Physical Structure (Handbook of Surface Science), Vol. 1

Handbook of Surface Science Volume I Handbook of Surface Science Volume I Handbook of Surface Science Series editors...

Author: W.N. Unertl

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Handbook of Surface Science Volume I

Handbook of Surface Science Volume I

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

9 LAUSANNE

9 NEW YORK

9 OXFORD

9 SHANNON

9 TOKYO

Volume I

Physical Structure

Volume editor W.N.

UNERTL

Laboratory for Surface Science and Technology Sawyer Research Center University of Maine Orono, ME 04469 USA

1996 ELSEVIER AMSTERDAM

9 LAUSANNE

9 NEW YORK * OXFORD

9 SHANNON

9 TOKYO

ELSEVIER SCIENCE B.V. Sara B u r g e r h a r t s t r a a t 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands

L l b r a r y oF Congress C a t a l o g i n g - I n - P u b l i c a t i o n

Data

Physical structure / volume editor, W.N. Unertl. p. cm. -- (Handbook of surface science ; v . 1) Includes bibliographical references and indexes. ISBN 0-444-89036-X 1. Solids--Surfaces. 2. Surfaces (Physics) 3. Surface chemistry. 4. Energy-band theory of solids. I. Unertl, W. N. (William N.) II. Series QC176.8.S8P49 1996 530.4'17--dc21 96-44174 CIP

ISBN 0-444-89036-X (Volume 1) ISBN 0-444-82526-6 (Series) 9 1996 Elsevier Science 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, Elsevier Science B.V., Copyright & Permissions Department, P.O. Box 521, 1000 AM Amsterdam, The Netherlands. Special regulations for readers in the U . S . A . - This publication has been registered with the Copyright Clearance Center Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside the U.S.A., should be referred to the copyright owner, Elsevier Science B.V., unless otherwise specified. 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. This book is printed on acid-free paper. Printed in The Netherlands.

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 molecular sensing industries. In social terms, an improved understanding of surfaces will facilitate the development of better catalysts and sensors for improvement of our environment. 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. This first volume is devoted to determining, by measurement and calculation the geometry of the surface. This is probably the oldest field of surface science investigation, dating back to the original experiments of Davisson and Germer employing electrons, and Estermann and Stern using atomic He beams to determine the crystal lattice constants in the surface region. In Volume 1, a wide variety of contemporary techniques are presented that now enable one to determine the location of atoms in the substrate surface as well as those in adsorbed molecules. This has enabled investigators to shed light on such issues as, how molecules can deform in the adsorbed state, and, the role of genuine surface relaxations both for a clean surface as well as for one covered by foreign species. The contributions deal with a wide variety of surface types, metals, insulators, ceramics etc. all of which play a role in real-world technological applications.

Volume 2 deals with the determination of the electronic properties of surfaces. As previously acknowledged, the separation is arbitrary and it is accepted that once it is known where the electrons are, then by calculating the total energy of the system, the geometry is uniquely determined simply by minimization. Well, these are early days and, while we believe that within the next decade this procedure will (hopefully) be routine, the electronic spectroscopies applied to surfaces and interfaces form a section of their own and are reviewed in depth in Volume 2. Volume 3 will address the dynamical aspects of surfaces and surface processes, reflecting in detail current understanding of energy exchange at surfaces and the part this plays in the adsorption and reaction of atoms and molecules at surfaces. Surface dynamics is, of course, intimately related to the geometric and electronic properties of the surface and while Volumes 1 and 2 address the equilibrium geometry and ground state electronic properties, Volume 3 pays particular attention to the response of surfaces to external stimulation. Taken as a set, the three volumes of this series aim to provide an in-depth introduction to the world of surface and interfacial science and would be most suited for scientists having obtained a first degree in the natural sciences.

N. V. Richardson and S. Holloway Surface Science Research Centre Liverpool, UK

Preface to V o l u m e 1

The primary goal of this book is to summarize the current level of accumulated knowledge about the physical structure of solid surfaces with emphasis on well-defined surfaces at the gas-solid and vacuum-solid interfaces. The intention is not only to provide a standard reference for practitioners, but also to provide a good starting point for scientists who are just entering the field. Thus, the presentation in most of the chapters assumes that the typical reader will have a good undergraduate background in chemistry, physics, or materials science but little, or no, knowledge of surface science. At the same time, coverage is comprehensive and at a high technical level with emphasis on fundamental physical principles. It is appropriate that this first volume of the Handbook of Surface Science is devoted to the physical structure of surfaces since knowledge of the atomic positions is often essential for complete understanding of electronic properties or dynamical processes that are the topics of the following two volumes. This volume is divided into four parts. Part I describes the equilibrium properties of surfaces with emphasis on clean surfaces of bulk materials. Chapter 1 defines the terminology and notations used to describe the crystallography of surfaces of crystals. Chapter 2 then outlines the relationships between the atomic structures and the macroscopic thermodynamic properties of surfaces. The major theoretical approaches that are used to calculate the structure of surfaces are presented and critiqued in Chapters 3 and 4. Chapter 3 emphasizes metals and elemental semiconductors and Chapter 4 emphasizes insulators. Chapters 5 and 6 are devoted to crystalline ceramics and to elemental and compound semiconductors, respectively. They demonstrate the complexity that can occur in multicomponent materials. Part II provides an introduction to some of the primary experimental methods that are used to determine surface crystal structures. Chapter 7 covers the three main diffraction methods. Chapter 8 describes techniques that provide more direct images. Spectroscopic techniques will be covered in a future volume of this Handbook. Part III provides an overview of the vast topic of the structure of adsorbed layers. A systematic treatment of the rich variety of structures formed by chemically adsorbed species is given in Chapter 9. Chapter 10 is an in-depth treatment of the structures that occur in physically adsorbed layers. Chapter 11 covers the complex topic of theoretical prediction of the interactions between adsorbed atoms.

vii

Part IV concludes the book. Chapter 12 covers the important topic of defects in surface structures. Chapter 13 is a comprehensive treatment of phase transitions that covers aspects of structure that go beyond local interactions. Surface science is a broad and interdisciplinary subject. This volume should be viewed as a general introduction rather than a comprehensive treatment. Nearly every chapter could easily be expanded into a full volume. Thus, some specific topics that are closely related to surface structure, such as surface magnetism and surface lattice dynamics, have not been included. Look for these in future volumes of the Handbook of Surface Science. I want to express my thanks to the Handbook editors and the publisher for offering me the opportunity to organize this volume. I am especially grateful to Patricia Paul and Barbara Deshane for their assistance in bringing this project to completion.

William N. Unertl Orono, ME, USA

viii

Contents of Volume 1 General Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preface to Volume 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contents o f Volume 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contributors to Volume 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List o f Symbols and Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . .

iv vii ix xi xiii

Part L Basic aspects of the structure of crystalline surfaces 1. W.N. Unertl Surface crystallography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. E.D. Williams and N.C. Bartelt Thermodynamics and statistical mechanics of surfaces

3 ............

51

3. C.T. Chan, K.M. Ho and K.P. Bohnen Surface reconstruction: metal surfaces and metal on semiconductor surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

101

4. J.P. LaFemina Theory o f insulator surface structures

137

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

5. R.J. Lad Surface structure of crystalline ceramics . . . . . . . . . . . . . . . . . . .

185

6. C.B. Duke Surface structures of elemental and compound semiconductors . . . . . . .

229

Part IL E x p e r i m e n t a l methods to study surface structure 7. E.H. Conrad Diffraction methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

271

8. W.N. Unertl and M.E. Kordesch Direct imaging and geometrical methods

361

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

Part III. Structure of adsorbed layers 9. H. Over and S.Y. Tong Chemically adsorbed layers on metal and semiconductor surfaces . . . . .

ix

425

10. J. Suzanne and J.M. Gay The structure o f physically adsorbed phases . . . . . . . . . . . . . . . . .

503

11. T.L. Einstein Interactions between adsorbate particles . . . . . . . . . . . . . . . . . . .

577

Part IV. Defects a n d p h a s e transitions at surfaces 12. M.C. Tringides Atomic scale defects on surfaces

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

653

13. L.D. Roelofs Phase transitions and kinetics o f ordering . . . . . . . . . . . . . . . . . .

713

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

809 855

Contributors to V o l u m e 1

Norman C. Bartelt, Department of Physics, University of Maryland, College Park, MD 20742-4111, USA K.P. Bohnen, Forschungszentrum Karlsruhe, Institut fur Nukleare Festk6rperphysik, Karlsruhe, Germany Che Ting Chan, Physics Department, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong Edward H. Conrad, School of Physics, Georgia Institute of Technology, Atlanta, GA 30332-0430, USA Charles B. Duke, Xerox Webster Research Center, 800 Phillips Road 0114-38D, Webster, NY 14580, USA Theodore L. Einstein, Department of Physics, University of Maryland, College Park, MD 20742-4111, USA J.M. Gay, Facult6 des Sciences de Luminy, D6partement de Physique, Case 901, 13288 Marseille Cedex 09, France Kai Ming Ho, Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA Martin E. Kordesch, Department of Physics and Astronomy, Ohio University, Athens, OH 45701, USA Robert J. Lad, Laboratory for Surface Science and Technology, 5764 Sawyer Research Center, University of Maine, Orono, ME 04469-5764, USA John P. LaFemina, Environmental Molecular Sciences Laboratory, Pacific Northwest National Laboratories, P.O. Box 999, Richland, WA 99352, USA Herbert Over, Fritz Haber Institut der Max Planck Gesellschaft, Faradayweg 4-6, D- 14195 Berlin Dahlem, Germany Lyle D. Roelofs, Physics Department, Haverford College, Haverford, PA 19041, USA J. Suzanne, Facult6 des Sciences de Luminy, D6partement de Physique, Case 901, 13288 Marseille Cedex 09, France David S. Y. Tong, Laboratory for Surface Studies, University of Wisconsin-Milwaukee, P.O. Box 413, Milwaukee, WI 53201, USA

Michael C. Tringides, Department of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA William N. Unertl, Laboratory for Surface Science and Technology, Sawyer Research Center, University of Maine, Orono, ME 04469, USA Ellen D. Williams, Department of Physics, University of Maryland, College Park, MD 20742-4111, USA

xii

List of Symbols and Acronyms lengths of unit mesh vectors in real space unit cell and unit mesh vectors in real space reciprocal lattice unit mesh vectors area, unit mesh area in real space, Hamaker constant, aperture A function unit mesh area in reciprocal space A.>g scattered amplitude A(Q) Auger electron spectroscopy AES atomic force microscopy AFM angle resolved photoelectron spectroscopy ARPES bulk modulus B Burger's vector, step height b body centered cubic bcc centered lattice c specific heat, contact function C commensurate-incommensurate transition CIT commensurate site lattice CSL crystal truncation reciprocal lattice rod CTR interrow spacing of rows (hk) dhk distance, diameter, diffusion coefficient D dimer adatom stacking DAS electron charge e kinetic energy, electric field strength, identity transformation, edge E dislocation, effective modulus interaction energy between neighbors separated by i Ei Fermi energy EF embedded atom method EAM electron energy loss spectroscopy EELS extended X-ray absorption fine structure EXAFS face centered cubic fcc f(ko,ki)f(Q,E ) form factor, scattering length F Helmholtz free energy, embedding energy, force F(Q,E), F(Q) crystal structure factor ai, bi ai,bi,ci 9 !, * a i ,I) i ,C i

xiii

F{...} FIM FLAPW FWHM

Ghkl g

g G G(r) H h (hk) {hk} HREELS HWHM I

l(O) i

[i} ISS

J(Q)

Jk k k k,kB l L LCAO LDA LEED LEEM LEIS LEPD LMTO LRO M m mij

MBE MEIS MSHD N

Fourier transformation of {... } field ion microscopy full potential linear augmented plane wave full width at half maximum reciprocal lattice vector two-dimensional reciprocal lattice vector glide line Gibbs' free energy mean-square fluctuation enthalpy, Hamiltonian Planck's constant, external magnetic field, Miller index coordinates of a reciprocal lattice rod, indices of a diffraction beam, indices of a set of parallel rows indices of a form of sets of rows, indices of a form of diffraction beams from such sets of rows high resolution electron energy loss spectroscopy halfwidth at half maximum tunneling current scattered intensity imaginary unit ~/-1 sites on a lattice ion scattering spectroscopy measured diffraction signal exchange constant between neighbors separated by k wavenumber (27t/L), Miller index wavevector Boltzmann's constant Miller index terrace length linear combination of atomic orbitals local density approximation low energy electron diffraction low energy electron microscopy low energy ion scattering low energy positron diffraction linear muffin-tin orbitals long-range order transfer matrix, spontaneous magnetization, atomic or molecular mass, magnification electron mass, lattice mismatch, mirror reflection plane elements of transfer matrix molecular beam epitaxy medium energy ion scattering mean square height deviation number of sites on a lattice, number of atoms xiv

NA

N(E) 1l tli

P e(g) P Q

O q qst R R,r R(Q) RHEED RBS REM si S SCF-LCAO SIMS SOS SPALEED STM STS t

t(g) T T(Q) T TEM THEED U(E) U U

uhv U, V

[uv] UPS V V

W W

Avogadro's number distribution of secondary electrons index of refraction site occupation variable in lattice gas model pressure, polarization factor pair distribution function, Patterson function primitive lattice partition function scattering vector phonon wavevector isosteric heat gas constant, reliability factor, radius position vectors in real space reflectivity reflection high energy electron diffraction Rutherford backscattering spectroscopy reflection electron microscopy magnetic moment in the Ising model entropy, screw dislocation self-consistent field, linear combination of atomic orbitals secondary ion mass spectroscopy solid-on-solid model spatially analyzed LEED scanning tunneling microscopy scanning tunneling spectroscopy time, reduced temperature, thickness transfer function translation operator instrument response function temperature, tunneling probability transmission electron microscopy transmission high energy electron diffraction inner potential internal energy, vibrational amplitude displacement from equilibrium position ultra-high vacuum coordinates of two-dimensional lattice points expressed in terms of a and b as units indices of a direction in the real space net indices of a form of directions related by symmetry ultraviolet photoelectron spectroscopy volume, potential speed work of adhesion, 2W is the exponent of the Debye-Waller factor interface width XV

XPS x-,y-,z-

x,y Z t~

A 8 Z E

Y

y(hkt) F 11 A P kt V

0 0

s O)

~(R) ~3(Q)

X-ray photoelectron spectroscopy directions of crystallographic axes, x- and y- are parallel to the surface, z- is the outward normal coordinates on any point in the unit mesh, expressed as fractions of a and b units canonical partition function total scattering cross section, mirror transformation matrix critical exponent of specific heat, ratio of scattered amplitudes from successive layers, angular divergence 1/kaT, critical exponent of order parameter vs. T critical exponent of order parameter vs. conjugate field bond length conserving rotation resolution, uniform strain susceptibility binding energy surface energy, angle between unit mesh vectors, critical exponent of susceptibility surface energy pair correlation function, center point of Brillouin zone critical exponent of correlation function potential function, work function attentuation length for elastic scattering wavelength density, density of states chemical potential critical exponent of the correlation length one-half the scattering angle coverage, angle grand potential, solid angle angular frequency, tilt angle correlation length dislocaton line scattered wave amplitude, wave function corrugation function interference function reflectivity

xvi

Part I

Basic Aspects of the Structure of Crystalline Surfaces

This Page Intentionally Left Blank

CHAPTER 1

Surface Crystallography W.N. U N E R T L Laboratory for Surface Science and Technology Sawyer Research Center University of Maine Orono, ME 04469, USA

9 1996 Elsevier Science B.V. All rights reserved

Handbook o.f Su~. ace Science Volume 1, edited by W.N. Unertl

Contents

1.1.

S o m e basic concepts of bulk c r y s t a l l o ~ a p h y 1.1.2.

Structure o f the unit cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Surface structure and surface order

1.3.

Surface c r y s t a l l o g r a p h y

1.5.

!.6.

5 5

Lattices, directions, and planes

1.2.

1.4.

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

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

1.1.1.

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

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

8 8 9

1.3.1.

C r y s t a l l o g r a p h y of a plane

11

1.3.2.

Point and space group s y m m e t r y

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

16

1.3.3.

Isolated adsorbed atoms and m o l e c u l e s . . . . . . . . . . . . . . . . . . . . . . . . . .

20

1.3.4.

The reciprocal lattice

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

21

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

25

1.4.1.

W o o d notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

25

1.4.2.

Matrix notation

!.4.3.

Classification o f o v e r l a y e r m e s h e s

Unit m e s h t r a n s f o r m a t i o n

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

Vicinal surfaces, steps, facets, and the stereographic projection

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

31

1.5.1.

Stereographic projection

1.5.2.

Vicinal surfaces and u n i f o r m arrays of steps . . . . . . . . . . . . . . . . . . . . . . .

33

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

35

Disorder

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

27 30 31

1.6.1.

Pair distribution function

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

37

1.6.2.

Defects of the first kind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

39

1.6.3.

Defects of the second kind

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

39

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

A p p e n d i x : The 17 t w o - d i m e n s i o n a l space groups

43

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

This chapter introduces the basic terminology and notation that is used in this book. Symbols for physical quantities follow the recommendations of the International Union of Pure and Applied Physics (SUN Commission, 1978). SI units are used throughout, unless explicitly stated otherwise. The chapter begins with a brief review of some basic concepts of bulk crystallography, followed by an extension of these concepts to surfaces. Next, the crystallography of perfect systems in one and two dimensions is presented with emphasis on those concepts most relevant to the structure of surfaces. This leads naturally into a treatment of the relationships and transformations between surface and bulk structures. Then the stereographic projection, vicinal surfaces, and notations for steps and facets are described. Finally, various types of disorder are introduced.

1.1. Some basic concepts of bulk crystallography Many aspects of surface terminology and surface crystallography are simple extensions of those used to describe the structure of bulk materials. Therefore, this chapter begins with a review of the relevant concepts from bulk crystallography. More extensive introductory treatments can be found in solid state physics textbooks such the one by Burns (1985). Vainshtein (1981) gives an advanced treatment. 1.1.1. Lattices, directions, and planes

Crystal structures in three dimensions are defined in terms of their Bravais lattices. A Bravais lattice is an infinite regular array of points that fills all space. This lattice is made up from a basic building block called the unit cell, which is usually specified in terms of three unit cell vectors (a, b, c), whose lengths and directions define the size, shape and orientation of the unit cell. This notation is illustrated for the case of the simple cubic unit cell in Fig. 1.1. The volume of the unit cell is V = la.(b•

(1.1)

Each unit cell can be translated to the position of any other cell in the lattice by the operation of a translation vector T(hkl) = ha + kb + lc

(1.2)

where (h,k,l) are integers. Crystal lattices often have several other symmetry properties in addition to translational symmetry. The symmetry operations that occur in both two and

6

W.N. Unertl

J

lal

= Ibl

=lcl

Fig. 1.1. Unit cell vectors a, b, and c for a cubic unit cell. [ 111 ] is the unit cell diagonal.

three-dimensional crystallography are rotation, reflection and glide plane. Operating on the lattice with any of these, or with combinations of them, always leaves the lattice unchanged. These operations as applied to surface crystallography are discussed further in w 1.3 below. A specific direction in a lattice is specified by the vector R ( u v w ) = ua + vb + wc

(1.3)

which is often written in shorthand using the square bracket notation [uvw]. As examples, [001 ] is parallel to c and [ 111 ] is parallel to the unit cell diagonal R( 111 ) = a + b + c as shown in Fig. 1.1. A line drawn over a character indicates negative values; i.e., [uvw] - [-u,v,w]. In most lattices, several directions are equivalent since at least one of the symmetry properties of the lattice transforms them into each other. A group of such symmetry related directions is called directions o f a f o r m and is denoted by angle brackets; e.g., . For example, the simple cubic lattice, whose unit cell is shown in Fig. 1.1, is left unchanged when rotated about the a, b or c axes by integral multiples of rt/2. Thus, the specific directions [001 ], [001], [010], [0T0], [TOO] and [100] are all members of the form . Once the coordinate system is specified, the direction [uvw] is unambiguous. However, because the form represents more than one direction, its correct interpretation requires that both the unit cell vectors and the symmetry operations of the lattice have also been specified. These two types of notation for directions are n o t interchangeable. The reader should beware however because this important distinction is misused frequently in the scientific literature of surface crystallography. A lattice p l a n e is any plane that contains three or more non-collinear, lattice points. Families of equally spaced parallel lattice planes can be constructed so that together they contain all the points of the Bravais lattice. Each such family of lattice planes is denoted by the notation (hkl), where the indices h, k and l are the Miller indices. A general operational procedure for determining the Miller indices for a particular family of planes is" m

Su~. ace crystallography

7

~

~

1.5c

y =

1.75b

(a)

x

~F

(100)

(110)

{111)

(b)

(c)

(d)

Fig. 1.2. Examples of Miller index notation for lattice planes in a cubic system. 1. Choose the origin so that one of the members of the family passes through it. This is always possible since the location of the origin in the unit cell is arbitrary. 2. Set-up a coordinate system with axes parallel to the mesh vectors a, b and c. Note that this coordinate system will not always be a Cartesian coordinate system. Only three of the fourteen lattice types possible in three dimensions have Cartesian coordinate systems; e.g., orthorhombic, tetragonal, and cubic. 3. Determine the intercepts (x, y, z) of the plane in the family that is closest to the origin; x, y, and z are measured in units of a, b, and c, respectively, and may be either positive or negative. 4. Take the reciprocals: (l/x, l/y, 1/z). 5. Express in terms of the lowest common denominator. 6. Factor out and discard the denominator. Consider the following examples: 1. Figure 1.2 shows a segment of a plane that intersects the x-axis at 2a, the y-axis at 1.75b, and the z-axis at 1.5c. Step 3 above yields (2,1.75,1.5). The reciprocals are (1/2,4/7,2/3) and the lowest common denominator is 42. Factoring this out leaves (21,24,28) which are the Miller indices of the plane shown. 2. Figure 1.2b-d shows some of the planes most commonly discussed in cubic crystals and gives their Miller indices. The (100) plane, for example, intercepts the axes at (1,oo,oo); i.e., the plane is perpendicular to the vector a. As is the case for directions of a form, several families of lattice planes may be related to each other by the symmetry properties of the lattice. Such planes of a form are denoted with curly brackets; e.g., {hkl}. The (100), (010) and (001) planes of a cubic lattice are members of the { 100} form of planes. The direction [hkl] is perpendicular to the (hkl) family of planes.

8

W.N. Unertl

1.1.2. Structure of the unit cell The Bravais lattice provides a useful framework for the discussion of real crystal structures. However, the Bravais lattice should never be confused with the actual atomic structure of a real crystal. For example, Bravais lattices are strictly infinite in extent whereas real crystals are finite. For many problems in bulk crystallography, this distinction is unimportant as long as the region of interest is far enough from the surface. However, the mere presence of a surface means that translational symmetry does not exist in the direction of the surface normal since neighboring atoms that would have been present in the bulk are missing on one side at the surface. Furthermore, the mathematical points that make up the Bravais lattice do not have to coincide with the positions of atoms in the real crystal. In a real crystal, the basic structural unit is the unit cell. There may be more than one atom per unit cell; e. g., in a crystal of protein molecules, the unit cell may contains thousands of atoms. These unit cell atoms form the basis of the cell and their positions are written as (xyz) where x, y and z are measured in units of a, b and c respectively. Again, note that x, y and z need not refer to a Cartesian coordinate system. The location of the origin within the unit cell is completely arbitrary. However, a corner of the unit cell is often selected. The usefulness of the Bravais lattice concept is that it provides a mathematically well-defined geometrical framework which can be used to model the properties of a real crystal. In this book, the terms ideal crystal and ideal surface are used to describe defect-free idealizations of real crystals with translational and symmetry properties which can be completely described by a Bravais lattice with a basis. 1.2. Surface structure and surface order

When an ideal surface is prepared by making a planar cut through an ideal bulk crystal, the resulting surface is identified by the Miller indices of the bulk plane parallel to which the cut was made. Most studies of real crystal surfaces have been carried out on the low-index surfaces (i.e., small values of h, k and l) of crystals with four types of bulk unit cells: face-centered cubic (fcc), body-centered-cubic (bcc), diamond and hexagonal-close-packed (hcp). Figures 1.3 to 1.6 show their bulk unit cells and the ideal structures of some of their low-index planes. In the case of hcp crystals the lattice planes and surfaces are usually, but not always, described by a four digit notation (hktl) where t = - ( h + k ) . The index t contains no new information about the crystal and serves only to identify the structure as hcp. Since the bulk unit cell vectors do not necessarily lie in the surface plane (consider for example, the (111) plane), it is usually more convenient to use a new coordinate system to describe the surface. The coordinate system used throughout this book is shown in Fig. 1.7. The z-axis points outward from the surface and the x- and y-axes lie in the surface plane. Unit vectors along the x, y and z directions are denoted i, j, and k, respectively. The y-axis is usually chosen to lie along a low-index direction in the surface.


9

~z

t~

101oi

11101

0o0o|

.0 0 0 0 0 8~[

1~" o 0 o | ._ | 1 7 4 1 7 4 o| |174174 o0o|

aI

0 0

0o0o|

0

o o o o 9 0 0 0

o o

0 0

(I00)

9

[11Ol

o _ o ~ o ~ o ~ o

o?o?o?o?o

|174 o~o~o~o~o

o o

0

o o

0

o

: _

0

o

0 0 0 (II0)

9

0

o~o~o~o~o

I,Oy

O'O'O'O'O ~Ol~l

(111) Fig. 1.3. The face-centered-cubic unit cell and the (100), (110), and (111 ) surfaces of an ideal crystal. The cross-hatched circles represent atoms in the surface layer, the open circles are atoms in the second layer, and the filled circles are atoms in the third layer. Nearest neighbor atoms in the unit cell are shown connected by solid lines.

Directions are defined with respect to their polar and azimuthal angles. The polar angle 0 is measured from the surface normal and the azimuthal angle ~ is measured from the positive x-axis with the positive sense corresponding to a counter clockwise rotation when observed by looking into the positive z-axis.

1.3. Surface

crystallography

Surface crystallography is the study of the structure and symmetry properties associated with the atoms and molecules near the surfaces of a crystal. Surfaces may have completely different atomic structure than found in the bulk of the crystal. Even the composition may be different due to structural rearrangements or because atoms and molecules from the environment can interact with the surface. Chapter 2 discusses the role of thermodynamics and statistical mechanics in determining

10

W.N. Unertl

~

z

I

y

>

[OlO]

t

0 @ @

9 0

o o

@ @

@

--.

@ 0

o o

@ 0

@

9

a~

@

o

@

a v

o

0

o

@

0

0

0

~

0

@ o o 0

@

o

0

@ o

O o @ o O o

o

O

o

O o 0 O o O o

@

(100)

0---~ o 111Ol

O

( 11 O}

0 9

0

..oy~

|

0 9

0

~

@

0 9

0

@

[110] 9

0

o (111)

Fig. 1.4. The body-centered-cubic unit cell and the (100), (! 10), and (111 ) surfaces of an ideal crystal. The cross-hatched circles represent atoms in the surface layer, the open circles are atoms in the second layer, and the filled circles are atoms in the third layer.

surface structure. Chapters 3, 4 and 11 describe fundamental aspects of the theory of metal, semiconductor, and insulator surface structures. Experimental methods used to determine surface structures are reviewed in Chapters 7 and 8. Examples of the surface structures for various materials and adsorbed layers are given throughout this volume. The final two chapters address the topics of defects on surfaces and surface phase transitions. This chapter focuses primarily on some of the foz'mal aspects of surface crystalIography. A useful starting point for this discussion, which is the topic of this section, is a study of the crystallographic properties of a purely two-dimensional system, the plane. The plane has no thickness and any point (x,y) on it can be described with reference to a two axis coordinate system with x- and y-axes. The reader should keep in mind, however, that the plane provides only an idealized description of a real surface since, in even the simplest cases, a real surface extends into a third dimension and may even be curved.


Il

~

z

I

I

['~101

[010----~

i

~i 9" ~

9"0

"~ 9

8 8 8

9

9e. ~. ~. oo..eo..~o..~

~.~8~

(100)

,go

(110) C4

[110] 00000000 ~

;?;?;?;?;

GoOoOoOo G o~o_o_o--o ^~^~o~o @'@'@'@'@ O ~a' ~

~j

[,oTi/r

~ 0 x'~, (111)

Fig. 1.5. The unit cell of the diamond lattice and the (100), (l l 0), and (! l I) surfaces of an ideal crystal. The cross-hatched circles represent atoms in the surface layer, the open circles are atoms in the second layer, thc fillcd small circles arc atoms in the third.

1.3.1. Crystallography of a plane The basic crystallographic properties of planes can be described in terms of the s y m m e t r y properties of an ideal t w o - d i m e n s i o n a l Bravais lattice with most quantities defined in direct analogy with their three-dimensional analogs. The definitions and notation used in this book are selected to be consistent with the International Tables 3"or Crystallography (1959). The two-dimensional Bravais lattice is an infinite planar array of points in real space, each of which can be transformed into any other by the translation operation vector

T(hk) = ha I +

kG 2

(1.4)

where the indices h and k are integers. The basic structural unit of the lattice is the

unit mesh defined by the unit mesh vectors a~ and a 2. The convention for c h o o s i n g

12

W.N. Unertl

I

[lOOl

,o,o,

t ~ ~ a ~

@o@o

a

@o@o.

9

I

O.O.O.O.O

C

.O.O'O.O"

*'~176176176

9

@o@o@o@o@

I ,_._, 0 o

0 , 0 o 0 , 0 , 0

.0.0.0.0.

@o@o@o@o@

@o@o@o@o@

(2i3ol

(1230)

@

@

@

@

@----~

o@o@o@o@o

!11Ol

@0 @0 @0 @0 @ 0

d

@0 @0 @0 @0

@o@o@o@o@ ~o~ (ooo~)

m

~

Fig. 1.6. The hexagonal-close-packed unit cell, the (0001) surface, and (1230) and (2130) prism faces of ah ideal crystal. The cross-hatched circles represent atoms in the surface layer, the open circles are atoms in the second layer, the filled small circles are atoms in the third layer.

R.I. ' Rzi z'" \ "-. \ \

.If Fig. 1.7. Cartesian coordinate system used in this book; the z-axis is the outward pointing normal.

Sueace crystallography

13

a~ and a 2 is that a 2 is always longer than al (i.e., lall < Ja2J) and will be used in this book. l Reader beware! Other non-standard conventions are c o m m o n l y encountered in the literature, specifically la~l > la21. The unit mesh is the basic building block of the surface crystal structure. Thus, if one starts with a single unit mesh and applies the translation operator T(hk) to it, every point in the entire two-dimensional lattice will be produced. The unit mesh area is lal• There are only five classes of unit meshes that have the property that they can cover the entire plane of the Bravais lattice when operated on by the translation operator. In crystallography, these meshes are most c o m m o n l y selected to be simple geometrical figures as shown in Fig. 1.8 along with their c o m m o n names: oblique, square, hexagonal, rectangular, and centered rectangular meshes. Each of these meshes can be defined by vectors a~ and a2. For convenience, the angle 7 between a~ and a2 is also specified. The lattice symbol p (always a lower case italic letter) is an optional prefix used for the primitive meshes - - oblique, square, hexagonal and r e c t a n g u l a r - which have lattice points only at their corners; for three-dimensional systems a capital symbol is used. A primitive mesh is always the smallest unit mesh for a given lattice. The only two-dimensional unit mesh which is not primitive is the centered rectangular mesh which, in addition to lattice points at each corner, has one lattice point at its center; i.e., at the location (1/2,1/2). The symbol c (lower case, italic) is used to designate a two-dimensional centered structure. A centered mesh is never the smallest possible unit mesh for its lattice. The primitive mesh for the centered rectangular structure is given by the unit mesh vectors bi and b 2 where b~ = 0.5 (a~ - a2) and b2 = a2. The following convention is used to draw unit meshes for the two-dimensional Bravais lattices: 1. The origin of the mesh is chosen to be in the upper left-hand corner. 2. The vector a 2 is drawn horizontal and pointing to the right. The direction of a 2 and the y-axis, Fig. 1.8, are usually chosen to be the same. 3. The vector as is drawn downward and pointing to the lower left; i.e., the angle , / b e t w e e n a~ and a2 must always be greater than or equal to 90 ~. Reader beware! The opposite convention (7 < 90~ is sometimes used. Failure to realize this can have drastic consequences, particularly in interpretation of diffraction data. 4. la~l _< la21. This convention will be used throughout the book and, where ever possible, results quoted from other sources will be transformed to be consistent with it. Crystallographers tend to use simple geometrical shapes (e. g., squares, rectangles, etc.) for the unit mesh, but this is not necessary. Instead, the unit mesh shape can be constructed to be complex but must still meet the constraint that it fill the plane like a jigsaw puzzle with identical pieces. Additional constraints on the shape are provided by the symmetry properties of the lattice as discussed below. M.C. Escher

1 Anotherwidely used definition of the unit mesh vectors is a = a I and b = a 2.

14

W.N. Unertl

First Brillouin Zone

Reciprocal Space

Real Space

§

§

§

§ o

o

, a1

Oblique

o

~

--,-- lOll

/

+a,']

"x

[tol

9

Square

o

§

a2

~---~ a1

[Ol1

§

§

§

§

(0 1)

.

§

§

9

(l T)

,a2

(lo)

+

+ (Ti)

§

§

§

§

§

§

al +

" ~ l t tl

ltOl

(II) (lo)

§ o

o

o

§ §

§

§

§ §

§

§ §

o

o a ~

o.-,. IYll

IlOl

li ll

[oil

4

--- lo,!

.

+ a l d ~ ' 2 + Ii'll (io)4~ ~. (ol) (2o)§ "d 4{o2)

"v-

o

liOl

§

2

[o 11

o

Hexagonal

§

(z l)

a*

o

9 ai~---~

9

§

)

( 0 0 ~

+

Centered Rectangular

4.

o

1~ol

Rectangular

41-

11o) NF (ll)

§ 9

~(Ol)

41-

11 ll

o

§

(oT)+ + (~176

,

1111

§ §

§ §

§

(ll}

+

+ (Tl)

§

(lo) \ (tl)

Fig. 1.8. Unit meshes of the five two-dimensional Bravais lattices in real and reciprocal spaces.


15

Fig. 1.9. (a) A drawing of a two-dimensional p l lattice by M.C. Escher. From MacGillavry (1976). Two possible choices of unit mesh are shown just below the drawing.( 9 1996 M.C. Escher/Cordon Art, Baarn, Holland. All rights reserved.) (b) Top view of the unit mesh structure of the (7x7) surface reconstruction of Si(111 ). After Robinson et al. (1986). This unit mesh is divided in half (shaded and unshaded regions) by an anti-phase boundary. (MacGillavry, 1976) has created many examples of such complex meshes. One of these is reproduced in Fig. 1.9a. It is sometimes useful to subdivide the unit mesh into distinct smaller units. One example, shown in Fig. 1.9b, is the hexagonal unit mesh of reconstructed Si(111) (Robinson et al., 1986), known as the (7• surface m the notation (7• is described in w 1.4.1. The basis associated with this reconstruction, which involves

16

W.N. Unertl

atoms from the first three atomic layers, is shown superimposed on the mesh z. Since the basis is divided into halves by a stacking fault (see Chapters 6 and 12), the mesh can be thought of as composed of two inequivalent triangles. In the case of the Escher print, Fig. 1.9a, the mesh is subdivided into a boat and a fish. The intersection of a three-dimensional plane (h',k',l') with the surface is called a lattice line; it passes through two or more lattice points in the surface plane. Families of equally spaced parallel lattice lines can be shown to contain all the points of the lattice. Lattice lines are denoted by the two-dimensional Miller indices (hk) of the surface vector perpendicular to the line. Since, in general, the unit mesh vectors a~ and a 2 do not coincide with the bulk unit cell vectors a, b, and c, the (hk) will generally be different from any of the (h',k',l') of the plane from which the lattice lines were constructed.

1.3.2. Point and space group symmetry In addition to the translational symmetry described by Eq. (1.2), two-dimensional lattices have other symmetry properties. The possible operations in two dimensions are the identity transformation, rotations, reflections, and glide lines. Groups are made up of combinations of the individual symmetry operations. The groups of interest in this chapter are point groups and space groups. The discussion of groups given here is rudimentary. More information can be found in Ertl and Kuppers (1985), Vainshtein ( 1981 ), and Cotton (1990). A point operation is any symmetry operation which leaves at least one point in the lattice unmoved. Except for the identity transformation E, reflections turn out to be the only possible point operations in one-dimensional lattices. Both reflections and rotations are possible in two dimensions and, in three dimensions, rotations, improper rotations, reflections, inversions and their combinations are possible. The simplest point operation is the identity transformation E which leaves a point (x,y,z) unchanged; i.e.

(x',y',z') = E(x,y,z) = (x,y,z) where

E

(1.4)

/!0!/01

The other point operations in two dimensions are rotations and reflections.

For the purposes of this section, the structure can be thought of as being projected onto a plane. Because of its three-dimensional extent, this structure is not strictly described by two-dimensional crystallography.


17

Table 1.1 Two-dimensional symmetry operations Operation

Shorthand notation International

Symbol

Sch6nflies

One-fold rotation

1

C~

none

Two-fold rotation Three-fold rotation

2 3

C2

9

C3

A

four-fold rotation

4

C4

9

Six-fold rotation

6

C6

Mirror reflection

m

G

Glide line

g

In two d i m e n s i o n s , the only rotational axes that are p e r m i t t e d are p e r p e n d i c u l a r to the lattice planet. The angle of rotation about a rotation axis is a - 2rt/N w h e r e N is the order of the axis. The five possible rotations for c r y s t a l l o g r a p h i c s y s t e m s have N = 1,2, 3, 4, 6. T h e s e are listed in Table 1.1 along with their s y m b o l s . In the widely used Sch6nflies notation, rotation axes are specified by CN. In matrix form, a rotation axis passing through the origin is given by

o/COSs,n . . !/ 0 Thus, a general point (x, y, z) t r a n s f o r m s to a new position (x', y', z') as follows

ysin'/"

y' - CN Z'

-

sin0~N+ycoso~ N

(1.6)

Z

N o t e that the z-coordinate is not c h a n g e d by rotations in t w o - d i m e n s i o n a l systems. The lattice is left u n c h a n g e d when it is rotated about the rotation axis by an integral m u l t i p l e of a . As is the case in three d i m e n s i o n s , this r e q u i r e m e n t rules out the e x i s t e n c e of a Bravais lattice that can be t r a n s f o r m e d into itself by a five-fold or

1

In two-dimensional slab systems, i.e., those with finite extent perpendicular to the lattice plane, there may also be 2-fold rotation axes and mirror planes oriented parallel to the lattice plane.

18

W.N. Unertl

seven-fold rotation. The 4-fold rotational operation necessarily contains the 2-fold rotation and the 6-fold rotation contains both the 2- and 3-fold rotations. Mirror-reflection planes, denoted by m or o, are the third two-dimensional point transformation. For example, if the xz-plane is a mirror plane,

o =fo o ~0 0

(1.7) 1)

and

I X///IY/ Y

y' =CYxz Z'

(~.8)

=

For this example, operation on a general point (x,y,z) changes only the y-coordinate. Only mirror-reflection planes that are perpendicular to the surface are required in two-dimensional crystallography. The intersection of a mirror plane and the twodimensional lattice is symbolized by a heavy solid line as shown in Table 1.1. In two dimensions, the mirror-reflection plane is also often referred to as a mirror line. There are only ten unique combinations of the rotation and reflection point operations on a two-dimensional lattice. These are called the ten two-dimensional point groups and are listed in Table 1.2. Each of the two-dimensional point groups can be identified by a three component notation, called the international (or H e r m a n n - M a u g u i n ) notation, which is interpreted in the following way: 1. First position: a numeral indicating the rotation operation (N = 1,2, 3, 4, or 6). 2. Second position: a mirror plane perpendicular to the x-axis plus all other mirror planes related to it by the rotation operation. 3. Third position: any second form of mirror planes. It is not always necessary to specify the second and third positions. The symbols in parentheses in Table 1.2 give a shorthand notation which is sufficieni to uniquely Table 1.2 The ten two-dimensional point groups Bravais lattice Oblique Oblique Rectangular Rectangular Square Square Hexagonal Hexagonal Hexagonal Hexagonal

Point notation I 2 lm 2mm 4 4mm 3 3m 6 6ram

Group

(m) (ram) (4m)

(6m)

Sch6nfleis notation E or C i C2 o C2v C4 Cnv C3 C3v C6 C6v


19

identify each point group but does not contain enough information to reconstruct the group. The Schi~nflies notation is another widely used notation for point operations and is given as the third column of Table 1.2. In the Schtinflies notation, the 1-fold rotation is designated by E or by C,. An n-fold rotation axis is described by Cn. The mirror plane m is given by ~v where the subscript v indicates that the mirror plane is normal to the two-dimensional lattice. For two-dimensional point groups containing more than one operation, the notation Cnv is used. Only one other symmetry operation, a space group operation, which is constructed by successive application of a mirror and a translation, is possible in two dimensions. It is called a glide line and is symbolized by a heavy dashed line or the letter g. Specifically, the glide operation consists of a reflection followed by translation parallel to the mirror by one-half the unit mesh length. Note that the glide operation does not contain the mirror operation even though reflection is used in the construction of the glide line. The set of points generated by applying all the symmetry operations of a group to an arbitrary point (x,y) is called the regular point system (RPS). Figure 1.10 gives an example of the RPS for the group 2mm. If (x,y) happens to lie on one of the symmetry elements, it is a point of special position, otherwise it is called a point of general position. There are only seventeen unique combinations of the ten two-dimensional point groups and the glide line. These combinations are called the two-dimensional space groups and represent all the possible two-dimensional lattices which can fill the

t-~.y) $

(x,y)

m

e) (x.-y) m

Fig. 1.10. The regular point system for the two-dimensional space group 2mm.

W.N. Unertl

20

entire plane I. These space groups are fully tabulated in the International Tables of Crystallography (1959). A simplified version of these tables is presented in the Appendix to this chapter. The space groups are arbitrarily numbered from 1 to 17. The space groups numbered 4 and 7 have proven to be important for adsorbate systems are reproduced in Fig. 1.11. Across the top of each space group are listed the Bravais lattice type (rectangular in these cases), the short form symbol for the point group (m and ram), the full space group notation (p 1g 1 and p2mg), the number of the space group (4 and 7), and the short form symbol for the space group (pg and pmg). T w o diagrams are shown are shown for each case. For each, an x,y- coordinate system is assumed with the x-axis pointing along the direction of a~, the y-axis pointing to the right along a2, and the origin in the upper left-hand corner. The left-hand diagram illustrates one unit mesh and shows how a point of general position (x,y), shown by the circled comma, is transformed under application of the space group operations. The right-hand diagram also represents the unit mesh and shows the location of the symmetry operators within it. Centered directly under these diagrams is a statement giving the convention that has been used to locate the mesh origin with respect to the location of the symmetry operations since this location is in general not unique. In the middle of Fig. 1.11 is a drawing of the structure of CO adsorbed on the Pt(110) surface. Individual CO molecules adsorb at atop sites on this surface (see w 1.3.3). However, the molecules are slightly too large to adsorb at adjacent sites and still be vertical to the surface. Instead, alternate molecules tilt to opposite sides of the 1110] rows of Pt atoms. Lambert (1975) recognized that this zigzag structure is described by glide lines along the rows as shown and assigned the structure to the p l g l space group. However, the structure also has two-fold rotations on the glide lines and mirror lines perpendicular to the glide lines. Because of these additional symmetry properties, the structure actually belongs to space group p2mg. This example illustrates the care that must be used to determine the correct space group of a structure. The space group of the (7x7) reconstruction of Si(l l 1), Fig. i.9b, is No. 14, D

p3ml. 1.3.3. Isolated adsorbed atoms and molecules Point symmetry operations are useful in the analysis of vibration and electronic properties of adsorbed atoms and molecules. Individual atoms and molecules are often adsorbed at specific sites in the surface mesh. Figure 1.12 gives examples of adsorption sites on various surfaces of fcc and bcc crystals. On-top sites (A) are directly above a substrate atom. A-sites are also called atop and linear. Bridge sites

In the case of planar layers or slabs of finite thickness, there are 80 symmetrygroups called the diperiodic or layergroups. Additional symmetryelements are needed to transform objects on one side of a singular plane located at the center of the layer into symmetrically equivalent sites on the other side. These additional elements must lie in the singular plane; they include 2-fold rotation and screw axes and glide planes. The layer groups are discussed in detail in the book by Vainshtein ( 1981). The layer groups have not found much use for the description of surface structures.


Rectangular

21

m

plgl

Pg

No. 4

|

Origin on g

Rectangular

p2mg

mm

|

ping

No. 7

|

I

I

I

I ,t

I A

I A

| |

|

Origin at 2 Fig. 1.11. Top: The rectangular lattice space group p lgl. Center: An adsorbate structure with p2mg space group. Bottom: The rectangular lattice space group p2mg. (B) involve bonding to two substrate atoms. B-sites typically have two-fold or mirror point symmetry. On some substrates both long bridge LB and short bridge SB sites are present. Finally, there are c e n t e r e d sites (C) in which bonds are formed to 3 or more surface atoms. C-sites are also called h o l l o w sites. C-sites on surfaces with square unit meshes are often referred to as four-fold hollows. Similarly two-fold and three-fold hollows occur on surfaces with rectangular and triangular arrangements of surface atoms. Other more complex sites are possible including bonding to subsurface atoms; see Chapters 9, 11 and 12.

W.N. Unertl

22

fcc

(100)

bcc

B_

B.

i

SB C (110)

LB

SB

7~A C LB

@@_O_~~c-3 (111) A C-I

Fig. 1.12. Various types of adsorption sites on low-index fcc and bcc surfaces. Black dots represent adsorbed atoms. Sites labeled A are atop (on top) adsorption. Centered (Hollow) sites with high symmetry are indicated by C. Bridge sites are labeled B if all possible sites are equivalent, otherwise there are long-bridge LB and short-bridge SB sites depending on the relative distances between the substrate atoms.

In the crudest approximation, the symmetry of a site is determined only by the adsorbed species and those substrate atoms to which it is directly bonded. For example, an atom bonded in an on-top site has Coo symmetry. More generally, the combined s y m m e t r y of the adsorbate-substrate system must be used. Thus, an on-top site for a lattice with square symmetry is reduced from Coo to C4v.

1.3.4. The reciprocal lattice The translation vector, Eq. (1.2), defines an infinite set of points called the direct, or real space, lattice. Another lattice, called the reciprocal lattice, is also extremely useful for describing diffraction, electronic band structure, and other properties of crystals. The reciprocal lattice can be specified in terms of a set of reciprocal lattice vectors G that satisfy the equation

eiCr= 1

(1.6)

for all allowed combinations of the Miller indices h, k, I. Since each Bravais lattice is described by a different set of T, each will also have a different reciprocal lattice. G is measured in units of inverse length.


23

In a three-dimensional crystal, the basis vectors a*, b*, and c* that define the reciprocal lattice are related to the real space basis vectors by 2/1; a" - - ~ (b x c)

(1.7a)

b* = ~2~ (c x a)

(1.7b)

271;

c* - ~

(a x b)

(1.7c)

where V is the volume of the real space unit cell. (Note that some authors do not include the factor of 2re in the definitions of Eqs. (1.7).) The basis vectors of the reciprocal lattice have the properties a* . a = b * . b = c * . c = 2r~

(1.8)

and a* . b = a * . c = 0 b* . a = b * . c = 0

(1.9)

c* . a = c * . b = 0 The origin of the reciprocal lattice coincides with the origin of real space and the points of the reciprocal lattice are generated by the reciprocal lattice vectors Ghkt= ha* + kb* +/c*

(1.10)

where (hkl) are the Miller indices. In two dimensions, only two reciprocal mesh vectors are required. In analogy with Eqs. (1.7a-c), these are defined by a2•

a~ = 2rc ~ la I x a21 a 2 ==-2rt

(1.11)

c xa 2

la~ x a21

where c is the outwardly directed unit vector and la~ x azl is the area of the unit mesh. The mesh vectors have the properties a l 9a i = a~ . a 2 = 2rt (1.12) a~

9 a

2

-

a 2 9a~ = 0

W.N. Unertl

24

The two-dimensional reciprocal lattice vectors are Ghk = ha] + ka~

(1.13)

These Ghk define a set of lines that are perpendicular to the two-dimensional lattice plane. These lines are called reciprocal lattice rods and are very important for the description of diffraction phenomena, as described in Chapter 7. Figure 1.8 gives examples of the two-dimensional reciprocal lattices for each of the real space Bravais lattices shown in Fig. 1.8. Each reciprocal lattice point and several important directions are also labeled in Fig. 1.8. Note that the primitive unit mesh has been used for the centered rectangular lattice. The choice of the nonprimitive mesh results in different unit mesh vectors in both real and reciprocal space; e. g., a - a ~ + a 2 and b - -a~ + a 2. This has the consequence that the directions and reciprocal lattice points have completely different names; e.g., [ 10] ~ [ 1,1 ], [11] ~ [10], (11) ~ (10), (01) ~ (1/2,1/2), etc. The first Brillouin zone is a common choice for the primitive cell of the reciprocal lattice and is an essential element in the description of electronic band structure, lattice vibrations and diffraction. Electronic band structure and lattice vibrations are treated in subsequent volumes of this handbook; diffraction is discussed in Chapter 7. The first Brillouin zone, which is the analog of the W i g n e r Seitz cell in real space, is constructed as follows" first, reciprocal lattice vectors are drawn from the origin (00), to the nearby reciprocal lattice points. Next, lines are drawn perpendicular to these reciprocal lattice vectors at their midpoints. The smallest area containing (00) and enclosed by these lines is the called the first Brillouin zone. The line segments defining the edges of the Brillouin zone are called the Brillouin zone boundary. The right hand column in Fig. 1.8 illustrates the Brillouin zones for each of the five two-dimensional Bravais lattices. The shapes of the Brillouin zones for the oblique and centered rectangular lattices will vary somewhat depending upon the precise values of a~, a2, and y. Figure 1.13 shows the relationships between the Brillouin zones l;or the (111), (110), and (001) planes of the fcc and bcc lattices and their respective bulk Brillouin zones. Various points and lines in Brillouin zone are related to each other by the symmetry operations of the lattice listed in Tables 1.1 and 1.2. Thus, when describing the properties of a crystal, it is sufficient to consider only the smallest symmetry independent portion of the zone. The center of the each zone (k - 0) is denoted by the Greek letter F. Other points with high symmetry are denoted by upper case roman letters; e. g., H, K, L, M, N, P, X. Symmetry points of surface Brillouin zones are sometimes written with a bar to distinguish them from points in the bulk zones. The simple square lattice has space group p4 and the triangle 1-"X M is the smallest symmetry independent portion as can be demonstrated by sequential applications of the C4 rotation axis at the zone center and the diagonal mirror lines. See Burns (1985) for a more detailed discussion of Brillouin zones and the notational conventions used to describe them.

Sue'ace crystallography

25 10011

(111j

.

]

IX

...,

(a)

Io~

"~.

"~" x

11001

(001)

',

(b)

,

,

\ (111)

o)

Fig. 1.13. (a) First Brillouin zones for the bulk, (111), (110), and (001) planes of fcc crystals. (b) First Bfillouin zones for the bulk, ( 111 ), (l 10), and (001 ) planes of bcc crystals. (Reproduced with permission of W.N. Unertl.)

W.N. Unertl

26

1.4. Unit mesh transformation In practice, it is sometimes more convenient to use a unit mesh which is different from one of the five Bravais lattices. This is particularly true for ordered overlayers, many of which can have a unit mesh of different size, symmetry and orientation from the substrate surface; see Chapters 9 and 10 for examples. Several notational systems have been developed to name and facilitate transformations between various possible lattices. The Wood notation and the matrix notation are the only widely used of these.

1.4.1. Wood notation In the early 1960s, the noted X-ray crystallographer, Elizabeth A. Wood (1963), developed the first systematic notation for surface crystallography. In addition to formulating one of the most widely used notations for ordered surface structures, she discussed other important concerns for development of a uniform, unambiguous system for describing surface structures which is also consistent with the well established conventions of bulk crystallography, as presented in the International Tables of Crystallography (1959). In W o o d ' s system, the unit mesh of the ideal substrate lattice is taken as the reference for all other lattices and is called the ( l x l ) mesh. The conventions given in the preceding section are used to specify this mesh. The Bravais lattice type of the substrate must be clearly specified in W o o d ' s system. Any other mesh, for example that of an ordered adsorbate on the substrate, can then be described in terms of this ( l x l ) mesh. In the simplest case, the overlayer unit mesh vectors (b~,b2) are parallel to (a~,a2) but have different lengths; i.e. bl = real and b2 = ha2 where m and n are integers. Such an overlayer mesh is identified as an (mxn) structure. Figure 1.14a shows an example of a (3x2) structure on a rectangular lattice. It is extremely important that m, the coefficient ofa~, precede n. For example the (3x2) structure of Fig. 1.14a is obviously very different from the (2x3) structure shown in Fig. 1.14b. The overlayer mesh can also be rotated by an angle 13 with respect to the substrate. In this case n and m are no longer restricted to be integers. There seems to be no general convention for defining the direction of positive 13. In this book, we assume 13 to be the clockwise rotation required to align b~ and a~. The full notation for the rotated overlayer mesh is

(mxn)R~ where 13 is expressed in degrees. An example for a ( ~ - x ~ - ) R 2 2 . 5 7 overlayer is shown in Fig. 1.14c. The full Wood notation for a chemical species A adsorbed onto the (hkl) surface of a substrate of chemical composition S is:

Su~.ace crystallography

27

b2= 2 _

(~)

9

9

9

9

9

9

9

|

~

a2

1t 2

b. I

9 9

9

~ ,

(~)

9

|

.

9

o

o|

9

b 2--3 a 2

Q

v

|

A a2

It

~

~

9 |

|

9

9

,, =28. I w

9

. +

,,

9

|

9

|

. ~ o

_

| 1

|

Fig. 1.14. Examples of the Wood notation.

S( hkl)( m• )R~-rlA where 1"1 is the number of species A in the unit mesh. If the space group is known, it can be incorporated into the Wood notation as follows: S(hkl)(mxn)R~spacegroup-rlA; e.g., Pt(110)(2• Systems that have more than one type of adsorbate do not have a standard notation. We suggest S(hkl)(mxn)R~5-rlA~ where is the number of species B per unit mesh. Unfortunately the Wood notation is unable to describe all physically possible overlayer-substrate combinations. For example, in the case of a hexagonal overlayer on a square substrate lattice, the angles between (al,a2) and (bl,b2) are different. Furthermore, there are several very common structures which have more than one description in the Wood notation. The most widely known of these are the identical c(2x2) and (x/2-x'~t2-)R45 structures illustrated in Fig. 1.14d. The latter description derives from the primitive unit mesh of the overlayer. In spite of these shortcomings, the Wood notation is usually clear and very easily visualized. It is also a neatly compact notation. These attractive features have led to its wide adoption in surface science.

28

W.N. Unertl

1.4.2 Matrix notation

A matrix notation, first introduced into surface crystallography by Park and Madden (1968), overcomes some of the shortcomings of the Wood notation. It is not as easy to visualize as the Wood notation but it facilitates mathematical manipulations involving quantities related to the surface structure. The matrix notation for two-dimensional systems is a special case of that developed for three-dimensional systems as described in the I n t e r n a t i o n a l Tables o f C r y s t a l l o g r a p h y (1959). As was the case for the Wood notation, the substrate unit mesh (a~,a2) provides the reference for all other lattices. Any overlayer mesh (b~,b2) can be described in terms of this substrate mesh by a linear transformation bl = sllal + s12a2

(1.14a)

b 2 = s21a I + $22a 2

(1.14b)

and the inverse linear transformation al = tzlbl

+

tlzb2

(l.15a)

a 2 = t21b I + t22b 2

(1.15b)

where the unitless coefficients sij and t,j are elements of the matrices S and T, respectively. In matrix notation

s ilai I

1 $22

2

(1.16)

2

and

Since these transformations, when applied sequentially, must reproduce the starting mesh ST=I

(1.18)

where I is the unit matrix; i.e., the transformation must be reversible. T is the inverse of S; T = S -~. The matrix S provides a complete description of the new mesh (b~,b2) in terms of the reference mesh (a~,a2). Thus, once (a~,a2) is chosen, specification of S completely determines the overlayer mesh (b~,b2). Once S is given, the t,~ can be determined from Eq. (1.17):

Su~ace crystallography

29

S22 t j~

-

det S --$21

t~2 = det S --S 12

t21 = det S Sll

t22 - det S

( 1.1 9)

where det S is the determinant of S; det S is never negative unless a left-handed coordinate system has been used. The area of the overlayer mesh, Iblxbzl, expressed in units of the substrate mesh area, lal• is equal to det S. Similarly, det T is the ratio of the substrate unit mesh area to the overlayer unit mesh area; i.e. b2-------~l Ib z x - (det T )-~ = det S la~ x a21

(1.20)

The structures shown in Fig. 1.14 provide examples for the matrix notation. The (3x2) structure has b~ = 3a~ and b 2 "" 2 a 2. Thus

(30)

is the matrix equivalent to the Wood notation (3x2). Since det S - 6, the o v e r l a y e r unit mesh area is six times that of the substrate. The inverse matrix is

,t

~,0 2 )

The (,fffx,~-)R22.57 structure in Fig. 1.13c has bl = 2al + a 2 and b 2 --- ---aI + 2a2, so that S=

(')

_21 2

The overlayer unit mesh area is 5 times the substrate unit mesh area and 2 T~

-1

W.N. Unertl

30

In the matrix notation, only the primitive overlayer unit mesh can be described and it is not possible to define the centered rectangular unit mesh. Thus, the ambiguity between the c(2x2) and the (x/2-x'~-)R45 notations does not arise. In this case b~ = a~ + a 2 and b 2 = -gl + a2, so that

':(: '/ 1 1

det S = 2, and 1

-1

Ambiguities do occur in application of the matrix notation since S can have several equivalent forms related by symmetry. For example, each of the four matrices below describes the (x/2xq2-)R45 lattice:

(', (:,

:/

Each can be obtained fi'om the others by sequential application of the rotation operator C4; i.e., Eq. (1.5) with czN = 90 ~ The matrix notation is very useful for the analysis of diffraction patterns where the need to make transformations between the real and reciprocal space lattices becomes important. The Wood notation for overlayer or reconstructed meshes,

S( hkl)( mxn ) R ~3-rlA is modified to incorporate the matrix description by replacing

(mxn)R~ with

,2 I 21 S22

1.4.3. Classification of overlayer meshes Overlayer, or superlattice, meshes can be divided into three general classes. 1. Simple lattices. Simple lattices have all s o as integers and det S is also an integer. The four examples in Fig. 1.14 are all simple lattices. 2. Coincidence lattices. For coincidence lattices, the sij are all rational numbers and det S is a simple fraction. Consider the one-dimensional example shown in


31

Fig. 1.15a in which the points on the two lattices are spaced by a and b = (5/4)a. If the lattices coincide at any lattice point, they will coincide again after a distance of 5a or 4b. Simple lattices and coincidence lattices are also often grouped together as coherent or commensurate lattices because of their property that lattice points of the substrate and overlayer repeatedly coincide. 3. Incoherent or incommensurate lattices. If the sij contain irrational numbers, det S is an irrational number, and the two lattices can never coincide at more than a single lattice point. Such lattices are said to be incoherent or incommensurate. This type of lattice plays an important role in the theory of two-dimensional phase transitions as described in Chapter 10. A simple example is an undistorted hexagonal lattice superimposed on a square substrate lattice as shown in Fig. 1.15b. Here b~ = a~ and b2 = - s i n 3 0 a~ + cos30 a2 and

5a or 4b

-I

b 9

9

9

9

9

9

9

9

9

R

(a)

ax

b1

a2.

0

o 9

o .

0 9

0

/.o. o .

o 9

o

0

0

9 0

9 0

9 0

9 0

9 0

0

0

9 0

0

*

0

0

0

0

0

0

9 0

9 0

9 0

0

9

9

0

o .

0

.

0

o 9

0

,

o .

0

.

.

.

.

o

0

0

0

*

0

0

0

9 0

9 0

0

0

0 9

9

0

0

0

0 9

0

0

*

9

9

0

0

0

0 9

9

0

9

0

9

0 9

,

0

b2

9 0

0

0

*

9 0

0

0

*

9 0

0

0

(b) Fig. I. 15. (a) A one-dimensional example of a coincidence lattice with b = (5/4)a. (b) An i n c o m m e n s u r a t e h e x a g o n a l lattice (bl,b2) on a square lattice substrate (al,a2) for the case la I = Ibl.

32

W.N. Unertl

/'

~

S = -sin 30 cos 30 so that det S = cos 30. Reader beware! In the literature, it is common to confuse coincidence lattices with incommensurate structures. Thus, the reader must be cautious before accepting such a reported incommensurate lattice.

1.5. Vicinal surfaces, steps, facets, and the stereographic projection

1.5.1. Stereographic projection A useful method for presenting crystallographic information, including a description of vicinal surfaces, is provided by the stereographic projection. Imagine a small crystallite located at the center of a sphere of unit radius. The points at which low-index directions [uvw] intersect this sphere can be mapped onto a plane. Figure 1.16 shows the principle of the projection. The projection plane is chosen to be tangential to the unit sphere, usually at a point where one of the low index directions intercepts it. The projection point is chosen to be opposite this point. Any point p on the unit sphere is mapped into a point P on the projection plane by constructing the straight line passing through the projection point and p as shown. Figure 1.17a shows the resulting stereographic projection for a face-centered crystal with the tangent point chosen as the [001] direction. The projection of the normal to each major family of low-index planes is labeled by the Miller indices hkl of their normal. Because of the high symmetry of most crystal lattices, the complete projection contains redundant information. The stereographic triangle, Fig. I. 19b, is the smallest segment of the total projection which contains all of the information of a stereographic projection.

projection

Fig. 1.16. The stereographic projection.

33


__

210,

310

~]o

too

glo 310

~_lo

110,

tl0

-

,120 130,

,T3o ~Tso

t50q

o?o,

)010

1,~0,

150

1~o

30

~.0

ST0

100

510

310

(a) ! lOO] zone

(001)

(ol])

(012 I

o

(]22)

(lll)

(b) Fig. 1.17. (a) The stereographic projection for a face-centered-cubic lattice with the [001 ] direction as the center. (b) The stereographic triangle for (a).

34

W.N. Unertl

The surface of any crystal is represented on the stereographic triangle in two ways. In the first, the surface normal [hkl] is projected onto a point that is labeled by the Miller indices hkl of the family of lattice planes parallel to the physical surface. In the second, the great circle formed by the intersection of the surface plane with the unit sphere is projected onto the stereographic triangle. This projected line is called the [hkl] zone. Examples of both types of projections are given in Fig. 1.17b.

1.5.2. Vicinal surfaces and uniform arrays of steps A slightly misoriented crystal surface will have atomic scale steps on it whose structure is described by the terrace-ledge-kink model illustrated in Fig. 1.18 where individual atoms are represented as cubes. The large flat regions between individual steps are called terraces and are connected by step edges, or ledges. Kinks occur wherever the ledge changes direction or when there is a vacancy or adatom defect at the step edge. Numerous direct observations of the t e r r a c e - l e d g e kink structure have been made and examples are shown in Chapters 2 and 12. Ordered arrays of nearly identical, equally spaced steps can be formed by cutting a surface at a small angle from a low index plane. Such surfaces are called vicinal surfaces and have been extensively studied because diffraction methods can be used to characterize their average structural properties as discussed in Chapter 7. Figure I. 19 shows perspective sketches of some vicinal surfaces. In many cases, the vicinal surface consists of large terraces with a low index orientation separated by steps that are one or two atoms high. In Fig. 1.19, each vicinal surface is labeled by the Miller indices of the family of bulk lattice planes parallel to the surface. However, the Miller index notation does not provide a clear visualization of the orientation of the vicinal plane. For example, it is not obvious from the Miller indices that the (14,11,10) vicinal surface shown in Fig. 1.19 consists primarily of (111) terraces and is tilted only 8.29 ~ away from the (111) plane toward [725]. A more convenient

Fig. 1.18. The terrace-ledge-kink model of a solid surface with single and double atomic height steps on it.


fcc

35

(977)

rcc

17551

rcc

18331

!

fcc

(4431

fcc 114.11.101

fcc

fcc

(3321

( 0,8,7)

fcc

(3311

fcc ( 13. I 1.91

Fig. 1.19. Ball models of the atomic structure of some vicinal surfaces. Steps shown in the upper two rows do not have kinks whereas each step in the lower row has kinks. (After Somorjai, 1994.)

36

W.N. Unertl

notation, which is more descriptive than the Miller indices, gives (1) the Miller indices of the nearest low index plane, (2) the angle between that low index plane and the vicinal plane, and (3) the zone which contains the surface normals of both the vicinal surface and the nearby low index plane; i.e., the direction that the vicinal surface is tilted away from the low index plane. For the example above, the (14,11,10) vicinal surface becomes a (111)8.29~ plane in this notation. Similarly, the top row of vicinal surfaces in Fig. 1.19a are all tilted from (111) in the [211] d i r e c t i o n ; i.e., (977) is e q u i v a l e n t to ( 1 1 1 ) 7 . 0 1 ~ ( 7 5 5 ) to (111)9.45~ and (533) to (111)14.42~ Other notations are sometimes used to describe regularly stepped surfaces. These notations all p r e s u p p o s e a knowledge of the atomic structure of the steps information which is not always available. They also assume the steps to be perfect with no step edge wandering and no statistical spread in terrace widths. Furthermore, some uniformly stepped surfaces, such as those near (0001) surfaces of hcp lattices, may have two different step widths that alternate across the surface. For these reasons, it is recommended that such notations not be used for the description of real stepped surfaces. 1.6. Disorder Real surfaces are never perfect. Fortunately, however, it is often possible to prepare samples that are sufficiently ideal that the effects of their intrinsic defects on an experiment are of secondary importance. On the other hand, imperfections play essential roles in many important processes and are coming increasingly under study as the power of available instrumentation improves. Tringides and Lagally give a detailed description of surface defects in Chapter 12. Schematic examples of most types of surface imperfections are shown in Fig. 1.21. Many bulk imperfections also terminate at surfaces including screw and edge dislocations and grain boundaries. Steps are defects characteristic of surfaces and interfaces and are without bulk analogs. This section summarizes some of the ways used to describe the various types of surface imperfections. In thermodynamic equilibrium, at T > 0 K, even the most perfect low-index surfaces must have defects. The probability of forming a defect at a given temperature is determined by the Boltzmann factor for the energy required to form a single defect from the perfect crystalline lattice. Figure 1.20 shows the results of a computer simulation of the equilibrium distributions of vacancies and adatoms at the (100) surface of a simple cubic lattice. As shown in the figure, the number of imperfections increases dramatically as the temperature is raised. 1.6.1. Pair distribution function

One way to describe the positions of atoms at a surface is to specify the instantaneous position vector of every atom. This is convenient only when the number of atoms involved is small or if they are arranged on a perfect crystalline lattice so that the basis of the single unit mesh specifies the whole structure. However, for many


37

~0

Fig. 1.20. Some types of surface defects.

0

~

Fig. 1.21. Thermally induced disorder on a surface. The temperature given for each configuration is in units of the vacancy formation energy divided by Boltzmann's constant.

38

W.N. Unertl

important systems, it is neither possible nor useful to attempt to determine the instantaneous atomic positions. Liquids and gases and their surface analogs are obvious examples as are amorphous solids. In these cases, it is usually the statistical properties that are of interest anyway. Furthermore, many real solids and their surfaces, although for the most part highly crystalline, have important imperfections which break their ideal translational symmetry - - many of these imperfections are described in Chapter 12. Finally, several of the most important experimental methods are not able to determine the individual atomic positions; these include all diffraction techniques such as LEED, X-ray diffraction and atom diffraction. An alternative, statistical description of an atomic arrangement is useful in many of the above situations and is provided by the pair distribution function, P(R) ~. If p(r) is a function that describes the probability of finding an atom within dr of r, P(R) is defined as

P(R) - ~p(r)p(r + R)dr

(1.19)

P(R) gives the probability of finding an atom at position R if there is another atom located at r. That is, P(R) is the self-convolution of p(r). P(R) is also called the auto-correlation function or the Patterson function. Figure 1.22 shows some simple, one-dimensional examples. The first, Fig. 1.22a, is an infinite array of equally spaced atoms with spacing a. Once any of these is chosen as the origin, there is zero probability of finding another atom anywhere else except at distances which are integral multiples of a. The Patterson function is simply the average of all such probability distributions obtained when each scatterer in turn is taken at the origin. For this example, P(R) is unity at each integral multiple of a and zero otherwise as shown. The second example, Fig. 1.22b, is also for an infinite array. However, in this case, the array consists of pairs of identical atoms with characteristic spacings a and b. If the left hand member of each pair is chosen as the origin, other scatterers will be found at a, a+ b, 2a + b, 2a + 2b . . . . . Choosing the right hand scatterer at the origin yields the distances b, b + a, 2b + a, 2b + 2a . . . . . These are the only unique choices of origin, so that each contributes equally to P(R) yielding the result shown. Notice that the peaks in P(R), which are at integral multiples of (a + b), have twice the weight. The third example, Fig. 1.22c, is for a finite array of four identical, equally spaced atoms. If the left most one is chosen as the origin, the others are found at a, 2a and 3a. If the second is chosen, the relative locations of the others are - a , a and 2a; for the t h i r d , - 2 a , - a and a; for the f o u r t h , - 3 a , - 2 a and - a . Since each of these choices is equally likely, each contributes equally to produce the P(R) shown. Thus, if one of the atoms is chosen at random, the probability of finding a second at +a is 0.75, at +2a is 0.5 and at +3a is 0.25. In all cases, P(R) is symmetrical about

1 Otherdistribution functions such as the height-height distribution function are also useful, for example, to describe surface roughness. See Chapter 12 for additional discussion of distribution functions.


39

m(x) 1

o

o

o

T

0

(c)

~

a

9

2a

T

i

3a

F

4a

X

P(x) 1

--•

-,~{-- a Oe

Oe

Oe

Oe

--

0

b

2b

T

3b

x"'

4b

v~

X

(b) P(x) 1

"'"

0

0

0

0

0

0

0

0

"'"

0

a

2a

3a 4a

5a

(a) Fig. 1.22. One-dimensional linear arrays and their pair distribution functions. (a) An infinite array with equal spacings. (b) An infinite array with two spacings. (c) A finite array with four equally spaced points. the origin. This has important consequences for diffraction as is discussed in Chapter 7. The preceding examples used simple, highly ordered atomic arrangements to illustrate the physical meaning of P(R). P(R) is also extremely useful in the description of disordered systems, including fluids. Most types of defects on crystalline lattices can be divided into two classes. Long range correlations are preserved on average for defects of the first kind and structure is retained in P(R) for large R. For defects of the second kind the long range correlations are not preserved.

