fukuda, M., see Bjorkman, C.H. 847 Fukuda, T., see Tanaka, T. 830 Fukutani, K. 641, 643, 669 Fulde, P. 579, 609 Funk, L., see Rehn, L.E. 890 Funk, S., see Bonn, M. 483, 674, 900 Furjanic, M.J., see Engstrom, J.R. 834–837 Gaan, S., see Feenstra, R.M. 696, 746 Gädecke, W., see Karlsson, U.O. 543, 560 Gades, H. 887, 890 Gadzuk, J.W. 6–9, 11, 12, 14–16, 19–24, 240, 284, 450–452, 464, 468, 476, 482, 483, 616, 637, 641, 642, 650, 659, 729 Gadzuk, J.W., see Chakrabarti, N. 658 Gadzuk, J.W., see Metiu, H. 3, 4, 14, 433 Gadzuk, J.W., see Plummer, E.W. 528 Gadzuk, J.W., see Rhodin, T.N. 509 Gaffney, K.J. 558 Gaffney, K.J., see Wong, C.M. 563 Gahl, C. 558, 559, 564 Gajdos, M. 288 Gale, J.D., see Hobbs, C. 692 Galperin, M. 21 Gambardella, P. 302, 778, 781 Ganduglia-Pirovano, M.V. 395 Ganduglia-Pirovano, M.V., see Hennig, D. 293 Ganesh, R. 275 Ganteför, G.F., see Burgert, R. 508 Ganz, E., see Pearson, C. 776 Gao, S. 20, 458, 461, 468, 616, 637, 718 Gao, S., see Busch, D.G. 635 Gao, S., see Stipe, B.C. 719 Gao, S., see Strömquist, J. 637 Gao, S.W., see Stipe, B.C. 458, 462, 463 Garcia, M.E., see Jeschke, H.O. 902 Garcia, N., see Binnig, G. 551, 593 García, N., see Sáenz, J.J. 694, 718 Garcia, S.P. 821, 849 García-Lekue, A. 531, 532, 563, 564 Garcia-Lekue, A., see Hofer, W.A. 765, 766 Garcia-Rosales, C. 886

Author index

Garcia-Rosales, C., see Eckstein, W. 886 Gardeniers, J.G.E., see Bressers, P.M.M.C. 849, 850 Gardiner, C.W. 404 Gardner, P. 357 Gardner, P., see Miners, J.H. 371 Garfunkel, E., see Frank, M.M. 791 Garnier, M., see Hengsberger, M. 542 Garnier, M., see Purdie, D. 538, 539 Garot, C., see Boiziau, C. 473 Garrett, B.C., see Schenter, G.K. 234 Garrett, B.G., see Chatfield, D.C. 174 Garrison, B.J., see Adelman, S.A. 6 Garrison, B.J., see Srivastava, D. 831 Garrison, B.J., see Wucher, A. 891 Garrison, B.J., see Zhigilei, L.V. 903 Gartland, P.O. 497, 528 Gartland, P.O., see Slagsvold, B.J. 540, 541 Gartner, K., see Schuller, A. 41 Gastel, R.v. 777 Gates, G.A. 189 Gates, S.M. 830, 832–835, 843 Gates, S.M., see Imbihl, R. 834 Gates, S.M., see Jasinski, J.M. 828 Gates, S.M., see Koleske, D.D. 843 Gates, S.M., see Kulkarni, S.K. 831 Gatti, F.G., see Brouwer, A.M. 744 Gaudioso, J. 735 Gaunt, D.S., see Sykes, M.F. 767 Gauyacq, J., see Marinica, D.C. 563 Gauyacq, J.P. 7, 558, 565 Gauyacq, J.P., see Auth, C. 481, 502 Gauyacq, J.P., see Bahrim, B. 481, 502 Gauyacq, J.P., see Borisov, A.G. 7, 473, 474, 481, 502, 555, 557, 558, 565 Gauyacq, J.P., see Chulkov, E.V. 7, 20 Gauyacq, J.P., see Hotzel, A. 563 Gauyacq, J.P., see Makhmetov, G.E. 473 Gauyacq, J.P., see Marinica, D.C. 20 Gauyacq, J.P., see Olsson, F.E. 549, 733 Gauyacq, J.P., see Teiller-Billy, D. 473, 474 Gauyacq, J.P., see Teillet-Billy, D. 481, 502, 722, 723 Gawronski, H., see Henzl, J. 736 Gaylord, R.H., see Bartynski, R.A. 560 Gaylord, R.H., see Kevan, S.D. 540 Gazdy, B. 8 Ge, N.-H., see Lingle Jr., R.L. 564 Ge, N.-H., see McNeill, J.D. 563

Author index

Ge, N.-H., see Wong, C.M. 563 Geerligs, L.J., see Rogge, S. 695 Geerlings, J.J.C., see Los, J. 7, 8, 12 Gellman, A.J. 236 Gellman, A.J., see Lei, R.Z. 237 Gellman, A.J., see Paserba, K.R. 236 Gelten, R.J. 362 Geng, P., see Dietrich, H. 298, 300 Génin, F.Y., see Crouch, C.H. 827 Gentile, P., see Dujardin, R. 809 Gentili, M., see Kiskinova, M. 379 Gentry, W.R. 9 Geohegan, D.B. 826 Georgakis, C., see Vayenas, C.G. 349 George, S.M., see Koehler, B.G. 829 George, S.M., see Wise, M.L. 715, 830 George, T., see Vasquez, R.P. 847 Georgiou, S. 908 Gerber, C., see Binnig, G. 579 Gerber, Ch., see Binnig, G. 685, 701 Gerber, G., see Gerstner, V. 715 Gergen, B. 4, 15, 148, 334, 335, 455, 482, 489–492 Gergen, B., see Nienhaus, H. 148, 240, 489–491, 493 Gerhard, W. 891 Gerischer, H. 842, 845–847 Gerischer, H., see Allongue, P. 846, 849, 850 Gerischer, H., see Rappich, J. 850 Gerlach, A. 539 Gerlach, A., see Straube, P. 542, 560 Gerstner, V. 715 Gerth, G., see Schubert, L. 811 Gesell, T.F. 487 Geus, J.W., see de Jong, K.P. 807, 812 Geuzebroeak, F.H. 210 Ghicov, A., see Tsuchiya, H. 822 Giaquinta, D.M., see Cong, P.J. 313 Gibbs, J.W. 792 Gibson, K.D. 44, 49, 100 Gibson, K.D., see Isa, N. 44 Gielbert, A., see van Tol, M.F.H. 403 Giersig, M., see Fojtik, A. 845 Giesen, K. 553 Giesen, M., see Hannon, J.B. 776, 777 Giessel, T., see Böttcher, A. 465, 488, 489, 512 Gil, L., see Coullet, P. 411 Gillen, K.T., see Tenner, A.D. 38, 40, 41 Gilmore, F.R., see Bauer, E. 11

935

Gimzewski, J.K. 686, 720, 748 Gimzewski, J.K., see Joachim, C. 738, 739 Gimzewski, J.K., see Jung, T.A. 686, 689, 691 Gimzewski, J.K., see Langlais, V. 739 Giovanelli, L., see Papageorgiou, N. 733 Giovannini, M., see Bromann, K. 780, 781 Giovannini, M., see Brune, H. 779 Giovannini, M., see Hamilton, J.C. 781 Girard, C. 695, 714 Girard, C., see Martin, O.J.F. 714 Giusti-Suzor, A., see Charron, E. 701 Givargizov, E.I. 808–811 Gjostein, N.A., see Wynblatt, P. 775 Gladstone, D.J., see Li, Y.L. 482, 483, 493 Glaefeke, H., see Klar, F. 488 Gland, G., see Fisher, G. 443 Gland, J.L. 718 Gland, J.L., see Gorte, R.J. 371 Glans, P.-A., see Balasubramanian, T. 543, 560 Glas, F., see Harmand, J.C. 812 Glass, S. 491, 492, 509 Glass, S., see Nienhaus, H. 492 Glazov, L.G. 889 Gleeson, M.A., see Gou, F. 32 Glembocki, O.J. 850 Glen, M., see Sykes, M.F. 767 Glownia, J.H., see Prybyla, J.A. 634 Gnaser, H. 884 Godwin, P.D. 692 Gohndrone, J.M. 371 Gokhale, A.A., see Kandoi, S. 308, 309 Gokhale, A.A., see Schumacher, N. 314, 315 Gokhale, S. 735 Goldman, J.R., see Deliwala, S. 634 Goldmann, A., see Campillo, I. 539 Goldmann, A., see Echenique, P.M. 527, 531, 560, 563 Goldmann, A., see Gerlach, A. 539 Goldmann, A., see Matzdorf, R. 538 Goldmann, A., see Paniago, R. 540 Goldmann, A., see Straube, P. 542, 560 Goldmann, A., see Theilmann, F. 538, 539, 548 Goldmann, A., see Weinelt, M. 556, 557 Gole, J.L., see Oldenburg, R.C. 460 Golla, A., see Horn-von Hoegen, M. 834 Golovchenko, J.A., see Becker, R.S. 551, 685, 738 Golovchenko, J.A., see Bedrossian, P. 735 Gölzhäuser, A. 765

936

Gölzhäuser, A., see Kyuno, K. 782 Gomer, R. 323 Gomer, R., see Leung, C. 647 Gomer, R., see Menzel, D. 468, 623 Gomer, R.J., see Menzel, D. 699, 723 Gomez, S., see Belen, R.J. 827 Gomez-Rodriguez, J.M., see Brihuega, I. 694, 728 Gómez-Rodríguez, J.M., see Paz, O. 590 Gong, B., see Jo, S.K. 843 Gong, X.-Q. 367 Gonzalez, E.R., see Perez, J. 331 Goodman, D.W. 315 Goodman, D.W., see Rodriguez, J.A. 293 Goodman, F.O. 36, 44, 46, 58, 71, 113, 483 Goodstein, D.M., see DiRubio, C.A. 58 Goodwin, E.T. 528, 529 Gordon, M.S., see Mills, G. 290 Gorelik, S.S., see Bublik, V.T. 795 Goringe, C.M., see Mayne, A.J. 735 Gorochov, O., see Belaïdi, A. 847 Gorochov, O., see Safi, M. 847 Gorodetskii, V., see Ernst, N. 404 Gorodetskii, V.V. 397, 402–404 Gorostiza, P. 842 Gorte, R.J. 371 Gortel, Z.W. 120 Gortel, Z.W., see Hussla, I. 660 Gortel, Z.W., see Yakes, M. 256 Gortel, Z.W., see Zaluska-Kotur, M.A. 256 Gosalvez, D.B., see Li, Y.L. 482, 483, 493 Gosalvez-Blanco, D., see Tate, M.R. 482, 483 Gösele, U., see Lehmann, V. 846, 847, 849 Gösele, U., see Leonard, S.W. 822 Gösele, U., see Li, N. 810, 811 Gösele, U., see Müller, F. 822 Gösele, U., see Schmidt, V. 811 Gösele, U., see Schubert, L. 811 Gösele, U., see Tan, T.Y. 809–811 Gossmann, H.-J. 831 Gostein, M. 167, 183 Gotthold, M.P. 214 Gottschalk, N. 378, 385, 389 Gottschalk, N., see Bär, M. 385 Gottschalk, N., see Mertens, F. 389, 390 Gou, F. 32 Gourbilleau, F., see Botti, S. 845 Gourdon, A. 739 Gourdon, A., see Chiavaralloti, F. 748

Author index

Gourdon, A., see Grill, L. 750 Gourdon, A., see Gross, L. 740, 748, 751 Gourdon, A., see Kuntze, J. 739 Gourdon, A., see Langlais, V. 739 Gourdon, A., see Moresco, F. 686, 740, 741, 751 Gourdon, A., see Repp, J. 732 Gourdon, A., see Rosei, F. 739, 748 Gourdon, A., see Sadhukhan, S.K. 748 Gourdon, A., see Schunack, M. 739 Gourdon, A., see Soukiassian, L. 726, 736 Gowland, L., see Chubb, J.N. 130 Goyhenex, C., see Henry, C.R. 290 Goyhenex, C., see Rusponi, S. 779 Grabhorn, H., see Otto, A. 469 Grabow, L.C., see Schumacher, N. 314, 315 Graham, A.P., see Braun, J. 101 Graham, M.C., see Trail, J.R. 19, 148, 189, 493 Graham, M.D. 345, 385, 415 Gråkæk, L., see Clausen, B.S. 799 Grange, P., see Parvulescu, V.I. 319 Grant, A.W., see Johanek, V. 407, 408, 414 Gras-Martí, A., see Winterbon, K.B. 882 Grasselli, R.K., see Yaccato, K. 317 Grassman, T.J., see Poon, G.C. 459, 466, 475, 477, 478, 482, 488 Graves, D.B. 32 Graves, D.B., see Humbird, D. 843–845 Graves, J.S. 902 Grazioli, C., see Gambardella, P. 778, 781 Grazul, J., see Frank, M.M. 791 Greber, T. 4, 434, 435, 455, 456, 464–466, 471, 482, 487–489, 494, 496, 512, 515 Greber, T., see Böttcher, A. 464–466, 473–475, 477, 478, 488, 489, 512 Greber, T., see Brandt, M. 482, 488, 489, 512 Greber, T., see Grobecker, R. 482, 488, 512 Greber, T., see Hermann, K. 488 Greber, T., see Wider, J. 395 Greeley, J. 277, 291, 304 Greeley, J., see Abild-Pedersen, F. 295, 297 Greeley, J., see Andersson, M.P. 294, 316–318 Greeley, J., see Kandoi, S. 308, 309 Greeley, J., see Skúlason, E. 333 Green, M.L., see Frank, M.M. 791 Greene, J.E., see Bramblett, T.R. 835 Greene, J.E., see Lin, D.-S. 831, 835 Greene, J.E., see Lubben, D. 830, 835 Greene, J.E., see Suda, Y. 830, 835

Author index

Greenlief, C.M., see Gates, S.M. 830, 832–835, 843 Greenlief, C.M., see Klug, D.-A. 828 Greenlief, C.M., see Kulkarni, S.K. 831 Greenlief, C.M., see Liehr, M. 834 Gregoratti, L., see Günther, S. 379, 380 Grepstadt, J.K., see Slagsvold, B.J. 540, 541 Greuter, F., see Levinson, H.J. 560 Grey, F., see Stokbro, K. 701, 705, 706 Grgur, B.N., see Markovic, N.M. 333 Griffiths, K. 357 Grigoropoulos, C.P., see Ye, M. 908 Grill, L. 740, 750, 751 Grill, L., see Alemani, M. 736 Grimblot, J., see Luntz, A.C. 98 Grimley, T., see Brivio, G. 120, 125, 483, 484 Grimley, T.B. 284, 447, 450–452 Grimley, T.B., see Brivio, G.P. 3 Grimmelmann, E.K. 36, 46 Grimvall, G. 433, 533, 534, 545 Grinberg, I., see Mason, S.E. 285 Gritsch, T. 362 Gritsch, T., see Modl, A. 210 Gritsch, T., see Wintterlin, J. 685, 766 Grobeckeer, R., see Böttcher, A. 473–475 Grobecker, R. 482, 488, 512 Grobecker, R., see Böttcher, A. 464, 466, 477, 478, 488, 489, 512 Grobecker, R., see Greber, T. 464, 465, 494 Grobecker, R., see Hermann, K. 488 Grobis, M., see Lu, X. 610 Groeneveld, J.A., see McCormack, D.A. 179, 180 Groenveld, J.A., see Watts, E. 161, 167, 169, 183, 184, 189 Groß, A. 3, 9 Gross, A. 32, 145, 146, 148, 151, 159, 164, 165, 174, 176–178, 181–183, 189, 221, 239, 463, 498 Groß, A., see Bach, C. 19, 23 Gross, A., see Brenig, W. 166 Gross, A., see Dianat, A. 146 Gross, A., see Eichler, A. 182, 183, 311 Gross, A., see Kroes, G.J. 3, 32 Gross, A., see Roudgar, A. 290, 292 Gross, E.K.U., see Marques, M.A.L. 580 Gross, L. 740, 748, 751 Gross, L., see Chiavaralloti, F. 748 Gross, L., see Moresco, F. 740, 751

937

Grubbisic, A., see Burgert, R. 508 Grünning, U., see Lehmann, V. 822 Grunze, M., see Boszo, F. 325 Grunze, M., see Bozso, F. 299 Gruyters, M. 101, 259, 362, 374 Gruzdkov, Y.A. 633 Gruzdkov, Y.A., see Matsumoto, Y. 633 Gu, S.L., see Liu, J.L. 808 Guan, S.H., see Cong, P.J. 313 Guan, Y.F., see Pedraza, A.J. 825 Guantes, R. 254 Guantes, R., see Vega, J.L. 254 Guckenheimer, J. 351 Güdde, J., see Berthold, W. 554 Güdde, J., see Stépán, K. 638 Gudiksen, M.S. 809 Gudiksen, M.S., see Cui, Y. 808, 810 Gudiksen, M.S., see Lauhon, L.J. 813 Guenther, S., see Hinz, M. 398 Guezebroek, F.H. 162 Guggisberg, M., see Bennewitz, R. 732 Guisinger, N.P., see Hersam, M.C. 726 Gulari, E., see Ziff, R.M. 355, 357, 407 Gulari, K.E., see Fichthorn, K.A. 407 Gulding, S.J. 177 Gulding, S.J., see Hou, H. 178, 222, 334, 433, 512 Gumhalter, B. 15, 56, 64, 106, 108, 112, 113, 240 Gumhalter, B., see Braun, J. 101 Gumhalter, B., see Burke, K. 8, 64 Gumy, J.C., see Poon, G.C. 459, 466, 475, 477, 478, 482, 488 Gunnarsson, O. 8, 14, 189, 278, 281, 283, 433, 444, 450, 451, 456, 484 Gunnarsson, O., see Jones, R.O. 433 Gunnarsson, O., see Lundqvist, B.I. 3, 14, 274, 278, 279, 282, 284 Gunnarsson, O., see Schönhammer, K. 8, 14, 16–19, 98, 189, 446, 448, 461, 483 Günther, C. 780 Günther, S. 379, 380, 398, 417, 419, 767 Günther, S., see de Decker, Y. 259 Günther, S., see Esch, F. 417 Günther, S., see Marbach, H. 398–400 Günther, S., see Schaak, A. 374, 376, 390 Günther, S., see Schmidt, T. 377, 390 Günther, S., see Schütz, E. 417, 419 Güntherodt, H.-J., see Bennewitz, R. 732

938

Guo, H. 23, 623, 650, 658, 659 Guo, H., see Li, S. 663 Guo, X.-C. 357 Guo, X.-C., see Hopkinson, A. 373 Guo, X.-C., see Pasteur, A.T. 357 Gupta, J.A., see Heinrich, A.J. 686, 709, 742 Gupta, P., see Wise, M.L. 715, 830 Gurney, R. 450, 452 Gurney, T. 765 Gustafsson, K. 98 Gustafsson, T., see Bartynski, R.A. 543, 560 Gustafsson, T., see Frank, M.M. 791 Gutdeutsch, U., see Gokhale, S. 735 Guthrie, W.L. 641 Gutleben, H., see Cheng, C.C. 735 Gwathmey, A.T. 298 Haas, G. 415, 416 Haas, G., see Rotermund, H.H. 349, 381, 390, 409, 821 Haase, G., see Asscher, M. 513 Habenschaden, E. 764 Haber, F. 325, 466 Haberland, H., see Sesselman, W. 473 Haberland, H., see Woratschek, B. 459, 472, 473 Haddad, J., see Oster, L. 466 Haeni, J.H., see Schlom, D.G. 791 Hafner, J., see Eichler, A. 182, 183, 311, 497 Hafner, J., see Gajdos, M. 288 Hafner, J., see Mittendorfer, F. 497 Hagberg, A., see Meron, E. 389 Hage, F., see Giesen, K. 553 Hagelstein, M., see Ressler, T. 366 Hagemeyer, A., see Yaccato, K. 317 Hager, J., see Frenkel, F. 207 Hager, J., see Segner, J. 208, 209 Hager, J., see Vach, H. 220 Hagstrum, H.D. 460, 466, 467, 472, 473, 475, 499, 509 Hahn, E., see Röder, H. 775, 778, 779, 815 Hahn, J.R. 728 Haight, R. 554 Haken, H. 345 Hall, R.I., see Belić, D.S. 647 Haller, G.L., see Mantell, D.A. 672 Hallock, A.J., see Mathews, C.M. 513 Halsey, G.D. 792 Halstead, D. 166, 167

Author index

Hamann, D.R., see Tersoff, J. 590, 706 Hamers, R.J. 816, 831 Hamers, R.J., see Bronikowski, M.J. 835 Hamers, R.J., see Wang, Y. 831, 834, 835, 837 Hamilton, J.C. 781 Hamilton, J.C., see Hannon, J.B. 776, 777 Hamilton, P., see Wu, Y.M. 835 Hammer, B. 150, 154, 168, 189, 261, 277–281, 283, 286–290, 293, 297, 320, 328, 450, 452, 453, 498 Hammer, B., see Besenbacher, F. 296, 298 Hammer, B., see Diekhöner, L. 513 Hammer, B., see Gambardella, P. 302 Hammer, B., see Gross, A. 463, 498 Hammer, B., see Kinnersley, A.D. 168, 169, 174, 176 Hammer, B., see Logadottir, A. 299, 301, 326, 330 Hammer, B., see Mavrikakis, M. 291 Hammer, B., see Nilsson, A. 287 Hammer, B., see Nørskov, J.K. 302, 303 Hammer, B., see Ruban, A. 287, 293 Hampton, J., see Powers, J. 57, 59 Hamza, A.V., see Budde, F. 82, 631, 644 Han, N., see Dove, P.M. 820 Han, P., see Liu, J.L. 808 Han, S. 672 Hanbucken, M., see Venables, J.A. 816 Hand, M.R. 185 Hänggi, P. 8, 20 Hanggi, P. 237 Hanika, J., see Silveston, P.L. 411 Hanisco, T.F. 211–213, 216, 513 Hannon, J.B. 776, 777, 808, 810, 811 Hanrahan, C., see Lee, J. 473 Hansen, D.A., see Engstrom, J.R. 834–837 Hansen, E.D. 538 Hansen, E.W. 253 Hansen, E.W., see Neurock, M. 304 Hansen, F.Y., see Henriksen, N.E. 237 Hansen, L.B., see Hammer, B. 277, 320, 328 Hansen, L.B., see Nørskov, J.K. 302, 303 Hansen, M. 764 Hansen, P.L., see Helveg, S. 812 Hansma, P.K. 592 Hao, Y., see Schmidt, O.G. 840 Haochang, P., see Horn, T.C.M. 49 Harada, Y. 456, 457, 472, 474, 475 Harbich, W., see Vandoni, G. 769

Author index

Harikumar, K.R. 696 Harikumar, K.R., see Dobrin, S. 696 Harmand, J.C. 812 Harmand, J.C., see Dubrovskii, V.G. 810 Harris, 8, 9 Harris, A.L., see Chabal, Y.J. 828, 830, 847 Harris, A.L., see Hines, M.A. 850, 851 Harris, C.B., see Gaffney, K.J. 558 Harris, C.B., see Lingle Jr., R.L. 564 Harris, C.B., see McNeill, J.D. 563 Harris, C.B., see Wong, C.M. 563 Harris, J. 97, 99, 103, 150, 239 Harris, J., see Andersson, S. 100, 102, 104, 105, 113, 114, 120, 124, 157, 163, 462 Harris, J., see Hand, M.R. 185 Harris, J., see Nordlander, P. 99 Harris, J., see Persson, M. 13, 105, 106, 108, 111 Harris, S. 650, 666–668 Harris, S.M. 669 Harris, T.D., see Hines, M.A. 850, 851 Harrison, I., see Artsyukhovich, A.N. 672 Harrison, I., see DeWitt, K. 836 Harrison, I., see Kavulak, D.F. 832, 833 Harrison, I., see Ukraintsev, V.A. 633, 671 Harrison Jr., D.E., see Tenner, A.D. 38, 41 Harten, U. 102 Hartmann, N. 366, 368, 385, 415–417 Hartmann, N., see Christoph, J. 415, 417 Hartmann, N., see Schütz, E. 345, 415, 417, 418 Hartung, J., see Zhang, J. 828 Harutyunyan, A.R. 812 Hasan, M.-A., see Bramblett, T.R. 835 Hase, W., see Isa, N. 44 Hase, W.L., see Yan, T.Y. 41 Hasegawa, S., see Liu, C. 775 Hasegawa, Y. 545, 771 Hasegawa, Y., see Avouris, Ph. 546 Hashinokuchi, M., see Watanabe, Y. 44 Hass, K.C., see Bogicevic, A. 319–322 Hassan, H.H. 847 Hasse, G., see Föll, H. 823 Hassel, M. 130, 135 Hassel, M., see Andersson, T. 46, 101, 102, 104 Hasselbrink, E. 435, 455, 459, 464, 465, 468, 494, 495, 497, 500, 515, 623, 644–646, 650, 658, 661 Hasselbrink, E., see Ambaye, H. 74, 75

939

Hasselbrink, E., see Binetti, M. 73, 149, 462, 493–495, 499, 500, 508 Hasselbrink, E., see Bornscheuer, K.-H. 663 Hasselbrink, E., see Dippel, O. 828 Hasselbrink, E., see Ertl, G. 455, 489, 496 Hasselbrink, E., see Harris, S.M. 669 Hasselbrink, E., see Kolasinski, K.W. 829 Hasselbrink, E., see Komrowski, A.J. 73, 494, 495, 500 Hasselbrink, E., see Nessler, W. 660, 661 Hasselbrink, E., see Weik, F. 625, 626, 669 Hasselbrink, E., see Weiße, O. 72, 73, 175, 183 Hasselbrink, E., see Wolf, M. 625, 626, 647 Hasselbrink, E., see Wright, S. 633 Hasselbrink, E., see Zhu, X.-Y. 647 Hatch, S.R. 672 Hatje, U., see Ressler, T. 366 Haugh, H. 592 Hayami, Y., see Ogawa, H. 849 Hayashi, H., see Nakao, K. 240 Hayden, B.E. 31 Hayden, B.E., see Davies, J.C. 293 Hayes, D.J., see Kalkan, A.K. 828 Hayes, W.W. 44 He, R., see Hochbaum, A.I. 808 Head-Gordon, M. 19, 46, 122, 125–127, 448, 449, 463, 484, 485 Heberle, A.P., see Ogawa, S. 551 Hecht, S., see Alemani, M. 736 Hedegård, P., see Brandbyge, M. 19, 637 Hedegård, P., see Newns, D.M. 19 Hedin, A., see Johnson, R.E. 894 Hedin, L. 530, 537 Hedin, L., see Almbladh, C.-O. 537 Hedin, L., see von Barth, U. 433 Hedouin, M.F.G., see Sloan, P.A. 720 Heer, W.A.d., see Billas, I.M.L. 781 Hefty, R.C., see Holt, J.R. 843 Heilmann, P. 357 Heimann, P. 540 Heinrich, A.J. 686, 709, 742 Heinrich, A.J., see Hirjibehedin, C.F. 686 Heinrich, R., see Staudt, C. 890 Heinrichs, S. 783 Heinz, K., see Heilmann, P. 357 Heinz, T.F., see Bartels, L. 638, 715 Heinz, T.F., see Brandbyge, M. 19, 637 Heinz, T.F., see Budde, F. 634, 635 Heinz, T.F., see Dürr, M. 715, 828, 830

940

Heinz, T.F., see Höfer, U. 715 Heinz, T.F., see Misewich, J.A. 634–636 Heinz, T.F., see Newns, D.M. 19 Heinz, T.F., see Prybyla, J.A. 634 Heinz, T.F., see Reider, G.A. 829 Heinze, S. 374 Heinzmann, U., see Brandt, M. 482, 488, 489, 512 Helbert, J.N. 851 Hellberg, L. 464–466, 475, 477, 479–483, 488, 494, 500 Hellberg, L., see Strömquist, J. 475, 480–483, 494, 499, 500, 509 Heller, E.J. 6, 20, 22, 548, 686 Heller, E.J., see Fiete, G.A. 686 Hellman, A. 245, 257, 261, 274, 296, 304, 309, 325, 327, 328, 334, 433, 434, 463, 475, 476, 480, 496, 499–508, 510 Hellman, A., see Honkala, K. 263, 274, 277, 296, 301, 304, 308, 325–330 Hellner, L., see Comtet, G. 688 Hellner, L., see Lastapis, M. 602, 688, 744 Hellner, L., see Mayne, A.J. 696, 746 Hellner, L., see Soukiassian, L. 726, 736 Hellsing, B. 19, 148, 189, 257, 464, 484–486, 533, 535 Hellsing, B., see Chakarov, D. 482 Hellsing, B., see Chulkov, E.V. 529, 548, 564, 565 Hellsing, B., see Eiguren, A. 534, 535, 541, 542, 560–563 Hellsing, B., see Persson, M. 483–486 Helveg, S. 330, 331, 812 Helveg, S., see Bollinger, M.V. 330 Hemmen, R. 473 Hempel, H., see Rose, H. 362 Henderson, J.I., see Xue, Y. 735 Henderson, M.A. 647 Hendriksen, B.L.M. 366, 367, 369, 370, 380 Hendriksen, B.L.M., see Frenken, J.W.M. 717 Henglein, A., see Fojtik, A. 845 Hengsberger, M. 542–544 Hengsberger, M., see Purdie, D. 538, 539 Henkelman, G. 306 Henkelman, G., see Olsen, R.A. 306 Henley, S.J., see Claeyssens, F. 825 Hennies, F., see Föhlisch, A. 566 Hennig, D. 293 Henriksen, N.E. 237

Author index

Henriksson, K.O.E., see Nordlund, K. 896 Henry, B.E., see Schüth, F. 259, 346 Henry, C.R. 263, 290 Henry, M.R., see Kalkan, A.K. 828 Henry de Villeneuve, C., see Allongue, P. 846 Henzl, J. 736 Henzler, M., see Falta, J. 363, 378, 396 Hepburn, J.W. 433 Her, T.-H. 827 Her, T.H., see Deliwala, S. 634 Herein, D., see Werner, H. 370 Hergert, W., see Stepanyuk, V.S. 772, 774, 775 Hergert, W., see Tsivlin, D.V. 773 Hermann, K. 488 Hermann, K., see Greber, T. 464, 465, 494 Hermans, L.J.F., see Krylov, S.Y. 237 Hermse, C.G.M. 254, 262 Hermse, C.G.M., see Jansen, A.P.J. 262 Hermse, C.G.M., see van Bavel, A.P. 254 Hernández-Pozos, J.L., see Riedel, D. 827 Herrero, C.P. 715 Hersam, M.C. 726 Hersam, M.C., see Abeln, G.C. 726 Hersch, J.S., see Fiete, G.A. 686 Herschbach, D.R., see Struve, W.S. 464, 476, 482 Hertel, T. 553, 660, 663 Hertel, T., see Grobecker, R. 482, 488 Hertel, T., see Nessler, W. 660, 661 Hertel, T., see Wolf, M. 563 Hertz, H. 487 Herzenberg, A., see Birtwistle, D.T. 19 Heskett, D. 300 Hess, C., see Bonn, M. 483, 674 Hess, G., see Jo, S.K. 843 Heun, S., see Schmidt, T. 380 Heurich, J. 738 Hewson, A.C. 452 Heyde, M., see Sterrer, M. 730 Heyde, M., see Yulikov, M. 275 Hibino, H., see Takagi, D. 813 Higashi, G.S. 847 Higashi, G.S., see Becker, R.S. 685, 687, 726 Higashi, G.S., see Burrows, V.A. 847 Higashi, G.S., see Trucks, G.W. 846 Highland, M. 486, 487 Hilbers, P.A.J., see van Bavel, A.P. 254 Hildebrand, H., see Beranek, R. 822 Hildebrand, M. 398

Author index

Hildebrand, M., see Sachs, C. 345, 380, 397, 401, 403 Hilf, M.F., see Brenig, W. 829 Hilico, J.C., see Bronnikov, D.K. 204 Hillenkamp, F., see Georgiou, S. 908 Himes, D., see Celli, V. 56 Himes, D., see Manson, J.R. 62, 70 Himpsel, F.J. 817 Himpsel, F.J., see Giesen, K. 553 Hines, M.A. 214, 824, 849–851 Hines, M.A., see Flidr, J. 851 Hines, M.A., see Garcia, S.P. 821, 849 Hines, M.A., see Wind, R.A. 824, 850, 851 Hinnemann, B. 330, 331, 333 Hinz, M. 398 Hinz, M., see de Decker, Y. 259, 398 Hinz, M., see Liu, H. 400, 401 Hinz, M., see Makeev, A. 389–391 Hinz, M., see Marbach, H. 398 Hirano, M., see Wang, C.X. 810 Hiraoka, N., see Kitazoe, Y. 894 Hiraoka, T., see Tomii, T. 76, 79, 81 Hiraoko, T., see Yagu, S. 76, 78, 79, 81 Hiraoos, T., see Yagu, S. 76 Hiratsuka, A., see Imamura, K. 254 Hirjibehedin, C.F. 686 Hirlimann, C., see Shank, C.V. 824 Hirose, K. 696 Hirose, M., see Bjorkman, C.H. 847 Hirose, M., see Tanaka, T. 830 Hirschorn, E.S., see Lin, D.-S. 831, 835 Hirshorn, E.S. 728 Hirstein, A., see Jeandupeux, O. 549, 771 Hirstein, A., see Knorr, N. 609, 771–774 Hirva, P. 830 Hirvonen, J.K., see Nastasi, M. 892, 893 Hiskes, J.R., see Wimmer, E. 449 Hitzke, A. 767 Hitzke, A., see Günther, S. 767 Hjelmberg, H. 443, 450 Hjelmberg, H., see Gunnarsson, O. 278, 283, 444, 450, 451, 484 Hjelmberg, H., see Lundqvist, B.I. 3, 14, 274, 278, 279, 282, 284, 443, 450, 463 Hjelt, T. 256 Hla, S.-W. 686, 692 Hla, S.-W., see Iancu, V. 736 Hla, W. 4, 22 Hliwa, M. 718

941

Hliwa, M., see Ami, S. 738 Ho, K.M., see Schochlin, J. 478 Ho, M.Y., see Frank, M.M. 791 Ho, W. 468, 470, 601, 671, 686, 687, 707, 709 Ho, W., see Asscher, M. 644 Ho, W., see Busch, D.G. 635, 636, 638 Ho, W., see Dai, H.-L. 19 Ho, W., see Gaudioso, J. 735 Ho, W., see Hahn, J.R. 728 Ho, W., see Kao, F.-J. 634 Ho, W., see Lauhon, L.J. 594, 597, 598, 600, 610, 728 Ho, W., see Lee, H.J. 686, 692, 728 Ho, W., see Lorente, N. 597, 598, 710 Ho, W., see Mieher, W.D. 671, 718 Ho, W., see Mikaelian, G. 734 Ho, W., see Nazin, G.V. 687 Ho, W., see Persson, M. 135 Ho, W., see Qiu, X.H. 687, 733 Ho, W., see Stipe, B.C. 321, 458, 462, 463, 592, 594, 687, 709, 710, 719, 722 Ho, W., see Wu, S.W. 714, 717, 746, 747 Ho, W., see Ying, Z. 623 Ho, W., see Zimmermann, F.M. 623, 644, 655, 656 Hobara, R., see Liu, C. 775 Hobbs, C. 692 Hobbs, C., see Keeling, D.L. 692 Hochbaum, A.I. 808 Hodgson, A. 31, 167–169, 172, 177 Hodgson, A., see Murphy, M.J. 174, 183, 184 Höfer, H., see Wintterlin, J. 685, 766 Höfer, U. 555, 564, 715, 828 Höfer, U., see Berthold, W. 551, 553, 554, 563, 564 Höfer, U., see Bratu, P. 829 Höfer, U., see Dürr, M. 183, 715, 828–830 Höfer, U., see Echenique, P.M. 527, 531, 560, 563 Höfer, U., see Marinica, D.C. 563 Höfer, U., see Misewich, J.A. 634, 635 Höfer, U., see Raschke, M.B. 828, 842 Höfer, U., see Reider, G.A. 829 Höfer, U., see Reuß, Ch. 556, 557 Höfer, U., see Shumay, I.L. 551, 555, 563, 564 Höfer, U., see Stépán, K. 638 Höfer, U., see Weinelt, M. 556, 557 Hofer, W.A. 589, 765, 766 Hofer, W.A., see Dobrin, S. 696

942

Hofer, W.A., see Harikumar, K.R. 696 Hofer, W.A., see Piva, P.G. 696, 730 Hofer, W.O., see Littmark, U. 889 Hofer, W.O., see Urbassek, H.M. 891 Hoffman, R. 3 Hoffmann, F.M., see Dwyer, D.J. 346 Hoffmann, H.D., see Jandeleit, J. 826 Hoffmann, P., see Lewerenz, H.J. 847 Hoffmann, P., see Wehner, S. 413 Hoffmann, R. 274 Hofmann, F. 64 Hofmann, P., see Bare, S.R. 357 Hofmann, P., see Comelli, G. 374 Hogg, C., see Yu, C.-F. 102 Hohage, M., see Bott, M. 780, 782 Hohage, M., see Lehner, B. 250 Hohage, M., see Michely, T. 782, 783 Hohenberg, P. 277, 281, 282, 433 Hohenberg, P.C., see Cross, M.C. 382 Hohlfeld, J., see Bonn, M. 900 Hoinkes, H. 98 Holbert, P.A., see Gates, S.M. 832–834 Holloway, S., see Bird, D.M. 148, 189 Holloway, S., see Corriol, C. 175, 183 Holloway, S., see Darling, G..R. 498, 499 Holloway, S., see Darling, G.R. 145, 151–153, 157, 169, 172, 174–176, 179, 183, 187, 188, 213 Holloway, S., see Gadzuk, J.W. 11, 12 Holloway, S., see Gates, G.A. 189 Holloway, S., see Halstead, D. 166, 167 Holloway, S., see Harris, S. 650, 666–668 Holloway, S., see Harris, S.M. 669 Holloway, S., see Hasselbrink, E. 661 Holloway, S., see Kay, M. 151, 159, 183 Holloway, S., see Kinnersley, A.D. 168, 169, 174, 176 Holloway, S., see Lahaye, R.J.W.E. 32, 34, 36–39, 44, 47–49, 162–164 Holloway, S., see Mizielinski, M.S. 148, 189, 448, 493, 611 Holloway, S., see Richardson, N. 433, 449, 451 Holloway, S., see Saalfrank, P. 663 Holloway, S., see Savio, L. 158, 159 Holloway, S., see Schweizer, E.K. 38 Holloway, S., see Smith, R.J. 122 Holloway, S., see Trail, J.R. 8, 15, 19, 148, 189, 493

Author index

Holloway, S., see Wang, Z.S. 146, 159–161, 170, 171, 186, 187 Holloway, S., see Weiße, O. 72, 73, 175, 183 Holmberg, C. 468, 469 Holmberg, C., see Nordlander, P. 99 Holmes, P., see Guckenheimer, J. 351 Holmström, S. 453, 454 Holt, J.R. 843 Homma, Y., see Takagi, D. 813 Hone, D., see Bloss, H. 447 Hone, D.W., see Blandin, A. 8, 19 Hone Jr., D., see Blandin, A. 447, 448, 459 Hong, S., see Xue, Y. 735 Hong, Y.K., see Kim, W. 735 Honkala, K. 149, 263, 274, 277, 296, 301, 304, 308, 325–330, 498 Honkala, K., see Bligaard, T. 307, 328, 330 Honkala, K., see Díaz, C. 471, 513, 514 Honkala, K., see Hellman, A. 257, 261, 274, 296, 304, 309, 325, 327, 328 Honkala, K., see Vang, R.T. 298, 300, 781 Hoogers, G. 293 Hopkinson, A. 371, 373 Hopkinson, A., see Guo, X.-C. 357 Hopstaken, M.J.P., see van Bavel, A.P. 254 Horch, S., see Hinnemann, B. 330, 331, 333 Horch, S., see Jaramillo, T.F. 277, 330–333 Horino, H., see Imamura, K. 254 Horiuti, J. 305 Horn, F., see Bailey, J.E. 411 Horn, K., see Cranney, M. 745 Horn, T.C.M. 49 Horn, T.C.M., see Kleyn, A.W. 38, 41, 207 Horn, T.C.M., see Tenner, A.D. 38, 40, 41 Horn-von Hoegen, M. 798, 803, 834 Horn-von Hoegen, M., see Sokolowski-Tinten, K. 824 Hornekær, L., see Diekhöner, L. 511–514 Horsthemke, W. 404 Hosono, H., see Wang, C.X. 810 Hossain, M.Z. 696 Hotzel, A. 563 Hotzel, A., see Bartels, L. 4, 558, 687, 699 Hotzel, A., see Gahl, C. 558, 559 Hotzel, A., see Hertel, T. 553 Hotzel, A., see Knösel, E. 556, 563, 564 Hotzel, A., see Velic, D. 558 Hotzel, A., see Wolf, M. 555 Hou, H. 178, 222, 334, 433, 512