1.6.2. Defects of the first kind Thermal vibrations of atoms about their mean positions in the lattice is the most common example of a defect of the first kind. Other examples include localized defects such as vacancies, interstitials, and impurity atoms. The distribution of

40

W.N. Unertl

P(x) 1

0

a

2a

3a

4a

5a

Fig. 1.23. Pair distribution functions for crystalline and fluid phases.

defects of the first kind 9(r) can be described with respect to the atomic distribution of the average lattice ; i.e. p(r) - + Ap(r)

(1.20)

where Ap(r) is the deviation from the average lattice and = 0. The average can be an average over time as in the case of thermal vibrations or over space as in the case of static vacancies. Figure 1.23 shows the effect of thermal vibrations on the pair distribution function for an ideal one-dimensional lattice like that in Fig. 1.22a. P(R) still has equally spaced maxima at the average atomic locations. But the vibrational motion causes the atoms to have a finite root-mean-square displacement from equilibrium so that the peaks in P(R) are broadened. 1.6.3. Defects of the second kind

Defects of the second kind do not maintain long range correlation between the atomic positions. There are several important examples. A surface step is created if a bulk screw dislocation S intersects the surface as illustrated for a simple cubic crystal in Fig. 1.24. The screw dislocation in Fig. 1.24 is formed on the (100) surface of the simple cubic crystal by slipping the top and bottom parts of the crystal in opposite directions parallel to the [100] direction to reveal the small shaded ledge of (001) orientation. A screw dislocation can also be visualized as a spiral stacking of crystal planes about a dislocation line. In this example, the screw dislocation line is parallel to [100]. At any point along a dislocation line, the direction is specified by a unit vector ~. For the screw dislocation in Fig. 1.24, ~ points in the [100] direction. Screw dislocations are important in crystal growth because their surface steps provide continually available nucleation sites at which new atoms can be incorporated into the growing crystal. The other type of bulk dislocation is the edge dislocation E formed when an extra half plane of atoms is inserted into the crystal. The edge of this half plane forms the dislocation line. Edge dislocations are also characterized by the vector ~. For the example in Fig. 1.24, ~ is parallel to [010]. The surface defect created by the intersection of an edge dislocation with the surface has locally distorted surface


41

(ool)

J

J

J J

J J

J J

J J

J

jL ---'qJ r

f

f

I

(olo)

-i

~

TM

(!00)

b

f f

r

f

I

f

r

m

Fig. 1.24. A mixed dislocation ~ in a simple cubic crystal. It emerges at the (100) surface as a perfect screw dislocation S and at the (0T0) surface as a perfect edge dislocation E.

W.N. Unertl

42

structure and will have different mechanical, chemical and electronic properties than the perfect surface. A displacement called the Burgers vector, b, is associated with every dislocation line. A simple procedure for determining the Burgers vector is to compare an atom-to-atom closed circuit which encircles the dislocation line to a similar loop in the perfect crystal. The direction of this loop is chosen to be that which would advance a right-hand screw in the direction of ~,. When compared to the loop in the perfect crystal, the loop around the dislocation requires an extra segment to close it. For example, consider the loop labeled P in Fig. 1.24. A closed loop is obtained by following the path 1 to 2 to 3 to 4 to 5. A similar loop drawn to enclose the edge dislocation is not closed. The Burgers vector is defined as the vector, drawn from the start of the loop at 1 to its end at 5, that closes the loop. In this case b is 1 unit in the [100] direction, or 1[100]. A second example is given for the screw dislocation. When b.~ = 0, the dislocation is said to have edge orientation; i.e., b is perpendicular to ~. When b.~= b, where b is the magnitude of b, it has screw orientation. In this case b is parallel to ~. A single dislocation, such as the one connecting the screw and edge dislocations shown in Fig. 1.24, is characterized by the same value of b at each point along its length even though it has very different characteristics on the two surfaces at which it emerges. Between the two faces, 0 < b.~ < b, and the dislocation is said to be mixed. If more than one dislocation line is enclosed in the loop path, the resulting Burgers vector is the sum of the Burgers vectors of the individual dislocations. Burgers vectors are also useful in describing the diffraction from a crystal with dislocations (Jackson, 1991). A more in-depth discussion of dislocations can be found in Wert and Thomson (1970). Most "single crystals" actually consist of a number of very slightly misoriented crystallites. Such a structure is called mosaic. The lattice mismatch, which occurs at the boundaries between the crystallites, is taken up by arrays of dislocation lines except for the special case of a twin boundary in which the lattices of the two crystals, which make up the twin, are related by a rotation about a crystallographic axis [hkl] called the twinning axis and/or by mirror reflection across the twin boundary plane (hkl). A simple example is a (111) or [111] twin in an fcc lattice. In the perfect lattice, the (111) planes are stacked in a sequence which repeats every three atomic planes; e. g., ...ABCABCABCABC .... The stacking sequence at a twin boundary is altered: ...ABCABCBACBAC... where the boldface C indicates the location of the twin boundary which, in this case, is both a mirror plane and a rotation axis. The second kind of surface defect discussed so far has been that associated with the surfaces of bulk crystals. Other important imperfections occur in layers of adsorbed atoms or molecules. In real systems these ordered structures will not be perfect. Both vacancies and impurities can be present. Furthermore, defects such as

43


c~

c"

c~

","

C~

",;

,,

,;

.

"'L . . . . .

".z

c)

~

;

", ~

r

C.s~.

.,~

~,,i

II (I

C;

,","

(',

C;

,";

C~

Fig. 1.25. Domain boundaries for a (2•

C}

C)

C)

overlayer structure on a square lattice.

emerging dislocation lines, which are present on the substrate surface, will influence the structure of the ordered overlayer. Mixed phases also occur in adsorbed layers. For example, if there is an attractive interaction between the molecules, small islands of an ordered phase, which might form a simple, coincident, or an incoherent lattice with respect to the substrate, can nucleate and grow in a surrounding two-dimensional fluid phase. On a perfect substrate, nuclei will form randomly and ordered islands will grow around them in registry with the substrate as illustrated in Fig. 1.25. If any two of these nuclei form at substrate sites which cannot be connected by a translation vector of the overlayer lattice, it will not be possible to cover the entire surface with a single region of order. Instead, when the ordered islands grow together, their edges will not match and a defect, called a domain boundary or domain wall, is created. These boundaries are sometimes also called anti-phase boundaries because their presence can cause certain diffraction beams to be absent as discussed in Chapter 7. Figure 1.25 shows an example of domain boundaries for a (2• adsorbate structure formed on a square lattice substrate by adsorption at atop sites. In this case, there are four equivalent adsorption sites labeled 1, 2, 3, and 4 in the figure; i.e. these sites lie on four independent, interpenetrating (2• sub-lattices. Domain walls form between the various sub-lattices and limit the degree of long range order that can be obtained in an overlayer. The domain walls are said to be light domain walls if the local atom density is lower than that of a perfect surface and heavy domain walls if the density is higher. Domain walls play an important role in surface phase transitions as discussed in Chapters 11 and 13.

References Burns, G., 1985, Solid State Physics. Academic Press, Orlando, FL, Chapters 1-5. Cotton, F.A., 1990, Chemical Applications of Group Theory. Wiley, New York.

44

W.N. Unertl

Ertl, G. and J. Kuppers, 1985, Low Energy Electrons and Surface Chemistry. VCH, Weinheim, Chapter 11. International Tables for Crystallography, eds J.S. Kasper and K. Lonsdale, 1959, Kynoch Press, Birmingham. Jackson, A.G., 1991, Handbook of Crystallography for Electron Microscopists and Others. Springer-Verlag, New York. Lambert, R.M., 1975, Surface Sci. 49, 325. Leamy, H.J., G.H. Gilmer and K.A. Jackson, 1975, in: Surface Physics of Materials, ed. J.M. Blakely. Academic Press, New York, p. 121. MacGillavry, C.H., 1976, Fantasy & Symmetry, Abrams, New York. Park, R.L. and H.H. Madden, 1968, Surface Sci. 11, 188. Robinson, I.K., W.K. Washkiewicz, P.H. Fuoss, J.B. Stark, and P.A. Bennett, 1986, Phys. Rev. B33, 7013. Somorjai, G.A., 1994, Surface Chemistry and Catalysis. Wiley, New York, p. 49. SUN Commission of IUPAT, 1978, Physica 93A, 1. Vainshtein, B.K., 1981, Modern Crystallography I. Springer-Verlag, Berlin. Wert, C.A. and R.M. Thomson, 1970, Physics of Solids. McGraw-Hill, New York. Wood, E.A., 1963, J. Appl. Phys. 35, 1305.

Appendix: The seventeen two-dimensional space groups Oblique

1

pl

0

0

No. 1

pl

0

0

Origin on 1

Oblique 2

No. 2

p211

| |

| |

Origin at 2

|

p2


Rectangular

45

plml

m

No. 3

pm

| |

Origin on m

Rectangular

plgl

m

No. 4

Pg

Origin on g

Rectangular

clml

m

No. 5

|

Origin on m

Cm

46

W.N. Unertl

Rectangular

|

p2mm

mm

|

|

pmm

No. 6

| A

|

|

j |

|

|

|

Origin at 2ram

Rectangular

mm

p2 mg

v J

9

I

I

I

6

6~--i

v J

|

~)

pmg

No. 7

I

Origin at 2

Rectangular

mm

Pgg

No. 8

p2gg

L

9

i I , ----f V

Origin at 2

I

9

9

I

9

47

S u # a c e crystallography

Rectangular

C2 m m

mm

@ |

@ @ | |

@ |

cram

No. 9

&

A

A

@ | | | @ |

| |

@ @ | |

,

.

Origin at 2 m m

Square

4

p4

No. I 0

@ @

@ @ @ |

L r

@ @

@ @ @ |

k

@ |

9 9

Origin at 4

Square

4 mm

p4 mm

@ @ | | @@

@ @ | @| @

No. 1 1

'*

// @~) |

|

@ @ | | @@ Origin at 4 r a m

W.N. Unertl

48

Square

4ram

p4 gm

No. 12

A

T',,

9

I /wx---l-~/ m

m

|

i(--

Origin at 4

Hexagonal

pa

3

No. 13

p3

|

Origin at 3

Hexagonal

3 m

p3mi

g

Origin at 3 m 1

No. 1 4

p3ml


49

Hexagonal

3m

p31m

p31m

No. 15

@

%

% Origin at 31 m

Hexagonal

6

p6

@ |

@ | |

p6

No. 1 6

|

| @

@

@@

@@

Origin at 6

Hexagonal

6mm

p6mm

No. 17

J m

|

| v

Origin at 6ram

,

v


CHAPTER 2

Thermodynamics and Statistical. Mechanics of Surfaces E.D. WILLIAMS and N.C. B A R T E L T Department of Physics University of Maryland College Park, MD 20742-4111, USA

Handbook of Sueace Science Volume I, edited by W.N. Unertl


51

Contents

2.1.

Introduction

2.2.

Thermodynamic formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. The surface excess quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

64

2.3.

Thermodynamics offaceting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Thermodynamics oforientational phase separation . . . . . . . . . . . . . . . . . . .

66 68

2.2.2.

2.3.2. 2.4.

2.5.

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

Gibbs adsorption equation

Types offaceting transitions

53 55 60

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

73

Statistical mechanics ofvicinal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1. Simple stepped surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Expressions for step formation and interaction energies . . . . . . . . . . . . . . . . .

79 80 82

2.4.3.

90

Experimental determination of statistical mechanical parameters

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

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

96

52

2.1. Introduction Real solid surfaces are seldom in equilibrium, yet the study of surface thermodynamics provides us with a wealth of tools for understanding and working with surfaces. In part this is because we can often use the concept of local equilibration. What this means is that even though an entire sample has not equilibrated with its vapor, on some limited length scale mass transport has occurred (for instance by diffusion across the surface or through the bulk) to a sufficient extent to allow the surface to equilibrate. One example of this phenomenon is adsorption onto a non-equilibrium substrate. Even though the original surface may be far from equilibrium, the adsorbing particles may equilibrate on the arbitrary structure of the surface to minimize their thermodynamic free energy subject to the constraints of their environment. Another example is thermodynamic faceting. While in real thermodynamic equilibrium a solid will attain a crystal shape which minimizes the surface free energy, in most real situations mass transport is not sufficient to allow this to happen. However, on a local scale surfaces with an arbitrary net orientation will rearrange to expose the orientations that would be present on the equilibrium crystal shape. The resulting new facets will grow, reducing the surface free energy until their size becomes so large that mass transport across them becomes negligibly slow. An example of observation of such local facet formation is shown in Fig. 2.1, in which facet growth on a Pt covered W surface is shown to occur at larger and larger length scales following annealing at higher and higher temperatures. In addition to its use in describing equilibrated systems, thermodynamics provides us with the information needed to understand the evolution of surfaces under far from equilibrium conditions (Blakely and Mykura, 1962; Bonzel et al., 1984; Herring, 1951b; Keeffe et al., 1993; Mullins, 1961). Very simply this is because mass flow requires gradients in the chemical potential, which in turn is a thermodynamic quantity. Thus it is worth some effort to understand the thermodynamics of equilibrium systems, and in particular how the thermodynamic quantities vary with parameters of interest in mass transport, such as temperature, concentration, composition and surface orientation. As a result, a large amount of effort is expended in trying to prepare well characterized, thermally equilibrated systems and to measure their properties. The question of whether a system is in equilibrium, or can be described even partially by equilibrium thermodynamics remains difficult. Many non-equilibrium processes result in structures that can look very much like equilibrium structures. A traditional strong signature of thermal equilibrium is the observation of reversible phase transitions. More generally thermal equilibrium can be demonstrated by an experiment which shows that the structure observed is not dependent on the path (for instance, thermal cycling or growth rate) used to 53

54

E.D. Williams and N.C. Bartelt

Fig. 2.1. A W(I 11 ) surface is thermodynamically stable when clean. However addition of Pt changes the free energies to favor formation of {211 }-type facets. Limitations of mass transport prevent the surface from completely rearranging to form three large {211 } facets. Instead, a large number of small faceted mounds form, each exposing the preferred orientations. Annealing at increasing temperatures allows increasing growth of the facets. The figure show pairs of LEED (incident energy 101 eV) and STM patterns of the faceted surface following annealing at 900 K (top panel), 1200 K (middle panel) and 1400 K (bottom panel). The coverage of Pt on each surface is -- 1-2 ML. Figure provided by Prof. T.E. Madey of Rutgers (Madey et al., 1993; Madey et al., 1991; Song et al., 1990; Song et al., 1991 ).

Thermodynamics and statistical mechanics of surfaces

55

reach it. With the advent of powerful surface imaging techniques, it is now also becoming possible to observe thermal fluctuations of systems in equilibrium directly. Such measurements not only confirm the existence of equilibration, but provide a wealth of atomic-scale information about rates and mechanisms. A thermodynamic approach to such information provides a way of condensing out the aspects of the information that have macroscopic physical consequences. Surface thermodynamics and statistical mechanics has been the subject of many comprehensive reviews, from which we will here only condense a focused subset of information. Most of the formalism needed to discuss surface thermodynamics is contained in Gibbs' work (Gibbs, 1961), and the quantitative application of these ideas to problems of surface morphology and faceting was nearly complete by the time of Herring's work on surfaces (Herring 1951a; Herring 1951b; Herring 1953). Excellent reviews exist on the applications of thermodynamics to problems of clean solid surfaces (Blakely, 1973) and interfaces (Cahn, 1977). The fundamental statistical mechanical problem of surfaces, understanding the entropy of step wandering, was first addressed by Mullins (Gruber and Mullins, 1967), and consequences of this problem and the related problem of surface roughening were worked out thoroughly in the 1970s and early 80s (Weeks, 1980; Wortis, 1988). The problems of merging a thermodynamic and statistical mechanical approach to understanding surface processes remains a topic of current research (Nozieres, 1991). We will review this evolution of surface thermodynamics and statistical mechanics briefly, and with an emphasis on the problem of surface faceting. The formal introduction of classical surface thermodynamics and surface excess quantities in w 2.2 should serve as a starting point to allow further investigation of the applications of these ideas to other problems as well (Dash, 1975; Griffiths, 1980). The discussion of surface faceting as an orientational phase separation, which is presented in w 2.3, is a formal and detailed presentation of ideas not generally explicitly addressed in reviews of surface thermodynamics. The overview of surface statistical mechanics is a synopsis of useful forms for quantifying experimental observables in such a way that they can be used in a thermodynamic formalism to predict the stability of surfaces with respect to faceting. A brief overview of some recent experimental observations which demonstrate the ideas concludes the chapter.

2.2. Thermodynamic formalism Before one can discuss surface thermodynamics, one first must have some idea about what one means by a surface. While many geometric, microscopic pictures instantly spring to mind, from the viewpoint of thermodynamics a surface is only what separates two phases in thermal equilibrium. Thus, strictly speaking, surfaces only occur for the special values of pressure and temperature where two phases coexist. The fundamental quantity of surface thermodynamics is the surface tension ~,, which governs the amount of work required to create a surface. It is defined as the change in internal energy when the surface area between two phases in thermodynamic equilibrium is increased at constant entropy S, volume V and particle number of each of the chemical components N/:

56


= Y

~9U -~

. s.v.u,

(2.1)

In thinking about this equation, it is important to emphasize that the numbers being conserved are the total particle numbers in the system, which includes both the condensed and vapor phase as well as the intervening surface. In the light of the discussion which follows, it should also be emphasized that Y is independent of any microscopic ideas about where the surface is actually located. As a simple example of the meaning of the surface tension, consider a simple cubic solid with lattice constant a which is bound together by nearest neighbor bonds of strength E. If the solid is pulled apart at T = 0 K to create two surfaces of (100) orientation one bond is broken per unit cell of area a 2. Thus the energy cost per unit area of creating each of the two surfaces is just E/2a 2. In principle 7 for solid surfaces can be measured directly in a cleavage experiment by determining the work required to separate two parts of a crystal from each other. The difficulties of performing and interpreting such an experiment are described by Blakely (Blakely, 1973). Another way of determining the surface tension directly is the method of "zero creep" (Josell and Spaepen, 1993). In the "zero creep" method the sample, often a thin wire, is subjected to an external force. At high enough temperature that surface diffusion is readily possible, the sample will tend to change its shape to minimize the area. The applied external force which balances this tendency (i.e. for which the rate of change ~ or creep ~ of the surface area is zero) gives an absolute measure of the surface tension. These measurements have limitations, being only suitable to solids at high temperatures, and also where the surface tension does not depend greatly on surface orientation (Blakely, 1973). Alternatively, the surface tension can be obtained by theoretical calculation. Although great progress has been made in the ability to calculate surface energies, this remains a difficult problem due to the complexities of surface reconstructions. Moreover, most calculations provide values only for zero temperature. A survey of representative values of the surface tension from calculation and experiment is presented in Table 2.1. This provides a feeling for the range of uncertainty in the determinations of the surface tension, and of its general magnitude. In spite of limited knowledge of the absolute values of the surface tension of real materials, the concept of surface tension is useful experimentally because changes in the surface tension with temperature, adsorption and orientation govern a range of important processes. To see how this arises, we will consider the inclusion of the surface tension first in the bulk thermodynamic equations, and then we will introduce the concepts of surface thermodynamics. Including the definition of surface tension in the internal energy, we have: U = T S - pV + yA + ~_~ kt~N~

(2.2)

d U = TdS - pd V + ydA + ~_~ ~i dN~.

(2.3)

and

i


57

Table 2.1 Selected values of the surface tension determined experimentally and theoretically are shown here to illustrate the range of values observed. The surface tension is a decreasing function of temperature so the experimental values for the melting temperature should be lower than the theoretical values for absolute zero. More complete tabulations of experimental values can be found in (Bonzel, 1995; Kumikov and Khokonov, 1983; Tyson and Miller, 1977). Tabulations of theoretical values can be found in (Liu et al., 199 lb; Methfessel et al., 1992; Smith et al., 1991). Element

Experimental value: T = Tm (from Tyson and Miller 1977) (Jim 2)

AI Ag

1.02 1.09

Ni

2.08

Pd Mo

1.74 2.51

Theoretical values: T = 0 K Miller index LAPW calculation (from EAM Calculation (from Methfessel et al. 1992) Liu et al. 1991a) (J/m 2) (Jim 2) (100) (100) (111) (100) (111) (llO) (311) (100) (110) (100)

1.21 1.21

1.86 3.14 3.52

0.55 0.70 0.62 1.63 1.49 1.78 1.77 1.45

w h e r e kti is the c h e m i c a l potential o f the ith c h e m i c a l c o m p o n e n t . (The additional c o m p l e x i t y of "wall t e n s i o n " terms is discussed by Griffiths (Griffiths, 1980), but will not be i n c l u d e d here). By standard m a n i p u l a t i o n s a m o u n t i n g to L e g e n d r e t r a n s f o r m a t i o n s (Callen, 1985) we can define other t h e r m o d y n a m i c potentials with different i n d e p e n d e n t variables, such as the H e l m h o l t z free energy, F = U - T S , yielding: d E = - S d T - p d V + ~[dA + ~_~ Ixi dNi .

(2.4)

i

T h e c h a n g e of variables gives alternative definitions o f the d e p e n d e n t variables, such as the surface tension, yielding: OF ~[= - ~

9

(2.5)

r.v.N,

W h i l e the functional f o r m of Eq. (2.1) for the surface tension w o u l d be useful for c o n s i d e r i n g an adiabatic process, Eq. (2.5) w o u l d be useful in an i s o t h e r m a l process. Both Eqs. (2.1) and (2.5) apply to closed systems, that is s y s t e m s in w h i c h the total n u m b e r (mass) o f each c h e m i c a l c o m p o n e n t is constant. A n o t h e r potential w h i c h is useful u n d e r c o n d i t i o n s o f c o n s t a n t c h e m i c a l potential is the g r a n d therm o d y n a m i c potential f l = F - ,Y_,kt~Ni = - p V + ),A, yielding" l

58

E . D . W i l l i a m s a n d N. C. B a r t e l t

dEl = - S d T - pdV + ~ldA - ~_~ N~ d~,

(2.6)

i

and another definition of the surface tension: (2.7) T, V, lai

We once again emphasize that in using the bulk thermodynamic equations such as Eqs. (2.1)-(2.7), we must keep in mind that they refer to the complete system of the solid, the fluid (either gas or liquid) in equilibrium with the solid, and the interface between the two. Thus the quantities held constant (such as entropy, volume and number in Eq. (2.1)) refer to the entire system, not the solid, fluid or surface individually. Because the surface generally represents a small contribution to the total content of the system, it can be difficult to isolate the thermodynamic effects of the surface. To make these points concrete, we can consider as an example a cleavage experiment, which might be used to measure the surface tension. To make such a measurement under the conditions of Eq. (2.1), we would consider a solid in equilibrium with its fluid in a rigid (zero net volume change), closed (zero net mass change), thermally insulated (zero heat flow and thus zero net entropy change) container. The surface tension would be measured by measuring the force needed to separate the solid into two pieces, exposing new surfaces of total new area 2A, as described by Blakely (Blakely, 1973). During this process, gas might adsorb onto the new surfaces, or atoms or molecules might be displaced from the new surfaces into the fluid phase. Thus, the numbers in the solid and fluid phases individually would change. However, the total number in the system would remain fixed, so Eq. (2.1) remains applicable. Thus using the bulk thermodynamic equations allows us to ignore the atomiclevel behavior of the system due to the surface in describing the macroscopic behavior. Of course this also means that using the bulk thermodynamic equations alone does not allow us to develop predictive capabilities for the influence of the interface. A simple example shows this immediately. Suppose that we carry out the adiabatic cleavage experiment. We expect that there will be a temperature change when we do work on a system adiabatically, so we approach this problem using the standard approach (Zemansky, 1968) of writing the differential of the entropy given constant volume and number: o = y, t s =

3S dT + T-ff-~

igS A, V,N i

dA

(2.8a)

T, V,N,

and then using a Maxwell relationship derived from Eq. (2.4):

~S ~A

T. V,N~

~T

(2.8b) VJ,,N,

59


and rearranging to obtain"

as

by

CA,V. N dT = T - ~

A, V,N~

dA,

(2.9)

A,V,N~

where C is the heat capacity. This shows that we need to know the heat capacity of the system as a function of the surface area, and the variation of the surface tension with temperature. Rigorously, we would measure both parameters directly for the real system. For a first estimate, if the solid has compact shape and thus few atoms at the surface, we might assume that the heat capacity is the sum of the heat capacities of the fluid and solid phases, and neglect the influence of the area on the heat capacity. The variation of the surface tension with temperature can be measured, and is typically found to be about (Blakely, 1973): 1

c3y

=-2•

-5 K -l

(2.10)

y(Tm) aT where T(Tm) is the surface tension of the solid just at the melting point. For a specific system, let us consider a solid of 1 mole of Cu at 1200 K. The vapor pressure at this temperature is about 10 -I torr and the heat capacity of the solid is Cp = 30.6 J/mol-K. The surface tension of Cu at the melting point is 1.7 J/m 2. If we have initially comparable volumes of solid and vapor, the contribution of the vapor to the total heat capacity is negligible, and we estimate the change in temperature between the original (T o) and final (Tf) states as: ln(Tf/To) = - 10- 6 m -2 AA ,

(2.11)

where the area is measured in square meters. This illustrates that the cleavage of a compact piece of Cu, which will yield a change in surface area of about 1 cm z, results in a negligible change in temperature. This conclusion is inescapable so long as the conditions which allowed us to neglect the effect of the surface on the heat capacity hold. For an extreme example of a large surface area situation, consider the same mole of Cu arranged as a thin film only 10 atoms thick, which is to be cleaved to give two films each 5 atoms thick. The increase in the surface area in this case will be

Nav a2 _ 6x 1013

AA =---~

- ---if---- (2.6x10 -8 cm)2 = 8•

7 cm 2

(2.12)

and the temperature change would be a decrease of about 0.8% from the original temperature using Eq. (2.11). However in this case, the assumption that the heat capacity of the system is the same as that of bulk Cu is highly questionable. To deal with such problems where the surface is playing an important role, we need some way to address the contribution of the surface to the thermodynamic parameters such as the heat capacity explicitly.

60


2.2.1. The surface excess quantities The question of what is the surface, as opposed to the solid or vapor, contribution to any thermodynamic quantity depends on how one defines a surface atom. This is a microscopic question, and thus there is no unique thermodynamic approach to answering it. Thus some arbitrary decision must be made to define what is the surface. Then, once this decision is made, it is possible to go on and derive an internally consistent surface thermodynamics. One natural starting point for specifying surface functions (taken recently for example, by Nozi~res (Nozi/~res, 1991) is to break up the volume of the coexisting phases into three regions by introducing a pair of dividing surfaces. These dividing surfaces define three regions, the condensed phase, the fluid phase, and the intervening surface region. As originally suggested by Gibbs (Gibbs, 1961; Griffiths, 1980), there is a simpler (although slightly more abstract) way to define surface quantities. The approach is to define a single, flat dividing surface (the Gibbs dividing surface) which divides the system in to two phases c (the condensed phase) and f (the fluid phase). Once one has done this the volumes of the two phases are fixed, and satisfy I,x + Vf = V

(2.13)

This dividing surface is not in any way intended to describe the real surface. The real surface, defined roughly as those atoms whose properties are very different from the bulk phases, could be very broad. Regardless of the width of the physical surface, sufficiently far away from the dividing surface, the densities of all the extensive quantities such as energy, entropy, and atom numbers have the welldefined values of the bulk phases. This allows the extensive properties of each phase to be specified: for example the number of atoms in the condensed phase c is just ~ = 9~V~. In general, sums such as ~ + ~ are not the same as the total number N,. The differences are called the surface excesses. For instance for the number of atoms of chemical component i, the equation: + NI+ N~ = Ni

(2.14)

defines the surface excess N~ of the ith component Ni, and similar equations exist for the other chemical components and the entropy. At this stage, the position of the dividing surface is arbitrary. For a single component system, the choice of Gibbs is to place the dividing surface so that the value of N~ is zero for the one component of the system. In a multi-component system, one can choose any one of the components to set to zero in defining the dividing surface. In using the thermodynamic equations which result from defining surface excesses, two points are important: (1) The placement of the dividing surface defines the area A which is used in the thermodynamic equations, and (2) the placement of the dividing surface is not fixed geometrically. The dividing surface will move physically to maintain/W as the system changes thermodynamically. Thus, for the component chosen as the reference, the intuitive property of a one-to-one correspondence between the surface area and the number of atoms on the surface is lost.

Thermodynamics and statistical mechanics ~r

61

p-gas p-solid

0000 2.75A 1.0000 0000 0000

( ~ p(Z)

2.5A

Fig. 2.2. Schematic cross section of a solid-vapor interface illustrating the choice of the dividing surface. The hypothetical simple cubic solid with bulk lattice spacir~g of 2.5/~ has a top layer which is expanded by 10% in its vertical (normal to the surface) lattice spacing. The zero of the z coordinate system is arbitrarily set through the center of the top layer of atoms. The variation of the density of this "real" system with respect to z is illustrated on the graph to the right of the diagram. The density in the gas phase is effectively zero.

As an illustrative example of the definition of the dividing surface, consider a s o l i d - v a p o r interface for which the spacing between the top two layers of atoms is 10% larger than the spacing in the bulk as shown in Fig. 2.2. As a simple approximation, we treat the atomic density in the solid as a series of delta functions located at the centers of the atoms, We then place the zero of the z axis arbitrarily through the center of the top layer of atoms as shown in Fig. 2.2. Then the overall density profile can be.written as a discontinuous function: P = P,,,,iJ

for

-1.1a z --~

(2.15c)

1

P=~

P = Pgas

Pso,~J

-

2

where a is the bulk interplanar spacing, Pgas is the density of the gas phase, and Psolid is the density of the condensed (bulk solid) phase. The surface excess n u m b e r density can then be calculated for any arbitrary choice of the dividing surface ds, using:

62

E.D. Williams a n d N. C Bartelt

N~ = A

p(z)dz

P~olia dz -

-

Zmin

Z min

(2.16a)

pgas dz s

w h e r e Psolid and Pgas, the densities o f the solid and gas phases, are explicitly a s s u m e d to be constant, p(z) is the real density o f the s y s t e m as a function o f position, and A is the area o f the d i v i d i n g surface, which is p e r p e n d i c u l a r to the z axis. P u t t i n g the density profile o f Eq. (2.15) into Eq. (2.16a), we obtain Ns 2.75/~ - ~ = ( - 1.375 A, - ds) Psolid "1- 1.1 Ps,,,~a+ (ds - 1.375)9gas

(2.16b)

To obtain a zero surface excess in this model, the d i v i d i n g surface m u s t be p l a c e d at ds = 1.125 A,, slightly a b o v e the center o f the top layer o f atoms. T h e intuitive c h o i c e o f a d i v i d i n g surface, which w o u l d be t h r o u g h the centers o f the top layers, does not yield a zero surface excess number, and in general will not do so. N e g a t i v e surface e x c e s s e s are o b t a i n e d for dividing surfaces placed h i g h e r than 1.125 A, and p o s i t i v e surface e x c e s s e s are obtained for dividing surfaces placed l o w e r than 1.125 A,. T h e case o f an e x p a n d e d surface layer, used to d e m o n s t r a t e this point is not artificial. Real surfaces have contracted or e x p a n d e d interlayer s p a c i n g s as illustrated in Table 2.2. In m u l t i c o m p o n e n t systems there is no reason to e x p e c t that the density profile o f a s e c o n d c o m p o n e n t will be similar to that o f the first (Blakely and S h e l t o n , 1975). For instance, if the solid is a s t o i c h i o m e t r i c c o m p o u n d , then there is likely to be preferential s e g r e g a t i o n o f one of the other c o m p o n e n t s near the surface. Or, in a n o t h e r c o m m o n situation, one can introduce a large v a p o r pressure o f a s e c o n d c o m p o n e n t which has a negligible solubility in the solid, but which adsorbs in Table 2.2 The interplanar spacings of near-surface layers are often expanded or contracted from the values of the bulk interplanar spacings. Some experimentally determined values of these modified spacings are shown here. The value ~!12 is the relaxation of the first interplanar spacing expressed as a percentage of the bulk interplanar spacing. A positive value indicates an expansion of the first interplanar spacing, a negative value a contraction. The values ~123 and &t34, are the corresponding relaxations of deeper layers. The selected values are taken from (Rous, 1995), where a full tabulation, references, and detailed discussion can be found. Element Mo Mo Ni Ni Ni Ni Ag Ag

Miller index

&l 12

~123

(100) (110) (111) (100) (110) (311) (111) (100)

-9.5+2.0 -1.5+2.0 -1.2+1.2 -1.1+_2.0 -8.6x'-0.5 -15.9+1.0 0.0-&5.0 -7.6+3.0

1.0+_2.0

+3.5_+0.5

+4.2+3.0

~134

-0.4_+0.7


63

appreciable quantities at the surface. For such a multi-component system, there will be an equation like Eq. (2.14) to calculate the excessnumber at the surface. Except for special cases, it will not be possible to find a dividing surface where more than one of the surface excess numbers is zero. Once the extensive values have been defined in terms of their bulk (solid and fluid) components and the surface components as in Eqs. (2.13) and (2.14), we can subtract the contributions due to the bulk components from the thermodynamic potentials of Eqs. (2.2)-(2.6) to obtain thermodynamic functions which describe the surface alone. To do this, we begin with the integral form of the internal energy: U "- TS - p V =

T(S~ +

.-I-'yA + ~,~ ].l,iN i i S f + a s) -- p ( V c -~- V f + V s) +

(2.17)

yA +

~.Li(N~/+

N ~ / + N~/)

i

and subtract the bulk (condensed + fluid phase) contribution to the entropy, volume and number, giving

9 = rs"

(2.18)

vA + is l

In making this definition, we have explicitly (by setting the sum over i ~ l) indicated that the values of N~ are no longer independently variable because we have defined the dividing surface by setting N~ = 0. We can write the corresponding differential form: dU~= TdS"+ 7dA + ~ B,dN~

(2.19)

is 1

where the equation reflects the fact that dN~ is zero because N~ is a constant. The form of the surface excess for the Helmholtz and grand potentials then can be found from the standard expressions, F = U - TS and ~ = U - T S - ~ BiN i to obtain t

F ' = 7,4 + X l'tiN~

and

d F ' = - S" d r + "idA + X Bi dN~

is 1

~2~= yA

(2.20a,b)

ir l

and

d~S=-

S ~ dT + ydA - X ~

dl.t,

(2.21 a,b)

isl

Thus, the Helmholtz free energy per area is equal to the surface tension for the case of a single component system, when the dividing surface is defined to set the number at the surface to zero. However, the grand potential per area is equal to the surface tension in all cases, which makes it useful for addressing changes in the surface properties during adsorption or segregation.

64

E.D. Williant~ and N.C. Bartelt

Once we have defined the surface thermodynamic potentials, we then immediately have all the associated thermodynamic identities which can be used to identify relationships among the various quantities. For the Helmholtz and grand potentials we readily derive the Maxwell relationships relating to heats of cleavage and adsorption, and the effect of adsorption on the surface tension: Helmholtz

Grand

-~9Ss ba r,u~

~9y

~T a .,V~

-~)S s T.A.N~,,

~N~ A.T.N~.,

0~ti /)T

bA

-/9S s ba

~T a.~

-/)S s

,gN~

~[i

~9y

A

r.a,~%

~9T A,~,,~j

,T,~j,,

~A

(2.22a,b)

(2.22c,d)

(2.22e,f)

In all of these equations it is implicitly assumed that i ;el, because the number density of the component used to define the Gibbs dividing surface is not an independent variable. The application of such thermodynamic relationships is most commonly made to problems of adsorption on surfaces. In these cases, the reference component used for defining the dividing surface is the dominant component of the bulk material. It is generally assumed implicitly that the substrate is unperturbed by the adsorption of other components. Under this assumption the shape of the surface is unchanged during adsorption, and the derivatives with respect to area are equivalent to definitions of quantities per area. As we will show later, such assumptions are unnecessary, and the thermodynamic formalism described above is applicable to real processes in which the substrate changes as well.

2.2.2. Gibbs Adsorption Equation The final Maxwell equation shown above for the grand potential (Eq. 2.22f) is also known as the Gibbs Adsorption Equation. It is most commonly expressed as -~),,/

~kl'i A.T.il,,,

_ -3N~

OA

_ N~ - 19 M II,.T.Ilj A A

(2.23)

where the coverage 0 is defined as the ratio of the number of adsorbed particles to the number of available adsorption sites M. In deriving this equation, we again note that it is based on a definition of the dividing surface in which the number of surface atoms of the principle component NI is fixed at zero. Under many conditions it is a good approximation to make the assumption that the fluid phase is an ideal gas, for which the chemical potential is:

65

Thermodynamics and statistical mechanics of su~'aces

kt(T,p) = kt~

+ kTln(p/po)

(2.24)

Then the adsorption equation can be written as

~ln(p,/p,..)

= k T 19

Nmax

(2.25)

A

A ,T,~I,j~i

Thus the surface tension decreases as an increased pressure of the adsorbing gas causes more adsorption. To evaluate this expression quantitatively, we need to know the relationship between the surface coverage and the pressure (or chemical potential) of the adsorbing species. Such relationships cannot be predicted thermodynamically. They must be measured, or calculated using some microscopic model of adsorption. There are a large number of different models for the adsorbing layer (Clark, 1970; Dash, 1975; Hill, 1960; Payne et al., 1991). As an example of the application of Eq. (2.25) we will present just one, the Langmuir adsorption model. In the Langmuir adsorption model, we consider the case of strong adsorption (chemical adsorption) in which adsorption occurs at specific sites on the surface. We also make the simplifying assumption that there are no interactions between atoms at different sites. Then the adsorption of atoms changes the energy via the binding energy E per atom with thesurface, and it changes the entropy via the configurational entropy generated by distributing the adsorbed atoms among the available binding sites of the surface. We can deal with this problem simply in the Canonical Ensemble by writing first the partition function of an individual adsorbed atom (Hill, 1960): q = qo exp (-~kT)

(2.26)

where qo is the vibrational partition function of the atom trapped in the potential well of the binding site. We then use simple combinatorics to determine the number of configurations of N molecules distributed among the M sites of the surface, and write the partition function of the adsorbed layer: M!q(T) N Q(N,M,T) =

(2.27)

N ! ( M - N) !

Then using the standard definition: (2.28)

F(T,A,N) = - kTln Q(N,M,T)

we can derive an expression for the chemical potential, l.t = - k T

~)lnQ 'ON

T,M

=

t" )

(2.29)

66


where 0 = N / M as in Eq. (2.23). Using Eq. (2.26) for the individual atom partition function, we see that the binding energy of adsorption contributes to the chemical potential of the adsorbed species:

)o

(2.30)

Explicitly noting that the area of the surface is Ma, where a is the area per binding site, we can also calculate the contribution of the overlayer to the surface tension: /)lnQ

- _ kTln

vo=-kT Ma

a

T,N

M M-N

_ kTln a

1

(2.31)

1 -0

This contribution to the surface tension is the equivalent of the "spreading pressure" of the adsorbed layer. That is 7~dA is the work done in expanding an overlayer of coverage 0 from area A to area A + dA. We can alternatively derive Eq. (2.31) by integrating the Gibbs adsorption equation, Eq. (2.23), with the relationship between chemical potential and coverage of Eq. (2.30) inserted explicitly: 6)

d T = - - - dlaa

-kT

a

dO 1-0

(2.32)

and integrating to obtain kT

y - y , + ~ In (1 - 0) a

(2.33)

where the constant of integration T0 is the surface tension at zero coverage, i.e. the surface tension of the clean surface.

2.3. Thermodynamics of faceting The reasons for the faceting of surfaces and the mechanisms by which faceting occurs are problems of long standing in materials science and surface physics (Flytzani-Stephanopoulos and Schmidt, 1979; Herring, 1951 a; Moore, 1962). Faceting is defined as the break-up of a surface of some arbitrary macroscopic orientation into a "hill-and valley" structure which exposes surfaces of different orientations. Such a change in surface morphology is illustrated in Figs. 2.1 and 2.3. In recent years, unambiguous identifications of equilibrium faceted surfaces, in sufficient detail for thermodynamic analyses to be applied, have begun to appear (Drechsler, 1985; Dreschler, 1992; Heyraud et al., 1989; Ozcomert et al., 1993; Phaneuf and Williams, 1987; Phaneuf et al., 1988; Song and Mochrie, 1994; van Pinxteren and Frenken, 1992b; Wei et al., 1991; Williams et al., 1993; Yoon et al., 1994). Such

Thermodynamics and statistical mechanics ~su~. aces

67

Fig. 2.3. When clean, stepped Ag surfaces of orientation near the (110) are stable. However, exposure to oxygen changes the surface free energies, causing faceting to expose the Ag(l 10) orientation (with no steps) and another orientation containing a high density of steps. These STM images, each showing a 500 x 500 /~ area, show the evolution of the surface structure from one containing thermally lquctuating steps, through nucleation of facets, to the formation of large scale facet structure during exposure to 1• -9 torr of oxygen at room temperature. The panels show (a) the clean surface with an average step separation (terrace width) of 35/~, (b) after 62 min exposure to oxygen, (c) after 74 min large fluctuations in the terrace width become apparent, (d) after 96 min the nucleation of a facet has occurred, (e) after 102 min facets are growing linearly, (f) after 138 min facet sizes saturate and no further growth is observed. Figure provided by Prof. J.E. Reutt-Robey of the University of Maryland (Ozcomert et al., 1993; Ozcomert et al., 1994).

68

E.D. Williamsand N.C. Bartelt

observations provide an opportunity to obtain information about the anisotropy of the surface tension, and about how the anisotropy is influenced by adsorption, reconstruction and temperature. Furthermore, when such thermodynamic observations are combined with the results of direct imaging techniques, such as STM (Binnig and Rohrer, 1987), LEEM (Bauer, 1985), and REM (Osakabe et al., 1981), it becomes possible to understand faceting from an atomic point of view. In this section, we will describe equilibrium faceting as a thermodynamic phase separation, and develop the appropriate intensive variables to describe the resulting "orientational phase diagram". 2.3.1. Thermodynamics of orientational phase separation

Surface morphology arises on solids (as opposed to liquids) because the surface tension depends on the crystallographic orientation of the surface. As a result, the surface morphology of minimum area (i.e. a flat surface) is not necessarily the morphology which minimizes the free energy. Gibbs (Gibbs, 1961) recognized that surfaces will spontaneously rearrange to minimize their total surface tension, even if this involves an increase in surface area. Herring explicitly addressed the problem of the equilibrium morphology of a macroscopic surface of arbitrary orientation

Fig. 2.4. A surface will be unstable with respect to faceting if the total surface tension decreases in going from the upper panel to the lower panel. Notice that the projected areas Ai' are additive (analogous to volume in fluids, and in contrast to the total areas). The requirement of conservation of macroscopic orientation, given by Eq. (2.34), is illustrated in the insert.

69


(Herring, 195 l a). The important physics of his approach is illustrated in Fig. 2.4. A The requirements for a surface of a given macroscopic orientation no to break up A A into new orientations n a and n b are simply that the net orientation is conserved, and the total surface tension is reduced: AO A

no =

o

A

Aa A

A n,, + A t' nb a

A

(2.34) b

A

A Y(no) > A y(n b) + A Y(nb)

(2.35) A

where y is the surface tension and A i is the area of the surface of orientation n~. Several formalisms can be used to determine the morphology that minimizes the surface tension, the most common being the construction of a Wulff plot to determine the equilibrium crystal shape (Herring, 1951 a; Jayaprakash et al., 1984; Kern, 1987; Rottman and Wortis, 1984; Wortis, 1988). Unfortunately, the relationship between the orientational variation of the surface tension, and the surface stability is cumbersome to apply quantitatively. A more easily applied approach to evaluating the conditions for faceting is to describe faceting as a phase separation: in other words we identify the facets of different orientations (as shown in Fig. 2.4) as phases in equilibrium with one another. To make this analogy explicit, we need to define a free energy for which the standard convexity arguments familiar to phase separation in fluids apply (Bartelt et al., 1991 ; Chernov, 1961 ; Metois and Heyraud, 1989; van Pinxteren and Frenken, 1992a). We recall from Eqs. (2.20) and (2.21) that the surface tension y can be identified with either the Helmholtz free energy in a one-component system, F~(T,A)

= yA

and

F.~ y(T) - ~ (one component system)

(2.36)

or with the Grand potential in a system with any number of components. ~(T,A,I~,,~) = yA

and

y(T,p~,,) - A

(2.37)

We then see that Eq. (2.35) above is equivalent to requiring that either the Helmholtz free energy or the Grand Potential be minimized when a surface breaks into a faceted structure. In what follows, we will use the Grand potential as it allows generalization to multi-component systems. An identical development can be performed using the Helmholtz free energy for a single component system. In a phase separation, the minimization of the free energy is constrained by conservation equations on the extensive variables. (For instance for the case of a two component fluid, the volume and numbers of each of the two components are conserved.) For the case of faceting, this constraint is the constraint that the macroscopic orientation of the surface is conserved, as indicated by Eq. (2.34). The vector form of Eq. (2.34) represents three conservation requirements, which can be expressed in rectangular coordinates as: A z = A "z + A b

where

A

A z = z. A~

(2.38)

70

E.D. W i l l i a m s a n d N.C. B a r t e l t

where

A y "- Ay + A or

where

A x = A~x + A b

Ay = ~. A~t A

A

A x = x. An.

(2.39) (2.40)

Physically, in making these definitions we generally define the z-direction as perpendicular to a low index surface, which serves as the reference surface. We then choose the x- and y-directions as orthogonal high-symmetry directions within the reference surface. The z component of the area Of a facet of arbitrary orientation is then the projection of the area of that facet onto the reference surface. The x and y components give the projections of the area onto planes perpendicular to the reference surface. The magnitude of their vector sum ~/A~ + Ay2 is a measure of the total "rise" of the facet with respect to the reference surface; in other words tan~ = "~A 2 + AI~ /Az.

We now can write the grand potential explicitly in terms of these extensive variables. This in turn allows us to define corresponding intensive variables, which must be equal for different phases (facet orientations) which are in equilibrium: ~ "~ = ~ "~( T , A z , A x , A y , l.t i~ l )

(2.41)

yields

3Az

~pz ~

etc. T . A .A t , l t , ,

(2.42)

,

Here we use the symbol "Pz" to indicate an intensive variable analogous to pressure, and note that there will be two similar variables corresponding to the derivatives with respect to the x- and y-components of the area. The use of these intensive variables is, as in standard problems of phase separation, to define equations which can be used to discuss the criteria for phase separation" explicitly we would write 9 . . A A for co-existence of facets of orientations n,, and n b ,,p,, . . . . =

h,,

Pz

etc.,

(2.43)

However, the use of the rectangular coordinate system leads to unnecessary mathematical complexity in setting up the conditions for equilibrium. It is possible to obtain a much more physically meaningful and mathematically tractable set of equations by working in spherical coordinates. We define the polar angle ~ as the angle between the normal to the surface of interest and the vector normal to the A reference plane, z. (In Fig. 2.4, ~ is shown as an angle between the surface and a horizontal reference surface. This is equivalent to the angle between the normals.) We define the azimuthal angle 0 as the angle between the projection of the surface normal into the reference plane and the x axis. With these definitions, the components of the surface area can be written as:

Thermodynamics and statistical mechanics of su.rfaces

71

A z = A cos t~

(2.44)

Ax = A sin ~ cos 0

(2.45)

A x = A sin ~ sin 0

(2.46)

The physical meaning of the two angles as illustrated in Fig. 2.5 for a surface with an orientation close to (typically within 10 ~ of) the reference surface has been discussed precisely by Nelson et al. (Nelson et al., 1993). The tangent of the polar angle is equal to the density of steps on the surface, and the tangent of the azimuthal angle is equal to the mean one-dimensional density of kinks on a step edge (at T = 0 K). Precise applications of these ideas to real surfaces require careful attention to the crystallographic definition of the step (Eisner and Einstein, 1993; Nicholas, 1965; Van Hove and Somorjai, 1980). Because the tangent of the polar angle is a physically meaningful quantity, we now define an appropriate reduced free energy, which has a simple dependence on tan

f=^

~

I

AzA~Ay ] ~ _f2 s z . A~ - -A~ T , -A- z' -A- z' A~' ['l'i~l

(2.47)

which now contains an explicit dependence on two areal densities:

Px = Ax/Az = tanr

(2.48)

py A y / A z = tant~sin0

(2.49)

=

h

Fig. 2.5. A surface vicinal to a high index surface will be composed of a density of steps and kinks determined by the misorientation angles 0 and t~. The number density of steps tan~ = p =h/, where h = the step height and = the distance between steps. When we define a reference direction in the reference surface for which the azimuthal angle 0 equals zero, the number density of kinks tan0 = an/, where an is the kink depth and is the average distance between kinks. In thermal equilibrium there will be a distribution of step-step and kink-kink separations, leading to step wandering. Perfectly periodic spacings are shown in the figure simply for clarity.

72


The areal densities have the physical meaning of the projected step density perpendicular to the x and y axes as in Fig. 2.5. We then re-write Eqs. (2.34) and (2.35) as s

A~' = A z + Abz

(2.50)

o o b b Azf > Azf ~ + Azf

(2.51)

The convexity requirement thus applies to this reduced free energy just as to the original grand potential. This allows us to use the tie-bar formalism in a plot of reduced free energy vs. either the x- or y-component of the areal density (Eqs. (2.48) and (2.49)). We can illustrate this graphically as shown in Fig. 2.6. The resulting equations describing the phase separation between a flat surface of orientation n o A A and a hill-and-valley structure containing new orientations na and nb are" A~ _ p~ - pO _ py _ py Az

P]~- p~

(2.52)

o O r - p~

In making use of the reduced surface free energy, we retain the original definitions of the intensive variables conjugate to Ax and Ay a s usual"

"Px

,,

Of 2 s OAx

T~4 t.A

v,

B ,, t

O(Ax /Az)

T,A

v,la,,;

0p/

, etc.

(2.53)

T,py,B,,~

f(p)

o

,r

~b

~0 Xb

~a

~

Xa

Fig. 2.6. Orientational phase separation occurs when a "hill and valley" structure has a lower total surface tension than a flat surface, as in Eqs. (2.35) and (2.51). This translates into a convexity requirement on the "reduced surface tension" (which is analogous to a Helmholtz free energy) versus either component (Eq. (2.8)) of the step density, p. The figure illustrates a non-convex surface tension curve which would lead to faceting. The illustrated curve is schematic only: the form shown would occur physically only if there are attractive interactions between steps strong enough to overcome entropic and elastic interactions. The,phase separation is indicated by the tie bar connecting points a and b. For a macroscopic orientation ?io the relative areas of the orientations na and nb are found from 9

,

A

A

Eq. (2.52), as illustrated in the figure with x,, _ p,, - Po _ A'b Here we use the nomenclature illustrated Xb Po -- Pb A'a" in Fig. 2.4 where A'i = A~ in the nomenclature of Eqs. (2.44) and (2.52).

Thermodynamics and statistical mechanics of su~. aces

73

The physical equality of the intensive variables in the two phases (" Px~ " = " pb ,,, , , p y a , , = , , pby ") appears as the equivalent slope of the free energy curves at the points of tangency in Fig. 2.6. In order to obtain a mathematically convenient form of the third independent equation corresponding to the z-component of the "step pressure", " -z P " we can define a different reduced free energy to obtain a thermodynamic function which varies as tanS, where we define p = tans = ~/p2 + p2. We proceed by dividing the grand potential by a combination of the x- and y-components of the area:

Az Ax Az fF _ f~s T, ----------w ~l A x 2 + 2' ~l A 2 2' ~[ A 2 + Ay2' ~[A 2x+ Ay Ay x at- Ay

~L i~ `

)=f P

(2.54)

We then recover the intensive z-component of the step density via: /

~gf~ s

"Pz" -

/)Az T,A z "4 y ,la i,~I

+

aOtal/ # +

A2

~(f/p) T,O,B~,,I

~(1/p)

(2.55) T,O,I.t.,~

As illustrated in Fig. 2.5, the magnitude of thermodynamic density of Eqs. (2.48) and (2.49), p = ~/p2 + P~ = tan~, can be interpreted physically as the step density on a vicinal surface. The two components are the step density projected onto the high symmetry direction (x) and an orthogonal direction (y). The azimuthal angle 0 in this description is the angle between the average direction of the step edge and the direction perpendicular to a high symmetry reference direction on the facet. It is thus physically related to the density of kinks on the step edge (Bartelt et al., 1992). This physical interpretation of the density components becomes particularly useful in interpreting observed orientational phase diagrams in terms of atomic models of the surface.

2.3.2. Types of faceting transitions To make the application of Eqs. (2.43), (2.53) and (2.55) concrete, we will introduce here a statistical mechanical description of stepped surfaces which uses the step and kink densities as the fundamental units of the structure. This equation and its parameters will be discussed in detail in the following section. The statistical mechanical description of the variation of the reduced surface tension with orientation which we will use is (Jayaprakash et al., 1984): f(~,0,T) =if(T) + ~(0,T) Itan~l + g(0,T) [ tan~[ 3 h

(2.56)

wheref"(T) =f(0,0,T) can be seen from Eq. (2.47) to be equal to the surface tension of the reference plane, 13(0,T) is the free energy cost per unit length of creating an isolated step, h is the step height, and g(0,T)ltan~l 3 is the free energy cost per unit area due to step-step interactions. The variations of the step formation and step interaction terms with both temperature and azimuthal angle 0 are governed by the

74

E.D. Williams and N. C. Bartelt

kink energy e. For temperatures well b e l o w the r o u g h e n i n g transition o f the low-index surface, specific functional forms for 13 and g can be derived for specific atomic models of the stepped surface, and used for quantitative analysis of experimental observations. Using this form for the reduced surface tension, we will discuss in general the different conditions which can arise in evaluating phase equilibria i n v o l v i n g faceting. An orientational instability may arise from the intrinsic variation o f the surface tension with orientation, as is illustrated in Fig. 2.6. H o w e v e r , real vicinal surfaces are most often o b s e r v e d to be stable with respect to faceting (Somorjai and Van Hove, 1989; Williams and Bartelt, 1989), so we can expect that a m o n o t o n i c variation of free energy with angle, as described by Eq. (2.56), will usually be appropriate for clean surfaces. In contrast, orientational instability (faceting) is frequently induced by chemical adsorption or by structural phase transitions. As p r o p o s e d by Cahn (Cahn, 1982), this can be understood by considering that the additional process completely alters the orientational variation of the reduced surface tension, resulting in intersecting curves as shown in Fig. 2.7. In this case the phase separation occurs not only b e t w e e n different orientations, but also between different compositions (in the case of adsorption) or structures (in the case of phase transitions). In the following, we will present an analysis of the conditions g o v e r n i n g phase separation in the case of such intersecting curves. B e c a u s e Eq. (2.56) contains a cusp at the origin, there are some possibilities in orientational phase diagrams which do not arise in normal phase equilibria. In Fig. 2.7, we illustrate three types of phase separation that might occur for a surface of arbitrary polar and azimuthal angle (00, 00) near to a low index orientation (~ = 0). The three

Opposite" Fig. 2.7. A change in surface composition (e.g. due to adsorption or segregation) or structure

(e.g. due to a phase transition) can change the variation of the surface tension with orientation. If the curve for a "perturbed" surface, labeled b, intersects that of the "unperturbed" surface, labeled a, then the convexity requirement illustrated in Fig. 2.6 will lead to^orienta~ional phase separation. The new ^ surface orientations exposed will have the vector densities p,, and Ph. The surfaces of orientation na ^ will have the composition or structure of the "unperturbed" phase, and those of orientation nb will have the composition or structure of the "perturbed" phase. The figure illustrates schematically how intersecting free energy curves lead to faceting. In the left hand column are shown the variation of the projected surface tension with one component of the vector density, Eqs. (2.48) or (2.49). Because the intersecting curves lead to a projected surface tension curve which is not convex, there is a region of unstable orientations, leading to step rearrangement to form facets. The corresponding projections of the phase diagrams into the 0--9 plane are shown in the right hand column. The three general types of tie lines one might expect from the faceting of vicinal surfaces: (a) No cusps are involved. The convexity requirement is applicable to the variation in reduced surface tension with both components Px and py of the step density. (The variation with lay is shown.) Thus Eqs. (2.66) and (2.67) determine the misorientation angles of the two phases. (b) If there is a knife-edge cusp along a high symmetry direction, one of the ends of the tie line can intersect the cusp, removing one of the tangency requirement, and replacing it with the inequality of Eq. (2.68). (c) If there is a deep cusp in the surface tension at the low-index surface (~ = 0), one of the ends of the tie line can intersect the cusp, leading to phase separation in the polar angle, leaving the azimuthal angle fixed. In this case the convexity requirement on the free energy is expressed in terms of the magnitude of the step density, rather than its vector components, as in Eq. (2.69).


fl'py)

75

a

P~ po

p~ Oa

I/I ////

::,"

I

Py

'

I b I p, p~

', p~

Y

P~

Y

(a)

X

f(Py)

pa pO ,"" P~x Oo ,'" ,"'"

/

fa(O,l~a,T)fb(o,C'v T)-

O(bl// j lIt

t I p~ po p~

Py

P~

(b)

f~p)

X

a

pa

p~ Pb

I

Po

I

Oo

l

Pa

(c) Fig. 2.7. Caption opposite.

p~

p~

Ii

Y

76


cases are: Case 1, separation to two arbitrary orientations, ~,, 0a and Cb, 0b for which the reduced surface tension is smoothly varying (differentiable); Case 2, separation between an arbitrary orientation Ca, 0a and an orientation %, 0b = 0 where there is a singularity in the reduced surface tension for 0b -- 0; and Case 3, separation between an orientation ~,,, 0,, = 00 and the orientation of the low index surface Cb = 0 when there is a cusp in the reduced surface tension for the low index surface. The projection of the tie-bar into the tanr plane for each of the three cases, and the variation of the reduced surface tension of the two phases with the components of the step density are shown in Fig. 2.7.

Case 1 In order to have a tie-bar between points a and b when the reduced surface tension has no cusps, as shown in Fig. 2.7a, there must be a plane which is tangent to the two reduced surface tension curves at both points a and b. This requirement results in three equations which specify that the two components of the slope must individually be equal, and that a plane of the overall slope actually intersects both points. If we define Px and Or axes, corresponding to 0 = 0 ~ and 90 ~ respectively, as in Fig. 2.7a, the first two equations determined by these conditions are that the slopes (or the x- and y-components of the step pressures) at the two points are the same:

afo(oo,r

aL(oo,r p,: ap~

(2.57)

and

afo(oo,r Op,~,

o:

ap~

(2.58)

b

P,

The evaluation of these derivatives requires some care (Williams et al., 1993). If the free energy function is expressed in terms of variables tan~ and tan0, then the expressions to be used in evaluating the derivatives are:

af

af

Opx

Otanr

Pv

cos0 + tan0

-tan0 /

af 0tan 0

tanr

(2.59)

tanO cos 0

and

af

af

~f ~)Py

o,

Otanr

tan0

sin0 + 0tan0

'-"nO

/tant~ lc~

"

(2.60)

The requirement that the tie-bar actually intersects both points results in the third equation"

77


L(o~,,~) =L(O~162176 + (px~ - pa)

+ (py~- p~)

afa(Oa'r

2p~

afa(O~,r p~

(2.61)

The geometrically derived equation can be replaced with the thermodynamically derived requirement of equal z components of the "step pressure". (Arduous algebra confirms that the simpler form of the step pressure equation can be expressed as a linear combination of Eqs. (2.57), (2.58) and (2.61).) From Eq. (2.55), the equality of the z-"step pressures" is:

fo-po&

T,O,,,la i, i

=fb -- Pb ~--~p b

(2.62) T, Ob,lai, I

When we have a specific expression for the variation of the reduced surface tension with orientation, we can evaluate these expressions explicitly. Using Eq. (2.56) in Eqs. (2.57) and (2.58) allows us to obtain the relationships:

f3i = "~/cosOi + 3g i tanZt~i cosOi-

/

/r176 LcOSOi1 tan20)(' )

1 ()~._._.._+.L_atanO ()gi tan2t~i hi/)tanO

5P~ o. = ~'/sin0i + 3g, tan2~i sin0i + -hi ~c)tan0 + atan 0

cos O;

(2.63)

(2.64)

In the simplifying case where there is no change in the polar angle 0 during the phase separation, we can define the angle 0 = 0 to be orthogonal to the step edges, and then Eqs. (2.57), (2.58) and (2.61) give a very simple result:

& D

o~tanr

T,O,,,M ,,t

atanCb T.0h.la,,~

(2.65)

Upon inserting Eq. (2.56) for the angular dependence of the surface free energy, and rearranging, this results in: ~ h ( T ) - [5,(T) = 3g,(T)h, tan2~,- 3gb(T)h o tan2~b

(2.66)

Using Eq. (2.56) in Eq. (2.61) directly gives the general result: f,~'-f~' = 2g,(0,,,T)ltan~,l 3 - 2gh(0h,T)ltan~bl3

(2.67)

where the difference in the surface tension of the two phases at ~ = 0 is Aft(T) = fo"-f~'. These results show that, within the formalism of Eq. (2.56), Case 1 step phase separation can occur if a perturbing process causes changes in step interactions, which could arise for instance from adsorption-induced changes in surface stress. However these changes must match in rather restrictive ways the corresponding changes in the zero-angle reduced surface tension f o and the step free energies [3. Phase separation between two low-symmetry orientations has been observed for

78


vicinal Pb(111) (van Pinxteren and Frenken, 1992a; van Pinxteren and Frenken, 1992b). In this case, the perturbing process is surface melting, so that phase separation occurs between an unmelted surface of low step density, and a melted surface of higher misorientation angle. In this case, of course, Eq. (2.56) is not appropriate to describe the free energy of the melted surface. However, the driving mechanisms for the phase separation can still be understood in general physical terms as due to changes in the initial slope and curvature of the surface tension curve. Case 2 In the second type of phase separation, illustrated in Fig. 2.7(b), we imagine that the reduced surface tension for one of the phases, fb, has a knife-edge singularity along the high-symmetry direction, 0b = 0. In this case we cannot require that the y-component of the tie-bar be tangent to the reduced surface tension curve at Oh = 0, and we lose the requirement of Eq. (2.58). However, for phase separation to Oh = 0 to occur, the slope of the reduced surface tension curve as it approaches Oh = 0 must be greater than the slope of the tie-line. Thus Eqs. (2.58) can be replaced by an inequality corresponding to

(2.68)

c)P>,

b

b

p,pv ~ ()

Equation (2.57) remains valid. Because the slope of the tie-bar must still match the tangent of free energy curve for phase a the form of Eq. (2.67), which describes the difference in the surface tension between the two phases, is also unchanged. The derivatives of the reduced surface tension of phase b, Eqs. (2.63) and (64), are simplified in form as Oh = 0. Case 3 In the final type of phase separation, illustrated in Fig. 2.7(c), we imagine that the reduced surface tension for one phase has a cusp-like singularity at Ch = 0. Physically, this corresponds to phase separation between a uniformly stepped surface, and a surface with a hill-and-valley structure consisting of a low-index facet and a bunches of steps. In this case we have no information about the behavior of the system with azimuthal angle, so we are reduced to two independent equations describing the system, rather than three. Furthermore, because of the cusp, the requirement on the slope becomes an inequality which can be written most usefully as:

&

I a

/)tan r

(2.69)

I 0, = o, ,~ -, ,,

The physical meaning of this equation is that the initial slope of curve 2 must be steeper than the slope of the tie bar (which is in turn tangent to curve 1). The initial slope is, of course, set by the energy cost of isolated steps, so this mathematical requirement tells us how much the perturbing process must change the step energy in order to cause faceting. The resulting inequality, using the form of Eq. (2.56) for the reduced surface free energy, is:


132(T) - I]l(T) > 3gl(T)hl tan 2 ~1

79 (2.70)

where the step heights are the same in both phases. As in Case 1 phase separation, this shows that the magnitude of the step-interactions sets the energy scale for step bunching to occur. Equation (2.67) still remains valid because of the requirement that the slope of the tie bar matches the tangent to the free energy of curve a, but it is simpler in form since ~h = 0: Af" =f~' - i f , = 2ga(0a,T)ltan~al 3

(2.71)

The magnitude of the change in the surface tension and the step energy required to allow orientational phase separation can be estimated using the formalism described in the following section and calculated or measured energetic values of step or kink energies and the surface stress (see below). The result is that approximately a 0.1% change in the facet energy coupled with approximately a 10% change in the step energy is needed to cause phase separation between a flat surface and a step bunch. Such a mechanism has been proposed for the orientational phase separation of vicinal S i ( l l l ) (Williams and Bartelt, 1991), Pb(111) (Metois and Heyraud, 1989; Nozieres, 1989) and for Au(111) near the melting temperature (Bilalbegovic et al., 1992; Bilalbegovic et al., 1993; Breuer and Bonzel, 1992).

2.4. Statistical mechanics of vicinal surfaces The thermodynamic description above allows us to think about changes in macroscopic parameters such as orientation, and to discuss those changes in terms of related thermodynamic parameters such as the surface tension. Intellectual curiosity, as well as the hope of someday being able to predict behavior such as faceting from an understanding of atomic properties, cause us to want to understand the governing thermodynamic principles in terms of underlying atomic structure and energetics. The link between thermodynamics and atomic properties is statistical mechanics. To capture the essential physics of the role of entropy in determining the surface behavior above T = 0 K, we use a simple lattice model of the solid as the basis for the statistical mechanics of the surface. In this model, called the solid-on-solid (SOS) model, atomic positions are restricted to discrete lattice sites and interact with one another via pair-wise interactions. To obtain a qualitative understanding, we use the simplest possible model, in which we neglect atomic vibrations at the lattice sites and restrict the model to only near-neighbor interactions. A surface structure calculated using a Monte Carlo simulation within the solid-on-solid model is shown in Fig. 2.8a. The effect of finite temperature is to create disorder in the structure in the form of vacancies and extra atoms on the terraces, and in the form of "kinks" at the edges of the steps. Within this model we find that the natural parameters which appear in the thermodynamic equations, the step density tan~, and the kink density tan0, are also the natural atomic scale structures whose energies define the behavior of the stepped surface.

E.D. Williamsand N.C. Bartelt

80

I

a)

1.2

kT=O.8E

. . . .

'

. . . .

'

. . . . .

'

()j_

1.0 0.8

"'--

O

~- 0.6

0

0.4 4

0.2 0.0 0.0

. . . . . 0.5

1.0 kT/e

1.5

2.0

Fig. 2.8. (a) Monte Carlo simulation of a stepped surface in the solid-on-solid (SOS) model at kT = 0.8[;,

where • is both the nearest-neighbor interaction energy and the kink energy. (Figure from the work of Bartelt et al. (1991 ).) (b) Calculation of the variation of the surface tensionf ~ and step energy ~(T) (solid line) for the body centered solid-on-solid model, based on the work of van Beijeren (van Beijeren, 1977; van Beijeren and Nolden, 1987).

2.4.1. Simple stepped surfaces The first stage in d e v e l o p i n g a description of the surface tension based on the properties of steps is to a s s u m e that there are cusps in the surface tension at a relatively small n u m b e r of low index orientations, and that between these orientations,

Thermodynamics and statistical mechanics of suqaces

81

the surface tension varies smoothly. One can then describe changes in orientation in terms of changes in step density, and introduce the temperature dependence of the surface tension by treating the entropy of the steps correctly (Gruber and Mullins, 1967; Voronkov, 1968). The variation of the surface tension with step density, tanr should then have a leading term which is due to the contribution of the terraces between the steps, a second term due to the free energy cost of the steps, and a third term due to any step-step interactions (Wortis, 1988). The inadequacy of zero-temperature calculations of the surface tension is immediately apparent if one considers the temperature dependence of these terms separately. The energy cost of creating an excitation at a step edge is much lower than that of creating an excitation on a terrace. For the simple-cubic solid-on-solid model with only near neighbor interactions of energy ~, for instance, the energy cost of moving an atom from a position in a straight step edge to a site along the edge is 2E. However, to create a vacancy and an extra atom on the terrace requires energy 4~. As a result, at a given temperature (below the roughening temperature) thermal excitations at the edge of a step are more prevalent than those on a terrace, as illustrated in Fig. 2.8a. Correspondingly, the contribution to the temperature variation of the surface tension of vicinal surfaces due to the steps is much larger than that of the terrace, as shown in Fig. 2.8b. Thus, one cannot simply extrapolate relative zero-temperature energies for high- and low-index surfaces to non-zero temperature. The correct treatment of the problem of the entropy of steps as a function of step density is not obvious, and was first addressed explicitly by Gruber and Mullins (Gruber and Mullins, 1967). They noted that at non-zero temperature, a step can wander due to the thermal excitation of kinks, as illustrated in Fig. 2.8a, thus generating configurational entropy which lowers the free energy of the isolated step, as illustrated in Fig. 2.8b. However, when there is a train of steps, the wandering is constrained because one expects that a step crossing, which would create an overhang, is energetically unfavorable. Thus, the entropy of step wandering makes the largest favorable contribution to the surface tension when the step density is lowest, an effect which is referred to as the "entropic step repulsion". This entropic repulsion has the same effect as a true energetic repulsion which falls off in strength as the square of the distance between the steps. Subsequent work provided a quantitative description of how this entropic repulsion (as well as any energetic interaction which is not of shorter range) influences the surface tension (Jayaprakash et al., 1984; Nozieres, 1991). The result is that the step interaction term is proportional to the third power of the step density and the expansion for the reduced surface tension contains no second order term in step density. The form of the reduced surface tension thus is, as previously stated in Eq. (2.56) (Jayaprakash et al., 1984): f(O,T) - ~ ' ( T ) + 13(T)Itan~] + g(T)ltan~l 3

(2.72)

where ~~ is the surface tension of the terraces between the steps, 13(T) is the free energy cost per unit length of an isolated step, h is the step height, and g(T) Itan3~l is the free energy cost per unit area due to step-step interactions. It is worth

82


emphasizing the physical reason for the absence of a second order term in this expansion. The "entropic" interaction between steps is the entropy cost associated with the fact that adjacent steps cannot pass through each other (Fisher, 1984; Fisher and Fisher, 1982): Each time a step wanders next to another step, the number of choices of directions the step can wander is reduced by a factor of 1/2. Thus, for each step collision, the entropy of the surface is reduced by kin2. The density of step collisions, and thus the total entropy cost, can be estimated from simple random walk arguments (Fisher, 1984; Fisher and Fisher, 1982). The average distance between step collisions of one step is roughly given by the distance required for a step to wander a distance equal to the average distance l between steps. This distance is simply determined by the step diffusivity as apl2/b 2 (see Eq. (2.84) below). The number of step collisions per unit area is thus kTb2/13ap, and thus the total free energy cost due to the entropy loss of step collisions is kT(ln2)b2/13af, = kT(ln2)b21tan3t~l/(h3ap), reproducing the form of Eqs. (2.56) and (2.72). The simplicity of this argument suggests that the form of Eqs. (2.56) and (2.72) is very general: thermal wandering only contributes a term proportional to Itan~l 3 in the surface free energy. Within a lattice model for the surface, as illustrated in Fig. 2.8, exact expressions are available for the temperature dependence of both the step formation energy (van Beijeren and Nolden, 1987; Williams and Bartelt, 1991) and the step interaction energy (Jayaprakash et al., 1983) in terms of the kink energy (Jayaprakash et al., 1984). Thus given a tabulation of calculated zero-temperature energies for the facet, step and kink, as illustrated in Table 2.3, one can estimate both the thermal and orientational variation of the surface tension. Using the equations presented in the next section at temperatures typical of experiments, we find that the magnitudes of the three parameters are approximately 0.1 eV/]k 2 for the surface tension ~(T), approximately 0.05 eV//~ 2 for the step formation term fS(T)/h, and 0.02 eV/A 2 for the step interaction term g(T).

2.4.2. Expressions for step formation and interaction energies Extracting specific values for atomic energies from thermodynamic observations is always fraught with problems of uniqueness. For the particular case of stepped surfaces, the thermodynamic behavior is completely defined by the tendency of steps to wander, with the result that one can obtain unique information about the step "stiffness" (defined below) which characterizes step wandering (Bartelt et al., 1992). To proceed from a value of the stiffness to a determination of the characteristic energies of the system, one must choose a reasonable microscopic model using either assumptions or additional information available about the system. It is sensible to begin this process by invoking the simplest possible microscopic model that can produce the phenomena of interest. One then uses the observations to fit the microscopic energies of this model. In evaluating how well the energies thus determined describe the true physical energies of the system, consistency of all observations with the model is necessary of course, but not sufficient.

83


Table 2.3 The energetics of steps on Ag(100) and Ag(111) as obtained from the embedded atom method (EAM) (Nelson et al., 1993) and the equivalent crystal theory (ECT) (Khare and Einstein, 1994). The value listed first in each entry is the EAM value, the number listed second is the ECT value. The staircase direction is the vector perpendicular to the average step edge, pointing in the downhill direction. Surface orientation

(100)

Facet e n e r g y ](~ = 0) (meV//~2) 44 98.6

Step-staircase direction

{0111 {001}

Step energy 13(T= 0) (meV/A)

Kink energy

36 56 49

102 213

(meV)

m

(111)

39 76.0

m

{211} {211} {110}

65 170 66 161 76

102 213 99 255

m

In this spirit, we present a specific formulation of the parameters of Eqs. (2.56) or (2.72) in terms of a nearest-neighbor square lattice model, with the addition of a long-range repulsion between steps. There are three energetic parameters in this model: the energy cost 13(0,0) per unit length of creating a step at T = 0 in the high symmetry (0 = 0) direction, the energy e of creating a single kink of depth an (depth is defined normal to the step edge), and the magnitude of the step-step repulsions. More complex models, including kink corner energies (Bartelt et al., 1992; Swartzentruber et al., 1990), non-linear variation of kink energy with kink size (Bartelt et al., 1992), kink-kink interactions (Zhang et al., 1991), honeycomb symmetry, and different forms for the step-step interactions (Frohn et al., 1991; Joos et al., 1991 ; Redfield and Zangwill, 1992) can be considered with correspondingly more complex forms of the equations below. The leading term in the expression for the surface tension as a function of orientation, Eq. (2.72), is the linear variation of the surface tension with step density. The coefficient of this term, I3(0,T), is the free energy per unit length of an isolated step. For arbitrary temperatures, the step free energy can be calculated using forms for the interface energy for the Ising model with both isotropic (Rottman and Wortis, 1981) and anisotropic (Avron et al., 1982) near neighbor interactions. The use of isotropic interactions is equivalent to assuming a square lattice in which the zero-temperature energy cost of steps equals the kink energy per unit length, 13(0,0) = e/ak, where ak is the length of the kink edge (ak > an, with the equality occurring when the kink is perpendicular to the step edge). The square-lattice model is a reasonable zeroth-order approach for high-symmetry surfaces. However, for surfaces of lower symmetry, such as Si(100) (Dijkkamp et

84


al., 1990; Swartzentruber et al., 1990), the kinks and steps have different bonding configurations, and thus a rectangular lattice model, for which one would use the anisotropic calculation, would be more appropriate. The results of the Ising model analogy give the step free energy for steps in the high symmetry direction, 0 = 0, as (Avron et al., 1982) 1 3 ( 0 ' T ) = 1 3 ( 0 ' 0 ) - kapT l n / c ~

2 ~~ /

(2.73)

where ap is the minimum separation between kinks along the step edge. We can derive the low temperature limit of this expression rather simply. In the case where the temperature is low compared to the kink energy, we can safely assume that only kinks of depth one will be excited, and that these will be excited rarely enough that they can be treated as independent. In this case we write an independent particle partition function q for each element of the step edge. Because each element can be straight, kinked out, or kinked in, this partition function has three terms:

q(T) = exp(-ap~(O,O) /kT) [1 + 2exp (-e/kT)]

(2.74a)

where the energy cost of a straight step element is just the length of the element at, times the energy per length of the straight step at zero temperature [3(0,0), and the energy of a kinked element (either in or out) is the energy of the straight element plus the kink energy ~. We then assume a step of N elements, or length L = aN, write down the step Helmholtz free energy and derive the step free energy as the line tension of the system:

F = - k T In qN

13(o,T)-

(2.74b)

~)F

(2.74c)

Then using a series expansion of the remaining logarithmic terming assuming e > kT, one obtains:

2kT ~(0,T) = [3(0,0) - ~ exp ap

"_s (2.74d)

Analytical expressions for the variation of 13(0,T) with tan0 for all temperatures less than the Ising critical temperature (kT c -- 1.13c) can be obtained for the symmetrical case with the substitution of ~ = 2J in the formulas given in (Rottman and Wortis, 1981 ), and for the anisotropic case with the substitutions, [3(0,0) = 2J/ap and ~ = 2J x (Avron et al., 1982). The full isotropic formula is: _a_v_ 13(0,T) = Icos01sinh-~(a(0)lcos01)

kT

+ IsinOIsinh-~(cz(O)lsinOI)

(2.75a)

Thermodynamics and statistical mechanics ~'surfaces

85

where 0;(0) = _2 1 1 -c 2 c [. 1 + (sin20 + c 2 cos 2 20)~]

(2.75b)

and

c=

2sinh (s/kT)

(2.75c)

cosh 2(s/k 73

At T = 0, the step energy reduces to

13(0,0) = 13(0,0) cos0 + -

13

ap

Isin01

(2.76)

where 13(0,0) = 13/ak in the isotropic case. The form of Eq. (2.76) indicates a cusp in the step energy at 0 = 0. At any non-zero temperature the entropy contribution due to the random distribution of kink sites removes the cusp at 0 = 0. However, differentiation of Eq. (2.75) with respect to 0 shows that the transition from zero slope to a large positive slope occurs at extremely small values of 0 for values of 13/kT> 5. This behavior which appears experimentally very much like the knife edge singularity discussed in w 2.2 (Case 2) may be observed for large kink energies or low temperatures. The variation of the step free energy with temperature and with kink density, tan0, calculated using Eq. (2.75), is shown in Fig. 2.9. The slopes of the [3-tan0 curves, which are important in evaluating Eqs. (2.63) and (2.64), can be calculated numerically. The final term in the expression for the reduced surface tension is due to the interactions between steps. The form and magnitude of this term are determined by step wandering due to thermal excitation of kinks, as well as any energetic interactions between the steps such as elastic or dipolar repulsions. This wandering can be described by the "step diffusivity", b2(0,T), which can be calculated as the mean squared displacement of the step edge when there are no kink-kink interactions (Bartelt et al., 1992). The general expression for the diffusivity for a step of zero misorientation angle 0 is:

2a 2 y_~ n 2 exp [-e(n)/kBT] b2(T) =

"--~

(2.77)

I + 2 ~ exp [-s(n)/kBT] n=l

Closed forms for the step diffusivity can be calculated by summing Eq. (2.77) for special cases of the dependence of the kink energy on kink depth, 13(n). The results are shown in Table 2.4 for four cases where no overhanging kinks exist, the


86

I

1.2-

I

I

I

...... " " .

~0--

.........- , . .

0~

_ _ O= 1 0 ~

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

..... 0 - 2 0

~

1.0-

~'* (D v a3.

0.8-

o

0.6-

0.4-

(a) I

0.2

0.0

(a)

1.4-

1.3-

I

I

0.4

I

I

kT/E

I

0.6

I

0.8

I

E/kT = 2 ~ 5 .......... 10 ....... 20 . . . .

I

. ........ :."

. ...... ;.-:22...... : : - ' " .........

30 .......

...-:;':"

....

1.2--

" "

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

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

.__

P <E 1.10

1.0

0.9-

(b) 0.8-

(b)

o.o

oI,

oI~

oI~ tane

o14

oI~

Fig. 2.9. The free energy per unit length of a step 13(0,T) calculated for the square nearest-neighbor lattice model, using Eq. (2.75). (a) Step energy vs. temperature for several values of the net azimuthal angle. (b) Step energy vs. tangent of the azimuthal angle for several values of the temperature. When the temperature ishigh relative to the kink energy e, step wandering lowers the step free energy, and makes it relatively isotropic. When the temperature is low, there is a significant energy cost to changing the orientation away from the high-symmetry direction (Williams et al., 1993).

u n r e s t r i c t e d T S K ( t e r r a c e - s t e p - k i n k ) m o d e l in which the kink e n e r g y varies linearly with kink depth, for the T S K m o d e l with a extra e n e r g y cost for the c o r n e r of a kink, for a discrete G a u s s i a n model, and for a m o d e l in which kinks are restricted to a depth o f o n e unit only. For steps along the h i g h - s y m m e t r y direction 0 = 0, these

87


e q u a t i o n s a r e r e a s o n a b l y a p p l i e d for k T < e, t h e k i n k e n e r g y . W i t h i n the u n r e s t r i c t e d T S K m o d e l , the e x p r e s s i o n for the step d i f f u s i v i t y is s h o w n to b e ( B a r t e l t et al., 1992) b2(T) 2z0 ----r- = ~ a, (1 - z0) 2

(2.78)

w h e r e z0 = e x p ( - e / k T ) . ( T h i s e x p r e s s i o n is e q u i v a l e n t to t h e f o r m s h o w n in T a b l e 2.4 via the s i n h i d e n t i t y . ) T h e t e m p e r a t u r e d e p e n d e n c e o f t h e s t e p d i f f u s i v i t y is s h o w n in F i g . 2.10, for the T S K m o d e l , a n d the o t h e r m o d e l s o f T a b l e 2.4.

Table 2.4. Temperature dependence of the step diffusivity b 2 for different models of the energy cost of forming a kink as a function of the kink depth, E =fiE,n). The table is reproduced from (Bartelt et al., 1992). Model

E(n)

b2/a2

TSK

InlE

I sinh-2(e/2kBT) 2

InlE + (1 - 8(n,o))E c

b2sK/(1 + [exp(e./kBT)- 1] tanh(eJ'kst))

n2E

not a simple function

E(0) = 0, E(+I) = 1

2/[2 + exp(e/kBT)]

TSK + corner Discrete Gaussian Restricted

'

c4~ --.-t~..O0

2

'

'

I

.

.

.

.

1

'

"

'

9

!

.

~

0 0.0

.

.

.

,

, ";

............... ;"; 0.5

1.0

.

.

.

.

.

. 1.5

.

/

.

.

'

' "

. 2.0

kT/c

Fig. 2.10. Temperature dependence of the step diffusivity, b2(T) for the four models of kinks at step edges shown in Table 2.4. (a) solid curve: TSK model in which a kink of length na costs energy E(n) = E; (b) dotted curve: TSK model with an additional corner energy 3E; (c) short-dashed curve: discrete Gaussian model, E(n) = n2E; (d) dash-dotted curve: restricted model, with only n = 0, 1 allowed; (e) long-dashed curve: restricted model in which the step is misoriented away from the high-symmetry direction (Bartelt et al., 1992).


88

For steps with a misorientation angle 0, there is an intrinsic kink density proportional to tan0. Within the TSK approximation, the derivation of the step diffusivity including the dependence on azimuthal angle yields: b2(0,T) 4z~)+ (1 + z~) ~/4z~]+ (1 -z~) 2 tan20 a--------~= tan20 +

(2.79)

(1 - z~):

(Note that due to typesetting errors, incorrect versions of this equation have appeared previously (Bartelt et al., 1992; Williams et al., 1993). Also note that in the case where a~, ~ea, (see Eq. 2.73) tan8 should be replaced by (a~ / tan8 in Eq. 2.79.) This equation is valid only for small values of 8, as overhangs quickly become important for 8 > 5 ~ The diffusivity is related to another important quantity, the step-edge "stiffness" [3 (Bartelt et al., 1992; Fisher, 1984; Fisher and Fisher, 1982): -kTa, 13(0,T) = b2(0,T ) cos30

(2.80)

The step edge stiffness, which measures the free energy cost to deform a step, is determined by the orientational dependence of the step free energy 13(0,T) through (Fisher, 1984; Fisher and Fisher, 1982): --

I3(0,T) - 13(0,73 +

~9~13(0,T)

002

(2.81)

For steps with overhangs, Eqs. (2.90) and (2.81) generalize the definition of the diffusivity in Eq. (2.77). The step edge stiffness diverges at T ---) 0 for high-symmetry (0 = 0) steps, as the diffusivity becomes small. The variation of the step diffusivity with angle, calculated using Eqs. (2.75), (2.80) and (2.81), is shown in Fig. 2.11. Once an expression for the step stiffness or diffusivity is obtained, ?t can be used to evaluate the step-step interaction term, g(0,T) in Eq. (2.56) or (2.72). The general form for this term, valid when the step diffusivity is much smaller than the square of the average step separation is (Williams et al., 1994): I/2

[ 4Aa 1 'Jt

g(O,T) = rc2kTb2(O'T) I + 24ap h3 L kTb2(O, T)

2

(2.82a)

for steps of height h, and for step-step interactions in which the strength of the interaction falls of as the square of the distance between steps, U(x) = A/x z. The inverse square form of energetic step-step interactions is expected physically for interactions which are due to elastic interactions or to dipole-dipole interactions (Einstein, 1995; Marchenko and Parshin, 1980; Voronkov, 1968). The temperature dependence of this term depends on two characteristic energies, the kink energy, and the interaction energy. The variation of the step interaction term with kink

Thermodynamics and statistical mechanics o/'su~. aces I

89

I

I

I

I

1.0-

....-S.-

30

~

/

0.8-

i

0 -,0.6

-

% 0.4-

0.2-

0.0

I

I

I

I

I

0.1

0.2

0.3

0.4

0.5

tanO

Fig. 2.11. The step diffusivity b2(0,T) as a function of azimuthal angle, calculated from Eqs. (2.80) and (2.81 ), using the values of the step free energies shown in Fig. 2.9. The diffusivity increases with kink density and with temperature (Williams et al., 1993).

I 2.0

,

!