Author index

Hou, H., see Gulding, S.J. 177 Hou, H., see Huang, Y. 148, 189, 190, 221 Hou, J.G., see Zhao, A. 728 Houlton, M.R., see Canham, L.T. 847 Houmøller, A., see Nørskov, J.K. 443, 450, 453, 463 Houston, P.L., see Asscher, M. 644 Howe, P. 839, 841 Howe, P., see Le Ru, E.C. 839, 842 Howe, P.-T. 633, 656 Hoyer, R., see Günther, S. 398, 417, 419 Hoyer, R., see Kibler, L.A. 292, 293 Hoyle, R.B. 259, 361 Hoyle, R.B., see Irurzun, I.M. 362 Hsu, B.B., see Park, J.Y. 492 Hu, B.Y.-K., see Stokbro, K. 701, 705, 706 Hu, J.T. 809 Hu, P., see Gong, X.-Q. 367 Hu, P., see Liu, Z.-P. 302 Hu, P., see Michaelides, A. 304 Hu, Z., see Dürr, M. 715, 828, 830 Huang, C. 735 Huang, C., see Widdra, W. 735 Huang, C.-P. 551 Huang, L.J. 715 Huang, W.X. 320 Huang, Y. 148, 189, 190, 221, 222 Huang, Y., see Duan, X.F. 808 Huang, Y., see Hou, H. 222 Huang, Y., see Wodtke, A.M. 189 Huang, Y.C., see Flidr, J. 851 Huang, Y.H. 334, 433, 463, 511, 512 Huang, Y.H., see Hou, H. 334, 433, 512 Hubacek, J.S., see Lyding, J.W. 701, 703, 705, 706, 726 Hubler, G.K., see Chrisey, D.B. 908 Huc, V., see Bernard, R. 748 Hudda, F.G., see Ehrlich, G. 817 Hudson, A.D. 187 Hudson, J.L., see Graham, M.D. 385 Hudson, J.L., see Punckt, C. 821 Huels, M.A. 700 Huett, T., see Zhu, X.-Y. 659 Hüfner, S., see Eiguren, A. 534, 535, 541, 542, 562 Hüfner, S., see Nicolay, G. 540, 541 Hüfner, S., see Reinert, F. 540, 541, 543, 548, 560, 561 Hugenschmidt, M.B., see Hitzke, A. 767

943

Hugo, P. 345 Huisken, F., see Ehbrecht, M. 845 Hulbert, S.L. 553 Hulbert, S.L., see Balasubramanian, T. 542, 543 Hulbert, S.L., see Valla, T. 544 Hul’ko, O., see Patitsas, S.N. 726 Hull, R. 798 Hulpke, E. 56, 64 Hult, E. 99, 500 Humbird, D. 843–845 Humbird, D., see Graves, D.B. 32 Humphry, M.J., see Keeling, D.L. 691, 692, 751 Hupalo, M., see Yakes, M. 256 Hurkmans, A. 48 Hurkmans, A., see Trilling, L. 48 Hurst, J.E., see Janda, K.C. 82, 199 Husinsky, W., see Betz, G. 889, 891 Hussain, Z., see Junren, S. 545 Hussla, I. 660 Hutchinson, F., see Gurney, T. 765 Hwang, C., see Kim, W. 735 Hwang, R.Q., see Günther, C. 780 Hwang, S.-T., see Abeln, G.C. 726 Hyldgaard, P. 772 Hyldgaard, P., see Bogicevic, A. 320, 775 Hyldgaard, P., see Repp, J. 772, 773 Hyldgaard, P., see Thonhauser, T. 500 Hynes, J.T., see Carter, E.A. 306 Iancu, V. 736 Ibach, H. 795, 797, 799 Ibach, H., see Hannon, J.B. 776, 777 Ichihara, S., see Katano, S. 735 Iftimia, I. 44 Igarashi, H. 293 Igarashi, H., see Toda, T. 291, 293 Igel, T., see Pfandzelter, R. 41 Iguchi, K., see Makino, O. 842, 847 Ihm, G., see Vidali, G. 99 Ihm, J., see Choi, B.-Y. 736 Ikai, M., see Janssen, N.M.H. 374 Ikeda, T., see Ogata, Y.H. 849 Ikeuchi, T., see Tomii, T. 76, 79, 81 Imamura, K. 254 Imanishi, A., see Matsui, F. 735 Imbeck, I., see Böttcher, A. 466, 477, 478, 488, 489, 512 Imbeck, R., see Böttcher, A. 148, 466, 477, 478 Imbeck, R., see Díaz, C. 151, 155, 186

944

Imbihl, R. 259, 345, 346, 348, 359–364, 370, 371, 373, 379, 381, 396, 404, 405, 411, 415, 821, 834 Imbihl, R., see Bassett, M.R. 368 Imbihl, R., see Cox, M.P. 357, 359, 360 Imbihl, R., see Dath, J.P. 371, 412 Imbihl, R., see de Decker, Y. 259 Imbihl, R., see Evans, J.W. 385 Imbihl, R., see Falta, J. 363, 378, 396 Imbihl, R., see Fink, T. 371–373 Imbihl, R., see Hartmann, N. 366, 368, 415–417 Imbihl, R., see Irurzun, I.M. 411 Imbihl, R., see Khrustova, N. 410 Imbihl, R., see Ladas, S. 363, 364, 368, 369, 395, 396 Imbihl, R., see Lombardo, S.J. 374 Imbihl, R., see Makeev, A. 383, 389–391 Imbihl, R., see Meissen, F. 396 Imbihl, R., see Mertens, F. 374, 378, 381–383, 385, 387, 389, 390 Imbihl, R., see Monine, M.I. 363, 365 Imbihl, R., see Pavlenko, N. 407 Imbihl, R., see Pineda, M. 408, 409 Imbihl, R., see Pismen, L.M. 382, 385 Imbihl, R., see Rose, K.C. 396, 397 Imbihl, R., see Sander, M. 359, 361, 363, 365, 396 Imbihl, R., see Schaak, A. 368, 385, 390, 391, 393, 395 Imbihl, R., see Schütz, E. 412 Imbihl, R., see Shvartsman, S. 262 Imbihl, R., see Shvartsman, S.Y. 415, 417, 418 Imbihl, R., see Suchorski, Y. 404, 405, 407 Imbihl, R., see Swiech, W. 383 Imbihl, R., see Uecker, H. 374 Imbihl, R., see Veser, G. 371, 374, 375, 385, 388, 410 Imbihl, R., see Wei, H. 384, 396 Immink, G., see Verheijen, M.A. 810 Imry, Y., see Büttiker, M. 584 Inanaga, S., see Rahman, F. 843 Inglesfield, J., see Hedin, L. 537 Inogamov, N. 904 Inogamov, N.A., see Anisimov, S.I. 904 Inogamov, N.A., see Zhakhovskii, V.V. 903 Ipsen, M., see Hildebrand, M. 398 Irie, M., see Kobatake, S. 736 Irie, M., see Matsuda, K. 736

Author index

Irurzun, I.M. 362, 411 Irurzun, I.M., see Mola, E.E. 362 Isa, N. 44 Isa, N., see Gibson, K.D. 44, 49 Isawa, K., see Matsui, F. 735 Ishida, H. 449, 450 Ishikawa, K., see Ogawa, H. 849 Ishikawa, T., see Kobatake, S. 736 Ishioka, K., see Gahl, C. 558, 559 Ishioka, K., see Hotzel, A. 563 Ishitani, A., see Takahagi, T. 847 Isobe, C. 828 Ito, S., see Nakao, K. 240, 246 Ito, T., see Watanabe, S. 847 Ivanov, D.S. 899 Ivanov, D.S., see Leveugle, E. 905 Iwasaki, M., see Ogata, Y. 847 Jackiw, J.J., see Pascual, J.I. 592 Jacklevic, R.C. 593 Jackman, T.E. 357 Jackman, T.E., see Griffiths, K. 357 Jackson, B. 6, 8, 9, 13, 110 Jackson, B., see Koleske, D.D. 843 Jackson, B., see Shalashilin, D.V. 239 Jackson, D.P., see Jackman, T.E. 357 Jackson, J.M. 55 Jackson, K., see Perdew, J. 433 Jackson, K.A., see Blakely, J.M. 808, 810 Jacob, I., see Egelhoff, W.F. 767 Jacobi, K., see Dietrich, H. 298, 300 Jacobi, K., see Grobecker, R. 482, 488 Jacobi, K., see Gruyters, M. 101 Jacobi, K., see Sano, M. 672 Jacobi, K., see Völkening, S. 380, 401 Jacobs, D.C. 3, 6, 208, 215 Jacobs, D.C., see Qian, J. 11, 12 Jacobs, H., see Nolte, S. 906, 907 Jacobs, P.W. 414 Jacobsen, C.J.H. 294, 312 Jacobsen, C.J.H., see Bligaard, T. 307, 328, 330 Jacobsen, C.J.H., see Dahl, S. 311, 327 Jacobsen, C.J.H., see Logadottir, A. 299, 301, 326, 330 Jacobsen, C.J.H., see Nørskov, J.K. 302, 303 Jacobsen, J., see Brune, H. 767, 769, 775, 778, 782 Jacobsen, K., see Brune, H. 775 Jacobsen, K.W. 278, 282

Author index

Jacobsen, K.W., see Bollinger, M.V. 330 Jacobsen, K.W., see Brandbyge, M. 585 Jacobsen, K.W., see Christensen, A. 764 Jacobsen, K.W., see Clausen, B.S. 799 Jacobsen, K.W., see Hammer, B. 154, 168, 450, 452, 453, 498 Jacobsen, K.W., see Jónsson, H. 304, 306, 320 Jacobsen, K.W., see Wingreen, N.S. 9, 21 Jacobson, N., see Rydberg, H. 500 Jacquet, D., see Bouneau, S. 895, 897 Jaeger, N., see Slinko, M.M. 346, 348 Jaeger, N.I. 330 Jaeger, N.I., see Slinko, M.M. 259 Jaegermann, W. 330 Jaffe, R.L., see Drauglis, E. 273, 277 Jaffee, R., see Drauglis, E. 455 Jahn, H.A. 433 Jain, S.C. 796 Jaklevic, R.C. 529, 709 Jaklevic, R.C., see Davis, L.C. 545 Jakob, P. 846 Jakubith, S. 381, 385, 387 Jakubith, S., see Hasselbrink, E. 644, 646 Jakubith, S., see Rotermund, H.H. 381, 390, 392 Jakubowicz, J., see Jungblut, H. 847 Jakubowicz, J., see Lewerenz, H.J. 847 James, E.W., see Suchorski, Y. 404–406 Jamneala, T., see Madhavan, V. 608, 609 Janda, K.C. 82, 199 Janda, K.C., see Sinniah, K. 715, 829 Jandeleit, J. 826 Janev, R.K., see Nedeljkovic, N.N. 473, 474 Janowitz, C., see Gerlach, A. 539 Jänsch, H.J., see Zubkov, T. 302 Jansen, A.P.J. 250–252, 262 Jansen, A.P.J., see Gelten, R.J. 362 Jansen, A.P.J., see Hermse, C.G.M. 254, 262 Jansen, A.P.J., see van Bavel, A.P. 254 Janssen, N.M.H. 370, 374, 385, 390 Janssen, N.M.H., see Cobden, P.D. 374 Janssen, N.M.H., see Makeev, A.G. 374 Jansson, U. 830 Jansson, U., see Uram, K.J. 830, 835 Jaramillo, T.F. 277, 330–333 Jasinski, J.M. 828 Jastrabik, L., see Tarasenko, A.A. 256 Jauho, A.-P., see Frederiksen, T. 592, 595, 615 Jauho, A.P., see Haugh, H. 592

945

Jayanthi, C.S., see Armand, G. 66 Jeandupeux, O. 549, 771 Jeandupeux, O., see Bürgi, L. 546, 547, 549 Jebasinski, R., see Bahr, D. 797 Jena, P. 778 Jena, P., see Burgert, R. 508 Jennings, P.J. 501 Jenniskens, H.G. 628, 629 Jennison, D.R., see Bogicevic, A. 320, 775 Jennison, D.R., see Burns, A.R. 644, 660 Jennison, D.R., see Orlando, T.M. 718 Jensen, E., see Balasubramanian, T. 542, 543 Jensen, E., see Bartynski, R.A. 543, 560 Jensen, E., see Diekhöner, L. 434, 463, 511–514 Jensen, E., see LaShell, S. 543, 548 Jensen, E., see Mortensen, H. 82, 513, 514 Jensen, E.T., see Dixon-Warren, St.J. 630, 631, 720 Jensen, H. 548–550, 562 Jensen, H., see Crampin, S. 546–548, 551, 562 Jensen, J.A. 462, 493 Jensen, J.H. 113 Jensen, J.H., see Burke, K. 58 Jensen, L.E. 812 Jentzen, W., see Shelnutt, J.A. 733 Jeong, H.D. 735 Jeppesen, S., see Jensen, L.E. 812 Jerdev, D.I. 323 Jeschke, H.O. 902 Jesse, S., see Pedraza, A.J. 825 Jesson, D.E., see Munt, T.P. 797 Jesson, D.E., see Shchukin, V.A. 804, 806 Ji, J.-Y. 828 Jia, S.-L., see Shelnutt, J.A. 733 Jiang, G. 715 Jiang, J.C., see Schlom, D.G. 791 Jiang, P., see Kuntze, J. 739 Jiang, P., see Rosei, F. 739, 748 Jiang, P., see Schunack, M. 739 Jimenez Rodriguez, J.J., see Vicanek, M. 879, 889 Jimenez-Bueno, G. 750 Jimenez-Bueno, G., see Grill, L. 740, 751 Jin, S., see McAlpine, M.C. 808 Jing, Z. 715, 832 Jo, S.K. 647, 843 Jo, S.K., see Bramblett, T.R. 835 Joachim, C. 21, 738, 739

946

Joachim, C., see Ami, S. 738, 739 Joachim, C., see Chiavaralloti, F. 748 Joachim, C., see Duchemin, I. 738 Joachim, C., see Dujardin, G. 686, 693, 728 Joachim, C., see Fiurášek, J. 738 Joachim, C., see Gimzewski, J.K. 686, 720 Joachim, C., see Girard, C. 695 Joachim, C., see Grill, L. 740, 750, 751 Joachim, C., see Gross, L. 740, 748, 751 Joachim, C., see Hliwa, M. 718 Joachim, C., see Jung, T.A. 686, 689, 691 Joachim, C., see Kuntze, J. 739 Joachim, C., see Langlais, V. 739 Joachim, C., see Moresco, F. 686, 740, 741, 751 Joachim, C., see Repp, J. 732 Joachim, C., see Rosei, F. 739, 748 Joachim, C., see Saifullah, M.S.M. 739 Joachim, C., see Schunack, M. 739 Joachim, C., see Stadler, R. 739 Joannopoulos, J.D., see Payne, M.C. 797 Johanek, V. 407, 408, 414 Johannessen, T., see Andersson, M.P. 294, 316–318 Johannessen, T., see Sehested, J. 317, 318 Jóhannesson, G.H. 306 Johannsen, I., see Gimzewski, J.K. 720 Johansson, D.K., see Nørskov, J.K. 443, 450, 453, 463 Johansson, J. 810–812 Johansson, L.I., see Balasubramanian, T. 543, 560 Johansson, P., see Limot, L. 548, 560 Johansson, S., see Fridell, E. 319 John, G.C. 823 John, S., see Leonard, S.W. 822 John, S., see Pacheco, K.A. 836 Johnson, A.D., see Beckerle, J.D. 98 Johnson, M.J., see Korolik, M. 220 Johnson, P.D. 447 Johnson, P.D., see Hulbert, S.L. 553 Johnson, P.D., see Valla, T. 544 Johnson, R.E. 893, 894, 907 Johnson, R.E., see Anders, C. 896, 897 Johnston, T.W., see Vidal, F. 904, 905 Jones, M.E. 832, 833 Jones, M.E., see Xia, L.Q. 833, 835, 836 Jones, R.O. 433 Jones, R.O., see Jennings, P.J. 501 Jones, R.V., see Dobrin, S. 696

Author index

Jones, T.S., see Howe, P. 839, 841 Jones, T.S., see Le Ru, E.C. 839, 842 Jones, T.S., see Mayne, A.J. 735 Jones, T.S., see Williams, R.S. 842 Jongma, R., see Silva, M. 511, 512 Jonkman, H.T., see Lof, R.W. 747 Jónsson, H. 304, 306, 320 Jónsson, H., see Hellman, A. 304, 325, 327, 328 Jónsson, H., see Henkelman, G. 306 Jónsson, H., see Jóhannesson, G.H. 306 Jónsson, H., see Olsen, R.A. 306 Jónsson, H., see Skúlason, E. 333 Jordan, K.D., see Nachtigall, P. 829 Jordan, R.E., see Lingle Jr., R.L. 564 Jordan, R.E., see McNeill, J.D. 563 Jørgensen, K.P., see Hinnemann, B. 330, 331, 333 Jørgensen, K.P., see Jaramillo, T.F. 277, 330–333 Jorné, J., see Kang, Y. 846, 847 Joyce, B.A. 807, 837, 838 Joyce, B.A., see Foxon, C.T. 818, 837, 838 Joyce, B.A., see Mokler, S.M. 831 Joyce, B.A., see Neave, J.H. 817, 818 Joyce, B.A., see Turner, A.R. 828 Joyce, B.A., see Zhang, J. 828 Jung, T., see Himpsel, F.J. 817 Jung, T.A. 686, 689, 691 Jungblut, H. 847 Jungblut, H., see Lewerenz, H.J. 847 Junkes, H., see Flätgen, G. 821 Junren, S. 545 Just, G., see Haber, F. 466 Juurlink, L.B.F. 181 Kaburagi, M. 321 Kadanoff, L.P. 447 Kadodwala, M., see Jenniskens, H.G. 628, 629 Kagan, C.R., see Murray, C.B. 747 Kahn, S.U.M., see Bockris, J.O.M. 330, 331 Kahng, S.-J., see Choi, B.-Y. 736 Kaindl, G., see Bauer, A. 548 Kaindl, G., see Wegner, D. 548 Kaiser, A., see Rethfeld, B. 899 Kaiser, W., see Laubereau, A. 711 Kalamarides, A., see Misewich, J.A. 634, 635 Kaletta, D., see Imbihl, R. 363, 396 Kalinin, D.V., see Bronnikov, D.K. 204 Kalkan, A.K. 828

Author index

Kam, A.F., see Foley, E.T. 701, 705, 706 Kammler, M., see Sokolowski-Tinten, K. 824 Kamna, M.M., see Stranick, S.J. 771 Kampen, T.U., see Cranney, M. 745 Kanamori, J., see Kaburagi, M. 321 Kanaya, H. 849 Kandoi, S. 308, 309 Kandoi, S., see Schumacher, N. 314, 315 Kang, H.C. 233, 250 Kang, H.C., see Shi, J. 829 Kang, J.K. 832, 833 Kang, Y. 846, 847 Kanis, M., see Lewerenz, H.J. 847 Kanno, T. 747 Kantorovich, L., see Hobbs, C. 692 Kantorovich, L., see Keeling, D.L. 692 Kao, C.L. 49 Kao, F.-J. 634 Kapeliovich, B.L., see Anisimov, S.I. 899 Kapral, R. 377, 378 Kapral, R., see Carter, E.A. 306 Kapral, R., see Wu, X.-G. 362 Kapteyn, H.C., see Huang, C.-P. 551 Kara, A., see DePristo, A.E. 13 Kara, A., see Smith, R.J. 122 Karasawa, T., see Bramblett, T.R. 835 Kariotis, R., see Mo, Y.-W. 816, 831 Karlberg, G.S. 257, 259 Karlberg, G.S., see Skúlason, E. 333 Karlsson, U.O. 543, 560 Karo, A.M., see Wimmer, E. 449 Karolewski, M.A. 877, 878 Karpowicz, A., see Berdau, M. 356, 366, 383 Kasai, H. 473, 474 Kasai, H., see Diño, W.A. 145 Kasai, H., see Miura, Y. 165 Kasemo, B. 149, 434, 443, 459, 460, 463, 465, 466, 475–478, 481, 482, 487, 488, 496, 509, 512 Kasemo, B., see Andersson, D. 465, 476, 477, 482 Kasemo, B., see Chakarov, D. 482 Kasemo, B., see Harris, J. 239 Kasemo, B., see Hellberg, L. 464–466, 475, 477, 479–483, 488, 494, 500 Kasemo, B., see Hellsing, B. 257 Kasemo, B., see Komrowski, A.J. 506, 507 Kasemo, B., see Langhammer, C. 470 Kasemo, B., see Olsson, L. 261

947

Kasemo, B., see Oner, D.E. 498 Kasemo, B., see Österlund, L. 72, 73, 149, 245, 334, 455, 495, 496, 498, 507 Kasemo, B., see Strömquist, J. 475, 480–483, 494, 499, 500, 509 Kasemo, B., see Ternow, H. 507 Kasemo, B., see Zhdanov, V.P. 239, 251, 256, 259, 262, 263, 356, 383, 408 Kasi, S.R., see Liehr, M. 834 Katano, S. 735 Katano, S., see Kawai, M. 601, 617, 687, 712 Kato, H., see Watanabe, K. 633 Kato, H.S., see Hossain, M.Z. 696 Kato, H.S., see Katano, S. 735 Kato, K. 245, 508 Katona, T. 373 Katz, G. 148, 499, 507–509 Katz, G., see Binetti, M. 73, 149, 495, 499, 500 Katz, G., see Romm, L. 237, 513 Kaufman, J., see Brannon, J.H. 907 Kaulich, B., see Günther, S. 379, 380 Kavulak, D.F. 832, 833 Kawai, M. 601, 617, 687, 712 Kawai, M., see Hossain, M.Z. 696 Kawai, M., see Katano, S. 735 Kawai, M., see Kim, Y. 734 Kawai, M., see Komeda, T. 597, 724 Kawai, M., see Sainoo, Y. 592, 601, 617, 711, 712 Kawai, T., see Kanno, T. 747 Kawazoe, Y., see Wang, J.-T. 695 Kaxiras, E., see Copel, M. 803 Kaxiras, E., see Maragakis, P. 306 Kay, B.D. 223, 225, 662 Kay, B.D., see Lykke, K.R. 210, 211, 213, 513 Kay, M. 151, 159, 183 Kay, M., see Darling, G.R. 152, 153, 172, 183 Kazansky, A.K., see Borisov, A.G. 7, 481, 502, 555, 557, 558, 565 Kazansky, A.K., see Gauyacq, J.P. 565 Keck, J.C. 306 Keeling, D.L. 691, 692, 751 Keen, J.M., see Canham, L.T. 847 Keener, J.P., see Tyson, J.J. 385 Keil, W., see Wicke, E. 346, 347 Keinonen, J., see Bringa, E.M. 896 Keinonen, J., see Nordlund, K. 896 Kellerman, B.K., see Mahajan, A. 828 Kellog, G.L. 357

948

Kellogg, G.L. 767 Kellogg, G.L., see Feibelman, P.J. 765 Kelly, J.J., see Boonekamp, E.P. 849 Kelly, J.J., see Bressers, P.M.M.C. 849, 850 Kelly, J.J., see Xia, X. 849, 850 Kelly, J.J., see Xia, X.H. 849, 850 Kelly, K.F., see Shirai, Y. 751 Kelly, R. 893 Kelly, R., see Miotello, A. 893 Kelly, R.D., see Goodman, D.W. 315 Kempter, Z., see Bremten, H. 473 Kenny, S.D., see Godwin, P.D. 692 Kern, K. 101 Kern, K., see Barth, J.V. 4, 775 Kern, K., see Bromann, K. 778, 780, 781, 783 Kern, K., see Brune, H. 767, 769, 775, 778–780, 782 Kern, K., see Bürgi, L. 546, 547, 549 Kern, K., see Fischer, B. 775 Kern, K., see Gambardella, P. 302, 778, 781 Kern, K., see Hamilton, J.C. 781 Kern, K., see Jeandupeux, O. 549, 771 Kern, K., see Knorr, N. 609, 771–774 Kern, K., see Müller, B. 778 Kern, K., see Röder, H. 775, 778, 779, 815 Kern, K., see Rusponi, S. 779 Kern, K., see Stepanyuk, V.S. 772, 774 Kern, K., see Vitali, L. 546, 550, 561, 562 Kern, K., see Wahl, P. 550, 609 Kesmodel, L.L., see van Hove, M.A. 290 Kevan, S.D. 540, 541, 560 Kevrekidis, I.G., see Bangia, A.K. 415 Kevrekidis, I.G., see Bär, M. 415, 416 Kevrekidis, I.G., see Graham, M.D. 345, 385, 415 Kevrekidis, I.G., see Haas, G. 415, 416 Kevrekidis, I.G., see Hartmann, N. 415 Kevrekidis, I.G., see Li, X.J. 415, 417 Kevrekidis, I.G., see Liu, H. 400, 401 Kevrekidis, I.G., see Makeev, A.G. 373 Kevrekidis, I.G., see Rotermund, H.H. 390 Kevrekidis, I.G., see Shvartsman, S. 262 Kevrekidis, I.G., see Wolff, J. 419, 420, 821 Kevrekidis, Y., see Hartmann, N. 415–417 Kevrekidis, Y., see Schütz, E. 345, 415, 417, 418 Keyling, R. 560 Khanna, S.N., see Jena, P. 778 Khanom, F., see Rahman, F. 843

Author index

Khoo, K.H., see Lu, X. 610 Khrustova, N. 410 Kibler, L.A. 292, 293 Kibsgaard, J. 330 Kief, M.T., see Nyberg, G.L. 767 Kieffer, J.-C., see Vidal, F. 904, 905 Kiehlbauch, M., see Belen, R.J. 827 Kiejna, A. 449, 450 Kieling, V., see Allongue, P. 846 Kienzle, O., see Schmidt, O.G. 840 Kikkawa, J. 810 Killelea, D.R., see Juurlink, L.B.F. 181 Killelea, D.R., see Smith, R.R. 181 Kim, D., see Ye, S. 849 Kim, D.H., see Kim, W. 735 Kim, H., see Choi, B.-Y. 736 Kim, H., see Kim, W. 735 Kim, H., see Vidali, G. 99 Kim, H.W., see Choi, B.-Y. 736 Kim, J., see Jerdev, D.I. 323 Kim, J.W., see Cranney, M. 745 Kim, K.-J., see Suemitsu, M. 828 Kim, M. 413, 414 Kim, S., see Choi, B.-Y. 736 Kim, S., see Jeong, H.D. 735 Kim, S.H., see Mendez, J. 717 Kim, W. 735 Kim, Y. 734 Kim, Y., see Kawai, M. 601, 617, 687, 712 Kim, Y., see Komeda, T. 597, 724 Kim, Y., see Sainoo, Y. 592, 601, 617, 711, 712 Kim, Y.D., see Over, H. 370 Kimman, J. 85, 97, 212 Kimman, J., see Rettner, C.T. 82–84, 87, 88, 148, 183, 189, 190, 212, 220 Kimmel, G.A., see Marston, J.B. 7, 12 Kimura, K., see Moula, M.G. 240 King, D.A. 98, 172 King, D.A., see Bare, S.R. 357 King, D.A., see Borroni-Bird, C.E. 763 King, D.A., see Brown, W.A. 357, 764 King, D.A., see Gruyters, M. 259, 362, 374 King, D.A., see Hopkinson, A. 371, 373 King, D.A., see Hoyle, R.B. 259, 361 King, D.A., see Michaelides, A. 304 King, D.A., see Mola, E.E. 362 King, D.A., see Singh-Boparai, S.P. 371 King, D.S., see Buntin, S.A. 641, 646, 669

Author index

King, D.S., see Gadzuk, J.W. 468, 641, 642, 650, 659 King, D.S., see Richter, L.J. 641 Kinnersley, A.D. 168, 169, 174, 176 Kino, Y., see Yagu, S. 76, 78, 79, 81 Kinosita, K., see Noji, H. 744 Kinosita, Y., see Noji, H. 744 Kirakosian, A., see Himpsel, F.J. 817 Kiran, B., see Burgert, R. 508 Kirchner, E.J.J. 32, 33 Kirchner, R., see Betz, G. 889 Kirczenow, G., see Emberly, E.G. 738 Kirschner, J., see Tsivlin, D.V. 773 Kishimoto, D., see Shen, X.Q. 842 Kiskinova, M. 377, 379, 390, 398 Kiskinova, M., see Comelli, G. 377, 390 Kiskinova, M., see de Decker, Y. 259 Kiskinova, M., see Freyer, N. 357 Kissel, R., see Colla, T.J. 895 Kitazoe, Y. 894 Kitchin, J.R. 292 Kitchin, J.R., see Nørskov, J.K. 331–333 Kittel, C. 763 Kiyonaga, T., see Rahman, F. 843 Klamroth, T. 669, 670 Klamroth, T., see Bartels, L. 687, 688, 724 Klamroth, T., see Vazhappilly, T. 675 Klar, F. 488 Kleban, P.H., see Bondzie, V.A. 368 Klein, D.L. 747 Klein, J.R., see Vidali, G. 99 Kleyn, A.W. 3, 31, 38, 41, 48, 83–85, 159, 206–208, 215, 216, 239 Kleyn, A.W., see Berenbak, B. 42, 43, 45, 209 Kleyn, A.W., see Bonn, M. 35 Kleyn, A.W., see Bronnikov, D.K. 204 Kleyn, A.W., see Butler, D.A. 45, 209 Kleyn, A.W., see Geuzebroeak, F.H. 210 Kleyn, A.W., see Gou, F. 32 Kleyn, A.W., see Guezebroek, F.H. 162 Kleyn, A.W., see Horn, T.C.M. 49 Kleyn, A.W., see Jenniskens, H.G. 628, 629 Kleyn, A.W., see Kirchner, E.J.J. 32, 33 Kleyn, A.W., see Komrowski, A.J. 506, 507 Kleyn, A.W., see Kuipers, E.W. 210 Kleyn, A.W., see Lahaye, R.J.W.E. 32, 34, 36–39, 44, 47–49, 162–164 Kleyn, A.W., see Luntz, A.C. 215 Kleyn, A.W., see Raukema, A. 38

949

Kleyn, A.W., see Raukerma, A. 245 Kleyn, A.W., see Rettner, C.T. 6, 31 Kleyn, A.W., see Riedmüller, B. 45 Kleyn, A.W., see Tenner, A.D. 38, 40, 41 Kleyn, A.W., see Tenner, M.G. 210 Kleyn, A.W., see Ternow, H. 507 Kliewer, J. 527, 529, 531, 535, 540, 543, 546, 548–550, 560–562, 686 Kliewer, J., see Chulkov, E.V. 529, 548, 564, 565 Kliewer, J., see Corriol, C. 548, 564 Klitsner, T. 728 Kloeck, B., see Smith, R.L. 850 Klug, D.-A. 828 Klüner, T. 631, 632 Klüner, T., see Bach, C. 19, 23 Klüner, T., see Kock, C.P. 20 Klüner, T., see Mehdaoui, I. 631, 632 Klüner, T., see Pykavy, M. 632 Klünker, C., see Hannon, J.B. 776, 777 Knall, J., see Mayne, A.J. 735 Knauer, F. 55 Knoll, A., see Gerstner, V. 715 Knorr, N. 609, 771–774 Knorr, N., see Stepanyuk, V.S. 772, 774 Knösel, E. 556, 563, 564 Knösel, E., see Bartels, L. 4, 558, 638, 687, 699, 715 Knösel, E., see Hertel, T. 553 Knösel, E., see Wolf, M. 555, 563 Knotek, M.L. 468 Kobal, I., see Imamura, K. 254 Kobal, I., see Moula, M.G. 240 Kobatake, S. 736 Kobayashi, N.P., see Xie, Q. 839 Kobayashi, T., see Ogata, Y.H. 849 Kobayashi, Y., see Takagi, D. 813 Koch, E.E., see Karlsson, U.O. 543, 560 Koch, R. 799 Kock, C.P. 20 Kodambaka, S. 810 Kodambaka, S., see Hannon, J.B. 808, 810, 811 Koehler, B.G. 829 Koehler, B.G., see Wise, M.L. 715, 830 Koel, B.E. 623 Koel, B.E., see Bartram, M.E. 320 Koel, B.E., see Jerdev, D.I. 323 Koel, B.E., see Lei, R.Z. 237 Koestner, R.J., see van Hove, M.A. 290, 357

950

Koh, S.J. 773, 774 Kohl, P.A., see Propst, E.K. 845, 847 Köhler, J.S., see Seitz, F. 891 Köhler, U. 834 Kohler, U., see Andersohn, L. 834 Köhler, U.K., see Hamers, R.J. 816, 831 Kohn, W. 25, 106, 277, 281, 433 Kohn, W., see Burke, K. 58, 111 Kohn, W., see Hohenberg, P. 277, 281, 282, 433 Kohn, W., see Jensen, J.H. 113 Kohn, W., see Lau, K.H. 770 Kohn, W., see Woll, E.J. 484 Kohn, W., see Zaremba, E. 99, 100 Kohn, W., see Zaremba, Z. 500 Kokalj, A., see Imamura, K. 254 Koker, L. 846–848 Kolaczkiewicz, J. 764 Kolasinski, K.W. 145, 311, 791, 792, 808, 811, 825, 827–829, 845–848 Kolasinski, K.W., see DeWitt, K. 836 Kolasinski, K.W., see Jacobs, D.C. 208, 215 Kolasinski, K.W., see Koker, L. 846–848 Kolasinski, K.W., see Mills, D. 792, 827 Kolasinski, K.W., see Nahidi, M. 792, 846 Kolasinski, K.W., see Riedel, D. 827 Kolasinski, K.W., see Shane, S.F. 829 Kolb, D.M., see Kibler, L.A. 292, 293 Kolb, F.M., see Schubert, L. 811 Koleske, D.D. 843 Koltsov, D.K., see Saifullah, M.S.M. 739 Komeda, T. 592, 597, 601, 707, 724 Komeda, T., see Katano, S. 735 Komeda, T., see Kawai, M. 601, 617, 687, 712 Komeda, T., see Kim, Y. 734 Komeda, T., see Sainoo, Y. 592, 601, 617, 711, 712 Kompa, K.L., see Bratu, P. 829 Komrowski, A.J. 73, 494, 495, 500, 506, 507 Komrowski, A.J., see Binetti, M. 462, 494, 495, 500 Kondo, J. 608, 609 Kondo, T. 38, 76–78, 80, 81 Kondo, T., see Tomii, T. 76, 79, 81 Kondoh, H., see Nagasaka, M. 259 Kondoh, H., see Nakai, I. 257 Kong, M.J. 735 Konjhodzic, D., see Kravets, V.G. 845 Können, G.P. 891 Koo, J.Y., see Kim, W. 735

Author index

Kooij, E.S. 846, 847 Koopmans, B., see Lof, R.W. 747 Kordesch, M., see Rotermund, H.H. 821 Kordesch, M.E., see Engel, W. 379 Korolik, M. 220 Koroteev, Yu.M., see Wegner, D. 548 Kortluke, O. 373 Kortluke, O., see Kuzovkov, V.N. 362 Kose, R., see Brown, W.A. 357, 764 Kosevich, Y.A., see Andreev, A.F. 792 Kosloff, R. 148, 498 Kosloff, R., see Asscher, M. 513 Kosloff, R., see Binetti, M. 73, 149, 495, 499, 500 Kosloff, R., see Citri, O. 148 Kosloff, R., see Katz, G. 148, 499, 507–509 Kosloff, R., see Klüner, T. 632 Kosloff, R., see Kock, C.P. 20 Kosloff, R., see Romm, L. 237, 513 Kostur, V.N. 534 Kowalski, T., see Pfandzelter, R. 41 Kozhushnev, M.A. 486 Kramer, J. 487 Kramer, L., see Aranson, I.S. 382, 410 Kramers, H.A. 237 Kratzer, P. 320, 841 Kratzer, P., see Penev, E. 841 Kratzer, P., see Wang, L.G. 804, 807 Krausz, F., see Xu, L. 551 Kravets, V.G. 845 Krenos, I.R., see Struve, W.S. 464, 476, 482 Kress, W. 534 Kretzschmar, I., see Wang, W. 592, 600 Kreutz, E.W., see Jandeleit, J. 826 Kreuzer, H.J. 695 Kreuzer, H.J., see Brenig, W. 829, 830 Kreuzer, H.J., see Gortel, Z.W. 120 Kreuzer, H.J., see Hussla, I. 660 Kreuzer, H.J., see Li, W.X. 320 Kreuzer, H.J., see Niedermayer, T. 135, 136 Kreuzer, H.J., see Stampfl, C. 252, 253, 320 Kreuzer, H.J., see Wierzbicki, A. 304 Krieger, W., see Frenkel, F. 207 Krieger, W., see Segner, J. 208, 209 Krim, J. 486 Krim, J., see Daly, C. 486 Krim, J., see Dayo, A. 487 Krim, J., see Highland, M. 486, 487 Krischer, K. 361, 411

Author index

Krischer, K., see Eiswirth, M. 410 Krischer, K., see Flätgen, G. 821 Krischer, K., see Hartmann, N. 368 Krischer, K., see Imbihl, R. 373 Krishna, V. 485 Krishnamohan, G.P., see Díaz, C. 471, 513, 514 Kroes, G.J. 3, 32, 145, 146, 151, 168, 179 Kroes, G.J., see Bonn, M. 35 Kroes, G.J., see Busnengo, H.F. 146, 151, 186 Kroes, G.J., see Díaz, C. 151, 155, 156, 186, 471, 513, 514 Kroes, G.J., see Farías, D. 146, 154, 155, 191 Kroes, G.J., see Hellman, A. 304, 325, 327, 328 Kroes, G.J., see McCormack, D.A. 146, 151, 159, 169, 175, 177, 179, 180, 182 Kroes, G.J., see Nieto, P. 31 Kroes, G.J., see Olsen, R.A. 147, 306 Kroes, G.J., see Pijper, E. 146 Kroes, G.J., see Somers, M.F. 161, 162, 169, 184, 189 Kroes, G.J., see Watts, E. 161, 167, 169, 183, 184, 189 Kröger, J. 543 Kröger, J., see Corriol, C. 548, 564 Kröger, J., see Crampin, S. 546–548, 551, 562 Kröger, J., see Jensen, H. 548–550, 562 Kröger, J., see Limot, L. 549, 733 Kröner, D., see Klamroth, T. 669, 670 Kröner, D., see Mehdaoui, I. 632 Kr’stanov, L., see Stranski, I.N. 803 Krueger, M., see Pearson, C. 776 Kruel, T.-M., see Eiswirth, M. 410 Krug, J., see Michely, T. 763, 767, 890 Krukowski, S., see Zaluska-Kotur, M.A. 256 Kruse, N., see Heinze, S. 374 Kruse, N., see Visart de Bocarme, T. 403 Kruse, N., see Voss, C. 370, 403 Krylov, S.Y. 237 Ksendzov, A., see Vasquez, R.P. 847 Kubby, J.A. 706 Kubiak, C.P., see Xue, Y. 735 Kubiak, G.D., see McClelland, G.M. 207, 208 Kuga, T., see Sagara, T. 829 Kühlert, O., see Zubkov, T. 302 Kuhlmann, F., see Brandt, M. 488, 489, 512 Kuhnke, K., see Gambardella, P. 302 Kühnle, A., see Bartels, L. 687, 688, 724 Kuipers, E.W. 210 Kuipers, E.W., see Tenner, M.G. 210

951

Kuipers, H., see Campbell, C.T. 671 Kuipers, H., see Frenkel, F. 207 Kuk, Y., see Choi, B.-Y. 736 Kulginov, D. 98, 122 Kulkarni, S.K. 831 Kulkarni, S.K., see Gates, S.M. 834, 835 Kumar, P., see d’Agliano, E.G. 3, 14, 19 Kumar, P., see d’Angliano, E.G. 448, 449, 479, 483, 484 Kummann, P., see Wicke, E. 346, 347 Kummel, A.C. 210, 218, 219 Kummel, A.C., see Binetti, M. 462, 494, 495, 500 Kummel, A.C., see Hanisco, T.F. 211–213, 216, 513 Kummel, A.C., see Jensen, J.A. 462, 493 Kummel, A.C., see Komrowski, A.J. 73, 494, 495, 500, 506, 507 Kummel, A.C., see Poon, G.C. 459, 466, 475, 477, 478, 482, 488 Kummel, A.C., see Sitz, G.O. 82, 205–207, 210–212, 216, 218, 219 Kummel, A.C., see Ternow, H. 507 Kunimori, K., see Mantell, D.A. 672 Kunimori, K., see Nakao, K. 240, 246 Kuntze, J. 739 Kunz, R., see Gates, S.M. 830, 843 Kunz, R.R., see Gates, S.M. 832 Kuo, C.L. 811 Kuo, Y.T., see Lin, J.S. 832 Kuperman, M., see Hildebrand, M. 398 Küppers, J., see Dinger, A. 843 Küppers, J., see Habenschaden, E. 764 Küppers, J., see Sesselman, W. 456, 458, 473 Küppers, J., see Woratschek, B. 459, 472, 473 Kuramoto, Y. 382, 410 Kuroda, H., see Takahagi, T. 847 Kuroda, M., see Rahman, F. 843 Kurogi, Y., see Oehrlein, G.S. 827 Kurth, S., see Perdew, J.P. 433 Kurtz, S.R., see Ferm, P.M. 130 Kustarev, V.G., see Kozhushnev, M.A. 486 Kustov, A., see Andersson, M.P. 294, 316–318 Kustov, A.L., see Sehested, J. 317, 318 Kutana, A. 843 Kutschera, M., see Reuß, Ch. 556, 557 Kutschera, M., see Weinelt, M. 556, 557 Kuzovkov, V.N. 362 Kuzovkov, V.N., see Kortluke, O. 373