I

I

I

-

1.8-

1.6to

o 1.4-

~1.2 c-

"

~~~

1.0-

..

0.80.60.0

I--" -

Jl ...................................

..jJ~5.... ............. '."_~~~~---:-..::..:::.::::::: ............... 1

0.1

I

0.2

I

0.3

tonO

1

0.4

I

0.5

Fig. 2.12. The contribution of step interactions to the reduced surface tension coefficient, g(0,T) as a function of the azimuthal angle, calculated using Eq. (2.82) with the values of the diffusivity shown in Fig. 2.11. The strength of the step-step interaction used in the calculation is A/apE - 0.22. The magnitude of the interaction coefficient increases with increasing step wandering and with temperature (Williams et al., 1993).

density, tan0, is shown in Fig. 2.12 for a specific ratio of the step interaction strength A to the kink energy e. A useful limiting form for the s t e p - s t e p interactions for the case where there are only entropic interactions, i.e. when A = 0, is,

90

E.D. Williams and N. C Bartelt

g(0,T) =

rc~kTb~(0,T) 6aph 3

(2.82b)

For low temperature and misorientation angle, the step diffusivity goes to zero and another simple form results: rI:2A g(0 ---) 0,T--~ 0 ) = 6h 3

(2.82c)

Finally, we will also need to consider the variation of the step interaction term g(0,T) with tan0. If all the 0-variation enters through bZ(0,T) (in other words, if the step-step interaction strength A is independent of angle, as it should be for elastic interactions on surfaces with three-fold or higher symmetry), we find easily that ~)g ~~0at0=0. 0tan0

2.4.3. Experimental determination of statistical mechanical parameters The advent of experimental techniques which allow direct imaging of steps allows the statistical mechanical description outline above to be tested directly. In particular, it is now possible to determine a partition function experimentally by compiling statistical distributions of the kink structure and the terrace widths. From such measurements one can test whether the step-wandering is consistent with thermal equilibrium and deduce the energetics governing surface morphology. A particularly beautiful example, in which a static kink structure was measured with atomic resolution, was performed by Swartzentruber et al. (1990) for kinks on stepped Si(100). They showed that the kink structure obeys Boltzmann statistics, and deduced the kink energy. Such studies can be performed if the steps can be thermally equilibrated at temperatures where a large number kinks are present and then quenched to a temperature where there is no real time kink motion to confuse the STM imaging process. Under these conditions, by simply counting the number of times each different type of kink structure is observed, one can obtain the relative energies via the Boltzmann distribution:

N(n) 2N(0)

- exp (-e(n)/kT)

(2.83)

where N(n) is the number of observations of a kink of depth n, N(0) is the number of step sites where there is no kink, and E(n) is the energy cost of a kink of depth n. The factor of two arises when the edge of the step has a mirror symmetry plane perpendicular to the step so that in- and out- kinks are equivalent. For the simple TSK model, one expects E(n) = InlE. For the case of Si(100), the best fit to the experimental data included a corner energy, E(n) = InlE + ~c (Swartzentruber, 1993;


91

Swartzentruber et al., 1990). While direct measurement of kink energies is desirable for comparison with atomic scale calculations of surface energetics, all that is really needed to predict the thermodynamic behavior of steps is the step stiffness, or the diffusivity. Generally, even if steps cannot be imaged with atomic resolution, the diffusivity can be determined from direct measurements of the mean-square displacements of steps, g(y) = ([x(y) - x(0)] 2)

(2.84)

where, as in Fig. 2.4, x is the displacement perpendicular to the average step direction and y is the position along the step edge. In the limit of low temperature or small y, the mean square displacement varies linearly with the distance along the step edge according to (Bartelt et al., 1990):

g(y) = b2(T)y/a

(2.85)

where a is the unit cell length parallel to the step edge. Measurements of the diffusivity in principle involve taking instantaneous "snapshots" of the step edge configuration, and then calculating the mean-square displacement by suitable averaging along the step edge. Such measurements can be made using electron microscope techniques such as REM, as have been performed for steps on S i ( l l l ) (Alfonso et al., 1992). When using a scanning technique such as STM, however a static image can only be obtained if the surface is at a low temperature, where the thermal motion of the steps is so slow as to be unobservable. Such conditions are often found at room temperature for semiconductor surfaces. However, if the activation energy for step motion is low enough, thermal motion of the steps can occur at room temperature. In this case, the images of steps will appear "frizzled" because the step edge position will move due to fluctuations during repeated scans of the STM tip. Examples of such analyses are the STM measurements of "frizzled" step edges on Ag and Cu surfaces (Ozcomert et al., 1995; Poensgen et al., 1992), and on Au surfaces (Kuipers and Frenken, 1993). Direct measurements of the terrace width distribution contain information about both the thermal excitations of the step edge and the nature of the step-step interactions. As mentioned above, bringing steps near to one another decreases the amount of wandering, and thus the configurational entropy, leading to an effective entropic repulsion between steps, even when there is no direct energetic interaction. This entropic repulsion also manifests itself in the spatial distribution of the steps. A step which is midway between its neighbors has more configurational entropy than one that is near a neighboring step, and thus the distribution of step-separations will be peaked near the average step-separation (Joos et al., 1991; Kariotis, 1991; Wang et al., 1990). The distribution for the case where all the steps are wandering simultaneously can be solved exactly for moderate temperature, and is shown as the solid curve in Fig. 2.5 (Joos et al., 1991). This scaled curve is universal: in other words is will be the same for any system regardless of the average step separation and the kink energy. Deviations from this universal form thus can be used to deduce

E.D. Williams and N. C. Barter

92

. 4

,L

,

.

.

i

.

.

.

.

f

,

9

,

9

i

.

.

.

.

i

.

.

.

.

1.2 J

1.0 .

! is

~

~" 0.8

v~ " 0.6-

~,,,. ,,"

"/7

'

X'

0.4 0.2 0.0

'~,,,'"' '.J-.~-',

0.0

,

,

0.5

. . . . .

1.0

e/(e)

1.5

2.0

-

2.5

Fig. 2.13. Distribution of step-step separations calculated using the free-fermion approximation (figure from B. Joos et al. (1991)). P(/) is the probability of observing two steps separated by a distance l. The behavior of freely wandering steps is shown by the solid curve. The distribution for steps with a repulsive energetic interaction in addition to the entropic behavior is shown by the dashed curve. The distribution for steps with an attractive energetic interaction in addition to the entropic behavior is shown by the dash-dot curve. The distributions change only slightly with temperature up to temperatures near the roughening transition of the terraces.

the presence of true energetic interactions between the steps, as illustrated by the dashed curves in Fig. 2.13 (Joos et al., 1991). If there are attractive interactions (of insufficient strength to overcome the entropic repulsion and cause step coalescence), the distribution becomes broader and peaked at a value smaller than the average step separation. If there are repulsive interactions, the distribution becomes narrower, and the peak moves to slightly larger value than the average step separation. Quantitative measurement of the shape of the distribution function can thus provide information about the nature of the interaction. Unfortunately, except for a few special cases, analytical expressions relating the shape of the distribution to the strength of the interaction do not exist. However, for repulsive interactions, a gaussian distribution is an excellent approximation (Bartelt et al., 1990) to the analytical form. The result is generally true for any repulsive interaction (Alerhand et al., 1988; de Miguel et al., 1992; Swartzentruber, 1993), but assumes an especially simple form if the interaction between steps is of the type (2.86)

U(x) = Ax-"

where x is the distance between steps measured perpendicular to the average step edge direction. In this case, the distribution of step separations is approximately a Gaussian of width w -

8 n ( n + 1)Aa

I-7-

(2.87)


93

where l is the average step-step separation, b2(T) is the diffusivity which is measured from observations of the step wandering, and a is the minimum kink-kink separation (Bartelt et al., 1990). A measurement of the width of the distribution as a function of the average step-step separation thus in principle allows a determination of both the form (value of n) and magnitude (value of A) of the step interactions. Terrace width distributions have been measured for several systems using STM and REM. Distributions characteristic of repulsive step-step interactions (Alfonso et al., 1992; Rousset et al., 1992; Swartzentruber, 1993; Wang et al., 1990), attractive step-step interactions (Frohn et al., 1991), and non-interacting steps (Yang et al., 1991) have been observed. The most extensive data set has been obtained by Alfonso et al. for steps on Si(111) using REM (Alfonso et al., 1992), which allows a large range of step separations to be studied by virtue of its large (compared to STM) field of view. These results suggested that the repulsion between steps falls off monotonically with the square of the distance (n = 2 in Eqs. (2.4) and (2.5)) (Alfonso et al., 1992; Balibar et al., 1993), with a magnitude of A -- 0.2 eV-A. In contrast, there is also evidence for oscillatory interactions between steps, that is step separations whose form varies from repulsive to attractive as a function of the step separation (Pai et al., 1994). The physical origin of step-step interactions, and in particular the origin of oscillatory interactions is discussed in Chapter 11 of this volume (Einstein, 1996). While it is difficult to prove the form of the step-step interaction using Eq. (2.87), because the dependence on the exponent n enters as n/4, physical predictions for step-step interactions through stress (Blakely and Schwoebel, 1971 ; Marchenko and Parshin, 1980) or dipole form (Redfield and Zangwill, 1992" Voronkov, 1968) all suggest a repulsive inverse square relationship. Stress mediated interactions are expected to be comparable or larger in magnitude than dipole interactions, and this provides a useful method for estimating the magnitudes of step-step interactions. The physical basis for stress-mediated interactions is illustrated in Fig. 2.14. Due to the different number of neighboring atoms for surface and bulk atoms, the lattice constant which minimizes the energy will generally be different for the surface than the bulk. (This can be shown intuitively by considering a Lennard-Jones interaction (Shuttleworth, 1950; Wolf, 1990).) The surface atoms can accommodate this difference by relaxation in the direction perpendicular to the surface, but are forced geometrically to maintain the lateral periodicity of the bulk. As a result, the surface is under a stress, which is the effective force holding the atoms in registry with the bulk. When there is a step on the surface, the atoms near the step can relax towards the preferred lattice constant, creating a strain (or displacement) field as shown in Fig. 2.14. This relaxation lowers the energy of the surface. However, when a second step (in the same direction) is created on the surface, the atomic displacements which lower the energy, oppose the displacements of the first step, and the net gain of relaxation energy is reduced. Thus there is an effective step-step repulsion. While this is an atomic picture, the form and magnitude of the stress-mediated step-interactions can be calculated using bulk elasticity theory, and are accurate down to atomic length scales (Poon et al., 1990; Wolf and Jaszczak, 1992). The relationship between the step interaction strength and the elastic parameters is:

94


+ p /...~

.

9

+

e ...,

I i +.,.+,,.

i

9

+ ,

+

J

.,o.~.,., ,,,,i.,.,

9 .

4

:+

. , 0 , ~ 0 0 o o e e o e ~ 6 . o o , * * . , ~ o o . .

'''t

-~

'? +

+

_+

_

' I

I

t

'

t .....

! ....

I

Fig. 2.14. The atomic displacements near two steps on a surface, calculated in a simple bali and spring model of atomic interactions are shown (a) as real-space displacements magnified • for clarity, and (b) contours of constant log-displacement, showing the extent of the strain field into the bulk.

2(1 - 0 2) z2 h2 A= rtE

(2.88)

where c~ is Poisson's ratio, E is Young's modulus, x is the surface stress and h is the step height (Marchenko and Parshin, 1980). The values of Poisson's ratio and the Young's modulus are bulk parameters which can obtained from various tabulations (Frederiske, 1972). The surface stress depends on the details of the surface structure, and must be measured or calculated directly. Increasingly there are good calculations (Meade and Vanderbilt, 1989; Needs et al., 1991) of the surface stress, and measurements of changes in the surface stress during chemical adsorption or deposition (Martinez et al., 1990; Sander and Ibach, 1992; Sander et al., 1992; Schnell-Sorokin and Tromp, 1990). There is also the possibility of determining stress from direct observations of the strain field (Pohland et al., 1993; Sato et al., 1992). Some of the available values are listed in Table 2.5, along with the corresponding values of the step-interaction coefficient A. There is good agreement between values of the step interaction coefficient determined from measured step distributions (Alfonso et al., 1992; Wang and Lagally, 1979; Wang et al., 1990; Williams and Bartelt, 1992; Williaml et al., 1992) and values determined from the surface stress using Eq. (2.88). Thus preliminary estimates of the strength of step-step interactions can be obtained from the tabulations of surface stresses

95


Table 2.5 Calculated values of the absolute surface stress are listed in the upper section, along with the corresponding step-interaction coefficient (as in Eq. (2.88)). Measured changes in surface stress due to adsorption are listed in the lower section of the table. Surface

Stress (eV/]k)

Reference

Step interaction coefficient A, eV-A (calculated Eq. 2.88)

Si(l 11)(7x7)

0.186

0.18

Si( 111)( 1x 1)-As Si(ll 1)(2x2) AI(I 11) lr( 111 ) Pt( 111 ) Pb( 111 ) Au(l 11)

0.178 0.130 0.078 0.331 0.349 0.051 0.173

Martinez et al. (1990); Meade and Vanderbilt (1989) Meade and Vanderbilt (1989) Meade and Vanderbilt (1989) Needs et al. (1991) Needs et al. (1991) Needs et al. (1991) Needs et al. (1991) Needs et al. (1991)

Change in stress

Reference

--O.41 -0.47 -0.53 +0.02 -0.45

Sander et al. (1992) Sander et al. (1992) Sander et al. (1992) Sander and Ibach (1992) Sander and Ibach (1992)

Surface + Adsorbate Ni( 100)-c(2x2)-S Ni( 100)c(2x2)-O Ni( 100)(2x2)-C Si(100)-O Si( 111 )-O

0.16 0.09 0.04 0.08 0.26 0.05 0.13

which are b e c o m i n g increasingly available ( C a m m a r a t a , 1992; G u m b s c h and Daw, 199 l; M e a d e and Vanderbilt, 1989; N e e d s et al., 1991 ; Vanderbilt, 1987; Wolf, 1990; W o l f and Jaszczak, 1992). T h e f o r m for the step interaction s h o w n in Eq. (2.88) is the interaction e n e r g y for two isolated steps. W h e n there is a periodic array of steps o f finite spacing, the total interaction on any one step due to all o f its n e i g h b o r s is ~2/6 larger than this ( K o d i y a l a m et al., 1995). W h e n the steps are w a n d e r i n g thermally, the effective step interaction_ to_be used in Eqs. (2.82) and (2.87) will be a value A i n t e r m e d i a t e b e t w e e n Aand rtZA/6.

2.5. Summary Surfaces are far m o r e interesting than their simplest description as infinitely periodic t w o - d i m e n s i o n a l structures w o u l d suggest. The realization of this fact, and its i m p o r t a n c e in all real surface processes, dates back to the earliest days of surface science. T h e r m o d y n a m i c s provides the f o r m a l i s m to describe the c o m p l e x i t y o f surfaces within a f r a m e w o r k of m e a s u r a b l e m a c r o s c o p i c p a r a m e t e r s . Statistical m e c h a n i c s p r o v i d e s the f r a m e w o r k to interpret those p a r a m e t e r s in terms o f under-

96


lying atomic properties. As a result of many years of experimental and theoretical effort, it is now possible to measure and understand the complexity of surface morphology within a simple and quantitative formalism based on the physical properties of steps on the surface. An increasing physical understanding of the properties of surfaces now make it possible to estimate the relatively small numbers of important parameters governing step behavior. Using such estimates, it is possible to approach any given problem with a reasonable qualitative understanding of what role morphology and changes in morphology can be expected to play. Conversely, direct observations of the thermodynamic behavior of surfaces can be used to deduce the physical quantities governing step behavior, and use them to predict behavior under varying conditions.

Acknowledgements The authors thank the DOD, ONR, NSF-MRSEC and NSF-FAW for support during the preparation of this manuscript, and also very gratefully acknowledge useful discussions with Profs. J.D. Weeks, T.L. Einstein, Dr. Hyeong Jeong and Mr. S. Khare on the topics of the manuscript. We also gratefully acknowledge Prof. W.N. Unertl's help and patience as editor.

References Alerhand, O.L., D. Vanderbilt, R.D. Meade and J.D. Joannopoulos, 1988, Phys. Rev. Lett. 61, 1973. Alfonso, C., J.M. Bermond, J.C. Heyraud and J.J. Metois, 1992, Surf. Sci. 262, 371. Avron, J.E., H. van Beijeren, L.S. Schulman and R.K.P. Zia, 1982, J. Phys. A 15, L81. Balibar, S., C. Guthmann and E. Rolley, 1993, Surf. Sci. 283, 290. Bartelt, N.C., T.L. Einstein and C. Rottman, 1991, Phys. Rev. Lett. 66, 961. Bartelt, N.C., T.L. Einstein and E.D. Williams, 1990, Surf. Sci. 240, L591. Bartelt, N.C., T.L. Einstein and E.D. Williams, 1991, Surf. Sci. 244, 149. Bartelt, N.C., T.L. Einstein and E.D. Williams, 1992, Surf. Sci. 276, 308. Bauer, E., 1985, Ultramicroscopy 17, 51. Bilalbegovic, G., F. Ercolessi and E. Tosatti, 1992, Europhys. Lett. 18, 163. Bilalbegovic, G., F. Ercolessi and E. Tossati, 1993, Surf. Sci. 280, 335. Binnig, B. and H. Rohrer, 1987, Rev. Mod. Phys. 59, 615. Blakely, J.M., 1973, Introduction to the Properties of Crystal Surfaces. Pergamon Press, New York. Blakely, J.M. and H. Mykura, 1962, Acta Metall. 10, 565. Blakely, J.M. and R.L. Schwoebel, 1971, Surf. Sci. 26, 321. Blakely, J.M. and J.C. Shelton, 1975, Surface Physics of Materials. Academic Press, New York. 189 pp. Bonzel, H.P., 1995, Surf. Sci. 328, L571. Bonzel, H.P., E. Preuss and B. Steffen, 1984, Appl. Phys. A35, 1. Breuer, U. and H.P. Bonzel, 1992, Surf. Sci. 273, 219. Cahn, J.W., 1977, lnterfacial Segregation. American Society of Metals, Metals Park, OH. Cahn, J.W., 1982, J. de Phys. C6(suppl.), 43, 199. Callen, H.B., 1985, Thermodyamics and an Introduction to Thermostatistics. Wiley, New York. Cammarata, R.C., 1992, Surf. Sci. 279, 341. Chernov, A.A., 1961, Sov. Phys. Usp. 4, 1116.


97

Clark, A., 1970, The Theory of Adsorption and Catalysis. Academic Press, New York. Dash, J.G., 1975, Films on Solid Surfaces. Academic Press New York. De Miguel, J.J., C.E. Aumann, S.G. Jaloviar, R. Kariotis and M.G. Lagally, 1992, Phys. Rev. 46, 10257. Dijkkamp, D., A.J. Hoeven, E.J. van Loenen, J.M. Lenssinck and J. Dieleman, 1990, Appl. Phys. Lett. 56, 39. Drechsler, M., 1985, Surf. Sci. 162, 755. Dreschler, M., 1992, Surf. Sci. 266, 1. Einstein, T.L., 1996, in: Handbook of Surface Science, Vol. 1: Physical Structure, W.N. Unertl (ed.). Elsevier, Amsterdam, Chap. 11. Eisner, D.R. and T.L. Einstein, 1993, Surf. Sci. 286, L559. Fisher, M.E., 1984, J. Stat. Phys. 34, 667. Fisher, M.E. and D.S. Fisher, 1982, Phys. Rev. B25, 3192. Flytzani-Stephanopoulos, M. and L.D. Schmidt, 1979, Prog. Surf. Sci. 9, 83. Frederiske, H.P.R., 1972, American Institute of Physics Handbook. McGraw-Hill, New York. Frohn, J., M. Giesen, M. Poensgen, J.F. Wolf and H. Ibach, 1991, Phys. Rev. Lett. 67, 3543. Gibbs, J.W., 1961, The Scientific Papers of J. Willard Gibbs. Dover, New York. Griffiths, R.B., 1980, Phase Transitions in Surface Films. Plenum Press, New York. Gruber, E.E. and W.W. Mullins, 1967, J. Phys. Chem. Solids 28, 875. Gumbsch, P. and M.S. Daw, 1991, Phys. Rev. B44, 3934. Herring, C., 1951a, Phys. Rev. 82, 87. Herring, C., 1951 b, The Physics of Powder Metallurgy. McGraw-Hill, New York. Herring, C., 1953, Structure and Properties of Crystal Surfaces. University of Chicago Press, Chicago, IL. Heyraud, J.C., J.J. Metois and J.M. Bermond, 1989, J. Cryst. Growth 98, 355. Hill, T.L., 1960, An Introduction to Statistical Thermodynamics. Addison-Wesley, Reading, MA. Jayaprakash, C., C. Rottman and W.F. Saam, 1984, Phys. Rev. B30, 6549. Jayaprakash, C., W.F. Saam and S. Teitel, 1983, Phys. Rev. Lett. 50, 2017. Joos, B., T.L. Einstein and N.C. Bartelt, 1991, Phys. Rev. B43, 8153. Josell, D. and F. Spaepen, 1993, Acta Metallurgica et Materialia 41, 3007. Kariotis, R., 1991, Surf. Sci. 248, 306. Keeffe, M.E., C.C. Umbach and J.M. Blakely, 1994, J. Phys. Chem. Solids 55, 965. Kern, R., 1987, Morphology of Crystals. Terra Scientific Publishing Co., Tokyo, p. 79. Khare, S.V. and T.L. Einstein, 1994, Surf. Sci. 314, L857. Kodiyalam, S., K.E. Khor, N.C. Bartelt, E.D. Williams and S. Das Sarma, 1995, Phys. Rev. B 51, 5200. Kuipers, L. and F.W.M. Frenken, 1993, private communication Kumikov, V.K. and K.B. Khokonov, 1983, J. Appl. Phys. 54, 1346. Liu, C.L., J.M. Cohen, J.B. Adams and A.F. Voter, 1991a, Surf. Sci. 253, 334. Liu, C.L., J.M. Cohen, J.B. Adams and A.F. Voter, 1991b, Surf. Sci. 253, 334. Madey, T.E., J. Guan, C.-Z. Dong and S.M. Shivaprasad, 1993, Surf. Sci. 2771278, 826. Madey, T.E., K.-J. Song and C.-Z. Dong, 1991, Surf. Sci. 247, 175. Marchenko, V.I. and A.Y. Parshin, 1980, Soy. Phys. JETP 52, 129. Martinez, R.E., W.M. Augustyniak and J.A. Golovchenko, 1990, Phys. Rev. Lett. 64, 1035. Meade, R.D. and D. Vanderbilt, 1989, Phys. Rev. Lett. 63, 1404. Methfessel, M., D. Hennig and M. Scheffier, 1992, Phys. Rev. B 48, 16. Metois, J.E. and J.C. Heyraud, 1989, Ultramicroscopy 31, 73. Moore, A.J.W., 1962, Metal Surfaces. American Society for Metals, Metals Park, OH, p. 155. Mullins, W.W., 1961, Phil. Mag. 6, 1313. Needs, R.J., M.J. Godfrey and M. Mansfield, 1991, Surf. Sci. 242, 215. Nelson, R.C., T.L. Einstein, S.V. Khare and P.J. Rous, 1993, Surf. Sci. 295, 462. Nicholas, J.F., 1965, An Atlas of Models of Crystal Surfaces. Gordon and Breach, New York.

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Nozieres, P., 1989, J. de Phys. 50, 2541. Nozieres, P., 1991, Solids far from Equilibrium. Cambridge University Press, Cambridge, 1. Osakabe, N., Y. Tanishiro and K. Yagi, 1981, Surf. Sci. 109, 353. Ozcomert, J.S., W.W. Pai, N.C. Bartelt and J.E. Reutt-Robey, 1993, Phys. Rev. Lett. 72, 258. Ozcomert, J.S., W.W. Pai, N.C. Bartelt and J.E. Reutt-Robey, 1995, Mat. Res. Soc. Symp. Proc. 355, 115. Ozcomert, M.S., W.W. Pai, N.C. Bartelt and J.E. Reutt-Robey, 1994, J. Vacuum Sci. Technol. A12, 2224. Pai, W.W., J.S. Ozcomert, N.C. Bartelt, T.L. Einstein and J. E. Reutt-Robey, 1994, Surf. Sci. 307-309, 747. Payne, S.H., H.J. Kreuzer and L.D. Roelofs, 1991, Surf. Sci. 259, L781. Phaneuf, R.J. and E.D. Williams, 1987, Phys. Rev. Lett. 58, 2563. Phaneuf, R.J., E.D. Williams and N.C. Bartelt, 1988, Phys. Rev. B38, 1984. Poensgen, M., J.F. Wolf, J. Frohn, M. Giesen and H. lbach, 1992, Surf. Sci. 274, 430. Pohland, O., X. Tong and J. M. Gibson, 1993, J. Vacuum Sci. Technol.All, 1837. Poon, T.W., S. Yip, P.S. Ho and F.F. Abraham, 1990, Phys. Rev. Lett. 65, 2161. Redfield, A.C. and A. Zangwill, 1992, Phys. Rev. B46, 4289. Rottman, C. and M. Wortis, 1981, Phys. Rev. B24, 6274. Rottman, C. and M. Wortis, i984, Phys. Rep. 124B, 241. Rous, P.J., 1995, Cohesion and Structure of Surfaces. Elsevier, Amsterdam. Rousset, S., S. Gauthier, O. Siboulet, J.C. Girard, S. de Cheveigne, M. Huerta-Garnica, W. Sacks, M. Belin and J. Klein, 1992, Ultramicroscopy 4244, 515. Sander, D. and H. Ibach, 1992, Phys. Rev. B43, 4263. Sander, D., U. Linke and H. lbach, 1992, Surf. Sci. 272, 318. Sato, H., Y. Tanishiro and K. Yagi, 1992, Appl. Surf. Sci. 60/61,367. Schnell-Sorokin, A.J, and R.M. Tromp, 1990, Phys. Rev. Lett. 64, 1039. Shuttleworth, R., 1950, Proc. Phys. Soc. A63, 444. Smith, J.R., T. Perry, A. Banerjea, J. Ferrante and G. Bozzolo, 1991, Phys. Rev. B44, 644. Somorjai, G.A. and M.A. Van Hove, 1989, Prog. Surface Sci. 30, 201. Song, K.-J., R.A. Demmin, C. Dong, E. Garfunkel and T.E. Madey, 1990, Surf. Sci. 227, Song, K.-J., C.Z. Dong and T.E. Madey, 1991, The Structure of Surfaces III. Springer-Verlag, Berlin, p. 378. Song, S, and S.G.J. Mochrie, 1994, Phys. Rev. Lett. 73, 995. Swartzentruber, B.S., 1993, Phys. Rev. 47, 13432. Swartzentruber, B.S., Y.-W. Mo, R. Kariotis, M.G. Lagally and M.B. Webb, 1990, Phys. Rev. Lett. 65, 1913. Tyson, W.R. and W.A. Miller, 1977, Surface Scienc 62, 267. Van Beijeren, H., 1977, Phys. Rev. Lett. 38, 993. Van Beijeren, H. and I. Nolden, 1987, Structure and Dynamics of Surfaces II, W. Schommers and P. von Blanckenhagen (eds.). 259. Van Hove, M.A. and G.A. Somorjai, 1980, Surf. Sci. 92, 489. Van Pinxteren, H.M. and J.W.M. Frenken, 1992a, Europhys. Lett. 21, 43. Van Pinxteren, H.M. and J.W.N. Frenken, 1992b, Surf. Sci. 275,383. V anderbilt, D., 1987, Phys. Rev. B36, 6209. Voronkov, V.V., 1968, Sov. Phys.-Crytallogr. 12, 728. Wang, G.-C. and M.G. Lagally, 1979, Surf. Sci. 81, 69. Wang, X.-S., J.L. Goldberg, N.C. Bartelt, T.L. Einstein and E.D. Williams, 1990, Phys. Rev. Lett. 65, 2430. Weeks, J.D., 1980, Ordering in Strongly-Fluctuating Condensed Matter Systems. Plenum, New York, p. 293. Wei, J., X.-S. Wang, N.C. Bartelt, E.D. Williams and R.T. Tung, 1991, J. Chem. Phys. 94, 8384. Williams, E.D. and N.C. Bartelt, 1989, Ultramicroscopy 31, 36. Williams, E.D. and N.C. Bartelt, 1991, Science 251, 393.


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Williams, E.D. and N.C. Bartelt, 1992, Surface Disordering: Growth, Roughening, and Phase Transitions. Nova Science Publishers, Les Houches, p. 103. Williams, E.D., R.J. Phaneuf, N.C. Bartelt, W. Swiech and E. Bauer, 1992, Mat. Res. Soc. Symp. Proc. 238, 219. Williams, E.D., R.J. Phaneuf, J. Wei, N.C. Bartelt and T.L. Einstein, 1993, Surf. Sci. 294, 219. Williams, E.D., R.J. Phaneuf, J. Wei, N.C. Bartelt and T.L. Einstein, 1994, Surf. Sci. 310, 451. Wolf, D., 1990, Surf. Sci. 226, 389. Wolf, D. and J.A. Jaszczak, 1992, Surf. Sci. 277, 301. Wortis, M., 1988, Chemistry and Physics of Solid Surfaces, VII. Springer Verlag, Berlin, p. 367. Yang, Y.-N., B.M. Trafas, R.L. Seifert and J.H. Weaver, 1991, Phys. Rev. B B44, 3218. Yoon, M., S.G.J. Mochrie and D. M. Zehner, 1994, Phys. Rev. B 49, 16702. Zemansky, M.W., 1968, Heat and Thermodyamics. McGraw-Hill, New York. Zhang, Z., Y.-T. Lu and H. Metiu, 1991, Surf. Sci. 259, L719.


CHAPTER 3

Surface Reconstruction" Metal Surfaces and Metal on Semiconductor Surfaces

C.T. CHAN Physics Department Hong Kong University of Science and Technology Clear Water Bay, Hong Kong

K.M. HO Ames Laboratory-USDOE and Department of Physics and Astronomy Iowa State University Ames, Iowa 50011, USA

K.P. BOHNEN Forschungszentrum Karlsruhe Institut fiir Nukleare Festk6rperphysik Karlsruhe, Germany

Handbook (?fSu~ace Science Volume 1, edited by W.N. Unertl

9 1996 Elsevier Science B. V. All rights reserved

lOl

Contents

3.1.

Introduction

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

103

3.2.

M e t h o d s of c a l c u l a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

104

3.3.

3.4.

3.2.1.

Classical models

3.2.2.

Tight-binding models

3.2.3.

First p r i n c i p l e s m o d e l s

3.2.4.

Forces

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

110

3.2.5.

New advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

111

H o w to m o d e l a surface

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

104

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

106

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

107

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

112

3.3.1.

Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

112

3.3.2.

G r e e n ' s function

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

112

3.3.3.

Slab

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

113

W h a t c a l c u l a t i o n s can tell us . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

3.4.1.

Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

3.4.2.

Surface energy

3.4.3.

S u r f a c e band structure

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

114

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

114

3.4.4.

W o r k function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

115

3.4.5.

C h a r g e density

116

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

3.5.

C l e a n metal s u r f a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

116

3.6.

S e m i c o n d u c t o r surfaces

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

123

3.7.

Metal o v e r l a y e r s on s e m i c o n d u c t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

123

3.7.1.

Si( 111)(ff3-3•

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

125

3.7.2.

O t h e r p h a s e s of noble metals on Si or G e . . . . . . . . . . . . . . . . . . . . . . . .

127

3.7.3.

B,AI, Ga, In on Si(100) and S i ( l l l )

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

128

3.7.4.

As, Sb and S i ( l l l ) - B i

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

129

3.7.5.

Alkali metals on s e m i c o n d u c t o r s

Ag and Si(l 1 l ) - A u

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

130

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

131

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

133

3.8.

Epilog

102

3.1. Introduction The positions of the atoms at the surface is one of the most basic questions one can ask in surface science. The atomic geometry is an important factor affecting the various behaviors of the surface such as chemical reactivity, surface energy, surface work functions, surface vibrations, and surface electronic states. With the advent of ultra-high vacuum technology and the development of experimental methods sensitive to the surface atomic structure, it rapidly became apparent that the arrangement of atoms at the surface of a crystal can be much more complicated than previously realized: the surface atomic arrangement can exhibit complicated deviations from the situation one would expect if one simply removed half of the atoms of the crystal to form a surface. Such rearrangements, called surface reconstructions, are driven by the tendency of the surface to reform bonds broken by the formation of the surface. However, such atomic rearrangements often occur at the expense of disruption of the bulk bonding and the creation of surface stresses and the resultant surface geometry is given by a compromise between the two competing interactions. As such, surface reconstructions offer an interesting view into the microscopic interactions present at the surface, and our understanding of surface structures is intimately connected with our understanding of the microscopic behavior of the surface. Over the past two decades, impressive progress has been made in the experimental field in both the amount of information acquired as well as in the number of techniques developed from which we can obtain information about the detailed geometry of the surface atoms. This began with the refinement of electron diffraction techniques and culminated with the more recent development of surface X-ray diffraction. Especially remarkable is the development of the scanning tunneling microscope (STM) which has achieved atomic-scale imaging of the surface in real space. However, even with all the powerful techniques available to date, it is still not possible to determine the atomic geometry by any one method alone. Reliable information can only be obtained by piecing together information from different sources. These include various experiments, and in many cases, input from theoretical calculations is also essential in differentiating between competing structural models. The richness and variety of surface reconstruction behavior is an indication of the delicate balance between the various energies involved at the surface. Although various simple theories have been developed to model the interatomic interactions at the surface, the accuracy needed for a proper description and the subtlety of the interactions at work make it necessary to use quite sophisticated calculations to get reliable answers and to provide accurate data for use in more 103

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approximate theories. The progress in this direction in the last two decades has also been very substantial, fueled by the rapid advance in the computing power of modern computers and developments in computational techniques and numerical algorithms. A detailed discussion of these advances is given in the next section. In addition to yielding reliable data, these microscopic calculations provide detailed information on the surface electronic structure, electronic wavefunctions and charge density distribution at the surface allowing us to examine and understand the basic physical forces in operation on the various surfaces. Such information is very useful in predicting and understanding the behavior of more general systems including chemical trends in surface geometries and the influence of various external fields such as stress or electric charging effects on the surface geometry. The basic concepts of the theories that have resulted in this area are the subject of this chapter. The discussion in subsequent sections are organized into different areas, involving different types of surfaces, specifically: metal surfaces, semiconductor surfaces and metal-semiconductor interfaces. The rich and vast field of chemisorption will not be covered in detail in the present article; instead, we will focus on clean crystalline surfaces. Our discussion of adsorbed surfaces will be limited to the case of metal-semiconductor interfaces, which is still in an early stage of development, and for systems which are directly related to the physics of the clean surface, such as alkali-metal-adsorbed surfaces.

3.2. Methods of calculation In this and the following sections, we will discuss (a) the three most popular methods for modeling inter-atomic interactions, (b) the three most popular ways to model a surface, and (c) what kind of information we can extract from a surface calculation. Some of these methods are also discussed in the following chapter in the context of their application to surfaces of insulating materials. To study surface structure and surface phenomena at atomic length scales, we have to deal with interatomic interactions at the atomic level. Atomic interactions can be handled with a hierarchy of approaches: "classical" models, tight-binding models, and first-principles methods, in increasing order of accuracy and computational difficulty. 3.2.1. Classical models

In classical models, the total energy of the surface system is written in some simple (usually analytic) functional forms that depend on the bond lengths and bond angles and have a short interaction range, so that energy and forces can be evaluated very quickly. The parameters in the energy functions are usually determined by fitting to some known experimental quantities, such as the cohesive energy, bond lengths and elastic constants. Recently, it has become increasingly popular to fit the parameters to high quality theoretical results. The simplest form of classical potentials are of the Lennard-Jones type (Kittel, 1976), in which the energies are written as simple

105

S u e a c e reconstruction

functions of bond lengths. These simple models can only give generic, but not atom specific information for surface systems of interest. More sophisticated models have to include explicitly the physics of the bonding; e.g. changes in electronic states. For metals, the most popular and reasonably accurate classical method is the embedded-atom-model (EAM) (Daw et al., 1993). In this approach, the total energy of the system is written as:

Etota,= '~ F(pi)+ ~ ,(r~)' i

ij

p, = ~

f(rii )

(3.1)

j~i

where F(p;) is an analytic many-body term dependent upon the local electron density Pi, which is calculated from a tabulated function fir). rij is the distance between two atoms at sites i and j, and ~(r) is usually a simple two-body potential. We note that the form resembles an energy expression of the density functional formalism (see below) although the energy depends only on the "charge density" at the atomic sites, rather than the charge density in the whole space. The energy is a non-linear functional of the bond lengths, and is thus more sophisticated and flexible than simple pair-wise interaction potentials such as Lennard-Jones potentials. We note that the EAM potentials are not explicitly volume dependent, which makes them very suitable for surface calculations. The energetics of the system depends only on the distances between the atoms, but not on the bond angles. This class of theories are thus more applicable to metallic systems where the atoms are almost "spherical" such as transition metals and noble metals where the d-shells are essentially filled. Examples are Au, Ag, Cu, Pt, Pd, Ni. Other methods such as effective medium theories (Norskov, 1989; Stave et al., 1990), glue models (Ercolessi et al., 1986), and Finnis-Sinclair models (Finnis and Sinclair, 1984) are similar in spirit to the EAM method and have the same range of applicability. For systems with directional bonding, such as semiconductor surfaces, realistic classical models must allow for the explicit dependence of the total energy on the bond angles. The simplest approach is to include potential functions that have "three-body" terms of the form V(1,2,3) = V(rl2,rl3,cos(O)), where rl2, rl3 are the distance between atoms 1,2 and 1,3, and 0 is the angle formed by the bonds r~2, rl3 (see, e.g., Keating, 1966; Stillinger and Weber, 1985; Biswas and Hamann, 1985; and Tersoff, 1986). There are also methods such as the equivalent-crystal-theory (Ricter et al., 1984) that are intended to work for metals as well as semiconductors. The advantage of these classical models is that modern supercomputers can handle systems with up to millions of atoms on the dynamical level and can be used readily with simulation techniques such as molecular dynamics to model fairly complex surface phenomena. The disadvantage is that there is always the risk of extending these models beyond their intended range of validity. The electronic degree of freedom is not included explicitly, and in situations where band structure effects, Fermi surface effects and charge transfer effects are important, these models should not be expected to give reliable results.

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3.2.2. Tight-binding models In the tight-binding approach, the potential energy of a system is written in the form: occupied

Etota, = Z I~i-k-Z O(Pij) (3.2) i 0 where r o is the distance between atoms i and j; and ~(r) is usually a classical potential. The first term, ~ei, is the sum of the eigenvalues for the occupied states obtained by diagonalizing, a Hamiltonian matrix H: /4~ = e S ~

(3.3)

The matrix S is called the overlap matrix. If S = I (identity matrix), the tight-binding orbitals are assumed to be orthonormal, and the tight-binding model is called an "orthogonal" tight-binding model; otherwise, the tight-binding model is called non-orthogonal. The Hamiltonian and overlap matrix elements are generated from a set of empirical parameters that are obtained by fitting to the band structure of the elements under consideration. Although more sophisticated approaches are possible, tight-binding models are usually based upon the Slater-Koster two-center approximation (Slater and Koster, 1954). A comprehensive account of the tightbinding method can be found in the book by Harrison (Harrison, 1980). In surface calculations, we may need to supplement Eq. (3.2) with a term of the form ~i l/2Ui(qi _ Q)2, where U is frequently of the order of 1 eV, and qi is the charge of the atom at site i and is determined self-consistently in the calculation, and Oi is the number of valence electrons of that atom. As is obvious from its form, this term can control charge transfer in the system and is frequently important for surface systems if we need to maintain a certain degree of charge neutrality. Since the tight-binding model contains electronic structure information, it is good for semiconductors, where covalent bonding prevails, and in bcc transition metals where directional d-bonding governs the structural and cohesive properties. It is also superior to classical models whenever Fermi surface or band structure effects are important. Tight-binding models have enjoyed considerable successes when applied to semiconductor surfaces (see e.g., Chadi, 1994), and have also been used to describe the surface reconstruction of bcc transition metals such as Mo(001) and W(001). Tight-binding parameters for many elements are published (see, e.g., Papaconstantopoulos, 1986), but the parameters frequently require modification for surface situations. New and more sophisticated tight-binding models are fit to a broad data base and can describe the energetics of covalent systems quantitatively under different physical and chemical environments (see e.g., Goodwin et al., 1989; Xu et al., 1992). There are now also methods that can extract tight-binding parameters directly from first-principles linear muffin-tin orbitals calculations (Nowak et al., 1991; Stokbro et al. 1994). Tight-binding models are usually more difficult to implement and computationally more intensive than classical models. The computational cost is dominated by the need to diagonalize Hamiltonian matrices of the order N = Natom

Su~ace reconstruction

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)< Norbita b where Natom and Norbita ! a r e the number of atoms and the number of orbitals

per atom respectively. Since diagonalizing a matrix is an N 3 problem, we are limited to a few hundred atoms if the traditional diagonalization schemes are used. However, it has been realized that the energy and forces in a tight-binding model can be formulation as an N 1 problem with methods such as the recursion method and density matrix methods (see the discussion below). A few thousand atoms can now be handled dynamically with tight-binding models. Although first principles techniques, as will be described in the next section, have made progress in leaps and bounds in recent years, there is always an important place for empirical and semi-empirical techniques for surface physics, simply by virtue of their capacity to treat a much larger ensemble of atoms, and to provide answers for more complex systems in a more timely manner. This is especially true in situations where dynamical and finite temperature information are needed. A sophisticated LDA surface calculation can take months to complete. First principles techniques, on the other hand, can provide definitive answers for problems that require higher accuracy. Available experimental information seldom provides enough information for fitting uniquely the parameters of a microscopic model and first principles methods can fulfil the purpose of generating an otherwise unobtainable data base, with which we can fit more reliable and transferable empirical parameters.

3.2.3. First principles models In surface calculations, "first principles methods" usually mean the local-densityapproximation (LDA) to the density functional formalism (Hohenbergand Kohn, 1964; Jones and Gunnarsson, 1989; Kohn and Sham, 1965; March and Lundqvist, 1983; Phariseau and Temmerman, 1984) for almost all occasions. We seldom go beyond LDA because LDA provides results that are good enough for many purposes, and in most surface calculations, we cannot afford the computer time to go beyond that. LDA almost always gets the correct structural model, in the sense that the structural model found in experiments also has the lowest energy in LDA, and the bond lengths are usually good to a few percent. We should remark that chemists have more stringent requirements for terms such as "first principles" or "ab initio", although LDA is gaining more acceptance in that community too. This is because a new generation of exchange-correlation functionals incorporating gradient corrections can now treat small molecules more accurately (Langreth and Mehl, 1981; Perdew and Yue, 1986; Perdew, 1986). One nice feature about LDA is that it includes exchange and correlation energies in the mean field sense, but the complexity of the problem is only that of a Hartree-type calculation, and is more economical than Hartree-Fock calculations, which do not include correlation energies. It is thus computationally very attractive. As originally formulated, LDA is meant for systems that have smoothly varying charge densities, and charge density variations at surfaces cannot be considered smooth. However, over a decade of computation has shown that the results are very

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good and in quite a few cases, theoretical predictions actually preceded subsequent confirming experiments. Within the frame-work of the local density formalism, there are different implementations, such as LMTO (linear muffin-tin orbitals) (see, e.g., Fernando et al., 1986), FLAPW (full potential linear augmented plane wave) (Wimmer et al., 1981), pseudo-potential-plane-waves (Ihm et al., 1979; Pickett, 1989), and various forms of LCAO (linear combination of local orbital) approach (see, e.g., Chelikowsky and Louie, 1984). These approaches basically differ in the way the electronic wave-functions, the charge density, and the potentials are represented, and also in the way the core-electrons are treated. Most of the approaches that are applicable to bulk calculations are also applicable to surface calculations, although plane-wave based methods have had many early successes. Traditionally, pseudopotential methods are applied to semiconductor surfaces and simple metals, while methods with local orbitals like FLAPW are more frequently applied to transition metals. With rapid advances in algorithm developments, these traditional boundaries are fading away. For a surface LDA calculation, the reliability of the results should be judged by its "convergence" (completeness of the basis, number of k-points sampled), whether the atomic positions are fully relaxed, and whether any constraints are imposed on the form of the potential or charge density. In density functional theories, the total energy is expressed as a functional of the electronic charge density 9(r), and can be written as , tlxc + E.xc[9(r)] + E~w~,lj

-

(3.4)

i

where ] ~ is the sum of eigenvalues of an effective Hamiltonian (the index i goes over band and k-point indices), p(r) is the charge density, VinpUtis, .xc the input Hartree and exchange-correlation potential, with the suffix "input" emphasizing that the screening potential is the input potential that determines the Hamiltonian. The first two terms in the right-hand-side of Eq. (3.4) give the kinetic energy of the electrons and the potential energy due to the electron-ion interaction respectively. The third term, Euxo is the electron-electron interaction energy and is given by the sum of the Hartree energy and exchange-correlation energy and is a functional of 9(r). E~w,,~dis the Coulombic ion-ion interaction energy of the ionic cores. It is called the "Ewald" term because we usually compute this term with an algorithm suggested by Ewald (Ziman, 1972). The eigenvalues and charge densities in Eq. (3.4) are obtained by solving an effective Schroedinger equation, usually called the KohnSham equation,

]

+ Vi,,,(r)+ VHxc(r) ~gi(r)- ei~lli(r)"

-~m

p ( r ) - Z f/vi(r)~gT(r)

(3.5) (3.6)

i

wheref~ is the occupancy of an eigenstate, ka~ the ionic potential, and VHxc is the screening potential due to the electrons, and is given by Vnxc = V. + Vxc, where

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VH = ~ d3r p(r') . I r - r'l'

Vxc _ ~9Exc[p(r)] 5p(r)

(3.7)

If we do not make any approximation to the exchange-correlation energy Exc [p(r)], we can obtain the exact ground state energy for a many-electron system. However, the exact functional form of Exc[p(r)] is not known, and it is approximated by a local functional of the form, Exc[p(r)] - ~ d3r~xc[p(r)] p(r), where exc(p(r)) is the exchange-correlation energy density of a homogeneous electron gas with density p(r) (Ceperly and Alder, 1980). This is called the local-density approximation. We note that in the K o h n - S h a m equation, the potential VHxc depends on the charge, which in turns depends on the wavefunctions ~(r), so the solutions to Eqs. (3.5) and (3.6) have to be obtained self-consistently. Nearly all the computational effort in a LDA calculation goes into the solution of Eq. (3.5), and there are a few ways in which it can be achieved. The traditional method transforms the K o h n - S h a m equations into a Hermitian matrix eigenvalue equation and the matrix (of the same form as the tight-binding Hamiltonian problem in Eq. (3.2)) is then diagonalized to yield the eigenvalues and eigenfunctions. Very robust diagonalization routines are readily available for this purpose. However, as we have mentioned before, the complexity of the diagonalization procedure goes like N 3, where N is the order of the matrix and the storage goes like N 2. This puts severe limits on the number of atoms that can be handled. The situation here is much worse than the case with tight-binding calculations, where the number of basis functions per atom are much smaller, and hence have smaller matrices to diagonalize. First principles calculations can handle only about 50-100 atoms if we use matrix diagonalization, depending on the details of the method used. In some methods, particularly plane-wave based methods, the number of occupied eigenstates is only a small fraction of the order of the Hamiltonian matrix. In those cases, iterative diagonalization procedures such as the "Davidson" scheme (Davidson, 1975) can save a lot of computational effort if we only need a small percentage of the eigenfunctions from the Hamiltonian matrix. There are methods that bypass the diagonalization and the self-consistency procedures, and solve for the ~ ( r ) ' s by directly minimizing the LDA energy functional with respect to a set of trial wavefunctions, subject to the orthonormal constraints. A very efficient algorithm for this purpose is the preconditioned conjugate gradient scheme proposed by Teter and co-workers (Teter et al., Allan 1989). Another very important development is the Car-Parrinello method (Car and Parrinello, 1985), which treats both the electrons and ions as classical dynamical systems and recasts the electronic structure problem as a molecular dynamics problem. The equations of motions for the electronic and ionic degrees of freedoms are ~lli = - Hltli + E

AijlllJ "

Mi'gi= - ddR E[R,gtl

(3.8)

J

where the A's are Lagrangian multipliers needed to keep the wavefunctions orthonormal, ~ is a (fictitious) mass for the electronic degrees of freedom. These

C.T. Chan et al.

1 10

equations are integrated in the time domain by standard molecular dynamics methods. The Car-Parrinello scheme is the first scheme that allows us to perform first principles (LDA) molecular dynamics simulations. Although the details are not the same, these new algorithms (iterative methods, conjugate gradient, Car-Parrinello) rely on the fact that H ~ (a trial wavefunction ~ operated on by the Hamiltonian operator H) can usually be computed very efficiently. In many cases, the Hamiltonian matrix elements are not computed and need not be stored, H is treated as an operator. This is particularly the case for plane wave basis sets, where the operations can be performed partly in real space (potential energy) and partly in momentum space (kinetic energy). It is thus not surprising that the conjugate gradient and Car-Parrinello methods are mainly associated with pseudo-potential plane wave methods and most of the ab-initio molecular dynamics simulations on surface systems to date are performed with plane waves. The newer algorithms can handle a few hundred atoms if the system under consideration can be represented reasonably well with plane waves.

3.2.4. Forces For static structural problems (surface relaxation and reconstruction) and for dynamical calculations such as molecular dynamics simulations, the importance of calculating forces accurately and efficiently cannot be understated. For classical potentials, the force calculations are trivial to implement. When there is an electronic degree of freedom, the forces due to the electronic part of the total energy can be computed by the Hellmann-Feynman theorem (Hellmann, 1937; Feynman, 1939), which can be formally written as: OH (V, ~--R Vi)

F~-Z i

(3.9)

"-"'ta

Here, Fo is the electronic force acting on atom It, and the index i runs over the occupied bands. Equation (3.9) may look trivial, but it has important implications. It means that there is no need to move the atoms to compute all the forces, otherwise it will take M calculations to get the forces if there are M degrees of freedom. In tight-binding calculations, this can be evaluated very easily. In first principles calculations, the calculation of the forces by the Hellmann-Feynman theorem can be tricky. If the basis set used is independent of the atomic positions (such as plane waves), the formal form of Eq. (3.9) holds and computation of forces is straightforward. If the basis set contains functions that are centered on the atomic coordinates, extra terms must be included before the forces calculated represent the true gradient of the energy surfaces. These terms are related to dCp/dR, where ~ is a local orbital. They are called the Pulay forces (Pulay, 1969), and are well known in quantum chemistry. In physics, their importance and formal formulation have also been known for some time (Bendt and Zunger, 1983), although the implementation has proved to be non-trivial. The mixed-basis pseudopotential approach, which uses a mixed representation of plane waves and local orbitals, has these "Pulay" terms coded (Ho et al., 1992) and thus has been rather successful for the structural studies

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of transition metal surfaces. Recently, almost all local-orbital based methods can perform force calculations (Yu et al., 1991, Methfessel and van Schilfgaarde, 1993; Jackson and Pederson, 1990). In first principles self-consistent calculations, the total energy is a variational quantity so that it is rather forgiving in the self-consistency of the charge and the potential. This is not the case for the forces, which require highly self-consistent potentials and charges before they can be computed accurately. There are methods that can alleviate this problem (Chan et al., 1993). Of all the first-principles methods, the pseudopotential plane-wave methods have made early and important contributions to surface structural studies, especially in semiconductor surfaces. The main reason for this success is the ability to compute forces easily and accurately. In recent years, pseudopotential plane-wave techniques have been even more popular because of the introduction of conjugate gradient and Car-Parrinello methods that elevated LDA from the statics to the dynamical level. These methods are essentially plane-wave based. There are however some difficulties with pseudopotential plane wave methods. The plane wave basis and momentum space formulations are efficient for those elements which have smooth pseudopotentials and valence (pseudo-) wavefunctions that can be represented easily by a plane-wave expansion. Examples are alkali metals (Li, Na, K), simple metals (AI, Mg, Be), and semiconductors (Si, Ge). Systems like first row elements and transition metals (especially 3d metals) have fairly sharp local orbitals, and require many plane waves to converge, and hence demand substantially more computing time for reasonably converged results. Many authors have introduced "smooth" pseudopotentials to alleviate this problem (see, e.g., Rappe et al., 1990; Troullier and Martins, 1991; Vanderbilt, 1990). By "smooth", we mean that the Fourier transform of the pseudopotential decays rapidly in momentum space, and are thus easier to converge with plane waves. We can also supplement the basis with local orbitals, such as the mixed-basis technique (Louie et al., 1979). There have also been recent attempts to improve the efficiency of the plane wave representations by employing wavelets, multigrids and adaptive grids, and some of these techniques may soon find applications in surface calculations. 3.2.5. New advances

Recent advances in "order(N)" methods may have significant impact on theoretical surface studies in the near future. As mentioned in previous paragraphs, modern electronic structure algorithms (such as conjugate gradient and Car-Parrinello) circumvent the diagonalization of the Hamiltonian matrix by taking advantage of the fact that we just need some (the occupied states) but not all of the eigenvectors; but these algorithms are still N 3 methods because the number of occupied states has to be proportional to Natomand making the eigenvectors mutually orthogonal is 3 already an N~,tom process. Recently, it has been recognized that we do not need to know any of the individual eigenvalues or eigenfunctions to obtain energies and forces. We only need to compute objects like the Green's function (Resolvent), the density matrix, or a sub-space spanned by localized orbitals that are unitarily equivalent to the subspace spanned by the eigenvectors. These can be computed

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with operations proportional to N ~, and are thus called "order(N)" methods. These new algorithms are still under rapid development (see, e.g., Li et al., 1993; Daw, 1993; Ordejon et al., 1993; Baroni and Giannozzi, 1992), and have already showed good promise in tight-binding calculations.

3.3. H o w to m o d e l a surface

In numerical computations, we need to model a solid/vacuum interface on the computer. There are a few ways that it can be achieved, and we will discuss them in the following.

3.3.1. Cluster Cluster methods use a "large" cluster (large here means 50 or so atoms for first principles calculations) to construct a fragment of a solid which has a surface with a particular orientation (see, e.g., Ye et al., 1989). This method is most popular among chemists. The advantage of this approach is that quantum chemistry codes are readily usable. Quantum chemistry codes are generally more "standardized", better documented and also readily available from many sources; and they usually have the ability to go beyond LDA, so that correlation effects can be accounted for in a more controlled manner (if one can afford to do so). Cluster methods are good for modeling isolated atoms on surfaces or low coverage situations in chemisorption problems. In this approach, many atoms that are not intended to represent surface atoms are also exposed and are not bonded. These atoms are usually "passivated" by bonding H atoms to them. It is difficult, if not impossible, to obtain information pertaining to the extended nature of the surface, such as the surface states; and modeling an extended system with a cluster of atoms may run the risk of obtaining results that are qualitatively different from the truly extended system. The convergence of the results relative to cluster size is usually slow.

3.3.2. Green's function Surface properties can in principle be determined if the Green's function of the entire surface system can be obtained by surface Green function matching techniques. Such techniques have been used for obtaining surface electronic structures, surface states and other elementary excitations, especially for simplified model Hamiltonians (see, e.g., Garcia-Moliner and Flores, 1979). Although Green's function methods offer beautiful formal solutions and have a lot of nice analytic properties, it is always tedious to construct and manipulate the Green's function for real systems with real potentials. It is especially difficult if we need to solve the problem self-consistently and have to compute forces to take care of relaxation and reconstructions of the surface (or interface) atoms. However, recent efforts by a few authors, especially the work of Feibelman (Feibelman, 1986), have demonstrated that these difficulties can be overcome. Green's function methods will be very important for studying adsorbate effects in low coverage situations. It is better in

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principle than cluster approaches because, once the Green's function of the substrate is known, the effect of the embedding environment can be taken into account essentially exactly, while for the cluster approach mentioned in the last paragraph, the cluster used to represent the surface is not embedded in the correct medium and does not have the right boundary conditions. Green's function methods are difficult to implement. 3.3.3. Slab

The most popular method to date is to use a thin slab, which by definition is bounded by 2 surfaces, to model a surface. As a rule, results are more reliable with thicker slabs. However, meaningful results 6an sometimes be obtained with slabs as thin as 5 or 7 atomic layers. The slabs are either repeated (separated by vacuum) to regain 3D periodicity or not repeated, depending on the method chosen. In Fourier space based techniques, such as the pseudopotential approach, a repeated slab geometry is required, while for local orbital based methods, the repeated slab configuration is optional. Though not necessarily the most elegant formulation, this is by far the most popular method because it is straightforward to implement. In many cases, codes developed for bulk calculations can be adapted with little or no modification for surface calculations. We just need more memory and more computer time. The method is limited to systems with small surface unit cells, including high-coverage ordered surface overlayer structures.

3.4. W h a t c a l c u l a t i o n s c a n tell us

The results of a first-principles total energy calculation for a clean or adsorbate covered surface can tell a lot about the structural and electronic properties of a surface. Once the surface structure has been determined, the same calculation usually provides more information than just the positions of the atoms. We will attempt to give a brief account of these various types of information and how they can be obtained from an electronic structure calculation. This information should be viewed as an integral part of the theoretical surface structure determination, because in many cases, it is these results rather than the atomic coordinates that can be compared directly with experiments. More extensive discussions of these nonstructural aspects of surfaces are given in Volume 2 of this handbook. 3.4.1. Structure

Surface structural parameters such as bond lengths and interplanar distances can be obtained directly from the atomic coordinates after the atomic degrees of freedom are fully relaxed. These quantities can be directly compared with experimental data from techniques such as LEED, ion-scattering and surface X-ray scattering. Theoretical calculations determine surface atomic arrangements by searching for a "zero-force" configuration with the "lowest" energy. In reality, it is very difficult to tell whether a "zero-force" configuration is just a metastable state (local minimum)

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or a ground state (global minimum); and in some surface systems, metastable states can be observed, and are as important as the global minimum configuration. This is particularly true in semiconductor surfaces with strong covalent bonds where energy barriers are high, and the n-bonded chain Si(111)(2• system is a good example of a metastable surface configuration. We need subsidiary evidence, such as the surface band structure, work function change, etc., to compare with experimental information before a strong case can be established.

3.4.2. Surface energy The surface energy can be determined in a total energy calculation. From a slab calculation, it can be determined by either one of the two methods described below, with the former method somewhat more popular. In the first method, we need to compute the total energy of the slab, subtract the total energy of the corresponding number of atoms in an ideal bulk environment, and divide the difference by the total number of surface atoms on both sides of the slab. The only difficulty with this method is that we have to obtain the same convergence in the k-point sampling for both the surface and the bulk calculations. The second way is to calculate the total energy of a slab as a function of the slab thickness, and then fit the results in a least square manner to the equation E(n) = nE(b) + 2E(s); where E(n) is the energy of a slab n layers think, E(b) is the energy in the ideal bulk environment for the atoms in one plane, and E(s) is the surface energy. In this manner, no separate bulk calculations are required. However, since E(b) is much larger than E(s) in magnitude, even small numerical errors can result in larger errors in E(s). It also requires several slab calculations (at least 2) (Ricter et al., 1984). When we make comparison with experimental results, we should note that the numbers quoted in experiments are usually surface tension measured at elevated temperatures, and usually measured for a surface with a mixed orientation; whereas theoretical calculation gives the surface energy of a surface of a specified orientation at T-- 0. In principle, the surface energy can also be determined if we can define and calculate the total energy density E(r) at any point r; which will allow us to compare the energy of the surface region with the interior. This is possible (Chetty and Martin, 1992) but this method has seldom been used.

3.4.3. Surface band structure The surface band structure can be of great use in characterizing the surface if photoemission results are available. After a tight-binding or first-principles surface calculation is performed, the eigenvalues and the eigenstates for each k-point of the surface Brillouin Zone (BZ) can give the dispersion of the surface states and surface resonances. Surface states are those electronic states that are created by the formation of the surface, and their amplitudes are localized near the surface region, decaying exponentially into both the vacuum and the bulk. We need to distinguish the surface states from the bulk states in the calculation. For "thin" slab calculations, that distinction can sometimes be a fairly tedious procedure. There are usually


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three criteria from which we can distinguish the surface states from bulk states in a calculation. First, bona-fide surface states exist in gaps or symmetry gaps (on high-symmetry lines) of the bulk band structure projected onto the 2D surface BZ (called the projected band structure). Second, surface states in a slab calculation normally come in nearly degenerate pairs. This is because there are two surfaces in a slab and the surface state can be on either side. If the slab is infinitely thick, the two surface states should be exactly degenerate, but the degeneracy is lifted by their interaction for a slab of finite thickness. The third and the operationally most useful criteria is to examine the charge profile of the state along the "z-direction" (the direction parallel to the surface normal). A surface state, by definition, should be localized near the surface layers. There is no universally accepted criteria concerning the degree of localization of a state on the surface layer before we label it a surface state, but when combined with the first two criteria, surface states can usually be identified by examining their charge profile in the z-direction. For slab calculations, the thicker the slab, the easier it is to identify surface states and surface resonances. Once we obtain surface states and surface resonances, results can be compared with photoemission or inverse photoemission experiments. In some cases, such comparisons can either rule out or provide strong support for a structural model. This is especially true for chemisorption systems, since the surface states of the substrate are strongly perturbed. In the older days, where only electronic but not total energy information was available from calculations, the comparison between theoretical and experimental surface bands had been used to determine the site of adsorption of foreign atoms on metal surfaces (Louie, 1979). However, when we compare the eigenvalues from a LDA calculation with the results of a photoemission experiment, we should note that the eigenvalues from a density functional theory (a ground state theory) should not be used directly to interpret electron excitation energies. For metals, the comparisons of LDA with spectroscopic data are in general reasonably good. In many cases, the eigenvalues from a LDA calculation can be used directly to map out surface states, with good agreement with photoemission results. For systems with a band gap, the gaps are usually underestimated by the LDA eigenvalue spectrum but there are well-formulated remedies to this problem, such as the GW approximation to the self-energy operator (Hybertsen and Louie, 1985; Louie, 1994) which was successful in computing quasiparticle energies for semiconductors and insulators. 3.4.4. Work function

The work function is a very useful piece of information in chemisorption systems where large changes can be induced, and the work function is strongly dependent on the position and coverage of the adsorbate. The work function can be obtained by comparing the Fermi level with the vacuum level in a LDA calculation. If a repeated slab geometry is used, care should be taken to ensure a large enough vacuum region so that the vacuum level is well represented. In many cases, an accuracy of about 0.1 or 0.2 eV can be obtained from LDA calculations.

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3.4.5. Charge density The charge density from a fully self-consistent LDA calculation can give us insight into the bonding of surface atoms as well as offering information for a quantitative interpretation of certain experimental results. For example, the interaction between He atoms and a surface is approximately proportional to the charge density, and so, the calculated charge density outside the surface can be of use in simulating He-scattering experiments. We can also compute the charge density contributions from electronic states that fall within a specified energy window, using E+5

p(E,r) = f dEIt~e(r)l 2

(3.10)

E-8

where ~e(r) is an electronic state with energy E, and 5 is a small number (in practice, it is taken to be a fraction of an eV). In the lowest order approximation, the charge density contour of p(E,r) at a small distance (a few A) above the top atomic layer can be used to approximate STM images with a tip-to-sample bias potential equal to E-EF (Tersoff and Hamann, 1985). This can be of great use in interpreting STM images (see e.g. Ding et al., 1991), which represent a convolution of structural and electronic information, as discussed in Chapter 9. Direct computation of tunneling signals are also possible, although it is rarely done.

3.5. Clean metal surfaces

We begin our discussion with the surfaces of simple metals. Simple metals here refer to those metals that have s,p electrons as their valence electrons and are reasonably well described by free-electron models. Typical examples are alkali metals and AI. Most of the low index simple metal surfaces do not reconstruct. They reduce the surface energy primarily by relaxation, i.e. by changing the inter-layer distances in the first few layers, and keeping the same surface periodicity as an ideal truncated bulk crystal. Transition metal and noble metal surfaces may exhibit reconstruction in addition to relaxation. In most cases, there is a contraction of the top interlayer spacing, and this seems to be a general trend for metal surfaces, transition metal and simple metal surfaces alike. The relaxation of compact closepacked surfaces such as fcc(111) are usually very small, of the order of 1% of the ideal interlayer spacing. The relaxation of the interlayer distances are of bigger magnitude for more open surfaces. For example, the top layer inward relaxation of AI(110), Cu(110) and A g ( l l 0 ) are about 7-9%, and Au(110)(1• has a top layer contraction of about 18%. Similar behavior is observed in bcc systems. Relaxations are typically small for the more close-packed bcc(110) surface, while the more open (100) surface usually has substantial relaxation. For example, the top layer of Mo(100) contracts by about 10%. Relaxation can extend several layers into the bulk, usually exhibiting an oscillatory pattern of contraction and expansion, with


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diminishing magnitude in the change of interlayer distances as it go deeper into the bulk. Tables of surface relaxation from theoretical calculations and experiments for many metal surfaces can be found in other review articles (see e.g., Bohnen and Ho, 1993). For the more open surfaces, the inward top-layer relaxation and the oscillatory relaxation pattern can be qualitatively explained by the redistribution of electronic charge at the surface. Following the picture as first discussed by Smoluchowski (Smoluchowski, 1941; Finnis and Heine, 1974), imagine that the crystal is partitioned into Wigner-Seitz cells, and the electronic charge remains in the unrelaxed position in the Wigner-Seitz cells when a surface is formed. The electron distribution would then have a very rough profile, leading to high kinetic energy and the system can reduce its energy by bringing the electrons into a smoother distribution parallel to the surface. This redistribution would tend to relocate electrons away from the ridge regions to the hollow regions on the top layer. Thus the ridges become slightly positively charged and the hollows slightly negatively charged. This charge redistribution on the top layer will produce electrostatic forces on the inner layers. The directions of the forces depend on the stacking sequence of the layers and for most cases, will lead to a contraction of the top interlayer spacing and an oscillatory relaxation pattern. A schematic picture of this mechanism is shown in Fig. 3.1. The few exceptions occur only for densely-packed faces where the relaxation effects are usually small. Because the computational effort grows as the size of the surface unit cell, most LDA calculations are restricted to the low-indexed crystal surfaces, which have been studied rather comprehensively (Bohnen and Ho, 1993). The structure and the energetics of higher indexed surfaces and stepped surfaces have also been investigated with classical potentials (see, e.g., Rodriguez et al., 1994). In general, atomic relaxations of higher-indexed faces involve substantial lateral atomic displacements in addition to normal displacements found in low-indexed faces. In surface reconstructions, in addition to the loss of translation periodicity in the normal direction, the surface atomic rearrangements also reduce the surface periodicity in the surface plane. Quite a few noble and transition metal surfaces reconstruct. The surfaces of transition metals and noble metals have been much studied because of their involvement in various catalytic processes. In particular, the noble metals are very popular for experimental studies because of the ease of preparing and maintaining a clean surface, they are also important electrodes for electrochemical studies. In this section, instead of giving a comprehensive review

I2/

\.

I -/

N ~ I -/

\

- I -/

\

- I -/

\

- I -/

\

- ! -/

\

- I

Fig. 3.1. A schematic picture showing the electron redistribution mechanism of Smoluchowski smoothing.

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C.T. Chart et al.

[11]

t

Top View

f[ll]

Fig. 3.2. The top view and side view of the W(001) reconstruction. For the top view, the empty circles and the dark circles represent the positions of the surface W atoms before and after the reconstruction.

of the various reconstruction processes appearing on transition metal surfaces, we describe a few prototypical cases. These can be classified into reconstructions of the displacive type, the missing-row type, and the contractive type. In d i s p l a c i v e r e c o n s t r u c t i o n s , the surface periodicity is disturbed by static distortions of the atomic positions occurring in the surface and near surface layers. The most well-studied prototypical example of displacive reconstruction is the c(2• reconstruction of W(001 ), where the surface atoms are displaced in the [ 11 ] direction to form zig-zag chains (Debe and King, 1977, 1979). A schematic figure is shown in Fig. 3.2. The displacements are mostly confined to the top layer. A similar, although more complex reconstruction has been observed in Mo(001), where recent experiments give a c(792-x~/2-) unit cell (Smilgies and Robinson, 1993; Daley et al., 1993). First principles calculations for both W(001) (Fu et al., 1985" Fu and Freeman, 1988) and Mo(001) (Wang et al., 1988) found that the top layer atoms are indeed unstable with respect to a zig-zag type displacement on the surface, and the magnitude of the displacement (about 0.2/~) agrees well with the experimental observations (Debe and King, 1977, 1979). Tight-binding calculations (Wang and Weber, 1987" Wang et al., 1988) indicate there is a softening of a surface phonon branch along the direction for both W and Mo (001) surfaces. For W, the maximum instability of the surface phonon branch occurs at the zone boundary of the surface Brillouin Zone, and is thus consistent with a c(2x2) unit cell. For Mo, the maximum instability happens to be close to, but not exactly at the zone boundary, leading to a much larger surface unit cell. The difference in behavior of the W(001 ) and the Mo(001 ) surface can be traced to relativistic effects which lowered the s bands relative to the d bands in W, leading to different surface wavevectors for the Fermi-surface-nesting giving rise to the instability. The symmetry of the undistorted ( l x l ) surface means that it is equally probable for the distortion to occur along the [ 11 ] direction as in the [ 11 ] direction. Thus, one would expect to find different domains to exist on the reconstructed surface where the displacement would point in different directions. From symmetry arguments, the potential energy of the surface as a function of the two dimensional displace-


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ment amplitudes of the zig-zag surface phonon modes should exhibit a Mexicanhat-like behavior: maximum at zero displacements, minima with displacements along the [11] and equivalent directions and saddle points along the [10] and equivalent directions. For such a potential, one would expect the c(2x2) long-range order of the reconstruction to be destroyed by randomizing the directions of the displacements of the atoms at different points of the surface rather than by reducing the atomic displacements to zero. Thus, we expect the high-temperature ( l x l ) surface to correspond to a disordered reconstructed surface (Debe and King, 1979) rather than an unreconstructed surface. This is in agreement with observations that, even in the high-temperature ( l x l ) surface, surface atoms still have large displacements from the truncated bulk positions comparable with the reconstructed phase (Stensgaard et al., 1979; Robinson et al., 1989). Theoretical calculations also indicate that the energy difference between the reconstructed and unreconstructed surface is much larger than kB To, where T c is the transition temperature (Singh and Krakauer, 1986; Yu et al., 1992; Roelofs et al., 1989). Details of the order-disorder transition have been studied both with molecular dynamics (Wang et al., 1988) as well as Monte Carlo simulations (Han and Ying, 1993). We note that displacivetype reconstruction is more common in bcc metals where directional d-bonding dominates the energetics. In displacive reconstructions mentioned above, the density of surface atoms remains the same as in the ideal surface and the reconstructions are achieved by small local displacements of the surface atoms from their ideal positions, without any long-range atomic transport. This is not the case for m i s s i n g - r o w and c o n t r a c tive reconstructions found in fcc metal surfaces. In the missing-row reconstructions, which occur on the (110) face of some fcc transition metals and noble metals, alternate rows of atoms on the surface are removed to form a (lx2) reconstruction of the surface periodicity (Moritz and Wolf, 1985; Copel and Gustafsson, 1986). Prime examples are the (lx2) reconstruction on the Au, Pt, and Ir (110) surfaces. In these cases, the surface (110) layer become less dense than the bulk (110) layers, with half the atoms missing. The resulting structure can be regarded as exposing micro-facets of the (111) surface, which has a lower surface energy than the (110) surface. First principles theoretical calculations (Ho and Bohnen, 1987) found a (lx2) ground state geometry with a missing top layer configuration that has substantial first layer contraction, lateral displacements in the second layer and buckling of the third layer, all in very good agreement with available experimental data (Moritz and Wolf, 1985; Copel and Gustafsson, 1986). A schematic figure of the (lx2) reconstruction is shown in Fig. 3.3. The calculations found that the (lx2) is stabilized relative to the ideal ( l x l ) since the more open (lx2) geometry allows the lowering of the kinetic energy of the s and p electrons without disrupting the d-bonding. Analogy with the case of the jellium surface indicate a threshold electron density above which the missing-row reconstruction is favored. For the 4d and 3d fcc transition metals, the clean (110) surfaces do not reconstruct because the bonding is weaker and hence the interstitial electron concentration is below the threshold. However, the (lx2) reconstruction can be induced on these surfaces by increasing the surface electron density (Fu and Ho,

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1989). Experimentally, induced (1• reconstruction is observed with the deposition of a small amount of alkali metals (Behm, 1989). From this picture, we would expect the missing-row reconstruction to be further stabilized on the Au(110) surface when alkali atoms are added to the surface. This is in agreement with experimental observation (Behm et al., 1987). With higher coverages, the Coulomb repulsion between alkali ions destabilize the missing-row geometry in favor of a c(2• structure, observed when 0.5 monolayer of K is adsorbed on the Au(110) surface, which can be regarded as the formation of a surface alloy (Ho et al., 1989; Haberle and Gustafsson, 1989). In principle, larger facets of the (111) surfaces can be formed by removing more atomic rows, leading to (lxn) reconstructions of the (110) surface. Theoretical results (Bohnen unpublished) found that the ( I x3) reconstruction is very close in surface energy to the (1• Experimentally, (1• reconstructions are indeed observed upon the absorption of small quantities of alkali-metals (Behm, 1989) or in electro-chemical cells (Ocko et al., 1992). The disruption of the surface periodicity is even more severe with surfaces undergoing reconstructions of the contractive type. In this case, there.is a contraction of the surface layer atomic spacing leading to a denser incommensurate or almost incommensurate close-packed surface atomic layer over the bulk substrate layers. We will discuss below the prototypical cases of Au(100) (Fedak and Gjostein, 1967; Van Hove et al., 1981 ; Binninget al., 1984) and Au(111) (Van Hove et al., 1981; Perdereau et al., 1974; Melle and Menzel, 1978; Heyraud and Metois, 1980; Takayanagi and Yagi, 1983, Harten et ai., 1985; EI-Batanouny et al., 1987; Woll et al., 1989). If we imagine that we have a single layer of fcc metal atoms suspended in vacuum, the layer should prefer to be in a hexagonal structure because that is the close-packing arrangement in 2 dimensions. If we view the surface of an fcc metal as being composed of a top layer residing on top of a substrate of the same species, only the top layer in a (11 l)-oriented surface is in the preferred close-packing arrangement. Would the top layer of say the (100) surface of an fcc metal reconstruct from a square net to a more compact hexagonal arrangement? It turns out that such top layer rearrangement does happen on 5d transition and noble metal surfaces. The unreconstructed (100) surface of Au should be a square lattice, but


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experiments indicate that the top layer becomes a contracted quasi-hexagonal overlayer on top of a square substrate. A schematic figure is shown in Fig. 3.4a. Since the reconstructed top layer is no longer commensurate with the substrate layers, it takes a big unit cell to describe the reconstruction if we insist on giving a surface unit cell. Earlier experiments (Fedak and Gjostein, 1967) indicated a (1• unit cell and later LEED measurements suggested a larger c(26• unit cell. The main driving force behind the Au(100) reconstruction is the strong tendency for the top layer to go to a more compact arrangement, so strong that it can overcome the energy loss by losing registry with the substrate underneath. For Ag in the 4d series, theoretical calculations (Takeuchi et al., 1989) indicate that the top layer also wants to transform to the more compact hexagonal arrangement, but the energy gained in such a transformation is not large enough to overcome the substrate potential which pins the top layer, so the surface stays unreconstructed. The bigger gain in energy

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in the case of Au upon contraction is due mainly to a stronger participation of the d-orbitals in the bonding in Au than Ag, which in turn can be traced to the fact that Au has a higher atomic number, so that it has a bigger core and stronger relativistic effects (Takeuchi et al., 1989, Fiorentini et al., 1993). These differences are not specific to Ag and Au, hence it is not surprising that similar reconstructions occur in the (100) surfaces of the 5d fcc metals Ir and Pt, but not in the corresponding (100) surfaces of the 4d metals Rh and Pd. It is interesting to note that Au(111) reconstructs, and apparently, .it is the only fcc(111) clean metal surface that exhibits a reconstruction. We have mentioned that as a single 2-dimensional layer, the top layer would like to be hexagonal-closepacked. The (111) face of Au is already hexagonal but the top layer, as a single layer, would prefer to have a higher atomic density than that of the fcc (111 ) planes as dictated by the bulk lattice constant. Apparently for most fcc elements, the energy gained in the contraction from a hexagonal to a more compact hexagonal structure is not big enough to compensate for the loss of registry with the substrate lattice, but the energy gained by the Au 2-dimensional layer is large enough to overcome the energy loss due to the mismatch of the substrate. Under the influence of the substrate pote_ntial, the contraction is not uniform. The to__p layer atoms contract along the directions by about 4-5%, forming a (22x~/3) superlattice. A schematic figure is shown in Fig. 3.4b. Since there are 3 equivalent directions on the surface, domains with different orientations that are degenerate in energy can co-exist, and mesoscopic ordered domains called "herringbone" patterns have been observed directly by STM (Barth et 1990; Chambliss et al., 1991). These mesoscopic patterns are consequences of the anisotropic surface tensile stress and can be explained by simple models (Narasimhan and Vanderbilt, 1992). The above discussions are centered around several prototypical reconstructions that have been studied with LDA calculations. Relaxation and reconstruction for clean metal surfaces have also been studied comprehensively with classical force models such as the embedded atom method and the closely related effective medium method. For surface relaxation, the results are almost always qualitatively correct, and in many cases, the results compare rather well with experimental results and first-principles calculations. Some of these results have been tabulated in (Ricter et al. (1984) and Bohnen and Ho (1993). For the more demanding case of surface reconstruction, these models do not always yield results that are consistent with experimental findings. For example, EAM potentials that are fitted mainly to bulk properties have incorrectly predicted an instability of Ag(110) towards the missing row reconstruction (Einstein and Khare, 1994) and have failed to predict the contractive construction of Au(100) (Haftel, 1993). However, EAM or EAM-like potentials (such as the "glue" model) that are specifically fitted to consider surface properties give correctly the structure and the energetics of the Au(100) (Tosatti and Ercolessi, 1991 ; Haftel, 1993). These models are of course indispensable if we want to study more complex surface phenomena such as diffusion mechanisms, energetics of surface defects, steps and kinks, faceting, surface roughening, surface melting, and the kinetics and dynamics of surface reconstructions.


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3.6. Semiconductor surfaces

Because of its relevance to device physics, semiconductor surfaces have been intensively studied ever since the advent of ultra-high vacuum technology enabled experimentalists to obtain reproducible results on surfaces. Early theoretical calculations were focused on the (111) and (100) surfaces of Si, and the GaAs(110) surface. These systems have also attracted a lot of attention from theorists, and have been treated by various computational techniques from tight-binding methods to first principles calculations. A comprehensive review on these surfaces can be found in Duke's article (Chapter 6). Here, we briefly discuss some prototypical systems (Si(111) and Si(100)) which will be referred to in the section concerning metals on semiconductor. S i ( l l l ) is a favorite surface for experimental studies since it is the common cleavage face for silicon. The freshly cleaved surface exhibits a (2x1) reconstructed structure (Lander et al., 1963) which converts to a (7x7) reconstruction (Schlier and Farnsworth, 1959) upon annealing. Some of the earliest first-principles self-consistent surface calculations were performed for the Si(111) (2x l) surface. While the (2• reconstruction was initially supposed to be caused by a buckling of the surface layer (Haneman, 1961), this model was found to be in conflict both with experimental data and total energy calculation results. A number of possibilities were considered before the Tt-bonded chain model due to Pandey (Pandey, 1981) was generally accepted to give a good description of the surface geometry. This structure is shown in Fig. 6.3 in Chapter 6. It is interesting to note that this is one of the first examples in which a theoretical effort correctly predicts surface atomic arrangements. The (7x7) reconstructed surface is so complex, that full first principles calculations of its atomic geometry are possible only very recently using the latest development in both computational techniques and massively parallel computers with novel computer architectures (Brommer et al., 1992; Stich et al., 1992). The currently accepted DAS (dimer-adatom-stacking-fault) model (Takayanagi et al., 1985) has a massive reconstruction, with a (7x7) unit cell containing a stacking fault, 12 adatoms and 9 dimer bonds. The driving force behind the reconstruction is a compromise between strain relaxation and the healing of the broken bonds at the surface. The Si(100) surface is of considerable technological interest since this is the surface adopted for silicon wafers in chip and device fabrications. The clean surface is reconstructed with a (lx2) structure in which the atoms in the surface layer pair up to form dimers (Lander and Morrison, 1962), as illustrated in Fig. 6.6 in Chapter 6. The detailed geometrical structure of the surface has been investigated by several calculations. Both tight-binding (Chadi, 1979) and first principles calculations (Batra, 1990) have contributed significantly to the understanding of the geometry of these systems. 3.7. Metal overlayers on semiconductors

There is a lot of interest in the study of the metal-semiconductor interface (Batra, 1991; Batra, 1987) because of its importance in various technology-related issues