952

Kwak, H., see Shen, X.J. 564 Kwon, S.J. 810 Kyuno, K. 782 Laasonen, K., see Honkala, K. 149, 498 Lacasta, A.M., see Sancho, J.M. 254 Ladas, S. 363, 364, 368, 369, 395, 396 Ladas, S., see Imbihl, R. 362 Lægsgaard, E., see Besenbacher, F. 717 Lægsgaard, E., see Helveg, S. 330, 331 Lægsgaard, E., see Kibsgaard, J. 330 Lægsgaard, E., see Lauritsen, J.V. 331 Lægsgaard, E., see Morgenstern, K. 777 Lægsgaard, E., see Rosei, F. 739, 748 Lægsgaard, E., see Schunack, M. 739 Lægsgaard, E., see Vang, R.T. 298, 300, 781 Lagally, M.G., see Liu, F. 803, 804, 806, 816 Lagally, M.G., see Mo, Y.-W. 816, 831 Lagally, M.G., see Salling, C.T. 696, 697 Lagally, M.G., see Tersoff, J. 804, 839, 840 Lagoute, J., see Olsson, F.E. 549, 733 Lahaye, R.J.W.E. 32, 34, 36–39, 44, 47–49, 162–164 Lainé, D., see Scoles, G. 31 Laloë, F., see Cohen-Tannoudji, C. 586, 587 Lambe, J., see Jacklevic, R.C. 593 Lambe, J., see Jaklevic, R.C. 529, 709 Lambert, R.M., see Prince, R.H. 465, 477 Landau, D.P. 304, 435 Landau, L.D. 115, 597, 878, 879 Landau, M., see Belić, D.S. 647 Landauer, R. 584 Landauer, R., see Büttiker, M. 584 Lane, I.M. 471 Lang, N.D. 7, 12, 17, 18, 128, 278, 283, 447, 449–454, 467, 476, 565, 696, 738 Lang, N.D., see Andersson, S. 135 Lang, N.D., see Smit, R.H.M. 592 Langa, S. 822 Langhammer, C. 470 Langhout, W.Y., see Tenner, M.G. 210 Langlais, V. 739 Langlais, V., see Gimzewski, J.K. 720 Langmuir, I. 356, 441, 449 Langmuir, I., see Taylor, J.B. 449 Langreth, D.C. 9, 14, 17, 277, 433, 435, 447, 448, 480, 481, 483, 485, 500 Langreth, D.C., see Burke, K. 8, 64 Langreth, D.C., see Chakarova-Käck, S.D. 608

Author index

Langreth, D.C., see Dion, M. 608 Langreth, D.C., see Gumhalter, B. 64, 113 Langreth, D.C., see Hult, E. 99, 500 Langreth, D.C., see Plihal, M. 19 Langreth, D.C., see Shao, H. 502 Langreth, D.C., see Thonhauser, T. 500 Lankard, J.R., see Brannon, J.H. 907 Lannoo, M., see Priester, C. 803, 804, 841 Lapujoulade, J. 640 Lapujoulade, J., see Perrau, J. 102 Larsen, J.H., see Dahl, S. 296–298, 300, 326, 327, 330, 781 Larsen, K.E., see Andersson, M.P. 294, 316–318 Larsen, K.E., see Sehested, J. 317, 318 Larsson, M.W., see Persson, A.I. 810, 812 Lasaga, A.C. 820 LaShell, S. 543, 548 Lastapis, M. 602, 688, 744 Lastapis, M., see Comtet, G. 688 Lastapis, M., see Martín, M. 746 Lastapis, M., see Mayne, A.J. 696, 746 Laszlo, J., see Eckstein, W. 886 Latkin, E., see Vishnevskii, A.L. 362 Lau, K.H. 770 Lau, W.M., see Fan, X.L. 508 Lau, W.M., see Huang, L.J. 715 Laubereau, A. 711 Laughlin, K.B., see Li, Y.L. 482, 483, 493 Laughlin, K.B., see Tate, M.R. 482, 483 Lauhon, L.J. 594, 597, 598, 600, 610, 728, 813 Lauhon, L.J., see Cui, Y. 808, 810 Lauhon, L.J., see Gaudioso, J. 735 Lauhon, L.J., see Gudiksen, M.S. 809 Lauhon, L.J., see Lorente, N. 597, 598, 710 Laurent, G., see Nieto, P. 31 Laurin, M., see Johanek, V. 407, 408, 414 Lauritsen, J.V. 331 Lauritsen, J.V., see Bollinger, M.V. 330 Lauritsen, J.V., see Helveg, S. 330, 331 Lauritsen, J.V., see Kibsgaard, J. 330 Lauterbach, J. 368, 390, 417 Lauterbach, J., see Gorodetskii, V.V. 397, 404 Lauterbach, J., see Lele, T. 368 Lauterbach, J., see Rotermund, H.H. 390 Lavery, A.C., see Powers, J. 57, 59 Layet, J.M., see Papageorgiou, N. 733 Lazorcik, J.L., see Long, D.P. 592 Le Lay, G., see Papageorgiou, N. 733

Author index

Le Ru, E.C. 839, 842 Le Ru, E.C., see Howe, P. 839, 841 LeBeyec, Y., see Bouneau, S. 895, 897 Lee, A.B., see Wahnström, G. 494, 719 Lee, C., see Mun, B.S. 293 Lee, C.S., see Lee, S.T. 810 Lee, G., see Kim, W. 735 Lee, H.J. 686, 692, 728 Lee, J. 473 Lee, K., see Kim, W. 735 Lee, K.B., see Seo, Y.H. 847 Lee, K.H., see Lin, J.S. 832 Lee, M.H., see Lin, J.S. 832 Lee, R.B., see Saunders, W.A. 845 Lee, S.B., see Ertl, G. 325 Lee, S.B., see Paal, Z. 325 Lee, S.T. 810 Lee, T., see Wang, W. 592, 600 Lee, Y.-S., see Wang, X.Y. 900 Lee, Y.S., see Jeong, H.D. 735 Lefever, R., see Horsthemke, W. 404 Lefferts, L., see Bos, A.N.R. 233 Lega, J., see Coullet, P. 411 Legay, F. 711 LeGoues, F.K., see Horn-von Hoegen, M. 798, 803 Legsgaard, E., see Österlund, L. 246 Lehmann, J. 564 Lehmann, V. 822, 846, 847, 849 Lehmann, V., see Leonard, S.W. 822 Lehmann, V., see Müller, F. 822 Lehmann, V., see Ottow, S. 822 Lehner, B. 250 Lehnert, A., see Rusponi, S. 779 Lei, R.Z. 237 Leibsle, F.M., see Hirshorn, E.S. 728 Leigh, D.A., see Brouwer, A.M. 744 Lejay, Y., see Lapujoulade, J. 640 Lele, T. 368 Lele, T., see McMillan, N. 390 Lemoine, D., see Busnengo, H.F. 146, 186 Lennard-Jones, J.E. 55, 113 Lenzner, M., see Xu, L. 551 Leonard, S.W. 822 Leonardelli, G., see Schmid, M. 73 Leone, S.R., see Bernasek, S.L. 671 Leong, W.Y., see Canham, L.T. 847 LeRoy, R.J. 101 Lesik, A., see Yaccato, K. 317

953

Lesley, M.W. 373 Leslie, A.G., see Abrahams, J.P. 744 Lesnard, H. 599 Lesnard, H., see Bocquet, M.-L. 597, 599, 600 Lettieri, J., see Schlom, D.G. 791 Leung, C. 647 Leung, K.M. 8 Leveugle, E. 905 Leveugle, E., see Zhigilei, L.V. 903 Levi, A.C., see Bortolani, V. 56, 62–64 Lévi, S., see Dahan, M. 747 Levine, R.D. 664 Levine, R.D., see Meyer, H.D. 65 Levinson, H.J. 560 Lewerenz, H.J. 847 Lewerenz, H.J., see Campbell, S.A. 846 Lewerenz, H.J., see Jungblut, H. 847 Lewerenz, H.J., see Peter, L.M. 847 Lewerenz, H.J., see Rappich, J. 845, 847, 850 Lewis, L.B., see Sinniah, K. 715, 829 Lewis, L.J., see Perez, D. 903 Lewis, S.P. 615 Li, A.-P., see Leonard, S.W. 822 Li, A.H., see Liu, F. 803, 804, 806, 816 Li, C., see Pacheco, K.A. 836 Li, H., see Kalkan, A.K. 828 Li, J. 545–547, 608, 609 Li, L. 791 Li, L., see Höfer, U. 715 Li, N. 810, 811 Li, N., see Dobrin, S. 696 Li, N., see Tan, T.Y. 809–811 Li, Q., see Zhao, A. 728 Li, S. 663 Li, S., see Thonhauser, T. 500 Li, W.X. 320 Li, W.X., see Wang, J.G. 367 Li, X. 6 Li, X., see Burgert, R. 508 Li, X.J. 415, 417 Li, X.L. 6, 8, 19 Li, Y. 448 Li, Y.-C., see Li, Z.-H. 720 Li, Y.L. 482, 483, 493 Li, Y.L., see Tate, M.R. 482, 483 Li, Y.S., see Burns, A.R. 660 Li, Z.-H. 720 Liauw, M., see Emig, G. 411 Libuda, J. 263

954

Lichtman, D. 623 Lieber, C.M., see Barrelet, C.J. 808 Lieber, C.M., see Cui, Y. 808, 810 Lieber, C.M., see Duan, X. 808, 809 Lieber, C.M., see Duan, X.F. 808 Lieber, C.M., see Gudiksen, M.S. 809 Lieber, C.M., see Hu, J.T. 809 Lieber, C.M., see Lauhon, L.J. 813 Lieber, C.M., see Lu, W. 807 Lieber, C.M., see McAlpine, M.C. 808 Lieber, C.M., see Morales, A.M. 808, 809 Lieber, C.M., see Wang, D. 808 Liebsch, A. 7, 19, 100, 486, 532 Liebsch, A., see García-Lekue, A. 531, 532, 563, 564 Liebsch, A., see Harris, J. 99, 103 Liebsch, A., see López-Bastidas, C. 532 Liehr, M. 834 Lifshits, E.M., see Landau, L.D. 878, 879 Lifshitz, E.M., see Landau, L.D. 115, 597 Lifshitz, I.M. 775 Lightowlers, E.C., see Zhang, J. 828 Lilienkamp, G., see Marbach, H. 398 Lilienkamp, G., see Wei, H. 384, 396 Liljeroth, P. 734 Lim, A.K.L., see Klein, D.L. 747 Lim, Y.-S. 403 Lima, D., see Andrade, R.F.S. 362 Limot, L. 548, 549, 560, 733 Limot, L., see Crampin, S. 548 Lin, A.L. 412, 419 Lin, D.-S. 831, 835 Lin, J.-L., see Himpsel, F.J. 817 Lin, J.S. 832 Lin, K.H., see McAlpine, M.C. 808 Lin, M.C., see Modl, A. 210 Lin, T.-H., see Guthrie, W.L. 641 Lin, T.L., see Vasquez, R.P. 847 Linde, P. 46, 101, 102 Linde, P., see Andersson, T. 46, 101, 102, 104, 111 Lindenberg, K., see Sancho, J.M. 254 Lindenblatt, M. 448 Lindenblatt, M., see Bird, D.M. 448 Lindgren, S.A. 471, 529 Lindhard, J. 531, 875, 877, 879, 881 Lingle Jr., R.L. 564 Lingle Jr., R.L., see McNeill, J.D. 563 Linic, S. 308

Author index

Linik, S. 325 Link, S. 564 Link, S., see Rhie, H.-S. 563 Linke, U., see Dondi, M.G. 102 Lintz, H.-G., see Adlhoch, W. 371 Lisowski, M., see Zubkov, T. 302 Littmark, U. 889 Littmark, U., see Ziegler, J.F. 877, 880 Liu, C. 775 Liu, D.-J. 382, 407 Liu, D.-J., see Evans, J.W. 355, 356, 382, 383 Liu, D.J. 234 Liu, D.J., see Pavlenko, N. 407 Liu, F. 803, 804, 806, 816 Liu, F., see Ganesh, R. 275 Liu, F., see Pala, R.G.S. 815 Liu, H. 400, 401 Liu, I.O.Y., see Cant, N.W. 319 Liu, J.L. 808 Liu, K., see Hepburn, J.W. 433 Liu, L., see Guo, H. 650 Liu, P. 294 Liu, P., see Meier, J. 292 Liu, P., see Strasser, P. 293 Liu, S.H., see Gaffney, K.J. 558 Liu, S.H., see Wong, C.M. 563 Liu, W., see Cui, J. 130 Liu, W.K., see Mokler, S.M. 831 Liu, Z.-M., see Costello, S.A. 623 Liu, Z.-P. 302 Liu, Z.-P., see Michaelides, A. 304 Liu, Z.F., see Fan, X.L. 508 Liuan, D.A.K.Z., see Lane, I.M. 471 Lizzit, S., see Comelli, G. 374 Locatelli, A. 399, 400 Lock, A. 73 Lœgsgaard, E., see Morgenstern, K. 804 Loeher, T., see Slinko, M. 374 Lof, R.W. 747 Logadottir, A. 299, 301, 326, 330 Logadottir, A., see Bligaard, T. 307, 328, 330 Logadottir, A., see Dahl, S. 296–298, 300, 326, 327, 330, 781 Logadottir, A., see Hellman, A. 261, 274, 296, 304, 309, 325, 327, 328 Logadottir, A., see Honkala, K. 263, 274, 277, 296, 301, 304, 308, 325–330 Logadottir, A., see Jacobsen, C.J.H. 294, 312

Author index

Logadottir, A., see Nørskov, J.K. 302, 303, 331–333 Logan, R.M. 36, 46 Löhneysen, H.V., see Reichert, J. 738 Lombardo, S.J. 374 Long, D.P. 592 Long, L., see Schubert, L. 811 Longo, R.C., see Stepanyuk, V.S. 775 Longwitz, S.R., see Rusponi, S. 779 López-Bastidas, C. 532 López-Cartes, C., see Helveg, S. 812 Lopinksi, G.P., see Alavi, S. 616 Lopinski, G.P. 696, 735 Lopinski, G.P., see Patitsas, S.N. 726 Loppacher, C., see Bennewitz, R. 732 Lorente, N. 485, 595–598, 605, 613, 616, 617, 710, 724 Lorente, N., see Bocquet, M.-L. 597, 599, 600 Lorente, N., see Frederiksen, T. 592, 595 Lorente, N., see Lesnard, H. 599 Lorente, N., see Monturet, S. 588 Lorente, N., see Pascual, J.I. 595, 597, 615, 616, 688, 723 Lorente, N., see Paulsson, M. 597 Lorenz, S., see Behler, J. 149, 245, 334, 433, 496, 499, 501, 508, 510 Lorke, A., see Kravets, V.G. 845 Los, J. 7, 8, 12 Los, J., see Hurkmans, A. 48 Los, J., see Rasser, B. 467 Los, J., see Tenner, A.D. 38, 40 Los, J., see van Wunnik, J.N.M. 7, 12 Loskutov, A.Y., see Mikhailov, A.S. 404, 409, 410 Lou, L., see Akpati, H.C. 746 Loudon, R. 555 Louie, S.G., see Lu, X. 610 Løvvik, O.M. 292 Lowndes, D.H., see Pedraza, A.J. 827 Lox, E.S.J. 370 Loy, M.M.T., see Budde, F. 634, 635 Loy, M.M.T., see Misewich, J. 220 Loy, M.M.T., see Misewich, J.A. 634, 635 Loy, M.M.T., see Prybyla, J.A. 634 Loy, M.M.T., see Zheng, C.Z. 256 Lu, P.H. 719 Lu, Q., see Bramblett, T.R. 835 Lu, W. 807 Lu, X. 610

955

Lu, X.H., see Yayon, Y. 765, 766 Lubben, D. 830, 835 Lubben, D., see Lin, D.-S. 831, 835 Lubben, D., see Suda, Y. 830, 835 Lübke, M., see Gerischer, H. 845, 847 Lucas, A.A. 8, 9 Luccardini, C., see Dahan, M. 747 Luches, A., see Anisimov, S.I. 908 Luerssen, B., see Marbach, H. 398, 399 Luh, D.-A. 538 Lukkien, J.J., see Hermse, C.G.M. 254 Lukkien, J.J., see van Bavel, A.P. 254 Luk’yanchuk, B.S., see Anisimov, S.I. 824–826, 845, 908 Lundqvist, B.I. 3, 7, 12, 14, 16, 274, 278, 279, 282, 284, 443, 449–451, 454, 456, 463 Lundqvist, B.I., see Andersson, S. 274 Lundqvist, B.I., see Björkman, G. 484 Lundqvist, B.I., see Bogicevic, A. 320, 775 Lundqvist, B.I., see Chakarova-Käck, S.D. 608 Lundqvist, B.I., see Dion, M. 608 Lundqvist, B.I., see Gao, S. 20, 458, 461, 616, 637, 718 Lundqvist, B.I., see Gunnarsson, O. 278, 281, 283, 433, 450, 451 Lundqvist, B.I., see Hellberg, L. 464, 466, 475, 477, 479–482, 488, 494, 500 Lundqvist, B.I., see Hellman, A. 245, 334, 433, 434, 463, 475, 476, 480, 496, 499–508, 510 Lundqvist, B.I., see Hellsing, B. 464, 486 Lundqvist, B.I., see Hjelmberg, H. 443, 450 Lundqvist, B.I., see Holmberg, C. 468, 469 Lundqvist, B.I., see Hult, E. 99, 500 Lundqvist, B.I., see Kasemo, B. 149, 443, 463, 465, 466, 475, 476, 478, 482, 488, 509 Lundqvist, B.I., see Komrowski, A.J. 506, 507 Lundqvist, B.I., see Lang, N.D. 476 Lundqvist, B.I., see Makoshi, K. 473, 474 Lundqvist, B.I., see Nørskov, J.K. 12, 14, 334, 433, 443–445, 447, 450, 453, 459, 460, 463–467, 475, 476, 478, 479, 481–484, 488, 493, 499, 508, 509 Lundqvist, B.I., see Ovesson, S. 304, 319–325, 771, 775 Lundqvist, B.I., see Persson, M. 486 Lundqvist, B.I., see Stipe, B.C. 458, 462, 463, 719 Lundqvist, B.I., see Strömquist, J. 475, 480–483, 494, 499, 500, 509

956

Lundqvist, B.I., see Ternow, H. 507 Lundqvist, B.I., see Yourdshahyan, Y. 149, 498–500 Lundqvist, S. 500 Lundqvist, S., see Gunnarsson, O. 433 Lundqvist, S., see Hedin, L. 530, 537 Luntz, A.C. 83, 98, 102, 215, 435, 455, 461–464, 471, 484, 485, 493, 496, 513–515, 675 Luntz, A.C., see Brown, J.K. 48 Luntz, A.C., see Diekhöner, L. 511–514 Luntz, A.C., see Kleyn, A.W. 83–85, 206–208, 215, 216 Luntz, A.C., see Mortensen, H. 82, 513, 514 Luntz, A.C., see Wagner, S. 675 Luntz, A.C.A., see Diekhöner, L. 434, 463, 513 Luppi, M., see Hellman, A. 304, 325, 327, 328 Luppi, M., see Vazhappilly, T. 675 Lurssen, B., see Günther, S. 398 Lusakowski, A., see Zaluska-Kotur, M.A. 256 Luss, D. 348, 381 Lüth, H., see Frohnhoff, S. 849 Luther, R. 353 Lutter, R., see Abrahams, J.P. 744 Lutterloh, C., see Dinger, A. 843 Luttge, A., see Lasaga, A.C. 820 Lutz, C.P., see Crommie, M.F. 545, 686, 688, 694, 771, 775 Lutz, C.P., see Eigler, D.M. 456, 458, 461, 717 Lutz, C.P., see Fiete, G.A. 686 Lutz, C.P., see Heinrich, A.J. 686, 709, 742 Lutz, C.P., see Heller, E.J. 548, 686 Lutz, C.P., see Hirjibehedin, C.F. 686 Lutz, C.P., see Manoharan, H.C. 609, 686 Lutz, C.P., see Zeppenfeld, P. 689, 690 Lyding, J.W. 701, 703, 705, 706, 726 Lyding, J.W., see Abeln, G.C. 726 Lyding, J.W., see Avouris, Ph. 602, 616, 699, 701, 704–706 Lyding, J.W., see Foley, E.T. 701, 705, 706 Lyding, J.W., see Hersam, M.C. 726 Lyding, J.W., see Shen, T.-C. 701, 704–706 Lykke, K.R. 210, 211, 213, 513 Lyo, I.-W. 696, 735 Lyo, I.-W., see Avouris, Ph. 546, 696, 697 Lytken, O., see Abild-Pedersen, F. 299 Lyubinetsky, I., see Dohnálek, Z. 715 Lyubinetsky, I., see Mezhenny, S. 735 Lyubovitsky, J.G., see Kong, M.J. 735

Author index

Ma, J.-G., see Shelnutt, J.A. 733 Ma, Y., see Han, S. 672 Ma, Y.-R., see Moriarty, P. 691 Maass, K., see Schuller, A. 41 Maass, P., see Heinrichs, S. 783 Maass, P., see Rottler, J. 783 Macak, J.M., see Tsuchiya, H. 822 Macdonald, R.G., see Hepburn, J.W. 433 Machado, M., see Chulkov, E.V. 529, 560 Machado, M., see Echenique, P.M. 529, 562, 564 Mackay, R.S. 210 Madden, H.H., see Evans, J.W. 385 Madey, T.E., see Goodman, D.W. 315 Madey, T.E., see Tolk, N.H. 699 Madhavan, V. 608, 609 Madhukar, A., see Xie, Q. 839 Madix, R.J., see Arumainayagam, C.R. 31, 173 Madix, R.J., see D’Evelyn, M.P. 174 Madix, R.J., see Kao, C.L. 49 Madix, R.J., see Weaver, J.F. 31, 836 Maeda, M., see Sato, Y. 847 Maes, H.E., see Jain, S.C. 796 Magee, J.L. 464, 476, 487 Magnera, T.F., see Balaji, V. 893 Magtoto, N.P. 371 Mahajan, A. 828 Mahan, G.D. 8, 9, 14–16, 67, 609 Maity, N. 837 Maity, N., see Jones, M.E. 832, 833 Maity, N., see Xia, L.Q. 833, 835, 836 Majumdar, A., see Nienhaus, H. 148, 240, 489–491, 493 Mak, C.H., see Koehler, B.G. 829 Mak, K.M., see Banholzer, W.F. 371 Makarenko, B., see Kutana, A. 843 Makarov, D.E., see Benderskii, V.A. 4, 5 Makeev, A. 383, 389–391 Makeev, A.G. 373, 374 Makhmetov, G.E. 473 Makimura, T. 826, 845 Makino, O. 842, 847 Makoshi, K. 473, 474 Makoshi, K., see Mingo, N. 709, 710 Makoshi, K., see Yoshimori, A. 473, 474 Maksimov, E.G. 534 Malchow, H. 404 Malevich, A.E., see Adler, P.M. 256

Author index

Mamatkulov, M. 745 Mamatkulov, M., see Cranney, M. 745 Mamatkulov, M., see Martín, M. 746 Manoharan, H.C. 609, 686 Manoharan, H.C., see Fiete, G.A. 686 Manolopoulos, D.E. 510, 512 Manrubia, S.C. 410, 411 Manson, J.R. 55, 56, 59, 62, 65–67, 70, 199, 456 Manson, J.R., see Ambaye, H. 44, 74, 75 Manson, J.R., see Armand, G. 66 Manson, J.R., see Dai, J. 44, 59, 66 Manson, J.R., see Hayes, W.W. 44 Manson, J.R., see Hofmann, F. 64 Manson, J.R., see Iftimia, I. 44 Manson, J.R., see Moroz, I. 76, 77 Manson, J.R., see Muis, A. 57, 59 Manson, J.R., see Powers, J. 57, 59 Mantell, D.A. 672 Manthe, U., see Hellman, A. 304, 325, 327, 328 Manthe, U., see van Harrevelt, R. 146 Mantl, S., see Bahr, D. 797 Mantooth, B.A., see Long, D.P. 592 Manz, J. 20 Maple, M.B., see Sales, B.C. 366 Maple, M.B., see Yamamoto, S.Y. 381 Maradudin, A.A. 64, 841 Maragakis, P. 306 Marbach, H. 398–400 Marbach, H., see de Decker, Y. 259, 398 Marbach, H., see Günther, S. 398, 417, 419 Marbach, H., see Hinz, M. 398 March, N.H., see Lundqvist, S. 500 Marcos, G., see Blauw, M.A. 827 Marcus, R.A. 3 Marhawa, B., see Luss, D. 348 Marin, G.B., see Bos, A.N.R. 233 Marinica, D.C. 20, 563 Marinopoulou, A., see Zhang, J. 828 Marion, J.B. 9 Markovic, N., see Nagard, M.B. 44 Markovic, N., see Tomsic, A. 44 Markovic, N.M. 291, 333 Markovic, N.M., see Mun, B.S. 293 Maroni, P. 181 Maroni, P., see Beck, R.D. 181 Maroni, P., see Schmid, M.P. 181 Maroun, F., see Chazalviel, J.-N. 847 Maroutian, T., see Limot, L. 548, 560

957

Marques, M.A.L. 580 Marso, M., see Frohnhoff, S. 849 Marston, J.B. 7, 12 Mårtensson, T., see Johansson, J. 810–812 Martín, F., see Díaz, C. 151, 155, 156, 186 Martín, F., see Farías, D. 146, 154, 155, 191 Martín, F., see Vidal, F. 904, 905 Martin, I.T., see Williams, K.L. 844, 845 Martín, M. 746 Martín, M., see Comtet, G. 688 Martín, M., see Lastapis, M. 602, 688, 744 Martin, O.J.F. 714 Martin, O.J.F., see Girard, C. 714 Martin, R. 579, 581 Martin, R., see Gardner, P. 357 Martin, R.M., see Lee, J. 473 Martinez, K., see Lin, A.L. 412, 419 Maruyama, S., see Watanabe, Y. 44 Marzari, N., see Sit, P.H.L. 500 Mase, K., see Fukutani, K. 643, 669 Masel, R.I. 233 Masel, R.I., see Gohndrone, J.M. 371 Masin, M. 256 Masin, M., see Chvoj, Z. 256 Mason, S.E. 285 Massobrio, C., see Blandin, P. 778 Masson, L. 831, 834 Masson, L., see Albertini, D. 833, 834 Massoth, F.E., see Topsøe, H. 330, 331 Masuda, S., see Harada, Y. 456, 457, 472, 474, 475 Mathews, C.M. 513 Matsiev, D., see Chen, J. 511, 512 Matsiev, D., see Ran, Q. 471, 513 Matsiev, D., see White, J.D. 189–191, 334, 433, 466, 496, 511, 512 Matsiev, D., see Wodtke, A.M. 3, 435 Matsuda, A. 828 Matsuda, I., see Liu, C. 775 Matsuda, I., see Matsui, F. 735 Matsuda, K. 736 Matsui, F. 735 Matsui, T., see Asada, H. 205 Matsumoto, Y. 633 Matsumoto, Y., see Gruzdkov, Y.A. 633 Matsumoto, Y., see Watanabe, K. 633 Matsumoto, Y., see Watanabe, Y. 44 Matsumura, M. 847 Matsuno, T. 829

958

Matsushima, T. 240, 242, 513, 671 Matsushima, T., see Gumhalter, B. 240 Matsushima, T., see Han, S. 672 Matsushima, T., see Imamura, K. 254 Matsushima, T., see Moula, M.G. 240 Matsushima, T., see Sano, M. 672 Matsushima, T., see Zhdanov, V.P. 254, 255, 259 Mattera, L., see Chiesa, M. 102, 103 Mattera, L., see Dondi, M.G. 102 Mattera, L., see Luntz, A.C. 102 Matthews, J.W. 799 Matthias, E., see Preuss, S. 825 Matthiesen, J., see Bligaard, T. 294, 311, 312, 314 Matzdorf, R. 538, 548 Matzdorf, R., see Paniago, R. 540 Matzdorf, R., see Theilmann, F. 538, 539, 548 Mavrikakis, M. 291, 302 Mavrikakis, M., see Alcalá, R. 302 Mavrikakis, M., see Greeley, J. 277, 291, 304 Mavrikakis, M., see Kandoi, S. 308, 309 Mavrikakis, M., see Nørskov, J.K. 302, 303 Mavrikakis, M., see Schumacher, N. 314, 315 Mavrikakis, M., see Tripa, C.E. 290 Mavrikakis, M., see Zhang, J.L. 291 Mayer, J.W., see Nastasi, M. 892, 893 Mayne, A.J. 4, 22, 602, 688, 694, 696, 699, 701, 703, 726, 728, 735, 746 Mayne, A.J., see Baffou, G. 734 Mayne, A.J., see Bellec, A. 736 Mayne, A.J., see Bernard, R. 748 Mayne, A.J., see Comtet, G. 688 Mayne, A.J., see Cranney, M. 696, 746 Mayne, A.J., see Dujardin, G. 602, 686, 693, 701, 728, 729 Mayne, A.J., see Molinàs-Mata, P. 694, 728 Mayne, A.J., see Papageorgiou, N. 733 Mayne, A.J., see Riedel, D. 715 Mayne, A.J., see Soukiassian, L. 701, 703–706, 726, 736 Mayne, H.R., see Beauregard, J.N. 175 Mayor, M., see Reichert, J. 738 Maytorena, J.A., see López-Bastidas, C. 532 Mazur, E., see Crouch, C.H. 827 Mazur, E., see Deliwala, S. 634 Mazur, E., see Her, T.-H. 827 Mazur, E., see Shen, M.Y. 827 Mazur, E., see Sundaram, S.K. 824

Author index

Mazzoleni, C., see Pavesi, L. 845 McAlpine, M.C. 808 McCabe, R.W., see Shelef, M. 319 McClelland, G.M. 207, 208 McClelland, G.M., see Ferm, P.M. 130 McClelland, G.M., see Pearlstine, K.A. 130, 132 McCloskey, D.J., see Bushnell, J.C. 905, 906 McClure, S.M. 31 McCormack, D.A. 146, 151, 159, 169, 175, 177, 179, 180, 182 McCormack, D.A., see Kroes, G.J. 3, 32 McCormack, D.A., see Watts, E. 161, 167, 169, 183, 184, 189 McDougall, B.A., see LaShell, S. 548 McEllistrem, M.T., see Bronikowski, M.J. 835 McFadden, D.L., see Davidovits, P. 479 McFadden, D.L., see Struve, W.S. 464, 476, 482 McFarland, E.W., see Cong, P.J. 313 McFarland, E.W., see Cuenya, B.R. 491 McFarland, E.W., see Gergen, B. 4, 15, 148, 334, 335, 455, 482, 489–492 McFarland, E.W., see Glass, S. 491 McFarland, E.W., see Nienhaus, H. 148, 240, 489–491, 493 Mcgonigal, M., see Li, Y.L. 482, 483, 493 McMillan, N. 390 McNab, I.R., see Dobrin, S. 696 McNeill, J.D. 563 McNeill, J.D., see Gaffney, K.J. 558 McNeill, J.D., see Lingle Jr., R.L. 564 McNeill, J.D., see Wong, C.M. 563 Meade, R.D. 796 Medforth, C., see Shelnutt, J.A. 733 Medvedev, V.K. 403 Medvedev, V.K., see Suchorski, Y. 404 Mehdaoui, I. 631, 632 Mehl, M., see Langreth, D.C. 433 Mehl, M.J., see Langreth, D.C. 277 Mehlhorn, M., see Henzl, J. 736 Meier, C., see Kravets, V.G. 845 Meier, J. 292 Meir, Y. 592 Meissen, F. 396 Meister, G., see Matzdorf, R. 538 Meister, G., see Paniago, R. 540 Meister, G., see Theilmann, F. 538, 539, 548 Meiwes-Broer, K.H., see Klar, F. 488

Author index

Memmel, N. 527 Memmert, U. 834 Mendes, P.M. 851 Mendez, J. 717 Menges, M., see Mull, T. 646 Mentes, T.O., see Locatelli, A. 399, 400 Menyhard, M., see Süle, P. 893 Menzel, D. 468, 623, 699, 723 Menzel, D., see Berthold, W. 563, 564 Menzel, D., see Feulner, P. 558 Menzel, D., see Föhlisch, A. 566 Menzel, D., see Gokhale, S. 735 Menzel, D., see Niedermayer, T. 135, 136 Menzel, D., see Schlichting, H. 46, 111, 125, 126 Menzel, D., see Watanabe, K. 470 Menzel, D., see Wurth, W. 558, 566 Mérat-Combes, M.-N., see Harmand, J.C. 812 Merikoski, J. 256 Mermin, D., see Ashcroft, N.W. 582 Meron, E. 389 Meron, E., see Bär, M. 389 Merrick, M.L., see Fichthorn, K.A. 774, 775 Merschdorf, M., see Lehmann, J. 564 Mertens, A., see Schuller, A. 41 Mertens, F. 374, 378, 381–383, 385, 387, 389, 390 Mertens, F., see Gottschalk, N. 378, 385, 389 Mertens, F., see Veser, G. 388 Methiu, H., see Gadzuk, J.W. 483 Metiu, H. 3, 4, 14, 433 Metiu, H., see Freed, K.F. 711 Metiu, H., see Gadzuk, J.W. 7, 8, 12, 14, 15 Metiu, H., see Lee, J. 473 Metiu, H., see Leung, K.M. 8 Metiu, H., see Mills, G. 290 Metiu, H., see Sawada, S.I. 6 Meyer, E., see Bennewitz, R. 732 Meyer, G. 689, 690 Meyer, G., see Bartels, L. 4, 558, 686–691, 699, 724, 725 Meyer, G., see Feenstra, R.M. 696, 746 Meyer, G., see Hla, S.-W. 686, 692 Meyer, G., see Liljeroth, P. 734 Meyer, G., see Moresco, F. 686, 741 Meyer, G., see Olsson, F.E. 275, 610, 732 Meyer, G., see Repp, J. 688, 700, 731, 732, 772, 773 Meyer, H.D. 65, 106

959

Meyer-ter-Vehn, J., see Sokolowski-Tinten, K. 824 Meyerhof, W.E. 528 Meyerson, B.S., see Jasinski, J.M. 828 Mezhenny, S. 735 Mi, Z. 839 Micha, D.A. 57 Michaelides, A. 304 Michaels, J., see Vayenas, C.G. 349 Michaels, J.N., see Vayenas, C.G. 349 Michalke, T., see Gerlach, A. 539 Michalke, T., see Straube, P. 542, 560 Michelsen, H.A. 172, 175, 179 Michelsen, H.A., see Gulding, S.J. 177 Michelsen, H.A., see Rettner, C.T. 167, 172, 175, 177 Michely, T. 763, 767, 782, 783, 890 Michely, T., see Bott, M. 780, 782 Michiels, J., see Hedin, L. 537 Michl, J. 893 Michl, J., see Balaji, V. 893 Michl, J., see Urbassek, H.M. 893 Mieher, W.D. 671, 718 Mieher, W.D., see Deliwala, S. 634 Mies, F.H., see Charron, E. 701 Miesch, M.S., see Evans, J.W. 357 Mijatovic, D. 791 Mikaelian, G. 734 Mikhailov, A. 378, 389 Mikhailov, A., see Battogtokh, D. 382 Mikhailov, A., see Hildebrand, M. 398 Mikhailov, A., see Khrustova, N. 410 Mikhailov, A., see Mertens, F. 378, 382, 387 Mikhailov, A., see Rose, K.C. 388 Mikhailov, A., see Veser, G. 388 Mikhailov, A.S. 345, 353, 378, 383–385, 397, 404, 409–411, 413 Mikhailov, A.S., see Beta, C. 413 Mikhailov, A.S., see de Decker, Y. 259, 398 Mikhailov, A.S., see Hildebrand, M. 398 Mikhailov, A.S., see Khrustova, N. 410 Mikhailov, A.S., see Manrubia, S.C. 410, 411 Mikhailov, A.S., see Punckt, C. 821 Mikhailov, A.S., see von Oertzen, A. 368, 390, 393, 394 Miki, K., see Bandyopadhyay, A. 737 Mikkor, M., see Jaklevic, R.C. 529 Miller, A.D., see Gaffney, K.J. 558 Miller, A.D., see Wong, C.M. 563

960

Miller, D.R. 498 Miller, R.E., see Francisco, T.W. 204, 223, 224 Miller, R.E., see Wight, A.C. 223, 225 Miller, T., see Hansen, E.D. 538 Miller, T., see Lin, D.-S. 835 Miller, T., see Luh, D.-A. 538 Miller, W.H. 237 Mills, D. 792, 827 Mills, D., see Kolasinski, K.W. 846 Mills, G. 290 Mills, G., see Jónsson, H. 304, 306, 320 Min, O.Y., see Hu, J.T. 809 Mindt, W., see Gerischer, H. 846 Miners, J.H. 371 Miners, J.H., see Bonn, M. 483 Mingo, N. 709, 710 Minot, C., see Mamatkulov, M. 745 Miotello, A. 893 Miranda, P.B., see Ye, S. 849 Miret-Artes, S., see Guantes, R. 254 Miret-Artes, S., see Vega, J.L. 254 Miron, R.A., see Fichthorn, K.A. 237 Misewich, J. 220 Misewich, J.A. 634–636 Misewich, J.A., see Brandbyge, M. 19, 637 Misewich, J.A., see Budde, F. 634, 635 Misewich, J.A., see Newns, D.M. 19 Misewich, J.A., see Prybyla, J.A. 634 Mitrović, B., see Kostur, V.N. 534 Mittach, A. 325 Mittendorfer, F. 497 Mittendorfer, F., see Eichler, A. 311 Mityushev, V., see Adler, P.M. 256 Miura, Y. 165 Miyamoto, N., see Suemitsu, M. 828 Miyazaki, S., see Bjorkman, C.H. 847 Miyazaki, S., see Tanaka, T. 830 Mizielinski, M.S. 148, 189, 448, 493, 611 Mizielinski, M.S., see Bird, D.M. 448 Mizuseki, H., see Wang, J.-T. 695 Mizuta, T., see Makimura, T. 826, 845 Mo, Y.-W. 816, 831 Mo, Y.M. 695 Mocuta, D., see Shen, X.J. 564 Modesti, S., see Gimzewski, J.K. 748 Modl, A. 210 Mödl, A., see Budde, F. 82 Moeller, P. 362 Moeller, P., see Eiswirth, M. 346–348, 359, 378

Author index

Moeller, P., see Schwankner, R.J. 412 Moffat, D.J., see Lopinski, G.P. 735 Moffat, D.J., see Patitsas, S.N. 726 Mohammad, S.N. 810 Moiseyev, N. 6 Mokler, S.M. 831 Mokler, S.M., see Zhang, J. 828 Mola, E.E. 362 Molenbroek, A., see Besenbacher, F. 296, 298 Molinari, E. 239 Molinàs-Mata, P. 694, 728 Moll, N., see Wang, L.G. 804, 807 Möller, D., see Bartels, L. 638, 715 Molloy, J.E. 744 Momma, C., see Nolte, S. 906, 907 Mondia, J.P., see Leonard, S.W. 822 Monine, M. 363, 393 Monine, M.I. 363, 365, 390 Monot, R., see Henry, C.R. 290 Monot, R., see Vandoni, G. 769 Montano, M.O., see Somorjai, G.A. 492 Montroll, E.W., see Maradudin, A.A. 64 Monturet, S. 588 Mooney, P.M. 828 Moore, J.H., see Olthoff, J.K. 630 Moore, J.S., see Abeln, G.C. 726 Moore, M.H., see Long, D.P. 592 Moos, G., see Hotzel, A. 563 Mora, E., see Harutyunyan, A.R. 812 Morales, A.M. 808, 809 Morante, J.R., see Gorostiza, P. 842 Morawitz, H., see Rettner, C.T. 148, 189, 220 Moresco, F. 686, 740, 741, 751 Moresco, F., see Alemani, M. 736 Moresco, F., see Chiavaralloti, F. 748 Moresco, F., see Grill, L. 740, 750, 751 Moresco, F., see Gross, L. 740, 748, 751 Moresco, F., see Repp, J. 772, 773 Morgan Jr., G.A., see Zubkov, T. 302 Morgante, A., see Böttcher, A. 148, 465, 466, 473–475, 477, 478, 488, 489, 512 Morgante, A., see Greber, T. 464, 482 Morgenstern, K. 776, 777, 804 Morgenstern, K., see Henzl, J. 736 Morgenstern, M., see Bott, M. 782 Moriarty, P. 691 Moriarty, P., see Beton, P.H. 691 Moriarty, P., see Keeling, D.L. 691, 692, 751 Morikawa, Y., see Hammer, B. 293