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such as the quality and stability of metallic contacts in semiconductor devices, the formation of Schottky barriers, and heteroepitaxial growth on semiconductor substrates. Although much effort has been devoted to experimental studies of these systems, it is quite evident that our understanding of the metal-semiconductor interface lags considerably behind our understanding of the clean surface systems. The situation here is complicated by the intrinsic reactivity of the deposited metal layers with the semiconductor substrates, leading to various possibilities involving intermixing and compound layer formation. Because of the strength and the variety of chemical bonding in these systems (metal-metal, semiconductor-semiconductor, metal-semiconductor), various metastable phases can exist, the occurrence of which depends on details of the deposition procedures as well as on the initial condition of the substrate surface since these factors can influence the extent to which diffusion of atomic species is allowed to take place. Theoretical study of these systems by first-principles calculations is still in its infancy and the few available calculations are concentrated on cases with simple geometries: for example, when there is a sharp interface between the substrate and a silicide layer (Hamann, 1988), and for monolayer or submonolayer coverages of the metal overlayer in cases when intermixing or penetration of the metal atoms into the substrate can be ignored. We will focus on a few cases in which the surface structure is relatively well characterized experimentally and where reliable calculations have been performed. We also mention a number of systems whose structures have not yet been completely determined. These illustrate the level of complexity that is currently being attacked theoretically and experimentally. Chapter 6 contains additional examples. Since a relatively large proportion of the detailed experimental studies have been devoted to the adsorption of various overlayers on the S i ( l l l ) and Si(100) surfaces, it is not surprising that these are also the popular surfaces for theoretical studies. For the Si(l 11) surface, surface reconstructions with (~(3-x~-)R30 surface periodicity are very common among systems with adsorbed metal overlayers (Kono et al., 1994). Many calculations have been performed for this case, including the geometries for AI, Ga, In, Ag, Au, and Sb adsorption. The case of ordered vacancies has also been considered (Ancilotto et al., 1991). However, it should be cautioned that the coverage of the metal overlayers are not the same in all of these q3-x~- structures" the AI calculations considered a coverage of 1/3 monolayer while the Ag calculations considered both 2/3 and one monolayer. The Au and Sb calculations are for one monolayer coverage. Other ordered patterns such as (lx3) are also frequently observed. For the case of Si(100), the (lx2) dimerized periodicity of the clean surface is often carried over when a monolayer of metal overlayer is adsorbed. The Si dimers frequently remain intact upon metal adsorption. Theoretical investigations of adsorption on this surface include the cases of AI, As and alkali metals. Calculations for the adsorption of overlayers on GaAs have been restricted to the cases of AI and Sn on GaAs(110) (Batra, 1984; Pandey, 1989). These are still preliminary studies and much more work is required to obtain a clear and detailed picture of metal adsorption in the system. In general, it can be said that there is a big imbalance between the theoretical effort and the experimental effort in the

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investigation of metal-semiconductor interfaces. Much more theoretical work is needed to help interpret and organize the large amount of experimental data available on these systems and to obtain a good understanding of the physical mechanisms at work. It should be mentioned that in many cases, although the periodicity of the surface structure is known fairly accurately, the coverages of surface metal atoms are more uncertain and often form the subjects of controversy between different experimental groups.

3.7.1. Si(l l l)(~[3•

and Si(l l l)-Au

The cases of noble-metal adsorption on silicon and germanium surfaces have attracted a lot of interest because noble metals are thought to be less reactive than transition metals with unfilled d-bands and hence the interfaces should have simpler structures more amenable to analysis. A survey of earlier experimental results can be found in a comprehensive review by LeLay (Lelay, 1983). The (~/3-x43-)R30 structure is observed on both surfaces after deposition of the metal at high temperatures. The Ag/Si interface has been regarded as the prototype of an unreactive interface with sharp transition between the two materials. However, even so, the deduction of the atomic arrangements on these structures has proved to be quite non-trivial. For a long time, there were nearly as many models proposed as there were papers published on this system (Ding et al., 1991). There were controversies even on whether the metal coverage in the system is l or 2/3 monolayer. The geometry of these structures have been studied with a formidable arsenal of experimental techniques including STM, ion scattering, LEED, RHEED, and surface X-ray diffraction (van Loenen et al., 1987; Wilson and Chiang, 1987a,b; Nicholls et al., 1986; Takahashi et al., 1988; Vlieget al., 1989; Kono et al., 1986; Bullock et al., 1990). Numerous conflicting models were proposed, sometimes even from similar data information. Recent first principles total energy calculations show that most of these models are unsatisfactory in that the surface formation energies of the system are so high that the Ag layer would not wet the Si surface. It was found that most of the models were misled by the honeycomb pattern observed in STM experiments. The actual geometry deduced from calculations and also from X-ray diffraction data is shown in Fig. 3.5a. This so-called "honeycomb-chained-trimer" or HCT model consists of a top layer of Ag atoms with one monolayer coverage arranged in a so called "honeycomb-chained-triangle" pattern located on top of a "missing-top-layer" (MTL) Si(111) substrate. For a "missing-top-layer" substrate, the top layer Si atoms have three broken bonds per atom, and the top Si layer is distorted in a (~/3-x~-)R30 periodicity, forming trimers. With two of the broken bonds satisfied by the formation of the Si trimers, the remaining bond is satisfied by a metallic-type of bonding with the Ag atoms in the top layer. The calculations indicated that the honeycomb pattern observed by STM c a n be explained by tunneling into an empty surface band whose electronic wavefunction has a maximum between three Ag atoms, so that the bright spot in the STM pictures represent a trimer of Ag atoms, rather than an individual Ag atom. Tunneling into the

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(a) Ag/Si(111)

et al.

(b) Au/Si(111)

Fig. 3.5. Top views comparing (a) the "Honeycomb-Chained-Trimer" (HCT) model for Si(111)-Ag and the (b) "Conjugate-Honeycomb-Chained-Trimer" (CHCT) model for Si(111 )-Au. The large dark

dots are the metal atoms, and the rest are the Si atoms, with sizes decreasing from the surface to deeper layers. Note that for the case of Ag, the Si atoms form trimers and for the case of Au, the Au atoms form trimers. occupied states should yield a pattern reflecting the atomic geometry of the top layer. This has been verified by recent STM studies (Wan et al., 1992a). For the Au adsorbed surface, the surface structure obtained depends on the initial state of the clean Si(ll l) surface. If one starts from a (7x7) substrate and deposit the Au layer at high temperatures, the surface exhibits a (~r3-x~-) pattern. Intriguingly, STM pictures for this surface show a different image from the Si(11 l )-Ag case. Instead of observing a honeycomb pattern of two bright spots per ( ~ - x ~ ) unit cell, STM images for Si(111)-Au show a triangular pattern of one bright spot per (~x'4~-) cell. Medium energy ion scattering (MEIS) experiments (Chester and Gustafsson, 1991) indicated a Au coverage of one monolayer and a twisted-trimer geometry quite similar to that of Si(l 11 )-Ag. Surface X-ray scattering also found Au trimers (Dornisch et al., 1991). First principles total-energy calculations were performed and the results show that the ground state geometry consists of a monolayer of Au on a missing-top-layer S i ( l l l ) substrate as in the case of Si(11 l) (~/3-x~-)R30-Ag. However, the lateral arrangements of the Au and Si atoms are reversed from that of S i ( l l 1)-Ag: the Au atoms form trimers while the Si atoms form honeycomb-chained-trimers. Such an atomic arrangement in the S i ( l l l ) ('~-• is called the "conjugate" honeycomb-chained-trimer (CHCT) model (Ding et al., 1992). A top view is shown in Fig. 3.5b. Unlike the results from the MEIS experiment, first principles calculations found no twist in the orientations of the Au trimers. This finding is in agreement with the results of surface X-ray diffraction and LEED data which indicate the presence of a mirror plane perpendicular to the surface for the ground state geometry. It is possible that the analysis of the MEIS data is complicated by the presence of domains of different registry for the Au layer on the substrate. Recent analysis of LEED data provide


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strong support for the calculated ground state model (Quinn and Marcus, 1992). The different atomic geometries found in Ag and Au on Si(111) can be explained by the difference in energy of the metal-metal and metal-semiconductor bonds. In the S i ( l l l ) (,]3-x,~-)R30-Ag case, both the Ag-Ag and Ag-Si bonds are weaker than the Si-Si bonds. The primary reconstruction process thus involves the rebonding of the surface Si atoms to form trimers, which in turn prompts the Ag atoms into a honeycomb-chained-trimer arrangement. Two of the three Si dangling bonds are satisfied by the trimer formation, and the remaining dangling bond is satisfied by the Ag adatom. For the case of Au, we note that Au has a higher cohesive energy than Ag (by about 0.9 eV/atom). The Au-Si bond is also stronger than the Ag-Si bond. In this case, the primary process involves the trimerization of Au atoms, forming trimers with bond lengths close to the bulk values. In this way, the Au atoms can optimize their own bonding, while keeping commensurate with the substrate Si to form strong Au-Si bonds. 3.7.2. Other phases o f noble metals on Si or Ge

When the Si(111 ) (,~-x,13-)-Ag phase is annealed at a high temperature (T = 550~ so that the Ag partly desorbs, it undergoes a transformation to a (1 x3) phase, which converts to a (lx6) upon cooling to room temperature (Lelay, 1983; Wan et al., 1993). The atomic arrangement has not been completely resolved yet, although it is believed that the (Ix6) structure is very similar to the (lx3) structure. Most of the older models assume that the Ag coverage is 1/3, with rows of Ag atoms sitting on 3-fold sites of an ideal S i ( l l l ) surface, aligned in the directions. Recent experiments suggest that the Si substrate has a missing-top-layer arrangement, and the top layer Si atoms are probably forming r~-bonded chains; and the Ag coverage may actually be 2/3 monolayer (Wan et al., 1993). More experimental and theoretical work is needed to determine the exact stoichiometry of Si surface layers and the exact location of the Ag chains. A 1/3 monolayer model recently proposed for (lx3) alkali metal/Si(111 ) may also be applicable to this case (see the discussion in w 3.7.5 below). The (,13-x4-3-)R30 structure is also observed when Au and Ag are deposited on Ge(l 11), although these systems have received less attention than the corresponding systems with Si as the substrate. Recent experiments found that the saturation coverage for the metal is also about one monolayer, and the atomic arrangements are very similar to the case with Si(111) substrate. For Ge(l 11) (,~-x~4-3-)R30-Au, surface X-ray diffraction (Howes et al., 1993) found that the structure is similar to that of S i ( l l l ) (,~-x,~-)R30-Au and can basically be described by the CHCT model, with the metal atoms forming trimers. The case of Ge(11 l)(,]3-x,~-)R30-Ag is more controversial, with surface X-ray diffraction (Dornisch et al., 1991) favoring the CHCT model, while a very recent LEED analysis (Huang et al., 1994) favors the HCT model. The basic physics that governs the surface structure should be very similar to the case of Si(111)-Ag and the ground state structure depends on the relative strength of the Ge-Ge bond. Since the CHCT model is found for Ge(111) (4-fxx/3-)R30-Au, it indicates that the Au-Au and Au-Ge bonding is more important

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than the Ge-Ge bonding, just as in the case of Si(111)-Ag. The LEED results (HCT model) for Ge(111) (q-3-xf3-)R30-Ag suggests that the Ge-Ge bonding is stronger than the A g - A g bonding, while the surface X-ray data (CHCT model) for the same system suggests just the opposite. A first principles calculation would be useful in providing a second opinion. 3.7. 3. B,A l, Ga, In on Si(100) and Si(l 11)

The growth of the first monolayer of the Group III elements AI, Ga, In on Si(100) shows common behavior. These systems are regarded as well characterized and the agreement between theory and experiment is very good, especially for the case of A1 which has been studied carefully by first principles calculations (Northrup et al., 1991). Although the phase diagram seems complex, the structure can basically be accounted for by the following construction: the substrate Si(100) reconstruction is almost the same as the clean surface, characterized by Si dimers. The metal atoms form chains of metal dimers, with the chains orientated perpendicular to the Si dimer chains. First principles calculations indicate that the metal dimers should be aligned parallel to the Si dimers, as shown schematically in Fig. 3.6. The metal-dimer chains repel each other so that at low coverages, larger unit cells such as (2x3) are observed, reaching an "ideal" arrangement of a (2x2) phase as shown in Fig. 3.6 when the coverage reaches half a monolayer. The atomic arrangements in the (2x2) phase are such that all the metal atoms are bonded to 3 nearest neighbors and all the substrate Si dangling bonds are saturated. On the S i ( l l l ) surface, the geometry and surface electronic structure of the (~-xq-3-) phase observed in the adsorption of A1, Ga, and In are also well characterized. At 1/3 monolayer coverage, a (q3-xq-3-)R30 unit cell is observed. The substrate Si(11 1) is believed to be an ideal truncated Si (11 1) surface, with the AI ad-atoms forming a (q3-x'~-) overlayer. Two adsorption sites were discussed, both

AI

$i dimer

Fig. 3.6. A top view of Si(100)(2x2)-A1, the black dots are AI atoms and the empty circles are Si atoms.

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0

..

0

0

(Jm~..()

~ .m~--~ )

I

0

Fig. 3.7. A schematic figure showing the/43 and 7"4sites on a Si(11 1) surface. lying in the three-fold sites above the surface Si(111) layer: the 7'4 site that lies directly above a Si atom in the second Si layer, and the H 3 site that is laterally displaced so that the AI atom lies above a hollow in the second Si layer. These two adsorption sites are illustrated in Fig. 3.7. For all three elements, the ?'4 site was shown to be energetically more favorable than the H 3 site by first principles calculations (Northrup, 1984; Nicholls et al., 1985, 1987). The calculated surface band structures obtained agree well with subsequent angle-resolved photoemission and inverse photoemission measurements. Recent STM investigations for the Si(l 11) ('~-x'~-)-In structure also support the results of the L D A calculations. Note that complete saturation of the dangling bond is achieved by such an atomic configuration at 1/3 monolayer A1 coverage and the AI atoms are also bonded to 3 nearest neighbor Si atoms. The bonding configuration for the Group III metals on Si(100) (2x2) and Si(111 ) (~3-x'~-)R30 are thus very similar. The smallest of element of the group, B, shows different behavior. Although B also induces a (,~-x'~-)R30 phase with 1/3 monolayer coverage, LDA calculations found that it is most favorable for the B atoms to substitute for the Si atoms directly below Si adatoms at a 7"4 site (Bedrossian et al., 1989; Lyo et al., 1989), in agreement with many experiment observations. When the coverage is higher than 1/3 monolayer, AI/Si(I 11) shows other ordered structures such as (47--x47-) and (7x7). Ga and In can also induce a variety of complex reconstructions on Si(l 11). There are no detailed theoretical studies on these more complex systems yet.

3.7. 4. As, Sb and Bi/Si(l 1 1) As adsorption produces an ideal termination of the Si(111) surface. The As atoms substitute for the Si atoms in the top layer forming a ( l x l ) structure in which all the Si atoms in the system are four-fold coordinated. Each As atom is bonded to three Si atoms in the second layer, leaving a lone pair which fully occupies the dangling bond orbital at the surface. This passivation of the surface is so effective that the surface is very chemically inert, even to oxygen exposures. The geometry and electronic structure of the system has been well studied in first principles self-consistent pseudopotential calculations (Uhrberg et al., 1987). This system is also quite thoroughly examined experimentally because of its relevance in the epitaxial

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growth of GaAs on Si(111). The heavier Sb and Bi atoms induce ({3-x'~-) reconstructions at one monolayer coverage. These group V atoms are believed to form trimers (Martensson et al., 1990; Shioda et al., 1993) on top of the Si double layer. Si(111)-Bi also has a 1/3 ML coverage ('~-x~/3-) phase, which has one Bi atom per ('#3-x'~-) unit cell, absorbed at the T4 site on the Si double layer (Shioda et al., 1993). 3.7.5. Alkali metals on semiconductors

The adsorption of alkali metals on semiconductors has attracted a lot of experimental attention (see, e.g., Soukiassian and Kendelewicz, 1988) and has been studied as a prototypical system for the metalization of semiconductor surfaces and Schottky barrier formation. These systems also received early attention from first principles calculations (Soukiassian and Kendelewicz, 1988; Batra, 1987, 1991), mainly because these systems can be treated by plane wave pseudopotential approaches. Unfortunately, there are still controversies about many aspects of these systems, such as the nature of the metal-semiconductor bonding, the absorption sites and the saturation coverage and more experimental work is probably needed to clarify these systems. For alkali metals on Si(100), the dimer rows in the Si substrate are believed to remain intact and the dimers are slightly elongated upon alkali metal absorption. At high coverages, the alkali metal atoms form one-dimensional chains parallel to the Si dimer rows. However, consensus has not been reached concerning the exact absorption site and the saturation coverage (i.e. how many alkali metal atoms are there for each surface Si atom). There are several inequivalent sites in which the alkali metals can reside, and they are depicted in Fig. 3.8. It seems that theoretical results using the plane wave pseudopotential approach (Kobayashi et al., 1992; Zhang et al., 1991) agree that the coverage should be one monolayer for Na and K (which means one alkali metal atom per Si atom, or equivalently, two metal atoms per Si dimer), and two inequivalent sites (the so-called "pedestal" site and "valley bridge" site) are occupied. Since these two sites are at a different level (the pedestal sites are above the dimer rows and the valley bridge sites are in the troughs between the dimer rows), this class of model is called a double layer model (Abukawa and Kono, 1988). Cluster calculations favors absorption at the cave site with half monolayer coverage (Spiess et al., 1993). Experimental results are divided between one monolayer and half monolayer saturation coverage for the case of Na and K, and various absorption sites have been proposed (see, e.g., Soukiassian et al., 1992; Wei et al., 1992). A ( l x 3 ) structure is also observed when alkali metals are deposited on Si(111) (Daimon and Ino, 1985). STM experiments (Jeon et al., 1992; Wan et al., 1992b) observed ordered rows of double chains of bright spots, separated by 3 times the lattice constant of the ( 1• 1) surface unit cell. There are also controversies regarding the coverage of the alkali metals (although more recent results seem to favor 1/3 monolayer coverage (Weiteringet al., 1994)), and the structure of the substrate, and quite a few models have been proposed. First principles calculations (Morikawa, 1994) found none of the proposed (1• alkali-metal/Si(l 11) models to be satisfactory. The most promising model seems to be the new model that has 1/3 monolayer


131

~176 (

---

5-.O+(3

9 O+O

~ - bridge site X__ k___ pedestal site (H) cave site (B) valley bridge site (C) top site (T)

Fig. 3.8. A schematic top view of Si(100) showing various possible sites for alkali metal adsorption. The names of these sites are those most commonly used in the literature. alkali metal coverage with the Si substrate reconstructed similar to the Pandey n-bonded chain model, but with 5 and 7-fold Si rings separated by a 6-fold ring to form (Ix3) surface periodicity (Erwin, 1995). More theoretical and experimental study will be needed to characterize these systems.

3.8. Epilog Since the physics and the mechanism underlying various surface reconstructions are tied to the properties of the constituent elements, which differ drastically from one element to another, it is very difficult to give one single rule of thumb as to how a surface should reconstruct or why it should do so. There is perhaps only one common tautological reason: surfaces reconstruct to lower the surface energy, and different classes of systems reduce their surface energy in very different ways. In any case, we will try to present some general observations from a theorist's point of view. Semiconductor systems are by definition those that have a gap in the eigenvalue spectrum. In the process of surface formation, states will be introduced in the gap, generally leading to a higher band energy. Most of the surface relaxation and reconstruction phenomena in semiconductor surfaces are consistent with the reduction or removal of these "defect" states in the gap, leading to a lowering of the sum

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of the eigenvalue term in the total energy (see e.g. Eq. (3.4)). This can be achieved by rebonding or relaxation. On a more intuitive level, we may say that for semiconductor systems which have directional covalent bonding, dangling (broken) bonds are produced by cutting a crystal to form a surface and the main purpose of the surface reconstruction is to reduce the number of dangling bonds. The surface atoms can u n d e r g o m a s s i v e r e a r r a n g e m e n t s to rebond t h e m s e l v e s , with Si(111 )(7• being a famous example. Since the covalent bonding is usually strong, such massive rearrangements of surface atoms may require a sizable activation energy, and thus may not be attainable unless the surface is annealed. There may be metastable configurations where the atoms can reduce the number of dangling bonds to a lesser extent but which can be achieved more easily by local rearrangement of atoms. Si(111)(2x l) with the r~-bonded chain configuration is a good example. These concepts are discussed further by Duke in Chapter 6. For bcc metals such as Mo and W, the energetics are dominated by the bonding of the electrons in the d-shells which are only half-filled. The bulk density of states exhibit a dip or minimum at the Fermi level, and this fact is usually used to explain why half-filled d-shell systems prefer the bcc structure over the more compact fcc or hcp arrangement of atoms. When a low index surface is formed, especially for the (100) surfaces which amounts to cutting away half of the nearest neighbors, the local density of states of the surface atoms actually have a peak at the Fermi level and the reconstructions of these surfaces can be regarded as the rebonding of the surface atoms to reduce the density of states at the Fermi level. From this point of view, bcc metal surface reconstructions share the same mechanism as elemental semiconductors. The fcc transition and noble metal atoms are nearly "spherical" since their d shells are full or nearly full, they reconstruct by changing the density of the surface atoms. The main effect is to reduce the kinetic energy of the s,p electrons and increase the binding due to the d-electrons. The complex reconstructions of the 5d low index fcc metals surfaces can be rationalized this way: The missing-row type reconstruction reduces the kinetic energy of the s,p electrons without decreasing the d-electron binding; while the contractive-type reconstructions increases the d-electron binding without increasing the kinetic energy of the s,p electrons. For simple metals, which can be regarded as ion cores sitting in a sea of nearly-free valence electrons, there is little the system can gain by changing the symmetry of the surface unit cell, so these systems typically relax by changing the interlayer distances. The contraction of the top layer can be explained by the "Smoluchowski" smoothing effect. The situation is much more complex with adsorbate-induced reconstructions. For the special case of metal overlayers on semiconductors, there are a few competing factors: the system will try to minimize the number of the dangling bonds in the semiconductor substrate, maximize the metal-semiconductor interaction, and at high coverage, optimize the metal-metal bonding. The final atomic configuration, most favored adsorption sites and the saturation coverage depend very much on the relative strength of the metal-metal, semiconductor-semiconductor and the metal-semiconductor bonds.


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Acknowledgements We thank Drs. N. Takeuchi, T.C. L e u n g , Y.G. Ding, B.L. Z h a n g , and M. Y a m a m o t o for c o l l a b o r a t i o n in the area of surface physics. A m e s L a b o r a t o r y is o p e r a t e d for U.S. D e p a r t m e n t of E n e r g y by I o w a State U n i v e r s i t y u n d e r C o n t r a c t No. W - 7 4 0 5 E N G - 8 2 . O u r work on surface physics has been s u p p o r t e d by the D i r e c t o r of E n e r g y R e s e a r c h , Office of Basic E n e r g y Sciences, i n c l u d i n g a grant of c o m p u t e r time on the C r a y c o m p u t e r s at the National E n e r g y R e s e a r c h S u p e r c o m p u t e r C e n t e r at Livermore.

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CHAPTER 4

Theory of Insulator Surface Structures J.P. L A F E M I N A Environmental Molecular Sciences Laboratory Pacific Northwest National Laboratory* Richland, WA 99352, USA *Operated for the US Department o f Energy by Battelle Memorial Institute under contract DE-AC06-76RLO 1830

Handbook of Su~. ace Science Volume 1, edited by W.N. Unertl


137

Contents

4.1.

Introduction 4.1.1.

4.2.

4.3.

4.4.

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

Basic definitions

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

139 139

4.1.2.

F u n d a m e n t a l principles of surface structures . . . . . . . . . . . . . . . . . . . . . .

141

4.1.3.

Purpose, scope, and o r g a n i z a t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . .

142

Computational methods

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

144

4.2.1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

144

4.2.2.

Quantum mechanical methods

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

145

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

149

4.2.2.1.

SCF-LCAO methods

4.2.2.2.

Density functional methods . . . . . . . . . . . . . . . . . . . . . . . . .

151

4.2.2.3.

Tight-binding methods

153

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

4.2.3.

C l a s s i c a l potential m o d e l s

4.2.4.

C o m p u t a t i o n a l a p p r o a c h e s to surfaces

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

156

154

4.2.5.

C o m p a r i s o n of the m e t h o d s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

159

T h e structure of clean surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

160

4.3.1.

Diamond(Ill)

surface

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

160

4.3.2.

R o c k s a l t (001) surface

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

163

4.3.3.

Rutile (110) surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

166

4.3.4.

P e r o v s k i t e (100) surfaces

4.3.5.

C o r u n d u m surfaces

4.3.6.

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

169

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

171

Silica surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

174

T h e structure of surface defects

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

174

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

174

4.4.1.

Introduction

4.4.2

Defects on the rocksalt (001) surface . . . . . . . . . . . . . . . . . . . . . . . . . .

4.5.

T h e structure of adsorbates

4.6.

Discussion

175

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

176

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

177

4.6.1.

R e h y b r i d i z a t i o n of d a n g l i n g bond charge density

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

178

4.6.2.

S u r f a c e stress and the i m p o r t a n c e of surface t o p o l o g y . . . . . . . . . . . . . . . . .

178

4.6.3.

A r e a s for future research

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

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

138

179 180

4.1. Introduction

4.1.1. Basic definitions The Webster's II New Riverside University Dictionary (1988)defines an electrical insulator, or dielectric, as "a nonconductor of ... electricity". A more precise (and useful) definition can be found in Van Nostrand's Scientific Encyclopedia, which states that an insulator is a material which "when placed between conductors at different potentials, will permit a negligible current ... to pass through it." (Considine, 1976). (At sufficiently large potentials, however, the insulator will allow current to flow as the material undergoes a phenomenon known as dielectric breakdown.) Even with this definition, however, the difference between insulators and semiconductors, for example, is far from straightforward. How negligible need the current be before a material is classified as an "insulator"? The closest thing to a perfect insulator is a vacuum, and quantitative measures of the insulating nature of materials are usually referenced to the vacuum. For example, in an ideal parallel plate capacitor, the capacitance between the plates in vacuum, Co, can be written as ~oA/d, where to is the dielectric constant of a vacuum (also called the permittivity of free space), A is the area of the plate, and d is the distance between the plates. If the vacuum is replaced by an insulator, the capacitance, C, is now given by eCo where e is the dielectric constant, or dielectric permittivity, of the insulator (Considine, 1976; Lorrain and Corson, 1970). Many factors can affect the value of the dielectric constant, including temperature and humidity. As can be seen in Table 4.1, materials with a wide range of dielectric constants are available, and they span almost every type of material known, including crystalline, amorphous and glassy ceramics and porcelains, metal oxides, minerals, and organic polymers and resins. The industrial and environmental applications of the materials listed in Table 4.1 result from far more than their inability to conduct electricity. In fact, the chemistry that occurs at the surfaces of these materials is important in diverse areas such as catalysis (Campbell, 1988), corrosion (Fehlner, 1986), ceramic synthesis and chemistry, (Dufour and Nowotny, 1988; Nowotny and Dufour, 1988) the formation of insulating layers in microelectronic devices (Pantelides, 1978; Pantelides and Lucovsky, 1988), groundwater transport of contaminants (Hochella and White, 1990; Reeder, 1983), glass formation (Paul, 1982), and the design of nuclear waste storage material (Northrup, 1987). As a result, there is an enormous literature on the surface physics and chemistry of "insulating" materials. Unfortunately, the bulk of this work has been performed on poorly characterized samples, such as powders, amorphous, and polycrystalline materials. Consequently, the results of 139

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Table 4.1 The dielectric permittivityof some representative materials. Valuestaken from Van Nostrand's Scientific Encyclopedia (Considine, 1976). Material Alumina (ceramic) Alumina (crystalline) Pyrex Muscovite Calcite Magnesium oxide Barium titanate Polyethylene Nylon Cellophane

8.1-9.5 10.0 5.1 7.0-7.3 9.2 8.2 4100 2.3 4.0--4.6 6.6

these studies are difficult to interpret in terms of the relationships between the surface atomic structure, chemical bonding, and process chemistry. The basis for understanding these relationships, is a detailed knowledge of atomic structure of the surface or interface of interest. "...there is no doubt that the atomic geometry of a surface or interface is its most fundamental characteristic. Any serious scientific effort to determine the electronic structure and properties of an interface must start with a realistic description of its atomic geometry..." (Duke, 1982) This chapter will present a simple set of chemical and physical principles that can be applied to insulator surfaces, and used to understand the qualitative features of surface structure, including surface stoichiometries, relaxations, and reconstructions (Duke, 1992; LaFemina, 1992). These principles have emerged from many years of detailed theoretical and experimental work on semiconductor surfaces. For insulating surfaces, this process is in its infancy. As stated above, the vast majority of the experimental work has been performed on poorly characterized material (Henrich, 1985). Sample preparation difficulties, coupled with the fact that many insulator surfaces fracture rather than cleave, make the preparation of surfaces with well-defined stoichiometries difficult (Henrich, 1985). In addition, there are many problems, such as surface charging, and the decomposition of surfaces under charged-particle beams, associated with the application of the traditional surface structural probes to insulator surfaces. Computationally, because of the complexity of many of these materials relative to metals and semiconductors, much effort has gone into the formulation of classical potential models (Colbourn, 1992). These models, while making many contributions to the understanding of insulator surface structures, are inherently limited. The classical nature of these potentials precludes any knowledge of the surface electronic structure; knowledge critical to the elucidation of the driving forces behind surface reconstructions and to the formulation of surface structure-activity

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relationships. At the opposite end of the spectrum are the first-principles, or ab initio, methods that, even with the most recent advances in computer hardware and software, are impracticable for most of the systems which we will examine. In between these two extremes lie the semiempirical and empirical quantum-mechanical methods, such as the tight-binding models, which have been enormously successful in providing quantitative descriptions of semiconductor surfaces and interfaces. (See Chapter 6 and reviews by Duke (1992) and LaFemina (1992).) The development of these models for many insulating systems has been inhibited, however, by lack of the detailed experimental information needed for their formulation. At this point it is useful to review some of the basic nomenclature used in this chapter. A surface is "relaxed" if the surface atomic geometry exhibits the same symmetry as the truncated bulk solid. Relaxed surfaces are referred to as (1• even though the atoms at this surface may lie as much as an Angstrom away from the truncated bulk lattice sites. If the symmetry of the surface is different than that of the bulk, then the surface is "reconstructed", and the Wood notation described in Chapter 1 for surface overlayers is used. 4.1.2. Fundamental principles of surface structures

Insight into the structure of insulator surfaces can be obtained, as stated above, by applying a simple set of chemical and physical principles. In fact, these principles can be cast as a set of five simple rules (which roughly follow the presentation in Chapter 6). (1) Saturate the dangling bonds: The creation of the surface creates dangling bonds: that is, bonds which used to bind surface atoms to their, now missing, bulk neighbors. This is in an energetically unfavorable situation, because these dangling bonds are only partially filled with electrons. Therefore, a driving force exists at the surface to redistribute the dangling bond electrons (charge density) into a more energetically favorable configuration. In general, this occurs in a way which satisfies the local chemical valences of the surface atoms. (2) Form an insulating surface: The most energetically favorable way to eliminate the dangling bonds is to create an "insulating" surface (i.e., open a gap between the occupied and unoccupied surface states). This can be done by forming new bonds at the surface, either between surface atoms or between the surface atoms and adsorbates. The surface also can be insulating as a result of a surface structural rearrangement (i.e., a surface relaxation or reconstruction) that transfers electrons between surface atoms. Finally, the energy of the surface can be lowered electronically, through strong electron correlation effects which open a gap between the occupied and unoccupied surface states. The bottom line, however, is that the opening of a gap between the occupied and unoccupied surface states lowers the energy of the occupied states while raising the energy of the unoccupied states, resulting in a net energy lowering for the surface. (3) Do not forget about kinetics: In general, and especially for the fracture surfaces of the crystalline metal oxides, the structure exhibited by any surface is dependent upon the processing history of the sample. That is, the "structure ob-

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J.P. LaFemina

served will be the lowest energy structure kinetically accessible under the preparation conditions" (Duke, 1993). This applies to the cleavage surfaces as well, where the exhibited surface structures are "activationless" in the sense that the activation energy for the relaxation or reconstruction is less than the energy provided by the cleavage process. (4) Form a charged neutral surface: The most energetically favorable way of pairing up dangling bond electrons is to fully occupy the anion dangling bonds and completely empty the cation dangling bonds. Alternatively, this can be thought of as completely filling the valence band orbitals while completely emptying the conduction band orbitals, which is, of course, what happens in the bulk. This rule, also referred to as autocompensation, or electron-counting, is simply a consequence of the fact that a surface which carries a net charge produces long range electric fields which cost a significant amount of energy. Anything which eliminates this surface charge will also greatly reduce the surface energy. This includes the formation of non-stoichiometric surfaces through the creation of surface and subsurface defects, and/or the adsorption of atoms or molecules onto the surface. This is an enormously powerful principle since it allows for the identification of a few, most likely, surface stoichiometries. (5) Conserve bond lengths: All of the factors described above provide potential driving forces for the surface atoms to move away from their bulk atomic positions and lower the surface energy. If this movement creates local strain in the surface or subsurface region, (that is, strain which results from the distortion of the local bonding environment) then the surface energy will be raised and the movement of the surface atoms resisted. Not all surface strains are created equal, however. For example, distortions of bond angles typically cost an order of magnitude less energy than distortions in near-neighbor bond lengths. Consequently, surface atomic motions which move the atoms into electronically favorable conformations, while (nearly) conserving near-neighbor bond lengths are most favorable. It is the topology, or atomic connectivity, of the surface that controls which atomic motions will be energetically favorable. Hence, it is the balance between the surface energy lowering due to the elimination of surface dangling bonds and the creation of an insulating surface, and the energy cost due to induced local strain that primarily determines the nature of surface relaxations and reconstructions.

4.1.3. Purpose, scope, and organization In the remaining sections of this chapter these five principles will be applied to a variety of insulator cleavage, fracture, and growth surfaces. Looking back at Table 4.1, a complete discussion of all of the material types listed is clearly beyond the scope this chapter. What this chapter will do, however, is to focus on the subset of these materials for which the most detailed understanding of surface atomic structure exists, namely the surfaces of diamond and of the crystalline metal oxides. In many cases, the knowledge gained from examining these surfaces is directly applicable to other materials of the similar structural types (e.g., the crystalline metal sulfides and selenides, as well as the rocksalt structure alkali halides). In

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143

addition, other chapters in this book will cover some of the materials not examined in this chapter. In Chapter 6 the structures of elemental and compound semiconductor surfaces are examined from both an experimental and computational point-ofview using these same principles as a guide. In Chapter 5 the structures of crystalline ceramics and insulators (oxides, carbides, nitrides, and halides) are examined from an experimental viewpoint. Other relevant chapters include Chapter 3, which details the computational methods used in surface structure computations, Chapters 7 and 8, which review the experimental methods of surface structure determination referred to in this chapter, and Chapters 10-13 which will examine the structure of adsorbed layers and surface defects. As indicated by the title, the emphasis of this chapter is primarily on the computation of the atomic structure of insulator surfaces. Most importantly, a major goal of this chapter is to demonstrate the applicability of the simple chemical and physical principles, summarized above, to insulator surfaces. Experimental surface structure determinations by low-energy-electron-diffraction (LEED) (Van Hove et al., 1986) and low-energy-positron-diffraction (LEPD) (Canter et al., 1987) intensity analyses, X-ray photoelectron diffraction (XPD) (Chambers, 1992), and ion scattering (van der Veen, 1985) will be used, where available, to evaluate the computational predictions of surface and interfacial atomic structure. In addition, the results of X-ray photoemission spectroscopy (XPS), ultraviolet photoemission spectroscopy (UPS), and electron-energy loss spectroscopy (EELS) will be used to evaluate the electronic structure predicted for the structural models that emerge from the computations (Woodruff and Delchar, 1986). The litmus test of success is that the same model predicts both the surface atomic and electronic structures in agreement with experimental measurements. The presentation in this chapter is such that readers familiar with fundamental concepts in solid state atomic structure and chemical bonding at the following level should have no difficulty: "Introduction to Solid State Physics" by C. Kittel (Kittel, 1976), "Chemistry in Two Dimensions: Surfaces" by Gabor A. Somorjai, (Somorjai, 1981) and "Physical Chemistry" by P.W. Atkins (Atkins, 1978). For readers looking for more in-depth treatments of these topics, several excellent graduatelevel texts are available: "Physics at Surfaces" by Andrew Zangwill (Zangwiil, 1988), "Solids and Surfaces: A Chemist' s View of Bonding in Extended Structures" by Roald Hoffmann (Hoffmann, 1988), "Atomic and Electronic Structure of Surfaces: Theoretical Foundations" by M. Lanoo and P. Friedel (Lanoo and Friedel 1991), and "Electronic Structure and the Properties of Solids: The Physics of the Chemical Bond" by W.A. Harrison (Harrison 1989). This chapter is organized as follows. In w 4.2 the various computational models commonly used for the computation of insulator surface structures are explored; the basic assumptions outlined, and the relative strengths and weaknesses evaluated. In w 4.3, the results of surface structure computations for the most common surfaces of insulators are examined. Sections 4.4 and 4.5 look at the structure of surface defects and surfaces with chemically adsorbed atoms respectively. Finally, w 4.6 contains a discussion of the major issues both clarified and raised by the computations and the chapter concludes with some speculations about promising areas for future research.

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4.2. Computational methods 4.2.1. Introduction This section describes various computational methods used to determine the atomic structure of insulator surfaces. The approximations associated with each of the methodologies will be presented and their advantages and disadvantages discussed. The methodologies can be roughly classified into two groups: those that use quantum-mechanical potentials and those that use empirically determined classical potentials. The quantum-mechanical methods can be further sub-divided into three major types w self-consistent field, linear-combination of atomic orbitals (SCFLCAO); density functional; and tight-binding B depending upon how the interactions between the electrons are treated. The empirical classical-potential models have primarily been developed, and applied to, "ionic" insulators because of the presumed predominance of the Coulombic interactions between the ions in the lattice (Madelung, 1909, 1910a,b). The various applications these methods to the study of surfaces also will be discussed. These techniques can be subdivided into three categories as well: Green's function techniques, slab calculations, and cluster calculations. The main difference between these methods is the boundary condition used to model the semi-infinite surface. The Green' s function techniques treat the semi-infinite nature of the surface exactly (and as a result are the most complex). Slab models treat the infinite (in the two dimensions parallel to the surface) two-dimensional nature of the surface properly, but have a finite thickness in the third dimension perpendicular to the surface. Finally, the cluster methods model the surface with a finite set of atoms. This section will emphasize the basic terminology; the goal is to enable the reader to evaluate computational results. For the details of a particular method, the reader is referred, primarily, to the review literature and to graduate-level texts. One feature common to all of these methods is the way in which the equilibrium surface atomic geometry is obtained. That is, once the method is chosen, and the total-energy functional specified, atomic forces (Feynman, 1939) are computed, and the total energy (or atomic forces) minimized as a function of the nuclear coordinates. The methods differ, of course, in the way in which the total-energy functional is specified. It is this specification, along with the associated assumptions, that will be addressed in the remainder of this section. The section is organized to proceed from the quantum to the classical; moving approximately from ab initio, or first-principles, methods to the empirical methods through the introduction of successive approximations. Section 4.2.2 begins with a description of the fundamentals common to all quantum mechanical methods. The two different ab initio approaches that have been developed B SCF-LCAO and density-functional B are then examined in w 4.2.2.1 and 4.2.2.2, while the most common (for surface structure computations) empirical quantum-mechanical method w tight-binding m is described in w 4.2.2.3. Section 4.2.3 covers the empirical classical potential models, which are, by far, the most widely used. The

Theory of insulator su~. iwe structures

145

Table 4.2 Summary of the advantages and disadvantages of the various theoretical methods discussed in w4.2 Method (Section)

Advantages

Quantum-Mechanical Methods SCF-LCAO (w 4.2.2.1) Excited state properties can, in principle, be computed.

Disadvantages

Unscreened exchange at HartreeFock level; correlation can only be included a posteriori; very computationally intensive (tens of atoms).

Density Functional (w 4.2.2.2)

Both exchange and correlation included in zeroth-order; first principles yet less computationally intensive than LCAO-SCF; can deal with tens of atoms.

Ground state theory, no excited state properties can be computed. Exchange and correlation interactions are approximated with no clear way to improve.

Tight-Binding (w 4.2.2.3)

Computationally simple; able to deal with tens to hundreds of atoms; description of bonding in terms of atomic-like orbitals; quantitative descriptions of surface structure possible.

All interactions are parameterized; non-self consistent; needs to be benchmarked against experiment or first principles computations.

Classical Potentials ( w4.2.3)

Computationally very simple; able to deal with hundreds of atoms.

Non-quantum mechanical; all interactions are parameterized.

different c o m p u t a t i o n a l a p p r o a c h e s to surface m o d e l i n g are d i s c u s s e d in w 4.2.4, and a c o m p a r i s o n of all of the m e t h o d s is given in w 4.2.5. Finally, a s u m m a r y of this i n f o r m a t i o n is given in Table 4.2. 4.2.2. Q u a n t u m m e c h a n i c a l m e t h o d s All of the q u a n t u m m e c h a n i c a l m e t h o d s c o n s i d e r e d in this section begin by m a k i n g the B o r n - O p p e n h e i m e r (Born and O p p e n h e i m e r , 1927) a p p r o x i m a t i o n , which ass u m e s that the m o t i o n of the electrons is fast c o m p a r e d to the m o t i o n of the nuclei, or ion cores. In this way the motion of the nuclei may be n e g l e c t e d when c o m p u t i n g the e l e c t r o n i c structure, in effect, separating the electronic and nuclear d e g r e e s of f r e e d o m . In this a p p r o x i m a t i o n the t o t a l - e n e r g y of a s y s t e m of N atoms, w h o s e n u c l e a r positions are g i v e n by R ~N~, can be written as the sum of an e l e c t r o n i c (EEl.) and a n u c l e a r (ENuc) c o n t r i b u t i o n . The nuclear c o n t r i b u t i o n , n o w i n d e p e n d e n t of the e l e c t r o n c o o r d i n a t e s , is the repulsion b e t w e e n ion cores ~t and v with c h a r g e s Z, and Z,,, s e p a r a t e d by a distance R~v

N ENUCIR(N)Ile2 It'"I-- 4--~C(I ~' ~,vY--~Z~ZVR~v

(4.1)

J.P. LaFemina

146

The electronic energy is determined by solving the many-electron, time-independent Schr6dinger (1926a-d) equation A

H~(r)

=

E~P(r)

(4.2)

where ~P(r) is the electronic wavefunction of the system of n electrons whose positions are given by r. At the set of nuclear coordinates R (N), tP(r) is a function not only of the electron positions but also of the electron spin. E is energy of the system. H is the Hamiltonian operator for the system, and represents the kinetic, T, and potential, V, energies of the system of n-electrons moving in the field of fixed nuclei. The kinetic energy term, then, represents the kinetic energy only of the n-electrons, and can be written as n

T-

vy

(4.3)

i=1

where h is Planck's constant, m is the mass of the electron, and V~ is the gradient operator applied to electron wavefunction i. The potential energy term

n

v-

N

-ZZ-+Z

n

•r o

/ (4.4)

describes the electrostatic interaction between the electrons and the fixed nuclei, or ion cores (an attractive interaction given by the first term in Eq. 4.4) and the electrostatic interaction between electrons (a repulsive interaction given by the second term in Eq. 4.4). The Hamiltonian described by Eq. (4.3) and (4.4) is incomplete,.omitting the effects of electron spin. These effects are typically small compared to the energies involved in determining surface structures, although, as we shall see in w 4.3, they may be important for some diamond surfaces. Also, because most insulators (and semiconductors) possess an even number of electrons (i.e., a closed shell), the ground state electronic structure is well described by pairing electrons of different spin. Consequently, the presentation of the various formalisms will be for closedshell, spin-paired systems, with no explicit consideration of electron spin. This Hamiltonian is also non-relativistic, and, hence, is not appropriate for systems which contain high atomic number atoms where the velocities of the electrons can approach the speed of light. In practice, it is the second term in Eq. (4.4), the electron-electron interaction, that makes Eq. (4.2) so difficult to solve. This is a many-body problem since the wavefunction depends upon the coordinates of all of the electrons in the system. This problem is often simplified by recasting it as an effective single-body problem, in which the total wavefunction for an n-electron system is expressed as a product

Theory of insulator su~ace structures

147

of n-wavefunctions, gt,,(r), (one for each electron in the system). This leads to a system of n one-electron SchrOdinger equations A

Hqt.(r) = E. gt.(r)

(4.5)

where E,, are the one-electron eigenvalues of tff,,(r). Each of these n-wavefunctions depend only upon the coordinates of a single electron (hence the name one-electron wavefunctions). It is in the specification of this effective single-particle potential (resulting from the one-electron wavefunctions) that the various quantum-mechanical methods differ. Before we discuss the differences between the methods, however, there are many features they have in common. Typically, the one-particle wavefunctions are expressed as a linear combination of some set of basis functions, ~i,

qt,(r) = ~_~ c,,,(r)

(4.6)

i

where c~ are the expansion coefficients (Lowe, 1978). Using Eq. (4.6), the system of n one-electron SchrOdinger equations given by Eq. (4.5) can be rewritten in matrix form as He = SeE

(4.7)

where E is the (diagonal) matrix of one-electron eigenvalues and c is the matrix of expansion coefficients (which define the eigenvectors). S is the overlap matrix, accounting for the spatial overlapping of the basis functions, whose elements are given by

So - I d?~(r) t~j(r ") dr dr"

(4.8)

Finally, H is the Hamiltonian matrix, whose elements are given by Ho- I r

HCj.(r ') dr d r '

(4.9)

The idea is to approximate the spatial character of the one-electron wavefunctions with as few functions as possible to minimize the computational expense. The proper choice of basis set is one of the critical aspects of practical computations. For example, the choice of an orthonormal set of basis functions can greatly simplify the problem since, for orthonormal functions, the overlap matrix is simply the identity matrix. Many different sets of basis functions have been used. They can be Slater-type orbitals, as in the empirical and semiempirical SCF-LCAO methods (Pople and Beveridge, 1970); contracted Gaussian functions as in the ab initio SCF-LCAO (Hehre et al., 1986; Pisani et al., 1988) and density-functional (Dunlap et al., 1990) methods; plane waves as is typical of the density-functional methods at all levels (Cohen and Chelikowsky, 1988); or finally a numerical basis derived from sophis-

J.P. LaFemina

148

ticated atomic computations (Delly, 1986, 1990). Considerable effort has gone into investigations on the construction of basis sets, and the literature available on this subject is enormous (Feller and Davidson, 1990). For the empirical tight-binding methods, the question of basis set is, in practical terms, a question of parameter set since these methods have no basis set p e r se. That is, basis functions are never actually specified, and Hamiltonian matrix elements are never computed. Instead, the "basis" is typically assumed to be a set of valence atomic-like orbitals which are orthonormal, eliminating the overlap matrix. The Hamiltonian matrix elements for this "basis" are then treated as adjustable fitting parameters. Once specified, Eq. (4.7) is solved for the eigenvector coefficients. Although the determination of these parameter/basis sets is dependent upon the details of the particular system under investigation, two particular parameterizations have emerged as the most widely used. The reader is referred to these for more information (Harrison, 1976, 1981, 1989; Vogl et al., 1983). Quantum mechanical computations can be all-electron (i.e., computations which take explicit account of every electron in the system) or valence-electron only. The valence-electron only calculations, assume that the valence-shell electrons dominate the interesting chemical and physical processes in the system. A further approximation is then made to the single-particle potential as these valence electrons are then considered to move in an effective, or pseudo-, potential that includes the interactions of the nuclei and core electrons. The pseudopotential can be empirical, semiempirical, or ab initio, depending upon the way in which it is derived. Empirical pseudopotentials are not self-consistent and are derived from fitting to experimental data. Self-consistent, semiempirical psuedopotentials assume a parametric form for the ionic core potential a priori and are then fit to either experimental data or the results of an all-electron computation. A b initio psuedopotentials, on the other hand, assume no a p r i o r i form for the potential, but construct the potential to most closely match the all-electron wavefunctions. More detailed discussions on the construction and use of pseudopotentials are available in Szasz (1985) and Zunger (1979). As stated previously, the differences between the various quantum mechanical methods lie in the way in which the many-body electron-electron interactions are reduced to effective one-particle, or one-electron, interactions. Perhaps the most straightforward way of illustrating these differences is to partition the one-electron electronic energy into component parts as follows: E E L I 'IR~NII -- F. lh,~N~l + rz IP~N~I + E X [I~ '" " I --KE-el t'" I ~ " ' t t L''~ I

1

+ rz IO~N~I " - " C t' ' ~ J

(4.10)

ERE_el is the one-electron kinetic energy and electron-ion attraction which results from Eq. (4.3) and the first term in Eq. (4.4). The remaining terms in Eq. (4.10) arise from the second term in Eq. (4.4), the electron-electron interaction. EH represents the two-electron (because it depends upon the coordinates of two electrons) Hartree (or Coulomb) interaction; Ex represents the two-electron exchange (or Fock) interaction, which arises from the indistinguishability of the electrons, which must be reflected by the wavefunctions; and Ec is the electron correlation energy. The

Theory of insulator su~ace structures

149

correlation energy corrects for the neglect of the correlated nature of the electron motion in recasting the many-electron problem into an effective one-electron problem. In the following subsections, the way in which the SCF-LCAO, density-functional, and tight-binding methods determine each of the terms in Eq. (4.10) will be presented, discussed, and contrasted. It is important to note that the presentation in the following sections will be for the ab initio variant of the SCF-LCAO and density functional methods. That is, the interactions that are included by the method will be assumed to be computed explicitly. Empirical and semiempirical variants, for which some subset of the interactions (i.e., Hamiltonian matrix elements) in the system are either neglected or parametrized, also exist for each method type. The details of these variants can be found in the following references (Pople and Beveridge, 1970; Parr and Yang, 1989; Cohen and Chelikowski, 1988). 4.2.2.1. SCF-LCA O methods Typically, SCF-LCAO methods simplify the computation of the electronic energy by restricting the one-electron wavefunction to a single Slater determinant (Slater, 1929). This requirement is one way to assure that the wavefunction is antisymmetric and describes electrons as indistinguishable particles. It also results in the neglect of electron correlation effects (although a limited amount of correlation is included since the method also obeys the Pauli exclusion principle). This level of calculation is called the "Hartree-Fock" level. In molecular systems, electron correlation effects can be included a posteriori in a variety of ways (Hehre et al., 1986). Unfortunately, these same methodologies are not applicable to periodic systems. (Some estimate of the effect of electron correlation on the energy of the system, however, can be obtained from electron density functionals.) At the Hartree-Fock level, then, Eq. (4.10) reduces to

IR(N~I

IR(N~IJ + 1= I.(N~IJ + E X/'" I.(N,I l'Ht'"

Evl. . t-- ; - EKE-el,'"

I

(4.11)

where all of the terms on the right-hand side of Eq. (4.11) are now evaluated explicitly. To do this, the one-electron wavefunctions are computed using the effective one-electron Hamiltonian operator (referred to in this context as the Fock (1978) operator) obtained from the variational principle A

F-

A

A

HKE_ei 4-

A

H. + Hx

(4.12)

where HKE_el represents the one-electron kinetic energy and electron-ion (nuclear) attraction, N

HKE-el =

87z2m

4rtc0

I r - R.I Ia=l

H , represents the two-electron Hartree screening (or Coulomb) interaction

(4.13)

J.P. LaFemina

150

HHVi(r) =

92

4nr

dr' y__, J=,

Ivj(r )1 vi(r) Ir-r'l

(4.14)

and Hx represents the two-electron exchange interaction dr' Z Vj (r)Vi(r )Vj(r) j=l Ir - r'l

Hx~i(r) = 4rr176

(4.15)

Note that the one-electron wavefunctions, ~i(r), appear in the definition of the Fock operator, requiring an iterative, or "self-consistent" solution. Once the one-electron wavefunctions are found, the energy terms in Eq. (4.11) can be evaluated as follows: n/2 E

Ii?(N)I_/ e2 ) KE-eiI'" I 4riCo

n/'2

Z i=1

n/2

2Ei--y___, Z i=1 j=l

(2dij - Kij )

(4.16)

where E; is the one-electron eigenvalue of ~i(r), Jij represents the two-electron Hartree (or Coulomb) interaction J,j - f dr dr"

I~(r)12 I~j(r')12 Ir - r'l

(4.17)

and Ki/represents the two-electron exchange interaction, 9(

9

t

i

K~j = ~ dr dr' ~d, r)vj (r )~dj(r)~di(r ) Ir - r'l

(4.18)

The remaining terms in Eq. (4.11) can be expressed in terms of the Coloumb and exchange interactions as b?(N)~ =

E"F" '

rt/2

n/2

"=

j=i

n/2

n/2

e2

4~Eo

(4.19)

and EXl/~N)It--

,

( e2 / 4rt~ ~

~_., Z

-Kij

i=1 1=!

yielding the total electronic energy as

(4.20)

Theory of insulator surface structures

9"ELt'" ,

/

4II:E:,, E

I 2Ei -- E

i=' L

151

q /

(2Jo- KO) I

(4.21)

One interesting aspect of Eq. (4.21) is that the electronic energy is not simply the sum of the occupied one-electron eigenvalues, Ei. (The factor of 2 accounts for two electrons, one spin up, the other spin down, in each occupied orbital.) This sum overestimates (or "double counts") the electron-electron interactions. This extra energy, given by the second summation term in Eq. (4.21), must be removed. To summarize, the SCF-LCAO methods neglect electron correlation effects by limiting the wavefunction to a single Slater determinant. The remaining energy terms of Eq. (4.11) m the kinetic, electron-ion, Coulomb, and exchange energies are then evaluated explicitly.

4.2.2.2. Density functional methods The density functional methods differ from the SCF-LCAO methods in that the electron density, p(r), is used as the variable of interest (Jones and Gunnarsson, 1989). It has been shown (Hohenberg and Kohn, 1964) that the ground state electronic energy can be expressed as a functional of the external (or nuclear) potential, v(r), and the electron density as,

/e2/(

E[p(r)] = 47t13o Tip(r)] + I p(r)v(r)dr + Vr162

(4.22)

The first term in Eq. (4.22), T[p(r)], is the kinetic energy. The second (integral) term is the electron-ion energy. The third term, Vee, represents all of the electron-electron interactions, including the Hartree, exchange, and correlation energies. In this way, Eq. (4.22) is simply a restatement of Eq. (4.5). The only constraint on the electron density is that it be "N-representable" (i.e., that it can be obtained from an antisymmetric wavefunction) and thereby have the electrons behave as indistinguishable particles. Most importantly, this electron density can be a "one-electron" density, that is, an electron density constructed from some set of one-electron functions. This is a remarkable result since it exactly transforms the many-body problem into a one-electron problem provided that the terms in Eq. (4.22) can be evaluated. This, of course, is the difficult part: the exact form of the kinetic energy and electron-electron interaction terms are unknown. The most commonly used approach is the Kohn-Sham method (Kohn and Sham, 1965). In this approach, the kinetic energy term of Eq. (4.22), T[p(r)], is replaced by the kinetic energy of a system with no electron-electron interactions, T.,[p(r)], but at the same ground state electron density of the original system (with electronelectron interactions). In this way the (newly defined) kinetic and electron-ion interaction energies can be computed from the one-electron eigenvalues, ei, of a system noninteracting electrons moving in the new external potential, v.,(r), of the noninteracting system,

15 2

J.P. LaFemina

n/2

L ' K E - e l I*"

J

4rter

T,[p(r)] +f "

9(r)v~(r)dr = "

4xE~

.=

2e i

(4.23)

with p(r') v~(r) = v(r) + S irr'i,,dr' +

Vxc(r)

(4.24)

The electron density is computed from the associated one-electron eigenfunctions, ~i(r), n/'2

9(r)-

2 ~

(4.25)

I~i(r)l z

i=1

and the Hartree energy (also referred to as LCAO methods,

E,I,RCN) I -

J[p(r)])

e2 o1 . -2-1 f Ir-r'l Jl o(r) ] - I 9(r)p(r')drdr'4rte

is computed as in the SCF-

(4.26)

This leaves only the exchange and correlation energies to be evaluated, along with the correction needed to account for the neglect of the electron-electron interactions in the kinetic energy computation. All of these terms are collected together as an effective one-electron term, referred to as the "exchange-correlation" energy, and given by Ex~,l-- i - T i p ( r ) ] - T, lp(r)! + V ~ l p ( r ) l - Jlp(r)l

f Vxcip(r.)lp(r)dr (4.27)

where Vxclp(r)] is the exchange-correlation potential. The specification of the exchange-correlation potential is, arguably, the most difficult aspect of applying the Kohn-Sham density functional method. One widely used form of this functional, based on the uniform-density electron-gas model, is 1/3

I/3

(4.28)

If or is set to 2/3 then Eq. (4.28) is the K o h n - S h a m exchange potential (Kohn and Sham, 1965), and contains no electron correlation effects. If or is set to 1, then Eq. (4.28) is the Slater exchange-correlation potential (Slater, 1974) which contains both average exchange and correlation interactions. The parameter o~ can also be

Theory of insulator su~. ace structures

153

treated as an adjustable parameter during the computation as is done in the so-called Xc~ methods (Parr and Yang, 1989). This form for the exchange-correlation potential has worked well for many systems. The development of new, sophisticated exchange-correlation potentials has also continued. The details of these developments can be found in the review of Salahub and Zerner (1989). To summarize, the density functional methods use the electron density, rather than the wavefunction, as the system variable of interest. As a result, it is possible to exactly transform the many-body problem into an effective one-electron problem. In this effective potential, the kinetic energy is computed for a system of non-interacting electrons. The kinetic energy correction (due to the fact that real electrons interact) is then lumped together with the exchange and correlation interactions into an effective, one-electron exchange-correlation potential. The Hartree energy is computed explicitly as in the SCF-LCAO methods.

4.2.2.3. Tight-binding methods The spirit of the empirical tight-binding methods (Slater and Koster, 1954) is simple: none of the terms in Eq. (4.10) are evaluated explicitly. Instead, the SchrOdinger equation (Eq. 4.2) is recast in matrix form (Eq. 4.7), and the elements of the Hamiltonian matrix (Eq. 4.9) are treated as adjustable parameters, fit at the high symmetry points of the first Brillouin zone to either experimental information, or the results of ab initio calculations for the bulk system. These parameters are then assumed to be transferable for use in computing the properties of surfaces. The range of these interactions is usually assumed to be nearest-neighbor or next-nearest-neighbor only, and the interaction matrix elements are assumed to have some parametric dependence upon the internuclear separation d (commonly a d -2 dependence for sp-bonded semiconductor systems) (Harrison, 1976, 1981). The assumption of transferability (from bulk to surface) for the Hamiltonian matrix elements will be valid provided that the charge density at the surface is not significantly different than the bulk. The success of this assumption (see reviews by LaFemina (1992) and Duke (1992)) for the covalently bonded semiconductor systems is an a posteriori justification for its use. For the more "ionically" bonded insulating systems, this assumption should be equally valid. The most important aspect of the empirical tight-binding methods, however, is that the electron-electron interactions are never computed, but are included empirically through the parameterization of the Hamiltonian matrix elements. This has consequences in the way the total energy is evaluated, since the electronic energy can only be expressed as the sum of the one-electron eigenvalues. As we have seen in the previous sections (Eq. 4.21), this sum overestimates the electronic energy by double-counting the Hartree and exchange interactions. Because this extra energy cannot be explicitly accounted for, it is usually lumped together with the nuclearnuclear repulsion, and the total energy is rewritten as E TOT

where

I~

J -" z..~b,~lll

j"~"

l

J

(4.29)

Ebs is the sum of the occupied one-electron eigenvalues (commonly termed

154

J.P. LaFemina

the "band structure" energy) and U is a pair potential (i.e., a potential that depends only upon the pairwise interactions between the atoms in the system) representing the nuclear repulsion and electron-double counting terms. Many forms have been proposed for this pair potential, but the most widely used is the form proposed by Chadi (1978, 1979, 1983, 1984), based on a harmonic, short-range, force-constant model, UIR t'."-(N~I J = ~ (U IE/j + U2c2)

(4.30)

i~j

where Eij is now defined to be the fractional change in the internuclear distance between atoms i and j. The constants U~ and U2 are determined by imposing the following conditions on the total energy functional (Eq. 4.29), 0EToT Ov

- 0

(4.31)

and c)2E,ro y

(4.32)

V - - - B OV 2

where V is the volume and B is the bulk modulus. What these conditions require, in essence, is that the minimum in the total energy functional (Eq. 4.29) occur at the correct bulk lattice constant, and that the curvature of the total energy functional near the minimum be correct. To summarize, in the empirical tight-binding methods none of the interactions in the system are computed explicitly, in contrast to the SCF-LCAO and density functional methods. Instead, they are included empirically, through the parameterization of the Hamiltonian matrix and the pair potential, U. 4.2.3. Classical potential models

The empirically determined classical potential models are, by far, the most widely used in the study of "ionic" insulator surface and interface structure (Colbourn, 1992). Typically, these potentials are central force, two-body potentials composed of three parts; a Coulombic interaction, a short-range interaction, and an interaction that accounts for the polarizability of the atoms. In some cases (for example, zeolites) a three-body term describing bond angle distortions has been found to be important and added to the potential (Colbourn, 1992). The Coulomb potential is simply the interaction between ions g and v with charges Z, and Zv

le /

Vc ...."'mb{R(N)}-- 4g~,,

N

Z laity

Ru v

(4.33)

Theory ofinsulator sud'ace structures

155

The short-range potential is typically given by the Buckingham potential which comprises a Born-Mayer repulsive, and a van der Waals attractive term (Baetzold et al. 1988; Born and Mayer 1932, Colbourn 1992) lh?(N)~= y__~ A~ exp

Vsh,,,. t

-

(4.34)

where A0v, P0v, and C~,vare empirical constants to be fit to either experimental data or the results of more sophisticated computations. Finally, the polarizability of the atoms is usually represented by a shell model (Dick and Overhauser, 1958), in which the atom is modeled as a spherical shell of negatively-charged electron density that is harmonically coupled, with a force constant k (also empirically determined), to a positively charged ionic core. The polarization occurs when a differential displacement, W, occurs between the core and the charged shell. The polarization potential is then given by N ....

=

Z

1

2

(4.35)

IJ~v

In theory the sums in Eqs. (4.33) and (4.34) are over all atom pairs in the system. In practice, however, the short-range potential (Eq. 4.34) is typically limited to nearest- or next-nearest-neighbors only (Colbourn, 1992). The long-range character of the Coulomb potential (Eq. 4.33), however, is more problematic since the Madelung (1909, 1910a,b) sum shown in Eq. (4.33) is conditionally convergent. That is, the answer you get depends upon how you truncate the summation. This can be avoided, however, by using the summation techniques of Ewald (1921) and Parry (1975, 1976). Nevertheless, the computation of the Coulombic interactions is easily the most computationally intensive process for these models. The question of how best to determine the empirical parameters that appear in Eqs. (4.33)-(4..35) is an interesting one. Equation (4.33) is included in this discussion since the issue of whether it is best to use formal charges (i.e., +2 for MgO) or some other "effective" charge remains open. There are essentially two approaches that can be taken in determining the parameters for these classical potentials: fitting to bulk experimental data (the more widely used) or fitting to the results of some "more sophisticated" computations for the bulk material. As in the empirical quantum-mechanical models, these parameters are then assumed to be transferable to the computation of surface properties. In the first approach, a variety of experimental information can be used, but typically the potentials are required to reproduce the correct bulk lattice parameters, along with selected bulk elastic constants, and phonon (crystal vibration) energies and dispersion curves (Colbourn, 1992). Even with the impressive agreement with bulk experimental data for these potentials, the results for surface properties remain very sensitive to both the exact form of the potential and the procedure used for its parameterization. This point will be discussed further in w 4.3.

156

J.P. LaFemina

The second approach, fitting the potential parameters to the result of "more sophisticated" computations, has also had mixed success. This is, in part, because the "more sophisticated" computations were not sufficiently accurate to provide an adequate potential. Typically, these computational results emerged from some variant of the electron-gas model, and had, at best, only moderate agreement with experimental data (Colbourn, 1992). Recently, however, there have been efforts to fit the empirical potentials to the results of ab initio computations (Colbourn, 1992; Kunz ! 988). While the preliminary results of these efforts are promising, much work remains to be done before this becomes a proven method for parameterizing classical potentials.

4.2.4. Computational approaches to surfaces There are three commonly used approaches for computing the properties of surfaces: Green's function (or embedding) methods, slab computations, and cluster models. As stated in the introduction to this section, the main difference between these methods is the boundary condition used to model the semi-infinite surface. An ideal surface is infinite (extending out forever) in the two spatial dimensions parallel to the surface. The surface is also infinite in the third dimension (perpendicular to the surface) but only in one direction (from the surface into the bulk). Thus the surface is said to be semi-infinite. The surface is also periodic in these directions (again only in one of the directions perpendicular to the surface). That is, some collection of atoms (called the surface unit cell) can, through translations, map out the entire semi-infinite system. The Green's function, or embedding, techniques treat the semi-infinite nature of the surface exactly (and as a result are the most complex). Slab models treat the infinite (in the two dimensions parallel to the surface) two-dimensional nature of the surface properly, but have a finite thickness in the third dimension perpendicular to the surface. Finally, the cluster methods model the surface with a finite set of atoms. In principle, any of these approaches may be used in conjunction with any of the methods, quantum or classical. In practice, however, the Green's function techniques are usually used with quantum mechanical potentials, while the slab and cluster models have been used with both quantum and classical potentials. One embedding (though non-Green's function) technique that has been extensively used with the classical potentials in the study of defects is the Mott-Littleton approach (Mott and Littleton, 1938). In this method, the system of interest is divided into regions that are treated at various levels of approximation. Figure 4.1 illustrates one example of this type of approach. In Region I, which contains the defect (or surface, or step, or whatever feature is being modeled) the atomic interactions are computed explicitly for a given potential. Region II contains ions whose positions are fixed, but are allowed to polarize in response to the feature in Region I (Baetzold et al., 1988). Finally, the material in Region III is treated as a continuum dielectric. The main point, however, is that only the atoms in Region I are treated with the full potential and allowed to relax in response to this potential. Region I typically numbers several hundred atoms, which is why this approach is almost exclusively used with classical potentials.

T h e o ~ ~?i' insulator su~. ace structures

157

Region III (Continuum Dielectric) F!g. 4. I. Schematic indication of the Mott-Littleton approach for the computation of detect structures using empirically-determined classical potentials. Region I contains the defect of interest, and the atomic interactions in this region are computed explicitly for a given potential. Region II contains ions whose positions are fixed, and whose distortion from the ideal lattice positions is determined by the defect induced polarization. In region III the material is treated as a continuum dielectric. Green's function methods (Applebaum and Hamann, 1976; Fisher, 1991; Ellis et al., 1991), are typically a more complicated form of embedding techniques, in which the properties of some region containing the feature to be modelled, are computed with the explicit inclusion of the boundary conditions, in this case the semi-infinite nature of the surface. The Green's function contains all of the information about the eigenfunctions and eigenvalues of the Hamiltonian. Recalling the one-electron Schr(~dinger equations (Eq. 4.5), /X

H ~ , , ( r ) - E,,~,,(r)

(4.36)

where ~,,(r) are the one-electron eigenfunctions of the Hamiltonian, H, and E,, are the one-electron eigenvalues of ~n(r), the Green's function is defined as /k

HG(r,r',E) - E G(r,r',E) - 8(r,r')

(4.37)

It can be shown that the solution of Eq. (4.37) is

G(r,r',E) - ~

vn(r)v~(r') E - E.

(4.38)

n

Using complex analysis, the eigenvalues (or the poles of the Green's function) and eigenvectors (or the residues of the Green's function) can be obtained. Hence,

158

J.P. LaFemina

anything that can be learned from the eigenvectors and eigenvalues can be obtained from the Green's function. The Green's function contains all of the information about all of the possible solutions to the differential equation (Eq. 4.37). The great advantage of this approach is that the relevant subspaces of the Green's function can be manipulated, isolated, and solved for. As a result, complex boundary conditions (such as a semi-infinite surface) can be dealt with. These methods are, by their very nature, computationally complex (a feature which has limited their application). For surface studies, however, they allow for the unambiguous identification of surface bound states and resonances; i.e., those states which have their electron density localized at the surface. In slab computations, a finite number of atomic layer, periodic within the layer, are used to simulate the surface (Fig. 4.2) (Hirabayashi, 1969; Lieske, 1984). The slab, by definition, has two surfaces (at the top and the bottom of the slab) which are typically made equivalent by symmetry. The slab thickness is a critical parameter for determining surface properties. Because the slab has two surfaces (top and bottom), the computed surface properties are converged when the addition of layers in the center of the slab does not affect the results. This critical thickness will also be different for different properties since not all properties converge at the same rate. For example, surface energies can typically be computed with slabs 3-4 atomic layers thick. For quantum mechanical potentials, the computation of surface states (states whose electron density is localized at the surface), typically requires on the order of 8 atomic layers. This minimum thickness is defined as the number of atomic layers needed to make the surface states on the top and bottom surfaces non-interacting. The equilibrium surface structures are computing by relaxing the atoms (i.e., computing the forces on each atom from the potential and then moving the atoms so as to minimize the forces) in the vicinity of the surface. Depending upon the complexity of the potential, this may include only the atoms in the top few atomic layers or every atom in the slab. Finally, in cluster computations, the surface is modeled by a finite cluster of atoms, with no periodic boundary conditions whatsoever (Tsukada et al., 1983). The advantage to this approach was that the absence of translational symmetry, i.e., having a finite system, greatly simplified the computation and allowed for the use Infinite

and Periodic./"" ,, . /

/

Surface

~

~

-

-

z ~ . L/"y

iiii~i~ii~iii~i.~.~.~ii~liiiiiii-xiiiiiiiiiiiii~ii!i~i!iiii~i Surface

Fig. 4.2. Schematic illustration of the slab model used to simulate a surface. The slab is infinite and periodic in the two directions parallel to the surface (x and y) and finite in the direction perpendicular to the surface (z). Note that the slab has, by construction, two surfaces which are typically made symmetry equivalent.

Theory of insulator su.rface structures

159

of ab initio quantum-chemical potentials, for which computer codes already existed. Over the past decade, however, ab initio density-functional and SCF-LCAO codes which explicitly incorporate the effects of translational periodicity, have been developed. Therefore, although clusters are interesting in their own right, their use as models for surfaces is no longer state-of-the-art. Other disadvantages to this approach include the presence of edge effects (i.e., artifacts in the computation resulting from how the edges of the cluster are terminated) which makes the examination of the surface electronic structure nearly impossible. 4.2.5. Comparison o f the methods

Of all the methods described in this section the use of empirical classical potentials (Eqs. 4.33-4.35) is, by far, the most widespread. The potential is sufficiently simple so that large numbers of atoms (on the order of hundreds) may be treated explicitly, with several hundred more treated implicitly through the use of Mott-Littleton type approaches. As a result, these potentials have been used to study a wide variety of complicated defects, grain boundaries, interfaces, stepped surfaces, and adsorbed surfaces. Moreover, these potentials have been used in molecular dynamics simulations to investigate the time-dependent nature of these systems. The primary disadvantage of these potentials is their obvious neglect of the quantum-mechanical nature of chemical bonding. No electronic states are computed, and no localized (either at surfaces or at defects) electronic states can be identified. In addition, the sensitivity of the computed results to the form and parameterization of the potential makes the general applicability of these potentials questionable. The empirical tight-binding method, however, is based upon quantum mechanics. It is also computationally efficient, and able to treat a large of number of atoms (on the order of one hundred atoms). It is also sufficiently transparent, being formulated in terms of atomic-like orbital interactions, so that the description of surface atomic movements in terms of how these interactions change at the surface is possible. This, in turn, permits the examination of surface reconstruction mechanisms across homologous systems. And because the method is parametrized to bulk optical and photoemission data, the extrapolation from bulk to surface properties can be both reliable and quantitative (Duke, 1992; LaFemina, 1992). The major disadvantages of this method are that it is not self-consistent, and therefore its predictions must be calibrated against more rigorous methods which explicitly include electron-electron interactions; and that the investigation of new systems requires the determination of new basis/parameter sets. This can be particularly troublesome when the system of interest contains interactions for which there are no bulk experimental data (such as in the study of adsorbates), or for which no usable higher-level computation exists. In these cases, the empirical d -z scaling law is usually invoked to obtain a first-order approximation to the parameters. The remaining methods, semi-empirical and ab initio local-density functional and SCF-LCAO, are all self-consistent. As such, however, they all suffer from the problem of being computationally intensive because of the need to explicitly compute the electron-electron interactions. This typically results in the treatment of

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J.P. LaFemina

systems with a limited number of atoms (on the order of tens of atoms). This implies either the use of a thinner slab, in which case the identification of the surface bound states and resonances becomes more complicated, or the use of a Green' s function technique or cluster calculation. The advantage to these methods is that, while computationally more difficult, they provide a first-principles determination of the surface structure independent of possible parameterization biases.

4.3. The structure of clean surfaces

4.3. I. Diamond (l I 1) surface Cubic carbon (i.e., diamond) has a band gap of approximately 5.5 eV, making it a good insulator. Diamond is also unusual because it exhibits covalent bonding, yet has a large bandgap. The study of diamond surfaces also provides an interesting contrast to the other Group IV elemental surfaces (i.e., Si and Ge) because of its propensity for forming multiple, Tt-bonds, which, as described in Chapter 6, are a common structural motif in semiconductor surface structures (Duke, 1992; LaFemina, 1992). Hence, understanding the surface and interface properties of diamond may yield fundamental and general insights into the factors which control semiconductor surface structure and chemistry (Pate et al., 1990). From a practical point of view, the structural and chemical properties of diamond surfaces are important to the growth of synthetic diamonds, and attempts to incorporate diamond interfaces into working electronic devices (Pate et al., 1990). The detailed study of diamond surfaces has been inhibited by a variety of experimental difficulties, including problems associated with the preparation of clean surfaces (Haneman, 1982). As most of the experimental and computational work has focused on the diamond (111) cleavage surface, we also will focus on this surface. Following cleavage and low temperature (900~ annealing, a (2x2) LEED pattern is observed which has been interpreted as resulting from a superposition of patterns from the three distinct (2x l) orientations (Pate, 1986; Sowa et al., 1988). As indicated above, the formation of 7t-bonds, in conjunction with sp2-hybrid ized chains, is common to the (2• structures on the (100) and (111) surfaces of Si and Ge (see Chapter 6). Moreover, similarities in the surface electronic structure as probed by angle-resolved photoemission spectroscopy and angle-resolved twophoton spectroscopy suggest a common (2x l) structure for the Group IV (111) surfaces (Himpsel et al., 1981; Kubiak and Kolasinski, 1989; Pate, 1986). The stability of these structures is easily understood using Prihciple 1 of w 4.1: Saturate the dangling bonds. The formation of these structures allows the surface atoms to saturate their valences by forming three bonds to neighbors in the surface and near

Theory o.f"insulator surface structures

161

surface region (sp2-hybridization), and then forming a double, or r~-bond, between the surface atoms. In addition, Principle 3, F o r m a c h a r g e n e u t r a l s u r f a c e , is satisfied trivially by this elemental surface. Two different r~-bonded chains, shown schematically in Figs. 4.3 and 4.4, have been proposed [originally for Si( 111 )] and investigated for the C(111 )(2x 1) surface (see Derry et al., 1986; Iarlori et al., 1992; Zheng and Smith, 1991). The primary difference between these structures is that they originate from different ways in which the cleavage process is proposed to occur. Figure 4.3 illustrates the Pandy rt-bonded chain which arises from cleaving the material in the (l 1 l) plane that contains a single bond per unit cell; the so-called "single-bond scission" (Pandy, 1981). If, however, the surface is formed by cleaving along the plane which contains three bonds per unit cell (the "triple-bond scission"), the resulting rtbonded chain is illustrated in Fig. 4.4 (Haneman, 1987).

C (111)- (2 x 1) Single-Bond Scission

Fig. 4.3. Ball-and-stick model of the C(111)2xl Pandy r~-bondedchain structure resulting from single bond scission. From Duke (Chapter 6).

C (111)-~2x 1) Triple-Bond ~cission

Fig. 4.4. Ball-and-stick model of the C(I 1l)2xl Haneman7t-bonded chain structure resulting from triple bond scission. From Duke (Chapter 6).

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J.P. LaFemina