Author index

Morin, S., see Allongue, P. 846 Morita, S., see Oyabu, N. 694 Moritz, W., see Fery, P. 357 Mormiche, C., see Hayden, B.E. 31 Moroz, I. 76, 77 Morrison, S.R., see Matsumura, M. 847 Mortensen, H. 82, 513, 514 Mortensen, H., see Diekhöner, L. 434, 463, 511–514 Mortensen, K., see Bedrossian, P. 735 Moryl, J., see Hodgson, A. 167, 168 Moses, P.G., see Hinnemann, B. 330, 331, 333 Motooka, T., see Suda, Y. 830, 835 Mott, N.F., see Cabrera, N. 275, 497 Mott, N.F., see Jackson, J.M. 55 Mottier, L., see Brouwer, A.M. 744 Moula, M.G. 240 Moula, M.G., see Beta, C. 413 Moulas, G., see Rusponi, S. 779 Mowrey, R.C., see Kroes, G.J. 146, 168, 179 Mowrey, R.C., see McCormack, D.A. 146, 159, 169, 175, 179, 180 Mowrey, R.C., see Watts, E. 161, 167, 169, 183, 184, 189 Mross, W.D. 398 Mrozek, I., see Otto, A. 469 Muehlhoff, L., see Bozso, F. 735 Muhler, M., see Over, H. 370 Muhlhausen, C.W. 85 Mühlig, A., see Bauer, A. 548 Muis, A. 57, 59 Mull, T. 646 Müller, B. 778 Muller, C.J., see Reed, M.A. 738 Muller, D.A., see Frank, M.M. 791 Müller, E.W. 380, 402, 404 Müller, F. 822 Müller, F., see Leonard, S.W. 822 Müller, K., see Heilmann, P. 357 Müller, P. 794, 795, 799, 800, 803, 821 Müller, R., see Gerlach, A. 539 Muller, R.S., see Williams, K.R. 850 Müller, V., see Bremten, H. 473 Müller-Hartmann, E. 4, 7, 8, 14, 107 Müller-Hartmann, T. 446 Mullins, C.B. 48, 49, 122, 833 Mullins, C.B., see Head-Gordon, M. 46, 122, 125–127 Mullins, C.B., see McClure, S.M. 31

961

Mullins, C.B., see Pacheco, K.A. 836 Mullins, C.B., see Rettner, C.T. 48, 49, 105, 125, 126 Mullins, C.B., see Sitz, G.O. 201 Mun, B.S. 293 Munakata, T. 558 Münder, H., see Frohnhoff, S. 849 Munford, M.L. 846 Munt, T.P. 797 Munt, T.P., see Shchukin, V.A. 804, 806 Murakami, K., see Makimura, T. 826, 845 Murata, Y., see Aruga, T. 475 Murata, Y., see Fukutani, K. 641, 643, 669 Murnane, M.M., see Huang, C.-P. 551 Murota, J., see Sakuraba, M. 830 Murphy, M., see Chen, J. 511, 512 Murphy, M.J. 174, 183, 184 Murray, C.B. 747 Murray, J. 345, 378 Murray, R., see Howe, P. 839, 841 Murray, R., see Le Ru, E.C. 839, 842 Murrell, C., see Lewerenz, H.J. 847 Murzin, D.Y. 261 Muscat, J.P. 284 Musenich, R., see Chiesa, M. 102, 103 Musgrave, C.B., see Kang, J.K. 832, 833 Musiani, M.M., see Cattarin, S. 849 Müssig, H.-J., see Dabrowski, ˛ J. 792 Muto, H., see Kobatake, S. 736 Myshlyavtsev, A.V. 256 Nachtigall, P. 829 Nagano, H., see Ogawa, S. 551 Nagano, H., see Petek, H. 539, 558, 565 Nagano, N., see Ogawa, S. 558, 559, 565 Nagard, M.B. 44 Nagasaka, M. 259 Nagasaka, M., see Nakai, I. 257 Nagasawa, Y., see Takahagi, T. 847 Nagasawa, Y., see Tanaka, T. 830 Nahidi, M. 792, 846 Nahidi, M., see Kolasinski, K.W. 846 Nahm, K.S., see Seo, Y.H. 847 Naitoh, Y., see Rosei, F. 739 Naitoh, Y., see Schunack, M. 739 Nakabayashi, S., see Misewich, J.A. 635 Nakai, I. 257 Nakajima, T., see Saalfrank, P. 650 Nakamoto, K. 76

962

Nakamura, H. 3 Nakamura, T., see Kanno, T. 747 Nakao, K. 240, 246 Nakayama, K.S. 844 Nakayama, N., see Watanabe, S. 847 Nakayama, T., see Sakurai, M. 701, 705, 706 Nakayama, T., see Thirstrup, C. 701 Nambu, A., see Nakai, I. 257 Namiki, A. 828, 829 Namiki, A., see Geuzebroeak, F.H. 210 Namiki, A., see Matsuno, T. 829 Namiki, A., see Rahman, F. 843 Namiki, A., see Sagara, T. 829 Nannarone, S., see Chiarotti, G. 528 Naschitzki, M., see Lim, Y.-S. 403 Nastasi, M. 892, 893 Nathel, H., see Huang, C.-P. 551 Naumovets, A.G. 254 Nazin, G.V. 687 Nazin, G.V., see Qiu, X.H. 687, 733 Nazin, G.V., see Wu, S.W. 747 Neave, J.H. 817, 818 Neave, J.H., see Williams, R.S. 842 Neddermeyer, H., see Heimann, P. 540 Nedeljkovic, Lj.D., see Nedeljkovic, N.N. 473, 474 Nedeljkovic, N.N. 473, 474 Nedelmann, L., see Müller, B. 778 Needels, M., see Payne, M.C. 797 Needs, R.J., see Payne, M.C. 797 Negulyaev, N.N., see Stepanyuk, V.S. 775 Nelson, J.S. 534 Nelson, J.S., see Feibelman, P.J. 765 Nelson, M.M., see Engstrom, J.R. 843, 844 Nessler, W. 660, 661 Nessler, W., see Bornscheuer, K.-H. 663 Nessler, W., see Kolasinski, K.W. 829 Nettesheim, S. 385, 386 Nettesheim, S., see Bär, M. 385 Nettesheim, S., see Hasselbrink, E. 644, 646 Nettesheim, S., see Wolf, M. 647 Neubrand, T., see Marbach, H. 398, 400 Neuendorf, R., see Koker, L. 847 Neufeld, M., see Falcke, M. 388, 393 Neumann-Spallart, M., see Hassan, H.H. 847 Neumark, D.M., see Manolopoulos, D.E. 510, 512 Neurock, M. 304 Neurock, M., see Hansen, E.W. 253

Author index

Neurock, M., see van Santen, R.A. 277 Newns, D.M. 19, 278, 283, 284, 447, 451 Newns, D.M., see Brako, R. 7, 8, 10, 12–14, 59, 63, 65, 73, 106, 148, 447, 448, 483 Newns, D.M., see Brandbyge, M. 19, 637 Newns, D.M., see Gumhalter, B. 15 Newns, D.M., see Hewson, A.C. 452 Newns, D.M., see Makoshi, K. 473 Newns, D.M., see Misewich, J.A. 636 Newns, D.M., see Muscat, J.P. 284 Newns, D.M., see Nørskov, J.K. 460, 464–466, 476, 479, 481, 482, 484, 488, 499, 509 Newns, D.M., see Walkup, R.E. 458, 616, 694, 696 Newns, D.W. 88 Newrock, M., see Pallassana, V. 301 Newton, T.A., see Flidr, J. 851 Nicolasen, G., see Tenner, M.G. 210 Nicolay, G. 540, 541 Nicolay, G., see Eiguren, A. 534, 535, 541, 542, 562 Nicolay, G., see Reinert, F. 540, 541, 543, 548, 560, 561 Nicolis, G. 345 Niebergall, L., see Stepanyuk, V.S. 775 Niedermayer, T. 135, 136 Niehaus, H., see Böttcher, A. 488 Niehus, H. 357 Nielsen, G., see Abild-Pedersen, F. 299 Nielsen, J.H., see Hinnemann, B. 330, 331, 333 Nielsen, J.H., see Jaramillo, T.F. 277, 330–333 Nielsen, O.H., see Hammer, B. 289, 290 Nielsen, V., see Lindhard, J. 875, 877, 879 Niemantsverdriet, J.W., see Chorkendorff, I. 277, 308 Niemantsverdriet, J.W., see Hermse, C.G.M. 254 Niemantsverdriet, J.W., see van Bavel, A.P. 251, 254 Niemantsverdriet, J.W., see van Santen, R.A. 233, 304, 307, 310, 311 Nieminen, R.M., see Echenique, P.M. 484 Nieminen, R.M., see Puska, M.J. 484 Nienhaus, H. 4, 15, 148, 240, 434, 435, 442, 455, 489–493, 509, 512, 515 Nienhaus, H., see Cuenya, B.R. 491 Nienhaus, H., see Gergen, B. 4, 15, 148, 334, 335, 455, 482, 489–492 Nienhaus, H., see Glass, S. 491, 492, 509

Author index

Nieto, F. 254 Nieto, F., see Tarasenko, A.A. 256 Nieto, P. 31, 455 Nieto, P., see Farías, D. 146, 154, 155, 191 Nieuwenhuys, B., see de Wolf, C.A. 377 Nieuwenhuys, B.E., see Carabineiro, S.A.C. 377 Nieuwenhuys, B.E., see Cobden, P.D. 374, 410 Nieuwenhuys, B.E., see de Wolf, C.A. 370, 377 Nieuwenhuys, B.E., see Imamura, K. 254 Nieuwenhuys, B.E., see Janssen, N.M.H. 370, 374, 385, 390 Nieuwenhuys, B.E., see Makeev, A.G. 374 Nieuwenhuys, B.E., see Peskov, N.V. 259 Nieuwenhuys, B.E., see van Tol, M.F.H. 403 Nihonyanagi, S., see Ye, S. 849 Niida, T., see Matsuno, T. 829 Niinistö, L. 791 Niki, H., see Ogata, Y. 847 Nikitin, E.E. 3, 8, 9, 243, 433 Nilekar, A.U., see Zhang, J.L. 291 Nilius, N., see Sterrer, M. 730 Nilius, N., see Watanabe, K. 470 Nilsson, A. 287 Nimiki, A., see Guezebroek, F.H. 162 Nishihara, K., see Anisimov, S.I. 904 Nishihara, K., see Zhakhovskii, V.V. 903 Nishijima, M., see Yoshinobu, J. 735 Nishinaga, T. 815, 837, 842 Nishinaga, T., see Shen, X.Q. 842 Nitzan, A. 738 Nitzan, A., see Galperin, M. 21 Nitzan, A., see Sawada, S.I. 6 Nix, R.M., see Wu, Y.M. 830, 834, 835 Noat, Y., see Smit, R.H.M. 592 Noguchi, N. 846 Nogueira, F., see Marques, M.A.L. 580 Noji, H. 744 Nolan, P.D., see McClure, S.M. 31 Nolte, S. 906, 907 Nordlander, P. 99, 447, 448, 480, 481, 502, 565 Nordlander, P., see Akpati, H.C. 746 Nordlander, P., see Avouris, Ph. 699, 701, 704–706 Nordlander, P., see Dutta, C.M. 480 Nordlander, P., see Johnson, P.D. 447 Nordlander, P., see Langreth, D.C. 447, 448, 480, 481, 500 Nordlander, P., see Shao, H. 502

963

Nordlund, K. 896 Nordlund, K., see Bringa, E.M. 896 Nordlund, K., see Süle, P. 893 Nørskov, J.K. 3, 8, 12, 14, 16, 247, 302, 303, 331–334, 433, 434, 441, 443–445, 447, 448, 450, 453, 459, 460, 463–467, 475, 476, 478, 479, 481–484, 488, 493, 499, 508, 509 Nørskov, J.K., see Abild-Pedersen, F. 295, 297, 299 Nørskov, J.K., see Andersson, M.P. 294, 316–318 Nørskov, J.K., see Andersson, S. 274 Nørskov, J.K., see Besenbacher, F. 296, 298 Nørskov, J.K., see Bligaard, T. 277, 278, 286, 289, 294, 307, 310–312, 314, 328, 330 Nørskov, J.K., see Bollinger, M.V. 330 Nørskov, J.K., see Brune, H. 775 Nørskov, J.K., see Christensen, A. 764 Nørskov, J.K., see Christensen, C.H. 234, 261 Nørskov, J.K., see Clausen, B.S. 799 Nørskov, J.K., see Dahl, S. 296–298, 300, 326, 327, 330, 781 Nørskov, J.K., see Díaz, C. 471, 513, 514 Nørskov, J.K., see Gadzuk, J.W. 6, 7 Nørskov, J.K., see Greeley, J. 277 Nørskov, J.K., see Gunnarsson, O. 444 Nørskov, J.K., see Hammer, B. 150, 154, 168, 189, 277–281, 283, 286–290, 320, 328, 450, 452, 453, 498 Nørskov, J.K., see Hellman, A. 261, 274, 296, 304, 309, 325, 327, 328 Nørskov, J.K., see Helveg, S. 330, 331, 812 Nørskov, J.K., see Hinnemann, B. 330, 331, 333 Nørskov, J.K., see Hjelmberg, H. 443, 450 Nørskov, J.K., see Honkala, K. 263, 274, 277, 296, 301, 304, 308, 325–330 Nørskov, J.K., see Jacobsen, C.J.H. 294, 312 Nørskov, J.K., see Jacobsen, K.W. 278, 282 Nørskov, J.K., see Kasemo, B. 149, 463, 466, 476, 478, 482, 488, 509 Nørskov, J.K., see Kitchin, J.R. 292 Nørskov, J.K., see Lang, N.D. 7, 12, 17, 18, 476 Nørskov, J.K., see Lauritsen, J.V. 331 Nørskov, J.K., see Liu, P. 294 Nørskov, J.K., see Logadottir, A. 299, 301, 326, 330

964

Nørskov, J.K., see Lundqvist, B.I. 3, 14, 274, 278, 279, 282, 443, 450, 463 Nørskov, J.K., see Mavrikakis, M. 291, 302 Nørskov, J.K., see Meier, J. 292 Nørskov, J.K., see Nilsson, A. 287 Nørskov, J.K., see Ovesen, C.V. 314 Nørskov, J.K., see Ruban, A. 287, 293 Nørskov, J.K., see Sehested, J. 317, 318 Nørskov, J.K., see Skúlason, E. 333 Nørskov, J.K., see Stoltze, P. 311, 326 Nørskov, J.K., see Strasser, P. 293 Nørskov, J.K., see Tripa, C.E. 290 Nørskov, J.K., see Vang, R.T. 298, 300, 781 Northrup, F.J., see Hepburn, J.W. 433 Norton, P.R., see Ertl, G. 357 Norton, P.R., see Thiel, P.A. 357, 358 Nourtier, A. 19 Nourtier, A., see Blandin, A. 8, 19, 447, 448, 459 Novicki, M., see Sterrer, M. 730 Nozières, P. 15, 792, 800 Nozières, P., see Pines, D. 899 Nozières, P., see Wolf, D.E. 792 Nuvolone, R., see Boiziau, C. 473 Nyberg, G.L. 767 Nyberg, M., see Lauritsen, J.V. 331 O’Brien, S., see Somashekhar, A. 847 Ochs, R., see Reichert, J. 738 O’Connell, R.F., see Li, X.L. 6, 8, 19 Oechsner, H., see Gerhard, W. 891 Oehrlein, G.S. 827 Oellig, E.M., see Ciraci, S. 829 Offenberg, M., see Liehr, M. 834 Offergeld, G., see Roose, R.F. 465 Offermans, W.K., see Jansen, A.P.J. 252 Ogasawara, H., see Katano, S. 735 Ogata, Y. 847 Ogata, Y.H. 849 Ogawa, H. 849 Ogawa, N., see Mikaelian, G. 734 Ogawa, N., see Wu, S.W. 714, 717, 746 Ogawa, S. 551, 558, 559, 565 Ogawa, S., see Petek, H. 539, 553, 558, 565 Ogg, R.A. 487 Ogitsu, T., see Aizawa, H. 844 Ohlsson, B.J., see Jensen, L.E. 812 Ohlsson, B.J., see Persson, A.I. 810, 812 Ohno, Y., see Imamura, K. 254

Author index

Ohno, Y., see Kikkawa, J. 810 Ohno, Y., see Moula, M.G. 240 Ohno, Y., see Sano, M. 672 Ohono, T., see Sasaki, T. 498 Ohta, T., see Matsui, F. 735 Ohta, T., see Nagasaka, M. 259 Ohta, T., see Nakai, I. 257 Ohtaka, K. 15 Ohtani, N., see Mokler, S.M. 831 Okawa, T., see Sainoo, Y. 601, 617, 711, 712 Okiji, A., see Diño, W.A. 145 Okiji, A., see Kasai, H. 473, 474 Olander, D.R. 830 Olander, D.R., see Farnaam, M.K. 833 Oldenburg, R.C. 460 Olin, H., see Wahlström, E. 772 Oliva, A., see Sigmund, P. 889 Olsen, R.A. 147, 306 Olsen, R.A., see Busnengo, H.F. 146, 186 Olsen, R.A., see Díaz, C. 471, 513, 514 Olsen, R.A., see Hellman, A. 304, 325, 327, 328 Olsen, R.A., see Løvvik, O.M. 292 Olsen, R.A., see McCormack, D.A. 146, 159, 169, 175, 179, 180 Olsen, R.A., see Nieto, P. 31 Olsen, R.A., see Pijper, E. 146 Olsen, R.A., see Somers, M.F. 161, 162, 169, 184, 189 Olsen, R.A., see Watts, E. 161, 167, 169, 183, 184, 189 Olsson, F.E. 275, 549, 610, 732, 733 Olsson, F.E., see Karlberg, G.S. 257 Olsson, F.E., see Repp, J. 688, 700, 731, 732 Olsson, L. 261, 323 Olthoff, J.K. 630 O’Malley, T.F. 3, 20, 433, 435, 436, 438, 441 Onchi, M., see Yoshinobu, J. 735 Ondarcuhu, T., see Saifullah, M.S.M. 739 Oner, D.E. 498 Onishi, H., see Nakao, K. 240 Ono, S., see Sakuraba, M. 830 Ono, T., see Esashi, M. 851 Oosterkamp, T.H., see Frenken, J.W.M. 717 Oparin, A., see Sokolowski-Tinten, K. 824 Oparin, A.M., see Anisimov, S.I. 904 Oppenheimer, R., see Born, M. 433 Oral, A., see Fishlock, T.W. 691 Organ, L., see Punckt, C. 821

Author index

Orlando, T.M. 718 Osgood, A.J., see Shirai, Y. 751 Osgood Jr., R.M. 529 Osgood Jr., R.M., see Shen, X.J. 564 Osma, J., see Chulkov, E.V. 564 Osma, J., see Echenique, P.M. 529, 562, 564 Osma, J., see Sarría, I. 564 Ossicini, S., see Bisi, O. 845 Oster, L. 466 Österlund, L. 72, 73, 149, 245, 246, 334, 455, 495, 496, 498, 507 Österlund, L., see Chakarov, D. 482 Ostwald, W. 775, 804 Otero, R. 4 Otto, A. 469 Ottow, S. 822 Ottow, S., see Müller, F. 822 Ouesen, C.V., see Clausen, B.S. 799 Oughaddou, H., see Papageorgiou, N. 733 Over, H. 370 Overbosch, E.G., see Hurkmans, A. 48 Ovesen, C.V. 314 Ovesson, S. 304, 319–325, 515, 771, 775 Ovesson, S., see Bogicevic, A. 320, 775 Owen, J.H.G., see Self, K.W. 735 Oyabu, N. 694 Ozaki, H., see Harada, Y. 456, 457, 472, 474, 475 Ozanam, F. 847 Ozanam, F., see Belaïdi, A. 847 Ozanam, F., see Cattarin, S. 845, 847 Ozanam, F., see Chazalviel, J.-N. 847 Ozanam, F., see Dubin, V.M. 847 Ozanam, F., see Hassan, H.H. 847 Ozanam, F., see Rao, A.V. 847 Ozeki, K., see Yagu, S. 76 Paal, Z. 325 Paavilainen, S., see Olsson, F.E. 275, 610, 732 Paavilainen, S., see Repp, J. 732 Pacchioni, G., see Sterrer, M. 730 Pacheco, K.A. 836 Pacheco, K.A., see Mullins, C.B. 833 Pachioni, G., see Yulikov, M. 275 Pagano, S.A.S.P., see Bressers, P.M.M.C. 850 Paggel, J.J., see Luh, D.-A. 538 Pakkanen, T.A., see Hirva, P. 830 Pal, A.K., see Bera, S.K. 845 Pala, R.G.S. 815

965

Pala, S., see Ganesh, R. 275 Pallassana, V. 301 Palmer, C., see Kalkan, A.K. 828 Palmer, R., see Kern, K. 101 Palmer, R.E. 19, 20, 700, 704, 720 Palmer, R.E., see Riedel, D. 827 Palmer, R.E., see Salam, G.P. 616, 706, 710 Palmer, R.E., see Sloan, P.A. 687, 720, 724 Paltauf, G. 904 Pan, S., see Zhao, A. 728 Pan, X., see Zimmermann, F.M. 829, 830 Pan, X.Q., see Schlom, D.G. 791 Panczyk, T. 249 Panczyk, T., see Rudzinski, W. 249 Pandelov, S., see Nørskov, J.K. 331–333 Paniago, R. 540 Paolucci, F., see Brouwer, A.M. 744 Papageorgiou, N. 733 Papageorgopoulos, D.C., see Beck, R.D. 181 Papageorgopoulos, D.C., see Berenbak, B. 42, 43 Papageorgopoulos, D.C., see Maroni, P. 181 Papageorgopoulos, D.C., see Riedmüller, B. 45 Papathanasiou, A.G., see Wolff, J. 419, 420, 821 Paramonov, G.K. 20 Parenteau, L., see Huels, M.A. 700 Pareto, V. 317, 318 Parhikhteh, H., see Gostein, M. 167 Parilis, E.S., see Bitensky, I.S. 894 Park, J.-G., see Kwon, S.J. 810 Park, J.Y. 491, 492 Park, J.Y., see Somorjai, G.A. 492 Park, Y.O., see Banholzer, W.F. 371 Parkhutik, V. 823 Parrinello, M., see Ancilotto, F. 829 Parrinello, M., see Car, R. 499 Parry, G., see Zhang, J. 828 Parvulescu, V.I. 319 Pascual, J.I. 592, 595, 597, 615, 616, 688, 723 Pascual, J.I., see Lorente, N. 595, 596, 616, 617 Paserba, K.R. 236 Paserba, K.R., see Gellman, A.J. 236 Pasquier, N., see Skalicky, T. 720 Passardi, G., see Benvenuti, C. 130 Pasteur, A.T. 357 Pasteur, A.T., see Gruyters, M. 374 Patchett, A.J., see Meissen, F. 396 Patitsas, S.N. 726

966

Patitsas, S.N., see Alavi, S. 616 Patriarche, G., see Harmand, J.C. 812 Patrick, D.L. 822 Patrykiejew, A. 246 ´ Patthey, F., see Cavar, E. 687, 688, 746 Patthey, F., see Pivetta, M. 540, 560 Patthey, F., see Silly, F. 770, 772–774 Paul, D.J. 796, 828 Paulsson, M. 585, 597 Pausson, M., see Frederiksen, T. 615 Pautrat, M., see Bouneau, S. 895, 897 Pavesi, L. 845 Pavesi, L., see Bisi, O. 845 Pavlenko, N. 407 Pawlik, S., see Aeschlimann, M. 899 Pawlik, S., see Bauer, M. 558, 565 Payne, M., see White, J.A. 154, 168 Payne, M.C. 797 Payne, M.C., see Ciacchi, L.C. 498, 499, 503 Payne, M.C., see White, J.A. 498 Payne, S.H., see Brenig, W. 829, 830 Payne, S.H., see Li, W.X. 320 Payne, S.H., see Niedermayer, T. 135, 136 Payne, S.H., see Stampfl, C. 252, 253, 320 Paz, O. 590 Pearlstine, K.A. 130, 132 Pearlstine, K.A., see Ferm, P.M. 130 Pearson, C. 776 Pearson, G.L., see Shockley, W. 528 Pearson, R.G. 3, 274 Pedersen, M., see Perdew, J. 433 Pedersen, M.O., see Österlund, L. 246 Pedraza, A.J. 825, 827 Pegg, D.J., see Davies, J.C. 293 Pehlke, E., see Bird, D.M. 448 Pehlke, E., see Dabrowski, ˛ J. 829 Pehlke, E., see Limot, L. 549, 733 Pehlke, E., see Lindenblatt, M. 448 Peierls, R.E. 433 Peiner, E. 845 Pelak, R.A., see Busch, D.G. 635 Pelz, J.P., see Pivetta, M. 540, 560 Pelz, J.P., see Silly, F. 770, 772–774 Pelzel, R.I., see Self, K.W. 735 Pemble, M.E., see Turner, A.R. 828 Pena, F., see Colen, R.E.R. 368 Peñabla, M., see Arnau, A. 467, 468 Pendry, J.B., see Echenique, P.M. 529 Penev, E. 841

Author index

Penev, E., see Kratzer, P. 841 Penn, D.R. 509 Penno, M., see Wight, A.C. 225 Perdew, J. 433 Perdew, J.P. 277, 320, 328, 433 Péré-Laperne, N., see Harmand, J.C. 812 Perel’man, T.L., see Anisimov, S.I. 899 Peremans, A., see Fukutani, K. 643, 669 Perera, A. 373, 386 Perera, A., see Chavez, F. 362 Perez, D. 903 Perez, J. 331 Perez-Murano, F., see Stokbro, K. 701, 705, 706 Pério, A., see Dutartre, D. 828 Perrau, J. 102 Persson, A.I. 810, 812 Persson, A.I., see Jensen, L.E. 812 Persson, B.N.J. 19, 129, 148, 189, 469, 471, 479, 483–486, 615, 616, 746 Persson, B.N.J., see Andersson, S. 135 Persson, B.N.J., see Bonn, M. 483 Persson, B.N.J., see Komeda, T. 724 Persson, B.N.J., see Ueba, H. 4, 22, 617, 713 Persson, H., see Olsson, L. 323 Persson, M. 13, 105, 106, 108, 111, 115, 117, 135, 461, 462, 466, 471, 483–486 Persson, M., see Andersson, S. 100, 102–105, 113, 114, 120, 124, 135, 150, 157, 163, 462 Persson, M., see Andersson, T. 104 Persson, M., see Bird, D.M. 148, 189 Persson, M., see Forsblom, M. 486 Persson, M., see Gao, S. 20, 458, 461, 616, 718 Persson, M., see Hassel, M. 130, 135 Persson, M., see Hellsing, B. 19, 148, 189, 464, 484–486 Persson, M., see Hyldgaard, P. 772 Persson, M., see Jackson, B. 110 Persson, M., see Karlberg, G.S. 257 Persson, M., see Kulginov, D. 98, 122 Persson, M., see Langreth, D.C. 483, 485 Persson, M., see Lorente, N. 485, 595–598, 605, 613, 710 Persson, M., see Luntz, A.C. 463, 471, 485, 493, 496, 514, 675 Persson, M., see Mizielinski, M.S. 148, 189, 448, 493, 611 Persson, M., see Olsson, F.E. 275, 549, 610, 732, 733

Author index

Persson, M., see Persson, B.N.J. 19, 148, 189, 469, 471, 479, 483–486 Persson, M., see Repp, J. 688, 700, 731, 732, 772, 773 Persson, M., see Salam, G.P. 616, 706, 710 Persson, M., see Shalashilin, D.V. 239 Persson, M., see Sloan, P.A. 720 Persson, M., see Stiles, M.D. 123 Persson, M., see Stipe, B.C. 458, 462, 463, 719 Persson, M., see Teillet-Billy, D. 722, 723 Persson, M., see Trail, J.R. 8, 15, 19, 148, 189, 493 Persson, M., see Wilzén, L. 103, 163, 462 Perunin, V.V., see Mortensen, H. 82 Pesce, L., see Saalfrank, P. 658 Pescia, D., see Weber, W. 781 Peskov, N.V. 259 Petek, H. 539, 553, 558, 565 Petek, H., see Ogawa, S. 551, 558, 559, 565 Peter, L.M. 847 Peter, L.M., see Cattarin, S. 845, 847 Peter, L.M., see Stumper, J. 847 Peters, M.V., see Alemani, M. 736 Pethica, J.B., see Fishlock, T.W. 691 Peticolas, L.J., see Hull, R. 798 Petroff, Y., see Smith, N.V. 538 Petrov, V. 412, 419 Petrov, Y.V., see Anisimov, S.I. 904 Petrova, N.V. 254 Petrovykh, D.Y., see Himpsel, F.J. 817 Petrunin, V., see Diekhöner, L. 434, 463, 511–514 Petrunin, V., see Mortensen, H. 513, 514 Pettersson, J.B.C., see Nagard, M.B. 44 Pettersson, J.B.C., see Tomsic, A. 44 Pettersson, L.G.M., see Nilsson, A. 287 Pettinger, B., see Flätgen, G. 821 Pfandzelter, R. 41 Pfeiffer, W., see Gerstner, V. 715 Pfeiffer, W., see Lehmann, J. 564 Pfnür, H., see Li, W.X. 320 Pfnür, H., see Stampfl, C. 252, 253, 320 Pforte, F., see Straube, P. 542, 560 Philippe, L., see Jenniskens, H.G. 629 Phillips, J.A.C., see Wei, T.C. 363 Phipps, C.R. 908 Pickel, M., see Roth, M. 555, 556, 564 Pickering, C., see Canham, L.T. 847 Piercy, P., see Hussla, I. 660

967

Pijper, E. 146 Pijper, E., see Busnengo, H.F. 146, 151, 186 Pijper, E., see Nieto, P. 31, 455 Pineda, M. 408, 409 Pines, D. 899 Pinhas, S., see Büttiker, M. 584 Pirug, G., see Freyer, N. 357 Pismen, L., see Monine, M. 393 Pismen, L., see Monine, M.I. 363, 365 Pismen, L.M. 345, 351, 382, 385, 409 Pismen, L.M., see Monine, M. 363 Pitarke, J.M., see Campillo, I. 539 Pitarke, J.M., see Chulkov, E.V. 564 Pitarke, J.M., see Echenique, P.M. 529, 530, 562, 564 Pitarke, J.M., see García-Lekue, A. 531, 532, 563, 564 Pitarke, J.M., see Gerlach, A. 539 Pitarke, J.M., see Sarría, I. 564 Pitters, J.L., see Piva, P.G. 696, 730 Piva, P.G. 696, 730 Pivetta, M. 540, 560 ´ Pivetta, M., see Cavar, E. 687, 688, 746 Pivetta, M., see Silly, F. 770, 772–774 Pletcher, T.D., see Lauterbach, J. 368 Plihal, M. 19 Plihal, M., see Gadzuk, J.W. 22 Plummer, E.W. 528 Plummer, E.W., see Bartynski, R.A. 543, 560 Plummer, E.W., see Heskett, D. 300 Plummer, E.W., see Junren, S. 545 Plummer, E.W., see Levinson, H.J. 560 Plummer, E.W., see Tang, S.-J. Ismail 544, 560 Poelsema, B., see Rosenfeld, G. 817, 819 Pohl, D.W., see Dürig, U. 689 Poizat, J.C., see Bouneau, S. 895, 897 Polanyi, J.C., see Carrington, T. 460 Polanyi, J.C., see Dixon-Warren, St.J. 630, 631, 720 Polanyi, J.C., see Dobrin, S. 696 Polanyi, J.C., see Harikumar, K.R. 696 Polanyi, J.C., see Hepburn, J.W. 433 Polanyi, J.C., see Jiang, G. 715 Polanyi, J.C., see Lu, P.H. 719 Polanyi, M. 464, 476 Polanyi, M., see Evans, M.G. 299, 308, 311 Polanyi, M., see Eyring, H. 435 Polanyi, M., see Ogg, R.A. 487 Pollak, E. 237

968

Pollak, E., see Guantes, R. 254 Pollak, E., see Shushin, A.I. 254 Pollard, I.E., see Chubb, J.N. 130 Pollmann, M. 416, 417 Pollmann, M., see Bertram, M. 413 Pollmann, M., see Kim, M. 413, 414 Pollmann, M., see Li, X.J. 415, 417 Pollmann, M., see Rotermund, H.H. 390 Pollock, H.M., see Singer, I.L. 486 Polyakov, A.Y., see Bublik, V.T. 795 Ponec, V., see Toolenaar, F.J.C.M. 276 Pons, S., see Peter, L.M. 847 Poojar, D.M., see Cong, P.J. 313 Poon, G.C. 459, 466, 475, 477, 478, 482, 488 Popova, I., see Zhukov, V. 73 Poppa, H., see Bauer, E. 764 Poppe, A., see Xu, L. 551 Powers, J. 57, 59 Poydenot, V., see Dujardin, R. 809 Preece, J.A., see Mendes, P.M. 851 Press, W., see Bahr, D. 797 Pressley, L.A. 628, 629 Preston, J.S., see Sipe, J.E. 827 Preston, J.S., see Young, J.F. 827 Preston, R.K., see Tully, J.C. 10 Preuss, S. 825, 826, 906, 907 Priester, C. 803, 804, 841 Prince, R.H. 465, 477 Prince, R.H., see Bourdon, E.B.D. 465, 477 Prins, R. 331 Priolo, F., see Pavesi, L. 845 Proctor, M.R.E., see Hoyle, R.B. 259, 361 Proctor, M.R.E., see Irurzun, I.M. 362 Propst, E.K. 845, 847 Provenier, F., see Dutartre, D. 828 Prybyla, J.A. 634 Pshenichnikov, M.S., see Baltuška, A. 551 Pullman, D.P., see Li, Y.L. 482, 483, 493 Pullman, D.P., see Tate, M.R. 482, 483 Punckt, C. 412, 821 Pupo, M., see Vattuone, L. 172 Purdie, D. 538, 539 Purdie, D., see Hengsberger, M. 542–544 Puska, M.J. 484 Puska, M.J., see Jacobsen, K.W. 278, 282 Puzder, A., see Thonhauser, T. 500 Pykavy, M. 632 Pykavy, M., see Mehdaoui, I. 632 Pylant, E.D., see Pressley, L.A. 628, 629

Author index

Qian, F., see Wang, D. 808 Qian, J. 11, 12 Qiu, X.H. 687, 733 Qiu, X.H., see Nazin, G.V. 687 Qiu, X.H., see Wu, S.W. 747 Quaade, U., see Stokbro, K. 701, 705, 706 Quate, C.F., see Bryant, A. 701 Queeney, K.T. 849 Quinay, E.B., see Sano, M. 672 Quinn, J.J. 530, 531 Quyang, Q., see Petrov, V. 412, 419 Rabalais, J.W., see Kutana, A. 843 Radelaar, S., see Werner, K. 835 Radojevic, A.M., see Shen, X.J. 564 Rafti, M. 374 Rafti, M., see Uecker, H. 374 Raghavachari, K., see Burrows, V.A. 847 Raghavachari, K., see Chabal, Y.J. 828, 830, 847 Raghavachari, K., see Higashi, G.S. 847 Raghavachari, K., see Jakob, P. 846 Raghavachari, K., see Queeney, K.T. 849 Raghavachari, K., see Trucks, G.W. 846 Rahman, F. 843 Rahman, T.S. 534 Raizer, Y.P., see Zel’dovich, Y.B. 905 Rajagopal, A., see Yilmaz, M.B. 828–830 Ramakrishhnan, V., see Müller-Hartmann, T. 446 Ramakrishnan, T.V., see Müller-Hartmann, E. 4, 7, 8, 14, 107 Ramseyer, C., see Marinica, D.C. 20, 563 Ramsier, R.D. 468 Ran, Q. 471, 513 Rao, A.V. 847 Rao, B.K., see Jena, P. 778 Rapenne, G., see Grill, L. 740, 751 Rapenne, G., see Jimenez-Bueno, G. 750 Rappe, A.M., see Lewis, S.P. 615 Rappe, A.M., see Mason, S.E. 285 Rappich, J. 845, 847, 850 Rappich, J., see Xia, X. 849, 850 Raschke, M.B. 828, 842 Raschke, M.B., see Dürr, M. 829 Ra¸seev, G. 723 Ra¸seev, G., see Gauyacq, J.P. 558, 565 Rasmussen, P.B., see Lauterbach, J. 417

Author index

Rasser, B. 467 Rastomjee, C.S., see Rausenberger, B. 384 Rastomjee, C.S., see Swiech, W. 383 Rastomjee, C.S., see Tammaro, M. 383 Ratner, M., see Aviram, A. 738 Ratner, M.A., see Galperin, M. 21 Ratner, M.A., see Joachim, C. 21, 738 Ratner, M.A., see Long, D.P. 592 Ratsch, C. 321 Ratsch, C., see Ruggerone, P. 320 Raukema, A. 38 Raukerma, A. 245 Rauscher, H. 828 Rauscher, H., see Himpsel, F.J. 817 Rausenberger, B. 384 Rausenberger, B., see Swiech, W. 380 Raval, R., see Gong, X.-Q. 367 Ravindra, N.M., see Fiory, A.T. 845 Rawlett, A.M., see Chen, J. 735 Ray, D.S., see Banerjee, D. 237 Rayez, J.C., see Busnengo, H.F. 172, 182, 186 Raymond, T.D., see Kay, B.D. 223, 225, 662 Razaznejad, B., see Hellman, A. 245, 334, 433, 434, 463, 475, 476, 480, 496, 499–504, 506–508, 510 Razaznejad, B., see Komrowski, A.J. 506, 507 Razaznejad, B., see Ternow, H. 507 Razaznejad, B., see Yourdshahyan, Y. 149, 498–500 Rebentrost, F., see Berthold, W. 563 Rebitzki, T., see Ehsasi, M. 368, 369 Reboul, G. 466 Redhead, P.A. 468, 623, 699, 723 Redinger, J., see Hofer, W.A. 589 Reed, M.A. 738 Reed, M.A., see Chen, J. 735 Reed, M.A., see Wang, W. 592, 600 Rehmus, P. 411 Rehn, L.E. 890 Reichert, C. 384, 408, 409 Reichert, C., see Starke, J. 409 Reichert, J. 738 Reichman, D.R., see Maragakis, P. 306 Reichman, M.I., see McClure, S.M. 31 Reider, G.A. 829 Reifenberger, R., see Xue, Y. 735 Reihl, B., see Binnig, G. 551 Reimann, C.T. 891, 893, 894 Reinert, F. 540, 541, 543, 548, 560, 561

969

Reinert, F., see Eiguren, A. 534, 535, 541, 542, 562 Reinert, F., see Nicolay, G. 540, 541 Reinhard, P.-G., see Boger, K. 555 Reisler, H., see Korolik, M. 220 Rekoske, J.E., see Dooling, D.J. 261 Rekoske, J.E., see Dumesic, J.A. 233, 305, 310 Remediakis, I.N., see Hellman, A. 261, 274, 296, 304, 309, 325, 327, 328 Remediakis, I.N., see Honkala, K. 263, 274, 277, 296, 301, 304, 308, 325–330 Rendulic, K.D. 183 Renken, A., see Thullie, J. 411 Rennagel, H.G., see McClelland, G.M. 207, 208 Renner, R.L. 487 Renzas, J.R., see Park, J.Y. 492 Repp, J. 688, 700, 731, 732, 772, 773 Repp, J., see Liljeroth, P. 734 Repp, J., see Olsson, F.E. 275, 610, 732 Ressler, T. 366 Restelli, P., see Vattuone, L. 172 Rethfeld, B. 824, 899, 903 Rethfeld, B., see Anisimov, S.I. 900 Rettner, C.T. 6, 31, 48, 49, 56, 82–84, 87, 88, 105, 125, 126, 148, 164, 167, 172, 175, 177, 183, 189, 190, 212, 220, 221, 455, 459 Rettner, C.T., see Berenbak, B. 45, 209 Rettner, C.T., see Gulding, S.J. 177 Rettner, C.T., see Head-Gordon, M. 46, 122, 125–127 Rettner, C.T., see Hou, H. 178, 222, 334, 433, 512 Rettner, C.T., see Huang, Y. 148, 189, 190, 221, 222 Rettner, C.T., see Huang, Y.H. 334, 433, 463, 511, 512 Rettner, C.T., see Kimman, J. 85, 97, 212 Rettner, C.T., see Kulginov, D. 98, 122 Rettner, C.T., see Michelsen, H.A. 172, 175, 179 Rettner, C.T., see Mullins, C.B. 48, 49, 122 Rettner, C.T., see Schweizer, E.K. 38 Rettner, C.T., see Winters, H.F. 49, 50 Reuff, J.E. 469 Reuß, Ch. 553, 556, 557 Reuß, Ch., see Fauster, Th. 556, 557 Reuß, Ch., see Höfer, U. 555, 564 Reuß, Ch., see Shumay, I.L. 551, 555, 563, 564

970

Reuß, Ch., see Weinelt, M. 556, 557 Reuter, K. 304, 308, 319, 324, 325, 366, 370, 515 Reuter, K., see Behler, J. 149, 245, 334, 433, 496, 499–501, 507, 508, 510 Reuter, K., see Ganduglia-Pirovano, M.V. 395 Reuter, M.C., see Copel, M. 803 Reuter, M.C., see Horn-von Hoegen, M. 798, 803 Reuter, M.C., see Kodambaka, S. 810 Reuter, M.C., see Ross, F.M. 810 Reynolds, A.E., see Imbihl, R. 363, 396 Rezaei, M.A., see Stipe, B.C. 321, 458, 462, 463, 592, 594, 687, 709, 710, 719, 722 Rezeq, M., see Piva, P.G. 696, 730 Rhallabi, A., see Blauw, M.A. 827 Rhie, H.-S. 563 Rhodin, T.N. 509 Ribeiro, F.H., see Jacobs, P.W. 414 Rice, S.A., see Tannor, D.J. 6, 20 Richardson, C., see Marston, J.B. 7, 12 Richardson, H.H., see Magtoto, N.P. 371 Richardson, N. 433, 449, 451 Richter, L.J. 641 Richter, L.J., see Buntin, S.A. 641, 646, 669 Richter, L.J., see Gadzuk, J.W. 468, 641, 642, 650, 659 Richter, L.J., see Struck, L.M. 483 Riedel, D. 715, 827 Riedel, D., see Comtet, G. 688 Riedel, D., see Lastapis, M. 602, 688, 744 Riedel, D., see Martín, M. 746 Riedel, D., see Mayne, A.J. 4, 22, 688, 699, 726 Rieder, K.-H., see Alemani, M. 736 Rieder, K.-H., see Bartels, L. 4, 558, 686–691, 699, 724, 725 Rieder, K.-H., see Braun, K.-F. 549, 550, 561, 562 Rieder, K.-H., see Chiavaralloti, F. 748 Rieder, K.-H., see Feenstra, R.M. 696, 746 Rieder, K.-H., see Grill, L. 740, 750, 751 Rieder, K.-H., see Gross, L. 740, 748, 751 Rieder, K.-H., see Henzl, J. 736 Rieder, K.-H., see Hla, S.-W. 686, 692 Rieder, K.-H., see Hla, W. 4, 22 Rieder, K.-H., see Meyer, G. 689, 690 Rieder, K.-H., see Moresco, F. 686, 740, 741, 751 Rieder, K.-H., see Repp, J. 700, 732, 772, 773