The qualitative, n-bonded chain, nature of the C(111)(2• 1) surface is generally agreed upon. And based upon comparisons with Si(111)(2• and Ge(111)(2• surfaces it is generally believed that the single-bond scission model is correct (see Chapter 6 for a more complete discussion). The details of this structure, however, are contentious. In particular, whether (and to what extent) the n-bonded chain is either dimerized, buckled, or both dimerized and buckled (Iarlori et al., 1992). These issues arise because for the unrelaxed n-bonded chain models (i.e., where all of the C atoms in the chain lie in the same surface plane and all C-C bond lengths within the surface chain are equal to the bulk C-C bond lengths) the surface is metallic, contrary to the photoemission data (Himpsel et al., 1981; Pate, 1986) and to Principle 2: Form an insulating surface. Buckling of the chain results in a charge transfer between chain atoms and opens a gap in the surface state eigenvalue spectrum. Dimerization, in which the C-C bond lengths along the chain alternate in length (... short-long-short-long ...) will also make the surface insulating and is analogous to the Peierls distortion in one-dimensional systems (Peierls, 1955). Lastly, strong electron correlation effects (i.e., antiferromagnetic ordering of the chains n-electrons) can open a gap ion the surface state eigenvalue spectrum with neither buckling nor dimerization. Carbon behaves differently in small molecules (and polymers) from the other Group IV elements, favoring the formation of multiple bonds and disfavoring large charge transfers. Hence, an unbuckled and dimerized n-bonded chain is expected for the C ( l l l ) ( 2 x l ) surface. This is in contrast to the S i ( l l l ) and G e ( l l l ) ( 2 x l ) n-bonded chains which are thought to be buckled and undimerized (Duke, 1992; LaFemina, 1992). Experimentally, the structure of the C(111)(2• surface has been limitedly probed by both LEED intensity analysis (Sowa et al., 1988) and by medium-energy ion scattering (Derry et al., 1986). In both of these analysis only the Pandy n-bonded chain was considered, and the LEED analysis considered only a limited range of chain relaxations. The results of these studies are inconclusive, with the LEED analysis indicating a slightly dimerized n-bonded chain, and the ion-scattering results indicating a strongly dimerized chain. Computationally, this issue is no clearer. Ab initio density functional pseudopotential calculations (Vanderbilt and Louie, 1984a,b) find unbuckled and undimerized (and metallic) chains, while recent ab initio density functional pseudopotential molecular dynamic computations (Iarlori et al., 1992) find an unbuckled and dimerized chain. Semiempirical Hartree-Fock computations find either the unbuckled dimerized chain (Dovesi et al., 1987) or the buckled dimerized chain (Zheng and Smith, 1991) to be most stable. Finally, self-consistent tight-binding computations (Chadi, 1989) find buckled and undimerized chains, while non-selfconsistent tight-binding computations (Chadi, 1989) find unbuckled and strongly dimerized chains. In most of these studies, with the exception of the C-C dimer bond length, near-neighbor bond lengths are found to be within 10% of their bulk values, in agreement with Principle 5: Conserve bond lengths. Lastly, we briefly discuss the C(111)(2x l) surface in the context of Principle 3: Don't forget about kinetics. The Si(ll l)(2xl) and G e ( l l l ) ( 2 x l ) structures are metastable (Duke, 1992; LaFemina, 1992). They form because the atomic motions


163

necessary for their formation are kinetically accessible during cleavage conditions. Upon annealing at moderate temperatures (ca. 400~ however, these surfaces reconstruct to form what are believed to be their thermodynamically stable structures, the Si(l 11)(7x7) and the Ge(111)c(2x8). For the C(111) surface, however, the (2x 1) structure is believed to be the thermodynamically stable surface, provided the surface is free of hydrogen. This difference is also believed to be related to the propensity for C to form multiple bonds and avoid large charge transfers, since both the Si(l 11)(7x7) and Ge(111)c(2x8) structures involve large charge transfers between surface and near-surface atoms. In summary, while the qualitative, rt-bonded chain, nature of the C ( l l 1)(2x1) structure is established, and in accord with our five principles of surface structure, much work, both experimental and computational, is needed before the structure is unambiguously and quantitatively known.

4.3.2. Rocksalt (001) surface The rocksalt lattice is perhaps the most simple crystal structure. It is also one of the most prevalent, with the alkali and silver halides and pseudohalides; alkaline earth oxides, sulfides, selenides, and tellurides; and a variety of transition metal monoxides crystallizing in this lattice (Wyckoff, 1963). In addition, many other materials (e.g., II-VI and III-V compounds) exhibit the rocksalt structure as a high-pressure phase (Liu and Bassett, 1986). Unfortunately, the amount of detailed experimental or theoretical information available about these surfaces is scarce. Several excellent reviews by Henrich, (1983, 1985, 1989) are available for oxides while the state of surface structure computations has been recently reviewed by Colbourn (1992) and LaFemina (1994). In the bulk rocksalt structure the atoms are octahedrally coordinated. The most stable surface of this lattice is the nonpolar (001) cleavage face shown schematically in Fig. 4.5. The rocksalt structure materials present an interesting problem in the theoretical interpretation of the chemical bonding. These materials are presumed to be ionic, that is comprising a lattice of positively charged cations and negatively charged anions, held together by Coulombic forces. The reason for this conceptual framework is that it is not possible to construct two-electron bonds for these materials using only the valence electrons. If we consider MgO, for example, the Mg atom contributes two electrons per MgO unit while the O atom contributes its six valence electrons for a total of eight electrons per MgO unit. Each MgO unit in the crystal contains six bonds, leaving each bond with only 4/3 electrons. In the ionic bonding framework, however, the Mg atom's two electrons are transferred to the O atom giving it a filled shell, "Ne-like", electronic structure and leaving the Mg atom also with a "Ne-like", filled-shell electronic structure. In this framework, concepts such as "dangling bond" charge density would seem to be inappropriate. Yet, recently, these concepts have been shown to be useful in understanding a wide range of oxide surface structures (LaFemina, 1994). The reason is simply that in the limit of a "completely ionic" material (defined here as one for which two-electron bonds cannot be constructed) the "dangling bond" charge density

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J.P. LaFemina

Rocksalt (001) Surface

T

[OOl]

Fig. 4.5. Ball-and-stick model of the rocksalt (001) surface relaxation. The large open circles are anions and the smaller filled circles are metal cations. The surface rumpling shown has been exaggerated for clarity.

is localized on the anion, which has a filled shell. The results of numerous electronic structure computations bear this out (LaFemina, 1994). Consequently, for the nc}n-polar faces, the dangling bond saturation and insulating surface principles are satisfied. The non-polar faces are charge neutral and autocompensated (satisfying Principle 4). The driving force for surface relaxation, or reconstruction, then comes from the energy stabilization associated with the rehybridization of the anion charge density in response to the lowered coordination at the surface. This can only {}ccur, however, if the surface atoms can move into more electronically favorable hybridizations without significantly distorting local bond lengths (Principle 5). At the rocksalt (001) surface (Fig. 4.5) the atomic coordination is reduced from six to five, with each surface atom bonding to four neighbors in the surface plane and to one atom in the plane directly below the surface. The symmetry of the (001) surface dictates that any relaxation (preserving the (1• symmetry of the surface) of the surface atoms occur perpendicular to the plane of the surface. It has long been thought (Verway, 1946) that because the anions and cations in the lattice have different polarizabilities (i.e., the ease with which the charge densities surrounding the cations and anions are able to distort is different) the surface should rumple. This rumple would consist of the anions and cations exhibiting a differential relaxation perpendicular to the surface plane. Because the surface atoms are fivefold coordinated, they cannot move very far before some near-neighbor bonds become strained. As a result of this inability of the surface atoms to undergo approximately bond-length conserving motions, the relaxation is inhibited. Very few rocksalt structure materials have had the structure of the their (001) surfaces determined quantitatively. The MgO (001) surface, however, has been extensively studied, both experimentally and theoretically and the results for this surface are representative of this class of material. Table 4.3 contains a compendium of the experimental and theoretical results for MgO (001). The relaxation parameters in Table 4.3 are given only for the surface layer.

165


Table 4.3 Comparison of the structural parameters for the relaxed MgO(001) surface. R is the rumple, or differential displacement of the surface anion and cation in the direction perpendicular to the (001) surface. A positive R indicates the surface anion is displaced outward from the surface while the surface cation is displaced inward towards the surface (as shown in Fig. 4.5). C is the contraction (if negative) or expansion (if positive) of the first interlayer spacing. Units are percent of the ideal interlayer spacing (2.1/~,). R +3 +1 +5 to +9 +2 to +3 +ll +3 0 to +5 0 to +5 +2 _+ 2 0 0 to +5 +6 0 +8 _+ 1 +0.5 _+ 1 (a) (b) (c) (d) (e)

C

Method (year)

Reference

-2 0 0 -2 to 0 +1 -0.5 -3 to 0 -3 to 0 0+ 1 0+2 0 _+ 1 0 0 to +3 -15 + 3

TBTE (a) (1991) ab initio HF (b) (1986) Shell Model (1978) Shell Model (1979) Shell Model (1985) Shell Model (1985) LEED (c) (1976) LEED (1979) LEED (1982) LEED (1983) LEED (1991 ) RHEED ~d) ( 1981 ) RHEED (1985) He Diffraction (1982) ICISS Ce)(1988)

LaFemina and Duke Caus?a et al. Welton-Cook and Prutton Martin and Bilz Lewis and Catlow de Wette et al. Kinniburgh Prutton et al. Welton-Cook and Berndt Urano et al. Blanchard et al. Gotoh et al. Maksym Reider Nakamatsu et al.

Tight-binding total energy computation. Ab initio Hartree-Fock computation. Low-energy-electron-diffraction intensity analysis. Reflection high-energy electron-diffraction. Impact-collision ion-scattering spectroscopy.

The first thing that is clear from results in Table 4.3 is that the MgO(001 ) surface relaxation - - a surface rumpling and a change in the first interlayer spacing is small; typically on the order of a few percent of the lattice spacing. The second observation is that it is difficult experimentally to determine changes in surface atomic positions on this order; the uncertainties in the measurements being the same order of magnitude as the displacement. Finally, the data in Table 4.3 reveals the sensitive dependence of the computed surface relaxation parameters computed via classical potential (i.e., "shell") models on the form and parameterization of the potential (Colbourn, 1992), in particular the short-range part of the potential (Tasker, 1979). This is precisely the part of the classical potential that suffers most from the neglect of the quantum mechanical nature of the bonding. Consequently, there is a need for both reliable, quantum-mechanically derived, short range potentials for use in classical dynamics simulations of complex structures (such as grain boundaries), as well as for reliable, quantum mechanical computations of equilibrium surface structures. Of course, the need for quantitative, experimental information is the most pressing, and its gathering is likely to be the rate-determining step in the entire process of developing quantum mechanical descriptions of these materials.

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J.P. LaFemina

4.3.3. Rutile (110) surface The rutile crystal structure is tetragonal, comprising octahedrally coordinated cations and three-fold coordinated anions. The most widely studied futile materials are TiO2 and SnO 2. Both materials have wide-ranging technological uses for which surface interactions play a dominant role, including catalysis (Berry, 1982; Wold, 1993) and chemical sensor (G6pel et al., 1988; Semancik and Cox, 1987) applications. Consequently, the discussion in this section will focus on these two materials. The most stable, and well-characterized, rutile surface is the ( l l 0) surface, and we will direct the discussion in this section towards this surface. Unfortunately, rutile crystals, unlike diamond and rocksalt materials, do not cleave, but fracture (Henrich, 1985). As a result, the surface stoichiometry and structure are highly dependent upon the processing conditions used to prepare the surface (Principle 3). Low-energy-electron diffraction and Auger spectroscopy studies of the SnO 2 (110) surface indicate that, depending upon the annealing temperature, the surface O/Sn ratio can vary from 0.49 to 0.76, and the surface can display p ( l x I ), c( I x I ), p(4x2), p ( 4 x l ) and amorphous structures (de Fr6sart et al., 1983). Recipes, however, for preparing stoichiometric, nearly perfect SnO2 surfaces have been published (Cox and Fryberger, 1990). The (110) surfaces of rutile TiO2, on the other hand, forms stc~ichi~metric, p( Ix l) structures under a variety of experimental conditions (Henrich, 1985). As a result, much more work has been performed on single-crystal TiO 2 (I 10) surfaces. A schematic indication of the (110) surface is given in Fig. 4.6. Two different p ( l • ) terminations have been studied. The stoichiometric surface, shown in panel (a) of Fig. 4.6 has a stoichiometric (two oxygens for every tin) surface unit cell. The surface anions, commonly referred to as the "bridging" anions (see Fig. 4.6), are two-fold coordinated with one dangling bond. There are two kinds of surface cations in the surface unit cell, one of which is five-fold coordinated with one dangling bond. The other surface cation maintains its bulk-like octahedral coordination. The surface is insulating (Principle 2) and autocompensated (Principles I and 4). The topology of the surface, however, does not allow the atoms to move significantly away from their bulk positions without straining near-neighbor bonds (Principle 5). Consequently, the p( 1• ) relaxation is expected to be small, as was found by the tight-binding total energy computations of Godin and LaFemina (1993) for the stoichiometric SnO 2 (110) surface, illustrated in panel (a) of Fig. 4.7. No other computational or experimental surface structural determinations for these surfaces have been published. The second termination considered in the literature is commonly referred to as the "reduced", or oxygen deficient, surface. This surface results from the bombardment of the stoichiometric surface with argon ions, which preferentially removes the oxygen atoms (Henrich, 1985; Rohrer et al., 1992). p ( l x l ) LEED patterns have also been obtained from this surface, which is believed to comprise the stoichiometric surface minus the bridging oxygen atoms. This reduced surface termination is illustrated in panel (b) of Fig. 4.6. There are several problems with this simple interpretation of the reduced surface. In this interpretation half of the surface


167

[i 10][001] S= ['110]

(a)

[li~ [ool] v _~[ilo]

(b) Fig. 4.6 . Ball-and-stick models of the truncated bulk (a) stoichiometric and (b) reduced rutile (110) surface. Open circles are anions, filled circles are metal cations. From Godin and LaFemina (1993).

cations are four-fold coordinated with two dangling bonds, and half are five-fold coordinated with one dangling bond. The surface anions, however, retain their bulk, three-fold coordination, with no dangling bonds. As a result the cation dangling bonds are partially occupied, violating the autocompensation principle (Principle 4). As a result, the surface is not charge neutral, but carries a net negative charge since it is oxygen deficient. The question is then: Why is surface stable? The answer

168

J.P. LaFemina

[iI~[ooI] v _[iio] -

(a)

[iI~[ooI] v _-[iio]

(b) Fig. 4.7. Ball-and-stick models of the relaxed (a) stoichiometric and (b) reduced rutile (110) surface. Open circles are anions, filled circles are metal cations. From Godin and LaFemina (1993).

is that the structure shown in panel (b) of Fig. 4.6 is simplistic, and that the true structure of the surface which results from argon ion bombardment most likely contains subsurface defects which autocompensates the surface making it charge neutral. This idea is supported by experimental data which find an appreciable density of occupied states in the bandgap for the reduced TiO2 (110) (Henrich, 1985; Zhang et al., 1991) surface. Data for the reduced SnO2 (110) surface, however, shows no density of occupied states in the gap (Cox et al., 1988; Egdell et al., 1986). No computations of defected "reduced" rutile (110) surfaces have been published.

Theory of insulator sueace structures

169

4.3.4. Perovskite (100) surfaces The cubic perovskite crystal structure has the general formula ABC3 (e.g., SrTiO 3, or KMnF3). It comprises small A atoms surrounded by twelve C atoms, and larger B atoms octahedrally coordinated by the C atoms. The most studied surface is the (100), which, for the perovskite oxides, is widely used as a substrate for thin films (Hikita et al., 1993; Liang and Bonnell, 1993). Materials in this structure, like the rocksalt structure materials, are ionic in the sense that it is not possible to form two-electron bonds from the valence electrons of the constituent atoms. The (100) surface has two possible terminations, illustrated in Fig. 4.8. Both terminations are non-polar (Principle 4) and insulating (Principle 2). As a result the valences of the surface atoms are satisfied, saturating the dangling bonds (Principle 1). These stoichiometric, (1• terminations are typically the result of vacuum fracture, followed by annealing in oxygen and so depend upon the processing history of the sample (Principle 3). The exact nature of the surface relaxation or reconstruction therefore will depend upon whether the atoms can move into some electronically more favorable local bonding environment (due to their reduced surface coordination) while approximately conserving near neighbor bond lengths (Principle 5). The two (100) terminations are classified by the type of atoms which reside in the surface plane. The first termination, shown in panel (a) of Fig. 4.8 is the BC2 or Type I termination. It comprises five-fold coordinated B cations (four neighbors in the surface plane and one in the plane directly below the surface) with one dangling

Fig. 4.8. Schematic illustration of the ideal ABC3 perovskite (100) surfaces. Two {100} steps to (100) terraces are shown. The uppermost terrace represents the Type II AC (100) surface, while the second uppermost terrace represents the Type I BC2 (100) termination. Large open circles are type C anions, large shaded circles are type B cations, and small circles are type C cations. The degree of shading indicates the depth of the atoms below the surface plane. From Henrich (1985).

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J.P. LaFemina

bond, and four-fold coordinated C anions (two B ligands in the surface plane and two A ligands in the plane directly below the surface) with two dangling bonds. The second termination is the AC or Type II termination, and is illustrated in panel (b) of Fig. 4.8. It comprises eight-fold coordinated A cations (four neighbors in the surface plane and four in the plane directly below the surface) with four dangling bonds, and five-fold coordinated C anions (four A ligands in the surface plane and one B ligand in the plane directly below the surface) with one dangling bond. It is clear from the topology of these surfaces that there are no approximately bond-length-conserving motions of the surface atoms. Consequently, we expect the surface to undergo small atomic motion relaxations. Moreover, the symmetry of the surface dictates that ( I x I ) relaxations be restricted to motions perpendicular to the surface plane, hence we expect a rumpling of the surface analogous to the rocksalt (001) surface relaxation. This is consistent with recent reflection high-energy electron diffraction (RHEED) studies of the TiO2 (Type I) and SrO (Type II) terminations of SrTiO3 (100). The RHEED determined structures for both surface terminations are shown in Fig. 4.9. As indicated in Fig. 4.9, for both the TiO2 and SrO terminated (100) surfaces the relaxations comprises a movement of the surface anions out from the surface and a movement of the surface cations in towards the bulk. A similar relaxation has been computed by Reiger et al. (1987), using a classical potential model, for the (100) surfaces of KMnF 3 and KZnF 3. Because the stoichiometry and structure of these surfaces are dependent upon the sample processing techniques used to prepare them, a variety of structures a

o.lo A t--

1.95 + 0.07/~[ 1.95 + 0.05/~,m (a) Sr 9 9 Ti 0.16 A

1.95 + 0.10 ' ~ I

)

1.95 + 0.05/~ 1 (b)

Fig. 4.9. Ball-and-stick models of the relaxed (a) TiO2 and (b) SrO terminated (100) surfaces of perovskite structure SrTiO3. From Hikita et al. (1993).

Theory qfinsulator su~. ace structures

171

Fig. 4.10. Ball-and-stick model (top view) of the (2• ordered oxygen vacancy-Ti 3§ structure on the TiO2 terminated (100) surface of perovskite structure SrTiO3. From Matsumoto et al. (1992).

possible. For SrTiO3, a (2x2) structure has been identified on the TiO 2 terminated (100) surface and associated with the formation and ordering of surface defects of the type Q-Ti3+-O (Q: oxygen vacancy) as illustrated in Fig. 4.10 (Cord and Courths, 1985; Henrich et al., 1978; Matsumoto et al., 1992). The change in the charge state of the two surface Ti atoms is a direct result of surface autocompensation (Principle 4). The topology of this reduced surface still prohibits any bond-length-conserving motions of the surface atoms, so the relaxation is expected to be small (Principle 5).

4.3.5. Corundum surfaces The corundum oxides, led by corundum itself, ~-AI203, are perhaps the most technologically important class of oxide materials. Some are excellent catalysts, while cz-alumina is one of the most widely used substrates for the growth of thin metal, semiconductor, and insulator films for both basic research and microelectronic applications. The trigonal corundum lattice, M203, comprises cations, M, in a distorted octahedral environment, and oxygen anions in a distorted tetrahedral environment. Ilmenite, ABO3, is a related structure in which cations A and B replace the metal ions, M, equally. Of the corundum (or Ti203, V203, ~-Fe203) and ilmenite (LiNbO3, LiTaO3) oxides that have had their surface properties examined, only the surfaces of or-alumina have been studied in any detail both experimentally and computationally. The discussion in this section, therefore, will focus on the ct-alumina surfaces, although the conclusions concerning surface relaxations should be generally applicable. A variety of ct-alumina surfaces, exhibiting a variety of surface structures, have been studied experimentally (Guo et al., 1992; Henrich, 1985). These structures are

172

J.P. LaFemina

of course a function of the sample processing conditions (Principle 3). For example, the (0001) basal plane exhibits a (1• structure in both air and vacuum when annealed at low temperatures (

,.

:

,v = (N,) 2 el- ~Q,,)/4/t]

(7.13)

where (N~) is the average number of unit cells per domain. Equation (7.13) is a

278

E. Conrad

G a u s s i a n with a F W H M of QaFwHM= 2n: 0.939/(N,), w h i c h is slightly b r o a d e r than the sin 2 function. If the crystal is infinite, the interference function reduces to a series o f delta functions, ~3(Q) = 8 ( Q ~ a - 2rch) 8(Q J , - 2n:k) 8 ( Q z c - 2rtl) Thus the total scattered intensity, is only a p p r e c i a b l e w h e n the m o m e n t u m transfer vector is equal to (7.14)

Q = k f - k i = Ghkl

w h e r e Gj, kt is the r e c i p r o c a l l a t t i c e v e c t o r defined in C h a p t e r 1. (7.15a)

Ghk z -- ha* + kb* + lc*

and

a*-

2rt(b • )

2 rc(c xa) , - b * = ~ ,

a . (bxc)

2rt(axb )

c * = ~

a . (bxc)

(7.15b)

a . (b•

It is easy to s h o w n that a . a * , b . b * , and c.c* - 2ft.

9

9

9

9

9

9

9

9

9

9

9

i" 9

i l l i i i i '/ i J" i

9

9

9

9

9

9

9

9

9

" 9

9

Fig. 7.3. Ewald construction. For an incident wave vector, ki, ending at an arbitrary reciprocal lattice point, the Ewald sphere is constructed with radius Ikl centered on the origin ofk~. Since the momentum transfer, Q, must be equal to a reciprocal lattice vector, non-zero diffraction peaks will only be observed if the final momentum, kf, ends on a reciprocal lattice point that lies on the sphere.

Diffraction methods

279

The condition that the momentum transfer must equal G in order to give a measurable scattered intensity is known as the Laue condition for diffraction. Since Ghk I a r e a set of translation vectors in reciprocal space corresponding to the real space crystal structure given by R(hkl) (see Chapter 1), the Laue condition simply states that the maximum scattered intensity occurs at the reciprocal lattice points of the real space crystal structure! The Laue condition can be conveniently visualized by the Ewald construction. The first step in the construction is to draw the 3-d reciprocal lattice. The incident wave vector is drawn with its tip at any reciprocal lattice point. Because the scattered wave vector for elastic scattering has the same magnitude as ki, its tip must lie on a sphere of radius Ikel with its origin at the tail of k~ (see Fig. 7.3). This sphere is called the Ewald sphere. The Laue condition is satisfied whenever the Ewald sphere intersects a reciprocal lattice point; since by construction Q = G. Of course, the magnitude of the scattered intensity still depends on the crystal structure factor IF(E,20,Q)I 2 that modulates the interference function.

7.1.2. Diffraction from surfaces To extend the previous section to include diffraction from surfaces requires detailed consideration of the solid-probe radiation interaction. How a truncated infinite lattice influences the diffraction depends on whether the incident radiation is penetrating (X-rays and neutrons), attenuated (electrons), or perfectly reflecting (He atoms). Each of these cases will be treated separately in following sections but for now a general approach to surface diffraction, common to all of these techniques, will be discussed. To begin with, assume that the radiation is attenuated as it passes through the solid and that the kinematic approximation is still valid. Consider two cases; a surface derived from an ideally terminated bulk crystal and a surface whose top layer has a (2• reconstruction. The elastic mean free path, A, is first defined in the conventional way 1 = 1oe-~A),

(7.16)

where x is the path traveled through the solid. A takes into account all losses, elastic and inelastic, that remove particles from the beam. This will be discussed in more detail in w 7.4.1. Because of these losses, radiation penetrating deeper into the surface will be more attenuated and have a smaller contribution to the radiation collected at the detector. If the inelastic mean free path for the radiation is assumed uniform in the solid, the attenuation coefficient for the relative amplitude scattered from deeper planes can be defined in the following way. Consider a particle incident on the surface at an angle 0~ to the normal that scatters off the nth layer of atoms and emerges into an angle Of. It will travel a total distance through the solid of n(c/cosO~ + c/cos0f). Then the ratio of the amplitude scattered from the nth and (n+l)th layer is

An

-e

-c~.

(7.17)

E. Conrad

280

For simplicity, the scattered intensity from the (001) surface of a simple cubic lattice that has an ideal bulk termination is calculated first. Once again c is parallel to the surface normal and the basis vectors (a~,a2) lie in the plane of the surface. Starting from Eq. (7.6), the total amplitude scattered from the crystal is N I ,N 2-1

N z- 1

eiQ'(a'h+a2k) Z

~

A(Q)=

hk=O

1

O~I eiQ3cl--e i-~(Q2a~N2+Q'aN')

I=0

•

sin(l/2NiQlal) sin(l/2Qlal)

sin(l/2N2Q2a2) 1 - o~u~ e i NzQzc sin(l/2Q2a2)

1 - o~eiQzc

(7.18)

The crystal structure factor is assumed to be 1.0. The scattered intensity is

I(Q) =

sin 2 (1/2 NiQial) sin 2 (1/2 N2Q2a2) 1 + o~zu` - 2o~N~cos (NzQzc) sin2 (1/2 Qlal) sin 2 (1/2 Q2a2) 1 + 0:2 - 2cz cos (Qzc) "

(7.19)

For the case of perfect absorption, o~ = 0, only the top layer contributes to the scattered intensity, and I(Q) does not depend on Q~. If the crystal is again allowed to be infinite parallel to the surface (i.e., N~ and N2 ---) co). the condition for a maximum in the scattered intensity is relaxed from the 3-d Laue conditions. Instead the maximum diffraction intensity occurs for all values of Qz as long as the momentum transfer parallel to the surface, Qal, is equal to the surface reciprocal lattice v e c t o r Ghk Qj; -

Ghk = ha~ + ka'2 .

(7.20)

The reciprocal space picture with attenuation now consists of a series of surface diffraction rods perpendicular to the surface instead of points (see Fig. 7.4), with the position of these rods given by Eq. (7.20). The diffraction rods are also referred to as crystal truncation rods (CTR). This is because the interference function for a surface can alternately be derived by considering the surface as a product of an infinite solid truncated by a step function | (Andrews, 1985). The form of I(Q) derived from the step function is the same as Eq. (7.18) in the limit that N2 ---) oo and that cz < 1.0 (Robinson, 1986). The CTR terminology is most commonly found in surface X-ray scattering literature. The interpretation of Eq. (7.19) is that the periodicity perpendicular to the surface has been lost and, therefore, the third Laue condition, Qz c = 2rtl, is no longer strictly valid,.The normally sharp 3-d diffraction peaks remains sharp in directions parallel to the surface but are broadened normal to the surface because the attenuation only allows the probe radiation to see a finite distance into the bulk. This has essentially the same effect as a finite bulk crystal has on the 3-d peak width. The bulk diffraction points are elongated perpendicular to the surface so that all values of Qz are allowed, subject only to the condition given by Eq. (7.20). Using the Ewald construction, the diffraction condition for surface scattering occurs when the sphere intersect a rod.

D~ffraction methods

281

[0011 (20)

(00)

(10)

(io)

(20)

/ Or'/

.

_

~Qzl

0~ /Q

\ k

QII=Q1

[lOOI

Fig. 7.4. Surface diffraction rods normal to the (QI,Q2) plane. For the parallel-piped geometry discussed in the text, the rods are separated by 2trial in theal direction. The rods are indexed according to the notation in Chapter 1. Diffraction occurs when the Ewald sphere intersects a rod at Oil = 2r~h/a~.

The picture of diffraction rods does not change if the attenuation is allowed to be non-zero. The intensity along a rod, however, is modulated. In the simple case of a bulk truncated surface the modulation is given by the last term in Eq. (7.19). Relaxation, reconstruction, and surface defects will also affect l(Qz) and will be discussed later in this chapter. Assuming that the attenuation length is much smaller than the finite domains in the crystal, then N2 in Eq. (7.19) can be taken to infinity with little error. The result is that the intensity along the rod is modulated by [1 + 122_ 212cosQzc]-~ (see Fig. 7.5). As in the 3-d cases, the intensity is still peaked when Qzc is equal to 2rtl, i.e. at the Laue condition. This is a general result: the intensity modulation along the rod gives information on the vertical spacing of atoms in the surface region. Note that the width of the diffraction peaks along the rod (i.e., measured in the Q~ direction) are broader for smaller 12. This is one way in which the attenuation can be measured. From Eq. (7.19) the peak to background ratio, l(Qzc - O)/l(Qzc = nrt), is [(1+12)/(1-12)] 2. Remember that 12 depends on Qz as well through Eq. (7.17) so that calculations of 12 from the peak to background ratio must be used with some caution.

7.1.3. Diffraction from a reconstructed surface Consider the reconstructed p(lx2) surface of a parallel-piped crystal shown in Fig. 7.6. Every other row of atoms in the y direction has been removed. The scattered amplitude can be written by breaking the sum in Eq. (7.9) into terms that contain only bulk and only surface atoms.

E. Conrad

282

1.2 =0.0

1.0 0.8 .~

0.6

=

0.4

0.5

0.2 0.0 0

2

6 2r~

4

8

10

Qzc Fig. 7.5 The intensity distribution along the (00) surface diffraction rod for the (001) surface of a simple cubic crystal with finite attenuation. The diffraction intensity is still peaked at the momentum transfer corresponding to the third Laue condition.

Fig. 7.6. A p( I •

reconstruction of the (001 ) surface of a simple cubic lattice. The top layer has also been relaxed outward by Ac.

N t -

1

N z -

1

A(Q) = f ~_~ e' e~,,h ~ h=O

N I -

e i Q,a2k Z

k=O

1

~t eiQc,

I=(1

(7.21) N I -

1

N2/2-

I

+ f ~_~ e iQ~a'h ~_a e iQr2azk [ e i q ~ A c ] h=()

k=O

The s e c o n d term is a sum o v e r only the surface atoms. S q u a r i n g Eq. (7.21) and letting N 3 ~ oo gives the scattered intensity

D~ffractionmethods

283

1.8 A i l x 2 ) ' " (10) ro~ /,. \

f . . . . . 1.6 Bulk Terminated

", (lO)rod

N .,-,4

O

. ,...q

/,; ",,\~

1.4

(b)

1.2 1.0 0.8 (0 89

0.6 0.4

i

. . . . . . 2 4

o

6

8

10

Qz c Fig. 7.7. (a) The reciprocal space picture of the p( 1x2) surface of Fig. 7.6. (b) The intensity distribution along the (0,0) (solid curve) and ( 1/2,0) rods (heavy solid line) are shown. For comparison the intensity profile for a bulk terminated surface is also shown (dotted curve). The effect of the relaxation is to add a modulation to the intensity profile that slightly shifts the peaks. Data is for c = 2.5 ~, o~= 0.2 and Ac = 0.8 A.

sin ( 1 Q2a221

I(Q) =/Bulk

+ Isurf,~ + 41S,,-.,:~

sinl2Q2a21 (7.22a)

Qza2- Qz Ac -

cos

l+ot"

otcos

~-Qla,

2o~co

(Qzc)

- Qz[Ac c]

where 9 2

1

.2

1

/Bulk

s,n

12Q2a2J

1 + c~2 - 2o~ cos

(Qzc)

(7.22b)

284

E. Conrad

and

sin212N,Q,a,i sin2 l 1 - ~ Q22a2/ Isu~-~,cc=If 12 sin2(1Q,a,!. sin~(1Q22a2),

(7.22c)

Equations (7.22a-c) show that the total scattered intensity is the incoherent sum of the intensity scattered from the ideal bulk crystal (with a nonzero attenuation coefficient) plus the intensity scattered from the surface layer and an additional interference term between the two. The reciprocal space picture has twice as many rods in the a2 direction, which from Eq. (7.22c), are given by instead of 2nrt (see Fig. 7.7). The extra rods come purely from the surface term, l,,rf. This could have been anticipated since the surface reciprocal lattice vector is twice as small in this direction compared to the underlying substrate. These additional rods, associated with the reconstruction, are called The relative importance of these three terms in Eq. (7.22a) depends on the attenuation. For o~ near 1.0 (penetrating radiation) the ratio of the bulk to surface contribution (Eqs. (7.22b) and (7.22c)) is about equal to N 3 with the bulk rod widths determined by the size of the bulk crystallites (at least for large N3). Also, the interference term (third term in Eq. (7.22a)) only contributes at the bulk truncated rod positions, = 2nrt, with its intensity determined in part by o~. Since the interference term is zero everywhere along the superlattice rods, the intensity of the these rods as a function of is constant. This is a general result" within the single scattering model, the superlattice rods only contain information about the structure of the surface layer. Had two atoms per surface cell been included, Eq. (7.22c) would contain a term (proportional to cos[Q,Az], where A. is the vertical height difference between the two surface atoms). This term would modulate the intensity along the superlattice rods. In terms of analyzing diffraction data from a reconstructed surface one need only consider the atoms in the surface cell when comparing calculated diffraction spectra to the experimental intensity distribution measured along the superlattice rods. This interpretation is no longer correct if multiple scattering is included.

Q2a2=nrt

superlatticerods.

Q2a2

Qz

7.1.4.Thermaleffects So far the scattering intensity has been calculated assuming that the lattice is static and that even zero point motion is neglected. At elevated temperatures the atoms vibrate, so that at any instant in time an atom will be displaced from its T = 0 K position by an amount ui(T) (see Fig. 7.8). The position of the ith atom will be ~; = r, + u;, where r i is the static lattice position of the ith atom, Eq. (7.8). To calculate the scattered intensity remember that the ui's are changing in time and that the incident wave sees a time dependent distribution of atomic positions. To handle the time dependent structure it is common to assume that the collision time between the incident radiation and the lattice is short compared to a typical vibration frequency.

Diffraction methods

285

Fig. 7.8. Instantaneous position of an atom (filled circles) from their T= 0 K positions (r~) (open circles).

Then the lattice can be considered "frozen" during a single scattering event and the scattered intensity can be found by averaging I(Q) over a large number of such scattering events for different possible ui's. This is referred to as a thermal average (James, 1962). This assumption is valid for electrons and photons, but the longrange helium-substrate interaction means that it is questionable for atom scattering. Nevertheless, it is found that many of the results that will be derived below are applicable to atoms as well as electrons and photons. Assuming f = 1.0, the average scattered intensity can be written using Eq. (7.7) by inserting the instantaneous atomic position ~i = ri + u~ and taking the average N

l~ve(a) = ~

etiQCr'-r)l

(eliQ'Cu'-u)l)

(7.23)

ij

The first part of Eq. (7.23) is the T = 0 K intensity distribution lo(Q). The average can be written in a convenient form by expanding the exponential 1 )2 i (e iQ ~u,-u,i) = (l + i Q . (u i - uj) . 2! (Q " ( u i - ui) --~. (Q . ( u , - uj)) 3 + . . . -l = exp -2- ((Q" ( u i - uj))2)

(7.24)

Equation (7.24) follows from the fact that all terms to odd powers of Au must average to zero by symmetry. To proceed, the displacement of each atom is written as a sum over normal vibrational modes. The displacements have contributions from both surface and bulk phonons. Because the amplitude of bulk modes decays rapidly towards the surface, only penetrating radiation will scatter from atoms with significant amplitudes from these modes. Atom scattering will see vibrations almost entirely from surface modes. Regardless, the effect of both surface and bulk modes leads to qualitatively the same effect on the scattered intensity as well as on the diffraction lineshapes (McKinney et al., 1967). A derivation of the effect of finite temperatures on diffraction is outlined below assuming only surface phonons (Conrad, 1987). A general discussion of phonon diffraction effects can be found in James (1962) and a similar derivation including both bulk and surface phonons is given by McKinney (McKinney et al., 1967).

E. C o n r a d

286

In terms of surface phonons the instantaneous displacement of the ith atom can be written as (Maladudin, 1971)

Uq~i= E Uq~eq~ sin(q 9ri, + Oaq~t + ~qr,) e-e tqt,, c.

(7.25)

q~

Here K: is one of the two polarization directions in the surface plane that have a polarization vector eq~. UqK is the amplitude of the qrcth normal mode. The terms COq,, and 8q,~ are the frequency and phase of the qx:th normal mode, respectively. The momentum of a surface phonon, q, only has a component in the plane of the surface (i.e., q = q,). The spacing between layers is c, and ni is the number of the layer in which the atom i is located. The quantity c is a measure of the damping of the phonon vibrational amplitude into the bulk. The scattered intensity is calculated by substituting Eq. (7.25) into Eqs. (7.24) and combining with Eq. (7.23). While the calculation is straightforward, it is rather tedious and will not be presented here (see Conrad, 1987). The result is found to be 2 I(Q) = e -2M I,,(Q) + e -2M y_, Uq~ (Q . eq~)2 I,,(Q +_q) +...

(7.26)

qtr

where N

/,,(Q)- ~

e lio'r'-r')l.

ia

The Debye-Waller exponent, 2M(T), is defined as l

2M--}- ~

2

2

Uq,~ ( Q . e q O .

(7.27)

qw,

The diffracted intensity is the sum of two terms. The first term in Eq. (7.26) is the normal Debye-Waller effect. To first order the thermal vibration of each atom is independent of the others. In that case ( ( u i - uj)) in Eq. (7.24) can be replaced by ~2, the mean squared vibrational amplitude. The first term in Eq. (7.26) is also referred to as the zero phonon contribution since it is derived assuming that no correlations exist between vibrating atoms (i.e., no phonons). From equipartition theory ~2 must be proportional to T for T > | where OD is the bulk Debye temperature. The Debye-Waller exponent in Eq. (7.27) for a bulk solid is usually written as (Warren, 1990):

2M -

2B(T) sin20 ~2 ,

where

12h2T 2 B ( T ) - mkBO------~

(7.28)

In general B(T) depends on the crystal geometry through the different phonon propagation vectors. Also, the expression for B(T) in Eq. (7.28) will be modified

D~ffraction methods

287

near the surface, since the surface Debye temperature will be substantially different from the bulk. To first order then, the effect of atomic vibrations is to decrease the scattered intensity by a term proportional to exp(-T) without changing the shape of the diffraction peaks. Another way of viewing this result is that the uncorrelated vibrations act only as point defects so that the total number of scatterers remains the same. Since the diffraction widths are proportional to 1/N, the lineshapes remain constant. The second term in Eq. (7.26) is the contribution to the scattered intensity due to the creation or annihilation of a single phonon (two and three phonon effects are included in the higher order terms that are not shown in Eq. (7.26)). From equipartition theory the mean squared amplitude of a surface phonon is kT2~c

UqK 2 =~

t 2

1

-

(7.29)

mNcq~ q

where N' is the number of atoms per surface plane (each plane separated by c). It has been assumed that the dispersion relationship for the surface phonons is C0q,:= qc,i ~, where cue is the speed of sound of the qK:th mode. For simplicity let the scattered intensity from the static lattice, Io, be represent by a delta function rod (i.e. 1o- 8 ( Q , - G,) where G, is a surface reciprocal lattice vector). Substituting Eq. (7.29) into Eq. (7.26) and assuming an isotropic surface ()f area A gives I(Q) = e -2M 6(Q - G,.) + e -2M

kTQ 2 A v. c m n 2 N pc, 2q,, IG, + Q,I

(7.30)

Equation (7.30) shows that the phonon contribution to the scattered intensity adds a term proportional to Q2Te-ZM. More importantly this term broadens the diffraction lineshapes by contributing a tail that decays as I / Q , (see Fig. 7.9). The form of the decay is independent of temperature. Note that the ratio of the zero and one-ohonon intensities is proportional to T. The equation for the one-phonon contribution is only valid near a diffraction rod. As can be seen, it diverges at Q, = Gll because the phonon lifetime has not been included in the calculation. A similar expression for the phonon broadened lineshape is also obtained using bulk phonons as long as the intensity variations along the rod (i.e. l(Qz)) are not too dramatic (McKinney et al., 1967). Two important points should be made about one-phonon scattering. First, the calculation of the one-phonon scattered intensity was made independent of the incident type of wave. In other words the inelastic scattering contribution in the kinematic approximation is the same for electrons, X-rays, or helium atoms and is only a function of temperature and total momentum transfer. The phonon contribution can be minimized by taking diffraction measurements in a scattering geometry that minimizes Q. This will be discussed in w 7.4.2.

288

E. Conrad

I(Q u)

-- Qll Fig. 7.9. The zero (dotted line) and one-phonon (solid line) contributions to a diffraction lineshape due to surface phonon scattering.

The second important consideration is that at high temperatures the one-phonon scattering gives a significant contribution to the diffraction lineshape. This must be kept in mind when analyzing any particular diffraction profile. It will play an important role in the ability to distinguish various type of surface disorder as discussed in the next section. The relative importance of inelastic scattering can be estimated by simply knowing the value of 2M. Barnes et al. have estimated the total intensity scattered into one Brillouin zone for each contribution to the inelastic scattering; i.e., zerophonon, one-phonon, and multi-phonon terms (Barnes et al., 1968). The integrated intensity over the Brillouin zone, normalized to the total scattered intensity is found to be 1.0 "~~ 0.8 "~

0.6

'~

0.4

0.0

0

1

2

3

4

5

2M Fig. 7.10. The integrated intensity over the Brillouin zone of the zero-, one-, and multi-phonon contribution to the total scattered intensity.

D~ffraction methods

289

I z e r o ( 2 M ) = e -2M

lone(2M) = 2Me -2M

(7.31 )

lmum(2M) = e-2~e Mw- 1 - 2M) The three contributions are drawn in Fig. 7.10. Note that the one-phonon term maximizes at 2M = 1.0. Also note that all three components are nearly equal when 2M = 1.0. Even though the multi-phonon component is larger than either the zero or one-phonon term above 2M = 1.0, the zero and one-phonon terms are peaked near G,. The multi-phonon term on the other hand is roughly uniform throughout the zone. Therefore, near a diffraction peak the intensity measured in the detector is heavily weighted towards the elastic and one-phonon contributions.

7.2. Diffraction from disordered structures So far the discussion of diffraction has focused on a perfectly ordered array of two-dimensional unit cells. Perfect surfaces are, however, the exception rather than the rule in surface physics and defects such as vacancies, adatoms, atomic steps, and mosaics are usually present on real surfaces. If the unit cell structure is to be determined, it is at least necessary to be able to separate or account for changes in the scattered intensity due to the loss of long-range order. On the other hand, many interesting physical phenomena such as enhanced catalytic activity, the instability of surfaces at high temperatures, disorder transitions, and others (see further volumes in this series) are directly related to the defect distribution on an otherwise perfectly ordered surface. For both reasons one would like to have a method of analyzing both the concentration and distribution of these defects on surfaces. From w 7.1.1 there are two contributions to the scattered intensity: the crystal structure factor and the interference function. Each is sensitive to different types of defects. The interference function is sensitive to partially ordered defects that reduce the periodicity of the surface. These will be discussed in the next section. Defects which change the structure of an individual unit cell affect I(Q) through either the atomic scattering factor, fiE,| or through the crystal structure factor, F(E,| This type of disorder is referred to as a point defect (e.g., vacancies, adatom, or substitutional atoms). If such defects are assumed to be distributed randomly throughout the surface and are therefore not correlated with one another, the scattered intensity can be calculated from the average of I(Q). For the case of one atom per unit cell, the scattered intensity (from Eq (7.9)) is equal to N-I

N-I

N-I

i4

i=j

i~j

(7.32)

where the intensity has been broken into two sums, one over identical atoms with i j and the other with i c j .

290

E. C o n r a d

Consider the case of a surface with n random vacancies and N - n undisturbed crystalline sites. The vacancy form factor will be assumed to be zero while all other atoms have a form f a c t o r f 2. For the first sum, i = j , there are N-n pairs of identical atoms each contributing an intensity f2. F o r i ~: j there is a probability (N-n)/N of having an atom at site i and the same probability of having another atom at a site j. The total intensity becomes (Henzler, 1979; Cowley, 1984)

I(Q) = ( N - n)f 2 +

f2 Z

eiQ(R'-~}

(7.33)

i,~j

Equation (7.33) can be written as an unrestricted sum by subtracting the contribution from the N pairs of surface atoms, i.e., N[(N-n)/N]Zf"z. N-I

I(Q) = n(N - n) f2 + Uun 12 ~_~ e iQ, . , - ~) N

(7.34)

ij

The last term in Eq. (7.34) is the diffraction from a perfect sample but with a reduced atomic scattering factor. Because the interference function itself remains unchanged, the width of the diffraction peaks for the disordered surface are the same as for the perfect surface. The other effect of random point defects is the addition of a uniform background (the first term in Eq. (7.34) which does not depend on Q). As an example, consider the case of a cubic primitive lattice with f = 1.0. If a concentration of 10% vacancies is present in an ordered surface containing 1000 atoms, the peak intensity would be reduced by 20%. The background on the other hand, which is the true indicator for the presence of the vacancies, would only amount to 0.15 eV 2~ 0.25 ~ 0.002~ ~ 1% 0.25 eV 150-500 ,~

0.25 eV 500-20,000

can reduce the apparent acceptance angle, df~, of the detector over the geometric acceptance angle by a factor of 4. Because of the small entrance aperture in this type of detector ( 2 0 - 4 0 ktm), magnetic field requirements are much more stringent. Fields as low as 10 m G can move the diffracted beam out of the detector even when the electron energy changes by as little as 10 eV. Standard electron guns suitable for L E E D are commercially available with typical operating parameters given in Table 7.2. They can produce very large incident currents but have poor spatial properties, i.e., large beam diameters and large angular divergence. When high Q-resolution performance is required a more elaborate system is required. Three types of electron guns meet the higher standards of spatial coherence. Henzler has designed a gun that uses a BaO2 cathode and a magnetic lens to produce a small well collimated electron beam. This gun is c o m m e r c i a l l y available and is an integral part of a commercial Spatially Analyzed LEED system ( S P A L E E D ) (Gronwald and Henzler, 1982). The S P A L E E D system has a fixed sample and uses four pairs of electrostatic deflection plates to change the incident angle. A sample manipulator provides course positioning of a diffracted beam into the detector. Instead of mechanically scanning the detector, the S P A L E E D sweeps the diffracted beam across a fixed detector aperture by two sets of deflection plates as shown in Fig. 7.31. This arrangement allows a rapid 2-d peak profile to be collected.

D~ffraction methods

315

Sample

Deflection Plates

Electron Gun Channeltron I

Computer Fig. 7.31. SPALEED system of Gronwald and Henzler (1982).

Several investigators have developed low energy electron guns using field emission tip cathodes (Martin and Lagally, 1983; Williams et al., 1984). The object of the field emission design is to have a small electron source (~1 ~m). Starting from such a small source the beam can be demagnified with an electrostatic lens to improve the angular divergence and still keep the spot size at the sample less than 100 ~tm. Field emission sources require an extraction voltage greater than 1 kV meaning that the emitted electrons must be decelerated to typical LEED energies. This requires that the gun be refocused when the energy is changed. Martin and Lagally have found that periodic Cs coating of the field emission tip not only stabilizes the beam current but allows extraction voltages as low as a few 100 eV. Although the brightness of these sources was originally thought to be able to produce intense electron beams even for small beam diameters, the total current from the tip is not really a factor. The maximum current the gun can produce is limited by the space charge in the beam, which is only a function of the electron energy and the beam divergence half angle, 13 (Klemperer and Barnett, 1971) ]max(~lA) -- V3/21~2

(7.65)

Since it is the divergence and not the beam diameter that is usually the important parameter in Q-resolution (see w 7.4.1.5), /max is usually determined by [3. The constraint on 13for Q-resolution and the typical brightness of LaB6 and BaO2 makes these cathode materials equivalent to field emission sources for most LEED applications. Cao and Conrad have designed a high resolution unipotential electron gun using an indirectly heated LaB 6 cathode (Cao and Conrad, 1989). The cathode has a 15 ~tm diameter tip. The source image is demagnified in the same way used by field

316

E. Conrad

emitter designs. The advantage of this gun is that the unipotential design means that electron energy can be changed without refocusing. It is also inexpensive, relatively small, and does not require constant Cs recoating. 7.4.1.3. Qualitative L E E D

The most common use of LEED is to determine the symmetry of the surface structure. As already discussed in w 7.4.1.1, the spot pattern on the LEED screen is an image of a portion of 2-d reciprocal space (the size of this image is determined by the Ewald sphere diameter, see Fig. 7.26). Without any detailed analysis, this picture gives an immediate description of the surface symmetry. As an example consider the LEED pattern from clean Ni(001) and the same surface covered with a partial coverage of oxygen, giving a p ( 2 x l ) structure (i.e., a Ni(001)-p(2xl)-O in Wood notation). For the clean unreconstructed surface the lattice vectors are shown in Fig. 7.32. The reciprocal lattice vectors are a~ =

2n ^ a~

and

2n ^ a2 = ,-S-Ta2 lu21

(7.66)

The reciprocal lattice for the clean surface is shown in Fig. 7.33a (which is also the pattern observed on the LEED screen when the incident beam is at normal incidence). If oxygen is adsorbed on Ni(001) to form a p(lx2) pattern as in Fig. 7.32, the new lattice vector, b2, is twice as long in the a2 direction. The corresponding reciprocal lattice vector, b~, is therefore half as small as a2. This means that the diffraction pattern will have twice as many spots in the a~ direction (see Fig. 7.33b). In general when the surface has a lower symmetry than the bulk, more than one type of domain can exist. In the example of oxygen, both p ( 2 x l ) and p ( l x 2 ) overlayers are equally likely (see Fig. 7.32). The actual LEED pattern is just the incoherent sum of the individual LEED patterns from the different domains (see Fig. 7.33c).

Fig. 7.32. The Ni(001 ) surface (open circles)'with adsorbed oxygen (shaded circles). Two 90 ~rotated domains are shown. The (lx2) domain are shown adsorbed on 4-fold hollow sites, and the (2xl) domain on 2-fold bridge sites.

Diffraction methods

317

(01)

b~ ~ _ ~ (01)

9

'-'1

(oo) 9 (1o) .

9

9

9

(01) 9

9

p(lx2)

(oo) 9

9

9

9

9

9

9

(60) 9

9

9

(Ol) 9

9

9

9

9

9

9

9

9

9

a~ (lxl)

(0o)

9

9

9

9

9

9

p(2xl) (a)

(b)

(c)

Fig. 7.33. (a) the LEED pattern from an unreconstructed Ni(001) surface. (b) The LEED pattern for both a p(2xl ) and p(1 x2) structure. (c) The LEED pattern for a surface with both p(2xl) and p(1 x2) domains. Note also that the type of adsorption site does not change the symmetry of the reciprocal lattice. The IV profiles, however, will be different. To analyze a LEED pattern the first step is to identify the possibility of rotated domains. Once this is done, possible choices for the new surface reciprocal lattice vectors ( b ] , b 2 ) are identified. From these vectors the surface symmetry relative to the unreconstructed bulk surface can be determined. While this is simple in the system already described, it is sometimes extremely difficult with higher order periodicities. The transfer from reciprocal to direct space can be made more tractable using a matrix method (Ertl and Kuppers, 1985). Let a~ and a2 be the bulk terminated pr!mitive lattice vector. The primitive lattice vectors of the reconstructed surface, b l and b2, can be written as a linear combination of the a ' s b~ = m] la~ + m~a (7.67) b ~= m 2 la ! + m22a Likewise the reciprocal lattice vectors of the reconstructed surface can be written as a linear combination of the bulk terminated reciprocal lattice vectors. = m~a~ + ml2a 2

(7.68) b2

= m'z~a~* + m22a * 2*

It can be shown that the coefficients of the reciprocal and real space vectors are related:

318

E. Conrad

liml'm'21_ l (m*z2-m:'i det M" ~-m;2 m~l

where M* =

(m! "/ l m~2

(7.69)

Since the mij s are known from the diffraction pattern, the mij s can be determined. From the diffraction pattern for the p ( 2 • surface in Fig. 7.33, m~l = 1/2, mlz = 0, m21 = 0, and m~2 = 1.

7.4.1.4. Quantitative LEED Quantitative information about either the structure of the surface unit cell or the ordered arrangement of the unit cells requires a more detailed understanding of LEED. As discussed in w 7.1.2, information about the unit cell structure is contained in the intensity variations of the reciprocal lattice rods as a function of the perpendicular momentum. In order to use the experimental data, the relationship between experimental geometry and reciprocal space must be understood. The standard LEED geometry with a movable detector is shown in Fig. 7.34. With reference to Fig. 7.20, this geometry corresponds to X = - 9 0 ~ so that the ~ axis is in the scattering plane (it is assumed that the surface normal is co-linear with the axis). In this geometry the components of Q perpendicular and Q parallel to the surface are

Sample Normal

ilo

i

~

kf ,fkf

ki

Sample

(0o)

(0o)

(0o)

k2

"',

k

k2>kl

(a)

~

(b)

v

(c)

Fig. 7.34. (a) LEED IV profile, (b) thermal diffuse scan (or rocking curve), and (c) a detector scan that gives I vs. 20.

D~ffmction methods

319

Q• = k(cos0f + cos0i) (7.70) Q, = k(sin0e- sin0 i) where k = 2~/~, is the magnitude of the electron wave vector. In all diffraction experiments the quantity of interest is the intensity scattered as a function of the momentum transfer vector (or a component of Q). In addition to the IV profile already discussed in w 7.4.1.1 (see Figs. 7.26 and 7.34a), there are several other important types of scans through reciprocal space that are mentioned below. The first type of scan is called the thermal diffuse scan (TDS). In this geometry the crystal angle is tilted while the gun energy and detector position are kept constant (see Fig. 7.34a). The scattering angle 20 is fixed meaning that f(E,20) is held constant and does not affect the shape of the measured diffraction peak. Note also that the detector path is symmetric in Qll for the specular rod (0i = Of) and nearly symmetric for the off-specular rod. In some cases the contribution off(E,20) must be known to correct the actual diffraction line shape. This is done by scanning the detector angle (scanning 20) with 0~ and electron energy constant (see Fig. 7.34c). As the detector scans through 20, the momentum transfer vector is changing so that both f(E,20) and I(Q) contribute to the intensity variations. The effect of I(Q) can be eliminated by preparing the surface in a completely disordered state (either by sputtering, quenched deposition, or heating to high temperatures). The electron wavelength in Eq. (7.64) was calculated in vacuum. As the electron moves into the solid, however, it experiences the periodic potential due to the lattice. The spatial average of this potential is called the inner potential U. The wavelength of the electron in the solid is decreased 150.4 Z,(h) =

(7.71)

E(eV) + U

While the inner potential is a function of E, it is typically of the order of 5-15 eV. The inner potential has an important effect on LEED analysis. The component of electric field in the plane of the surface of the incoming radiation must be continuous across the surface. On the other hand the electrons are accelerated normal to the surface by the inner potential thus causing the beam to be refracted. This means that, while the momentum transfer parallel to the surface is the same inside and outside the surface, the momentum transfer perpendicular to the surface changes. This is illustrated in Fig. 7.35. Inside the surface the perpendicular momentum transfer is (Webb and Lagally, 1973) Q~, =

2____~_~{Ecos2 0~ + U] '/z + [Ecos 2 0f + U] '~} (7.72) 4150.4 At low energies or grazing angles (i.e., low Qz) the contribution to the momentum transfer from the inner potential is very large. This means that instead of in collecting J(Qz) in an IV profile, the measured signal is j (Qz). As an estimate of AQ z = Qzi,-Qz, consider a solid with U = 15 eV. A 100 eV electron incident on this solid

320

E. Conrad

(10)

(oo)

(I0)

Of

ki-in

y

v

QJi Fig. 7.35. The effect of the inner potential on the electron momentum measured inside (superscript "in") the sample. Solid arcs are portions of the Ewald sphere. Note that QII is the same with or without the inner potential.

at an angle, 0~ = 45 ~ will scatter into the specular rod with a Qz shifted by AQz = 1 ~-J. This is half of the reciprocal unit cell size (i.e., Ghk "- 2-3 ,/k-~) and cannot be ignored! 7.4.1.5. LEED resolution

The resolution function in the surface plane (AQll) and perpendicular to the surface (AQz) is found by using the standard scanning geometry in Fig. 7.4. Differentiating Eqs. (7.70) with respect to the incident and scattered angle gives AQI*i =

k(cos0f)Aot,

f ~

(7.73a)

AQII = k(cos0i)Act~

(7.73b)

AQ~ = k(sin0f)A~f

(7.73c)

AQ~z= k(sin0i)At~

(7.73d)

where the At~'s are those defined in Eq. (7.61). The contributions to At~ and Actf in a LEED apparatus are dominated by three terms; the angular dispersion in the incident beam, the angular acceptance of the detector, the finite size of the beam on the sample. The largest contribution to LEED's resolution function is Act i, and it is dominated by the divergence of the incident electron beam, Ate. The minimum electron divergence angle is usually determined by the space charge in the beam and is sometimes referred to as the source extension. As a rule of thumb, the more

D!ffraction methods

321

Detector

Fig. 7.36. Beam diameter elongation on the sample. collimated the incident beam is, the larger the spot size on the sample. Too large a diameter beam on the sample will increase Acxf as discussed below. The best L E E D systems have A~v-- 0.04 ~ although typical values are an order of magnitude worse (see Table 7.2). The AQ, resolution of the instrument can be improved by going to a grazing angle geometry to reduce the cos0f term in Eqs. (7.73a and b). Because the electron beam has a finite diameter, its projected size on the sample will depend on the incident angle and give rise to an error in the scattered angle (see Fig. 7.36). Collimating slits, such as solar slits used in X-ray scattering, in principle could remove this problem. However, strong electron scattering from the slit material and space limitations in the vacuum make them unusable in LEED. For a zero divergence incidence beam of diameter d and a point detector, Ao~fwill have a contribution from the spot size of cos (Or) d AOtBD= ~ --, COS(0i) D

(7.74)

where D is the distance between the sample and the detector. Equation (7.74) shows that AC~nDcan be minimized at normal incidence as expected. The detector aperture also contributes to 8orr. The acceptance angle of the detector is given by AcxD= s/D, where s is the effective aperture diameter discussed in w 7.4.1.2. Typical values are listed in Table 7.2. Assuming that the spot size and detector acceptance angle contribute to the response function as Gaussian variables, the total contribution to the instrument function will be the square root of the sum of the squares of each term (i.e., Ac~ = A~ZD + ACX~D). The total resolution function, including the finite energy resolution (Eq. 7.60a), is found by adding up the contributions to the resolution in quadrature. The combined resolution width of the system is 2

=

2+

LI,DJ LkD ;

2

+

kcos(0,)

l+ (~--~-j Q~

(7.75a)

J + (,-2-E-J Qz2

(7.75b)

J

r + ' ~'-~(0i) D

322

E. Conrad

7.4.1.6. Dynamic LEED and multiple scattering While LEED has historically been the most often used diffraction method for surface structural analysis, it suffers from one serious drawback: LEED is not accurately described by the kinematic approximation. The strong electron interaction with the solid necessary to insure surface sensitivity also leads to strong multiple scattering. Simply put, the plane wave description of the outgoing scattered wave is incomplete. Spherical waves diffracted near the surface in turn diffract from other scatters as they pass through the solid. Each multiple scattered beam contributes to the collected intensity. Therefore, structural determination of the unit cell using LEED must include a full quantum mechanical treatment of the problem. Complex computational techniques have been developed for this purpose. In general they require an assumed structure for the atoms in the unit cell. As a first guess, the diffraction data can be treated as kinematic. Using this first guess structure the diffracted intensity is calculated and compared to the experimental data. A slightly new structure is guessed from this comparison and the intensity recalculated. This procedure is iterated to a desired degree of accuracy. The process becomes extremely complicated for large unit cells. Surface cells containing more than a 10 atom basis set are rarely calculable. There has been a good deal of advancement in full multiple scattering treatments of the LEED scattering problem (Van Hove et al., 1986). Fast algorithms for handling the scattering matrix problem as well as perturbation techniques to optimize structural parameter searches allow larger unit cell structures to be determined. A complete treatment of dynamic LEED calculations is beyond the scope of this book. Instead, a common approach to these calculations is outlined below. It includes a discussion of the accuracy of these calculations both from the standpoint of critical assumptions in the calculations and on the errors associated with experimental data collection. The problem of calculating the scattered LEED intensity from a known atomic structure can be broken into two parts. First, a self consistent scattering potential including band effects, thermal vibrations, inner potential, and adsorption must be known. Second, given this potential, the scattering matrix for a single atom can be calculated. The transition matrix takes the incident plane wave and scatters it into a given diffraction beam. The greatest improvement in calculating LEED intensities has been in determining the scattering matrix. The basic problem is to solve the wave equation

i-~mV2+k21~(r)=O,

k2 = --~ 2m (E- V(R))

(7.76)

The difficulty in obtaining a solution to Eq. (7.76) is that the amplitude scattered from the ith atom depends not only on the incident plane wave but also on the amplitude of the outgoing scattered wave from every other atom in the unit cell. At the same time the amplitude scattered from all the other atoms in the surface depends on the amplitude scattered from the ith atom. This requires the solution of a set of self consistent equations.

323

Diffraction methods

There are several methods for solving for the scattered amplitude. Most of these are based on a two level solution. First the surface is divided into vertical planes (with their normals co-linear with the surface normal); each plane contains one or more atoms. The scattering matrix for a plane wave incident on each of these planes is then calculated. In the second step, the inter-planer scattering matrix is then calculated for the collection of planes, including a set of bulk planes. A powerful example of this approach is the Combined Space Method of Van Hove and Tong (1980). In this method a spherical wave representation for the scattering is used within the plane and a plane wave representation for the scattering between planes. In general the layer scattering matrix is the difficult part of the calculation. Several methods are commonly employed for its solution: Renormalized Forward Scattering (RFS) (Pendry, 1971), Reverse Scattering Perturbation (RSP) (Zimmer and Holland, 1975), or the Beeby matrix inversion (Kinniburgh, 1975). After the intensity is calculated from the model structure, the results must be compared with the experimental LEED I-V profiles. This means that the scattering matrix must be calculated for a set of energies between 10 and 200 eV. For accurate comparison the matrix must be calculated for a large set of diffraction rods. The comparison is quantified by use of reliability or R factors. In X-ray scattering the Rx factor is defined as (Robinson, 1991) Z [IFi'call 2 -IFi.exol 2 I

Rx = i

(7.77) Z IFi.~xpI 2 i

where F is the crystal structure factor defined below Eq. (7.9). In LEED the intensity modulations along a rod are much richer than in X-ray diffraction (in large part due to the multiple scattering). Information about the structure is contained in both the shape of a peak in the IV profile as well as in the position and intensities of their minima and maxima. To take this into account a LEED RL factor was developed by Zanazzi and Jona (1977) RL=A

' - l~pI " I~I " ,:,i - l"e,,pI I~I~,,, dE

II'r

Ima,, + I l'~xo I

(7.78)

where A = 37.04/j" le,,pdE and ' and " represent first and second derivatives, respectively. The weighting factor in this analysis is purely for normalization

I] = :b

lexo dE (7.79)

J Ic.,l dE Zanazzi and Jona have concluded that R L > 0.5 is a bad fit and R L < 0.2 is a good fit.

324

E. C o n r a d

Once the calculation and comparison is done for a trial structure, a new guess for the structure must be made, the calculation performed again, and a new comparison with the data is made to see if the new structure improves the fit. Since the computing time goes as the cube of the number of experimental CTRs, and because derivatives of the calculated IV profile must be made for the R factor determination, this task is extremely time consuming. This is especially true because no algorithm is available for iterating the guess for the next trial structure. Recently Pendry has developed a method called Tensor LEED analysis that provides a means to speed up the iteration process (Rous et al., 1986). Starting from a base structure, the full dynamic calculations are performed. Using the transfer matrix for the base structure, the scattering amplitude can be calculated for small perturbations of the atomic positions without recalculating the transfer matrix. The perturbation technique is good as long as atomic coordinates are not change by more than about 0.4 A,. The change in amplitude for N displaced atoms is N

~A (k,,,,. k~,,) = Y_, ~ i

Tiv (k,,,, ki,,)tz, &~t~L,(Sty)

(7.80)

LL"

where T~.v is the transfer matrix for the base structure. T,.,, (k,,,,, k~,,) = (k,,,,IVvV ( r - r,) I k~,~) (7.81) 8ZLL. (Sri) = e -iO" ~". 0 are allowed (referred to as open channels). For a soft wall potential the atom wave function can exist in the classically forbidden region of the potential. This gives rise to diffraction channels with k~ < 0 (closed channels). These additional channels can have a significant contribution to the total multiple scattering component of I(Q). The other drawback to the hard wall potential is that it does not take into account the attractive part of the well. In particular, under the correct energy and momentum conditions it is possible for the incoming helium atom to be temporarily trapped in a bound state of the helium-surface potential well. The coupling of the incoming wave to the scattered wave through these bound state resonances is very large especially at low momentum transfers, and gives rise to large dips and spikes in I(Q) that cannot be accounted for in a hard wall model. A full treatment of a coupled channel calculation is beyond the scope of this chapter. The method, however will be outlined. A 2-d periodic potential can be written as

(7.117)

V(R,z) = ~ vc(z) e i ~ "R G

If V(R) is known, the Fourier coefficients, vc, can be found from the inverse Fourier transform of the potential. Likewise the wave function is written as ~13(R,z) = ~ ~ol~(z) exp[i(Q + G). R ]

(7.118)

G

Substituting Eqs. (7.117) and (7.118) into the Schr6dinger equation (Eq. (7.76)) and integrating over the unit cell gives the set of coupled second order differential equations 2m

[k2 -- (Q + G)2] +

t~t:;l~(Z) = E VG'6'(Z) t~g'fI(z) G'

The boundary conditions for the soft potential are now

(7.119)

349

Diffraction methods

0,

when z --->+ oo and

v(R,z) =

k2
AXIS W (100) 2.0

MEV

HE+

-

u.l_ 6 0 0 >-

5OO 4OO 500

-

-

200 -

1.52

1.60

1.68

1.76

1.84"

ENERGY (MEV)

Fig. 8.23. Typical experimental backscattering spectra for incidence along a channeling direction and a non-channeling direction. (Reprinted from Feldman, 1094, with kind permission from Elsevier Science NL, Sara Burgerhartstraat 25, ]022KV Amsterdam, The Netherlands.)

the surface; i.e., channeling suppresses backscattering from the bulk. Thus, a measured energy spectrum of ions back scattered through the same 0 consists of a "surface peak" due to direct backscattering from atoms at the ends of strings plus a background due to ions scattered from deeper in the crystal. Figure 8.23 shows an example for 2.0 MeV He + ions scattered from a W(001) surface. The open circles are the back scattered spectrum for incidence along the [001 ] direction. The surface peak occurs near 1.8 MeV and its width is limited by the energy resolution of the detector. The area of the surface peak is directly proportional to the number of atoms per string visible to the ion beam. If the ion beam direction is a few degrees from an aligned direction, called a random direction, channeling cannot occur and bulk scattering is not suppressed. The solid dots in Fig. 8.23 show a backscattering spectrum measured along a non-channeling direction. R is comparable in magnitude to the thermal vibrational amplitudes P of atoms in real materials. Thus, more than one atom along a string will be visible to the incoming ions, even for perfect alignment, and the intensity of the surface peak is increased over the value expected for a rigid lattice. Stensgaard et al. (1978) used computer simulations to show that the increase is a universal function of p/R as illustrated in Fig. 8.24.

394

W.N. Unertl and M.E. Kordesch

6.0

Ni (110) - < 101> P I (111)

- c il6>

9- FOM

- Il - F O M

Si (001) - C O O l > - a b - SUNY 5.0 -

Si (OOl)

PI (001)

cO01 > -

-

W (0011-

W (001)-

-

- II -

Pt (111) - c l i O > Aq(ttt)

~4.0

9 - BTL

-

9

~/

8.TL

/

8.T.L

/

9 - CHALK RIVER

> -e,-

-

- ~ l l - O.R.N.L

PI (111) - C l 0 0 >

- q r - AARHUS

8/

k--

:r taJ

tJ

~- 3 . 0 D ~o

2.0

w

0!2

II

L

9

L ,l 0.4

L

0.6

t

I

0.8

~1

1.0

2.0

/o/R M

Fig. 8.24. Intensity of the surface peak (measured in atoms per row) as a function of the universal parameter p/R (Feldman, 1980).

The deviation of the scattering potential from the simple Coulomb form and the importance of thermal vibrations require that computer simulations be used for quantitative analysis of the data. This is done using Monte Carlo techniques based on an approach originally developed by Barrett (1971). See Feldman (1989) for additional references. The ion-solid potential is usually taken to be a linear superposition of the individual atomic potentials. Since the backscattering at high energies is dominated by events with very small impact parameters, any potential which accurately describes the inner shell electron distribution is suitable for determination of corrections to the Rutherford cross section. However, the precise flux distribution at the edge of the shadow cone is determined by the small angle scattering and is sensitive to the choice of potential. The Moliere potential is most often used. The probability that an atom on a string will extend beyond the shadow cone is calculated assuming that thermal vibrational amplitudes have a Gaussian distribution. The effects of correlated vibrational motion and variations near the surface must also be included in determination of p (Unertl, 1982). Monte Carlo

Direct imaging and geometrical methods

395

simulations of the double alignment geometry (Tromp and van der Veen, 1983) are additionally complex because each ion trajectory must be followed both on the way into the crystal and on the way out. Equipment for high energy ion scattering studies of surface structure is expens i v e - in the one million dollar r a n g e - and therefore is available at only a few laboratories. The ion accelerator must be capable of producing ions with energies between 0.1 and 2 MeV at beam currents in the 0-100 nA range with an angular divergence less than 0.1 ~ Beams with these properties can cause severe damage to the sample and precautions must be taken to minimize the extent and effects of the damage. Special shielding and licenses are required to use some types of beam because of radiation hazards. The accelerator must be interfaced to an ultra-high vacuum chamber. The crystal sample holder must have at least two rotation axes an azimuthal axis normal to the surface and a tilt axis normal to it. Angular orientations must be reproducible to at least 0.02 ~. In addition the sample must be accessible for cleaning and analysis by other surface sensitive techniques. The capability to vary the temperature over a wide range is essential for many experiments. In the single alignment geometry, the detector can be as simple as a stationary Si surface barrier detector. Although absolute measurements are possible because the scattering process is well understood, calibration standards are also available (L'Ecuyer et al., 1979). Once an apparatus has been calibrated using a standard, only relative measurements are required. In the case of the double alignment technique, the detector must have the capability to detect the spatial distribution of the back scattered ions and, consequently, the apparatus is more complex (Tromp and van der Veen, 1983; van der Veen, 1985). More detailed information about the experimental aspects of high energy ion scattering can be found in the references (Tesmer and Nastasi, 1995; Feldman, 1989). High energy ion scattering measurements have made many significant contributions to the understanding of surface crystal structure of clean and adsorbate covered systems. We briefly mention a few here. Ion scattering can accurately measure deviations from a simple bulk termination of a crystal and a number of important determinations of surface relaxations have been made (van der Veen, 1985; Bohnen and Ho, 1993). In most cases, these measurements are in good agreement with the results of LEED studies which not only gives confidence in the results, but provide a check on the reliabilities of the two techniques. Chapter 3 contains a more detailed discussion of surface relaxations. Very accurate determinations of the number of atoms displaced from bulk positions as viewed along various crystallographic directions is a quantity directly determined by high energy ion scattering. Thus, it has played a very important role in structure determination because knowledge of the number of displaced atoms places severe restrictions on possible structural models. Even though the ion scattering data by itself cannot uniquely determine the structure, it reduces the number of models that need to be tested by other techniques. Some of the most important examples are for surface reconstructions such as those on low index faces of noble metals, as discussed in Chapter 3, and more complex systems (Copel and Gustafsson, 1986; Fenter and Gustafsson, 1988). Adsorbate induced reconstructions


396

Z 0 I,-Z

Z

i

\

5 0 mV

n~ 0

r

_ 1

1

1

I

I

1.0 o

0.8

t,,,. ~

0.6

0 0 t.) z 0 om

0.4

U,,I ,~

0.2 0.0

m t~

1.0 ~ N

lad

09

"~"

08

a:: 0 Z

07 O6

'

4__~g___~i_~r_.~.~ ~,, L

400

l

450

I

500 TEMPER~,TURE

;

550 (K)

L

600

Fig. 8.25. Correlation of ion scattering data with CO coverage and work function changes during the hex-to-( I x l ) phase transition on Pt(001 ). (From Jackman et al., 1983.)

such as the H-induced displacive reconstruction on W(001) (see Chapter 13) and the CO induced h e x - t o - ( l x l ) transition on Pt(100) (Jackman et al., 1983) are important examples. Figure 8.25 shows typical data for the latter case. Interfacial structures developed during the initial stages of thin film growth are also accessible to high energy ion scattering as demonstrated by the early work of Feldman on the epitaxial growth of Ag on Au(111) (Feldman, 1989). This is likely to be one of the most important areas of application of ion scattering. 8.3.3.3. Low energy ion scattering Low energy ion scattering most often refers to ions with energies in the 1-10 keV range although both lower and higher energies are used. The technique is also often called ion scattering spectrometry (ISS). A historical review of the development of ISS as a useful tool for surface crystallography has been given by Rablais (1994). The shadow cone is much broader for low energy ions and more sensitive to screening by the valence electrons than is the case for higher energy ions. Thus, simulations of ISS scattering are more sensitive to the choice of scattering potential. Because the radius of the shadow cone is comparable to typical interatomic spacings, the ions are able to penetrate only the outermost few layers of a crystal and


397

channeling is usually unimportant. Also, because the scattering cross section and shadow cone radii are large, multiple scattering events are relatively much more important for ISS than for RBS or MEIS. Neutralization and sputtering are additional effects that are important for ISS. Because a low energy ion moves relatively slowly, there is a significant probability that it will pick up an electron and be neutralized during its interaction time with the surface (Woodruff, 1982; Souda et al., 1985; Beckschulte and Taglauer, 1993). Thus only a fraction of the backscattered ions will be charged. Many ISS systems, particularly older ones, use electrostatic analyzers that have high energy resolution but are unable to detect the neutral ions. More modern systems use time-of-flight methods that have lower energy resolution but are able to detect all of the back scattered ions. The neutralization probability is often significantly lower for alkali ions (Hagstrum, 1977) and this has lead to the widespread use of Li § and Na § beams, especially when electrostatic analyzers are used. The impact of a low energy ion with a surface also causes substrate atoms to be ejected (Robinson, 1981; Sigmund, 1981). In fact sputtering by low energy inert gas ions is one of the most widely used methods to clean a surface. Since sputtering can cause significant damage to a surface, special care must be used to insure that the surface structure is not significantly altered during the time required to record a data set. The simulation of ISS experiments has reached a rather sophisticated level in the sense that reasonable approximations to the scattering potentials are available and several standard computer codes, including ones that can be run on personnel computers, are available to carry out the necessary Monte Carlo simulations (Robinson, 1981 ; Teplov et al., 1994). Figure 8.26 gives examples of how the surface structure influences the back scattered intensity for the case of 2 keV Ne ions incident on a reconstructed Pt (110) (Ix2) surface. The data show the backscattered intensity for fixed_scattering angle as a function of the incidence angle for incidence along , , and azimuths (6 = 0, 35.3, and 90 ~ respectively) as shown in the plan view of the surface. The right hand part of the figure shows the scattering events that are responsible for each labeled feature (A, B, C, D) in the backscattered intensity profiles. The ions first scatter from a surface atom and, whenever the edge of the shadow cone intercepts another atom, a second scattering takes place causing the steep rises in backscattered intensity that are indicated by the open circles. Clearly substantial information about the geometrical arrangement of the substrate atoms can be extracted from the locations of the steep rises and a knowledge of the shape of the shadow cone. However, complete quantitative evaluation of structural models, including refinements of atom locations, requires modeling as described in the previous paragraph. Equipment for low energy ion scattering is compact enough to be incorporated as part of an ultra-high vacuum surface analysis system. The primary components are an ion source, a high quality manipulator capable of varying the incidence and azimuthal orientation of the sample, and a detector capable of energy or velocity analysis. Figure 8.27 shows the basic components of an instrument with time of

398


Fig. 8.26. Low energy ions scattering data from a Pt(110) (lx2) surface for a scattering angle of 149~ and three different azimuths 5. The ball models on the right show the scattering trajectories that cause the major features in the backscattering data. (Reprinted from Rablais, 1994, with kind permission from Elsevier Science NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.) flight detector for velocity analysis (Rablais, 1994). The ion source provides a collimated beam of a few keV energy and must usually be differentially pumped to maintain ultra-high vacuum in the sample chamber. If time of flight detection is used, the beam must also be pulsed.

8.4. Electron

microscopy

Electron microscopy and surface science have developed as separate fields due to several significant differences in their goals and methods. Electron microscopes evolved along a course dictated by image resolution; the result was high voltage, high intensity electron beam microscopes, with sample size and access dictated by the focal length of compact, solid lens bodies, and poor vacuum. The latter because commercial microscopes based on non-bakeable construction and materials with diffusion-pumped vacuum were firmly established for thirty years before ultra high vacuum technology became known or semi-practical.


399

c..cl,,,,,,!

Ip~. A

B

C

O

[ !

E C

F

/

PULSE ] GENERATOR

AZIMUTHAL f- 'l~

I

I~ TIME-TO -

ANGLE

B

ANg

I

I~

AMPLITUDE

ANALYZER

,NCII 'NT

sTART

CONVERTER

.'U.SE.CIGNT

]1

"~SC~T~L~R~NG

I~ ~OM,'UTE.

l

{ "

SWITCHING Lt,

. . . .

CIRCUIT .j

I STE,',;ING 3PLOTTERI ' MOTOR .

ICONTROLLERJ

Fig. 8.27. Schematic diagram of the major components of a low energy ion scattering system with time of flight detection. A is the ion source, B is a Wien filter to monochromate the beam, C is an electrostatic lens for focusing, plates D are used to pulse the beam, E is an aperture, F are deflection plates to steer the beam, G is the sample, H is an electron multiplier with an energy prefilter, and I is an electrostatic deflector used to remove ions from the detected beam. (Reprinted from Rablais, 1994, with kind permission from Elsevier Science NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.) Surface science began with ultra high vacuum and was primarily a spectroscopic science ("non-direct methods") until the advent of scanning Auger spectroscopy and other imaging spectroscopies and the STM and AFM (Duke, 1994). The need for a low penetration depth by the electron probe and the primary goal of surface sensitivity in surface studies has led to low energy electrons as probes or signal in surface spectroscopy and microscopy. Ultra high vacuum microscopes are essential for the comparison of microscopy and surface science results. The goal is for UHV electron microscopy "to show LEED, Auger, electron energy loss, transmission electron diffraction and transmission electron microscopy results all from the same surface" (Marks, 1991a). Today one might include STM or AFM, and substitute LEEM or REM for TEM. The UHV versions of these microscopes are relevant to surface science, and may bridge the gap between direct (imaging) and non-direct methods (spectroscopy and diffraction). The term "direct" imaging, used loosely in this chapter to distinguish methods that produce an image of the surface rather than a spectrum or diffraction pattern alone, is more often applied to "parallel" imaging as opposed to sequential imaging. In parallel imaging methods, the entire image is acquired at one instant: no scanning or sequential image composition occurs. Generally, only "direct" meaning "parallel" imaging methods will be examined here. A comprehensive survey of all types of surface imaging has been made recently by Hubbard (1995).

400


There are literally hundreds of books on electron microscopy. A personal selection from elementary to extremely specialized is: Wischnitzer (1962), beginners; Watt (1985), broad based introduction; Busek et al. (1988), comprehensive, including surfaces; Spence (1988), practical TEM and related techniques; Zuo and Spence (1992), microdiffraction; Bethge and Heydenreich (1987), application to solid state physics; Septier (1967) and Hawkes and Kasper (1989), electron optics for microscopy. The history of electron microscopy can be found in Ruska (1980) and Hawkes (1985). The journal Ultramicroscopy is a valuable reference, often devoting a single volume to one topic with state of the art discussion by experts. There are Internet newsgroups and list-servers, as well as World Wide Web sites from manufacturers, individual laboratories and professional societies for electron microscopy.

8.4.1. Image formation in the electron microscope Image formation in electron microscopes is analogous to image formation in light optics. Much of the same terminology and concepts are used in both. The detailed discussion of image formation is expertly handled in the book by Spence (1988). An extended discussion of contrast and image interpretation in the many variants of electron microscopy will not be attempted here. The basic ideas necessary to understand how the wealth of accumulated knowledge in electron microscopy can be applied to the imaging of surfaces will be touched upon below, with a guide to further reading. Yagi ( 1988, 1989) has reviewed the progress in surface imaging up to 1989 with several methods. Bonevich and Marks (1992) have made a short and readable progress report up to 1992. The idealized process of image formation in an aberration-free optical system where the optical transfer is linear and the small angle approximation is valid consists of a sequence of two Fourier transforms that combine the Fourier analysis of the object wavefunction into the diffraction pattern and the Fourier synthesis of the diffraction pattern into the image. In the wave optical treatment, the specimenbeam interaction results in an object wavefunction (written in one dimension for simplicity) ~o(X) at the exit or reflection plane of the specimen. The electrons described by ~o have either undergone scattering after transmission through the object (TEM), or reflection at glancing (REM) angles or normal reflection (LEEM, EEM), and have been altered by interaction with the notential of the object or object surface. The action of the objective lens is mathematically equivalent to a Fourier transform of gto that results in a Fraunhofer diffraction pattern at the back focal plane of the objective lens. The wave function in the back focal plane, ~ ( u ) , where u is in reciprocal space, is again transformed to the real space image at the Gaussian image plane: gtd(U) = F[~o(X)] and ~i(x)= F-l[~d(U)]

(8.27)

where F represents the Fourier transformation operation. The degree to which the


401

sample characteristics and the optical system allow faithful transformation of the object wavefunction into the image wavefunction defines the essence of electron microscopy. When non-ideal optics and specimen are considered, lens aberrations such as spherical aberration Cs, defocus value Af, finite aperture size, specimen and illumination characteristics must all be taken into account. For coherent, axial illumination, the amplitude ("Scherzer") contrast transfer function describes the phase shift of the electron wave due to spherical aberration and defocus, and is the first step in calculating image parameters such as resolution. For a scattering angle | = u)~, C(|

= exp[(-2xi/~,){ (Af/2)(~ 2 - (C,]4)(~4},]

(8.28)

defines the Scherzer contrast function, with Af the defocus. Thus in a real optical system: ~j.(u) = ~d(u)C(u)A(u) = ~d(u) exp(ix)A(u) = { (C,]2)~,3u 4 - A f L u z }

(8.28) (8.29)

A(u) is the aperture function and includes chromatic aberration, Cc, and beam characteristics and C(u) is the contrast transfer function. The intensity distribution in the image is proportional to [tt/il2, that of the diffraction pattern is proportional to I~d,I2. The contrast transfer function can be oscillatory and have several zeros. The conditions for optimum resolution are derived from the behavior of the real and imaginary parts of C(u), often expressed as the sine and cosine of X. The manipulation of ted' with an aperture in the back focal plane is important for bright and dark field imaging, high resolution TEM, and critical to surface specific imaging in some instances; these effects arise fiom amplitude contrast associated with the real part of the contrast transfer function. Phase contrast is associated with the imaginary part. For a full treatment with extensive mathematical detail and references to several other books, see Spence (1988).

8.4.2. Transmission electron microscopy 8.4.2.1. Physical principles of operation Surfaces are generally imaged in two ways in UHV-TEM: in plan view, using standard TEM methods such as dark field imaging or with surface diffraction beams or HRTEM to achieve sufficient contrast and resolution, and in profile. UHV-TEM instruments are further divided into microscopes where the surface analysis and preparation is made in the microscope, or made in an attached preparation chamber and subsequently transferred through UHV. Some instruments are only UHV in the sample region. Instruments where the surface is prepared and transferred through air are not considered.