Author index

Riedmüller, B. 45 Riedmüller, B., see Berenbak, B. 42, 43, 45 Riedmuller, B., see Berenbak, B. 209 Riess, H.J., see Giesen, K. 553 Riffe, D.M., see Wang, X.Y. 900 Risse, T., see Sterrer, M. 730 Ritchie, R., see Arnau, A. 467, 468 Ritchie, R.H. 530 Ritchie, R.H., see Echenique, P.M. 484 Riveau, B., see Dahan, M. 747 Rivière, P., see Farías, D. 146, 154, 155, 191 Rizk, M., see Ward, C.A. 249 Rizk, R., see Botti, S. 845 Rizzo, T.R., see Beck, R.D. 181 Rizzo, T.R., see Maroni, P. 181 Rizzo, T.R., see Schmid, M.P. 181 Robert, O., see Dujardin, G. 686, 693, 728 Roberts, J.K. 55 Roberts, N., see Payne, M.C. 797 Robota, H., see Modl, A. 210 Robota, H., see Segner, J. 208, 209 Rocca, M., see Luntz, A.C. 102 Rocca, M., see Savio, L. 158, 159 Rocca, M., see Vattuone, L. 31, 150, 172 Rockett, A. 831 Rod, T.H., see Logadottir, A. 299, 301, 326, 330 Rodberg, L.S. 57, 62 Röder, H. 775, 778, 779, 815 Röder, H., see Bromann, K. 778, 783 Röder, H., see Brune, H. 775, 780 Rodriguez, J.A. 293 Roffia, S., see Brouwer, A.M. 744 Rogers, D., see Jiang, G. 715 Rogers, D.J., see Lu, P.H. 719 Rogge, S. 695 Rohrer, H., see Binnig, G. 551, 579, 593, 685, 701 Roloff, H.F., see Heimann, P. 540 Romero, A.H., see Sancho, J.M. 254 Romm, L. 237, 513 Rönnebeck, S., see Lehmann, V. 822 Roop, B., see Costello, S.A. 623 Roose, R.F. 465 Rösch, N., see Gokhale, S. 735 Rose, F., see Dujardin, G. 602, 686, 693, 701, 728, 729 Rose, F., see Mayne, A.J. 602, 694, 701, 703, 728

Author index

Rose, H. 362 Rose, K.C. 388, 396, 397 Rosei, F. 739, 748, 828 Rosei, F., see Otero, R. 4 Rosei, F., see Schunack, M. 739 Rosén, A., see Ding, F. 810, 813 Rosenfeld, G. 817, 819 Rosenfeld, G., see Morgenstern, K. 776, 777, 804 Ross, F.M. 810 Ross, F.M., see Hannon, J.B. 808, 810, 811 Ross, F.M., see Kodambaka, S. 810 Ross, J., see Eiswirth, M. 352 Ross, J., see Metiu, H. 3 Ross, J., see Rehmus, P. 411 Ross, J., see Tsarouhas, G.E. 412 Ross, J., see Vance, W. 411, 412 Ross, P.N., see Markovic, N.M. 291, 333 Ross, P.N., see Mun, B.S. 293 Rossi, A.R., see Avouris, Ph. 602, 616, 699, 701, 704–706 Rossi, S.D., see Crampin, S. 529 Rossmeisl, J., see Hellman, A. 304, 325, 327, 328 Rossmeisl, J., see Skúlason, E. 333 Rost, M.J., see Frenken, J.W.M. 717 Rostaing, P., see Dahan, M. 747 Rostrup-Nielsen, J.R., see Helveg, S. 812 Rotermund, H.H. 259, 345, 349, 379, 381, 390, 392, 409, 821 Rotermund, H.H., see Bär, M. 385, 390, 392, 415 Rotermund, H.H., see Engel, W. 379 Rotermund, H.H., see Gorodetskii, V.V. 397, 404 Rotermund, H.H., see Jakubith, S. 381, 385, 387 Rotermund, H.H., see Lauterbach, J. 390 Rotermund, H.H., see Nettesheim, S. 385, 386 Rotermund, H.H., see Pollmann, M. 416, 417 Rotermund, H.H., see Punckt, C. 412, 821 Rotermund, H.H., see von Oertzen, A. 368, 390, 393, 394 Rotermund, H.H., see Wolff, J. 419, 821 Roth, J., see Eckstein, W. 886 Roth, J., see Garcia-Rosales, C. 886 Roth, M. 555, 556, 564 Roth, M., see Boger, K. 555 Roth, R., see Klein, D.L. 747

971

Rottler, J. 783 Rottler, J., see Heinrichs, S. 783 Rottman, C. 799 Roudgar, A. 290, 292 Rous, P.J. 20 Rous, P.J., see Palmer, R.E. 19, 20, 700, 704 Rousseau, R., see Alavi, S. 616 Roussel, J. 473, 509 Roussel, J., see Boiziau, C. 473 Ruban, A. 287, 293 Ruban, A.V., see Christensen, A. 764 Rubinstein, B.Y., see Monine, M.I. 390 Rubinstein, B.Y., see Pismen, L.M. 382, 385 Rubio, A., see Campillo, I. 539 Rubio, A., see Dubois, M. 745 Rubio, A., see Echenique, P.M. 530 Rubio, A., see Gerlach, A. 539 Rubio, A., see Marques, M.A.L. 580 Rubio, A., see Silkin, V.M. 542, 543, 560 Rubio-Bollinger, G., see Agrait, N. 591 Rudd, D.F., see Dumesic, J.A. 233, 305, 310 Rüdenauer, F., see Betz, G. 889 Rudge, W.E., see Eigler, D.M. 456, 458, 461, 717 Rudolph, P., see Leung, K.M. 8 Rudzinski, W. 249 Rudzinski, W., see Panczyk, T. 249 Ruestig, J., see Ertl, G. 357 Ruggerone, P. 320 Ruppender, H., see Ehsasi, M. 368 Rurali, R., see Lorente, N. 724 Rusina, G.G., see Sklyadneva, I.Yu. 534 Rusponi, S. 779, 781 Rusponi, S., see Cren, T. 780 Rusponi, S., see Gambardella, P. 778, 781 Russ, R., see Brenig, W. 166 Russell, N.M., see Mahajan, A. 828 Rust, H.-P., see Pascual, J.I. 592, 615, 616 Rust, H.-P., see Sterrer, M. 730 Rust, H.P., see Pascual, J.I. 688, 723 Rust, H.P., see Yulikov, M. 275 Rutkowski, M., see Wagner, S. 675 Rutkowski, M., see Wetzig, D. 178 Rutledge, J.E., see Renner, R.L. 487 Ryali, S.B., see Mantell, D.A. 672 Ryberg, R. 469, 483, 485 Ryberg, R., see Persson, M. 483, 485 Rydberg, H. 500 Rydberg, H., see Dion, M. 500, 608

972

Rydberg, H., see Hult, E. 99 Ryu, S., see Jeong, H.D. 735 Rzenicka, I., see Imamura, K. 254 Saalfrank, P. 19, 22, 23, 624, 637, 650, 658, 659, 663, 670 Saalfrank, P., see Bartels, L. 687, 688, 724 Saalfrank, P., see Bornscheuer, K.-H. 663 Saalfrank, P., see Finger, K. 23, 653, 657, 659 Saalfrank, P., see Guo, H. 23, 623 Saalfrank, P., see Hasselbrink, E. 661 Saalfrank, P., see Klamroth, T. 669, 670 Saalfrank, P., see Paramonov, G.K. 20 Saalfrank, P., see Vazhappilly, T. 675 Saarloos, W.v., see Gastel, R.v. 777 Sabatier, P. 313, 315 Sabella, M., see Tammaro, M. 355, 407 Sacchi, M., see Maroni, P. 181 Sacher, E. 846 Sachs, C. 345, 380, 397, 401, 403 Sachtler, J.W.A. 293 Sadhukhan, S.K. 748 Sáenz, J.J. 694, 718 Safi, M. 847 Safi, M., see Belaïdi, A. 847 Safi, M., see Chazalviel, J.-N. 847 Sagara, T. 829 Saifullah, M.S.M. 739 Sainoo, Y. 592, 601, 617, 711, 712 Sainoo, Y., see Kawai, M. 601, 617, 687, 712 Sainoo, Y., see Komeda, T. 597 Saito, T., see Ye, S. 849 Sakata, K., see Makino, O. 842, 847 Sakka, T., see Ogata, Y. 847 Sakka, T., see Ogata, Y.H. 849 Sakong, S., see Dianat, A. 146 Sakuraba, M. 830 Sakurai, M. 701, 705, 706 Sakurai, M., see Stokbro, K. 701, 705, 706 Sakurai, M., see Thirstrup, C. 701 Salam, G.P. 616, 706, 710 Salem, L. 3 Salemink, H.W.M., see Rogge, S. 695 Sales, B.C. 366 Salin, A., see Busnengo, H.F. 146, 147, 151, 172, 182, 185, 186 Salin, A., see Crespos, C. 172, 186 Salin, A., see Díaz, C. 151, 155, 156, 186 Salin, A., see Farías, D. 146, 154, 155, 191

Author index

Salin, A., see Olsen, R.A. 147 Salling, C.T. 696, 697 Salmeron, M., see Cerda, J. 585 Salomon, E., see Papageorgiou, N. 733 Saltsburg, H. 199, 200 Saltsburg, H., see Zuburtikudis, I. 414 Salvan, F., see Albertini, D. 833, 834 Salvan, F., see Binnig, G. 551 Salvo, C., see Chiesa, M. 102, 103 Samartsev, A.V. 895 Samson, P., see Hodgson, A. 169 Samuelson, L., see Jensen, L.E. 812 Samuelson, L., see Johansson, J. 810–812 Samuelson, L., see Persson, A.I. 810, 812 Sanche, L.J., see Huels, M.A. 700 Sanchez-Castillo, M.A., see Kandoi, S. 308, 309 Sánchez-Portal, D., see Chulkov, E.V. 7, 20 Sánchez-Portal, D., see Corriol, C. 548, 564 Sánchez-Portal, D., see Föhlisch, A. 566 Sancho, J.M. 254 Sander, D. 797 Sander, M. 359, 361, 363, 365, 396 Sander, M., see Falta, J. 363 Sander, M., see Imbihl, R. 363 Sanders, D.E. 767 Sanders, D.E., see Evans, J.W. 767 Sano, M. 672 Sanz, F., see Gorostiza, P. 842 Sarría, I. 564 Sarría, I., see Chulkov, E.V. 564 Sasahara, A., see de Wolf, C.A. 377 Sasaki, K., see Zhang, J.L. 291 Sasaki, M., see Yagu, S. 76 Sasaki, T. 498 Sasaki, T., see Kondo, T. 76–78, 80, 81 Sathyamurthy, N., see Chakrabarti, N. 658 Sato, S. 237 Sato, Y. 847 Saúl, A., see Müller, P. 794, 795, 799, 800, 803, 821 Saunders, W.A. 845 Sautet, P., see Busnengo, H.F. 185, 186 Sautet, P., see Cerda, J. 585 Sautet, P., see Filhol, J.S. 292 Sautet, P., see Girard, C. 695 Sauvage, J.P. 744 Savage, D.E., see Mo, Y.-W. 816 Savio, L. 158, 159

Author index

Savio, L., see Gross, L. 740, 751 Savio, L., see Vattuone, L. 31 Savrasov, D.Yu., see Maksimov, E.G. 534 Savrasov, S.Yu., see Maksimov, E.G. 534 Sawabe, K., see Gruzdkov, Y.A. 633 Sawabe, K., see Matsumoto, Y. 633 Sawabe, K., see Watanabe, K. 633 Sawabe, K., see Watanabe, Y. 44 Sawada, S.I. 6 Sawatzky, G.A., see Lof, R.W. 747 Sawin, H.H., see Kulkarni, S.K. 831 Saxon, R.P., see Tenner, A.D. 38, 41 Scagnelli, A., see Yulikov, M. 275 Scammon, R.J., see Dingus, R.S. 905, 906 Schaak, A. 368, 374, 376, 385, 390, 391, 393, 395 Schaak, A., see Hartmann, N. 385 Schaak, A., see Janssen, N.M.H. 374, 385, 390 Schaak, A., see Monine, M.I. 390 Schaak, A., see Schmidt, T. 377, 390 Schäfer, A. 552, 553, 560, 564 Schäfer, C. 905–907 Schaich, W., see d’Agliano, E.G. 3, 14, 19 Schaich, W., see d’Angliano, E.G. 448, 449, 479, 483, 484 Schaich, W.A. 483, 485 Schaich, W.L. 19 Scharff, M., see Lindhard, J. 875, 877, 879, 881 Scheffler, M. 278 Scheffler, M., see Behler, J. 149, 245, 334, 433, 496, 499–501, 507, 508, 510 Scheffler, M., see Bonn, M. 674 Scheffler, M., see Dabrowski, ˛ J. 829 Scheffler, M., see Eichler, A. 182, 183, 311 Scheffler, M., see Fichthorn, K.A. 323, 774, 775 Scheffler, M., see Ganduglia-Pirovano, M.V. 395 Scheffler, M., see Gross, A. 146, 151, 159, 164, 165, 177, 181–183, 463, 498 Scheffler, M., see Hammer, B. 154, 168, 450, 452, 453, 498 Scheffler, M., see Hennig, D. 293 Scheffler, M., see Kratzer, P. 320, 841 Scheffler, M., see Kroes, G.J. 3, 32 Scheffler, M., see Li, W.X. 320 Scheffler, M., see Penev, E. 841 Scheffler, M., see Ratsch, C. 321 Scheffler, M., see Reuter, K. 304, 308, 319, 324, 325, 366, 370, 515

973

Scheffler, M., see Ruggerone, P. 320 Scheffler, M., see Stampfl, C. 252, 253, 320 Scheffler, M., see Wang, L.G. 804, 807 Scheffler, M., see Wilke, S. 498 Scheit, S., see Moiseyev, N. 6 Schenter, G.K. 234 Schiff, L.I. 132 Schiffrin, D.J., see Baum, T. 849, 850 Schiffrin, D.J., see Campbell, S.A. 850 Schillinger, R., see Zubkov, T. 302 Schimansky-Geier, L., see Malchow, H. 404 Schimansky-Geier, L., see Pineda, M. 408, 409 Schimansky-Geier, L., see Rose, H. 362 Schimtz, S., see Veigel, C. 744 Schiøtz, J., see Clausen, B.S. 799 Schiøtz, J., see Meier, J. 292 Schlachetzki, A., see Peiner, E. 845 Schlegel, H.B., see Li, X. 6 Schlichthörl, G., see Cattarin, S. 845, 847 Schlichting, H. 46, 111, 125, 126 Schlichting, H., see Niedermayer, T. 135, 136 Schlittler, R.R., see Gimzewski, J.K. 748 Schlittler, R.R., see Joachim, C. 738 Schlittler, R.R., see Jung, T.A. 686, 689, 691 Schlittler, R.R., see Langlais, V. 739 Schlom, D.G. 791 Schluter, M. 792 Schmatloch, V., see Heinze, S. 374 Schmeisser, D., see Lewerenz, H.J. 847 Schmeisser, D., see Wehner, S. 413 Schmid, M. 73 Schmid, M.P. 181 Schmid, M.P., see Beck, R.D. 181 Schmidt, L.D., see Flytzani-Stephanopoulos, M. 363 Schmidt, L.D., see Gorte, R.J. 371 Schmidt, L.D., see Lesley, M.W. 373 Schmidt, L.D., see Schüth, F. 259, 346 Schmidt, L.D., see Takoudis, C.G. 373 Schmidt, O.G. 840 Schmidt, S., see Nicolay, G. 540, 541 Schmidt, S., see Reinert, F. 540, 541, 543, 548, 560, 561 Schmidt, T. 377, 380, 390 Schmidt, V. 811 Schmitz, R., see Sheintuch, M. 346 Schmuki, P., see Beranek, R. 822 Schmuki, P., see Sieber, I.V. 822 Schmuki, P., see Tsuchiya, H. 822

974

Schnadt, J., see Vang, R.T. 298, 300, 781 Schneider, A.M., see Knorr, N. 771–774 Schneider, F.W., see Field, R.J. 345 Schneider, M.A., see Knorr, N. 609 Schneider, M.A., see Stepanyuk, V.S. 772, 774 Schneider, M.A., see Vitali, L. 546, 550, 561, 562 Schneider, M.A., see Wahl, P. 550, 609 ´ Schneider, W.-D., see Cavar, E. 687, 688, 746 Schneider, W.-D., see Li, J. 545–547, 608, 609 Schneider, W.-D., see Pivetta, M. 540, 560 Schneider, W.D., see Silly, F. 770, 772–774 Schneider, W.F., see Ovesson, S. 304, 319–325 Schnöckel, H., see Burgert, R. 508 Schochlin, J. 478 Schoenlein, R.W. 455, 470, 483, 551 Scholte, P.M.L.O., see Rogge, S. 695 Schön, G., see Heurich, J. 738 Schön, G., see Leung, K.M. 8 Schöne, W.D., see Keyling, R. 560 Schönhammer, K. 8, 14, 16–19, 98, 189, 446, 448, 461, 483 Schönhammer, K., see Bönig, L. 19 Schönhammer, K., see Gunnarsson, O. 8, 14, 189, 456 Schoonmaker, R.C., see Spencer, N.B. 326 Schou, J. 908 Schou, J., see Johnson, R.E. 893 Schrieffer, J.R. 433, 609 Schröder, E., see Chakarova-Käck, S.D. 608 Schröder, E., see Dion, M. 500, 608 Schröter, L. 178 Schubert, D.S., see Young, R.D. 765 Schubert, L. 811 Schulberg, M.T., see Li, Y.L. 482, 483, 493 Schuller, A. 41 Schuller, A., see Winter, H. 38, 41 Schultz, G.J. 475, 503 Schultz, J.A., see Koleske, D.D. 843 Schulz, G., see Werner, H. 370 Schumacher, N. 314, 315 Schunack, M. 739 Schunack, M., see Rosei, F. 739, 748 Schuppler, S., see Fischer, R. 553 Schuster, H.G. 409, 410 Schuster, R., see Sander, M. 365 Schuster, R., see Wintterlin, J. 321, 671, 719 Schüth, F. 259, 346, 371 Schütz, E. 345, 412, 415, 417–419

Author index

Schütz, E., see Esch, F. 417 Schütz, E., see Shvartsman, S.Y. 415, 417, 418 Schwankner, R., see Eiswirth, M. 359 Schwankner, R.J. 412 Schwarzwald, R., see Fukutani, K. 641, 643 Schwegmann, S., see Mertens, F. 374, 390 Schweizer, E.K. 38 Schweizer, E.K., see Eigler, D.M. 601, 685, 688, 694, 738 Schweizer, E.K., see Rettner, C.T. 48, 49, 105, 125, 126 Schweizer, S., see Jungblut, H. 847 Schwenke, D.W., see Chatfield, D.C. 174 Schwöbel, R.L. 817 Sclittler, R.R., see Gimzewski, J.K. 720 Scoles, G. 31, 56, 203 Scott, B.A., see Imbihl, R. 834 Scott, B.A., see Jasinski, J.M. 828 Scully, J.R., see Punckt, C. 821 Sebastian, K.L. 447 Sebastian, K.L., see Grimley, T.B. 447 Sedlmeir, R. 106 Seets, D.C., see McClure, S.M. 31 Segner, J. 208, 209 Segner, J., see Campbell, C.T. 671 Segner, J., see Frenkel, F. 207 Segner, J., see Modl, A. 210 Segovia, P., see Hengsberger, M. 542–544 Sehested, J. 317, 318 Sehested, J., see Dahl, S. 311, 327 Sehested, J., see Helveg, S. 812 Sehested, J.S., see Bligaard, T. 294, 311, 312, 314 Seidel, C., see Ehsasi, M. 368 Seideman, T. 22, 616 Seideman, T., see Alavi, S. 616 Seideman, T., see Guo, H. 23, 623 Seifert, W., see Johansson, J. 810–812 Seitsonen, A.P., see Over, H. 370 Seitz, F. 891 Seki, H., see Hussla, I. 660 Self, K., see Cong, P.J. 313 Self, K.W. 735 Sellers, J.R., see Veigel, C. 744 Selloni, A., see Ancilotto, F. 829 Selloni, A., see Vittadini, A. 829 Senderens, J.B., see Sabatier, P. 315 Senz, S., see Schmidt, V. 811 Seo, Y.H. 847

Author index

Seravalli, L., see Franchi, S. 837, 838, 840 Sercel, P.C., see Saunders, W.A. 845 Sesselman, W. 456, 458, 473 Sesselmann, W., see Woratschek, B. 459, 472, 473 Sexton, B., see Fisher, G. 443 Sexton, B.A., see Gland, J.L. 718 Sexton, J.Z., see Komrowski, A.J. 73, 494, 495, 500, 506, 507 Sexton, J.Z., see Ternow, H. 507 Shackman, L.C. 167, 171, 183, 189 Shaikhutdinov, S., see Hartmann, N. 385 Shaikhutdinov, S., see Schaak, A. 385 Shalashilin, D.V. 239 Sham, L.J., see Kohn, W. 277, 281, 433 Shane, S.F. 829 Shane, S.F., see Jacobs, D.C. 208, 215 Shane, S.F., see Kolasinski, K.W. 829 Shanes, F.C., see Black, J.E. 107 Shank, C.V. 824 Shank, C.V., see Downer, M.C. 824 Shao, H. 502 Shapira, Y., see Lichtman, D. 623 Sharp, J.W., see Eres, G. 828 Shashidhar, R., see Long, D.P. 592 Shchukin, V.A. 797, 803, 804, 806 Shchukin, V.A., see Munt, T.P. 797 Shcütz, E., see Shvartsman, S. 262 Sheehy, M., see Shen, M.Y. 827 Sheintuch, M. 346, 387 Sheintuch, M., see Luss, D. 348, 381 Shelef, M. 319 Shelnutt, J.A. 733 Shen, M.Y. 827 Shen, T.-C. 701, 704–706 Shen, T.-C., see Avouris, Ph. 699, 701, 704–706 Shen, T.-C., see Ji, J.-Y. 828 Shen, T.-C., see Lyding, J.W. 701, 703, 705, 706, 726 Shen, T.C., see Avouris, Ph. 602, 616 Shen, W., see Davis, L.C. 545 Shen, X.J. 564 Shen, X.Q. 842 Shen, Y.-R., see Ye, S. 849 Shen, Z.X., see Junren, S. 545 Sheng, J., see Dai, J.Q. 175 Shenvi, N. 6 Shenvi, N., see Cheng, H. 6, 10, 12 Shenvi, N., see Cheng, H.Z. 471

975

Sherman, M.G., see Sinniah, K. 715, 829 Sherman, N.K., see Corkum, P.B. 636 Sherratt, P.A.J., see Koker, L. 847 Shi, H., see Grobecker, R. 482, 488 Shi, J. 829 Shi, Y., see Liu, J.L. 808 Shibataka, T., see Sagara, T. 829 Shigekawa, H., see Sainoo, Y. 711, 712 Shigeno, M., see Watanabe, S. 842, 847 Shimada, T., see Nakai, I. 257 Shinjoh, H., see Takahashi, N. 319 Shipsey, E.J., see Schwöbel, R.L. 817 Shirai, Y. 751 Shirykalov, E.N., see Chulkov, E.V. 542 Shluger, A., see Hofer, W.A. 589 Shobatake, K., see Watanabe, Y. 44 Shockley, W. 528 Shockley, W., see Brattain, W.H. 528 Sholl, D.S. 10 Short, K.T., see Hull, R. 798 Showalter, K., see Kapral, R. 377, 378 Showalter, K., see Mikhailov, A.S. 345, 411, 413 Shub, B.R., see Kozhushnev, M.A. 486 Shulga, V.I. 889 Shulga, V.I., see Glazov, L.G. 889 Shumay, I.L. 551, 555, 563, 564 Shumay, I.L., see Berthold, W. 563 Shumay, I.L., see Fauster, Th. 556, 557 Shumay, I.L., see Höfer, U. 555, 564 Shumay, I.L., see Reuß, Ch. 556, 557 Shumay, I.L., see Schäfer, A. 552, 553, 560, 564 Shumay, I.L., see Weinelt, M. 556, 557 Shushin, A.I. 254 Shuttleworth, R. 792, 800 Shvartsman, S. 262 Shvartsman, S.Y. 415, 417, 418 Siadati, M.H., see Chianelli, R.R. 330, 331, 333 Sibener, S., see Yu, C.-F. 102 Sibener, S.J., see Cowin, J.P. 164 Sibener, S.J., see Gibson, K.D. 44, 49, 100 Sibener, S.J., see Isa, N. 44 Sibirev, N.V., see Dubrovskii, V.G. 810 Sibold, D. 908 Siders, J.L.W. 85–87, 211 Siders, J.L.W., see Watts, E.K. 221 Sieben, B. 404 Sieber, I., see Tsuchiya, H. 822

976

Sieber, I.V. 822 Siegman, A.E. 827 Siekhaus, W.J., see Olander, D.R. 830 Siekhaus, W.J., see van Buuren, T. 849 Siera, J. 373, 374 Siera, J., see Cobden, P.D. 374, 410 Siera, J., see van Tol, M.F.H. 370, 374 Sievers, A., see Chabal, Y.J. 483, 485 Sievers, J., see Link, S. 564 Sigmund, P. 879, 885, 886, 889, 891, 893 Sigmund, P., see Falcone, G. 887 Sigmund, P., see Glazov, L.G. 889 Sigmund, P., see Vicanek, M. 879, 889 Sigmund, P., see Winterbon, K.B. 882 Signore, G.D., see Chiarotti, G. 528 Silkin, V.M. 529, 542, 543, 560, 561 Silkin, V.M., see Balasubramanian, T. 543, 560 Silkin, V.M., see Berthold, W. 551, 553, 554, 563 Silkin, V.M., see Borisov, A.G. 558, 565 Silkin, V.M., see Chulkov, E.V. 7, 20, 529, 534, 542, 548, 560, 561, 564, 565 Silkin, V.M., see Corriol, C. 548, 564 Silkin, V.M., see Echenique, P.M. 529, 562, 564 Silkin, V.M., see Eiguren, A. 534, 535, 541, 542, 562 Silkin, V.M., see Kliewer, J. 527, 529, 531, 535, 540, 543, 546, 548, 560, 561 Silkin, V.M., see Link, S. 564 Silkin, V.M., see Schäfer, A. 552, 553, 560, 564 Silkin, V.M., see Vitali, L. 546, 550, 561, 562 Silly, F. 770, 772–774 Silly, F., see Pivetta, M. 540, 560 Silva, M. 511, 512 Silveston, P.L. 411 Silvestrelli, P.L. 735 Silvi, S., see Badjic, J.D. 744 Simon, D., see Filhol, J.S. 292 Simon, G., see Rethfeld, B. 899 Singer, I.L. 486 Singh, D., see Perdew, J. 433 Singh, V.A., see John, G.C. 823 Singh-Boparai, S.P. 371 Sinniah, K. 715, 829 Sipe, J.E. 827 Sipe, J.E., see Young, J.F. 827 Sirotna, K., see Tsuchiya, H. 822 Sit, P.H.L. 500

Author index

Sitz, G.O. 82, 83, 201, 205–207, 210–212, 216, 218–220 Sitz, G.O., see Gostein, M. 167, 183 Sitz, G.O., see Gotthold, M.P. 214 Sitz, G.O., see Kummel, A.C. 210, 218, 219 Sitz, G.O., see Luntz, A.C. 493 Sitz, G.O., see McClure, S.M. 31 Sitz, G.O., see Shackman, L.C. 167, 171, 183, 189 Sitz, G.O., see Siders, J.L.W. 85–87, 211 Sitz, G.O., see Watts, E. 161, 167, 169, 183–186, 188, 189 Sitz, G.O., see Watts, E.K. 221 Sjölander, A. 57, 59 Sjölander, A., see Björkman, G. 484 Skalicky, T. 720 Sklyadneva, I.Yu. 534 Skoglundh, M., see Fridell, E. 319 Skoglundh, M., see Olsson, L. 323 Skottkeklein, M. 512 Skriver, H.L., see Christensen, A. 764 Skriver, H.L., see Ruban, A. 287, 293 Skúlason, E. 333 Skúlason, E., see Hellman, A. 304, 325, 327, 328 Slagsvold, B.J. 540, 541 Slagsvold, B.J., see Gartland, P.O. 528 Slater, J.C. 500, 905 Slezak, J., see Schmidt, T. 380 Slinko, M. 374 Slinko, M., see Lombardo, S.J. 374 Slinko, M.G. 346 Slinko, M.M. 259, 346, 348, 410 Slinko, M.M., see Makeev, A.G. 374 Slinko, M.M., see Peskov, N.V. 259 Slinko, M.M., see Slinko, M.G. 346 Sljivancanin, Z., see Gambardella, P. 302 Sljivancanin, Z., see Nørskov, J.K. 302, 303 Sloan, P.A. 687, 720, 724 Sloan, P.A., see Dobrin, S. 696 Sloan, P.A., see Harikumar, K.R. 696 Slyozov, V.V., see Lifshitz, I.M. 775 Smadici, S., see Shen, X.J. 564 Smardon, R.D. 832, 836 Smedler, G., see Fridell, E. 319 Smentkowski, V.S. 827 Smit, R.H.M. 592 Smith, C.W., see Mayne, A.J. 735 Smith, D.C., see Gudiksen, M.S. 809

Author index

Smith, D.P.E., see Bryant, A. 701 Smith, F.T. 436 Smith, N.V. 538, 700 Smith, N.V., see Kevan, S.D. 560 Smith, N.V., see Rhie, H.-S. 563 Smith, P.V., see Craig, B.I. 829 Smith, R., see Godwin, P.D. 692 Smith, R.J. 122 Smith, R.J., see Lahaye, R.J.W.E. 44, 49 Smith, R.L. 820, 823, 850 Smith, R.R. 181 Smith, R.R., see Juurlink, L.B.F. 181 Smith Jr., J.N., see Saltsburg, H. 199, 200 Snively, C., see McMillan, N. 390 Sokolov, I.M., see Sancho, J.M. 254 Sokolowski, S., see Patrykiejew, A. 246 Sokolowski-Tinten, K. 824 Sokolowski-Tinten, K., see Rethfeld, B. 824, 903 Soler, J.M., see Paz, O. 590 Solina, D.H., see Krim, J. 486 Somashekhar, A. 847 Somers, M.F. 161, 162, 169, 184, 189 Somers, M.F., see Busnengo, H.F. 146, 186 Somers, M.F., see Díaz, C. 151, 155, 156, 186 Somers, M.F., see Farías, D. 146, 154, 155, 191 Somers, M.F., see Kroes, G.J. 145, 146, 151 Somers, M.F., see Olsen, R.A. 147 Somfai, E., see Gastel, R.v. 777 Somorjai, G.A. 246, 277, 278, 290, 298, 310, 330, 414, 492 Somorjai, G.A., see Guthrie, W.L. 641 Somorjai, G.A., see Katona, T. 373 Somorjai, G.A., see Park, J.Y. 491, 492 Somorjai, G.A., see Prins, R. 331 Somorjai, G.A., see Sachtler, J.W.A. 293 Somorjai, G.A., see Spencer, N.B. 326 Somorjai, G.A., see Strongin, D.R. 326 Somorjai, G.A., see van Hove, M.A. 290 Sondag, A.H.M., see Boonekamp, E.P. 849 Song, X.-Z., see Shelnutt, J.A. 733 Song, Y.J., see Choi, B.-Y. 736 Song, Z., see Pascual, J.I. 592, 615, 616, 688, 723 Sonnet, Ph., see Baffou, G. 734 Sonnet, Ph., see Cranney, M. 745 Sonnet, Ph., see Mamatkulov, M. 745 Sonnet, Ph., see Martín, M. 746 Sorensen, M.R., see Brandbyge, M. 585

977

Sørensen, T., see Andersen, H.H. 889 Sosa, C., see Nachtigall, P. 829 Sosolik, C., see Powers, J. 57, 59 Soukiassian, L. 701, 703–706, 726, 736 Soukiassian, L., see Mayne, A.J. 696, 726, 746 Sowa, E.C., see Nelson, J.S. 534 Spangler, J.D., see Rahman, T.S. 534 Spanjaard, D., see Desjonquères, M.-C. 605, 606, 799 Spencer, N.B. 326 Spielmann, C., see Xu, L. 551 Spiller, G.D.T., see Venables, J.A. 816 Sprik, M. 306 Springsteen, L.L., see Asscher, M. 644 Sprunger, P.T., see Junren, S. 545 Sprunger, P.T., see Tang, S.-J. Ismail 544, 560 Squier, J., see Sokolowski-Tinten, K. 824 Srinivasan-Rao, T., see Corkum, P.B. 636 Srivastava, D. 831 Srivastava, G.P., see Smardon, R.D. 832, 836 Stach, E.A., see Hull, R. 798 Stadler, R. 739 Staemmler, V., see Klüner, T. 631, 632 Stahlbush, R.E., see Glembocki, O.J. 850 Stair, P.C., see van Hove, M.A. 290, 357 Stamenkovic, V., see Mun, B.S. 293 Stampfl, C. 252, 253, 320 Stampfl, C., see Bonn, M. 674 Stampfl, C., see Li, W.X. 320 Stampfl, C., see Scheffler, M. 278 Stampfli, P. 824, 902 Stark, K., see Manolopoulos, D.E. 510, 512 Starke, J. 409 Starke, J., see Reichert, C. 384, 408, 409 Starodub, D., see Frank, M.M. 791 Starr, D.E., see Stuckless, J.T. 763, 765 Staudt, C. 890 Stauffer, L., see Baffou, G. 734 Stauffer, L., see Cranney, M. 745 Stauffer, L., see Mamatkulov, M. 745 Stauffer, L., see Martín, M. 746 Stechel, E.B., see Burns, A.R. 644, 660 Stechel, E.B., see Orlando, T.M. 718 Stedile, F.C., see Frank, M.M. 791 Steen, K., see Langa, S. 822 Steijns, M.H.G.M., see Bos, A.N.R. 233 Steinbüchel, C., see Leung, C. 647 Steiner, P., see Nicolay, G. 540, 541 Steinmann, W., see Fauster, Th. 529, 553, 556

978

Steinmann, W., see Fischer, R. 553 Steinmann, W., see Giesen, K. 553 Steinrück, H.-P., see Gokhale, S. 735 Steinrück, H.P., see D’Evelyn, M.P. 174 Stensgaard, I., see Besenbacher, F. 296, 298, 717 Stensgaard, I., see Helveg, S. 330, 331 Stensgaard, I., see Österlund, L. 246 Stensgaard, I., see Rosei, F. 739, 748 Stensgaard, I., see Schunack, M. 739 Stenström, S., see Persson, A.I. 810, 812 Stenum, B., see Andersen, H.H. 889 Stépán, K. 638 Stepanov, A.A., see Myshlyavtsev, A.V. 256 Stepanyuk, V.S. 772, 774, 775 Stepanyuk, V.S., see Tsivlin, D.V. 773 Stephenson, J.C., see Struck, L.M. 483 Stern, O. 55 Stern, O., see Estermann, I. 55 Stern, O., see Knauer, F. 55 Sterrer, M. 730 Sterrer, M., see Yulikov, M. 275 St¸es´licka, M., see Davison, S.G. 529 Stewart, H.B., see Thompson, J.M.T. 351, 352, 409 Stich, I., see White, J.A. 154, 168, 498 Stickney, R.E., see Yamamoto, S. 36, 46 Stiles, M.D. 113, 123 Stimming, U., see Meier, J. 292 Stimming, U., see Nørskov, J.K. 331–333 Stingl, A., see Xu, L. 551 Stipe, B.C. 321, 458, 462, 463, 592, 594, 687, 709, 710, 719, 722 Stoddart, J.F., see Badjic, J.D. 744 Stoffel, N.G., see Kevan, S.D. 560 Stojkovic, S., see Grill, L. 750 Stojkovic, S.M., see Chiavaralloti, F. 748 Stojkovic, S.M., see Grill, L. 751 Stojkovic, S.M., see Gross, L. 748 Stojković, S.M., see Repp, J. 732 Stokbro, K. 701, 705, 706 Stokbro, K., see Thirstrup, C. 701 Stokes, S.T., see Burgert, R. 508 Stolte, S., see Berenbak, B. 42, 43, 45, 209 Stolte, S., see Butler, D.A. 45, 209 Stolte, S., see Geuzebroeak, F.H. 210 Stolte, S., see Guezebroek, F.H. 162 Stolte, S., see Komrowski, A.J. 506, 507 Stolte, S., see Kuipers, E.W. 210

Author index

Stolte, S., see Lahaye, R.J.W.E. 32, 34, 36–39, 44, 47–49, 162–164 Stolte, S., see Tenner, M.G. 210 Stolte, S., see Ternow, H. 507 Stoltze, P. 234, 256, 311, 326 Stoltze, P., see Brune, H. 775 Stoltze, P., see Christensen, A. 764 Stoltze, P., see Ovesen, C.V. 314 Stoltze, P., see Ruban, A. 287, 293 Stoop, F., see Toolenaar, F.J.C.M. 276 Strachan, C. 55 Strachan, C., see Lennard-Jones, J.E. 113 Stranick, S.J. 771 Stranski, I.N. 803 Strasser, P. 293 Strasser, P., see Eiswirth, M. 352 Strasser, P., see Yaccato, K. 317 Straube, P. 542, 560 Strömquist, J. 475, 480–483, 494, 499, 500, 509, 637 Strömquist, J., see Bogicevic, A. 320 Strömquist, J., see Gao, S. 637 Strömquist, J., see Hellberg, L. 464, 466, 475, 477, 479–482, 488, 494, 500 Strömquist, J., see Wahnström, G. 494, 719 Strongin, D.R. 326 Stroscio, J.A. 686, 688, 706, 765, 767 Stroscio, J.A., see Persson, M. 135 Stroscio, J.A., see Whitman, L.J. 695 Struck, L.M. 483 Struve, W.S. 464, 476, 482 Stückelberg, E.C.G. 441 Stuckless, J.T. 763, 765 Stuke, M., see Preuss, S. 825, 826, 906, 907 Stumper, J. 847 Stumper, J., see Cattarin, S. 845, 847 Stumper, J., see Peter, L.M. 847 Stumpf, R., see Hamilton, J.C. 781 Stutzki, J. 113 Stutzmann, M., see Herrero, C.P. 715 Suchan, M.M., see Korolik, M. 220 Suchorski, Y. 404–407 Suchorski, Y., see Medvedev, V.K. 403 Suda, Y. 830, 835 Suemitsu, M. 828 Suemune, I., see Noguchi, N. 846 Sugawara, Y., see Oyabu, N. 694 Sugita, Y., see Watanabe, S. 849 Suhl, H., see d’Agliano, E.G. 3, 14, 19

Author index

Suhl, H., see d’Angliano, E.G. 448, 449, 479, 483, 484 Suhl, H., see Langreth, D.C. 435 Süle, P. 893 Sumanasekara, G.U., see Chandrasekaran, H. 810, 813 Sun, J.W., see Wang, F.D. 807 Sun, Q., see Gazdy, B. 8 Sun, Y.K., see Campbell, C.T. 513 Sundaram, S.K. 824 Sundberg, R.L., see Heller, E.J. 20 Sundell, P.G. 237, 255 Sundqvist, B.U.R., see Johnson, R.E. 893, 894 Šunjić, M., see Gadzuk, J.W. 14, 15 Šunjić, M., see Lucas, A.A. 8, 9 Sunkara, M.K., see Chandrasekaran, H. 810, 813 Surko, C.M., see Yamamoto, S.Y. 381 Sutcu, L.F., see D’Evelyn, M.P. 830 Suzuki, M.T., see Ogawa, H. 849 Suzuki, S., see Takagi, D. 813 Svensson, C.P.T., see Johansson, J. 810–812 Svensson, K. 101, 128 Svensson, K., see Hassel, M. 130, 135 Swartzentruber, B.S., see Becker, R.S. 551, 685, 738 Swartzentruber, B.S., see Mo, Y.-W. 816, 831 Swiech, W. 380, 383 Swiech, W., see Rausenberger, B. 384 Swinney, H.L., see Petrov, V. 412, 419 Sykes, M.F. 767 Szabò, A. 844 Sze, S.M. 735 Tabata, H., see Kanno, T. 747 Taborek, P., see Renner, R.L. 487 Tachibana, A., see Makino, O. 842, 847 Takagi, D. 813 Takahagi, T. 847 Takahashi, N. 319 Takami, S., see Kobatake, S. 736 Takeda, S., see Kikkawa, J. 810 Takoudis, C.G. 373 Talker, P., see Pollak, E. 237 Talkner, P., see Hänggi, P. 8, 20 Talkner, P., see Hanggi, P. 237 Tamm, I.E. 528 Tammaro, M. 355, 373, 382, 383, 385, 386, 407