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8.4.2.2. Resolution Resolution in TEM is directly related to the contrast transfer function (Eq. 8.28), since the optimum focus and aperture size for optimum resolution are calculated from this function. There are several choices for how, and in what plane, resolution is calculated. Sometimes a disk of confusion is used. A point object is imaged as a disk of diameter d. Here d = Cfc~ 3, w h e r e f i s the focal length of the objective lens, ~ is the aperture and C.~.is the spherical aberration. The necessary use of small apertures and focal lengths to increase resolution is clear from this formulation. For the optimum defocus and aperture values Afopt = -1.22(C,~,)'/2, O~op t "-

1.4(L/C,.)!/4

(8.30) (8.31 )

the often-derived value for point-to-point resolution is 8 = 0.66C1/49~ 3/4 .~.

(8.32)

Tsuno (1993) has reviewed experimental progress in reaching the resolution limits in TEM, 0.1 nm is now relatively common. Sarikaya 1992 reviewed "Resolution in the Microscope", covering TEM, REM and EEM, and several more.

8.4.2.3 Instrumentation UHV TEM instruments are usually commercial instruments that are special order, one-of-a-kind products, built using the combined expertise of the manufacturer on more commonly produced microscopes and the customer. The cost for one of these sophisticated microscopes is in the million dollar range, and can climb much higher as "extras" are added. Skilled operators and maintenance are also a requirement. The "home-made" UHV TEM is not competitive. A recent example of the state-of-the-art UHV TEM is shown in Figure 8.28; a photograph of the Hitachi UHV H-9000 microscope attached to the Sample Preparation Evaluation Analysis and Reaction (SPEAR) system at Northwestern University (Collazo-Davila, 1996). A schematic diagram of this system is shown in Figure 8.29. A list of microscopes from various manufacturers is given in Bonevich and Marks (1992). Usually options such as electron energy loss spectroscopy, Auger spectroscopy and X-ray fluorescence spectrometers can be attached to the TEM. A combination instrument called MIDAS is described by Liu and Cowley (1991).

8.4.2.4. Samples for surface TEM Three types of specimen are common in UHV TEM. Noble metals that can be prepared by heating and have low adsorption coefficients, such as gold. Semiconductor surfaces that can be cleaned by sublimation, and metals deposited on clean semiconductor surfaces. Metals are usually thinned mechanically, dimpled, and ion milled to 10 nm thickness or less for profile imaging.


403

Fig. 8.28. A photograph of the Hitachi UHV H-9000 microscope attached to the Sample Preparation Evaluation Analysis and Reaction (SPEAR) system at Northwestern University (Collazo-Davila, 1996).

Fig. 8.29. A schematic diagram of the SPEAR system. Surface studies are usually made at 300 kV or less, at higher voltages, displacement of the atoms by the beam becomes possible. Haga and Takayanagi (1992) have imaged individual Bi atoms on Si in HRTEM. The It(001) surface has been studied in UHV TEM (Dunn et al., 1993). Profile imaging is demonstrated on CdTe by Smith et al. ( 1991).

404


Fig. 8.30. UHV Transmission electron diffraction TEM diffraction pattern from a region with predominantly a single domain of the Si(l 11 )(5• structure. This pattern was indexed in terms of a centered (10x2) unit cell, thus the arrow points at a strong surface beam with (h,k)= (13,2). (Marks and Plass, 1995).

Fig. 8.31. Near Scherzer defocus, noise filtered, off-zone HREM image of the Si(11 l)(5x2)-Au surface. Clearly visible are two (arrows) rows of dark features which correspond to gold atoms. (Marks and Plass, 1995).

Figure 8.30 s h o w s a U H V T E M diffraction pattern and Fig. 8.31 a high resolution i m a g e from the Si(1 1 l ) ( 5 x 2 ) - A u structure ( M a r k s and Plass, 1995).

8.4.3. Reflection electron microscopy R e f l e c t i o n electron m i c r o s c o p y ( R E M ) is similar to T E M , so much so that the s a m e m i c r o s c o p e can be used for R E M as for T E M . H i g h voltage b e a m s are used as for T E M . T h e R E M m e t h o d bases its surface sensitivity on very s h a l l o w i n c i d e n c e and reflection angles; it is closely related to R H E E D , much as L E E M is related to


405

LEED. The utility of REM is also much like RHEED, surface steps and the morphology of growing surfaces are the ideal specimens for study. Image formation in REM is in principle the same as for TEM, most of the techniques associated with TEM have analogous REM cousins: bright and dark field imaging, interference fringes, SREM, convergent beam-REM, REM holography, and others. Because of the shallow incidence angle, REM images show severe foreshortening effects; only a narrow band of the image perpendicular to the incidence direction of the electron beam is in focus. The imaging of steps is possible mainly through phase contrast. Resolution is typically about 1 nm. Some sample preparation methods are similar to TEM studies, i.e. semiconductors cleaned by sublimation in UHV. An effective technique in REM, not strictly UHV, is to observe the surfaces of very fine melted wires. The cooled wires are terminated by recrystallized, facetted spheres suited to glancing incidence illumination and easy rotation and alignment of the incidence azimuth for diffraction (Lehmpfuhl and Uchida, 1993). A REM example from the study of such gold spheres is shown in Fig. 8.32, which shows the reconstructed Au(l 11) surface with its (23• superstructure. In this particular example, the surface reconstruction was

Fig. 8.32. (a) REM image of the reconstructed Au(ll 1) surface with (23• superstructure recorded with the intensity enhanced (666) rellection near the [ 112] zone axis. Periodicity of the structure is 6.6 nm. The horizontal line of exact locus is shown by the two bars. The dark area is another domain of reconstruction rotated by 60~, where the superstructure is not resolved because of foreshortening. (b) Diffraction pattern with fundamental and superstructure spots (Wang et al., 1992).

406


Fig. 8.33. (a) REM image of the reconstructed Au(100) surface showing in two domains the twisted ccll of reconstruction with 1.4 and 4.0 nm. The image was recorded with the intensity enhanced 10.00 rcflection near the [ 110] zone axis. (b) Diffraction pattern and with the fundamental spots and the superstructure spots of the two lattices (Wang et al., 1992).

maintained after passing the specimen through air by heating the specimen jn the microscope to about 200~ A technique specific to REM due to the b e a m - s p e c i m e n geometry is the excitation of surface resonance scattering parallel to the specimen surface. Intensity enhancement of surface Bragg diffracted beams that meet the resonance conditions (see e.g. Lehmpfuhl and Uchida, 1993; or Wang and Bentley, 1991 ) is observed and is beneficial for surface imaging. The image in Fig. 8.32 and the one in Fig. 8.33 were recorded using surface resonance intensity-enhanced reflections. Buried interfaces have also been imaged in an unusual type of REM (Spence, 1994). The more open specimen geometry in REM allows versatility in additional spectroscopy or measurement in these microscopes. STM and REM have been combined in one instrument (Lo and Spence, 1993). Spectroscopy for chemical analysis can also be added. Such a combination is shown for MgO surfaces in Fig. 8.34, from Crozier and Gajardziska-Josifovska (1993) with EELS of O and Mg included with the REM image. Due to the long working distance and possibilities for

Fig. 8.34. (a) A REM image from the freshly cleaved (100) surface and the accompanying RHEED pattern showing the (800) specular reflection at surface resonance. (b,c) REELS spectra and acquired from the (800) resonance condition from the freshly cleaved (100) surface showing the presence of oxygen (b) and Mg (c) K-edges. (Crozier and Gajdardziska-Josifovska, 1993).

Opposite:


407

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3.20

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i

b

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2.40 O r..)

1.60

0.80

0.00

I

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ENERGY LOSS [eV] x 104 5.50

I

i

i

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4.40 Mg-K 3.30 2.20 1.10

0.00

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ENERGY LOSS leVI Fig. 8.34. C a p t i o n opposite

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408

elemental analysis some forms of REM may be applied to in situ evaluation of film growth in the future. REM has been comprehensively reviewed by Peng and Kuo (1993) with a bibliography of the technique.

8.4.4. Emission electron microscopy Emission microscopy usually refers to electron microscopy using an immersion objective lens or "cathode lens". An acceleration voltage of several kV is applied between the specimen and first objective lens element, a distance which is commonly several mm. There are many types of emission microscope (see Kordesch, 1995): thermionic emission, photoelectron emission, soft X-ray and X-ray absorption near edge structure microscopes based on synchrotron radiation illumination, and low energy electron microscopy, using low energy electrons and commonly known as LEEM. Because LEEM is an electron probe microscopy, most of the discussion here will concern LEEM and Mirror Electron Microscopy (in the case when the specimen is not a single crystal). Other emission microscopes and variants will be discussed in w 8.4.5.

a. T H E R M I O N I C

d. SECONDARY

g. LEED

b. PEEM

r XPEEM

e. AUGER

f. MIRROR

h. D I F F R A C T I O N

J. D A R K FIELD

"~///)/ff/////~ j. FRESNEL

It. I N T E R F E R E N C E

L SPLEEM

Fig. 8.35. Contrast mechanisms available in a LEEM microscope (Veneklasen, 1992).


409

LEEM uses electrons in the energy range of 2-100 eV for specimen illumination; the LEEM geometry also allows observation of solid, thick single crystal specimens. These facts make LEEM images directly comparable to other surface science methods. The history of what is justifiably called the "Bauer LEEM" has been reviewed by Bauer himself (Bauer, 1994a). Results obtained with the several generations of this instrument have also been reviewed by Bauer (1994b, 1995). The design parameters and technical details of LEEM have been presented by Veneklasen (1992). Recent advances in emission microscopy in general are assembled in Griffith and Engel (1991 ).

8.4.4.1. Principles of operation The LEEM is similar in function to TEM, because both a diffraction pattern and real space image are formed by the optical system, but very different in practice. The illumination source and Gaussian image are on the same side of the specimen, making the geometric arrangement of a LEEM very different from the straight columns usual for TEM. Also, normal electron lenses function very poorly at the 2-100 eV used for LEEM. As noted above an immersion objective lens is used. The contrast in LEEM images is also very different from TEM, because it depends on diffraction and electron reflectivity of the specimen surface at very low electron energies compared to TEM. Data known as VLEED, very low energy electron diffraction, phenomena are related to image contrast in LEEM. Veneklasen (1992) has diagrammatically listed the contrast mechanisms available in LEEM; these are shown in Fig. 8.35.

8.4.4.2 Resolution in LEEM The resolution of emission microscopes is commonly calculated (as a rough estimate) with the Recknagel formula, which relates the resolution of a cathode lens to the ratio of the starting voltage of the emitted electrons (Vo) to the electric field strength at the specimen surface (E): ~5 = Vo/E. In this formulation, the resolution limit is due to the limitation of the accelerating field. The field strength is practically limited to about 100 kV/cm. The energy spread of the emitted electrons depends on the illumination method. The Recknagel formula was derived without including an aperture or control of the energy width of the illumination source. An energy distribution of the emitted electrons was assumed from thermodynamic relations of source temperature and a Maxwellian distribution appropriate to thermionic emission and photoelectron emission. In LEEM, contrary to what is expected for thermionic and ultraviolet photoelectron emission microscopy where the Recknagel formula applies, Bauer proposed that a narrow energy spread in the illuminating beam and an optimum aperture could allow LEEM resolution to approach the 5 nm range, and possibly less with further instrument development (Skoczylas et al., 1991). In terms of the discussion of TEM and REM, the resolution in LEEM is calculated differently, due to the inclusion of the electron emission parameters and the action of the homogeneous field between the sample surface and the lens that constitutes the "cathode" lens.

410


The aberrations of the objective lens system is added to these considerations to arrive at the total resolution limit. The principle, however, is the same: a calculation for optimum aperture and energy spread for a final energy of the source is made (See Bauer, 1994b). With the cautionary note that real and theoretical emission microscopes are very different, and that resolution calculations are also very different for different purposes (Rempfer, 1992), the theoretically attainable resolution in an emission microscope can be written as a combination of the aberrations of theelectric field used to accelerate the emitted electrons and the aberrations of the objective lens. The aberrations will depend on both the electron emission energy e Vo and the electron emission energy spread eAVo, and the accelerating field E = V/a, where a is the distance between the specimen and cathode lens. The spherical and chromatic aberration can be written (Engel, 1968):

8x s = [ Vo/E + Cs( VJV)3/2]sin317~,

(8.33)

6Xc = [ V~IE + Cc(V~IV)3/2](AVolVo) sin oq,

(8.34)

where ~0 is the electron emission angle relative to the surface normal, and Cs and Cc are the aberration coefficients of the objective lens (see Engel (1968) for a calculation for a magnetic objective lens in PEEM; Rausenberger (1993) for LEEM with magnetic objective lens; and Veneklasen (1992) and Rempfer and Griffith (1989) for electrostatic objectives). From Eqs. (8.33) and (8.34) the strong dependence of the resolution on emission energy and emission energy spread is clear. Rempfer and Griffith make explicit comparisons of emission microscope resolution with the Scherzer formula, the interested reader is referred to the discussion in Rempfer and Griffith (1989), and Rempfer (1992). The discussion in Rempfer (1992) is a clear and concise introduction to the general question of resolution in the electron microscope. Bauer has commented that resolution without sufficient intensity and contrast for focusing is useless (Bauer and Telieps, 1988). Detailed calculations of electron illumination intensity, expected image intensity and problems of specimen damage, etc., have been made by Veneklasen 1992 for most modes of LEEM operation.

8.4.4.3 Instrumentation There are less than a dozen LEEM instruments in the world today, about half are actively producing LEEM data. Bauer and Tromp offer commercial instruments, some are still "home-made". The commercial instruments are comparable in cost to UHV TEM instruments. It is necessary to separate the illumination and imaging beam paths. The LEEMs in use today are all based on a 60 ~ deflection of the incident and exit beams using a magnetic prism before and after they reach the specimen at normal incidence and exit angles (see Fig. 8.36). The LEEM optics operate at a modest 15-20 kV throughout most of the beam path, so that no special high voltage technology is necessary. At the sample, the electrons are slowed to < 50 eV; in the region of the sample and objective lens,


411

Fig. 8.36. A photograph of the IBM LEEM. The electron column is on the left, the imaging column on the right-hand side. The sample chamber and objective lens are at the top center, above the magnetic sector deflection magnet in the center of the photograph (Tromp and Reuter, 1992).

compensation of the earth's magnetic field, shielding of non-uniform static fields, and dynamic compensation of AC magnetic fields is often necessary. The magnetic lenses other than the objective are external to the vacuum system. An exception is Rempfer (Skoczylas et al., 1991), who has built a compact, electrostatic LEEM assembled completely on an a electron optical bench. Specimen chambers and sample manipulators can be adapted to LEEM from standard UHV equipment. The original Bauer-Telieps LEEM had a provision for flipping the specimen 90 ~ from imaging to "sputtering" position, other LEEM instruments have sputter guns at glancing incidence. Most LEEMs are equipped with sample heating to 1500 K or more, some with cooling to LN2 temperature, adsorbate dosing arrangements for UHV surface chemistry and a few with fast sample transfer or load lock capability. The IBM LEEM developed by Tromp is shown in Fig. 8.36. The "Y"-type of geometry is characteristic of present-day LEEM instruments.

8.4.4.4. Samples for LEEM The samples for LEEM are the same as for any standard surface science experiment: single crystal disks a few mm thick and 10-20 mm diameter, evaporated films, semiconductors, and some thin insulators; some of the insulators can be imaged by simultaneous illumination with UV light to promote photoconductivity. In order to make use of the diffraction contrast mode, single crystals or epitaxial film sub-

412


Fig. 8.37. (Top) An AFM image of an epitaxial Ag island on Si(l 11). A LEEM image of the circled region is shown (bottom). The field of view in the LEEM image is 4 lam. Arrows: a, interface steps, b, surface steps c, stacking faults. (Tromp et al., 1993). strates are necessary. It is also necessary for LEEM samples to be flat and smooth for best results, because the emission microscope has relatively poor depth of field, and irregularities on the surface cause distortions in the accelerating field. An image recorded with the IBM LEEM by Tromp (Tromp et al., 1993) is shown in Fig. 8.37. In this figure an AFM image is also shown with the LEEM micrograph. Both interfacial atomic steps and surface steps can be identified in the LEEM image of silver islands grown on S i ( l l l ) . This particular example is chosen from an already large LEEM literature because a direct comparison is made between the available information from LEEM, AFM and plan view TEM, including sub-surface features, purely surface phenomena and some features which were observed in real time, in situ with LEEM (features marked c) that resulted from cooling of the Ag film after growth was stopped. The interplay of several imaging techniques is elegantly displayed in this study.


413

While the majority of L E E M work to date has concerned silicon or metals adsorbed on other semiconductors or metals, some surface chemical reactions that can be understood only with real time in situ observation of both adsorbate distribution on a surface a n d the dynamics of the underlying surface crystal structure have also been investigated with LEEM. The oxidation of CO to CO2 on platinum surfaces is a problem of surface chemistry that has been investigated by several generations of surface scientists; it may also fall to direct imaging techniques as did the Si(111)(7x7) reconstruction to TEM and later STM. An immense body of literature has been produced with L E E M , P E E M and new optical microscopy techniques (see below) that have solved many of the questions related to the dynamics of the CO oxidation reaction. Rausenberger (1993) has studied the oxidation of CO on the Pt(100) surface in situ and in real time with L E E M and Mirror Electron Microscopy (MEM). Figure 8.38 shows a sequence of LEED patterns recorded during the reactive desorption of CO from the Pt(100)c(4x2)-CO surface with oxygen at 300 K. The

Fig. 8.38. A sequence of LEED patterns observed recorded during the reactive desorption of CO from the Pt(100)c(4x2)-CO surface with oxygen at 300 K. The reaction diffusion front passed through the illuminating electron beam during observation. The succession of LEED structures: (a) c(4x2), (b) c(4x2) and C~-2-x3"~-)R45, (c) (~-x3"~-)R45, (~'~-x'~-)R45 and (5xl), (d) (~-x'~-)R45 and (5xl), (e) (5xl) and (f) (2xl) are clearly visible. The electron energies for (a)-(e) are 27 eV, (f) 54 eV. (Rausenberger, 1993).

414


Fig. 8.39. A LEEM image at 23 eV (a,b) of a similar event as is in Fig. 8.38 is shown with a schematic representation of the surface structures (c,d) that give rise to the LEEM image contrast. (Rausenberger, 1993). reaction diffusion front passed through the illuminating electron beam during observation. The succession of c(4x2), c(4x2) and ('#-2-x3~r2-)R45, (~-x3"~r2)R45, (~2-x'(2-)R45 and (5x l), and other complex structures is observed over several seconds as the front passes. In Fig. 8.39, a LEEM image of a similar event is shown with a schematic representation of the surface structures that give rise to the LEEM image contrast. A M E M image of a reaction diffusion front crossing through the field of view is shown in Fig. 8.40. The determination of CO structures involved with the reaction diffusion front allowed the interpretation of previous PEEM results that showed work function contrast in the reaction diffusion front, and the selection between models of the CO-oxidation reaction. In a related measurement on Pt(110) (Rose et al., 1996), the spiral patterns that have become well known from PEEM results (Ertl, 1991; Rotermund, 1993) have been imaged with mirror microscopy in a LEEM-type instrument. Spiral reaction fronts that were pinned to surface d~fects and travelling spirals were observed in real t i m e , in situ, at partial pressures of the reactants of 10 -5 Pa and temperatures of 4 3 0 - 4 5 0 K. An example of this imaging mode is shown in Fig. 8.41.


415

Fig. 8.40. A MEM image of a reaction diffusion front crossing through the field of view, similar to that in Figs. 8.38 and 8".39. (Rausenberger, 1993).

Fig. 8.41. A sequence of MEM images showing an elliptically shaped target pattern during the catalytic oxidation of CO on Pt(110). Dark bands represent CO-dominated areas, light bands are O-dominated areas. (Rose et al., 1996).

416


8.4.5. R e l a t e d m e t h o d s

There are several direct and scanning imaging methods that may yet prove to be invaluable to surface scientists. In most cases, the beneficial aspect of the imaging mode may not be purely resolution. In some microscopies, the compatibility with reaction conditions such as high pressure and temperature may be deciding factors; possibly the detection of a single monolayer or less with reduced lateral resolution would be sufficient for some purposes. Figure 8.42 shows a PEEM image of a bulk-diffusion mediated surface reaction-diffusion front on Mo(310) at elevated temperature, 950 K (Kordesch, 1995b; Garcia and Kordesch, 1995). In principle, PEEM can achieve resolution in the lateral direction similar to or better than that of LEEM (because the optics involved are identical) (Kordesch, 1995a). Practically, however, a sufficiently intense source must be provided that allows high magnification PEEM and image acquisition in a reasonable time frame. Polycrystalline surfaces which do not fully exploit the LEEM contrast mechanisms may also be observed in other emission modes. Some very new optical methods, based on ellipsometry, have also been reported (Rotermund et al., 1995), in one case operating at atmospheric pressure. Figure 8.43 shows a Reflection Anisotropy Microscopy image of the CO oxidation reaction on Pt(110). There are new imaging methods based on synchrotron radiation that can detect magnetic layers (Stoehr et al., 1993), and of course spin polarized low energy electron microscopy has also been realized (Pinkvos et al., 1993). Spectromicroscopy is another area of growth, where an image is passed through the microscope with an energy analyzer, or microspectroscopy, where a small region of the image is analysed with a non-imaging method. Depth of field in rough specimens may require a scanning probe. Scanning LEEM (Kirschner et al., 1986) and UHV

Fig. 8.42. A PEEM image of carbon-sulfur deposition on oxygen covered Mo(310) at 950 K, from 10 -6 Torr 5% methane in hydrogen. The field of view is 260 lam. (Kordesch, 1995b).


417

Fig. 8.43. A set of Reflection Anisotropy Microscopy images recorded during the catalytic oxidation of CO on Pt(ll 0), partial pressures: oxygen 4 • 10-n mbar, carbon monoxide 6.2 • 10-5 mbar, T = 494 K. These conditions are similar to those in Figs. 8.39-41. The image sizes are 3.1 • 3.9 ram2! (Courtesy of H.H. Rotermund.)

scanning low energy electron microscopy (Muellerova and Lenc, 1992) are also being investigated. There is also an effort under way to correct the aberrations in emission microscopes, pioneered by Rempfer (Rempfer and Mauck, 1992; Rempfer, 1990), and developed further by Tonner (Tonner, 1990), Rose (Rose and Priekzas, 1992) and Engel in Berlin. These efforts are aimed at increased resolution, but also for increased intensity at synchrotron sources by enlarging apertures while not reducing resolution. Acknowledgements:

M E K would like to thank Dr. Wilfried Engel of the Fritz Haber Institute in Berlin for many helpful discussions, and those who have generously provided figures for this work: A.M. Bradshaw, EA. Crozier, W. Engel, M. Gajdardziska-Josifovska, G. Lehmpfuhl, L.D. Marks, B. Rausenberger, H.H. Rotermund, R.M. Tromp and Y. Uschida.

418


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Part III

Structure of Adsorbed Layers


CHAPTER 9

Chemically Adsorbed Layers on Metal and Semiconductor Surfaces

H. O V E R Fritz-Haber-lnstitut der Max-Planck-Gesellschaft Faradayweg 4-6 D-14195 Berlin, Germany

S.Y. T O N G Department of Physics and Laboratory of Surface Studies University of Wisconsin-Milwaukee Milwaukee, WI 53201, USA

Dedicated to Prof. Dr. G. Ertl's 60th birthday



425

Contents

9.1.

Introduction

9.2.

Adsorption of carbon monoxide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2.1. Relationship between chemistry and surface science: Blyholder model . . . . . . . .

9.3.

9.4.

9.2.2.

Structural results

9.2.3.

Kinetic oscillations: CO-oxidation reaction

9.6.

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

427 428 429 434 440

O x y g e n adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

446

9.3.1.

Adsorption on close-packed surfaces: fcc(l 1 l ) , h c p ( 0 0 0 1 )

445

9.3.2.

Adsorption on open surfaces: f c c ( l l 0 ) , hcp(1010) . . . . . . . . . . . . . . . . . . .

449

9.3.3.

Developing of oxides

456

Alkali-metal/metal systems 9.4.1.

9.5.

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

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

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

L a n g m u i r - G u r n e y model and recent theoretical results

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

459 461

9.4.2.

Initial growth and adsorption geometry . . . . . . . . . . . . . . . . . . . . . . . . .

465

9.4.3.

Coadsorption R u ( 0 0 0 1 ) - C s - O and Ru(0001)-Cs--CO . . . . . . . . . . . . . . . . .

477

Metal/semiconductor Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

486

9.5. I.

Adsorption of metals on Si(111) and Ge(111) substrates . . . . . . . . . . . . . . . .

486

9.5.2.

C o m m o n l y found models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

487

References

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

426

493

9.1. Introduction In this chapter we concentrate on chemically adsorbed layers on metal and semiconductor surfaces. Because of the volume of prior works and ongoing research, it is necessary to focus on a few topics. We have chosen to cover four topics. Each area has seen an influx of new ideas over the past several years. Even though none of these areas are completely u n d e r s t o o d - in fact, many of the microscopic details of chemisorption processes remain a mystery - nevertheless, a number of interesting trends have emerged. A further restriction is that we shall only discuss processes occurring on single-crystal substrates. Studies of chemisorption, whether on metal or semiconductor surfaces, are propelled partly by the interesting scientific questions they pose and partly by the technological importance of these areas. For example, the study of chemisorption on metal substrates is relevant to the understanding of catalytic reactions and the search for improved catalysts. Studies of chemisorption on semiconducting materials are of great interest to the electronic industry, in its ever increasing demand for miniaturization of devices. However, apart from noting some examples of technological applications, this chapter will draw attention to the basic physical and chemical properties of chemisorption. We shall concentrate on some illustrative models and discuss concepts which are relevant to a broad area of chemical bonding. The first section is concerned with the chemisorption of CO on transition metals. One may view CO as a paradigm of surface science, much like the hydrogen is in atomic physics. Many experimental and theoretical studies have been performed to elucidate the underlying adsorption process of CO. Carbon monoxide is one of the simplest molecules to a chemist and a molecule of great complexity to a physicist. The CO molecule exhibiting many interesting features is a stepping stone to the investigation of more complex molecular adsorbate systems. The study of CO on metal substrates also provides a link between carbonyl chemistry and solid-state physics because similar bonding mechanisms exist. In the second section we will concentrate on oxygen adsorption and the initial stage of oxidation. While on close-packed surfaces, the adsorption process seems to be quite simple, it is no longer true for the more open surfaces, e.g. fcc(110). With the latter systems the adsorption process is frequently accompanied by a reconstruction of the surface, indicating strong metal-oxygen bond formation. Oxidation, on the other hand, represents a final step in the chemisorption process and is associated with the incorporation of oxygen in the substrate layers to form, at least locally, oxide crystallites. The underlying process is often connected with a dramatic re-arrangement of substrate atoms during which the morphology is considerably changed.

428

H. Over and S. Y. Tong

In the third section we will look at the adsorption of alkali-metal atoms on metal surfaces. This system has been regarded as a simple prototype to study the bonding of adsorbate atoms to the substrate. However, recent experimental and theoretical results belie this simplicity by showing that this system has curious properties which are inconsistent with the simple picture. Fortunately, theoretical calculations are tractable because the chemisorption process is predominantly determined by a single s-valence electron of the alkali-metal atoms. Therefore, the calculations are able to clarify the underlying mechanism as exemplified by the alkali-metal/AI(111) system. It is shown that the interrelation between the electronic and structural properties is the key to this mechanism. Another interesting adsorption system represents alkali-metal atoms on Ru(0001). Here, depending on the coverage, the adsorption site changes. A more complicated system consists of the coadsorption of oxygen and Cs on Ru(0001 ). This system exhibits several ordered structures reflecting the complex Cs-oxygen chemistry. We will compare this coadsorbate system with CO on Ru(0001)-Cs. The latter system is governed by an ionic interaction between the species. The last section of this chapter discusses the chemisorption of metal atoms on semiconductor substrates. A general principle is the desire of the interface to reduce the number of unpaired electrons in the adsorbate-substrate system. The balance between electronic and stress energies results in a number of models favored by different metals of the Periodic Table. We will present the electronic and structural properties of these models and explore the reasons why certain metals favor or reject particular models.

9.2. Adsorption of carbon monoxide CO oxidation and the Fischer-Tropsch reaction (catalytic CO hydrogenation) are of great practical importance, both in the chemical industry for making basic chemicals and for such environmental applications as automotive exhaust control. Manufacture of hydrocarbons and of other basic chemicals from coal is of particular interest since coal is an abundant mineral resource in many countries. Hot coal can be gassified by exposing it to water steam which forms a mixture consisting mainly of water and CO. This mixture can then be catalytically synthesized to different chemical compounds. Carbon monoxide, although a small molecule, provides a number of interesting features that qualify it as a stepping stone for the investigation of more complex molecular adsorbate systems. No other adsorbate has been as well studied as CO" hundreds of experiments and dozens of calculations have been performed on it, see for example the reviews of Campuzano (1990) and Hoffmann (1988). One important aim of surface science is to gain insight into the elementary steps of heterogeneous catalysis (Christmann, 1991 ). Although this objective is far from being achieved, the first steps in that direction have been taken, such as determining the geometric structure of small molecules at single-crystal surfaces. In addition, there are interesting analogies between surfaces and cluster complexes, like metal carbonyis, which append to our understanding of the bonding of molecules to metallic surfaces. Such comparisons will be used to understand the chemisorption process of CO on transition-metal surfaces (cf. w 9.2.1 and 9.2.2).

Chemically adsorbed layers on metal and semiconductor su~. "aces

429

A simple example of a chemical reaction on surfaces is presented in the last section (9.2.3) where we focus on the CO-oxidation reaction with special emphasis on the occurrence of kinetic oscillations. This reaction is readily catalyzed by transition metals, especially with platinum-group metals.

9.2.1. Relationship between chemistry and surface science: Blyholder model Let us first consider the molecular orbitals (MO) of CO for the simple case that the MOs are constructed by linear combination of atomic orbitals of C and O; the atomic orbitals are denoted in Fig. 9. l a on the right and left-hand side. Only those atomic orbitals with similar energies can be combined to give MOs. The full set of MOs are then filled with electrons of C and O, starting from the lowest energy level (1o) towards higher energies, according to Hund's rule. The most important MOs determining the reactivity of CO are the 5o and the 2n (frontier) orbitals. The 5o orbital, the highest occupied MO (HOMO, acting as the donor state) of CO, is a C-2pz-derived state which forms a lone-pair orbital concentrated at the carbon end of CO, while the 4o-bonding state results from a hybridization of O-2pz and C-2s orbitals; the z axis is directed along the molecule axis. The 2rt and l rt MOs are symmetric or anti-symmetric combinations of C-2px and O-2px or C-2py and O-2py states, giving rise to the bonding l rt and the anti-bonding 2rt orbitals. The antibonding 2rt MO of CO is empty and represents the lowest unoccupied MO (LUMO, acting as the acceptor state). The symbol o is used to denote an MO that has its charge density concentrated along the internuclear axis of the molecule, while the symbol rt is used for MOs that have zero-charge density on the internuclear axis. The bonding of CO on transition-metal surfaces is conventionally viewed in terms of a donor-acceptor mechanism (Blyholder model) analogous to that found in metal carbonyls (Blyholder, 1964). Upon chemisorption onto a metal surface the 5 ~ - C O molecular orbital (MO) interacts and hybridizes with the band states of the

Fig. 9.1. (a) Schematic molecular orbital diagram for a CO molecule; (b) conventional Blyholder scheme for the chemisorption of CO on transition metals.

430


metal substrate exhibiting proper symmetry, e.g. d,, states in Fig. 9. lb. Thus, the bonding 50. and the anti-bonding 50.* state of the combined metal-CO system are formed. This process is accompanied by electron donation from this MO to the metal which provides a part of the metal-carbon bond. In addition to this 0. donation, there occurs a back-donation of p and d electrons from the metal again with proper symmetry (e.g. d,~ in Fig. 9. l b) into the anti-bonding 2rt MO of the CO which strengthens the metal-CO bond and weakens the internal C - O bond. The two effects of 0. donation and 2rt back-donation are coupled because the more electrons are transferred into the 2rt MO of CO, the more electrons are donated from the 50. level of CO to the metal to maintain charge uniformity. It may be noted that the resulting dipole moment is rather small (around 0.5 Debye) suggesting nearly electroneutrality. In addition, both these CO MOs (50. and 2rt) have a larger amplitude on C, so the binding occurs with the C side down. The donation and back-donation of electrons have opposite effects on the strength of the C - O bond. The removal of charge from the 50. level slightly strengthens the C - O bond due to a more uniform distribution in the l rt level, while addition of charge to the anti-bonding 2rt level weakens the C - O bond (Hoffmann, 1988). As a net effect, the C - O bond is weakened. This can be seen from the vibrational spectra of metal-carbonyl complexes where the species with the highest metal-carbon stretching frequencies (strongest CO-metal bond) show the lowest carbon-oxygen stretching frequency (Emmett, 1965). Ultraviolet photoelectron spectroscopy (UPS) is a standard technique for studying electronic properties of (molecular) adsorption phenomena on solid surfaces. Figure 9.2 shows photoelectron-emission energy distribution curves of CO adsorbed on Ru(0001) (Menzel, 1975) in comparison with the spectra obtained for Ru:~(,CO)~2 and free CO molecules in the gas phase (Plummer et al., 1978). The UP spectrum of free CO molecules shows the 40., Ix, and 50. states expected from the molecular orbital diagram (Fig. 9.1); note that the 2rt MO of CO in the gas phase is empty and is therefore not seen in UPS. The spectrum obtained from Ru3(CO)12, however, exhibits only two broad peaks below the Ru-derived d levels. The easiest way to explain these data is that either the bonding interaction has lowered the energy of the 50. level to overlap the It t-derived state or Ix undergoes a bonding shift to overlap the 40. Theoretical studies of a Ni-CO cluster have demonstrated that 50. and l rt bands overlap and that bonding is achieved through hybridization of the 50. orbital (Ellis et al., 1977; Hermann and Bagus, 1977). Experimental verification of the orbital assignment is possible using angle-resolved UPS (ARUPS), as demonstrated with the system Ru(0001)-CO (Fuggle et al., 1975). The two spectra of Ru carbonyl and CO adsorbed on Ru(0001) (Fig. 9.2) are very similar. This means that the features characterizing the electronic properties of CO attached to only a small number of metal atoms (metal carbonyls) are essentially identical to the situation where CO is adsorbed on Ru(0001). The localized character of chemisorption, at least for this ligand, is strongly supported by this experimental result. For a systematic comparison of UP spectra of transition-metal carbonyl complexes to corresponding spectra of adsorbed CO the reader is referred to Plummer et al. (1978).

431


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I

Er

Fig. 9.2. Comparisonof UV (40.8 eV) photoemission spectra of free CO (a), Ru3(CO)12(b) (Plummer et ai., 1978), and CO adsorbed on Ru(0001) (c) (Menzel, 1975). The adsorption of CO on Ni(100) was studied with an all-electron local-densityfunctional method which confirms and refines the simple Blyholder model (Wimmer et al., 1985). In Figs. 9.3a and b the charge densities of the free molecule 5cy orbital and the free molecule 2~ orbital are presented. The changes in charge density that occur upon CO adsorption onto a Ni(100) surface are shown in Fig. 9.3c, from which a depletion of charge associated with the 5(~ orbital and a gain of charge density with 2~ character are evident to the eye. The B lyholder model predicts a strong correlation between the CO chemisorption energy and the degree of backbonding, i.e., with increasing back-donation the CO-chemisorption energy increases. The back-donation is favored by a large d-electron concentration close to the 2rt MO of CO (cf. Fig. 9.4). In Table 9.1 some selected initial heats of adsorption of CO on various metal single crystals are presented. The observed heats range from 58 to 160 kJ/mol, indeed revealing a strong variation of d-electron density close to the 2~ MO level. For Cu, e.g., the d-electron density near 2~ is small, hence back-bonding is weak associated with a small value of chemisorption energy. From the Blyholder model a further general feature can be deduced: the stronger the metal-carbon bond, the weaker the internal C - O bond, and therefore, the higher the tendency to dissociate. As depicted in Table 9.2 (Broden et al., 1976), at room temperature the adsorption of CO molecules onto transition metals in the left-hand region of the Periodic Table is characterized by dissociation. For metals on the right-hand side of this table, molecular adsorption takes place. It should be noted

432


Fig. 9.3. (a) Charge density of the free 5~ orbital of CO; (b) charge density of the free 2n orbital of CO; (c) change in charge density due to chemisorption of CO on Ni(001) (Wimmer et al., 1985). that at sufficiently low temperatures CO adsorbs molecularly on most transition metals on the left, while at higher temperatures CO dissociates on all of these metals. Furthermore, it is assumed that the back-bonding effect becomes more pronounced as the coordination number of the CO molecule on the metal surface

433


Fig. 9.4. The energetic position of the center of the d band for the first transitions series. The positions of the CO 5~ and 2re levels are superimposed; after Hoffmann (1988). Table 9.1 Initial heats of adsorption (qst) of carbon monoxide on various metal surfaces Surface

qst (kJ/mol)

Reference

128 125 106 111 52 60 160 184 161 142 159

Bridge et al. (1979) Tracy (1972) Klier et al. (1970) Christmann et al. (1974) Tracy (1972) Horn et al. (1977) Pfntir et al. (1978) Brennan and Hayes (1965) Behm and Christmann (1980) Ertl and Koch (1970) Comrie and Weinberg (1976)

Co(0001 ) Ni(100) Ni(110) Ni(! 11) Cu(100) Cu(ll 0) Ru(0001 ) Rh( 111 ) Pd(100) Pd(l 11) It( 111 )

Table 9.2 Tcndency of CO to dissociate on transition metals at room temperature. Rapid dissociation takes place on the non-shadowed region. (Broden et al., 1976) IIIB

Sc Y La

IVB

VB

VIB

VIIB

VIII

VIII

Vlll

IB

Ti Zr Hf

V Nb Ta

Cr Mo W

Mn Tc Re

Fe Ru Os

Co Rh lr

Ni Pd Pt

Cu Ag Au

i n c r e a s e s , i.e., the 2rt o r b i t a l o f a C O m o l e c u l e in a h o l l o w or a b r i d g e site finds m o r e o c c u p i e d d states o f a p p r o p r i a t e s y m m e t r y (d,0 with w h i c h to i n t e r a c t . S i n c e the f r e q u e n c y o f the C O - s t r e t c h i n g v i b r a t i o n r e p r e s e n t s a g o o d i n d i c a t o r o f the d e g r e e to w h i c h this b o n d w e a k e n s (cf. T a b l e 9.3), H R E E L S a n d I R A S can f r e q u e n t l y be u s e d to a s s i g n the c o o r d i n a t i o n n u m b e r o f the a d s o r p t i o n sites.

434


Table 9.3 Frequencies of C-O stretching vibration in correlation with the coordination number of adsorption for various metal surfaces Surface

Adsite

Frequency (cm-l)

Reference

Ni(100) c(2• Ni(l 11), 0 < 0.2 0 = 0.3

on top 3-fold 2-fold

2070 1817 1910

Campuzano and Greenler (1979)

Andersson (1977)

Ru(0001 )(~f3-x'f3-)

on top

1980

Thomas and Weinberg (1979)

Rh( 111 )(~-•

on top

2070

Dubois and Somorjai (1980)

Rh(111) 0 > 1/3

on top bridge

2070 1870

Dubois and Somorjai (1980)

Pd( 111)(~-x~/3-)

fcc

1840

Bradshaw and Hoffmann (1978)

Pt(l 11) 0 > 1/3

on top bridge

2100 1875

Hayden and Bradshaw (1983)

Pt(ll 0) on ( I x I ), 0 < 0.3; on ( I x2), 0 < 0.3

on top on top

2094 2080

Bare (1982) Hofmann (1982)

9.2.2. S t r u c t u r a l results

On most of the transition metals detailed studies of the adsorption process of C O have been p e r f o r m e d . For an o v e r v i e w the reader is referred to C a m p u z a n o (1990) and H o f f m a n n (1983). In Table 9.4 the structural results of C O adsorbed on various metal surfaces are listed and partly c o m p a r e d to results k n o w n for metal c a r b o n y l s and C O on clusters. This c o m p a r i s o n reveals a l m o s t identical m e t a l - C and C - O bond lengths in adsorption systems and c o r r e s p o n d i n g c l u s t e r s / c a r b o n y l s . Striking is the variety of adsorption sites of C O on different metal surfaces" on-top, bridge and hollow sites; see Fig. 9.5 for an illustration of these adsorption sites. The adsorption sites vary not only from metal to metal, but also from one face to a n o t h e r face of the s a m e metal, and also from one c o v e r a g e to another c o v e r a g e on the s a m e face of the s a m e metal. T h e r e f o r e , it is clear that predicting the site o c c u p a t i o n of

Fig. 9.5. Illustration of CO adsorption in on-top, bridge, and hollow sites for the case of a densely packed surface (fcc(l I 1), hcp(0001)).

435

Chemically adsorbed layers on metal and semiconductor su~. aces

Table 9.4 CO adsorption structures on various metal (Me) surfaces in comparison with the results for respective metal carbonyls or clusters Structure

Site

Me-C bond

C-O bond

(A)

(A)

Reference

Ni(100)c(2x2)-CO

on top

1.7_-_-t-0.1

1.13+0.1

Kevan et al. (1981)

Ni(CO)4

on top

1.82

1.15

Sutton (1977)

Cu( 100)c(2x2)--CO

on top

1.9-~. 1

1.13+0.1

Andersson and Pendry (1980)

Cu cluster

on top

1.79-1.93

1.04-1.16

Chini et al. (1976)

Pd(100) (2~x ~-)R45-2CO

bridge

1.93_+0.07

1 . 1 5 + 0 . 1 Behm and Christmann (1980)

Pd cluster

bridge

1.85-1.94

1.09-1.17

Chini et al. (1978)

Pd( 110)(2xl )p2mg

bridge

1.94+0.02

1.1

Huang et al. (1993)

Pd(l 11) ('~-•

fcc

2.05_+0.05

1.15_+0.05

Ohtani et al. (1987)

Ru(0001) (ff-33xxf3-3)R30-CO

on top

1.90-&5).05 1.93_+0.04

1.16_+0.03 1.15_+0.05

Michalk et al. (1983) Over et al. (1993a)

Ru3(CO)i 2

on top

1.91

1.14

Mason and Rae (1968)

Rh( 1I 1) (~t3x'~.5,-)R30-2CO

on top

1.95+0.1

1.07+0.1

Koestner et al. ( 1981 )

Rh( 111 )(2x2)-3CO

on top

1.94+0.1

1.15+ 0.1

Van Hove et al. (1983)

and

bridge

2.03_+0.07

1.15+0.1

Rh(110)(2xl )p2mg

bridge

1.97_+0.09

1.13_+0.09

Batteas et al. (1994)

Rh6(CO)i 6

on top

1.96

1.10

Corey et al. (1963)

1.99

1.36

and

bridge Pt(l 11)c(4•

on top bridge

1.85_+0.10 2.08_+0.07

1.15_+0.05 1.15_+0.05

Ogletree et al. (1986)

C O on a p a r t i c u l a r m e t a l s u r f a c e is difficult, so that e a c h p a r t i c u l a r s y s t e m n e e d s its o w n c o m p l e t e s t r u c t u r e a n a l y s i s . T h e fine b a l a n c e b e t w e e n the a d s o r b a t e - s u b s t r a t e and a d s o r b a t e - a d s o r b a t e i n t e r a c t i o n s is d e m o n s t r a t e d in the d i v e r s i t y o f s t r u c t u r e s f o r m e d . E v e n on m e t a l s w h i c h e x h i b i t v e r y s i m i l a r c h e m i s t r y and are v e r y c l o s e in the P e r i o d i c T a b l e , s u c h as Pd and Rh, the C O a d s o r p t i o n site is d i f f e r e n t . T h e d e v i a t i o n of t e r m i n a l l y b o n d e d C O m o l e c u l e s f r o m 180 ~ for m e t a l c a r b o n y l s ( C - O - M e b o n d is not l i n e a r in this c a s e ) p o i n t s to b e n t b o n d i n g w h i c h m a y o c c u r a l s o on c o r r e s p o n d i n g C O - m e t a l a d s o r p t i o n s y s t e m s . F o r the c a s e o f Ru3(CO)i2 this a n g l e was f o u n d to be a b o u t 170 ~ ( M a s o n and R a e , 1968), see Fig. 9.6. T h i s m a y be the first r e f e r e n c e to b e n d i n g - v i b r a t i o n a l m o d e s , as d i s c u s s e d b e l o w .

436


Fig. 9.6. Molecular stereo-chemistryof Ru3(CO)I2with molecular symmetry not significantly different from D3h; after Mason and Rae (1968). In this section we will focus on only two of these CO-adsorption systems, namely Ru(0001)-CO and Pt(110)-CO, illustrating how different techniques in surface science provide vital contributions to determine the local adsorption geometry. Structural properties of the Pt( 110)-CO system are used in w9.2.3 for an understanding of kinetic oscillations during the CO-oxidation reaction on Pt(110) surfaces. Ru(0001)-CO has been studied with almost every surface-analytical technique, and, therefore, a wealth of information is available. Exposure of the Ru(0001) surface to CO at 110 K forms two ordered LEED structures, namely a (ffxff3-3)R30 at about 0 = 0.3 and a (2q-fx2q3-)R30 at 0 = 7/12 (Williams and Weinberg, 1979). A more detailed LEED study revealed that the (ff3-x{-f)R30 structure exists over a wide range of coverage due to the repulsive interaction between nearest-neighbor CO molecules and attractive interaction between next-nearest-neighbor molecules (Pfntir and Menzel, 1984). Using vibrational spectroscopy (HREELS), Thomas and Weinberg (1979) found only one vibrational band related to CO linearly adsorbed on top of Ru atoms (cf. Table 9.3: C - O stretch frequency of about 2000 cm -~ are assigned to CO in on-top position). The frequency of this C - O stretching mode shifts from 1984 cm -~ to 2061 cm -~ with increasing coverage (Pfntir et al., 1980). This shift was ascribed to dipole-dipole coupling of the CO molecules. Thorough UPS studies of Ru(0001)-CO (Steinkilberg et al., 1975; Hofmann et al., 1985; Heskett et al., 1985) showed that CO is adsorbed upright, even at high coverages. ESDIAD investigations by Madey (1979) arrived at similar conclusions. He found that the angular distributions for O + and CO + ions produced by electron impact are consistent with CO molecules centered about the surface normal with their widths being temperature-dependent. More specifically, the half angles at half


437

Fig. 9.7. Structure model (static) of Ru(0001)-(~x~/3-)R30-CO and the structural parameters for the best-fit arrangement. Small shaded circles: O atoms; small filled circles: C atoms. maximum of the ion cones (reflecting the average inclination of the molecular axis of the adsorbed CO molecules) are found to be about 16 ~ at 290 K and 12 ~ at 90 K. This finding was considered as evidence for the existence of a bending mode. The (4-3-3x4-3-3)R30 ~ structure of Ru(0001 ) - C O was also subjected to conventional LEED structure analyses (Michalk et al., 1983; Over et al., 1993a). These studies clearly favored that CO molecules are adsorbed through their carbon atoms over single Ru atoms (i.e. in 'on-top' position) with their molecular axes being parallel to the surface normal. The structural characteristics are summarized in Fig. 9.7: the Ru atoms coordinated with CO are displaced outwards by 0.07 + 0.03 ]k. The R u - C (1.93 + 0.04 A) and C - O (1.10 + 0.05 ]k) distances are within the expected range" cf. also Table 9.4. Under the influence of thermal excitation the oxygen atoms are expected to have more freedom to move parallel to the surface than perpendicularly, see Fig. 9.8. In fact, such anisotropic temperature effects have shown to be important in surface crystallography and were simulated in LEED by temperature-dependent distributions of O and C positions, according to the harmonic oscillator approximation (Gierer et al., 1996). This analysis indicated a bending-mode vibration of CO with an excitation energy of 5 + 1 meV (cf. Fig. 9.8); note that such low-frequency modes are related to large root mean-square displacements of oxygen by about 0.4 A at 150 K. Since these low-frequency modes are usually not so easily accessible with conventional techniques of vibrational spectroscopy, this represents a promising side application of LEED. The found value of 5 + 1 meV for the bending-mode vibration is also in nice agreement with the value found for a related system P t ( I l l ) - C O (CO adsorbed also in on-top position): the excitation energy of the CO-bending mode was determined to be 6 meV by means of He-atom scattering (Lahee et al., 1986).

438


Fig. 9.8. Illustration of the CO bending-mode vibration on Ru(0001) (Gierer et al., 1996).

Fig. 9.9. Phase transformation from the bulk-truncated Pt(l I0) surface into the missing-row (Ix2) reconstructed Pt(ll 0) surface. We now turn to the system Pt( 110)-CO. This substrate is qualitatively different from Ru(0001) since its thermodynamically stable surface consists of a (Ix2) missing-row structure (cf. Fig. 9.9). Exposure of the Pt(110)(Ix2) surface beyond 1 L CO at 300 K results, however, in a ( 1x I ) LEED pattern exhibiting a high degree of disorder. This indicates that CO lifts the reconstruction above a critical CO coverage of about 0.2 (Imbihl et al., 1988) and now renders the ( l x l ) phase energetically more favorable. With STM (Gritsch et al., 1989) the elementary step for this process has shown to be the creation of local ( I x l ) nuclei (holes) which required atomic motion of Pt only over a few lattice sites (activated process), as indicated in Fig. 9.10. This finding is also consistent with the observation of the adsorbate-induced transformation of the (2xl) into the ( l x l ) even at 250 K (Bonzel and Ferrer, 1982) which is too low to allow surface diffusion of Pt atoms over longer distances. Below 250 K this transformation is completely inhibited (Jackman et al., 1982). With increasing CO exposure the density of holes increases continuously up to about 3 L, creating a highly distorted P t ( 1 1 0 ) ( l x l ) surface. At elevated temperature (350 K), this transformation takes place via the movement of longer [ 110] strings in [001] direction, see Fig. 9.11. This leads to the formation of

Chemically adsorbed layers on metal and semiconductor surfaces

439

Fig. 9.10. At 300 K the elementary step for the CO-induced transformation of the (2x 1) into the ( 1x I ) phase consists of the creation of small holes (with lengths of the order of 10-15 ]~), whereby atoms are shifted only over short distances to missing-row sites. (a) Magnified STM image of one of the 'holes' and (b) corresponding ball model (Gritsch et al., 1989).

Fig. 9.11. At elevated temperatures Q50 K) the transformation of the (2xl) into the (lxl) phase proceeds through shifting of longer [ 110] rows in the [001 ] direction by one lattice constant. (a) STM image and (b) corresponding ball model (Gritsch et al., 1989). larger patches of the (1 x 1) d o m a i n s with typical d i m e n s i o n s of 2 0 - 5 0 ,~ in the [001 ] direction and 1 0 0 - 3 0 0 A in the [ITO] direction, thus giving a better ordered (1• structure as e v i d e n c e d by L E E D .

440


Pt(110)-(2xl)pg-CO .[OOl]

l-

giide plane --... [ l l O l

Fig. 9.12. The top view of the Pt(110)(2xl)p2mg--CO structure. The CO molecule is represented by closely spaced small circles. The glide plane is indicated. In order to form a well-ordered CO overlayer on this surface, one must heat the sample to about 500 K in 10 -7 mbar of CO resulting in a ( 2 x l ) p 2 m g structure with coverage 0 - 1 (Comrie and Lambert, 1976); a top view of real space model of this structure is shown in Fig. 9.12. Analogous to the system Pd(110)-CO and Ni( 110)CO (Pangher and Haase, 1993), the presence of the glide symmetry points towards the presence of zigzag chains along the [110] direction. Angular-resolved UPS studies (Hofmann et al., 1982) have indicated that the zigzag chains come from the CO molecules being tilted by about 26 ~. The on-top position for CO used in this model has been derived from RAIRS and HREELS measurements (Bare et al., 1982" Hofmann et al., 1982), cf. Table 9.3. Static tilting is frequently found for densely packed CO molecules, as for example for N i ( 1 1 0 ) ( 2 x l ) p m g - 2 C O (Hannaman and Passler, 1988; Wesner et al., 1988). The evolution of the Pt(110)(2xl)p2mg-2CO phase at T - 300 K has been studied by means of TDS, UPS, LEED, RAIRS, and EELS (Bare et al., 1982; Hofmann et al., 1982). At low coverages (0 = 0.1 ), isolated CO species are adsorbed in on-top sites on the reconstructed Pt(110)(Ix2) surface as found by infrared spectroscopy. UPS shows that these molecules are adsorbed with their axes perpendicular to the surface. For 0.1 < 0 < 0.3, the isolated CO molecules coexist with islands of CO adsorbed in on-top sites on the ( l x l ) surface. This finding has been confirmed by the inverse photoemission results of Ferrer et al. (1985) who found that the empty 2rt level of CO adsorbed on the ( Ix l) surface occurs at 5 eV above the Fermi energy, while this level for CO adsorbed on the (lx2) surface occurs at 3.4 eV above the Fermi level. With increasing coverages, islands grow in size (EELS), and the molecules in the islands are adsorbed with their axes tilted (UPS) due to the high density of molecules in the islands. At coverages greater than half a monolayer, the islands begin to coalesce, and eventually the ( l x 2 ) to ( l x l ) transition is completed. This transition was also the subject of a recent quantitative RHEED study (Schwegmann et al., 1995). 9.2.3. Kinetic oscillations: CO-oxidation reaction

The catalytic CO oxidation reaction on Pt-group metals proceeds through the Langmuir-Hinshelwood mechanism (Engel and Ertl, 1979) where both reactants are adsorbed on the surface before they react to form the product. Oxygen molecules

Chemically adsorbed layers on metal and semiconductor su~'aces

441

adsorb dissociatively requiring two neighboring, unoccupied sites. By contrast, CO molecules adsorb molecularly and tend to form densely packed layers which block further uptake of oxygen. As described in the last section, the clean Pt(110) surface is 'missing-row' reconstructed which transforms upon CO adsorption into a ( l x l ) structure when the CO coverage exceeds a critical value of about 0.2 (Imbihl et al., 1988). Oxygen, on the other hand, forms a relatively open network which allows CO to coadsorb into. The recombination of adsorbed CO with adsorbed O forms CO2 which is immediately released into the gas phase under typical reaction conditions. Usually, the rate of CO2 formation is stationary, and a large coverage of CO results in a low reaction rate. Under certain reaction conditions, however, the CO2-reaction rate is not stationary but exhibits an oscillatory behavior called kinetic oscillations. The crucial point for their occurrence is that the sticking probability of oxygen is about twice as high on a ( l x l ) phase as on a (lx2) phase. The origin of these oscillations has been explored in great detail with the Pt(110) surface and can be traced back to a close coupling between structure and reactivity of a surface (Eiswirth et al., 1986, 1989). A detailed and comprehensive report on kinetic oscillations can be found in Ertl (1990, 1991) and Imbihl (1989). The complete oscillation cycle will be described in the following. We start with a clean (lx2) surface which is exposed to a mixture of CO and 02 for conditions under which adsorption of oxygen is rate-limiting. Due to the larger sticking coefficient of CO compared to O2, CO is preferentially adsorbed and removes the (Ix2) reconstruction when it exceeds the critical CO coverage. Such an adlayer does not completely inhibit O2 adsorption, mainly due to the inevitable presence of surface defects. At these sites the sticking coefficient of 02 is considerably higher, so oxygen atoms adsorb and react off the adsorbed CO, resulting in two empty adsorption sites on the ( l x l ) phase. In an autocatalytic process, the oxygen uptake and hence the CO2 rate increase. This process continues until the CO concentration on the surface is depleted to such an extent that the reconstructed Pt(110)(Ix2) is restored and one reaction cycle is completed. Clearly, rate maxima occur after lifting the ( l x 2 ) reconstruction due to the larger amount of oxygen available on the surface for the CO2 reaction, and minima are related to the ( l x l ) structure. In Fig. 9.13 the CO2 reaction rate shows oscillatory behavior when the CO and oxygen partial pressures are kept fixed at a certain value. The work function A~ is proportional to the oxygen coverage and parallels the reaction rate. Temporal oscillations can be modeled by the numerical solution of (three) coupled non-linear differential equations, describing the temporal variations of the CO and O coverages (rate equations) as well as the fraction of the surface present as (lx2) or ( l x l ) phase (Krischer et al., 1991, 1992). All input parameters are taken from independent measurements. Corresponding results in the limit t ~ ,,o (socalled limit sets) are presented in Fig. 9.14 which agree qualitatively with data of Fig. 9.13. Another aspect consists in spatial self-organization which synchronizes the behavior of different local regions on a macroscopic scale and is therefore a necessary precondition for the occurrence of overall temporal oscillations. The

442


.

.

.

.

! .....

i

'!

_

el

{ 2o 100~01- . . . . . 0

, 200

- - -

~ . . . . 400

Time

~ -600

Is)

Fig. 9.13. Kinetic oscillations during the CO/O2 reaction at a Pt(110) surface. The CO2 partial pressure is proportional to the reaction rate, and the work function A ~ changes parallel to the oxygen coverage. Control parameters: substrate temperature T = 480 K and partial pressures: oxygen p(O2) = 2.1 x 104 mbar and p(CO) = 6.8• -~ mbar (Eiswirth et ai., 1989). "G' r

E

3 01

"~

0.7

\

:0i: _Q

4

-.

06[

~

0.4

.J

.,/" ""

/", ,,,.. "

--/,,, ""

/\~x~/,, -,,, ,.-

, /',,- , - :,',, "

"

"

""

...'.. .... ""

"

0

r

0.2

06

'

,o

20

3o Time

~o

50

6o

70

(s)

Fig. 9.14. Time series calculated by integrating the corresponding rate equations modeling the kinetics of CO oxidation on Pt(110). Particular control parameters: T= 540 K, p(O2) = 6.7• -5 mbar and p(CO) = 3.0• -5 mbar (Krischer et al., 1992).


443

Fig. 9.15. (a) Spatial-temporal patterns of the standing wave type associated with harmonic temporal oscillations during the CO oxidation on Pt(110), cf. Fig. 9.9. The images are recorded in intervals of 0.5 s on a section of0.3x0.3 mm 2. Control parameters: T = 550 K, p(O2) = 4.1x10 -4 mbar and p(CO) = 1.75x10 -4 mbar. (b) Growth of a spiral wave. Width of the images are 0.2 mm recorded at 0, 10, 21,39, 56 and 74 s. Control parameters: T = 4 3 4 K, p(O2) = 3.0x10 ~ mbar and p(CO) = 2.8x10 -5 mbar (Jakubith et al., 1990).

444


through surface diffusion. In Fig. 9.15 two types of spatial patterns during the CO oxidation are depicted. To make local variations visible in the coverages of adsorbed species, a photoemission electron microscope (PEEM) is applied (Rotermund et al., 1990; Engel et al., 1991). This method, which yields a spatial resolution of about 1 l.tm, is based on the principle that the yield of photoelectrons depends sensitively on the local work function as one illuminates the sample with photons from a deuterium discharge lamp. The lateral intensity distribution of the photoemitted electrons is imaged through a system of electrostatic lenses onto a fluorescence screen. Since the work function of a clean Pt(110) surface increases by 0.3 eV and 0.5 eV when saturated with CO and oxygen, respectively, the areas covered with O appear dark in the images, while those covered by CO are brighter. The continuously growing spiral wave in Fig. 9.15b is diffusion-controlled (Jakubith et al., 1990). Its shape is elliptic, with the long axis along the [110] direction of the substrate single crystal for which the anisotropy of surface diffusion is responsible and in turn affects the propagation velocities of the fronts of the 'chemical waves'. The velocities in the two proper directions are 3.3 and 1.2 mm/s, respectively. Another type of pattern is related to CO2 rate oscillations (cf. Fig. 9.12) which consist of standing rather than propagating waves (Fig. 9.15a). The coupling mechanism proceeds through the gas phase, as can be seen from Fig. 9.12. The small modulation in the CO partial pressure ( 300 K initiates a second structural transformation during which a c(6x2) structure is formed. This structure, with a local oxygen coverage of 2/3, represents the counterpart to the p(3x I ) found with Ni( 110)-O and should therefore have a similar adsorption geometry. With a combination of STM, surface X-ray diffraction, and theoretical methods this complex structure was solved (Feidenhans'l et al., 1990). This solution has been confirmed also by a recent LEED study (Liu et al., 1996). A basic part of this structure consists again of Cu-O chains lying on top of a nearly undistorted substrate; these Cu-O chains are arranged with (3x l) periodicity similar to the high-coverage N i ( 1 1 0 ) p ( 3 x l ) - O phase. In addition to that, two 'super' Cu atoms are sitting above the chains, linking and stabilizing them, cf. Fig. 9.22. The c(6x2) structure nucleates preferentially at step edges and grows almost isotropically above 300 K, unlike the (2xl) configuration which nucleates at flat terraces and grows anisotropically in the [001] direction in the form of Cu-O chains. While the adsorption of oxygen on fcc(110) surfaces has been studied extensively, much less work has been done on the 'equivalent' (1010) surfaces of hcp metals, such as Co( 1010). Adsorption of half a monolayer of oxygen on this surface at 150 K and subsequent annealing at 230 K < T < 350 K result in the appearance of an ordered c(2x4) structure which transforms irreversibly into a p ( 2 x l ) structure upon warming to 350 K < T < 580 K (Schwarz et al., 1990b). The occurrence of an activated process and similar chemical properties of Ni and Co suggest that the p ( 2 x l ) structure may be related to a reconstruction of the substrate similar to that in N i ( 1 1 0 ) ( 2 x l ) - O . A structure analysis of the p ( 2 x l ) phase (Over, 1991b) based on LEED I - V curves clearly indicates that an added-row model can be excluded. Note that the Co(1010) surface has two different possible terminations due to the ABAB... stacking, whereby only the termination with a smaller corrugation represents the stable atomic configuration (Lindroos et al., 1990; Over et al. (1991 a); cf. I

!

454


Fig. 9.22. Structure model of the reconstructed Cu(110)c(6x2)-O phase; (a) side view and (b) top view. Grey circles represent the 'super' Cu atoms, and the small black circles indicate the oxygen atoms. A c(6x2) unit cell is shown (Feidenhans'l et al., 1990). Fig. 9.16a. A missing-row model, on the other hand, is not very likely because the formation of this structure would need to remove Co atoms from a tightly bound double layer (layer spacing between the atomic plane in this double layer is only 0.7 A). It is worth mentioning that added row and missing row are no longer equivalent for this surface, as opposed to fcc(l 10) surfaces. From STM studies it has been indicated that this phase is formed by a double-layer missing-row reconstruction (Koch et al., 1993, 1994), which is illustrated in Fig. 9.23. The last system in this section which we focus on is oxygen adsorption on fcc-Rh(l 10). This system has gained some attention due to the ability of rhodium

Fig. 9._23. A hard-sphere model (top view) of the double-layer missing-row structure for the Co( 1010)(2x I )-O surface. Co atoms' large circles, Oatoms: small black circles. The arrows indicate the movements of Co atoms to transform the clean Co(1010) surface into the (2• 1)-reconstructed surface; after (Koch et al., 1993).

Chemically adsorbed layers on metal and semiconductor su.rlaces

455

to reduce NOx and has thus made Rh an important constituent of the three-way automotive exhaust catalyst. Upon interaction with oxygen this surface shows a series of LEED patterns, namely (2xl)pmg, (2x2)pmg, c(2x8), np(2x3), c(3x6), c(3x8), and complex streaked c(2x2n) patterns, depending on parameters such as sample temperature, O coverage and reduction cycles (Tucker, 1966; Schwarz et al., 1990a; Comelli et al., 1992a,b; Dhanak et al., 1992). Evidence was found that the oxygen interaction is not only restricted to the outermost Rh atoms, but may also include deeper layers of the crystal (Schwarz et al., 1990a; Comelli et al., 1992a,b; Wohlgemuth, 1994). Further discussion will be confined to the structures (2x 1)p2mg and (2x2)p2mg. Exposure of Rh(110) to 1 L oxygen at low temperatures (125 K) results in a (2xl)pmg structure which transforms irreversibly upon annealing (470-970 K) into a (2x2)pmg structure (Comelli et al., 1992a). This transformation into the (2x2)pmg phase is thermally activated (the activation temperature is similar to the case of Ni(110)(2xl)-O), suggesting that its formation might be associated with a reconstruction process. Reacting the (2x2)pmg phase with CO or H2 at elevated temperatures (,

L~

:- I

I

__j:_LL 0.1

I/I L.

h,

~

O -

__

=

-5

-I

0.3

I

0.4

0.5

Fig. 9.33. Experimental phase diagram for the system Ru(0001)--Cs.The cross-hatched area represents a qualitative result (Bludau, 1992; Over et al., 1992a).


469

revealed that Cs resides in on-top position with a Ru-Cs bond length of 3.25 + 0.08 ~, corresponding to a hard-sphere radius of 1.9 /~ for the adatom, closer to its Pauling ionic radius (1.69/~) than to its covalent radius (2.35 ,~). The adsorption process in the p(2x2) phase induces a shift of 0.10/~ in the Ru atom coordinated with Cs towards the bulk. Also for the p(2x2) structures formed by Cu(11 l)-Cs (Lindgren et al., 1983), Ni(111)-K (Fisher et al., 1992), AI(111)-K (Stampfl et al., 1992), and AI(111)-Rb (Nielsen et al., 1994) adsorption in on-top sites was determined by several methods, see Table 9.6. With increasing temperature, a transition from the (2x2) phase to disorder proceeds through a range characterized by a ring-like diffraction pattern (cf. Fig. 9.33) intersecting the (1/2, 0) position (melting transition). This indicates that the dipole-dipole interaction is still sufficient to force the Cs atoms into a constant interatomic spacing, while the azimuthal ordering by the corrugation of the substrate is weaker and disappears first. The transition to the (f3-x4-3-)R30 structure (saturation coverage) takes place via a phase regime denoted as 'rotated structures' with a continuous spot splitting as a function of coverage. The decrease in length of the unit-cell vectors in real space and the corresponding increase in coverage indicate a fully relaxed Cs overlayer. The repulsive lateral interactions between the adatoms are strong enough to prevent the Cs atoms from locking in at highly symmetric adsorption sites. In contrast to liquid-like structures, the A-S interaction is still able to align the overlayer unit cell with a particular (coverage-dependent) angle with respect to the unit cell of the substrate. A similar phenomenon of 'rotational epitaxy' (predicted by the theory of Novaco & McTague (1977)) was also found in other systems, such as Li and Na on Ru(0001) (Doering and Semancik, 1984, 1986), P t ( I I I ) - K (Pirug and Bonzel, 1988) and Rh(100)-Cs (Besold et al., 1987). Near saturation coverage of the monolayer, the polarization of the alkali atoms has almost disappeared, and the formation of an adlayer with a metallic binding c h a r a c t e r can be o b s e r v e d (e.g. as s u r f a c e - p l a s m o n e x c i t a t i o n ) . The Ru(0001 ) ( ' ~ - x f f ) R 3 0 - C s structure provides the completion of the first monolayer. It can be observed up to desorption temperature. During the multilayer adsorption the long-range order is destroyed. In the (~-x'~-)R30 overlayer (cf. Fig. 9.34b), Cs occupies the hcp site, inducing no relaxations of the substrate as derived from a LEED structure analysis (Over, 1992a). The derived Cs-Ru layer spacing of 3.15 + 0.03 ]k corresponds to a hard-sphere radius of the adsorbed Cs atom of 2.17 + 0.02 ]k, probably reflecting a more metallic bonding. Alkali ions have effective radii typically 1 /~, smaller than the effective radius of the respective alkali-metal bulk value. This observation might offer the prospect of finding an increase in bond length with increasing coverage which could point to a transition from more 'ionic' to more 'metallic'. In comparison with the Cs radius found with the (2x2)-Cs system (1.90 ~), the increase of the effective radius of the adparticle with coverage might therefore be superficially interpreted as such a transition; this issue will be addressed in more detail below. These data are in qualitative agreement with the structural parameters for the Ag(111)-Cs system as determined by SEXAFS (Lamble et al., 1988); note that the corresponding adsorption sites were not determined in this study.

470


Rul0001)/Cs-I~]• ~

~

Ru{0001)/Cs

,

~

~

(2,2) '

0.0z,~

1 2.1z,• 0.05/~

0.07~

2.13~.07A

Cs - rodius" 2.2~

- rodius: 1.95 A

I

.

f Fig. 9.34. Structural models lot Ru(O(X)l)(x/3-3xx/-f)R30--Cs (a) and p(2x2) (b) with optimal structural parameters as found by LEED analyses (Over et ai., 1992a). The results of the analyses of both ordered Cs structures are summarized in Fig. 9.34. Essentially, two important structural features have emerged: apart from the variation of the Cs radius with coverage, the Ru(0001)-Cs system exhibits a switching of adsorption sites from on top to hcp (hollow) as the coverage increases from 0.25 to 0.33. For the (~3-x,f3-)R30 structure, the corresponding dipole moment of Cs (-- 2.8 Debye) is significantly lower than for the p(2x2) phase (= 2.0 Debye), due to mutual depolarization of Cs atoms as indicated by an increase in the work function. Consequently, a change in coverage is paralleled by a modification of the electronic properties of the adatoms and therefore the interaction between adsorbate and substrate. In general, when the coordination number increases, the bond length increases, and the strength per bond decreases. For ionic bonding (Kittel, 1976), a switching from coordination number 3 (hcp) to 1 (on top) should result in a decrease in bond length by about 0.3 /k, so that the observed change in bond length for the system Ru(0001)-Cs (and we expect the same for Ag(l 1 l)-Cs) is more related to the local adsorption site than to the degree of ionicity. This conclusion is supported by structural analyses of the systems Ru(0001)-K (Gierer et al., 1992), Ru(0001)-Na (Hertel et al., 1994a) and Ni(100)-K (Wedler et al., 1993) for which the coordination numbers of the adsorbed particles are independent of the coverage and no variation of the alkali-metal radii has been found. Furthermore, the structural analysis of the coadsorbate system Ru(0001)(,f3-xff)R30-Cs-O (Over et al., 1992b) exhibits a


471

salt-like structure in which the effective Cs radius reduces only by 0.1 /~ with reference to the clean Ru(0001)-('~-x~4~-)R30-Cs phase. It should be noted that the geometry of the Cs-adsorption site is not altered and that in this ionic Cs-O structure the Cs adatom donates easily its 6 s-valence electron density to the substrate and to the electronegative O atom. This result might thus be regarded as an upper limit for the variation of effective Cs radii expected as a function of coverage. However, even a constant bond length does not preclude the possibility of an ionic to a metallic transition, as this conclusion would necessitate identifying bond length with ionicity. It appears that a simple comparison of hard-sphere radii with ionic, covalent or metallic Pauling radii is not sufficient to fully characterize the nature of bonding. For example, the covalent radius of Cs is given by the bond length of Cs2, while the bonding of Cs on Ru(0001) results from a complicated interplay of A - A and A-S interaction. The behavior of occupying different adsorption sites in the (2• and (4-f3xg3-)R30 ~ structures is more peculiar. Both phases are commensurate with the substrate lattice and would permit identical adsorption site geometries without affecting the unit cell if the A - A and A-S potentials are uncoupled. As mentioned before, the A - A interaction is dominated by dipole-dipole repulsion. Generally, the interaction energy between two dipoles decreases with an increase of electron densities between the Cs ions. This electron density could, for instance, be supplied by the substrate. Figure 9.34 shows that, in the (2x2) phase, better screening between the dipoles can be achieved if neighboring Cs atoms have a substrate atom directly between them, as occur at the on-top sites but not at hollow sites. The fact that these substrate atoms between adatoms are raised by 0.1/~ (and thus enhancing the screening ability) is consistent with this model. As can be seen from Fig. 9.34, with the (f3-3x'~r3-)R30 phase, occupation of the on-top sites would no longer improve the screening, and instead, the hollow sites are preferred. The energy difference between high-symmetry sites is small for a Cs atom due to its large size, which implies that it experiences a rather small substrate electrondensity corrugation (Neugebauer, (1992a). To study this latter effect, Over et al., 1995f) analyzed the temperature dependence of LEED I-V curves by applying the concept of 'split positions' (Over et al., 1993a). This technique has been shown to be sensitive to lateral (harmonic) discursions of adparticles around their equilibrium positions associated with small excitation energies. On the one hand, the resulting 'split positions' as a function of the temperature can be compared with the mean-square deviation derived from the harmonic-oscillator approximation (twodimensional and isotropic), where only the excitation energy is used as a fitting parameter. The corresponding value turns out to be 1.2 + 0.3 meV. On the other hand, when the energy potential relief is dominated by the dipole-dipole (D-D) interaction (for the (2x2) phase the dipole moment is about 2.8 Debye), the curvature of the D - D energy-potential surface at the equilibrium position (evenly dispersed (2x2) overlayer) is directly correlated to the excitation energy of this vibrational mode. The corresponding excitation energy turns out to be 1.0 meV. The result of 1.2 + 0.3 meV (split position) in comparison with 1.0 meV (D-D i~ateraction) may yield an estimate for the contribution of the A-S interaction to the

472


excitation energy of roughly 0.1-0.5 meV, which indeed is consistent with a very small A - S corrugation. A similar analysis can also be carried out for Cs motions perpendicular to the surface. The resulting excitation energy of about 9 meV is in very good agreement with an energy loss found in HREEL spectra (7.8 meV) for the Ru(0001)(2x2)-Cs system (Jacobi et al., 1994). The actual adsorption site occupied in alkali-metal systems will be a result of a sensitive balance between corrugation of the substrate potential, magnitude of the dipole moment, interatomic spacings, and electrostatic screening. Therefore, no prediction of the adsorption site can be made, as demonstrated, for example, by the system Ru(0001)-K. For this system the same two ordered overlayers as those for Ru(0001)-Cs have been observed, namely a p(2x2) and a ('~-x,~-)R30 structure, for which LEED analyses were performed. In the ( ' ~ x , ~ - ) R 3 0 phase the K atoms reside in threefold hcp sites, while in the p(2x2) phase the fcc site is favored. In both phases the K hard-sphere radii are nearly equal and close to the covalent Pauling radius, cf. Table 9.7. In contrast to the Ru(0001)p(2x2)-Cs system, potassium d o e s n o t occupy an on-top site. The effective corrugation of the substrate potential is expected to be larger due to the smaller K radius, so that the influence of the repulsive dipole-dipole interaction might be not as dominant as with the Ru(0001 )-Cs system, and screening via the substrate should play only a minor role. This explanation is supported by the fact that the initial dipole moment for Cs with about 10.5 Debye (Hrbek, 1985) is larger than for K with 7.5 + 0.3 Debye (Uram et al., 1986). Concerning the occupation of the different threefold-hollow sites, it should be noted that the difference in the binding energy per adatom for adsorption on the hcp and the fcc sites is presumably very small. For example, total energy calculations Table 9.7 Structural results found for alkali-metal adsorption on Ru(0001). The hard-sphere radii are compared with corresponding Pauli radii (Over et al., 1995d). Pauling radii (]k)

Alkali metal/Ru(0001): LEED analysis Alkali

Structure

Site

Radii (A)

Ionic

Covalent

Metallic

on top hcp

1.90 2.17

1.69

2.35

2.67

Cs

(2x2) ('f3-•

Rb

(2• (q-3-xq-3-3)R30

fcc hcp

2.03 2.03

1.48

2.16

2.48

K

(2x2) (,fJ-3x,/3-)R30

fcc hcp

1.94 1.98

1.33

2.03

2.35

Na

(2• (q-J-x,,/3-)R30

fcc hcp

1.58 1.58

0.95

1.54

1.90

Li

(,]3-xq-J-)R30

hcp

1.39

0.60

1.23

1.55


473

performed for ordered structures of the AI(111)-K system (Neugebauer and Scheftier, 1992b) led to the conclusion that the binding energies per adatom for the two types of threefold-hollow sites are practically degenerated. Nevertheless, a switching of large domains from one type of adsorption site to another one represents a phase transition. In this case, energy differences between two islands occupying exclusively fcc and hcp sites of the order of kT would suffice to stabilize one phase over the other. It is worth noting that the occupation of different adsorption sites during the formation of a crystal layer has been observed even for a homoepitaxial system (Wang and Ehrlich, 1989): investigations with field ion microscopy of the It( 111 )-Ir system demonstrated that single Ir adatoms favor the hcp sites, while for clusters of Ir atoms the probability of occupying the bulk-like fcc sites rises rapidly with increasing cluster size, reaching nearly 100% for clusters with seven Ir atoms. The self-adsorption of Ir is governed by strong lateral attraction (as reflected by the nucleation in islands at low coverages) which accounts for a coupling of the adsites of neighboring adparticles. For ordered alkali-metal/metal systems, such a correlation is likewise given by dipole-dipole repulsion, so that the site switching determined for Ru(0001)-K is not surprising, although it is not completely understood. An alternative view could be that a cluster of seven Ir atoms builds up a metallic state. The transition from p(2x2) to ,f3-x,f3 is also accompanied by a transition from more 'ionic' to more 'metallic', so that one can speculate that the 'expected' site (fcc site for fcc(l l l) and hcp site for hcp(0001)) will always be occupied if metallization occurs. In this line of reasoning, one would expect that alkali metals smaller than potassium, namely sodium and lithium, should occupy hollow sites. Indeed, structural analyses by LEED (Gierer et al., 1992; Hertel et al., 1994a) confirm this speculation. As with K adsorbed on Ru(0001 ), Na atoms occupy threefold-fcc sites in the p(2x2) phase, while for the ('~-x4-3-)R30 phase the hcp site is favored. The hard-sphere radii found in this study are nearly constant (1.58 ~) and close to the covalent Pauling radius. For the system Ru(0001 )-Li Gierer et al. (1995) could only observe one ordered structure, namely the ('43-x'43-)R30 phase; Doering & Semancik (1986) have reported a p(2x2) structure in addition to the (,13-x,~-)R30 phase. A quantitative LEED analysis of the (,]3-x,f3-)R30 structure indicated that Li atoms reside also in hcp sites with a hard-sphere radius of 1.39/~. Besides the two phases of Ru(0001)-Na mentioned above, the high-coverage (3x3)-4Na overlayer was analyzed by Hertel et al. (1994a). They found that the nearest-neighbor distances are almost uniform and Na is not adsorbed in high-symmetry sites. This demonstrates a strong adsorbate-adsorbate interaction, presumably attractive, which dominates over the small 'substrate' corrugation potential. Last, we briefly describe the results of the system Ru(0001)-Rb which provides a connecting link of Cs and K in various respects (Hertel et al., 1994b). The radius of Rb as well as the initial dipole moment are between the values given for Cs and K. Especially for the p(2x2) structure, the question where the Rb atom will reside is of particular interest. A LEED study revealed, similar to Ru(0001)-K, a twisting of adsorption sites from fcc for p(2x2) to hcp for (~r3-x,f3-)R30, so that the on-top adsorption on Ru(0001) is restricted to Cs. For the Rb(2x2) phase, also an analysis

474


of the lateral vibrations was performed, applying the concept of split positions. It turned out that the excitation energy of this lateral vibration is 2.0 + 0.5 meV which is in agreement with values found for the Rb-graphite system employing the technique of inelastic He scattering (2.8 meV) (Cui et al., 1993). A comparison of different alkali metals on Ru(0001) shows that the hcp site is always favored for the (~f3-x~/-3-)R30 phase; cf. Table 9.7. This site represents the 'expected' site if growing of Ru on metallic Ru(0001) is considered. One might speculate that the favored adsorption site in metallized alkali films should be the 'expected' one. A more complicated situation arises for the (2• phase where the fcc site is preferred for all alkali metals but Cs (on top). This means that each of these alkali-metal/metal systems undergoes a switching in adsorption site with varying coverage. In order to resolve the underlying mechanism, especially for the transformation from hcp to fcc sites, extended a b - i n i t i o calculations as a function of the overlayer density are required. Another property shared by all the investigated systems is that the alkali-metal radii found by experiment, although not subject to a deeper understanding of the type of bonding, nicely reflect the tendency of corresponding Pauling radii. In all these cases, the alkali-metal radii found by LEED are independent of the coverage. Several conclusions can be drawn by comparing the local geometry of alkali metals adsorbed on different metal substrates. Top sites have been observed only on hexagonal surfaces, and the effective radius of alkali-metal adatoms in on-top positions is less than when adsorbed in higher coordination. This effect can be traced back to the general experience that with increasing coordination number the bond length increases. All alkali radii are close to the respective covalent Pauling radii if corrected for the effect of different coordination numbers, and, last, the adsorption of alkali metals on 'open' surfaces generally takes place in highly coordinated sites, presumably due to a stronger corrugation potential of the substrate. We will now briefly review the structural properties of alkali metals adsorbed on AI(111), for which results from different experimental techniques as well as theoretical investigations are available. Theoretical calculations of the adsorption geometry and electronic structure from first principles for A! (s-p band metal) are in general less demanding, compared to alkali adsorption on transition-metal surfaces where d electrons of the substrate play an important role. For alkali-metal adsorption on metal surfaces it is commonly assumed that alkali atoms reside on the surface (see Table 9.6) and that no intermixing occurs with the substrate. This seems plausible since alkali metals have a low solubility in most metals (Miedema and Niessen, 1988). Recently, however, the formation of a Na-AI surface alloy has been reported. Using polarization-dependent SEXAFS measurements, Schmalz et al., 1991) showed that the Na atoms for coverages of 0 = 0.16-0.33 occupy an unusual sixfold-coordinated substitutional site on AI( 111 ) at room temperature (cf. Fig. 9.35a). Subsequently, a b - i n i t i o density-functional calculations have elucidated the underlying mechanisms (Schmalz et al., 1991; Neugebauer and Scheffler, 1992b), proving that this adsorption geometry is indeed energetically favorable due to the small energy for the formation of surface vacancies (0.41 eV) and a better substrate-mediated screening of the direct A - A repulsion. The explanation for the


475

Fig. 9.35. (a) The atomic structure of AI(111)(~3-x~-)R30-Na room temperature phase. The Na atom substitutes the AI atom; (b) the underlying vacancy structure of AI(111). low vacancy formation energy for the (43-x43-)R30 structure is that for this geometry the group III A1 substrate can create a favorable spZ-bonded surface layer (cf. Fig. 9.35b). The Na adatoms at the substitutional site behave similarly to isolated adatoms, and the ionic type of bonding can therefore develop more strongly than on the unreconstructed surface. High-resolution core-level spectroscopy clearly supports the occurrence of intermixing between Na and AI (Andersen et al., 1992). This technique utilizes the fact that the core-level binding energy of an atom is depending on its coordination sphere, and thus it is possible to determine the number of different sites involved in this system. Once a special feature of a core-level binding spectrum can be assigned to an adsorption site (by an alternate technique like LEED or SEXAFS), this method can be used for a fingerprinting. Another intriguing system, AI(I 1 I)-K, was analyzed by LEED (Stampfl et al., 1992) which shows that the adsorption of K on AI(I 1 1) at 90 K and 300 K forms a (~/-f3• structure. However, the adsorption sites are significantly different; both structures are displayed in Fig. 9.36. At 90 K the adatoms occupy on-top sites

Fig. 9.36. The atomic geometry of low temperature (a) and the room temperature adsorption structure of K on AI( 111 ). For low temperatures, K occupies the on-top site, while for room temperature the substitutional site is occupied.

476


on a buckled surface. The A1 atom coordinated with K is displaced downward by 0.25 A! These sites convert irreversibly to a configuration with K residing in a substitutional site on warming to 300 K (activated process). The on-top site, metastable phase, thus serves as a precursor for the equilibrium adsorption site. Neugebauer and Scheffler (1992b) showed that in contrast to Na, the on-top position might become favorable. This result was explained as a consequence of the energy gain due to substrate relaxation for the on-top geometry and the bigger size of K compared to Na. Andersen et al. (1993) performed a comprehensive study of Na, K, Rb, and Cs adsorbed on AI(111) at 100 K and at room temperature using high-resolution core-level photoemission spectroscopy in combination with qualitative LEED. They found that island formation is more the rule than an exception for alkali adsorption on AI(111), a conclusion which is at odds with the commonly accepted picture of alkali-metal adsorption (compare, e.g., the growth properties of alkali metal/Ru(0001)). More specifically, at 100 K, for small coverages, a dispersed phase for Na, K, and Rb was observed which transformed beyond a coverage of about 0.1 in island growth. This indicates that the net alkali-alkali interaction changes from repulsive to attractive. For Cs adsorption at 100 K no island formation was found at any coverage. This is attributed to a strong repulsive interaction between Cs atoms which is reasonable due to their large dipole moments and bigger sizes. Theoretical studies by Neugebauer and Scheffler (1993) confirmed that for AI(11 l)-Na at 0 > 0.1 the island formation (with a metallic, attractive adsorbate interaction) is energetically advantageous. For the dispersed phases they predicted the threefold-hollow position. Brune et al. (1995) obtained images of the island formation and the substitutional site for Na by STM. Similar results were obtained with the system AI(111)-K. Again, at very low coverages, calculations Neugebauer and Scheffler (1993) predicted that K atoms adsorb in hollow sites, so that the island formation has to be accompanied by a switching of the adsites from threefold to ontop position. In the on-top geometry the AI atom coordinated with K moves down by about 0.25/~, (Stampfl et al., 1992) and thus makes screening work better (this is presumably the driving force of this process). Andersen et al. (1993) pointed out that all the ordered structures formed by alkali adsorption at room temperature involve removal of the AI atoms where the vacancies are occupied by alkali-metal atoms. It is also interesting to note that for all the systems there exists a threshold alkali coverage which has to be exceeded before alloy formation can begin. In this section we have demonstrated that the adsorption of alkali-metal atoms on metal substrates reveals a variety of interesting physical and chemical properties which are beyond a simple adsorption system suggested by the Langmuir-Gurney picture. The drastic increase of recent publications clearly demonstrates that this issue is of current interest. The main difference between Ru and A1 is that AI represents a 'soft' material which allows removal of Al-surface atoms during the adsorption process and hence the formation of surface alloys. Several observations especially on AI(111) ~ could be explained either by simple physical pictures or by involved theoretical investigations. Nevertheless, many questions remain to

477


be answered. For AI(111), e.g., no method has yet been found which predicts the details of the reaction path through which the surface A1 atoms are kicked out in the process of substitutional adsorption. While in the last few years much of the theoretical work has been done for the A1 substrate, theoretical studies on transition metals are almost non-existent. It therefore appears that a generalized theoretical view on the entire variety of alkali-metal/metal adsorption systems is required. 9.4.3. Coadsorption Ru(OOO1)-Cs-O and Ru(OOOI ) - C s - C O