979

Tammaro, M., see Evans, J.W. 355, 356, 382, 383 Tan, T.Y. 809–811 Tan, T.Y., see Li, N. 810, 811 Tan, T.Y., see Schubert, L. 811 Tanabe, Y., see Ohtaka, K. 15 Tanaka, H., see Kanno, T. 747 Tanaka, K., see Sagara, T. 829 Tanaka, K., see Siera, J. 373, 374 Tanaka, T. 830 Tang, H., see Bernard, R. 748 Tang, H., see Dujardin, G. 686, 693, 728 Tang, H., see Gimzewski, J.K. 720 Tang, H., see Gross, L. 748 Tang, H., see Joachim, C. 739 Tang, H., see Jung, T.A. 686, 689, 691 Tang, H., see Kuntze, J. 739 Tang, H., see Langlais, V. 739 Tang, H., see Lorente, N. 724 Tang, H., see Moresco, F. 686, 740, 741, 751 Tang, R., see Wang, F.D. 807 Tang, S.-J. Ismail 544, 560 Tang, S.J., see Junren, S. 545 Tanimura, K. 723 Tannor, D., see Heller, E.J. 20 Tannor, D.J. 6, 20 Tannor, D.J., see Qian, J. 11, 12 Tarasenko, A.A. 256 Tarasenko, A.A., see Nieto, F. 254 Tarasevitch, A., see Sokolowski-Tinten, K. 824 Tarus, J., see Nordlund, K. 896 Tasch, A., see Mahajan, A. 828 Tate, M.R. 482, 483 Tate, M.R., see Holt, J.R. 843 Tautermann, C.S., see Hellman, A. 304, 325, 327, 328 Taveira, L., see Tsuchiya, H. 822 Taylor, A.G., see Turner, A.R. 828 Taylor, H.S. 298 Taylor, J.B. 449 Taylor, P.A., see Cheng, C.C. 735 Tcheließnig, R., see Schmid, M. 73 Teichert, C., see Michely, T. 890 Teichert, C., see Tersoff, J. 804, 839, 840 Teiller-Billy, D. 473, 474 Teiller-Billy, D., see Borisov, A.G. 473, 474 Teiller-Billy, D., see Makhmetov, G.E. 473 Teillet-Billy, D. 481, 502, 722, 723 Teillet-Billy, D., see Auth, C. 481, 502

980

Teillet-Billy, D., see Bahrim, B. 481, 502 Teillet-Billy, D., see Borisov, A.G. 481, 502, 565 Teillet-Billy, D., see Marinica, D.C. 20, 563 Tempea, G., see Xu, L. 551 Tenner, A.D. 38, 40, 41 Tenner, M.G. 210 Tenner, M.G., see Geuzebroeak, F.H. 210 Tenner, M.G., see Guezebroek, F.H. 162 Tenner, M.G., see Kuipers, E.W. 210 Teplyakov, A.V., see Kong, M.J. 735 Teracuda, K., see Ishida, H. 449 Terakura, K., see Kato, K. 245, 508 Terminello, L.J., see van Buuren, T. 849 Ternes, M., see Silly, F. 770, 772–774 Ternov, H., see Hellman, A. 245 Ternow, H. 507 Ternow, H., see Hellman, A. 334, 433, 434, 475, 480, 496, 499, 501–503, 506, 508 Ternow, H., see Komrowski, A.J. 506, 507 Ternow, H., see Oner, D.E. 498 Tersoff, J. 590, 706, 804, 839, 840 Tersoff, J., see Daruka, I. 799, 804 Tersoff, J., see Kodambaka, S. 810 Tersoff, J., see Ross, F.M. 810 Teshima, R., see Gortel, Z.W. 120 Teshima, R., see Zaluska-Kotur, M.A. 256 Thaler, R.M., see Rodberg, L.S. 57, 62 Theilmann, F. 538, 539, 548 Theilmann, F., see Weinelt, M. 556, 557 Theis, C.D., see Schlom, D.G. 791 Theis, W. 775, 776 Thibaudau, F., see Albertini, D. 833, 834 Thibaudau, F., see Masson, L. 831, 834 Thiel, P.A. 357, 358 Thiel, P.A., see Evans, J.W. 763, 767 Thiel, S., see Pykavy, M. 632 Thirstrup, C. 701 Thirstrup, C., see Sakurai, M. 701, 705, 706 Thirstrup, C., see Stokbro, K. 701, 705, 706 Thiry, P., see Smith, N.V. 538 Thomann, U., see Höfer, U. 555, 564 Thomann, U., see Reuß, Ch. 556, 557 Thomann, U., see Shumay, I.L. 551, 555, 563, 564 Thomann, U., see Weinelt, M. 556, 557 Thomas, J.M. 233, 277 Thomas, L.H. 500 Thomas, W.-J., see Thomas, J.M. 277

Author index

Thomas, W.J., see Thomas, J.M. 233 Thompsett, D., see Hoogers, G. 293 Thompson, D.S., see Abeln, G.C. 726 Thompson, J.M.T. 351, 352, 409 Thompson, M.W. 888 Thomson, J. 459 Thon, A., see Gerstner, V. 715 Thon, A., see Lehmann, J. 564 Thonhauser, T. 500 Thönissen, M., see Frohnhoff, S. 849 Thullie, J. 411 Tian, R., see Balaji, V. 893 Tian, W., see Schlom, D.G. 791 Tian, Z. 470 Tiginyanu, I.M., see Langa, S. 822 Tilak, B.V., see Conway, B.E. 330, 331, 333 Timmerman, R.H., see Rogge, S. 695 Tip, A., see Können, G.P. 891 Tischler, M.A., see Collins, R.T. 845 Toader, O., see Leonard, S.W. 822 Toda, T. 291, 293 Todd, G., see Bauer, E. 764 Toennies, J.P. 56 Toennies, J.P., see Benedek, G. 56 Toennies, J.P., see Braun, J. 101 Toennies, J.P., see Brusdeylins, G. 55, 61 Toennies, J.P., see Celli, V. 56 Toennies, J.P., see Harten, U. 102 Toennies, J.P., see Hofmann, F. 64 Toennies, J.P., see Lock, A. 73 Toigo, F., see Silvestrelli, P.L. 735 Tok, E.S., see Shi, J. 829 Tokune, T., see Harutyunyan, A.R. 812 Tolk, N.H. 699 Tom, H.W.K., see Prybyla, J.A. 634 Tomellini, M. 239 Tomellini, M., see Molinari, E. 239 Tomii, T. 76, 79, 81 Tomii, T., see Kondo, T. 38 Tomishige, K., see Nakao, K. 240, 246 Tomkiewicz, M., see Glembocki, O.J. 850 Tommasini, F., see Dondi, M.G. 102 Tommasini, F., see Luntz, A.C. 102 Tompkins, J., see Manson, J.R. 55, 66 Tomsic, A. 44 Tong, W.M. 814 Toolenaar, F.J.C.M. 276 Topsøe, H. 330, 331 Topsøe, H., see Clausen, B.S. 799

Author index

Topsøe, H., see Helveg, S. 330, 331 Topsøe, H., see Jacobsen, C.J.H. 312 Topsøe, H., see Kibsgaard, J. 330 Topsøe, H., see Lauritsen, J.V. 331 Törnqvist, E., see Dahl, S. 296–298, 300, 311, 326, 327, 330, 781 Törnqvist, E., see Kasemo, B. 149, 463, 466, 476, 478, 482, 488, 509 Tosatti, E., see Eiguren, A. 534 Tossell, J.A., see Olthoff, J.K. 630 Toulouse, G., see Müller-Hartmann, E. 4, 7, 8, 14, 107 Toulouse, G., see Müller-Hartmann, T. 446 Tour, J.M., see Chen, J. 735 Tour, J.M., see Long, D.P. 592 Tour, J.M., see Reed, M.A. 738 Tour, J.M., see Shirai, Y. 751 Trail, J.R. 8, 15, 19, 148, 189, 493 Trail, J.R., see Bird, D.M. 148, 189 Tran, P., see Celli, V. 56 Trasatti, S. 331, 332 Traum, M.M., see Tolk, N.H. 699 Travers, L., see Harmand, J.C. 812 Traversaro, P., see Hodgson, A. 167, 168 Treusch, H.G., see Jandeleit, J. 826 Trevino, A.A., see Dumesic, J.A. 233 Treviño, A.A., see Dumesic, J.A. 305, 310 Trevisi, G., see Franchi, S. 837, 838, 840 Tribollet, J., see Dujardin, G. 602, 701 Tributsch, H., see Jaegermann, W. 330 Triller, A., see Dahan, M. 747 Trilling, L. 48 Tringides, M.C., see Yakes, M. 256 Trioni, M., see Brivio, G. 446 Tripa, C.E. 290, 672, 673 Trischberger, T., see Gokhale, S. 735 Triviño, A.A., see Dumesic, J.A. 326 Troisi, A., see Long, D.P. 592 Tromp, R.M., see Copel, M. 803 Tromp, R.M., see Hamers, R.J. 831 Tromp, R.M., see Hannon, J.B. 808, 810, 811 Tromp, R.M., see Horn-von Hoegen, M. 798, 803 Tromp, R.M., see Rotermund, H.H. 821 Tromp, R.M., see Theis, W. 775, 776 Trost, G., see Zambelli, T. 298, 330 Trost, J. 771 Trost, J., see Brune, H. 73, 493, 494 Trost, J., see Zambelli, T. 298, 330

981

Trout, B.L., see Chu, J.W. 306 Trucks, G.W. 846 Trucks, G.W., see Higashi, G.S. 847 Truhlar, D.G., see Chatfield, D.C. 174 Truhlar, D.G., see Schenter, G.K. 234 Tsarouhas, G.E. 412 Tsarouhas, G.E., see Vance, W. 411, 412 Tsekouras, A.A., see Li, Y.L. 482, 483, 493 Tsekouras, A.A., see Tate, M.R. 482, 483 Tsivlin, D.V. 773 Tsivlin, D.V., see Stepanyuk, V.S. 772, 774 Tsong, T.T. 694, 695, 771 Tsong, T.T., see Müller, E.W. 380, 402, 404 Tsu, R., see Lin, D.-S. 831, 835 Tsu, R., see Lubben, D. 830, 835 Tsuchiya, H. 822 Tsuda, H., see Yoshinobu, J. 735 Tsukada, M., see Hirose, K. 696 Tsuneyuki, S., see Aizawa, H. 844 Tsurumaki, H., see Matsuno, T. 829 Tsurumaki, H., see Rahman, F. 843 Tu, X.W., see Mikaelian, G. 734 Tucker, A.S., see Ward, C.A. 249 Tucker, J.R., see Avouris, Ph. 602, 616, 701, 705, 706 Tucker, J.R., see Lyding, J.W. 701, 703, 705, 706, 726 Tucker, J.R., see Shen, T.-C. 701, 704–706 Tufton, P.J., see Campbell, S.A. 850 Tully, C., see Wodtke, J. 4, 6, 19 Tully, J. 498 Tully, J., see Cheng, H.Z. 471 Tully, J.C. 3, 4, 6, 10, 25, 41, 42, 442, 447, 449 Tully, J.C., see Chabal, Y.J. 828, 830, 847 Tully, J.C., see Cheng, H. 6, 10, 12 Tully, J.C., see Grimmelmann, E.K. 36, 46 Tully, J.C., see Head-Gordon, M. 19, 46, 122, 125–127, 448, 449, 463, 484, 485 Tully, J.C., see Johnson, P.D. 447 Tully, J.C., see Kimman, J. 85, 97, 212 Tully, J.C., see Krishna, V. 485 Tully, J.C., see Kummel, A.C. 210, 218 Tully, J.C., see Li, X. 6 Tully, J.C., see Muhlhausen, C.W. 85 Tully, J.C., see Nordlander, P. 447, 448, 480, 481, 502, 565 Tully, J.C., see Rettner, C.T. 6, 31, 212 Tully, J.C., see Schlichting, H. 46, 111 Tully, J.C., see Shenvi, N. 6

982

Tully, J.C., see Sholl, D.S. 10 Tully, J.C., see Sitz, G.O. 210, 216, 218 Tully, J.C., see Tolk, N.H. 699 Tully, J.C., see Wodtke, A.M. 240, 434, 435, 496, 497, 512, 515 Tully, J.C., see Yan, T.Y. 41 Tünnermann, A., see Nolte, S. 906, 907 Turing, A.M. 355, 378 Turner, A.R. 828 Turner, D.R. 846 Turner, H.W., see Cong, P.J. 313 Turner, H.W., see Yaccato, K. 317 Turner, J.E., see Sales, B.C. 366 Turski, L.A., see Zaluska-Kotur, M.A. 256 Tüshaus, M., see Gardner, P. 357 Tyson, J.J. 385 Uberuaga, B.P., see Henkelman, G. 306 Uchida, H., see Igarashi, H. 293 Uchida, H., see Toda, T. 291, 293 Uda, T., see Kato, K. 245, 508 Ueba, H. 4, 22, 616, 617, 713 Ueba, H., see Komeda, T. 724 Ueba, H., see Paulsson, M. 597 Ueba, H., see Persson, B.N.J. 616 Ueba, H., see Tanimura, K. 723 Uebing, C. 256 Uebing, C., see Myshlyavtsev, A.V. 256 Uebing, C., see Nieto, F. 254 Uebing, C., see Tarasenko, A.A. 256 Uecker, H. 374, 387 Uecker, H., see Rafti, M. 374 Uetsuka, H., see Nakao, K. 240 Ukharskii, A.A.V.P.N., see Slinko, M.M. 410 Ukraintsev, V.A. 633, 671, 715 Ullrich, C.A., see Marques, M.A.L. 580 Unterwald, F.C., see Hull, R. 798 Untiedt, C., see Agrait, N. 591 Untiedt, C., see Smit, R.H.M. 592 Uosaki, K., see Ye, S. 849 Upward, M.D., see Moriarty, P. 691 Uram, K., see Jansson, U. 830 Uram, K.J. 830, 835 Urbasch, G., see Jandeleit, J. 826 Urbassek, H.M. 891, 893, 894 Urbassek, H.M., see Aderjan, R. 896 Urbassek, H.M., see Anders, C. 896, 897 Urbassek, H.M., see Balaji, V. 893 Urbassek, H.M., see Betz, G. 889

Author index

Urbassek, H.M., see Colla, T.J. 895, 896 Urbassek, H.M., see Conrad, U. 890 Urbassek, H.M., see Gades, H. 887, 890 Urbassek, H.M., see Schäfer, C. 905, 907 Urbassek, H.M., see Sibold, D. 908 Urbassek, H.M., see Waldeer, K.T. 889, 892 Urbassek, H.M., see Winterbon, K.B. 882 Uschmann, I., see Sokolowski-Tinten, K. 824 Ustinov, V.M., see Dubrovskii, V.G. 810 Usuda, K., see Kanaya, H. 849 Utsumi, Y., see Akazawa, H. 830 Utz, A.L., see Juurlink, L.B.F. 181 Utz, A.L., see Smith, R.R. 181 Utzny, C., see Bär, M. 389 Vach, H. 220 Vahala, K.J., see Saunders, W.A. 845 Vaidyanathan, N., see Paserba, K.R. 236 Valadez, L., see DeWitt, K. 836 Valbusa, U., see Luntz, A.C. 102 Valbusa, U., see Scoles, G. 31 Valbusa, U., see Vattuone, L. 150, 172 Valla, T. 544 van Bavel, A.P. 251, 254 van Bavel, A.P., see Hermse, C.G.M. 254 van Breugel, Y., see Cobden, P.D. 374 van Buuren, T. 849 van de Ven, J., see Boonekamp, E.P. 849 van den Berg, A., see Mijatovic, D. 791 van der Avoird, A., see Fassaert, D.J.M. 279 van der Drift, E., see Blauw, M.A. 827 van der Merwe, J.H., see Frank, F.C. 799, 802 van Driel, H.M., see Leonard, S.W. 822 van Driel, H.M., see Sipe, J.E. 827 van Driel, H.M., see Young, J.F. 827 van Essenberg, W., see Jenniskens, H.G. 628, 629 van Hardeveld, R. 290, 297 van Harrevelt, R. 146 van Harrevelt, R., see Hellman, A. 304, 325, 327, 328 van Hemert, M.C., see Smit, R.H.M. 592 van Hove, M.A. 246, 290, 357 van Hove, M.A., see Cerda, J. 585 van Kampen, N.G. 404 van Montfoort, A., see van Hardeveld, R. 290, 297 van Noort, W.D., see Carabineiro, S.A.C. 377 van Overstraeten, R., see Jain, S.C. 796

Author index

van Ruitenbeek, J., see Smit, R.H.M. 592 van Ruitenbeek, J.M., see Agrait, N. 591 van Santen, R.A. 233, 277, 304, 307, 310, 311 van Santen, R.A., see Ciobîcã, I.M. 302 van Santen, R.A., see Gelten, R.J. 362 van Santen, R.A., see Hermse, C.G.M. 254 van Santen, R.A., see Riedmüller, B. 45 van Stralen, J.N.P., see McCormack, D.A. 179, 180 van Stralen, J.N.P., see Watts, E. 161, 167, 169, 183, 184, 189 van Tol, M.F.H. 370, 374, 403 van Veenendaal, M.A., see Lof, R.W. 747 van Wunnik, J.N.M. 7, 12 Vanag, V.K. 378 Vance, W. 411, 412 VanDenHoek, P.J., see Horn, T.C.M. 49 Vanderbilt, D. 796 Vanderbilt, D., see Meade, R.D. 796 Vandoni, G. 769 Vang, R.T. 298, 300, 781 Vanmaekelbergh, D., see Kooij, E.S. 846, 847 VanWunnik, J.N.M., see Rasser, B. 467 Varandas, A.J.C., see Hellman, A. 304, 325, 327, 328 Varga, P., see Schmid, M. 73 Vargoz, E., see Rusponi, S. 779 Vasquez, R.P. 847 Vassell, W.C., see Jaklevic, R.C. 529 Vattulainen, I., see Hjelt, T. 256 Vattulainen, I., see Masin, M. 256 Vattuone, L. 31, 150, 172 Vattuone, L., see Savio, L. 158, 159 Vayenas, C.G. 349 Vazhappilly, T. 675 Vega, J.L. 254 Vega, J.L., see Guantes, R. 254 Vegard, L. 795 Veigel, C. 744 Veigel, C., see Molloy, J.E. 744 Velic, D. 558 Velic, D., see Bartels, L. 4, 558, 687, 699 Velic, D., see Wolf, M. 555 Venables, J.A. 771, 775, 816 Venugopalan, V., see Vogel, A. 904 Verdasca, J. 385 Verheijen, M.A. 810 Veronese, M., see Gambardella, P. 778, 781 Veser, G. 371, 374, 375, 385, 388, 410

983

Veser, G., see Imbihl, R. 348, 411 Veser, G., see Khrustova, N. 410 Vestergaard, E.K., see Vang, R.T. 298, 300, 781 Viala, C., see Sadhukhan, S.K. 748 Vicanek, M. 879, 889 Vicanek, M., see Rethfeld, B. 899 Vicente, J.L., see Irurzun, I.M. 411 Vicente, J.L., see Rafti, M. 374 Vicente, L., see Chavez, F. 362 Vicente, L., see Perera, A. 373, 386 Vickers, J.S., see Klitsner, T. 728 Vidal, F. 904, 905 Vidali, G. 99 Vieira, S., see Agrait, N. 591 Vielhaber, W., see Modl, A. 210 Vielhaber, W., see Segner, J. 208, 209 Viernow, J., see Himpsel, F.J. 817 Viescas, A.J., see Johnson, P.D. 447 Villullas, H.M., see Perez, J. 331 Vincent, J.K., see Díaz, C. 471, 513, 514 Vincent, J.K., see Hellman, A. 304, 325, 327, 328 Virojanadara, C., see Balasubramanian, T. 543, 560 Visart de Bocarme, T. 403 Vishnevskii, A.L. 362 Vitali, L. 546, 550, 561, 562 Vitali, L., see Wahl, P. 609 Vittadini, A. 829 Vogel, A. 904 Vogel, W., see Hartmann, N. 366 Vogelgesang, R., see Wahl, P. 550 Voigtlander, B., see Andersohn, L. 834 Völkening, S. 257, 380, 401 Volkening, S., see Sachs, C. 345, 380, 397, 401, 403 Voll, S., see Lehmann, J. 564 Volmer, M. 803 Volpe, A.F., see Yaccato, K. 317 von Barth, U. 433 von der Linde, D., see Rethfeld, B. 824, 903 von der Linde, D., see Sokolowski-Tinten, K. 824 von Hofe, T., see Corriol, C. 548, 564 von Neumann, J. 439 von Niessen, W., see Kortluke, O. 373 von Niessen, W., see Kuzovkov, V.N. 362 von Oertzen, A. 368, 390, 393, 394 von Oertzen, A., see Nettesheim, S. 385, 386

984

von Oertzen, A., see Rotermund, H.H. 381, 390, 392 von Weizsäcker, C.F. 500 Vondrak, T. 558, 715 Vosko, S., see Perdew, J. 433 Voss, C. 370, 403 Voter, A. 304, 306, 322 Vrijmoeth, J., see Günther, C. 780 Vukmirovic, M.B., see Zhang, J.L. 291 Vvedensky, D.D., see Joyce, B.A. 807, 837, 838 Wacaser, B.A., see Johansson, J. 810, 812 Wachman, H.Y., see Goodman, F.O. 36, 44, 46, 71, 483 Wade, C.P. 849 Wagner, C. 775 Wagner, H., see Ciraci, S. 829 Wagner, R.S. 808, 809 Wagner, S. 675 Wagner, S., see Luntz, A.C. 675 Wahl, M. 888, 891 Wahl, P. 550, 609 Wahl, P., see Vitali, L. 546, 550, 561, 562 Wahlström, E. 772 Wahnström, G. 494, 719 Wahnström, G., see Engdahl, C. 494, 719 Wahnström, G., see Karlberg, G.S. 257, 259 Wahnström, G., see Li, Y. 448 Wahnström, G., see Ovesson, S. 771, 775 Wahnström, G., see Sundell, P.G. 237, 255 Wakayama, Y., see Bandyopadhyay, A. 737 Wako, S., see Moula, M.G. 240 Walch, S.P. 842 Waldeer, K.T. 889, 892 Walecka, J.D., see Fetter, A.L. 612 Walker, J.E., see Abrahams, J.P. 744 Walkup, R.E. 458, 616, 694, 696 Walkup, R.E., see Avouris, Ph. 546, 602, 616, 699, 701, 704–706 Walkup, R.E., see Dujardin, G. 601, 687, 699, 718 Walkup, R.E., see Shen, T.-C. 701, 704–706 Wallace, R.M., see Cheng, C.C. 735 Wallace, R.M., see Wilk, G.D. 797 Wallauer, W., see Höfer, U. 555, 564 Wallauer, W., see Reuß, Ch. 553 Wallauer, W., see Shumay, I.L. 551, 555, 563, 564

Author index

Walldén, L., see Andersson, D. 465, 476, 477, 482 Walldén, L., see Kasemo, B. 434, 459, 460, 465, 466, 475–477, 482, 512 Walldén, L., see Lindgren, S.A. 471, 529 Walldén, L., see Wahlström, E. 772 Wallenberg, L.R., see Persson, A.I. 810, 812 Wallis, R.F., see Black, J.E. 107 Wallis, R.F., see Maradudin, A.A. 841 Waltenburg, H.N. 828 Walther, H., see Frenkel, F. 207 Walther, H., see Segner, J. 208, 209 Walther, H., see Vach, H. 220 Wang, B., see Zhao, A. 728 Wang, C., see Grill, L. 740, 751 Wang, C., see Shen, T.-C. 701, 704–706 Wang, C.L., see Lauhon, L.J. 813 Wang, C.S. 512 Wang, C.X. 810 Wang, D. 808 Wang, D.-S., see Wang, J.-T. 695 Wang, D.L., see Cui, Y. 808 Wang, E.G., see Wang, J.-T. 695 Wang, F., see Bartels, L. 638, 715 Wang, F., see Liu, J.L. 808 Wang, F., see Veigel, C. 744 Wang, F.D. 807 Wang, H. 808 Wang, J., see Boger, K. 555, 556 Wang, J., see Gudiksen, M.S. 809 Wang, J., see Roth, M. 555, 556, 564 Wang, J.-T. 695 Wang, J.F., see Cui, Y. 808, 810 Wang, J.G. 367 Wang, L.C., see Kreuzer, H.J. 695 Wang, L.G. 804, 807 Wang, N., see Lee, S.T. 810 Wang, S.C. 765, 768 Wang, W. 592, 600 Wang, W., see Zhao, A. 728 Wang, W.-N., see Li, Z.-H. 720 Wang, W.U., see Cui, Y. 808 Wang, W.U., see McAlpine, M.C. 808 Wang, X.Y. 900 Wang, X.Y., see Osgood Jr., R.M. 529 Wang, Y. 831, 834, 835, 837 Wang, Y., see Bronikowski, M.J. 835 Wang, Y., see Perdew, J.P. 277, 328 Wang, Z.S. 146, 159–161, 170, 171, 186, 187

Author index

Wang, Z.S., see Darling, G.R. 151, 152, 187, 188 Waqar, Z., see Dobrin, S. 696 Ward, C.A. 249 Warren, P., see Dutartre, D. 828 Warrender, J.M., see Crouch, C.H. 827 Wassermann, B., see Hla, S.-W. 686 Watanabe, F., see Ehrlich, G. 765, 771 Watanabe, K. 470, 633 Watanabe, K., see Gruzdkov, Y.A. 633 Watanabe, K., see Matsumoto, Y. 633 Watanabe, M., see Igarashi, H. 293 Watanabe, M., see Toda, T. 291, 293 Watanabe, S. 842, 847, 849 Watanabe, Y. 44 Watson, R.E., see Weinert, M. 293 Watts, E. 161, 167, 169, 183–186, 188, 189 Watts, E., see Gostein, M. 167, 183 Watts, E.K. 221 Watts, R.K. 851 Wayner, D.D.M., see Allongue, P. 846 Wayner, D.D.M., see Lopinski, G.P. 696 Weaver, J.F. 31, 836 Weaver, J.H., see Aldao, C.M. 844 Weaver, J.H., see Nakayama, K.S. 844 Webb, M.B., see Mo, Y.-W. 816, 831 Weber, A., see Volmer, M. 803 Weber, H.B., see Reichert, J. 738 Weber, W. 781 Weckesser, J., see Barth, J.V. 775 Wegner, D. 548 Wegner, D., see Bauer, A. 548 Wehner, S. 413 Wei, H. 384, 396 Wei, H., see Marbach, H. 398 Wei, T.C. 363 Wei, Z., see Baltuška, A. 551 Weibel, E., see Binnig, G. 579, 685, 701 Weida, M.J., see Petek, H. 558, 565 Weide, D., see Budde, F. 631, 644 Weide, D., see Mull, T. 646 Weigand, P., see Misewich, J.A. 635 Weik, F. 625, 626, 669 Weinberg, H.W., see Strasser, P. 293 Weinberg, W.H., see Campbell, C.T. 513 Weinberg, W.H., see Cheng, C.C. 735 Weinberg, W.H., see Cong, P.J. 313 Weinberg, W.H., see Gergen, B. 4, 15, 148, 334, 335, 455, 482, 489–492

985

Weinberg, W.H., see Huang, C. 735 Weinberg, W.H., see Kang, H.C. 233, 250 Weinberg, W.H., see Mayne, A.J. 735 Weinberg, W.H., see Mullins, C.B. 48, 49, 122 Weinberg, W.H., see Nienhaus, H. 148, 240, 489–491, 493 Weinberg, W.H., see Rettner, C.T. 48, 49 Weinberg, W.H., see Self, K.W. 735 Weinberg, W.H., see Sinniah, K. 715, 829 Weinberg, W.H., see van Hove, M.A. 246 Weinberg, W.H., see Widdra, W. 735 Weinberg, W.H., see Yaccato, K. 317 Weinelt, M. 556, 557, 563, 564 Weinelt, M., see Boger, K. 552, 555, 556 Weinelt, M., see Fauster, Th. 556, 557 Weinelt, M., see Reuß, Ch. 556, 557 Weinelt, M., see Roth, M. 555, 556, 564 Weinelt, M., see Schäfer, A. 552, 553, 560, 564 Weinert, M. 293 Weinert, M., see Hulbert, S.L. 553 Weinert, M., see Jennings, P.J. 501 Weisker, T., see Adlhoch, W. 371 Weiss, G.H., see Maradudin, A.A. 64 Weiss, M., see Boszo, F. 325 Weiss, M., see Bozso, F. 299 Weiss, M., see Ertl, G. 325 Weiss, N. 782 Weiss, N., see Cren, T. 780 Weiss, N., see Rusponi, S. 781 Weiss, P.S. 765 Weiss, P.S., see Pascual, J.I. 592 Weiss, P.S., see Stranick, S.J. 771 Weiße, O. 72, 73, 175, 183 Weiße, O., see Ambaye, H. 74, 75 Weiße, O., see Binetti, M. 73, 149, 462, 494, 495, 499, 500 Weiße, O., see Komrowski, A.J. 73, 494, 495, 500 Weisskopf, V.F., see Blatt, J.M. 3 Weldon, M.K., see Queeney, K.T. 849 Welland, M.E., see Saifullah, M.S.M. 739 Wellegehausen, B., see Nolte, S. 906, 907 Weller, M.R., see Weller, R.A. 892 Weller, R.A. 892 Wellershoff, S.-S., see Bonn, M. 900 Welling, H., see Nolte, S. 906, 907 Wellner, A., see Koker, L. 847 Wells, M.G., see King, D.A. 172 Wenzel, W., see Heurich, J. 738

986

Werner, H. 370 Werner, H.-J., see Manolopoulos, D.E. 510, 512 Werner, K. 835 Werner, P., see Fan, H.J. 807 Werner, P., see Schubert, L. 811 Wesenberg, C., see Ambaye, H. 74, 75 Wesenberg, C., see Weiße, O. 72, 73, 175, 183 Westerberg, B., see Fridell, E. 319 Wethekam, S., see Schuller, A. 41 Wetli, E., see Wider, J. 395 Wetzig, D. 178 Wetzl, K., see Eiswirth, M. 346–348, 359, 378 Wetzl, K., see Moeller, P. 362 Weyers, S.J., see Gergen, B. 4, 15, 148 Weyers, S.J., see Nienhaus, H. 491 Whaley, K.B., see Yu, C.-F. 102 Wharton, L., see Becker, C.A. 671 Wharton, L., see Cowin, J.P. 164 Wharton, L., see Janda, K.C. 82, 199 White, J.A. 154, 168, 498 White, J.D. 189–191, 334, 433, 466, 496, 511, 512 White, J.D., see Chen, J. 511, 512 White, J.M., see Costello, S.A. 623 White, J.M., see Hatch, S.R. 672 White, J.M., see Huang, W.X. 320 White, J.M., see Jo, S.K. 647, 843 White, J.M., see Koel, B.E. 623 White, J.M., see Mahajan, A. 828 White, J.M., see Pressley, L.A. 628, 629 White, J.M., see Wolf, M. 625, 626, 647 White, J.M., see Zhou, X.L. 623 White, J.M., see Zhu, X.-Y. 647, 649, 659 Whitesides, G.M., see Metiu, H. 3 Whitlow, H.J., see Andersen, H.H. 889 Whitman, L.J. 695 Whitten, J.L., see Jing, Z. 715, 832 Wicke, E. 346, 347 Wicke, E., see Beusch, H. 345 Wicke, E., see Schüth, F. 371 Widdra, W. 735 Widdra, W., see Gokhale, S. 735 Widdra, W., see Huang, C. 735 Widdra, W., see Self, K.W. 735 Wider, J. 395 Wiechers, J., see Brune, H. 73, 493, 494 Wiechers, J., see Wintterlin, J. 685, 766 Wiersma, D.A., see Baltuška, A. 551 Wierzbicki, A. 304

Author index

Wiesenekker, G., see Kroes, G.J. 168, 179 Wiets, M., see Schäfer, A. 552, 553, 560, 564 Wiggers, H., see Kravets, V.G. 845 Wight, A., see Hodgson, A. 169 Wight, A.C. 223, 225 Wight, C.A., see Benderskii, V.A. 4, 5 Wigner, E. 304, 305, 435 Wigner, E.P. 3, 21 Wigner, E.P., see von Neumann, J. 439 Wilcox, J.P., see Chianelli, R.R. 330, 331, 333 Wilk, G.D. 797 Wilk, G.D., see Frank, M.M. 791 Wilke, S. 498 Wilke, S., see Gross, A. 146, 159, 463 Wilkins, J.W., see Stiles, M.D. 113, 123 Wilkins, J.W., see Wingreen, N.S. 9, 21 Williams, A., see Lang, N.D. 449–454 Williams, A.R., see Binnig, G. 551 Williams, A.R., see Lang, N.D. 278, 283, 565 Williams, K.L. 844, 845 Williams, K.R. 850 Williams, L.R., see Muhlhausen, C.W. 85 Williams, R.S. 842 Williams, R.S., see Tong, W.M. 814 Wilson, E.B. 76 Wilzén, L. 103, 163, 462 Wilzén, L., see Andersson, S. 102, 104, 105, 113, 114, 120, 124, 150, 157, 163, 462 Wilzén, L., see Persson, M. 105, 117 Wimmer, E. 449 Wind, R.A. 824, 850, 851 Wind, S.J., see Jacobs, P.W. 414 Windham, R.G., see Bartram, M.E. 320 Winfree, A.T. 353 Wingreen, N.S. 9, 21 Wingreen, N.S., see Madhavan, V. 608, 609 Wingreen, N.S., see Meir, Y. 592 Winkler, A., see Rendulic, K.D. 183 Winter, H. 38, 41 Winter, H., see Auth, C. 481, 502 Winter, H., see Borisov, A.G. 565 Winter, H., see Pfandzelter, R. 41 Winter, H., see Schuller, A. 41 Winterbon, K.B. 882 Winters, H.F. 49, 50, 827, 843–845 Wintterlin, J. 321, 401, 402, 671, 685, 719, 766, 828 Wintterlin, J., see Brune, H. 73, 493, 494, 765, 766

Author index

Wintterlin, J., see Mendez, J. 717 Wintterlin, J., see Trost, J. 771 Wintterlin, J., see Völkening, S. 257 Wintterlin, J., see Zambelli, T. 298, 330, 718 Wio, H., see Hildebrand, M. 398 Wise, M.L. 715, 830 Wiskerke, A.E., see Geuzebroeak, F.H. 210 Wiskerke, A.E., see Guezebroek, F.H. 162 Wittich, G., see Wahl, P. 609 Wittig, C., see Korolik, M. 220 Wittmaack, K. 886, 887, 889 Wll, Ch., see Celli, V. 56 Wodtke, A.M. 3, 189, 240, 334, 433–435, 455, 496, 497, 512, 515 Wodtke, A.M., see Chen, J. 511, 512 Wodtke, A.M., see Gulding, S.J. 177 Wodtke, A.M., see Hou, H. 178, 222, 334, 433, 512 Wodtke, A.M., see Huang, Y. 148, 189, 190, 221, 222 Wodtke, A.M., see Huang, Y.H. 334, 433, 463, 511, 512 Wodtke, A.M., see Ran, Q. 471, 513 Wodtke, A.M., see Silva, M. 511, 512 Wodtke, A.M., see White, J.D. 189–191, 334, 433, 466, 496, 511, 512 Wodtke, J. 4, 6, 19 Wojciechowski, K.F., see Kiejna, A. 449, 450 Wolf, D., see Fery, P. 357 Wolf, D.E. 792 Wolf, D.E., see Nozières, P. 792, 800 Wolf, M. 555, 563, 625, 626, 647 Wolf, M., see Bartels, L. 4, 558, 687, 688, 699, 724 Wolf, M., see Bonn, M. 483, 674, 900 Wolf, M., see Denzler, D.N. 675 Wolf, M., see Frischkorn, C. 19, 22, 470 Wolf, M., see Gahl, C. 558, 559 Wolf, M., see Hasselbrink, E. 644, 646, 661 Wolf, M., see Hertel, T. 553, 660, 663 Wolf, M., see Hotzel, A. 563 Wolf, M., see Knösel, E. 556, 563, 564 Wolf, M., see Luntz, A.C. 675 Wolf, M., see Misewich, J.A. 635 Wolf, M., see Velic, D. 558 Wolf, M., see Wagner, S. 675 Wolf, M., see Zhu, X.-Y. 647, 659 Wolff, J. 419, 420, 821 Wolff, J., see Dicke, J. 381

987

Wolff, P.A., see Schrieffer, J.R. 609 Wolkow, R.A. 829 Wolkow, R.A., see Alavi, S. 616 Wolkow, R.A., see Lopinski, G.P. 696, 735 Wolkow, R.A., see Mezhenny, S. 735 Wolkow, R.A., see Patitsas, S.N. 726 Wolkow, R.A., see Piva, P.G. 696, 730 Wöll, C., see Braun, J. 101 Wöll, C., see Fuhrmann, D. 496 Wöll, C., see Harten, U. 102 Wöll, Ch., see Lock, A. 73 Woll, E.J. 484 Wong, C.M. 563 Wong, C.M., see Gaffney, K.J. 558 Wong, C.M., see McNeill, J.D. 563 Woodruff, D.P. 293 Woodward, L.A. 76 Woodward, R.B., see Hoffmann, R. 274 Woratschek, B. 459, 472, 473 Woratschek, B., see Sesselman, W. 456, 458, 473 Wortis, M., see Rottman, C. 799 Wöste, L., see Manz, J. 20 Wouda, P.T., see van Tol, M.F.H. 403 Wright, S. 633 Wright, S., see Dippel, O. 828 Wu, B., see Junren, S. 545 Wu, C., see Her, T.-H. 827 Wu, S.W. 714, 717, 746, 747 Wu, X.-G. 362 Wu, X.D., see Zhang, J. 828 Wu, X.L., see Balasubramanian, T. 542, 543 Wu, Y., see Barrelet, C.J. 808 Wu, Y.M. 830, 834, 835 Wucher, A. 891 Wucher, A., see Samartsev, A.V. 895 Wucher, A., see Staudt, C. 890 Wucher, A., see Wahl, M. 888, 891 Wulff, G. 799, 842 Wurpel, G.W.H., see Brouwer, A.M. 744 Wurth, W. 558, 566 Wurth, W., see Föhlisch, A. 566 Wyatt, R.E. 3 Wyld, H.W. 68 Wynblatt, P. 775 Xia, L.Q. 833, 835, 836 Xia, L.Q., see Engstrom, J.R. 834–837 Xia, L.Q., see Jones, M.E. 832, 833

988

Xia, L.Q., see Maity, N. 837 Xia, X. 849, 850 Xia, X.H. 849, 850 Xiang, H., see Zhao, A. 728 Xiao, X., see Zhao, A. 728 Xiao, X.D., see Zheng, C.Z. 256 Xie, Q. 839 Xie, X.C., see Stokbro, K. 701, 705, 706 Xu, G.Q., see Cao, Y. 720 Xu, G.Q., see Dixon-Warren, St.J. 630 Xu, L. 551 Xu, Y., see Nørskov, J.K. 302, 303 Xue, Y. 735 Yaccato, K. 317 Yagu, S. 76, 78, 79, 81 Yagu, S., see Tomii, T. 76, 79, 81 Yagyu, S., see Kondo, T. 38 Yakes, M. 256 Yakovkin, I.N., see Petrova, N.V. 254 Yamada, K., see Kanaya, H. 849 Yamaguchi, H., see Watanabe, Y. 44 Yamamoto, S. 36, 46 Yamamoto, S., see Kondo, T. 38, 76–78, 80, 81 Yamamoto, S., see Tomii, T. 76, 79, 81 Yamamoto, S., see Yagu, S. 76, 78, 79, 81 Yamamoto, S.Y. 381 Yamamoto, T. 306 Yamamura, Y., see Kitazoe, Y. 894 Yamanaka, T., see Sano, M. 672 Yamazaki, H., see Makino, O. 842, 847 Yamazaki, S., see Liu, C. 775 Yamazaki, T., see Bjorkman, C.H. 847 Yan, C., see Hanisco, T.F. 211, 213, 216 Yan, C., see Jensen, J.A. 462, 493 Yan, C., see Self, K.W. 735 Yan, T., see Isa, N. 44 Yan, T.Y. 41 Yang, C., see Wang, D. 808 Yang, J., see Zhao, A. 728 Yang, J.J., see Li, Y.L. 482, 483, 493 Yang, J.J., see Tate, M.R. 482, 483 Yang, J.S.Y., see Dobrin, S. 696 Yang, P., see Hochbaum, A.I. 808 Yang, P.D., see Hu, J.T. 809 Yang, S.H., see Dixon-Warren, St.J. 630 Yang, W.L., see Junren, S. 545 Yang, Y.L., see D’Evelyn, M.P. 830 Yao, Y., see Long, D.P. 592

Author index

Yasada, R., see Noji, H. 744 Yaskolko, V., see Oster, L. 466 Yates, J.T., see Bozso, F. 735 Yates, J.T., see Dohnálek, Z. 715 Yates, J.T., see Goodman, D.W. 315 Yates, J.T., see Sinniah, K. 715 Yates, J.T., see Tripa, C.E. 290 Yates, J.T., see Ukraintsev, V.A. 715 Yates, R.T., see Ramsier, R.D. 468 Yates Jr., J.T. 290, 298 Yates Jr., J.T., see Cheng, C.C. 735, 830 Yates Jr., J.T., see Mezhenny, S. 735 Yates Jr., J.T., see Sinniah, K. 829 Yates Jr., J.T., see Tripa, C.E. 672, 673 Yates Jr., J.T., see Waltenburg, H.N. 828 Yates Jr., J.T., see Zhukov, V. 73 Yates Jr., J.T., see Zubkov, T. 302 Yayon, Y. 765, 766 Ye, M. 908 Ye, S. 849 Yelenin, G.G., see Berdau, M. 356, 366, 383 Yelon, A., see Sacher, E. 846 Yen, R., see Shank, C.V. 824 Yeom, H.W., see Matsui, F. 735 Yeung, C.K., see Zheng, C.Z. 256 Yeyati, A.L., see Agrait, N. 591 Yi, I., see Oyabu, N. 694 Yi, S.I., see Widdra, W. 735 Yilmaz, M.B. 828–830 Ying, S.C., see Ala-Nissila, T. 254 Ying, S.C., see Merikoski, J. 256 Ying, Z. 623 Yingling, Y.G., see Zhigilei, L.V. 903 Yoffe, A.D. 803, 837 Yokota, R., see Nakai, I. 257 York, R.L., see Somorjai, G.A. 414 Yoshimori, A. 473, 474 Yoshimori, A., see Makoshi, K. 473, 474 Yoshinobu, J. 735 Young, J.F. 827 Young, J.F., see Sipe, J.E. 827 Young, R.D. 765 Young, R.D., see Gurney, T. 765 Young, S., see Dunstan, D.J. 799 Younkin, R., see Shen, M.Y. 827 Yourdshahyan, Y. 149, 498–500 Yourdshahyan, Y., see Hellman, A. 245, 433, 434, 475, 480, 496, 499, 501–503, 506, 508 Yourdshayan, Y., see Hellman, A. 334

Author index

Yu, C.-F. 102 Yu, C.F., see Cowin, J.P. 164 Yu, H., see Wang, F.D. 807 Yu, M.L. 459, 467 Yu, Y., see Liu, J.L. 808 Yuan, Z., see Langhammer, C. 470 Yulikov, M. 275 Zacharias, H. 178 Zacharias, H., see Budde, F. 634, 635 Zacharias, H., see Misewich, J. 220 Zacharias, H., see Schröter, L. 178 Zacharias, H., see Wagner, S. 675 Zacharias, H., see Wetzig, D. 178 Zacharias, M., see Fan, H.J. 807 Zaera, F. 234 Zaitsev, A.A., see Bublik, V.T. 795 Zakharov, N.D., see Schubert, L. 811 Zaluska-Kotur, M.A. 256 Zaluska-Kotur, M.A., see Yakes, M. 256 Zambelli, T. 298, 330, 718 Zambelli, T., see Trost, J. 771 Zanette, D.H., see Manrubia, S.C. 410, 411 Zangwill, A. 125, 449 Zare, R.M., see Mathews, C.M. 513 Zare, R.N. 20 Zare, R.N., see Hines, M.A. 214 Zare, R.N., see Jacobs, D.C. 208, 215 Zare, R.N., see Kolasinski, K.W. 829 Zare, R.N., see Kummel, A.C. 210, 218, 219 Zare, R.N., see McClelland, G.M. 207, 208 Zare, R.N., see Michelsen, H.A. 175 Zare, R.N., see Oldenburg, R.C. 460 Zare, R.N., see Shane, S.F. 829 Zare, R.N., see Sitz, G.O. 82, 205–207, 210–212, 216, 218, 219 Zaremba, E. 99, 100, 127, 129 Zaremba, E., see Chizmeshya, A. 99–102 Zaremba, Z. 500 Zazzera, L. 847 Zboray, S., see Berenbak, B. 42, 43 Zeifman, M.I., see Zhigilei, L.V. 903 Zeiri, Y. 672 Zeiri, Y., see Asscher, M. 3, 50 Zeiri, Y., see Binetti, M. 73, 149, 495, 499, 500 Zeiri, Y., see Katz, G. 148, 499, 507–509 Zel’dovich, Y.B. 905 Zeller, R., see Dederichs, P.H. 500 Zeller, R., see Gambardella, P. 778, 781

989

Zener, C. 439–441 Zeng, H.C., see Dixon-Warren, St.J. 630 Zeppenfeld, P. 686, 689, 690 Zeppenfeld, P., see Lehner, B. 250 Zewail, A.H. 20 Zgrablich, G. 254 Zhakhovski, V.V., see Anisimov, S.I. 904 Zhakhovskii, V.V. 903 Zhang, C.J., see Michaelides, A. 304 Zhang, G., see Celli, V. 56 Zhang, J. 828 Zhang, J., see Mokler, S.M. 831 Zhang, J., see Neave, J.H. 817, 818 Zhang, J., see Shi, J. 829 Zhang, J., see Turner, A.R. 828 Zhang, J.L. 291 Zhang, J.Z.H., see Dai, J.Q. 175 Zhang, J.Z.H., see Wyatt, R.E. 3 Zhang, R., see Liu, J.L. 808 Zhang, Z., see Junren, S. 545 Zhang, Z., see Li, Y.L. 482, 483, 493 Zhang, Z.Y., see Naumovets, A.G. 254 Zhao, A. 728 Zhao, H., see Hodgson, A. 167, 168 Zhao, Y., see Shirai, Y. 751 Zhdanov, V.P. 233, 234, 236, 238, 239, 241, 244–251, 254–263, 323, 334, 356, 373, 374, 383, 386, 408, 411, 433, 475, 499, 508 Zhdanov, V.P., see Chakarov, D. 482 Zhdanov, V.P., see Hellsing, B. 257 Zhdanov, V.P., see Myshlyavtsev, A.V. 256 Zhdanov, V.P., see Olsson, L. 261 Zhdanov, V.P., see Uebing, C. 256 Zheng, C.Z. 256 Zheng, Y.D., see Liu, J.L. 808 Zhigilei, L.V. 826, 903 Zhigilei, L.V., see Ivanov, D.S. 899 Zhigilei, L.V., see Leveugle, E. 905 Zhigilei, L.V., see Schäfer, C. 905, 907 Zhong, Q., see Gahl, C. 558, 559 Zhong, Z., see Wang, D. 808 Zhong, Z.H., see Cui, Y. 808 Zhou, C., see Reed, M.A. 738 Zhou, X.J., see Junren, S. 545 Zhou, X.L. 623 Zhu, Q., see Zhao, A. 728 Zhu, X.-Y. 647, 649, 659 Zhu, X.-Y., see Hatch, S.R. 672 Zhu, X.-Y., see Vondrak, T. 558, 715

990

Zhu, X.-Y., see Wolf, M. 625, 626 Zhu, X.-Y., see Zhou, X.L. 623 Zhu, X.Y. 558 Zhu, Y., see Igarashi, H. 293 Zhukov, V. 73 Zhukov, V.P. 539 Zhukov, V.P., see Chulkov, E.V. 7, 20 Zie, M.H., see Zhang, J. 828 Ziegler, J.F. 877, 880–882, 884 Ziff, R.M. 355, 357, 407 Ziff, R.M., see Fichthorn, K.A. 407 Zikovsky, J., see Piva, P.G. 696, 730 Zimmermann, F.M. 623, 644, 655–657, 829, 830 Zimmermann, F.M., see Asscher, M. 644 Zimmermann, F.M., see Yilmaz, M.B. 828–830 Zöphel, S., see Meyer, G. 689, 690

Author index

Zorić, I., see Hellman, A. 245, 334, 433, 434, 475, 480, 496, 499, 501–508 Zorić, I., see Komrowski, A.J. 506, 507 Zorić, I., see Langhammer, C. 470 Zorić, I., see Oner, D.E. 498 Zorić, I., see Österlund, L. 72, 73, 149, 245, 334, 455, 495, 496, 498, 507 Zorić, I., see Ternow, H. 507 Zubkov, T. 302 Zubkov, T.S., see Tripa, C.E. 290 Zuburtikudis, I. 414 Zuelicke, C., see Bär, M. 356, 383, 384 Züger, O., see Dürig, U. 689 Zülicke, L., see Nikitin, E.E. 3 Zupan, A., see Perdew, J.P. 433 Zwartkruis, A., see Fondén, T. 467 Zykov, V.S., see Mikhailov, A.S. 385 Zyrianov, P.V., see Bronnikov, D.K. 204

Subject index 1-decanethiol 49 1-decanethiol monolayers 1D chains 773 3d series 287, 292 4d series 287 5d series 287 SCF method 502

activator–inhibitor models 352 active site 327, 328, 330 activity 276, 314, 327, 331 adatom 548, 890, 896 adatom formation energy 763 adiabatic – account 498 – approximation 433, 437 – aspects 441 – descriptions 433 – PES 435, 498, 499, 504 – PES’s questioned 511 – region 438 – representation 3, 6, 7, 448 – state 435, 439 – transition 6, 7, 442 adiabatic, adiabaticity 433, 435–437 adlevel resonance 480 adlevel-Fermi-level crossing mechanism (strong NA) 479 adsorb 275 adsorbate 282, 325, 326, 579, 581, 588, 593, 599–601, 608, 609, 612, 617 – electron structure change 442, 444 – incident on a metal surface the NA way 493 – interaction 320 – levels broadened 442 – levels shifted 442 – resonance crossing Fermi level 444, 464 – states 565 adsorbate-induced – electron structure 277, 441, 453 – local density 454 – surface phase transition 357 – vibration structure 441 adsorbate-resonance model 451 adsorbate-substrate system 278 adsorbate-valence level 283 adsorbed species 278

44

ablation 792, 793, 809, 824–827, 845 ablation threshold 904–907, 910 ablation yield 906 absolute rate 304 absorption of photons 463 abstracted Cl atoms identified by REMPI 483 abstraction 462, 493, 498, 515 – channel 495 – – dominates 494 – of an oxygen atom 502 – scenario for O2 495 abstraction and ballistic motion 494 abstraction contribution about 70 per cent 495 abstraction energies measured 483 abstraction for about 50 per cent 495 abstraction is favored for end-on molecules 494 abstractive chemisorption 515 abundance of neutral O atoms 495 access to more emitted electrons 491 accumulating evidence that coupling of largeamplitude molecular vibration to metallic electron degrees of freedom can be much stronger even at the lowest accessible incidence energies 512 activated 278 – adsorption of N2 on Ru(0001) 496 – dissociation 463 – state 234, 246, 252, 258 activation barrier 304, 311, 333, 441 activation energy 235–237, 247, 249, 270, 278, 295, 297, 299, 300, 310, 319–322, 326, 328, 330, 442

991

992

adsorption 31, 45, 47, 235–240, 244–250, 252– 258, 260, 261, 263, 277, 278, 284, 304, 320, 323, 332, 433, 797, 808, 814, 820, 822, 827– 837, 842, 845, 847 – dynamics 441 – energetics 313 – energy 270, 279, 281, 282, 286–288, 290–292, 294, 299, 304, 313, 314, 333, 334, 763 – mechanisms 449 – structure 320 – system 278 adsorption–desorption equilibrium 239, 248, 256, 258, 263 adsorption-induced resonance crossing Fermi level 485 adsorption-mechanism – for ionic alkali adsorption 451 – for polarized-atom alkali adsorption 451 – for resonance alkali adsorption 451 affinity and Fermi level crossing 474 affinity level of an atom outside a metal 453 affinity resonance derived from 4σ ∗ 469 affinity-level variation 480 Ag(111) 32, 38, 546, 549, 550, 625, 628–630, 633 agree with experimental damping-rate data 486 alkali 548, 563, 565 alkanes 236 alloy 278, 298 alloying 275, 276, 290 almost complete conversion of the singlets 473 aluminum metal 497 ammonia 649, 659–661, 664 – productivity 329 – synthesis 261, 263, 295, 296, 308, 311, 313, 317, 324, 325, 327, 328 amplitude turbulence 411 AN process 473 Anderson orthogonality 13 angle-resolved photoelectron spectroscopy 535 angular distribution 72–79, 89 angular momentum 209, 215 – alignment of 215–217 – orientation of 217–219 – polarization 215–219 anisotropic etching 792, 824, 844 anisotropy 378, 388 – of the polarizability tensor 102, 103 another abstraction channel 495 antibonding 280, 285, 288 – density of states 452 – LUMO resonance 479 – state 442, 558 Antoniewicz mechanism 469

Subject index

Antoniewicz model 624 apparent Arrhenius parameters 235, 237 appreciable at larger atom-surface distance 473 area coverage 332 Arrhenius expression 310 Arrhenius form 307 associative desorption 300, 434, 463 – at higher coverage measures desorption from the terrace 513 – of H2 from Cu(111) adiabatic 485 – of N2 from Ru(0001) nonadiabatic 485 asymmetry 504, 505 atmospheric chemistry 456 atom climbing ladder 458 atom surface collisions 55 atom–surface – interaction 442 – potential 456 – scattering 63, 64, 66 atom-beam scattering (ABS) 456 atom-caused emission of molecule 458 atomic – adiabatic level 479 – adsorption 462 – adsorption energy 276 – affinity level rapidly down-shifted 482 – chains 773 – collisions 436 – defects 890 – desorption 462 – oxygen 491 – scattering 462 – switch 458, 461, 514 – time scale 318 – vibrations in Xe potential well 458 atomic and molecular scattering from surfaces 72 atomic and molecular surface scattering 60 atomic-scale experiments on friction 471 atomic-scale friction 486 atoms adsorbed close to each other 494 attachment barriers 776 attempt frequency 307 Au(111) 49, 546 Auger decay 443, 465 Auger deexcitation (AD) 457, 461, 476 – of empty state derived from affinity level 488 Auger neutralization (AN) 457, 461, 472, 479 Auger process 479 Auger rate 467 Auger transition 454, 460, 478, 489 Auger-type process (I) 474 autodetachment 475 average separation of about 80 Å 494

Subject index

avoided crossing

993

438, 463

B5 site 327 backfolded Brillouin zone 565 background subtraction 56, 89 ballistic transport 491 ballistic travelers from metal film surface to Schottky interface 490 band structure calculations 562 Bardeen approximation 588 barrier for dissociation 294, 315 basis functions 437, 438 Baule result 105 Be(0001) 543 Be(1010) 543 below superconductivity transition temperature EHP energy gap 486 benchmark values 502 benchmark-type experiments 455 Benjamin–Feir instability 410 benzene 558 BEP plot 302, 303 BEP relation 300–302, 309, 314–316 better detectability 491 beyond BOA 431, 434 bi-metallic core-shell islands 781 bifurcation 351 – diagram 352 binary collisions 36 binding energy 279, 288, 442 – of reaction intermediate 332 bioluminescent reactions 471 bistability 356, 383 bistable catalytic cycle 259 blackbody radiation 130 BOA 333 – breakdown 433, 434 – in surface chemistry? 496 BOA may break down 487 Boltzmann factor 307 Boltzmann plot 206 bond – breaking 281, 300, 302 – energy 281 – making 281 – making and breaking 434, 462, 463, 482, 515 – order 504 – strength 330 bond-cleavage reaction 302 bonding 280, 285 bonding–antibonding picture 450 bonds elongation 511 Boreskov–Horiuti–Enomoto rules 257 Born and Oppenheimer 439

Born–Oppenheimer approximation (BOA) 4, 25, 234, 240, 305, 431, 433–435, 437–439, 442, 443, 463, 484, 497, 502, 511, 650, 666, 676 bosonized 7 bounded electron gas 532 break bond 278 breakdown of nearly-adiabatic friction 493 Breit–Wigner lineshape 537 broadening of adsorbate level 278 broadening of vibrational levels of chemisorbed species 486 broader range of systems to study 491 Brønsted–Evan–Polanyi relationship 299, 333 Brønsted–Polanyi relation 249, 252 bulk diffusion 324 Burton–Cabrera–Frank theory 815 C–O stretch mode on Cu(100) 485 calculated and experimental results share key features 502 calculations of multidimensional effects on dissociation of N2 on Ru(0001) 514 called oxide model 366 cannot predict sticking of NO, a spin doublet 509 canonized Gadzuk resonance model 452 capture and loss processes 467 carbon monoxide methanation process 314 carbon uptake 296 cascade of EHP’s 509 catalysis 271, 273, 277, 304, 308, 334, 433 – informatics 313 catalyst 277, 278, 313, 325, 327, 330, 332–334 – design 313, 333 – development 333 – materials 271 catalytic – activity 324, 333 – converter 273 – growth 807, 813 – nanodiode 491, 492 – NO reduction 370 – processes 435 – rates 271 – reactions 233–235, 237, 252, 259, 262, 263, 295, 317, 319, 325, 433, 435 – reactivity 323 CCl4 630 Celli diagram 461 cellular patterns 387 central nearest neighboring force constant model 107 certain other thermal chemical reactions 515 chance for a chemicurrent 509

994

change in slope and threshold behavior 512 chaos 259, 260 chaotic oscillations 351 characteristic times for electronic and nuclear motions comparable 515 characterize extent of CT from the surface 507 charge carriers travel ballistically to interface 490 charge exchange in helium scattering 441 charge transfer (CT) 3, 4, 6, 7, 12, 14–16, 18, 20, 275, 447, 449, 558 charge-transfer induced particle emission 487 charge-transfer model with dissipation to EHP’s 499 charging 715, 730–732, 734 chattering 161, 163 chemical – activity 275 – coupling 484 – energy released in exothermic reactions at metal surfaces 490 – gas–surface reaction 463 – hole diving model 465 – potential 248, 249, 251, 252, 256 – process 275 – reaction 271, 324, 717, 726, 728 – – dynamics 434, 435 – – heat 471 – reactions at surfaces 496 – reactions in molecular assemblies 455 – reactions on metal surfaces 455 – surface reactions 496 – transition state 497 – trapping 319 chemically induced EHP’s 490 chemicurrent 15, 433, 434, 487, 491, 493, 502 – attenuation lengths 491 – correlation with turnover rate 492 – measurements 491, 496 – more effective than exoelectron emission 492 chemicurrents measure also electrons below workfunction 489 chemielectronics 491 chemiluminescence 455, 490 – spectra 460 chemiluminescence hard to observe 489 chemiluminescent reactions 471 chemisorbed oxygen atoms randomly distributed 494 chemisorption 278, 280, 334 – bond 278, 288 – energy 286, 291, 293–296 – region 453 chemistry involving excited electrons 435 chlorine on pure and sodium-doped Zr 477

Subject index

Chulkov 529 Cl atom ejection 482 Cl chemisorbed on surface 483 Cl ejected 483 Cl2 and Cl− 2 PES variation 478 Cl2 molecules impinging on an Na surface 476 Cl2 on Al(111) 478, 482 Cl2 on K 476, 483 Cl2 PEC’s 479 classical energy loss 110 classical model of sticking 105 classical molecular-dynamics calculations 498 classical multiphonon limit 65 classical regime of sticking 122 classical scattering 71 classical trajectory calculations 514 classical-mechanical multidimensional Langevin equation 449 closed-packed surface 289, 300, 303 cluster diffusion 779 cluster emission 885, 890, 891 cluster impact 876, 892, 895, 896, 909 CN and CO internal stretch modes 485 CO disproportionation 274 CO methanation 271 CO on Cu(100) 485 CO oxidation 239, 240, 242, 254, 257, 259, 260 CO2 methanation 271 collective excitations 563 collision 440, 456 – cascade 882–886, 889, 890, 892, 895 combination of DFT 515 combustion 455 common minimal entrance-channel energy barrier 498 competition between time scales 503 complete quantum dynamics 507 complex configurations 511 complex Ginzburg–Landau equation 382 complex reactions 271 composite surfaces 415 computational power 313 computational screening 271, 317 computational theory 330 computer-based method in search for catalyst 325 concept 278 conceptual description 451 conceptual framework 277, 278 conclusions 514 conditions at metal/silicon interface 491 configuration integral 305 configuration space 321

Subject index

conformation 733, 736, 741, 750 consequences of PES crossing 484 conservation of spin can lead to excited state 508 consistent with change in excitation mechanism 513 consistent with exit-channel nature of PES barrier 513 consistent with QMB data for Kr monolayer 486 constant velocity 476 contact process 273 continua of low-energy EHP excitations 434 continuous spectrum of EHP’s vs. just the lowest EHP 508 continuum 4, 11, 12, 14, 16, 18, 23 – of EHP’s directly responsible for S-shape 508 control of bond breaking at transition-metal surfaces 492 controlled by nozzle temperature 498 controlling – chaotic systems 413 – magnitude of chemicurrent yield 493 – patterns 419 – turbulence 413 controversial issues 498 conventional thermal catalysis 435 conversion of CO 313 conversion of molecular vibrational energies 466, 511 convert a molecule impinging on a surface into an ion 464 convert ionic adsorbate to a neutral K atom 482 coordination number 289 copper 314 core levels 14 correlation 277, 299, 314, 335, 447 correlation stretch-mode damped and level crossing 485 corrosion 497 “corrugated wall” 97 couple to the metallic electron degrees of freedom 496 coupling 284, 289 – between the two states 440 – between vibrations of Xe atom and EHP’s 458 – matrix element 288, 289 – parameter 533 covalent (Na + Cl) 439 covalent state 436 coverage 298, 319, 320, 322, 323, 333, 475 cracking of alkanes 273 crater formation 893, 896 creating EHP’s 493

995

critical size 384 critical thickness 798, 799, 807, 838, 839 criticism 487 cross correlation 634, 635 cross section 626, 628–633, 647–649, 659, 670, 675, 878–880, 882, 905 crossed molecular beam NA reaction 433 crossing with the Fermi level in reaction zone 479 crystal face dependence of S0 113 Cs 548 CT and discrete-PES’s models 508 CT model 500, 503, 508, 509 CT process 455, 480 Cu(001) 551 Cu(111) 546 cube model 32, 44, 46 current monitors Mg oxide island nucleation and growth 492 curve crossing 6, 12, 15, 438, 439, 493 curve seam 15 CVD 808, 811, 828–830, 834, 837 cyclohexene 493 d electrons 280, 531 d states 280, 290, 291, 295, 297, 333 d states at step 297 d-band 283, 288, 292 – center 287–289, 292–294, 296, 297 – effect 282 – metal 279 – model 271, 278, 280, 287, 288, 290, 295–297, 302 – shift 293, 297 damping – into EHP’s 513 – of nuclear motion 484 – rate of a vibration 484 database of computed materials 313 debate 514 debated topic 487 Debye model 534, 562 Debye–Waller factor 109, 111 decay – of a vibration in a weakly adsorbed molecule 484 – of admolecular modes into EHP’s 469 – of admolecular modes into phonons 469 – processes 502 – rate of hole 476 decomposition 322 deconvolution of the kinetic-energy distribution 472

996

deep Cl 3p hole 482 deexcitation 472 – by Auger transition 479 – channel 476 – mechanisms of a metastable atom 461 – process involving ET 466 – via electrons 456 – via ions 456 defect 463, 796, 798, 799, 810, 822, 825, 827, 831, 834, 844 – annealing 892 – distribution 883 – scattering 555 defect-mediated turbulence 411 dehydrogenation 302 delayed ET 497, 500 density and distribution of interfacial states in MOS device 491 density functional theory (DFT) 25, 271, 276, 277, 433, 451, 498, 500, 511, 579 density of states 280, 590, 596, 604, 607, 613, 614 – in the filled conduction band 472 dependence on incident energy 493 dependence on isotope 493 deposited energy 883, 893–895, 904 deposition 433 descriptor 277, 278, 314, 333 descriptors for adiabaticity 503 descriptors for metal catalysts 271 design by catalysis informatics 313 design of catalyst 296 – through modeling 271, 313 desorb 275, 315 desorption 3, 13, 19–24, 48, 234, 236, 237, 246– 252, 254, 257, 278, 313, 320, 322, 433, 456, 685, 687, 693, 695, 696, 699, 701–706, 715, 717, 723–726, 728, 814, 820, 826–832, 834, 835, 844 – dynamics 125, 660, 663–665 despite lack of energy barrier in entrance channel 502 detailed balance 125, 320 detect hot charge carriers 491 detected as a chemicurrent in diode 489, 490 detection efficiency 491 detection of O atoms in the gas phase sensitive 495 deterministic chaos 409 develop predictive understanding of surface reactivity beyond the BOA is an important challenges to current research 515 development of the LDOS in the course of dissociative molecular adsorption on Mg 450

Subject index

DFT 271, 276–278, 281, 287, 293, 298–300, 307, 314, 318, 319, 323–325, 327, 328, 330, 332, 333, 579–581, 588, 597, 605, 608, 613, 617, 618 – very important tool in surface science 515 DFT-based – SCF method 500 – calculation 485 – “locally-constrained” method 500 – “penalty-functional” method 500 DFT-calculated friction coefficients 513 DFT-KMC method 317–319, 322, 325 diabatic – descriptions 433 – PES 436, 466, 500, 504 – picture 476, 499 – representation 3, 6, 440, 441 – states 243, 436 – transitions 3, 7, 10, 18 diabatic curve crossing and tunneling 513 diagonal elements 440 diagonal representation 439 diatomic molecule 303 diatomic system 436 DIET 514 difference in trends and magnitudes 486 different GGA’s 501 differential conductance 545 differing time scales for ET and nuclear motion 500 diffraction 154–156 – and physisorption 156, 157, 163, 164 – rotationally elastic/in-elastic 164, 165, 168, 169 – vibrationally elastic/in-elastic 168, 169 diffusion 236, 238–240, 248, 254–256, 259, 260, 263, 320, 804, 808, 811–821, 825, 829, 831, 834, 835, 841–843 – barrier 322 – long chainlike molecules 256 diffusional coupling 348 DIMET 635, 636 dipole – activity 135 – excited transitions 128 – layers 449 – matrix elements 129 – moment 282 – – function 133 – – matrix elements 133 dipole–dipole interaction 247 direct – abstractive chemisorption 482

Subject index

– and indirect adsorbate interactions 319 – and precursor-mediated abstractive chemisorption 482 – contact 686, 688, 693, 700, 728 – conversion of vibrational to electronic excitation 513 – dissociation 462 – electronic excitation 482 – evidence of NA energy dissipation during adsorption 490 – infrared desorption 130, 132 – infrared photodesorption 128, 133 – measurement of reaction-induced hot electrons and holes 490 – NA processes 454 – observations of ‘hot’ electrons 511 – process 498 – – at high incident translation energy 482 – – at high surface temperature 482 discrete number of diabatic PES’s 499 discrete PES’s model 507 discrete set of PES’s 507 discrete-PES’s model 508 dislocation 798, 799, 802, 803, 838 disordered catalytically active surface 492 dispersed metal 274 displaced oscillator 8, 22 displacement trapping 49 dissipation 3, 7, 8, 10, 12, 13, 18, 455, 490, 612 – to EHP’s 463, 471 – to phonons 471 dissipative – forces 471 – process 455 dissociate 280, 288 dissociated 476 dissociation 209, 213, 302, 313, 315, 325, 326, 329, 483, 687, 692, 695, 707–720, 723–726, 728, 736 – by ET 482 – dynamics – – and molecular rotation 175–178 – – and molecular vibration 178–181 – – and quantized librations 174–176 – – and steering 172, 175 – – and trapping 172, 173 – – direct 172 – – energy scaling 173, 174 – – incidence angle dependence 173–175 – – microscopic reversibility 177, 178 – – surface temperature dependence 183–185 – – tunnelling 174, 176 – – vibrational efficacy 179–181 – of N2 297

997

– of oxygen molecule on Al(111) 497 – rate 312 – rate-determined model 309 – via abstraction 494 dissociation of oxygen molecules 498 dissociative – adsorption 3, 434 – – of O2 molecule 497 – – of simple molecules 495, 497 – attachment 436, 441 – chemisorption 315, 316 – – energy 286, 311, 328 – molecular adsorption 450 – recombination 441 – states 507 – sticking probability 498 distinct electronic signals for diverse species 492 distorted wave Born approximation 113 distribution – angular 199, 200, 209, 224 – internal state 203 – rotation state 205–208, 223 – rotational polarization, see angular momentum – translation 204 – velocity 200, 204 dividing surface 306 DOS near the Fermi level 475 DOS of valence bands 472 dynamic – bistability 385 – dipole polarizability 99 – electron transfer 455 dynamical – correction 306 – matrix 611, 614, 615, 617 – process 498 – – at metal surfaces 435 dynamically created hole 476 dynamics 435, 436 – of bond making and breaking at surfaces 463 – of ET processes 455 earlier exoelectron emission 490 edge – length 332 – of nanoparticle of MoS2 330 – row 327 – site 331 – structure 330 EELS 514 effective – electron potential at a jellium surface – ionization energy near surface 472

449

998

– ionization potential 472 – mass 36, 44, 306, 562 efficient electronic dissipation 500 EHP – contribution to the sticking probability 445 – created at Ag and Cu surfaces by adsorption 490 – excitation 502 – – limit lives of mechanical modes 483 – in substrate during one round trip 444 Ehrenfest theorem 6 Ehrlich–Schwoebel barrier 817, 819 eigenfunction 437 eigenstate 437 eigenvalue 435 ejected O-atoms 495 ejected O− ions 465 ejection – of K atoms 482 – of negative ions 464 – of neutral Cl atoms 482 elastic scattering 456, 462 – data 111 electric field 456 electrochemical cell 331 electrochemistry 330 electron – affinity 221, 222, 476 – damping absent 487 – density 281 – emission 466, 477, 480, 482, 496 – – varies between different stages 488 – emitters in TV tubes 449 – energy-loss spectrum 128, 131 – escape over surface barrier 472 – gas friction 19 – population 480 – processes at solid surfaces 431 – structure 277, 278, 281, 283, 287, 297, 304 – times 464 electron and ion spectroscopy according to Hagstrum 471 electron structural 271 electron tunneling – and molecular motion compete 504 – competes with nuclear motion of adsorbate 503 – time 505 electron–electron and electron–phonon scattering 527 electron–electron scattering 560 electron–hole pair 3, 4, 15, 16, 18 – excitation 188–191 electron–phonon 533, 560 – coupling 563, 900, 902

Subject index

– – parameter 538 – scattering 533 electron–vibration coupling 596, 597, 599, 616 electron-emission process 476 electron-energy transfer between nascent HF and Br 433 electron-energy-loss spectroscopy (EELS) 470 electron-hole pair (EHP) excitation 442, 444, 455, 457 electron-molecule collisions 436 electron-spectroscopical method tied to Hagstrum 471 electron-stimulated desorption 470 electron-vibration coupling 594 electronegative adsorbate 463 electronegativity 439, 503 electronegativity and Pauli repulsion determine curve crossing 504 electronegativity of molecule 504 electronic – damping in sticking 484 – density profile at the jellum surface 449 – device 491 – effect 276, 278, 295, 297, 300 – energies 438 – excitation 334, 484, 490, 687, 688, 696–701, 704, 715, 718, 728, 733, 736, 741, 744–746 – factor 274, 275, 333 – friction 484 – functions 438 – Hamiltonian 435–437 – heat conduction 900, 907 – mass enhancement 433 – matrix elements of He 437 – mechanism 485, 486 – motion 503 – NA and spin-polarized adsorbate 494 – NA effects 496, 515 – stopping 880–882 – surface states 528 – time scale 318, 334 – tunneling time vs. passage time 504 electronically adiabatic 98 – PES 274 electronically adiabatic mechanical mechanism 512 electrons and photons created by NA 483 electrons at and outside the surface 473 electrostatic energy 281, 282 elementary excitations 455 elementary process 319 elementary reaction steps 497 Eley–Rideal mechanism 239, 459, 462, 466

Subject index

Eliashberg coupling function 545 Eliashberg function 533, 534 ellipsometry 380 emission of – atoms (abstraction) 497 – electromagnetic radiation 459, 465 – electrons 457, 482, 487 – exoelectrons 466, 495 – exoelectrons and photons manifest NA 487 – ions 487, 497 – molecules (abstraction) 497 – neutrals 493 – photons 487 – photons by radiative decay 482 empirical friction parameters 449 empty site 296 endothermic 321, 323 energetic requirements for NA 487 energetically allowed on Al(111) 495 energetics 319 energy – accommodation 433 – barrier 307 – broadening 472 – dependence of the matrix element 472 – dependence of the sticking probability 498 – diagram 315 – dissipation 487, 499 – dissipation from vibrationally excited adsorbates 434 – distribution 466 – exchange to phonons 460 – landscape 434 – loss to EHP’s 484 – loss to lattice 513 – resolved spectra 72, 77, 81, 82 – transfer 8, 36, 55, 56, 58, 60, 62, 63, 65, 67, 72, 77, 80–82, 89, 90, 97 – – into EHP 483 – – into phonons 484 energy-dissipation channel 434 energy-level shifts 450, 451 energy-resolved spectra 61, 78, 80, 83 enhanced efficiency of EHP excitations 471 ensemble average of the occupation 482 ensemble effect 276, 293 enthalpy and entropy of activation 235 entrainment 411 entrance channel 442 entrance-channel activation barrier 498 entropy contribution 307 environmental problem 273 enzyme catalyst 334 epitaxial 792, 796, 797, 805, 828, 831, 838

999

– growth 319, 434, 497, 763 epitaxy 791, 828, 831, 835, 837, 838, 841 equilibrium crystal shape 799, 815 equipotential-energy curves 443 ESD 514 ET – at large molecule-surface separations 476 – between sp-band and frontier orbitals 499 – between the Xe atom and the surface and tip 458 – from a metal surface to an incident or adsorbed particle 482 – occurring much closer to surface 500 – provides a chance for neutralization 494 – this time called RI 476 – time 444 etching 433, 791–793, 819–824, 827, 828, 830, 842–851 ethanol reforming 302 evidence – by dynamical calculations 493 – for abstraction mechanism 494 – pointing at the ubiquitous contribution of NA effects to dynamics at 515 – reported from QCM experiments 487 ex situ experiment 330 example of abstraction 494 excess atmosphere of oxygen 319 excess energy about 2.8 eV 495 excess energy given to EHP’s 458 exchange 277, 335 – current density 331, 332 – of energy between molecular motion and EHP’s at metal surfaces 448 exchange-correlation energy 281, 282, 328, 329 excitation – mechanism 444 – of electronic system 490 – probability 445 – process 444 excited hole 480, 500 excited PES’s 500 excited states in solids 435 excitons 455 exhaust-gas cleaning 275 exit channel 658, 659, 662, 665, 666 exiting O− ions 465 exoelectron 477, 482, 487, 490, 502 – emission 333, 434, 455, 466, 476, 477, 479, 487, 488 – – at oxidation 489 – – upon adsorption 489 – emission/chemicurrents 514

1000

– yield dependence on incident velocity 479 exoelectron-emission experiments 466 exoelectrons hard to observe 490 exoemission 456, 487 exolectron-emission study of dimer dissociation for Cl2 on K 494 exothermic 321 – surface reaction 434, 476 experimental 508 – design 313 – findings 279 experimentally confirmed Hagstrum picture 509 experimentation study in predicting 316 explicit experimental proof of ET 459 explicit picture of the dissociation dynamics 480 exponential distance dependence 502 external and internal exoelectron emission 515 external potential 281 F2 on Si(100) 483 face dependence of workfunction 449, 451 facetting 362, 396 factors describable in adiabatic terms 431 failure of adiabatic dynamics 513 Fano line shape 22 Fano-like line shapes 124 far from equilibrium 256, 259 fast-ion scattering 457 fcc three-fold hollow site 320, 322, 323 femtochemistry 20, 462 femtosecond timescale 551 Fermi level 7, 16–18, 442, 443 – phase shift 15 Fermi-distribution function 480 Fermi’s Golden Rule 536 fertilizer 273 field electron microscopy 402 field emission microscopy (FEM) 380 field emitters 449 field ion microscopy (FIM) 380, 402 fields that affect surfaces 455 filling of adsorbate-induced hole state 443 filling of d states 286 film-slip time 486 final state 305 final-state energy 301, 302 final-state interactions 472 first ET occurs closer to the surface 500 first-layer electronic Raman scattering 470 first-principles estimates – of breakdown of BOA 485 – of friction 486 FitzHugh–Nagumo (FHN) model 353 flow-reactor experiment 319, 323

Subject index

fluctuation–dissipation theorem 484, 612 fluctuations 404 fluorescent 456 focus on PES of the triplet state of the O2 molecule 499 force constant 306 forced-oscillator model (FOM) 106, 493 forces 435, 436 form bond 278 formation 322, 323 – of ions 495 – of the stoichiometric aluminum oxide 497 four descriptors 503 four electronically distinguishable PES’s 507 fraction of incident molecules that become trapped 498 fractional occupancy of the MO resonance 481 fractions of bare ions (He++ ) 467 Franck–Condon factor 14, 15 Franck–Condon transition 643 Frank–van der Merwe growth 802 free rotor 216 frequencies of adsorbate vibrations 450 friction 3, 4, 8, 13, 19, 449, 484, 486, 488, 499 – approach 637 – coefficient 238, 240, 254, 448, 484 – coupling 484 – treatment 448 – values much larger than typical 513 friction-like 471 friction-type sticking (weak NA) 479 frictional damping 484 Friedel oscillation 546, 770 Friedel sum rule 17, 19 from former state to latter 500 front nucleation 383, 384 frontier orbitals 461 frozen-density approximation 282 fuel cell 293, 313, 330 full molecular wavefunction 440 fully quantum-mechanical treatments 498 function of translational energy 496 gallium arsenide (GaAs) 795, 796, 802, 803, 809, 811, 817, 822, 824, 829, 838–842 gas flow model 893 gas–surface – dynamics 462 – interactions 433, 452 – reaction 434, 463 gas-phase O atoms 495 gaseous oxygen 497 gasoline 273

Subject index

Gaussian 9, 10, 22 generalized-gradient approximation 282 geometric effect 276, 278, 297, 299, 300, 302 geometric factor 274–276, 333 germanium (Ge) 795, 796, 798, 799, 801, 803, 808, 811, 813, 822 GGA 320, 321, 329, 501 gist of hindered spin-flip model 509 glancing incidence and low surface temperature 483 global coupling 348, 386 global (subcritical) Hopf bifurcation 352 glow lamps 433 grain size 493 grand canonical distribution 248 graphite 44 great possibility for abstraction 500 Green’s function 284, 595, 603, 604, 613–615 ground-state adiabatic PES 442 ground-state O2 molecules 496 group orbital 285 group velocity 549 growth 433, 434, 471, 515 growth modes 792, 801, 803, 804, 806, 816, 817, 819, 820 GW 530 H and D atoms incident on Cu(111) surface 493 H2 at Mg(0001) 443 H2 dissociation on Cu(111) 498 He matrix 439 Hagstrum picture 499 halogen adsorption on group IIIA and IA 477 halogens on sodium 477 hard sphere scattering 36 hard-cube model 35, 219 harmonic – entrainment 411 – normal-mode analysis 326 – oscillations 359 – oscillator 7, 9, 12, 21 – transition-state theory 306, 325, 326 harpooning 17, 464, 476, 479, 481 hcp site 322 HDS 273 He∗ (21 S) 473 He∗ (23 S) 473 heating rate 249–251, 253, 255 HER 330, 331 heteroepitaxy 780 heterogeneous catalysis 233, 237, 269, 273, 275, 277, 278, 299, 304, 317, 319, 324, 330, 334, 434, 462, 497, 781 heterogeneous catalytic process 304

1001

high diffusion barrier for chemisorbed oxygen atoms 494 high number of trajectories of particles 498 high number of wavepackets 498 high predictivity for kinetic phenomena 515 high vibrational-friction value 513 high workfunction hinders all but very energetic electrons 490 high-resolution electron energy-loss spectroscopy 128 high-velocity neutral-atom ejection 482 higher value on the vertical affinity 503 highly excited NO molecules 497 highly exothermic surface reactions 490 hindered spin-flip model 509 – special for O2 509 hole 443, 502 – left behind in substrate 444 – survival probability 479, 500 – survives in outer parts 489 hole–decay 540 HOMO 461 HOMO-LUMO gap 463 homogeneous – catalysis 273, 334 – catalyst 330 – electron 531 homogeneously catalyzed 309, 311 Honda electrolytic cell 468, 514 Hopf bifurcation 351 hopping 454 – matrix element 447 hot electron 434, 468, 470, 482, 512 hot electrons measured with ultrathin metal-film Schottky diode detectors 490 “hot” precursors 239–241 hot-atom mechanism 494 hot-charge-carrier excitation and transport 491 hot-electron 4, 19, 20 – interpretation of DIET 470 HREELS 130, 131 HTST 306, 307 hydrodesulphurization 273 hydrogen 292, 332 – diffusion 237, 256 – evolution reaction 317, 330, 331 – production 302 hydrogenation 316, 317 hyperthermal 31 – atom/ion scattering 497 – Cl 482 hysteresis 356, 368

1002

ideally electronegative adsorbates on low-workfunction metals 488 IETS 592, 593, 595, 597–601, 615, 616 image – force 452, 453, 461 – plane 453 – potential 7, 11, 452, 553 – – of the Jennings saturated form 501 – shifts 453 image-force attraction 476 image-potential shift – of the electron-affinity level 453 – of the electron-ionization level 453 image-potential states 528, 550, 551, 563 implantation 49, 881, 889 impurities 463 in situ experiment 330 indirect infrared photodesorption 135 individual events 472 individual isolated O-atoms observed in STM 495 induced dipole moment 453 induced LDOS for alkali adsorption 454 induction period 364 industrial catalyst for HDS 330 industrial conditions 325, 327 industry 275 inelastic atom scattering 461 inelastic electron tunneling spectroscopy 593 inelastic scattering 202, 456, 462 – direct 199, 208, 214 – trapping-desorption 200, 208, 223 infrared spectroscopy 276 infrared-reflection spectroscopy (IRS) 469 initial electron detection sensitivity 334 initial state 305 initial sticking 334, 434, 497 – calculated in CT model 504 – of O2 on Al(111) 504, 509 – probability 495, 499 initiated by perturbation 456 INS 515 instantaneous adiabatic ground state 448 interaction between two adsorbates 296 interaction potential 32, 57, 62, 64, 70, 81, 85 interband decay 554 interband transitions 561, 563 intermediate – excited states 472 – localized surface plasmon 470 – positive-energy trapping 121 – state 23, 470 intermediate electronegativity values 506 intermediates 441

Subject index

intermode coupling 616, 617 internal degree-of-freedom 199, 220, 223 internal exoelectron emission 448 interplay between experiment and theory 277, 317, 330 interpolation model 294 interpolation principle 294 intra-molecular vibration 7, 12 intraband scattering 554, 560 intraband transitions 565 intrinsic resonance width 118 involve ET 454 ion – dissociates 480 – ejection 514 – emission 479 – neutralization 472 – scattering 434, 457 – source 449, 457, 459 ion-neutralization spectroscopy (INS) 333, 458, 460, 467, 471 ion-surface image force 479 ionic adsorption well outside surface 454 ionic and neutral adparticles 453 ionic (Na+ + Cl− ) 439 ionic state 436 ionization at hot tungsten surface 457 irreversibility 455 IRS 483 island growth 802, 803, 807, 837, 841 islands 797–799, 801–807, 814–817, 819, 820, 829, 831, 834, 837–839, 841 isotope effect 647–649, 653, 659–661, 675 – for deuterated CH3 O on Cu 485 isotope mass effect 509 Jahn–Teller effect 433 jellium-model calculations 451 K on graphite exposed to photons 482 kinematic effects 538 kinematic enhancements 124 kinematical factor 123 kinetic – analysis 463 – description 325 – effect 321, 323 – energy 281 – – distribution of ejected electrons 472 – model 489 – Monte Carlo simulations (KMC) 304, 435 – oscillations 259, 260 – phase transitions 259

Subject index

kink 289 KMC 304, 318, 320–323 Knotek–Feibelman mechanism 468 Kohn–Sham version 281 Kr monolayers solidifying on Au and Ag surfaces 486 Kramers model 238 lack of time dependence 515 “ladder climbing” 123 Landau–Zener 10 – analysis 508 – formula 441 – model 243–245 Landau’s quasiparticles 538 Langevin dynamics 19 Langevin equation 6 Langevin-type classical dynamics 484 Langmuir–Hinshelwood mechanism 239, 256, 462, 466 Langmuir–Hinshelwood scheme 356 lanthanide 548 Laplacian 437 large rearrangements in the adsorbate electronic structure close to the surface 499 large-amplitude – molecular vibrations that couple to EHP’s 515 – vibrations 510 – – of adsorbates can excite EHP’s 512 large-area, ultrathin-film, Schottky diode devices 490 laser – ablation 876, 898, 903, 904, 907 – excitation 22 – heating 419 – induced fluorescence 203, 208, 215, 220 – spectrometry 494, 497 laser spectroscopy 495 laser-induced associative desorption (LAAD) 513 lateral adsorbate interaction 320, 321 lateral interactions 233, 236, 245–247, 249–259 lattice constant 795, 804, 840 lattice kinetic Monte Carlo approach 304 lattice misfit 795, 799, 803 lattice-gas approximation 245, 247, 248 layer growth 817–819, 838 layer-by-layer growth 801–803, 817, 838 LDOS 473 lean-burn gasoline 319 level 493 levels are shifted and broadened 453 Liapunov exponent 409 LIF 640

1003

lifetime 481, 527, 529, 547, 601, 602, 606, 615 – estimates 484 – in CT model 502 – of an electronically excited state 502 lifetimes of surface states 552 ligand effect 276, 288, 292, 294 light absorption 898 lightworms 433 like Cl2 /Na 500 limit ω0 T h¯ 447 Lindhard energy 877 line shape 117 linear cascade 882, 895 linear coupling model 114 linear-collision cascade 873, 882, 885, 891, 892 linear-response theory 467, 468 local density of states (LDOS) 443, 454 localisation of surface states 549 localized potential 14–17 localized surface plasmons 434 locally-constrained DFT (LC-DFT) method 501 long-range interactions 770 Lorentzian-like line shapes 124 low energy electron microscopy (LEEM) 380 low probability for radiative transitions 490 low temperatures a regime where electronic mechanisms discerned 487 low thermal sticking probability 503 low-dimensional MDEF – with ab initio friction 464 – with first-principles friction 486 low-temperature plasmas 455 low-workfunction surface 463, 465, 473 – to enhance yield 513 lowering of workfunction by cesiation 449, 451 lowest order in screened electron–ion interaction 485 LUMO 462 MAES 456, 457, 473, 514 magnetism 781 magnitude of the deexcitation 493 magnons 455 making a Cl− 2 ion 481 making and breaking bonds at surface 463 manifold and great variety of NA processes at surface 514 manifold of dynamical manifestations of NA and ET 456 many-body effects 472 many-body picture 535 many-body theory 529, 536 many-state problem 436

1004

mapping of PES 498 margarine 273 mass 437 – ratio 201, 211, 214 – transport 319, 399 master equation 500 – for ET processes 447 mathematical modeling 381 matrix element 284, 437 Matthews equation 799 Maxwell–Boltzmann distribution 633, 641, 645, 656, 657 MBE 791, 798, 802, 808, 809, 811, 817, 831, 834, 837, 838, 842 MD trajectories followed until O–O bond stretched beyond 2.4 Å 508 MDEF, with parametrized electronic friction 513 mean free path 117, 119 mean number of phonons 109 mean-field microkinetic model 308, 325 mean-free method 304 measured – dependence on vibrational temperature 513 – exoelectron current 494 – kinetic-energy distribution 494 – velocity exceeds incidence velocity 483 mechanical work 456 mechanism (II): image force shifts occupation 474 mechanisms for ET 434 melting 876, 884, 896, 898, 902–904 Mentzel–Gomer–Redhead (MGR) mechanism 468 metal 276, 292, 434 – continuum of easily excited EHP’s 433 – film thinner than the electron mean free path 492 – films properly deposited and annealed 492 – oxidation 275 – particle 283 – surface 279 metal–semiconductor interface structure 492 metallic – adsorption at close distances 454 – catalyst 277 – electrons near the Fermi level 488 – nanoparticle 470 – surface 434 metastable atom electron spectroscopy (MAES) 456 metastable deexcitation spectroscopy (MDS) 459, 460 methanation 314, 316 methane 298

Subject index

– activation 296, 297 – dissociation 296, 297 – molecular beam 298 methanol decomposition 309 methanol synthesis 313 methyl nitrite 628, 630 Mg oxidation 492 MGR-model 623 microkinetic model 310–312 microstructured surfaces 413 minimum energy path 306, 320, 321 mirror electron microscopy (MEM) 380 misfit strain 795, 804, 838, 841 mixed-mode oscillations 359, 364 Mn 549, 550 Mo(110) 544 model 334 – for oxide film growth 498 – systems 463 molecular 455 – adiabatic level 479 – adsorbate 579, 588 – adsorption 462 – affinity level 453, 454 – beam 203, 223, 495, 497 – bonding 452 – chemisorption energy 276 – desorption 462 – dissociation 433, 461, 479 – dynamics 34 – – with electronic friction (MDEF) 449, 463, 484 – electronics 685, 686, 733, 738 – hydrocarbons 492 – internal vibration 67 – molds 726 – nanomachines 744 – orbital 281, 477 – orientation 209, 210, 221 – “precursor” 105 – reactions are NA 491 – resonance 558 – rotation 63, 83 – scattering 433, 462 – wavefunction 436 molecular vibration – excitation in scattering 220–222, 224, 225 – high levels of excitation 222 – relaxation in scattering 220, 226 – survival probability 220 molecular-beam scattering (MBS) 296, 462, 463, 465, 490, 495, 498, 514 molecular-dynamics simulation 304, 498

Subject index

molecularly chemisorbed state 498 molecule surface 60 molecule times 464 molecule–surface – collisions 54, 56, 72 – interaction 105 – – potential 55, 56, 88 – potential 442 – processes 453, 454 – reaction heat 466 – scattering 56, 60, 90 molecules vibrating with large amplitudes 496 moment 284 Monte Carlo simulation 249, 317, 326 Morse potentials 479 multi-phonon expansion 108 multilayer growth 818, 819, 822 multiphonon – degrees of freedom 90 – effects 89, 90 – exchange 65 – excitation 54, 56, 66, 89, 90 – limit 54, 89 – scattering 57, 89 – – from surfaces by atoms and molecules 56 – spectra 56, 89 – transfer 64–66, 89, 90 multiple sites 330 multitude of ET phenomena at surfaces 431, 456 N2 , O2 , and F2 on Al(111) 504 N2 dissociative adsorption 237 N2 O decomposition 254 NA 492, 510 Na 548 NA – coupling effects 435 – couplings important at transition states 497, 512 – deexcitation 487 – description necessary 495 – description with diabatic PES’s 504 – dynamics 508 – effects 499 – – minimal for H2 /Cu(111) system but quite important for N2 /Ru(0001) 514 – electronic processes 455, 481 – ET 482 – gas–surface reactions involve more than a single PES 488 – mechanism 513 – particle emission 497 – pathways 495, 497 – processes at surfaces 435, 455, 495, 511

1005

– processes in surface chemistry 515 – reaction routes 495 – resulting in decay without radiation 476 – resulting in emission of a photon 476 – surface process manifestations 515 – transition 442 – vibrational damping for H2 (D2 ) on Cu(111) 493 – vibrational damping for N2 from Ru(0001) 514 Na, K, and Al metals and halogen gases 475 NA as surface dynamical processes 514 Na + Cl system 436, 439 nanocatalyst 330 nanoparticle 331 nanoscale islands 397 nanostructures 548 nanotribology 486 nanotube 807, 808, 811–813 nanowire 807–813, 817 narrow reactivity zone 479 natural coordinates 664, 665 nature and dynamics of transition state 511 nearest-neighbor interaction 320 nearly adiabatic dynamics calculations 513 NEB 306, 320 negative ion 474 – resonance 627, 628, 666, 668 negative molecular ion 476 negative-particle emission 488 negligible excitation of EHP’s from He collisions 456 negligible temperature dependence 486 net rate 328 neutral atoms (He0 ) 467, 468 neutral diabatic PES of O2 /Al(111) system 500 neutral released O atoms 465 neutrals ejected 482 new chemical species 510 new H2 –Cu potential 462 new molecule-surface systems and phenomena 489 Newns–Anderson model 278, 284, 287, 486 Newns–Anderson–Grimley Hamiltonian 605, 617 Newns–Anderson-model approach 448, 451 Newns–Grimley–Anderson Hamiltonian 614 NEXAFS 646 next-near neighbor interaction 320 Ni(111) 49 nitric oxide 207, 215 nm-sized catalyst particles 262, 263 NNL model 466, 476–478, 482–484, 490, 499 – extended by trajectory calculations 479

1006

– for Cl2 on Na and K 477 – typical starting point 488 NO 629, 631, 632, 634, 635, 641–644, 646, 647, 657–659, 665–669 – dissociation 261 – molecules in a selected excited vibrational states 511 – molecules with N end-on and without steering 506 – reduction 370 – reduction by CO 240 NO + CO reaction 371 NO + H2 reaction 373 NO + NH3 reaction 373 NO2 with an Al(111) surface 495 Nobel Prize 433 noble metal 313, 553 noble-gas ions 472 non-equilibrium effects 237 non-Fickian diffusion 378 nonadiabatic – aspects 441 – corrections 484 – dynamics 6 – effects 240 – process 274 – processes at surfaces 471 – scattering 219 – – electron transfer 221, 222 – transitions – – between multiple states, a quantum case 446 – – between multiple states semiclassically 441 – – between two states 440 nonadiabaticity (NA) 271, 333, 334, 429, 431, 433, 435, 448, 455, 471, 489 nondiagonal elements 440 nonequilibrium methods 447 nonzero probability of no-loss, i.e. elastic, scattering 447 normal energy scaling 174 – of the sticking 117 normal-mode analysis 306, 307 nuclear – kinetic-energy operator 436 – motion 4–6, 19, 20, 25, 502 – stopping 880, 881, 887 – time scales 334 – trajectory calculation 480 – velocities 438 – wavefunction 437 nucleation 799, 807–809, 812–819, 821–823, 831, 834, 837, 841 nudged elastic band method 306

Subject index

number of detected electrons per incident reactant 334 number of hot electrons per product molecule 492 O2 adsorption 244 + 500, 501 O− 2 /Al(111) √ √ O2 molecules with a Cs( 3 × 3 ) structure on Ru(001) 495 O2 on Al(111) 494–496, 498, 500 occupancy 286 one-dimensional CT model, within DFT-based SCF framework 508 one-electron – description 281 – picture 444 – potential 529 – width 453 one-phonon – loss function 107 – processes 121 – regime 109, 113 one-state problem 438 one-step Auger process 473 opposite trends 508 ordinary dissociative chemisorption 494 orientational dependence of the initial sticking 506 ortho-D2 102 Ostwald ripening 775, 803, 804, 806, 807, 810, 816 outlook 514 overlayer 292, 321 – states 563 oxidation 308, 317, 318, 323, 324, 434, 497 oxide 276 oxygen 287, 292 – islands 257 – molecule that interacts with the Al(111) surface 494 oxygen-rich environment 321, 324 paints 433 pair potential 32, 772 pairs of adatoms 495 para-H2 102 parameter forcing 411 parameter modulation 411 Pareto plot 318 Pareto-optimal 316 partial surface dislocations 780 particle morphology 327 particle size distribution 327 particle–phonon coupling 116

Subject index

particle–surface collisions 455 particle–surface scattering 58 partition functions 234–236, 248, 307 partitioning of positive-energy trapped particles 122 passivated surface 213 pathways in between initial and final states in CT 455 pattern formation 259, 260 Pauli kinetic-energy repulsion 503 Pauli repulsion 99, 288 Pd oxide 368 Pd(111) 49, 553, 625, 626, 634–636, 644, 646– 648, 669 peak energies 469 PEC’s of ionic and covalent states 439, 440 Peierls transition 433 penetration 32 Penning ionization (PI) 461 Periodic Table 279, 281 permanently bound electronic states 435, 438 peroxide 507 Persson–Persson model 483 perturbed (displaced) bosons 446 PES 274, 307, 513 – and NA-coupling strength from DFT 493 – barrier must increase substantially with N coverage 513 – crossing 486 – for H2 dissociation over Cu(111) 452 – for O2 /Al(111) from SCF 501 – of O2 above Al(111) surface on fcc site 499 phase – explosion 876, 893–895, 903, 910 – relaxation length 549 – shift 4, 13, 15–17, 19, 484 – space 563 – – trajectory 23 – turbulence 411 phonon 55, 56, 58, 60–67, 69, 73, 75, 83, 88–90, 455, 456, 460, 466, 476, 490 – emission and absorption 119 – energy and crystal momentum 461 – energy loss distribution 107 – excitation 54, 185–188 – mechanism 486 – spectral functions 108 phonons inactive in energy dissipation 499 phosphorescent 456 photocatalysis 435 photochemistry 456, 462 photodesorbed K atoms 482 photodesorption 127, 624, 625, 632–634, 642, 646, 647, 649, 659, 660, 662–665, 670, 676

1007

photodesorption dynamics 659 photodissociation 489, 625, 629, 631, 644, 647, 670 photoelectric effect 536 photoelectron emission microscopy (PEEM) 379 photoelectron spectroscopy 535 photoemission equipment 449 photoemission spectroscopy (PS) 470, 515 photon and exoelectron emission 483 photon- and electron-stimulated desorption 497 photon-emission intensities 476 photon-stimulated desorption 467 photosynthesis 433 physisorbed particle couples electromagnetically to EHP’s 484 physisorption 99, 150, 156, 157, 163, 164, 334, 507, 632, 660 – interaction potentials for H2 103 – parameters 101 – potentials for He, Ne and Ar 100 plasma formation 898, 910 plasmon energies 563 plasmon-mediated chemistry 470 plasmons 455 plug flow reactor 327 poisoning 293 Poisson distribution 9 polarization curve 331 polyatomic molecules, scattering of 222 – rotationally inelastic 223, 224 – vibrationally inelastic 224–226 polyatomic system 436 population of vibrational states 498 pores 262 positive-energy trapped particles 116, 120 possible dissociation 502 possible mechanism for desorption in Honda cell 469 possible NA effects – in gas-surface dynamics 515 – in vibrational energy transfer 515 possibly strong-coupling case with the H affinity level crossing the Fermi 493 post ionization in field evaporation 457 potential energies for elastic motion 440 potential energy 18 Potential Energy Surface (PES) 147, 201, 233– 235, 239, 240, 242, 243, 245, 246, 274, 305, 433, 435, 476 – adiabatic 147–150 – anisotropy 209, 212, 213, 223–225 – corrugation 213, 217, 218 – diabatic 147–150

1008

– energetic and geometric corrugation 174, 175 – for rotations 153, 161, 163 – rotations plus translations 171 – spin diabatic 149 – surface site dependence 154, 168, 169, 174, 175 – vibrational “elbow potential” 166, 169 – well depth 201 potential for O2 /Al(111) 500 potential surface 3 potential-energy curve (PEC) 465 potential-energy diagram 295, 296, 325 potentials 875, 876, 878, 879, 909 powerful detection technique 490 pre-exponential factor 235, 236 precursor states 248 precursor-mediated 497 – association 462 – channel 482 – dissociation 462 – sticking in general strongly temperature dependent 497 prefactor 307, 322, 326 prerequisites 514 pressure & material gap 420 pressure gap 324, 366 pressure wave 905, 906 pressure-pulse model 894 probability – for a CT event 500 – for energy loss in adsorption well 446 – for hole state 476 – of a transition from one state to another 440 – of HCl incident on an Au(111) surface measured 512 probe molecule 276 process time scale 318 product 275 product region 305 production of two holes in valence band 472 productivity 328 projected density of states 284, 289, 290, 292, 604 prompt vertical electron harpooning 482 proposes abstraction 494 proximity 275, 434 PSD 514 pseudomorphic layer 796, 800 Pt(111) 42, 44, 48, 49, 632, 633, 636, 638, 641– 644, 646, 659, 660, 666, 669, 671, 672 pulsed molecular beam containing NO 511 pump-and-probe experiment 455, 470, 483, 514 pump-and-probe laser spectroscopy 486 PVD 330

Subject index

qualitatively wrong behavior 499 quantitative account of NA in dynamical processes at surfaces 515 quantitative study of chemically induced electronic excitations at metal surfaces 492 quantum – dot 803, 837–841 – mechanics 433, 446 – reflection 106 quantum versus classical dynamics 150–153, 174–177, 182, 183 quantum-beat oscillations 557 quantum-mechanical – approach 457 – interference 453 – many-particle system 436 – treatment 462 quantum-well states 528 quartz-crystal microbalance (QMB) 486 quasi-bound states 117 quasi-chemical approximation 249, 258 quasielastic approximation 533 quasiparticle 527 quasiperiodic behavior 411 quasistationary representation 436 quenching 623, 647, 648, 650–653, 656, 658, 659, 675 Quinn and Ferrell 530 radiative decay 443, 465, 478 radiative transitions 454, 465 radical differences 513 rainbow scattering 32, 38, 154, 155, 161, 164 rainbows 42 Raman pumping 498 random forcing 412 random phase approximation 530 range of femtoseconds 502 rapid growth with increasing separation 502 rare-gas metastable atom 472 rate 278, 304, 317, 320, 324, 327–329 – constant 304–306 – controlling 314 – determining 311–313 – limiting 326 – oscillations 346 Rayleigh surface mode 534 Rayleigh surface phonon 562 Rayleigh wave 542 reactant region 305 reaction 278, 320, 329 – (and ET) pathway 494

Subject index

– barrier 304, 434 – dynamics 434, 435, 483 – energetics 321 – energy 333 – enthalpy 466 – equilibrium 256, 257 – front 353, 383, 397 – kinetics 247, 256, 257, 261–263, 319, 321 – mechanism 275, 314 – of Cl2 with Al(111) 482 – path 275, 325, 441 – pathways 487 – rate constant 233, 234, 241 – system for simultaneous reaction-rate and current measurements 492 – times 487 – zone 476, 479, 489, 502 “reaction zone” close to surface 500 reaction–diffusion (RD) equations 349 reaction-induced facetting 363 reactive 325 – boundaries 417 – molecular-beam scattering 514 – phase separation 398 – processes at surfaces 434 – surface 463 reactivity 276, 278, 293, 313, 314, 323, 325, 330, 514 realistic models 382 recapture of charge on the escaping O− 494 recapture of escaping charge 465 recoil 8, 10 recombination 278, 315 – of electrons with molecules 436 reconstructed 290 reconstruction 327, 780 recrossings 306 redox mechanism 314 reduced mass 436 reduced scattering of hot electrons in metal 491 refinery 273 reflected from surface 476 regenerate 273 relative electron emission probabilities for the indicated vibrational states 512 relaxation 320, 487 – in luminescence 456 release of O− ions 494 remote dissociation 482 removal 317 REMPI 640, 675 REMPI and TOF-MS 482 repulsive lateral interaction 323 repulsive part 500, 501

1009

repulsive term 99 residence time 47, 127, 319 resonance – excitation 20, 21 – filling rate 481 – ionization (RI) 457, 461, 472, 475 – lifetime 4, 20, 21, 23, 24 – model of adsorbate electron structure 450 – scattering 21 – – measurements 102, 103 – to cross Fermi level 502 – tunneling 443, 465, 480 – – of a substrate electron to the LUMO 502 resonance enhanced multi-photon ionization 203, 216, 217, 220, 223 resonance-enhanced multiphoton-ionization (REMPI) 495 resonance-filling rate 481 resonant – cross section 21 – electron tunneling 509 – ET 460, 480 – ionization 475 – sticking 113, 123 resonator 459, 548 role of – electronic excitations 455 – hot electrons 493 – NA effects in chemical intuition of practitioners in field 515 – surface defects 493 – surface diffusion 493 – vibrational excitation 498 rotation 692, 695, 697, 710–712, 717, 718, 720, 722, 730, 735, 741, 742, 751 – of N2 514 rotational 483 – alignment 177, 178 – energy 63, 82–88, 90 – hindrance 495 – inelastic scattering 104 – level 435, 438 – rainbow 206, 211, 213, 214 rotational inelasticity 159–165 – and diffraction 164, 165 – and molecular vibrations 160, 169–171 – and translational motion 163–165 – in H2 /Cu(100) 162 – in NO/metals 162, 163 – surface temperature dependence 183–188 rough surface 256 roughness 493 Ru(0001) 42, 45

1010

S-shaped energy dependence 502 S0 at off-normal incidence 120, 121 S2T conversion of metastable He atoms 473–476 Sabatier analysis 309, 311 Sabatier principle 332 Sabatier volcano-curve 312, 313 saddle loop (sl) bifurcation 352 saddle node of periodic orbit (sniper) bifurcation 352 saddle-node (sn) bifurcation 351 sandwiched 3d-metals 291 scaling 302 scanning photoelectron microscopy (SPEM) 379 scanning photoelectron spectroscopy 374 Scanning Tunneling Microscope (STM) 456, 457, 491, 492, 579, 585, 617 scanning tunneling microscopy (STM) 380, 401, 545 scattering 7–10, 12–14, 20, 22, 31 – of metastable atoms 434 – of metastable He in the 1s2s singlet or triplet states 509 – resonance between atom and surface 485 schematic PES’s 442 schematics for ET 443 Schottky barrier 434 Schottky diodes with ultrathin metal films 490 screened Coulomb potential between surface electrons 473 screening 527, 561 – length 877 second electron filling antibonding MO of oxygen molecule 500 secondary-ion emission 457, 459, 514 seeding the gas 498 selective adsorption 101, 156, 157, 163, 164 – resonance 105 selective oxidation 325 selective reactive-gas sensors 491 selectivity 276 self-consistency in simple models 476 self-consistent electron-structure calculations 451 self-consistent one-phonon approximation 123 self-energy 529, 537 self-organization 345 semiclassic theory 447, 457 semiclassical approximation (SCA) 440, 449 semiclassical description of the NA energy transfer 493 semiclassical theory within a trajectory approximation 444 semiclassical trajectory approximation 480 semiclassically 443

Subject index

semiconductor 276 sequential transfer of two electrons 495 series of random uncorrelated small energy losses 484 SERS of nanostructures 470 SERS of single molecules 470 SERS of transition metals 470 SERS-active sites 470 shifted and broadened adsorbate levels 450 shifted downwards 480 shifting and broadening of the Xe affinity level upon adsorption 458 shifting of adsorbate level 278 shifts in energy levels at the surface 453 Shockley 528 Shockley surface states 528 Shockley-type surface states 540 short lifetimes 486 short time limit 656, 666 short-range nature 473 short-range repulsion 771 shortcomings 515 significance of early CT channels 495 significance of early ET channels 497 significant curve crossings 515 silicon (Si) 791–793, 795–799, 803, 805, 808– 813, 816, 820–837, 841–851 silver films on n-Si(111) 491 simple isotope-mass dependence 485 simple master equation 480 simultaneous CO and CO2 hydrogenation 271 single ionized ions (He+ ) 467, 468 single-molecule chemistry 462, 515 single-molecule dissociation 462 single-particle picture 536 singlet atoms converted to triplet before AD 474 singlet O2 molecules should have stronger sticking 510 singlet-to-triplet (S2T) conversion of H∗ 473, 475 singular friction coefficient at transition point 493 singularity index 17 size distributions 778 skin 291 skin depth 898 slave bosons 447 sliding friction 497 – from excitation of EHP’s 486 slow heavy-particle collisions 441 slowed down by heavier seed gas (antiseeding) 498 solar hydrogen production 330

Subject index

solid surface 275 sorption of oxygen atoms 498 sp band 283 sp electron 278 spatio-temporal chaos 410 specificity of SERS 469 spectral function 537 spectroscopic observation 435 spectroscopic surface methods 293 spectroscopy (STS) 545 spectrum of electronic excitations 493 speeded up by lighter seed gas (seeding) 498 spillover 263 spin conversion 244, 245, 508 spin of O2 molecule 508 spin-orbit interaction is too weak for conversion 508 spin-selection model 510 spin-triplet PES 502 spiral waves 384 spontaneous exoemission 471 spontaneous photon emission 132 sputter yields 885, 887, 893, 897, 909 sputtered atoms 885, 888, 889 sputtering 457, 459, 875–877, 882, 884–887, 889–898, 904, 907, 909 stable triplet state 508 standard model of chemical reactivity 435 standard use of the BOA 515 standing waves 387, 393 Stark 551 Stark contrast to the exoelectron results 483 Stark effect 548 Stark-shifted 550 state-dependent anisotropy 388 state-resolved experiments for associative desorption 513 state-selected, highly vibrationally excited molecules 334, 433 state-to-state scattering experiments on Cu(100) 493 static ET phenomena 449, 450 static properties 435 stationary – adiabatic representation 438, 439 – eigenfunction 439 – pattern 398 – Schrödinger equation 437 – states 436 statistical growth 765 statistical mechanics 324 statistical physics 515 statistics 498 steam reforming 313

1011

steering 158, 159 steering and dissociation 172, 175, 177 step 233, 236, 238, 256–261, 296, 299, 300, 327, 463 – atom 296 – edge 300 – effect 299 – site 325 step vs. terrace 493 step-flow growth 815, 817, 818, 821, 838 stepped surface 297, 303 steric consequence of late-barrier PES 513 sticking 3, 4, 12, 13, 104, 479, 497, 504, 506 – at low energy of incidence is dominated by dissociation at steps 513 – behavior for NO on Al(111) surface 508 – coefficient 47, 106, 236, 252 – – versus surface temperature 498 – curves embracing measured curve 506 – for H2 462 – initiated by first ET 500 – measurements 483 – of atoms 461 – of D2 on Cu(111) 513 – of N2 on Ru(0001) 513 – of N2 on Ru(0001) nonadiabatic 464 – of O2 and NO on Al(111) is NA 506 – of O2 on Al(111) 496 – of O2 on Al(111) surface NA 507 – of O2 on Al(111) surfaces 514 – of O2 on transition metals 498 – probability 296, 298 – – in CT model 502 – – values within range for measured data 502 stiffness 44 stimulated emission pumping 222, 511 stimulated exoemission 471 stimulated theoretical quantitative work 493 STM 330, 331, 579, 581, 585, 588, 590–597, 599, 601, 609, 610, 615–617 – image 300 – micrographs 493 – studies and laser spectroscopic analysis 494 STM observations 494 stochastic transitions 404 stoichiometric network analysis 352 stopping 875–877, 879, 880, 889, 899 – power 467, 514, 879, 880, 885, 886, 893, 896 strain 792, 794–807, 815–817, 821, 829, 837– 842, 844 – effect 292, 294 strained layers 795, 799 Stranski–Krastanov growth 803, 838

1012

stress 792, 794–797, 800, 805 strong adsorption of O atom 495 strong dependence on translational energy 498 strong enhancement by vibrational excitations 498 strong NA near transition point for H/Cu(111) system 493 strongly dependent on workfunction 477 strongly varying adlevel 479 structural effect 296 structure of film 492 structuring 275 subharmonic entrainment 411 substrate-mediated interaction 247 subsurface oxygen 368, 390, 393 sudden approximation limit 6 suggestion that NA deexcitation of vibration is the cause 514 sulfide 276 sum frequency generation (SFG) 492 supercell approximation 502 superconductivity 433 superharmonic entrainment 411 superheating 903 superlattices of atoms and molecules 770 superoxide 507 supersonic expansion trough a tiny nozzle 498 support 276 – material 273 supported Ru catalyst 327 surface 433, 434, 514 – binding energy 876, 886–888 – bonding 333 – catalyzed reaction 333 – chemical reaction 434 – chemiluminescence 333, 433, 434, 460, 466, 475, 479, 483, 496, 514 – chemistry 301 – diffusion 235, 248, 254–256, 433 – dynamics 433, 434, 551 – electron-structure features 431 – energy 327, 792, 797–801, 803, 805, 807, 814, 817 – evaporation 893, 894 – explosion 373 – harpooning 479 – heterogeneity 233, 249, 261 – ionization 459, 514 – lattice dynamical model 107 – molecular dissociation 514 – neutralization 457, 514 – phase transition model 357 – photochemistry 489 – plasmon 473

Subject index

– processes 434, 471 – reaction 271, 278, 462 – reactivity 295, 297 – – beyond BOA 497 – residence time 220 – scattering 55, 56, 58, 60, 63–67, 90 – – theory 62 – science 271, 273, 274, 276–278, 317, 324, 325, 330, 333 – state 553, 558, 770 – structure 289, 297 – tension 801 surface-catalyzed reaction 313 surface-dynamical processes 431 surface-enhanced – hyper-Raman scattering (SEHRS) 470 – Raman scattering (SERS) 469, 515 – resonance Raman scattering (SERRS) 470 surface-mediated chemical reaction 320 surface-modified electron affinities 479 surface-reaction dynamics 435 surface-resistivity data 487 surface-science heritage of understanding 271 surface-sensitive experiment 277 surfactant 798, 803, 824 survival probability 453 survival rate 480 symmetric charge exchange 436 Tafel plot 331, 332 Tamm 528 target patterns 384 TDDFT 485, 493 TEM 325, 327 temperature dependence 510, 562 temperature rise 456 temperature-programmed desorption (TPD) 249, 250, 513 temporary negative-ion state 479 terrace atom 296 tert-butyl nitrite 628, 629 TF approximation 500 theoretical description of friction 485 thermal – chemical reactions at surfaces 472, 513 – coupling 348 – desorption 124, 322, 471, 514 – dissociation rate 298 – sticking coefficient 299 – surface processes 431 – surface reaction 333 thermo-neutral 321 thermodynamic surface phase diagram 324

Subject index

thermoluminescent 456 thermomechanical spallation 902–905, 910 thin metal films 434 thin metal-film Si Schottky diodes 491 Thomas–Fermi energy 877 Thompson formula 888 three-component system 319 threshold sputtering 886 time characterizing passage of molecule 444 time of flight (TOF) 514 time scales 304, 463, 502 time to fill the resonance 443 time T to shift LUMO resonance past Fermi energy 505 time-dependent – Anderson model 447 – density-functional theory (TDDFT) 448 – force 9 – mean-field Newns–Anderson model 493 – Schrödinger equation 436 timescales for nuclei and electrons comparable 487 tip-enhanced Raman scattering (TERS) 470 TOF 310 TOF REMPI 513 topology of seam between physisorption and superoxide PES’s 508 total – diabatic O2 / Al(111) PES by matching VR to VvdW 500 – electron yield 472 – energy 281 – – scaling of sticking 117 – excitation probability 445 – reaction rate 328 total-energy picture 444 TPD spectra 250–255 trajectory 3, 5, 6, 8–10, 12, 13, 22, 23, 645, 646, 650–652, 654, 658, 662 trajectory approximation 113, 477 transfer of electrons (ET) 514 transfer of first electron 482 transfer-matrix technique 249 transient motion 765 transient switching 14 transition metal 276–278, 280, 283, 284, 286, 287, 289, 314, 333 – surfaces 271, 279, 280, 283 transition probabilities 472 transition probability of RN 474 transition rate 473 transition state 235, 236, 296, 300, 301, 305, 320, 434 – in activated associative desorption 463

1013

– in activated dissociative adsorption 463 – resonances 174–176 transition-density function 472 transition-state – energy 299, 301, 302, 324 – structure 303 – theory (TST) 234–238, 240, 247, 248, 254, 304, 305, 435, 515 – traversals 514 transitions from triplet to singlet and spin-selection rules 509 translational 483 – and vibrational degrees of freedom 237 – energies about 1/7 of the excess energy 495 – energy 56, 57, 60, 61, 63, 65, 71, 73, 75, 77–80, 82, 83, 85–88, 90, 498, 625, 629, 633–635, 637, 639–641, 643, 644, 646, 654–658, 660, 672 transmission coefficient 235, 237, 242, 243, 245 transmission electron microscopy 325, 327, 328 trapped 476 trapping 104 – probability 115 trapping-desorption fraction 126 traveling wave fragments 390 trend 277, 279 triangular reaction fronts 393 triplet model 508 TST 304–306, 320 tunneling 237, 246, 454 – currents 601, 617 turbulence 410 Turing pattern 378 turnover 333 – frequency 271, 274, 310–312, 315, 329 two temperature model 636, 637 two-dimensional electron gas 545 two-electron Auger-type transitions 472 two-photon electron spectroscopy (2PPE) 470 two-photon photoemission 551 two-pulse correlation 674, 675 two-state problem 436, 439 two-step resonance process 473 two-temperature model 898–900, 902, 909 typically statistical methods with adiabatic energy barriers 498 UHV 330 ultra-high-vacuum study 323 ultrafast laser pulses 898 ultrathin Cu films evaporated on Si(111) surfaces 491 ultrathin metal film Schottky diode 492

1014

ultraviolet-excited SERS (UV-SERS) 470 umklapp 115, 116 uncoupled states 439 universal models 382 universal relationship 271, 302 universality class 302 unreactive boundaries 415 unreconstructed 290 up-hill diffusion 397 upshift of d-band energies 286 vacancy diffusion 777 van der Waals 127 – attraction 97, 99 – interaction 247, 500 – model 129 – reference plane 100 – term 99 vapour–liquid–solid mechanism 807 variational transition-state theory 306 Vegard law 795, 838 velocity 305 – dependence of the frictional force 486 velocity-selected Cl2 impinging on K 479 vertical – affinity 482 – electron harpooning mechanism 482 – electron-affinity levels 479 – ET 482 – Franck–Condon transition 482 vertically resonant charge transfer 479 vibrational 483 – adiabaticity 166, 167 – – and diffraction 168, 169 – – and rotational inelasticity 169 – coupling to EHP’s 496 – damping 514 – – of adsorbates 479, 483 – – of molecules adsorbed on surface 484 – – of N2 on Ru by EHP’s 514 – deexcitation in associated desorption 514 – degrees of freedom 60, 63 – dependence of electron emission efficiency 512 – displacement 68 – dynamics 510 – energy 512 – – relaxation 434 – excitation 67, 70, 512, 687, 688, 699, 700, 706, 707, 710, 720, 734 – – in scattering 514 – inelasticity – – and electron–hole pairs 189–191 – – and molecular rotations 169–171 – – elbow PES 166, 169

Subject index

– – surface site dependence 167–169 – – with electron emission 190, 191 – level 435, 438 – mode 54, 58, 60, 63, 72, 76, 82, 87, 89, 90 – promotion of surface chemistry on metals 512 – selectivity of SERS 469 – states 63, 68, 76 vibrational, rotational and translational degrees of freedom 236 vibrational-damping rate 484 vibrationally – enhanced sticking 505 – excited molecules 503 – excited O2 molecules 496 – highly excited NO molecules 512 vibrations 435 – on surfaces 611 vicinal surface 556 virtual level 17 viscous damping 19 volcano – curve 311–313, 316 – plot 311, 312, 332 – relation 271, 309 – result 328 volcano-shaped 333 Volmer–Weber growth 803 W(110) 38, 48 washboard model 32, 41 water gas shift reaction 313 water splitting 330 wave fragments 389 wave packet propagation 6 wave-vector cut-off 115 wavefunction spatially more extended 473 weak variation of the photon emission yield with v(Cl2 ) 483 weak- and strong-coupling cases 484 well-characterized surfaces 495 well-established observations in surface science 455 WGS 313–315 width 469 – function 448 – of degenerate p band 472 – of valence band 472 – related to fluctuation in orbital occupation 486 Woodward–Hoffman reaction rules 274 workfunction 450, 459, 472, 475 – changes through cesiation 449 – lowering by K adsorption 473 – variation 475

Subject index

1015

yield 466 – of catalytic nanodiode Young equation 801

Wulff construction 327, 799, 842 Wulff polyhedra 328 X-ray edge 4, 15, 17, 19 X-ray photoelectron spectroscopy X-ray spectroscopy 14 xenon 491

491

492

ZBL potential 877, 878, 880 zero-coverage limit for sticking zero-point fluctuations 112 ZGB model 355

105


Dynamics, Volume 3 (Handbook of Surface Science)

Structural Dynamics, Volume 3

Water Science Reviews 3: Volume 3: Water Dynamics

Handbook of Econometrics, Volume 3

Dynamics of molecule surface interactions

Surface science

Handbook of Mathematical Fluid Dynamics, Volume 1

Handbook of Mathematical Fluid Dynamics: Volume 2

Handbook of Mathematical Fluid Dynamics : Volume 2

Handbook of Mathematical Fluid Dynamics : Volume 2

Handbook of surface and nanometrology

Handbook of Surface and Nanometrology

Handbook of Surface Plasmon Resonance

Business Plans Handbook, Volume 3

Handbook of Algebra, Volume 3 (Handbook of Algebra)

Handbook of Green Chemistry, Volume 3: Biocatalysis

Cereals (Handbook of Plant Breeding, Volume 3)

Handbook of Metalloproteins, 3 volume set

Handbook of Shock Waves, Volume 3

Handbook of Industrial Organization, Volume 3

Handbook of herbs and spices: Volume 3

Handbook of Herbs and Spices - Volume 3

Handbook of Dynamical Systems, Volume 3

Handbook of Magnetic Materials Volume 3

Handbook of Liquid Crystals (3 Volume Set)

Nonlinear Dynamics of Surface Tension-Driven Instabilities

The Science of Hysteresis : 3-volume set

Physical Structure (Handbook of Surface Science), Vol. 1

Spectroscopy for surface science

Surface volume 2011-11

The Cambridge History of Science, Volume 3: Early Modern Science

Dynamics, Volume 3 (Handbook of Surface Science)

Structural Dynamics, Volume 3

Water Science Reviews 3: Volume 3: Water Dynamics

Handbook of Econometrics, Volume 3

Dynamics of molecule surface interactions

Surface science

Handbook of Mathematical Fluid Dynamics, Volume 1

Handbook of Mathematical Fluid Dynamics: Volume 2

Handbook of Mathematical Fluid Dynamics : Volume 2

Handbook of Mathematical Fluid Dynamics : Volume 2

Handbook of surface and nanometrology

Handbook of Surface and Nanometrology

Handbook of Surface Plasmon Resonance

Business Plans Handbook, Volume 3

Handbook of Algebra, Volume 3 (Handbook of Algebra)

Handbook of Green Chemistry, Volume 3: Biocatalysis

Cereals (Handbook of Plant Breeding, Volume 3)

Handbook of Metalloproteins, 3 volume set

Handbook of Shock Waves, Volume 3

Handbook of Industrial Organization, Volume 3

Handbook of herbs and spices: Volume 3

Handbook of Herbs and Spices - Volume 3

Handbook of Dynamical Systems, Volume 3

Handbook of Magnetic Materials Volume 3

Handbook of Liquid Crystals (3 Volume Set)

Nonlinear Dynamics of Surface Tension-Driven Instabilities

The Science of Hysteresis : 3-volume set

Physical Structure (Handbook of Surface Science), Vol. 1

Spectroscopy for surface science

Surface volume 2011-11

The Cambridge History of Science, Volume 3: Early Modern Science

Recommend Documents