The properties of alkali-metal and oxygen overlayers coadsorbed on transition-metal surfaces have been the subject of numerous investigations (see, e.g. Surnev (1989), Bonzel (1988)), not only because of fundamental interest but also for the role these systems play as actual promoters in catalytic reactions such as, e.g., ammonia synthesis (Ertl, 1991 a). Surface analysis of a catalyst used in industrial ammonia synthesis (Ertl et al., 1983) revealed that its active surface is uniformly covered by a submonolayer of a composite K+O phase with nearly 1:1 stoichiometry. Oxygen is not reduced under reaction conditions, but strongly interacts with K atoms. This interaction thermally stabilizes potassium while preserving its promoter efficiency. In the 'electrostatic' model by NOrskov et al. (1984) the electronic field formed by the dipole of the 'ionic' alkali-adsorbate complex predominantly stabilizes an electronegative molecule (such as CO) adsorbed in its vicinity. In this section we will mainly be concerned with thin films formed by coadsorption of Cs and O atoms on a Ru(0001) surface, starting with multilayer-Cs films. In addition, a few comments will be made on the interaction of Cs with CO. This specific coadsorbate system is particularly useful since it allows to use most of what we have learned in the preceding three sections. But let us start with the discussion about the (Cs+O)-Ru(0001) system. The oxidized films nicely reflect the complex chemistry of corresponding Cs oxide-bulk materials whose structures had been studied extensively (Simon, 1971; Simon and Wasterbeck, 1977; Vannerberg, 1962). If the alkali overlayer is in the 'metallic' state, i.e. at higher coverages, interaction with other molecules lead frequently to compound formation where

3 =2 r

1 0

IA 0

IB 1

t~'~l 2 3

I 4

1 5

t 6

I C,, t 7 8

O2-Exposure[k]

9

10

2O

Fig. 9.37. Variation of the work function of a 3-4 ML thick Cs film as a function of 02 doses (Woratschek et al., 1987).The regions markedby A, B, C denote the rangesover which the oxides Csl iO3, Cs20, CsO2, respectively, are formed.

478


Fig. 9.38. Structural unit of cesium suboxide Cs1103 (Simon, 1971). solely the valence orbitals of the alkali atoms are involved. One of the pronounced attributes found in these systems is the lowering of the work function upon interaction with oxygen. This has several practical applications in manufacturing high-efficiency photocathodes. Figure 9.37 shows the variation of the work function of thick Cs films with 02 exposure. Different regions denote the ranges over which various oxides are formed as identified by UPS and MDS (Woratschek et al., 1987). De-excitation of metastable noble-gas atoms (e.g. He*) (MDS) occurs at the surface with low work function (such as Cs-covered surfaces) via Auger de-excitation. The excitation energy of He* (21.6 eV) serves to emit an electron from the target. In contrast to UPS which probes a layer of about 5-10/~ thickness, MDS is extremely surface-sensitive and probes only the valence-electronic levels at the outermost atomic layer (Ertl and K~ippers, 1987). The clean Cs surface exhibits an intense band in MDS from the 6s states just below the Fermi level whose intensity even increases with small 02 doses. This initial stage of oxidation corresponds to nucleation and growth of the cesium suboxide Cs~ ~O3 (the structural unit is depicted in Fig. 9.38). The combination of UPS and MDS studies give clear evidence for the penetration of doubly charged O 2- ions below the surface, while the topmost layer still consists of metallic Cs atoms. The enhanced Cs 6s emission observed in MDS (surprisingly since part of the formation of O2-consumes part of the 6s electrons) was interpreted as a confirmation of the quantum-size effect (Woratschek et al., 1986) predicted by Burt and Heine (1978), to explain the lowering in work function. The conduction electrons tend to avoid 02- ions and are thus confined in space. As a consequence of the uncertainty principle, their kinetic energy rises, i.e., the work function is reduced. In this case, the 6s wave function should 'leak' further into the vacuum, which should increase in turn the 6s-derived MDS signal. Continuing 02 deposition (about 0.4 L) causes a decrease in intensity of the Cs 6s emission (MDS) due to the onset of the formation of cesium peroxide (Cs202) at the surface; this compound consists of Cs § ions (with empty 6s levels) and peroxide ions O~-. This process is then followed by transformation into cesium superoxide CsO2 which contains Cs § and hyperoxide ions 02. Switching to submonolayer-Cs films, one cannot expect to find Cs-oxide compounds (but at least it might be possible to identify similar 'planar' structural elements). The coadsorption of Cs and O on Ru(0001) results in a formation of a


479

large wealth of mixed phases with long-range order (Bludau et al., 1995), see Fig. 9.39. The general property of stabilizing alkali-metal adlayers by the addition of oxygen is reflected by the substantial increase of the desorption temperature for Cs in the presence of coadsorbed oxygen which suggests strong interactions between both adsorbates. Furthermore, the sticking coefficient S (defined as the ratio of the number of actually adsorbed particles over the number of impinging particles) for oxygen adsorption is substantially enhanced by Cs pre-adsorption: S changes from 0.4 (0c, = 0) to 1 (0c., = 0.3) (Kiskinova et al., 1986; Bludau et al., 1995). It is generally assumed that the increase of the oxygen-sticking coefficient is due to the enhanced electron flow into the anti-bonding molecular orbital accompanied by a reduction of the activation barrier of dissociation. Starting with an ordered R u ( 0 0 0 1 ) ( ' ~ - • structure, i.e. one monolayer of Cs, already the addition of small doses of oxygen (0.05 L) leads to the appearance of a new incommensurate superstructure and a gradual disappearance

Fig. 9.39. Schematic phase diagram for the Ru(000l)-Cs-O coadsorption system; after Bludau et al. (1995).


480

of the ('~-x'~-)R30-LEED pattern. This clearly indicates a strong interaction between oxygen and the Cs-adsorbate layer. Oxygen exposures between 0.3 and 0.9 L and subsequent annealing give rise to an appearance of a new (~-3-x'~-)R30 structure. The corresponding stoichiometry is Cs:O of 1:1. A respective LEED structure analysis (Over et al., 1992b) has provided a model in which Cs and O atoms are located in hcp sites with respect to the Ru(0001)-substrate lattice; structural parameters are summarized in Fig. 9.40. Thus, a 'salt-like' structure is developed which represents the optimum arrangement allowing the greatest number of oppositely charged 'ions' to touch without requiring any squeezing together of 'ions' with the same charge. With respect to the positions of the Cs atoms, atomic oxygen resides in threefold sites below the plane formed by the alkali-metal adlayer, which is consistent with work function and MDS data (B6ttcher et al., 1991 ). The appearance of atomic oxygen instead of molecular oxygen (as contained in the stoichiometric equivalent Cs202 compound) is confirmed by HREELS measurements (Shi et al., 1992). Compared to the structures of the respective pure adsorbate phases, the Ru-O and Cs-Ru bond lengths are modified in a way consistent with a net transfer of electronic charge from Cs to O: the oxygen hard-sphere radius increases by 0.12 A, and the effective Cs radius decreases from 2.2 A in the clean Ru(0001)-Cs phase to 2.1 A with the coadsorbate phase (Over et al., 1992b). Moreover, oxygen atoms in the Cs-O layer at the hcp sites experience Ru (O001)/Cs/O -(~3 x~-3} R30 ~

b)

O- radius: 0.8/~

ygen

Fig. 9.40. Structural model for Ru(0001)('43-x'~-)R30-Cs-O with optimal structural parameters as found by a LEED analysis (Over et al., 1992b).

481


~

7.28

(3x2~/3)rect Cs-O / Ru(0001 )

CsO2-bulk

Fig. 9.41. Planar structure model for the system Ru(0001)(3x2"~-)rect-Cs-O. (a). The main structural element Cs-O2-Cs zigzag chain (b) is compared with the CsO2 bulk structure (c) (Bludau et al., 1995). an enhanced charge density induced by the Cs layer, as expected by comparison with calculated charge-density distribution of the related AI(I 11)(,f3-x,~-)R30-Na system with Na residing in a threefold-hollow site (Neugebauer, 1992a). This nicely reflects their electron affinity. As summarized in Fig. 9.39, the coadsorption of Cs and O leads to an overwhelming number of different ordered structures; for a comprehensive representation of the geometric structures, see Bludau et al. (1995). We will restrict ourselves here to two illustrative examples which point to similarities between Cs oxide-bulk structures and Cs-O surface species. Exposure of 02 to a Cs monolayer (0c~ = 0.33) lead to a (3x24-3-)-rect. structure at 0o = 0.68, corresponding to a stoichiometry Cs:O = 1:2. Both the appearance of a glide-plane symmetry and the dominance of certain LEED beams in the LEED pattern give evidence for a model exhibiting distinctive Cs-O2-Cs zigzag chains. The interatomic Cs spacing in these chains (7.2 ~ ) agrees well with that found in crystalline CsO2 (7.28 ]k) (Fig. 9.41); note that the stoichiometry in the (010) surface of CsO2 is also Cs:O = 1:2. A similar result was found with the ('(7-• structure which was prepared by oxidizing a submonolayer Cs film (0c., = 0.28). Again, the stoichiometry turned out to be Cs:O = 1:2. A comparison of this surface structure with the (010) surface of CsO2 is presented in Fig. 9.42 from which the structural element of now linear C s - O - O Cs chains becomes evident. The formation appears to be a compromise of attaining a Cs-O bond length comparable to CsO2 and the interaction of the Cs-O complex with the substrate. The higher density of Cs atoms in the (3x24-3-)-rect. structure (0.052 atoms//~, 2) compared to that in the (010) surface of CsO2 bulk (0.044 atoms/]k 2) causes a large strain in the Cs-O overlayer resulting in the formation of zigzag instead of linear C s - O - O - C s chains. The Cs density in the ~ structure (0.045 atoms/~ 2) is almost the same as in the corresponding oxide. Here, the strain induced by the substrate's corrugation only is uniaxially relieved: C s - O chains 2 and 3 (Fig. 9.42) are parallel-shifted by 3.6/~ (half length of the @-- periodicity) and compressed more tightly in the direction perpendicular to the chains. The most striking feature of the phase diagram (Fig. 9.39) represents the existence of the (x/-7-x@--)R19.1 structure over a wide range of the Cs coverage,

482


Fig. 9.42. Structure model for Ru(0001)(~--• 19. l-Cs-O in comparison with the (010) surface of CsO2 bulk material (Bludau et al., 1995).

while the respective optimum oxygen coverage is determined by the stoichiometry of two oxygen atoms per Cs atom. The presence of the (~-• 19.1 LEED pattern down to coverages as low as 60% of the nominal coverage indicates island growth and underlines the strong driving force to build up the CsO2 surface species. Upon raising the sample temperature, these (~r7--x'~--)R19.1 islands dissolve as observed in LEED by the transformation of a (~--x4-7-)R 19 ~ pattern into a ring-like pattern intersecting the nominal ~ spots (Trost et al., 1995). The ring-like LEED pattern indicates that still the ~ distance is abundant, and since Cs is a much stronger scatterer than oxygen, this might be attributed to the persistence of C s - O - O - C s clusters, while losing the azimuthal order. Even for coverages higher than the density of the CsO2 bulk (010) plane, the Cs-O layer forms a (3x2~f3-)rect. lattice, now exhibiting C s - O - O - C s zigzag chains instead of linear chains. Altogether these findings give strong evidence for the formation of two-dimensional Cs oxides with

483


ITDS

Cs/Ru(0001), CO/Ru(0001)

Oc~-0.2s

Oco~sat.=0.54

,~

a)

CO/Cs/Ru(0001) Oc~=0.25 |176

=~"

~.

[ I

b)

ffl co

400

600 800 1000 1200 temperature (K)

Fig. 9.43. Thermal desorption spectra of the systems: (a) Ru(0001 )-CO: saturation of CO at T = 300 K (0co = 0.54) and Ru(0001 )(2• (0c.,= 0.25); (b) Ru(0001 )(2x2)-Cs-CO: Cs precoverage = 0.25; CO coverage = 0.50 (Over et al., 1995c). a stoichiometry of two oxygen atoms per Cs atom. The commensurability of the qff and the (3x2q-3)rect phase, on the other hand, reflects the influence of the corrugation potential of the substrate on the formation of these structures. While the interaction of Cs with oxygen is dictated by the chemical reactivity between these species, as seen by the formation of structures which are consistent with a surface species of Cs oxide, the situation changes drastically when we proceed to the coadsorption of Cs and carbon monoxide on Ru(0001). The direct interaction between Cs and CO is very weak. However, when brought together onto a Ru(0001) surface, they interact strongly with each other, as demonstrated by the effect of thermal stabilization of each adsorbate. As indicated in the thermodesorption spectra (see Fig. 9.43) of singly adsorbed Cs and CO in comparison with the compound system, not only both species are thermally stabilized, but also a coincident desorption of both species takes place at about 600 K. This thermal stabilization can readily be explained by a substrate-mediated interaction between these species, i.e., the alkali metal /'lushes the surface with electron-charge density that the CO molecule in turn uses in order to form a stronger back-bonding (cf. w 9.2.1 ). This would explain why CO is more strongly bound to the substrate. Cs is also more strongly bound to the Ru(0001 ) surface since this mediated charge transfer from Cs to CO re-ionizes the Cs atom which strengthens the bond to the substrate according to the L a n g m u i r - G u r n e y model (cf. w 9.4.1). The coincident desorption, on the other hand, is rationalized by an autocatalytic reaction process: with the release of CO from the surface the cause of the stabilization of Cs also ceases (and vice versa).

484


This interpretation is supported by HREELS measurements which showed that indeed the C - O stretch frequency is reduced substantially from 252 meV (clean Ru(0001 ) surface) (Thomas and Weinberg, 1979) to 203 meV for the (2x2)-Cs-precovered Ru(0001) surface (Jacobi et al., 1994) consistent with an increased backdonation into the anti-bonding CO 2n-orbital. The re-ionization has also been demonstrated by HREELS by monitoring the Cs against substrate vibration. For very small Cs coverages this vibration showed up in HREEL spectra at about 8 meV. This vibration disappears, however, for higher Cs coverages beyond 0.25, thus indicating that the Cs layer becomes metallic and the dipole excitation is screened. If onto such a metallic (2x2)-Cs surface CO is coadsorbed, the Cs-Ru vibration reappears (Jacobi et al., 1994) consistent with the interpretation of an re-ionized Cs overlayer. Since the Cs-Ru vibration appears at nearly the same energy as for low Cs coverages, comparable bond strengths between Cs and Ru are expected, so that the addition of CO increases the effective strength of the Ru-Cs bond. What remains to be clarified is the actual adsorption geometry of this coadsorbate system. As we have already shown in the last sections, both CO and Cs sit in on-top positions on Ru(0001) when adsorbed separately. If Cs and CO form

Fig. 9.44. CO (a) and Cs (b) both reside in on-top position in their pure phases formed on Ru(0001 ). In the mixed (2x2) phase of these species it is not possible that both can retain on-top site. Either CO or Cs has to leave its accustomed adsorption site (c, d). The actual adsorption geometry found is model (c).


485

a mixed (2x2) phase, it is not possible that both species can retain their 'natural' adsorption site for steric reasons (cf. Fig. 9.44). A recent LEED analysis (Over et al., 1995b,c) revealed that Cs remains in on-top position, while CO switches from on-top (clean surface) to a threefold-coordinated hcp site (Cs-precovered surface). The corresponding structure is depicted in Fig. 9.45. The site change of CO can be explained if one recalls that high coordination sites are characterized by an improved back-donation, while on-top occupation favors the mechanism of ~ donation (cf. the discussion in w 9.2.1). Hence, the actual adsorption site of CO will be a competition between both effects determining the energetically lowest adsorption geometry. This mechanism is sufficient for a proper description of the energetics of this coadsorption system since it is known from theoretical studies of related alkali-metal/metal systems that the adsorption energy difference for Cs at different adsorption sites is only very small (about 20 meV) (Neugebauer and Scheffler, 1992b). Therefore, the total energy of the mixed (CO+Cs) system is determined by the adsorption site of CO. The presence of coadsorbed alkali-metal atoms improves the capability of back-donation, due to the enhanced electron charge density at the surface, and hence forces CO to change its adsite from on-top to high-coordination sites. Theoretically, this site switching has been proposed by MiJller (1993) for a related system CO+K on Pt(1 11 ). Another aspect of the bonding geometry consists in the enhanced Cs-induced buckling (0.18/~) found in the topmost Ru layer when compared to the clean (2x2)-Cs (0.10 A) surface. This effect can be traced back to the additional bonding of the remaining three Ru atoms in the (2• unit cell to the

Fig. 9.45. The atomic geometryof the coadsorbate phase Ru(0001)(2x2)-Cs--CO (Over et al., 1995b).

486


CO molecules, what in turn weakens the R u - R u bonding to the single Ru atom beneath the Cs atom and hence allows for an enhanced rumpling of the topmost Ru layer. Probably the most apparent feature of the ( 2 • structure is the 'saltlike' arrangement of Cs and CO: the interaction in the mixed overlayer of Cs and CO adsorbed on Ru(0001) seems to be described by a two-dimensional lattice of adsorbates interacting electrostatically. The charge transfer from Cs, likely to be mediated by the Ru(0001) substrate, to the 2n: CO anti-bonding orbital is responsible for the observation of a check pattern of anions and cations. The electrostatic interaction between CO and Cs dominating the lateral interaction is supported by the observed island growth of this mixed phase. If one starts with a liquid-like Cs overlayer (coverages 2-d liquid ---> 2-d solid transitions in the monolayer and 2-d phase diagrams analogous to their 3-d counterpart have been drawn. The existence phase domains are shown in pressure-coverage (Fig. 10.2), temperature-coverage (Fig. 10.3), or temperaturepressure (see further Fig. 10.15) diagrams. The pressure-coverage phase diagram can be directly deduced from sets of volumetric adsorption isotherms (see w 10.4.2.1) as shown in Fig. 10.2. When two 2-d phases coexist, the system is monovariant according to Eq. (10.6); this results in a vertical isotherm at a pressure P function of temperature. A triple point (coexistence of 2-d gas, 2-d liquid, and 2-d solid phases) is represented by a single vertical line in the pressure-coverage diagram at temperature T~D. For T < T~D, one can see the 2-d dilute phase (gas) --) 2-d dense phase (solid I) transition with increasing coverage. For T~t D < T < TED, two sharp transitions can be seen in the series of isotherms: the first step characterizes the 2-d dilute phase (gas) --) 2-d dense phase (liquid) transition, whereas the second small substep, is the signature of the 2-d dense phase (liquid) ---) 2-d dense phase (solid I) transition. At T > T~D, the first transition rounds up whereas the second remains vertical. ~D is the 2-d critical temperature. The phase diagram is classical

2d S

A ..I W O
solid II) was first observed between commensurate and incommensurate solids in the krypton/graphite system and is evidenced by a small substep in the isotherm at higher coverage. Since then, many transitions of this kind have been reported and studied in numerous other systems (see w 10.2 and 10.3). Volumetry measures the number of adsorbed molecules and the definition of the coverage is then conventional. For chemisorbed layers, coverage one is usually defined as that corresponding to the areal density that is equal to the density of substrate adsorption sites. In physisorbed systems, molecules do not necessary sit on lattice sites of the substrate surface. As a consequence, coverage one is specified with respect to a well defined dense monolayer phase. It results that, for a given substrate, the areal density of physisorbed monolayers at coverage one will depend upon the adsorbate under consideration. In the example of Fig. 10.2, coverage one corresponds to a layer of solid I just before the solid I-solid II transition. In most of the cases, when no substep is observed, the monolayer coverage is defined by the inflection point of the plateau after the step. When the 2-d solid phase structures are known, coverage one can be determined as that corresponding to a given phase. For ethane on graphite shown in Fig. 10.3, coverage one is assigned to the commensurate ("~3-x",f3) $3 phase.

1.2

S 3 . 2 nd

1.0

S2" S3 ~5

2 $1 S 2

0.8 .< c~

layer

I 12+

~J3

12

t.

S I T ' 21

_

L

-I"1,

.... S 1 0.6

o u

F 0.4

I 1 + 2D Gas

S 1+ 2D Gas

0.2

I

I

I

I

I

I

20

4Q

60

80

1 O0

120

Tic

I 140

TEMPERATURE (K)

Fig. 10.3. Phase diagram of ethane monolayer adsorbed on graphite. Coverage of unity corresponds to that of a completed $3 (commensurate ~-x~/-3-) solid phase. (From Gay et al. 1986b).

The structure of physically adsorbed phases

511

Perhaps, the clearest and most straightforward signature of 2-d behavior in an adsorbed monolayer comes from heat capacity measurements. It is expected from statistical thermodynamics that in 2-d, the lattice heat capacity C of a solid follows a C ~ T2 law whereas in 3-d solids a C ~ T 3 law prevails; e.g., the exponent of the temperature is equal to the dimensionality of the system. Indeed, measurements have shown that the heat capacity of a monolayer of 3He or 4He on graphite obeys the quadratic law (Goodstein et al., 1965; Dash, 1975; Hering et al., 1976; Van Sciver and Vilches, 1978). After experimental evidence of the existence of 2-d liquids from adsorption isotherm measurements (indirect) and mobility measurements using M0ssbauer spectroscopy (see w 10.4.2.4.), another convincing indication came from the shape of the liquid-vapor phase boundary near the critical point. In bulk neon and nitrogen (Pestak and Chan, 1984), the density difference between liquid and vapor along the boundary near Tc is described by (P~iq- P~,p) " (To - T) 1~

(10.7)

where 13 = 0.32 close to the theoretical value 0.315 (Stanley 1971). The density p of a 2-d system at the liquid 2-d-vapor 2-d boundary also varies as a power law, but with [3 = 1/8 in theory (Stanley, 1971). Measurements have been performed in the case of a methane monolayer adsorbed on graphite (Kim and Chan, 1984) and the value 13 = 0.13 has been found, again very close to the theoretical prediction. Another experimental measurement on the second layer of argon on cadmium chloride (Larher, 1979a) has given 13 = 0.16, also in pretty good agreement with the theoretical value.

10.1.2.2. Dynamics of 2-d phases Two kinds of motions animate an adsorbed atom: vibration and translation above the substrate surface. When molecules are concerned, rotations may also occur. Measurement of vibrational modes or translational and rotational motions is very useful to understand the microscopic properties of the system, particularly the adsorbate-adsorbate and adsorbate-substrate interactions. Average vibrational properties appear in the Debye-Waller factor which can be deduced from measurements of the temperature dependence of the elastic scattering of electrons (LEED), atoms (He), neutrons, or X-rays. The eigen frequencies of individual molecules or the density of states of collective excitations (phonons) can be determined through surface spectroscopies.

Individual vibrations of adsorbed molecules.

The D e b y e - W a l l e r factor gives the mean square displacement of adsorbed molecules. A typical value is around 0.015 A,2 for a xenon atom above the graphite surface at T = 60 K (Coulomb et al., 1974). This value is about five times smaller than the component of vibrations of surface atoms perpendicular to the (111) face of a bulk xenon crystal. This reduction indicates a partial hindering of the xenon vibration when adsorbed onto the graphite surface. Modeling of this effect gives interesting information about the potential and the force constant of the xenon-graphite interaction.

512

J. Suzanne and J.M. Gay

More important is the measurement of the vibrational frequencies which are directly related to the adatom (or molecule)-surface force constant through the second derivative of the potential. Here again, it is worth comparing the various modes of vibration of an adsorbed molecule to those occurring in the bulk. As an example, we consider the molecular vibrations observed in butane molecules adsorbed on graphite (Taub et al., 1977a, 1978). The incoherent inelastic neutron scattering spectrum (see w 10.4.2.2) obtained within the monolayer coverage range is shown in Fig. 10.4. It shows a rich structure with four well-defined peaks which are compared with model calculations. The first three peaks are due to intramolecular CH 3 and CH2 torsions of the bulk solid. There is no appreciable change in the frequencies due to physisorption of the butane molecules. The fourth peak at 112 cm -~ (i.e. 14 meV) is interpreted as a rocking surface mode about an axis parallel to both the surface and the hydrocarbon chain direction. Model calculations also predict two other modes consistent with the broad peak centered at 50 cm -~ (i.e. 6.2 meV). The first one is a rocking mode with an axis perpendicular to the hydrocarbon chain and the second is a bouncing mode of the entire molecule perpendicular to the surface. The models suggest that the butane molecule is adsorbed with its carbon skeleton parallel to the graphite surface. The important conclusion of these studies is that physisorption produces only a weak perturbation of the internal molecular modes. Besides, a bouncing mode specific to the adsorbed state appears. III

I

I

I

I

!

j

/

CH2-CH2 CH 3 TORSION

I

P,

~\tt~/'k~

t

C4HIo MONOLAYER ON CARBOPACK B T = 77K

,11 111 I i 400300 200

I

I I00

I 50 AE

I 25 (cm "1)

Fig. 10.4. Incoherent inelastic neutron scattering spectrum of a butane monolayer adsorbed on graphite. The background has been subtracted from the spectrum. The arrows indicate the energy of the three lowest-lying modes of the bulk solid. The vertical lines at the bottom of the figure are calculated modes. The inset shows the proposed orientation of butane with respect to the graphite basal plane. Only the four coplanar hydrogen atoms (O) on one side of the carbon skeleton have been included for clarity. (From Taub et al., 1977a).


513

Collective motion. An ideal 2-d monoatomic crystal exhibits two vibrational modes involving lateral displacements of the atoms within the plane of the crystal. The two modes are orthogonally polarized either in the direction of propagation or perpendicular to it; these are the longitudinal mode and the transverse mode respectively. One can get the dispersion relations c0(k) from standard models treated in solid state physics textbooks. The calculations are similar to those used in the model of the linear chain of atoms (Kittel, 1976). Furthermore, upon adsorption, a third mode appears due to vibrations of the adatoms perpendicular to the substrate surface. If the substrate is supposed to be rigid and smooth, the mode normal to the surface is simply given by

coI -

(10.8)

where k0 is the force constant of the adatom-substrate interaction. This mode is independent of the wavevector k parallel to the layer. It is referred to as an Einstein mode. In reality, a dynamical coupling occurs between the substrate (surface wave and bulk phonons) and the adsorbate. Atom scattering experiments on rare gas monolayers adsorbed onto graphite or on fcc metals (Mason and Williams, 1983; Gibson and Sibener, 1985; David et al., 1986), or on a methane monolayer on MgO(100) single crystal surface (Jung et al., 1991) have shown that the normal modes are nearly dispersionless. Experimental phonon dispersion curves are shown further in w 10.4.3.2 (see Fig. 10.31). The results of the theoretical models taking into account the adlayer-substrate dynamical coupling (see for instance, Hall et al., 1985) have shown an overall quantitative agreement with helium atom scattering experiments in the case of rare gases adsorbed on graphite (Toennies and Vollmer, 1989) and on Pt (111) surfaces (Kern et al., 1986b, 1987c; Zeppenfeld et al., 1990b). Translational and rotational motions. The existence of 2-d plastic and fluid phases is deduced from the observation of translational and rotational diffusion in adsorbed layers. Rotational diffusion is of course particular to molecular adsorbates. We will see in w 10.2 that adsorbed layers may form solids that are either commensurate or incommensurate with the underlying substrate lattice. Similarly, 2-d adsorbed fluids can behave as lattice fluid or as normal isotropic fluids. In the former case, the atoms or molecules are strongly correlated and keep a partial positional and orientational order. They perform a jump diffusion process between surface sites with a residence time in each site that is long compared to the time required for the jump between sites. In the latter case, the molecules display a translational brownian motion leading to a 2-d isotropic diffusion; that is, the fluid does not feel the substrate corrugation any longer. The system may pass from one type to the other by increasing the temperature. Diffusion has been investigated in various types of hydrocarbon molecules physisorbed on graphite: CH4 (Coulomb et al., 1981), C2H 6 (Coulomb and Bienfait, 1986), CzH 4 (Grier et al., 1984), and MgO substrates: CH4 (Bienfait et al., 1987b).

514


Table 10.1 Translational diffusion coefficient Dt (in 10--6cm 2 S-1) of CH4 adsorbed on MgO(100) and graphite(0001), and bulk methane. (From Bienfait et al., 1987b) Temperature

Coverage

CH4 bulk

(K) CH4/graphite 0.63 72 88 91.5 97

0.72

CH4/MgO 0.90

83 115

0.8 =0 5-6

=0 --0

12

30

40

Quasi elastic neutron scattering is a very powerful tool for these studies, as explained in w 10.4.2.2. Methane adsorbed on graphite and MgO, and ethane adsorbed on graphite provide two interesting examples described below. Methane on graphite and methane on MgO(lO0). Table 10.1 gives the translational diffusion coefficient D t o f C H 4 molecules on graphite and MgO (100) surfaces (2-d melting temperatures are 56 K and 82 K, respectively). It clearly appears that the 2-d solid is stabilized on the MgO substrate to temperatures higher than on graphite. Both methane solid layers are commensurate, ('~3-x'~-) on graphite and c(2x2) on MgO. This means that the substrate corrugation is larger for the MgO surface than for graphite. On MgO, the methane molecules in the 2-d fluid phase continue to experience the strong modulation of the square substrate and the motion is slowed down. Between 87 K and =100 K, jump diffusion takes place between equivalent sites 4.21 ]k apart. The mean residence time t on site has been experimentally evaluated to be t = l x l 0 -~~ s at 88 K and 4x10 -~ s at 97 K. The corresponding translational diffusion coefficient Dt (given in Table 10.1 by Bienfait et al., 1987b) can be deduced from t with Eq. (10.23) given in w 10.4.2.2. At 72 K, the methane molecules perform an isotropic rotation around their center of mass which remains located at surface sites. Ethane on graphite. Ethane on graphite also presents lattice liquid phases that will be described in more detail in w 10.3.3. Because of the rod-like shape of the molecule, various rotational motions can occur as found experimentally in quasi elastic neutron scattering studies (Coulomb and Bienfait, 1986). In the low density plastic phase I~ (see phase diagram in Fig. 10.3), ethane molecules are animated by rotational and translational motions. Analysis of the neutron scattering data shows that the rotational motion is isotropic around the center of mass of the molecule. Up to T = 87 K, the molecules perform a jump translational motion on the graphite sites in agreement with LEED experiments which show a strongly correlated fluid with a (2x2) commensurate structure (Gay et al., 1985). Table 10.2 gives the rotational and translational diffusion coefficients in the I~ phase versus temperature (Coulomb and Bienfait, 1986).


515

Table 10.2 Translational diffusion coefficient Dt and rotational diffusion coefficient Dr for the isotropic motion of the C2H6 molecule in the I1 phase for different coverages and temperatures.(From Coulomb and Bienfait, 1986) T (K)

66.4

71.4

76.2

84.1

87

122

Coverage

0.54

0.54

0.54

0.54

0.4 and 0.63

0.4 and 0.63

D r in 101~ -l D t in 10-6 cm2s -l

5-t-1 -

200

,,

9

"i

,

I

I

I

'

'I

I

100 nA is large enough to perturb appreciably most of the physisorbed species. The heaviest rare gases are probably the only adsorbates which allow the use of AES for measuring equilibrium adsorption isotherms. Molecular adsorbates such as methane or other hydrocarbons would suffer electron stimulated desorption or dissociation. 10.4.3.2. A t o m scattering

Light atom (H, He) beam scattering has become a powerful tool for investigations of the structure and dynamics of adsorbed layers. Major characteristics of He scattering are given in Chapter 7. Therefore, we mention only some particular properties of this technique that make it well suited to investigations of physisorbed layers. First, the low energy of the He atoms and their inert nature ensures that He scattering is a completely nondestructive probe, especially with delicate phases, like physisorbed layers. A thermal He nozzle beam has a wavelength (1.09-0.46 ,A,) comparable with the interatomic distances in the adsorbed layers and is therefore well suited for diffraction studies. In addition, the energy of the He atoms (17-100 meV) is in the same range as those of collective excitations in overlayers. The advantage of He scattering over inelastic neutron scattering (w 10.4.2.2) is in the use of single crystals and thus the capability to measure phonon dispersion curves in different crystallographic directions. An example of time-of-Ilight He spectrometer is shown in Fig. 10.28 (David et al., 1986). In such a set-up, the momentum resolution is about 0.01 A,-~ for ~ = 1 A,; this resolution is sufficient for most usual qualitative discussions. Energy resolution can be about 0.4 meV FWHM, for 20 meV He beam energy, but it can be brought to ~0.1 meV with lower beam energies. Due to the large atomic scattering cross-section (e.g., >110 /~2 for HeXe/Pt(l 11) at 18 meV after Poelsema et al. (1983) atom scattering is sensitive to very low coverages. This property is used for detecting impurities, like hydrogen or for studying the gas ---> (2-d solid+gas) transition in adsorbed layers. In a way similar to LEED isotherms, the attenuation of the specular beam upon adsorption changes at the onset of 2-d islanding. The coverage dependence of the attenuation is used to determine the lateral interaction energy between the adatoms (Poelsema et al., 1983). Figure 10.29 shows the specular intensity reflected from a Pt(111) surface upon condensation of Kr. Like any diffraction technique, H or He coherent elastic scattering allows determination of the real space lattice and, in particular, the symmetry and orientational epitaxy of adsorbed layers (Ellis et al., 1981; Chung et al., 1987; Kern et al., 1988). The power of the He diffraction technique is illustrated by studies by Kern et al. (1988). They investigated in detail the Xe monolayer physisorbed on Pt( 111 ). Using a well ordered defect-free single crystal surface, they identified unambiguously commensurate ---->striped incommensurate ---->hexagonal aligned incommensurate ----> hexagonal rotated incommensurate transitions with increasing incommensurability. Figure 10.30 shows the (1,2) diffraction lineshape for incommensurate Xe layers. The intensity analysis involves calculations of the elastic diffraction probability based on the He-adsorbed Xe layer potential (Ellis et al., 1981; Schwartz, 1987).


556

D ifferentiat pump stages

Nozzle chamber

O

\

/I

k

I

;i L,-,3

/

I\ C

S

Scattering chamber

Fig. 10.28. Schematic view of a high resolution He time-of-flight spectrometer used tar physisorbed layer studies. N is the nozzle beam source, S I and $2, and A I ..... A5 are skimmers and apertures, respectively. Gas is introduced by the gas doser (G) in front of the sample (T). (CMA) represents a cylindrical mirror analyzer for Auger spectrometry. The chamber is also equipped with an ion gun (IG) and a LEED (L). The scattered beam is detected by a quadrupole mass analyzer with channeltron (QMA). (C) is a chopper. (From David et al, 1986).

Ts- 54 K

y

0.96 o

!1

0.92

0.88 1

0

I 1 L, L I t &O 80 120 160 exposure time (secl

Fig. 10.29. Specularly reflected He beam intensity from a Pt(l 11) surface upon exposure of Kr (PK,- = 2. I x 10-9 Torr) at 54 K. (From Kern et al., 1988).


557

Fig. 10.30.3-d plot of the (1,2) He diffraction peak of incommensurate Xe layers on Pt(l 11) at various incommensurabilities corresponding to (a) a striped phase, (b) coexistence of a striped phase and a hexagonal aligned_phase and (c) a hexagonal rotated phase. Q denotes the reciprocal lattice vector in the F' K direction, while q~denotes the azimuthal angle. (From Kern, 1987a). Surface lattice dynamics (see w 10.1.2.2) has been studied by inelastic He scattering. The change of dynamical behavior from the monolayer to thicker layers is informative of the growth mode of the adsorbed films. Figure 10.31 shows the phonon dispersion curves of Kr overlayers (1, 2, 3 monolayer thick) on Ag(l 11). Very detailed studies can also be performed to reveal the adlayer-substrate dynamical coupling (Zeppenfeld et al., 1990a). The attractive part of the He-surface interaction potential is usually neglected (hard wall approximation), since it changes only weakly the corrugation shape of a surface. It is, however, related to the resonant scattering effect or selective adsorption. These resonances appear in azimuthal scans of the He specular reflection intensity. An example has been recently given by Jung et al. (1989) for the orientational configuration of the methane molecule adsorbed on MgO (100), see Figure 10.32. The resonance lineshapes allow the determination of the binding energies.

10.4.3.3. X-ray diffraction X-ray diffraction from layers physisorbed on single crystal substrates benefits from the advantages of X-ray scattering (see Chapter 7). The use of simple kinematic scattering theory for quantitative interpretation of the diffraction lines makes the technique particularly attractive. The high resolution of the experimental set-up enables accurate investigations, specially by the use of synchrotron radiation X-ray sources, although conventional sources can also provide sufficient flux to allow

558


I

-

v

I

v , "

I

v . / ,.._v,.. w

_

"

.

.

.

d~V

.

r

A

z

1 8UL

a

F

O. 25

13. 5

O/Omax

O. 75

I

M

Fig. 10.3 !. Phonon dispersion curves of mono-, bi-, and trilayers of Kr overlayers on Ag(l 11) from inelastic He scattering measurements. The dispersion curve of bulk krypton is also reported. (From Gibson and Sibener, 1985). valuable experiments on single crystals. The experimental studies using powdered substrates (see w 10.4.2.3) are more easily tractable, but they suffer from the lack of direct orientational structural information. Only single crystal studies can provide unambiguous results about the orientational structure and epitaxy of physisorbed layers. Single crystal studies require a surface substrate that is easy to clean and that can be made with relatively large uniform regions. Natural graphite (Specht et al., 1987; Hong et al., 1989) or silver single crystal (Greiser et al., 1987) have been used to date. The surface area is much smaller than in powder experiments, so that the surface coverage can dramatically change with a weak variation of pressure in the cell. This difficulty is overcome by putting a mass of powdered material in the cell that acts as a stabilizing ballast (Hong et al., 1989). Great care should be taken with the cleanliness of the surface. A common experimental set-up consists of a uhv chamber with a device allowing periodic cleaning; e.g., thermal flash, sputtering, etc. The sample is oriented in a way that the scattering vector lies nearly in the surface plane. This grazing incidence geometry yields a lower background level and higher peaks than a transmission geometry, as used with powders. The experimental studies of layers physisorbed on single crystals, reported to date in the literature, essentially deal with Kr (Specht et al., 1987) and Xe (Hong and Birgeneau, 1989; Hong et al., 1989) on graphite and Xe on silver (Greiser et

559


I

I

I

I

I

c

El GEl A Fi/;?~ A~'.b B

M/ V ~'" 'q/~a8 =71.5~

A

/

c=.

^

..

\

70.5

~

03

c--.

U~ v

>.. F--cO

Z W

t--

z_ C15. ., 60

"

~8o

o

~ 2ooo

(1)

E

c oll

,

i

,

l

,

O(A I)

I

I

I

!

,

!

I

74.00K

0

I

i

59.77 K

I~

'59.

'

50-

~ 1.6

I

2o

'~176176 / 1.5

,

40

i50 t-

~~176

I

1.?'

0

---

1__1__

1.62

~_l~_J___a_L_s

1.66 Q(A')

1.70

-0.4

0 0.4 ( degl ees )

0.8

Fig. 10.33. Comparison ot'diffraction lines from 1.16 layers of xenon adsorbed on vermicular (a)and single-crystal (b) graphite substrates. Azimuthal to-scans are significant only on a single-crystal. The commensurate structure ('~-3-x~/3-)R30is characterized by Q = 1.70 ~-~ and the azimuthal angle to = 0 ~ (From Hong et al., 1989). temperatures, the xenon monolayer is incommensurate (Q = 1.62 ,/k-~) rotated (co = + 0.6~ The rotation angle and the lattice parameter decrease continuously as the temperature is lowered. At 79.78 K, Q = 1.67/~,-~, the rotation angle goes to zero; then the realigned incommensurate phase appears. At comparable temperatures, the diffraction lines from the powder experiment are consistent with the single crystal study. Their typical sawtooth shape has been already explained in w 10.4.2.2 in the similar case of neutron scattering. The power of the single crystal technique is nicely demonstrated by this example. Besides the surface X-ray diffraction experiments that investigate the projection of the reciprocal lattice onto the surface, other studies were dedicated to measuring the Bragg rod profiles (Hong and Birgeneau, 1989). These latter experiments provide information about the growth of multilayer films.

10.4.3.4. Ellipsometry Ellipsometry is a non-perturbative tool for studying physisorbed layers. It measures changes in polarization upon reflection of a light beam from a surface. Its applicability has been considerably improved by the availability of uhv techniques and by

The structure o.]physically adsorbed phases

561

the computerized automation of the measurements. The principle of the technique has been described in numerous papers (Quentel et al., 1975; Bootsma et al., 1982; Nham and Hess, 1989). Experimental details have been given by Jasperson and Schnatterly (1969) and by Volkmann and Knorr (1989). Adsorption isotherms can be measured through the change of the ellipsometric parameters versus temperature at constant pressure: - A, the phase change between the incident and the reflected beam, - p, the ratio of the amplitude reflection coefficients for incident light linearly polarized parallel and perpendicular to the plane of incidence, i.e., s and p polarization. The Fresnel theory for reflection by plane parallel continuum layers can be used to relate these ellipsometric quantities to substrate and overlayer properties. Particularly, the surface coverage can be measured with a sensitivity of the order of 0.01 monolayer (Quentel et al., 1975; Bootsma et al., 1982). A great advantage of the technique lies in the weak interaction of the light with atoms or molecules which produces a negligible influence on the adsorbed monolayer. In situ measurements at high gas pressures are possible which gives the capability to explore the whole 2-d phase diagram on single crystal surfaces. Experiments have been performed on various adsorbed layers on graphite: xenon (Quentel et al., 1975; Faul et al., 1990), krypton (Faul et al., 1990; Nham and Hess, 1989), argon (Faul et al., 1990; Nham and Hess, 1989; Youn and Hess, 1990a), oxygen (Drir and Hess, 1986a; Youn and Hess, 1990b), nitrogen (Faul et al., 1990; Volkmann and Knorr, 1991), methane (Nham and Hess, 1989), tetrafluoromethane CF 4 (Nham et al., 1987) and ethylene (Drir and Hess, 1986a). Most of the recent experiments have studied multilayer adsorption of these adsorbates (see w 10.6.1) and have shown that ellipsometry is a powerful tool for studying growth modes and wetting behavior on single crystal surfaces. 10.5. R e v i e w o f e x p e r i m e n t a l results

Table 10.4 reports most of the physisorbed systems experimentally investigated in the monolayer range. It is certainly not exhaustive and the reader is invited to refer to the references given in the papers that are here mentioned. Some of the systems have been extensively characterized; only a few properties are known for others. The authors' intention is to provide the essential references on the experimental work available in the literature. For each system, the structural information indicates the presence of commensurate, incommensurate, etc. phases without any specification about their domain of existence. Similarly, the melting behavior lists only whether the transition is first order or continuous and does not include possible changes with coverage within the monolayer range. 10.6. T h e s t r u c t u r e o f m u l t i l a y e r f i l m s

While studies of physisorbed monolayers continue, recent years have seen a shift to multilayer films. The role of the third dimension of these systems, that is not equivalent to the two in-plane dimensions, has attracted the interest of many

562


theorists and experimentalists. The physics of molecular films evolves from two to three dimensions as the films thicken. The systematics of the 2-d ~ 3-d transition relies on how the bulk gas-solid (T < T~d) or gas-liquid (T~d < T) coexistence is approached. New phenomena emerge and they evolve, with increasing thickness, to become the surface transitions of bulk matter. Adsorbed multilayers are indeed useful as test systems for studies of some bulk surface properties such as surface melting and surface roughening. The multilayer behaviour is related to wetting. Excellent reviews of multilayer physisorption have been recently done by Dash (1985, 1988), Dietrich (1988), Schick (1990) and Hess (1991). 10.6.1. From 2-d to 3-d; wetting

The various growth modes can be classified as: (i) "complete wetting" or type 1; at low temperature, the adsorbed film deposits as a succession of distinct layers, detected in a stepwise adsorption isotherm. The layer-by-layer deposition continues ad infinitum as the equilibrium pressure rises to the bulk vapor pressure, i.e., the film grows asymptotically to bulk. This is shown in Fig. 10.34. (ii) "Incomplete wetting" or type 2; the thickness of the adsorbed film is limited until vapor pressure of bulk saturation is reached, at which point any additional adsorbate condenses into bulk crystallites or droplets. (iii) "Nonwetting" or type 3 is the particular case with a zero layer limited thickness. The wetting behavior was studied systematically by Pandit, Schick and Wortis (1982). This work presents a valuable classification scheme for a comprehensive picture of wetting. It relies on the relative strengths of the adsorbate molecule-substrate potential u and the adsorbate-adsorbate molecular potential v. It was theoretically predicted that complete wetting might be expected for sufficiently large u/v, whereas smaller u/v ratio would induce incomplete wetting. Experimental studies of various films adsorbed on graphite (Bienfait et al., 1984) showed reentrant incomplete wetting on the high u/v side, at low temperatures. Complete wetting would be therefore limited to a range of intermediate u/v ratios. Other theoretical works (Muirhead et al., 1984; Gittes and Schick, 1984; Huse, 1984b) have introduced substantial changes in the former theory. In particular, they consider the effect of structural mismatches between the first monolayer of solid adsorbate and any plane of the bulk crystal. The continuity of the growth depends strongly on how the bottom layers can restructure. Even if the lattice structure and molecular orientations in the monolayer and bulk solid are compatible, a strongly attractive substrate (large u/v) will compress the few first layers inducing stress energy that can prevent complete wetting. Complete wetting by a liquid film is much more common than by a solid film. Indeed, numerous systems exhibit a triple point wetting transition characterized by incomplete wetting of a solid film for T < T~d, but complete wetting by liquid for T > 7',3d. The thickness dc of the film versus temperature at coexistence follows a power law (Krim et al., 1984) i

d~

,

(10.27)


563

77.3 K

,.,,! ILl r,r'

ILl 0 0

!

1

0.2

!

1

0.4

I

I

0.6

I

0.8

1

P/Po .Fig. 10.34. Volumetric adsorption isotherm of krypton adsorbed on graphite at 77.3 K. P,, is the bulk vapour pressure ( 1.75 Torr) at 77.3 K. The layer-by-layer condensation causes the stepwise isotherm, up to 5 layers. (From Thorny and Duval, 1970a). The exponent (-1/3) has been predicted for long range interactions which is the case for van der Waals systems. Critical wetting when the temperature is increased at three phase coexistence has also been considered theoretically (Dietrich and Schick, 1985). In this case the thickness of the film at coexistence varies as

dc= (Tw- T)-'

(10.28)

which shows that complete wetting should occur at Tw the critical wetting temperature. Hess (1991) has summarized the wetting behavior of various adsorbates on graphite (see Table 10.7). Most of the experimental work has investigated wetting on graphite (see reviews by Suzanne (1986) and Hess ( 1991 )). Studies on gold ( 111 ) (Krim et al., 1984) should also be mentioned as well as on Ag (111) (Gibson and Sibener, 1985; Suzanne, 1986), Pt (111) (Kern et al., 1986b), Pd (Miranda et al., 1984) and MgO (100) (Gay et al., 1990).

10.6.2. Layering and surface roughening Adsorption of a thick film may proceed layer-by-layer (first order condensation) or continuously with increasing vapor pressure. The ellipsometric isotherms of argon/graphite reported in Fig. 10.35 (Youn and Hess, 1990a) illustrate first order and continuous condensation with the particular situation of reentrant first order layering transitions in this system. For each layer, a critical temperature is estimated


564

Table 10.7 Wetting properties of various adsorbates on graphite.The relative strength of the adsorbate-substrate and adsorbate-adsorbate interactions is measured by u/v. The wetting modes 1 or 2 are defined in the text. Parentheses indicate uncertain interpretations or extrapolations. (From Hess, 1991) Adsorbate

u/v

H20 CO 2 C2H 4 CF 4 O2 CzH 6 Xe Kr Ar N2 Ne CH 4 CO H2

Wetting mode

0.31 0.46 0.42-0.79 0.73 0.93 0.99 1.04 1.17 1.23 1.32 1.39 1.51 1.55 4.6

Low-T solid

Higher-T solid

Liquid

2 2 2 2 2 1 1 1 2 2 1 2 I

(1) 1 1 1 1 -

2 1 1 1 1 1 1 1 1 (1) 1 I (1)

. m

c'-

5

7'

t 0.8

t

>.., 'L

.,Q ,< v

79.6

K

72.7

K

68.8

K

66.9

K I

4 0.4

I

[ ~- [IJ,,o[ -1/3

I 1.2

(K) -1/3

Fig. 10.35. Ellipsometric isotherms of argon condensed on graphite. The ellipsometric signal is reported versus Ila-l.to1-1/3, where l.t and ~ are the multilayer chemical potential and the bulk chemical potential, respectively. Sharp layering transitions are observed up to 67 K and around 73 K (reentrant first order layering transitions). Smooth isotherms are indicative of continuous condensation. ( F r o m Youn and Hess, 1990a).


565

above which there is a continuous adsorption of the layer. The layer critical points may fall either below or above the bulk melting point. This is a major determinant of the character of multilayer film growth. This results, in the former case, in the existence of a range of temperatures below the melting point where the growth of solid films is continuous whereas layered liquid films may exist in a range above the melting point, in the latter case. For wetting solid films, it has been theoretically demonstrated that the sequence of the layer critical temperatures converges to the roughening temperature of the bulk adsorbate material (Nightingale et al., 1984). At the roughening transition of a semi-infinite crystal, thermally excited steps and kinks disorder the surface (see Chapters 2 and 13). The same can also occur in adsorbed multilayers when the surface of the film is delocalised, i.e. for temperatures above the critical points. It is noteworthy that the substrate field limits the surface disorder, except for very thick films. To date, the experimental studies of layering and surface roughening of thick films are based on measurements of the layer critical points, essentially for various adsorbates on graphite (Hamilton and Goodstein, 1983; Kim et al., 1986b; Larher and Angeraud, 1988; Nham and Hess, 1988; Zhu and Dash, 1988; Youn and Hess, 1990a).

10.6.3. Surface melting The surface melting phenomenon consists in the appearance of a liquid-like disordered surface layer on a solid in equilibrium at temperatures below the bulk melting point (Dash, 1989; Nenow and Trayanov, 1989). Like surface roughening, surface melting is basically a surface transition of a semi-infinite crystal (see Chapter 13). This phenomenon may be viewed as wetting of the crystal by its melt. Mean field theory has been developed predicting the occurrence of surface melting (Trayanov and Tosatti, 1988; Dash, 1988). Theoretical models also show a power law temperature dependence of the quasiliquid layer thickness for long range forces (van der Waals systems, for instance). Thick physisorbed films have been extremely valuable for surface melting investigations. As for surface roughening, the substrate field may hinder the surface disorder. In particular, substrate-induced freezing has been observed above the bulk melting point. This is an effect of the substrate field which stabilizes a solid layer at the substrate interface. Surface melting in thick solid films has been extensively studied in krypton, argon and neon films on graphite (Zhu and Dash, 1988; Gay et al., 1991; Pengra et al., 1991), in oxygen on graphite (Krim et al., 1987), in methane films on MgO (Bienfait et al., 1988; Gay et al., 1990) and on graphite (Bienfait et al., 1990; Gay et al., 1992), and in deuterium hydride HD on MgO (Zeppenfeld et al., 1990c).

10.6.4. Experimental Most of the experimental techniques described in w 10.4 and used for studies of physisorbed monolayers, have been employed in investigations of thick films as well. The quartz microbalance has been more particularly used in the latter cases. Its characteristics are briefly reported at the end of this section.

566


Fig. 10.36. RHEED patterns of xenon films adsorbed on graphite around 50 K. (From Venables et al., 1984). (a) Bilayer film giving rise to uniform streaks. (b) Uniform wetting 10 layer thick film responsible for the streak modulations.

The first evidences of film growth have been provided by volumetric isotherm measurements (Thomy and Duval, 1970a; Menaucourt et al., 1977; Ser et al., 1989). The resolution is limited to 4-5 layer thick films, above which the actual equilibrium pressure of the thick film becomes undistinguishable of that of bulk. In addition, as for all classical tools requiring powdered materials (calorimetry, neutron scattering, etc.), the volumetric technique is severely limited by capillary condensation; it has been observed in graphite and MgO powders beyond 4-5 layers (Set et al., 1989; Larese et al., 1989, Gay et al., 1990). This effect may completely hide or modify intrinsic multilayer phenomena (wetting, surface melting, etc.). Experimental techniques using single crystal substrate surfaces overcome this problem. Among them, LEED and RHEED (Venables et al., 1984; Krim et al., 1986; Gay et al., 1984, 1988), elastic and inelastic helium scattering (Gibson and Sibener, 1985; Kern et al., 1986b), ellipsometry (Nham and Hess, 1989; Faul et al., 1990) and quartz microbalance (Krim et al., 1984) are the most widely used techniques in multilayer studies. Figure 10.36 shows typical RHEED patterns for a xenon film deposited on graphite around 50 K; the uniform streaks for the bilayer evolve to strongly modulated streaks for a 10 layer thick film, indicating a uniformly wetting thick film.

Quartz micro-balance. High frequency microbalances have been used as thickness monitors for deposited thin films or to measure the amount of gas adsorbed on metal surfaces (Lu and Czanderna, 1984). The resonant frequency of a crystal oscillating in its thickness shear mode is lowered by mass loading. The frequency shift is proportional to the mass deposited. The sensitivity is adequate to detect coverage changes of less than one monolayer (Krim et al., 1984). The metal substrate also serves as the oscillator electrode. It is first evaporated onto the quartz. The result is not a single crystal surface. In the case of gold, it is polycrystalline with (111) faces parallel to the quartz substrate. Adsorption isotherms have been measured for various gases (rare gases and light molecules) onto this surface (Krim et al., 1984) to determine the growth habits of multilayer films of these molecules.


567

It has been shown that triple point wetting takes place. The thickness of the film at coexistence obeys Eq. (10.27). 4He films on Ag and Au(l 1 1) surfaces have also been studied by this technique (Migone et al., 1985). The results show incomplete wetting with a thickness at coexistence dc which varies according to Eq. (10.28). 10.7. Conclusion Physically adsorbed phases have provided the physico-chemist with a good understanding of the matter in reduced dimensionality. Unlike chemisorbed systems, physisorbed films can be studied under true thermal equilibrium because of the weak interactions. In the monolayer range, the adsorbed phase has a quasi 2-d behavior. Phase diagrams featuring 2-d gas, 2-d liquid and 2-d solid phases with coexistence regions, triple point and critical points have been constructed as in bulk. Commensurate or incommensurate solids are observed within the monolayer depending on the relative lattice mismatch between the adsorbate and the substrate. Furthermore, 2-d polymorphism can be associated with different orientational orders of molecules above the surface. Various first order and continuous phase transitions have been identified: commensurate-incommensurate transitions, solid l-solid 2 transitions, order-disorder transitions and 2-d melting transitions. The mechanisms of these transitions have been analysed and the role of domain walls, dislocations and disclinations has been emphasized. Model calculations are an accessible challenge due to the simple adsorbate-adsorbate and adsorbate-substrate interactions involved. This special feature of physisorbed phases has allowed the modelling of the statics and the dynamics of the monolayers leading to a better understanding of the mechanisms governing the monolayer behavior. Experiments have been performed either with single crystal surfaces under uhv conditions or with well characterized, uniform powders under less restrictive conditions. Techniques and probes normally devoted to the study of bulk matter, like neutron or X-ray scattering have been used with these latter systems. Finally, physisorbed films with increasing thickness can be used to study the change in the properties of the matter when going from 2-d to 3-d. The analysis and modelisation of the growth mode of films or wetting behaviors is an important challenge in many aspects of thin film processing. A new phase transition, namely surface melting, has been inferred from thin physisorbed film studies, and here again, model calculations could be performed and compared to experiments to test the role of the various parameters. References Abraham, F.F., 1983, Phys. Rev. B 28, 7338. Abraham, F.F., 1984, Phys. Rev. B 29, 2606. Abraham, F.F., S.W. Koch and W.E. Rodge, 1982, Phys. Rev. Lett. 49, 1830. Angot, T. and J. Suzanne, 1991, in: The Structure of Surfaces III, S.Y. Tong and M.A. van Hove, eds. Springer Series Surf. Science 24, Springer, Berlin, p. 671.

568


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CHAPTER

11

Interactions Between Adsorbate Particles

T.L. E I N S T E I N Department of Physics University of Maryland College Park, MD 20742-4111, USA

Handbook of Su.rface Science Volume 1, edited by W.N. Unertl


577

Contents

11.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I 1.2. General features of lateral interaction energies . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1. Fundamental ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2. Electronic indirect interactions in simple tight-binding model . . . . . . . . . . . . . 11.2.2.1. M o d e l H a m i l t o n i a n . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

589 581 581 586 586

11.2.2.2. C a l c u l a t i o n of c h a n g e in o n e - e l e c t r o n e n e r g i e s using G r e e n ' s functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2.3. S i m p l e r illustration: pairs on a ring 11.2.3.

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

Multisite interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.3.1. T h r e e - a d a t o m (trio) interactions . . . . . . . . . . . . . . . . . . . . . .

597 597

11.2.3.2. C o m p l e t e o v e r l a y e r s

598

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

11.2.4. Coulombic effects: self-consistency and correlation, and other improvements

11.3.

11.2.5.

Lattice indirect interactions: phonons and elastic effects

11.2.6.

Asymptotic form of the indirect interaction between atoms and between steps

Attempts to model real systerns

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

Embedded cluster model

! 1.3.3.

Effective medium theory and embedded atom method - - semiempiricism

11.3.4.

Empirical schemes

599 601

....

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

11.3.2.

607 611 611

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

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

613 615 619

11.3.5.

Field-ion microscopy, modern tight-binding, and more on semiempiricism . . . . . .

620

11.3.6.

Scattered-wave theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

628

Further implications of lateral interactions

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

11.4.1. Ordered overlayers and their phase Boundaries

11.5.

....

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

11.3.1. Tight-binding, jellium, and asymptotic-ansatz

11.4.

588 592

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631 631

11.4.2.

Local correlations and effects on chemical potential . . . . . . . . . . . . . . . . . .

632

11.4.3.

Surface states on vicinal and reconstructed fcc(110) surfaces

635

Discussion and conclusions

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

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

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

578

638 642

11.1. Introduction

Progress in computing the interactions between small numbers, even pairs, of chemisorbed atoms has been remarkably slow because of the very low symmetry of the problem. In contrast, the energetics of monolayers of adatoms, which have the full two-dimensional symmetry of the substrate, can now be characterized with impressive precision. However, even treatment of partially completed adlayers with (2xl) or c(2x2) symmetry doubles the size of the surface primitive cell, but quadruples the size of secular matrices, raising computer time requirements by a factor of order 43. At the other end of the scale, a single adatom (in a symmetric site) will at least have the point-group symmetry of the substrate. Associated with these symmetries are conserved quantities ("good quantum numbers") which make calculations simpler. As a result, a variety of elaborate many-body techniques have been applied successfully to these systems. For two adatoms on a surface, there is little or no symmetry, typically just a two-fold rotation or mirror plane (leading often to splittings of levels). Few systems have been treated in a satisfactory way. Sophisticated computations attempting to assess these interactions tend to resort to studies of ordered overlayers (Tom~inek et al., 1986). Desjonqu6res and Spanjaard (1993) signaled the difficulty of the problem by placing it as the final topic in their recent text. Reviews stressing various aspects of the problem have been presented by Einstein (1979a, 1991), Muscat (1987), March (1987, 1990), Braun and Medvedev (1989), Feibelman (1989a), and NCrskov (1993). This chapter will explore the many mechanisms by which chemisorbed atoms interact with each other. To set the stage early, it is useful conceptually to distinguish between direct and indirect interactions. Direct interactions would occur even if there were no substrate; they are, thus, sometimes called "through-space". Examples include van der Waals, dipolar, and electronic hopping (between the adatoms). The substrate, however, will generally provide at least some degree of perturbation. The alternative is indirect interactions, in which to lowest order there would be no interaction without the substrate. The coupling can be by electronic states (usually predominant), elastic effects, or vibrational coupling (usually insignificant). Since the coupling to the substrate is crucial, these are sometimes called "through bond". Special emphasis will be given to the indirect electronic ("pair") interaction between two light gas or transition series atoms on a (transition) metal. Moreover, we thoroughly explore a simple model of these interactions. The motivation is not so much to explain specific data but rather to give a theoretical framework in which to understand the relative magnitudes and qualitative behavior of the interactions. Without this sort of picture, it is difficult to make sense of the results that emerge from more realistic attempts to describe the adsorption systems. 579

580

T.L. Einstein

We also present a thorough summary of the methods that have been applied to make progress in understanding this general problem. As we fix ideas, it may be helpful to describe our problem in oversimplified terms by speaking of three characteristic energies: (1) Eat is the binding energy of an isolated atom to the most attractive site on the surface. Typically this is a high-symmetry site; e.g., on a square lattice, such sites (cf. Fig. 1.12) can be called A (atop, or linear, or on-top, the latter two inconsistent with the abbreviation), B (bridge, between two substrate atoms), and C (centered, or hollow, above the middle of a square). Identical terminology applies to substrates with triangular symmetry. Ed denotes the diffusion barrier, due to variations in the adatom-substrate potential, between adjacent most-favored sites. Usually this is a saddle point in a potential energy surface; e.g., if the C site is most attractive, one might expect Ed = E c - Ea, although substrate relaxation can sometimes lower this barrier significantly. The corrugation of the substrate potential provides an upper bound for Ed. Finally, E.,,, is the magnitude of the characteristic energy of interaction between nearby atoms. In physisorption, Ea, is comparable to E~, both being much greater than E0. (For dense overlayers, the actual diffusion barrier increases significantly due to adatom-adatom effects.) In contrast, in chemisorption E,~ > Ed >> E,a. Thus, in this ideal scenario, adatoms all sit in the most favorable site; their lateral interactions are relatively small. In this case it is fruitful to couch the discussion in terms of a classical lattice gas picture. (Cf. Chapter 13 by Roelofs.) In setting the stage for what we will find regarding electronic lateral interactions, it may be helpful to divide (somewhat artificially) the physics into a few regimes, depending on the separation between adsorbates. (1) In the near regime, the adatoms may be close enough to have non-negligible direct interactions. If not, they typically still "share" one or more substrate nearest neighbors, so that the bonding of one adatom to this substrate atom is strongly influenced by the presence of the second adatom. This regime is the most important for applications: in chemistry it determines the details of dissociative adsorption; in surface transport it enters problems of surface diffusion mechanisms. The strong impact of one adatom on the other may alter their binding sites, weaken bonding to the substrate, etc., as emphasized in a review by Feibelman (1989a). In short, in this regime Eaa may be comparable to E,.,. (2) In the intermediate regime, these effects fade and the lattice-gas approximation improves. The individual-adatom adsorption process is largely immune to the interactions. Most of the interesting physics can be isolated in the question of how the disturbance produced by a point defect at some position propagates to another. All the occupied band states in the substrate are involved in a complicated way. This regime is important in describing the formation of ordered fractional monolayers of adsorbates and in characterizing the chemical potential and the correlation functions of these adsorbates, even at higher temperatures at which there is little order. Thus, these interactions play a role in understanding thermal desorption spectra, vibrational line shifts, etc. (3) The asymptotic regime is reached when the adatoms are several spacings apart. The interaction is dominated by the substrate Fermi surface. Analytic expressions, albeit complicated, can be derived. Until recently, there was little evidence of experimental impact of

Interactions between adsorbate particles

5 81

this regime, but there may be implications for the interactions between steps on metal surfaces. We caution that as with most simple pictures of complicated phenomena, it is easy to point out ways in which the broad-brush rendition is oversimplified. For example, in the case of weak, non-directional bonding, the adatoms may slip out of high-symmetry binding sites even when a couple spacings apart, as suggested by Persson (1991) for some cases of CO adsorption. Moreover, for weak bonding, the Fermi-surface electrons may dominate at close spacings, inviting simple descriptions in terms of frontier orbitals (Hoffmann 1963, 1988). A major motivation for studies of pair interactions is to understand the origin of the wide variety of ordered overlayers at fractional coverages on metal surfaces (see Chapter 13). These have been tabulated by Ohtani et al. (1987), Van Hove et al. (1989), Watson (1987, 1990, 1992), and Watson et al. (1993). Consider a c(2z2), i.e. a checkerboard pattern on a square lattice (Chapter 1, Fig. 1.1). This pattern could arise simply because of a strong nearest-neighbor repulsion: E~ > 0 for an overlayer with about half the sites occupied. If additionally there is a next-nearestneighbor attraction, one finds islands of adatoms with c(2x2) symmetry at low temperatures and coverages. Sometimes these lie at temperatures so low that the equilibrium local configuration is not attained during the time of an experiment, at most several hours. But when islands are present, they provide strong evidence of an attraction. There are many more complicated phases. The explanation of most of their ground-state energies in terms of pair interactions is fairly obvious (cf. Suzanne and Gay, Chapter 10, and Roelofs, Chapter 13), but for troublesome cases, exhaustive tabulations have been published by Kaburagi and Kanamori (1974, 1978) and Kaburagi (1978). By attraction or repulsion here, we mean that the lattice site is favorable or unfavorable. There is no implication about the direction of a force acting on an adatom sitting in a lattice site; in the lattice gas picture, this force is assumed to vanish. In the near region, this assumption may often be questionable, but in the intermediate region it should be reasonable. Some experimental data on these systems appear in Chapters 10, 12 and 13 to which we shall refer. We will dwell mostly on theory. Progress in the field has come in the form of study of self-consistency and correlation effects (which seem to be less important than might be expected) and of multi-parameter-model attempts to describe real systems. Next we shall show in a single simple model how pair, three-adatom, etc., interactions combine to produce ordered overlayers. We will briefly consider changes in density of states (DOS) caused by two-adatom interactions from a similar viewpoint, and also show the more dramatic effects that arise when these combine to produce an ordered overlayer. In closing, we shall speculate on areas ripe for development.

11.2. General features of lateral interaction energies 11.2.1. Fundamental ideas

If chemisorbed atoms are sufficiently close to overlap each other, there will be a strong direct interaction. This interaction is essentially a chemical bond, compara-

582

T.L. Einstein

ble in strength to the chemisorption bond. (There is interesting physics in the degree to which these bonds are not simply the equivalent of bulk bonds. We shall explore these effects further in later sections.) For larger lateral interadatom distance R, the interaction falls off exponentially along with the overlap so that for R more than a few A, it is negligible. Most of the physics of this problem comes from the two adatoms and their substrate nearest neighbors; hence a cluster calculation can be appropriate. These interactions are important for supersaturated or even monolayercovered surfaces. They also arise in the problem of dissociation and reassociation of adsorbing molecules (e.g., the question of whether there is an activation barrier) which has been studied both schematically and in great detail. If the chemisorption b o n d involves charge transfer, electric dipole moments l.t will develop. Kohn and Lau (1976) showed that the non-oscillatory part of the dipole-dipole interaction energy on metals behaves as Edip_dip '~

2~all-ttCt4~EoR3

(11.1)

for large R. The novel aspect of this expression is the factor of 2, for which they give the following qualitative explanation: For either adatom, say a, ~ta is the product of the charge transfer qa to the adatom and the distance Za between the adatom and the surface/image plane, at which the induced charge of--qa lies. However, the potential experienced by a second adatom is determined by the first adatom and its image at-z~,, and so is 2~aZb(4~Eo)-lR-3. Hence, the work in bringing the second charge from z = +oo to z = Zb, and so Ed~p.d~~ contains the novel factor of 2. Inserting numbers, we find this interaction energy to be 1.25 eV times the two dipole moments in units of debyes divided by R 3 in A 3. N0rskov (1993) reviews the direct electrostatic interaction in some detail. The effect is generally larger for electropositive than electronegative adsorbates because the latter tend to bond closer to the substrate; consequently, they are better screened and so have a smaller dipole moment. For alkali adatoms, dipolar effects dominate the interactions which determine the 2D phase diagram (Bauer, 1983; Mtiller et al., 1989). Pre-adsorbed alkali-metal atoms increase both the binding energy and the dissociation rate of light gas dimers like CO, NO, N2, or 02 on metals, while preadsorbed electronegative atoms do the opposite: Typically adsorption of these dimers involves some charge transfer to them. (Back donation to the anti-bonding molecular orbital exceeds donation from the bonding orbital.) In the simplest approximation, the resulting energy is the product of the admolecule-induced dipole moment normal to the surface and the gradient of the electrostatic potential due to the preadsorbed atoms (or the extra charge times the potential itself). To support this picture, NCrskov et al. (1984/5) explored the form of this potential, for several different preadsorbed atoms on jellium, as a function of the height of the dimer above the surface. For the particular case of N2 on Fe( 111 ) with pre-adsorbed K, NCrskov (1993) finds an interaction of 0.08 eV, which can be used to account for most of the measured shift in adsorption energy due to predosing. The second-order correction, proportional to the square of the potential, is always attractive. Thus, for cases in which the charge transfer is from the dimer, he notes


583

that the long-range interaction with a pre-adsorbed alkali can be repulsive while the short-range interaction is attractive. In addition to producing static repulsions, dipole-dipole interactions can raise the vibrational frequency of adsorbed molecules. For example, Scheffler (1979), using just dipole-dipole coupling, accounts for the coverage dependence of the shift of the C - O stretch frequency of CO on Pd(100) and Pt(l 11) measured by IR absorption reflection spectroscopy. In the process, he derives a coverage-dependent (as well as frequency dependent) effective polarizability which depends significantly on the distance of the dipole from the reference (image) plane of the metal (assumed to be jellium). In focusing on the cases of CO on Cu(100) and Ru(0001), Persson and Ryberg (1981) advanced the treatment of these questions by treating the adsorbate polarization as a single entity rather than trying to split it into admolecule and an image; they, furthermore, used the coherent potential approximation (CPA) (Soven, 1966) to consider interactions for a dense but not ordered overlayer. They find that the dipole-dipole interaction is enough to account for the coverage-dependent frequency shift for the Ru substrate, but that on Cu there is a counteracting chemical shift of nearly the same magnitude. NCrskov (1993) discusses the shifts of dimer vibrational frequencies due to interactions between pre-adsorbed (non-neutral) atoms and the dimer. The van der Waals interaction always produces a weak attraction between two adatoms and is the dominant contribution in the case of physisorption. The leading term is the dipole-dipole contribution, which goes as -C/R6; C in turn is proportional to the square of the polarizability. According to Hirschfelder et al. (1954), C is roughly 30 eV-~ 6 for Ar, N 2, and 02, and five times as great for Xe. For physisorbed gases, this mechanism dominates the interaction, and hence the details of the interatomic potential have been studied extensively. To fit gas-phase data, one must go beyond a simple R -12 Lennard-Jones repulsion (to some exponential description) to avoid overestimating C by nearly a factor of two. Two higher-order gas-phase effects are non-negligible: (1) The R -8 dipole-quadrupole force which increases the depth of the well-minimum by roughly 10% and (2) the repulsive (in all important cases) R -9 triple-dipole (Axilrod-Teller (1943)-Muto (1943)) interaction, the magnitude of which is at most 3% (for Ar) to 5% (for Xe) of a pair interaction if all distances are set at their equilibrium values. While this effect is of little concern here, there have been interesting applications (Klein et al., 1986). A variety of calculations of rare gas adsorption onto jellium (Sinanoglu and Pitzer, 1960), continuous dielectrics (McLachlan, 1964), Xe crystals (MacRury and Linder, 1971), and graphite (Freeman, 1975) all show that physisorption reduces the gas-phase pair attraction by roughly 20%. As an example of the state of the art in this refined subject, Barker and Rettner (1992) produce an accurate "empirical" (actually more semiempirical, in the language we will use later) potential for Pt(111)-Xe as a "benchmark." For the lateral interactions, they include, in addition to the van der Waals potential, the "nonadditive" McLachlan modification, the interaction of adsorption-induced and image dipoles, and the triple-dipole term, citing as reference Bruch's (1983) clear and comprehensive discussion of the significant contributions. (This classic review of lateral interactions in physisorption,

584

T.L. Einstein

as well as of the single-atom holding potential, provides an account of the general features of this problem that is evidently still timely a decade later. March (1987, 1990) presents more recent reviews. Vidali et al. (1991) have produced a useful compilation of potentials for physisorption.) The substrate-mediated dispersion energy is the largest contribution to the lateral interaction at the intermediate separations of ordered overlayers, accounting for slightly over half the (repulsive) corrections to the gas-phase interaction for two sample Xe overlayers (Bruch, 1983). The effect of the substrate on the interadatom interaction was first tackled using perturbation theory by Sinanoglu and Pitzer (1960). McLachlan (1964) calculates in second-order perturbation theory the interactions of adatom dipoles and their images in the substrate, including a frequency-dependent response for the substrate. Explicit expressions for the substrate-mediated dispersion energy and tables of the attendant coefficients are given by Bruch's (1983) review; a key issue is determining the distance of the adatoms from the image plane. Freeman (1975) approaches the problem using the Gordon-Kim (1972) version of density functional theory (for Ar adatoms) and obtains fair agreement with the preceding formalism; similarly, Vidali and Cole (1980) apply both methods to He on graphite. The next largest contribution to the substrate-related interaction, perhaps half the size of the preceding, is the interaction of adsorption-induced dipoles. The role of the surface was noted earlier in Eq. (ll.1). Bruch (1983) reviews the many contributors to this subject. He and Phillips (1980) showed how to compute these effects for an overlayer lattice. Other effects include triple-dipole (Axilrod-Teller (1943)-Muto (1943))interactions within the overlayer and changes in zero-point energy. In this framework, lateral interactions can be computed to an accuracy that makes those working on chemisorption truly envious. Nonetheless, there are some differences between calculations on particular systems, e.g. the above-mentioned benchmark (Mtiller, 1990; Gottlieb and Bruch, 1991 ; Barker and Rettner, 1992). To apply the van der Waals perspective to chemisorption, we can invoke the surface molecule picture to posit that the interaction between, say, two chemisorbed O atoms (coupled to their substrate neighbors) is similar to that between two O2 molecules (although now the molecules are oriented), i.e. roughly -25 eV/~6/(R[/~])6. At second and third neighbor separations on Ni(100), for instance, this yields an interaction o f - 1 3 meV a n d - 2 meV, respectively, which is usually negligible compared to the electronic indirect interaction. For heavy adsorbates (e.g., W or Re) these numbers could possibly be several times greater; no firm data exists. A curious application, to Ni(100)-O, of van der Waals ideas by Gallagher and Haydock (1979) suggested that by virtue of large overlap with the attractive Ni potential of the substrate, the O 2p orbitals become larger and far more polarizable, dramatically increasing the associated interaction. There has been little follow-up work on this viewpoint. The first proposal that adatoms might interact indirectly was made by Kouteck)5 (1958). The essence of this interaction is seen in Fig. 11.1, taken from the pioneering work by Grimley (1967) on this problem, which even now begins most discussions. Consider two atoms, each with an atomic potential producing some (relatively high-lying) bound state. In free space (and at moderate separation), each


585

I I

Fig. 11.1. Classic schematic of the indirect interaction between pairs of adatoms. (a) Potential and wave thnctions Ibr two atoms in vacuum separated so far that there is no overlap and so no direct interaction. (b) The same atoms, now chemisorbed on a simple metal surface. From Grimley (1967a), with permission. of the bound-state wavefunctions will remain confined near its atomic site; the vacuum barrier is insurmountable. If, alternatively, they are adsorbed onto (or absorbed into) a metal, both atomic wavefunctions can tunnel through the narrow potential barrier to the metal and couple with propagating metal wavefunctions. Figure l l . l b shows how both atomic wavefunctions might couple to one such background eigenstate. If the coupling places the two atomic wavefunctions in (out of) phase, the interaction is attractive (repulsive), lowering (raising) the energy of the participants. From the oscillatory nature of the intermediate wavefunction, the electronic indirect interaction should be oscillatory in sign as a function of interadatom distance. It should be (two-dimensionally) isotropic if and only if the (surface of the) metal background is. Such isotropy is expected only for substrates which can be well approximated by free-electron or jellium models. Furthermore, the two adatomic orbitals can couple through not just one, but any of the occupied states (including surface states). As adatom separation increases, fewer substrate wavefunctions will match well with the atomic orbitals, causing a rapid decay in magnitude of the interaction energy. As discussed at the outset, our discussion assumes that Ed >> E~,, which should be a good approximation for strong chemisorption at low to moderate coverage. Under these circumstances, the most favorable adsorption sites will be filled or vacant, and when nearby sites are filled, the associated interaction energy will modify the total energy of the system. In this lattice gas picture, the Hamiltonian of the adatoms takes the form:

H=E, Zninj+E2 Z ninj+... (ij),

(6)2

(11.2) "~" Z ET Z llinjllk'Jr" Z EQ Z trlitlJ llkrlldr ''" T ((jk).r Q (ijkl)tI

Each site of the net of most-favored substrate sites (labeled i) can be occupied (ni = 1) or vacant (ni = 0). Here the pair interaction energies are denoted E,,, for mth

586

T.L. Einstein

neighbors; Ev is the "trio" (three-adatom, non-pairwise) interaction energy, with the index running over the possible trimer configurations; E o is the "quarto" energy; and so forth. For this formulation to be useful, the pair energies should fall off relatively rapidly in magnitude with increasing m, so that only a few need be considered. Furthermore, the multisite terms should be small; at worst, only a small number of the most closely spaced multiadatom terms should contribute. We shall see that the pair interactions do decay rapidly. The multisite terms are smaller but not always negligible. Moreover, there may be several different configurations with comparable magnitude. Nonetheless, cancellations typically occur such that the energies of ordered submonolayer overlayers are often adequately described by the pair energy of the closest pair(s) found in the overlayer. Before delving into specific simple models, it is worth stating the underlying philosophy motivating them. 11.2.2. Electronic indirect interactions in simple tight-binding model

To gauge roughly the relative magnitudes and general behavior of these interactions, it is convenient and customary to study a simple model, in this case a tight-binding model in which the substrate is a single-band, simple-cubic solid. (See LaFemina, Chapter 4, for a discussion of tight-binding models.) This model was adopted two decades ago (Einstein and Schrieffer, 1973) (hereafter ES) to embody the idea that the d-bands of the substrate were primarily responsible for the interactions and, unlike jellium, allowed one to consider the dependence of the interaction on the type of adsorption site in a simple way. 11.2.2.1. Model Hamiltonian The model, as well as many subsequent discussions of interactions between adatoms, is couched in terms of an Anderson (1961) (magnetic) model in which the adatoms are represented as dilute impurities at sites r (= a,b for pairs) in an unperturbed host: m

me,~,+ Z (H'~ + H'r)

__ n ~

(I 1.3)

r

The first term in the parentheses represents the atomic factors of adatom r, while the second is this atom's coupling to the metal. To include a direct interaction, one would add terms of the form H',~, coupling atoms a and b. Until recently, most work on the problem has amounted to taking progressively more realistic expressions for various of these terms and solving the resulting system to varying levels of approximation. To simplify notation, we assume that the adatoms are identical. Over the last decade the coadsorption problem has attracted some interest; it is straightforward to extend the formalism. Some of the simplicity of the above ansatz comes from the use of an atomic orbital picture. While this formulation makes it easy to do initial calculations, it neglects such effects as orbital deformation and local distortion, which may often be important. The adatom part of the Hamiltonian is


H'~ = G'~~ n~,~+ Unit n~,

587

(1 1.4)

(y

and similarly for H~. This expression can be generalized to include degenerate orbitals, multiple levels, etc. As a first approximation, one might set e] at the ionization level - I of the adatom and take as U the difference between -1 and the affinity level. For greater accuracy, e ~ should be raised and U reduced by correlation effects (screening and image charges). In a (restricted) Hartree-Fock approach, one neglects U entirely and replaces G~ by G,, = G~ + U(na`,) where (n~`, is the mean occupation of the adatom for either spin direction. For neutral chemisorption, (n,,`,) is 1/2, suggesting G,, be the (negative) average of the ionization and affinity energies, as in many chemical molecular orbital calculations (where this is called the Mulliken (1934) electronegativity) (cf., e.g., Pople and Beveridge, 1970) Using the idea of chemical transferability, Pandey (1976) adjusted the adatom and coupling parameters so that cluster computations of small molecules fit the levels found in photoemission experiments; presumably the same parameters carry over to the chemisorption system. Brenig and SchOnhammer (1974), Hertz and Handler (1977), and Bell and Madhukar (1976) went beyond Hartree-Fock in the case of single atom adsorption. The first group also showed in the pair problem that correlation effects are relatively unimportant, compared to the single atom case, as we shall discuss below. On the other hand, using self-consistent Hartree-Fock and resorting to mean-field theory, Gavrilenko et al. (1989) explored the parametric conditions for magnetic ordering of the adatoms. Davydov (1978) considers a similar question, including direct interactions for a chain of adatoms. The simplest approximation for the substrate assumes a single band of one-electron states with energy ek. (A band index would also be needed if more than a single band were considered.) Many-body effects could also be included by putting a diagonal Coulomb term like U on each substrate site. Since only the component of crystal momentum parallel to the surface is conserved for a slab or semi-infinite crystal, k merely labels the states in some suggestive fashion. It is usually convenient to work in a mixed representation of kll and a layer index. In general there can be a different coupling between each k state and the adatom. For most purposes, it is adequate to consider, in the case of bonding at an atop site, a single coupling constant V between adatom a (or b) and its nearest neighbor on the substrate: H,,s - - Vc+~,co,, + h.c

(11.5)

where c § and c are creation and annihilation operators, respectively, for electrons in the state indicated by the substrate. For bonding in a bridge or centered site, Co,, is replaced by a symmetric normalized combination of c-operators for the number of substrate neighbors of the adatom. In principle this coupling should also consider an overlap term between atoms and metal. This question has been discussed at length by, among others, Sch6nhammer et al. (1975), Grimley (1974), and Einstein (1973). The usual approach is to "renormalize" previously stipulated natural orbitals (and resulting energies) with L6wdin ( 1 9 5 0 ) o r Gram-Schmidt (Birkhoff and MacLane, 1965) schemes.

T.L. Einstein

588

11.2.2.2. Calculation of change in one-electron energies using Green's functions Our goal is to find interaction energies between chemisorbed atoms, which in a one-electron framework can be expressed in terms of the associated change in density of states Ap: El:

AW=2f

( E - EF) Ap(~) dE

(11.6)

The factor of 2 comes from spin degeneracy, and the use of ~ - eF rather than just s indicates that the number of electrons rather than the chemical potential is being fixed (Grimley, 1967; Newns, 1969; ES). (The contribution due to the integral over svAp(s) is the result of an infinitesimal shift in the Fermi energy.) In the calculation presented below, the essential idea is that the interaction between adsorbates can be obtained by finding the underlying shifts in the one-electron energies of the system. It is convenient to do so in terms of one-electron Green's functions. We present enough detail below to show that this procedure is not so daunting as novices might suspect. Nonetheless, we follow this subsection with an explicit simple illustration using a ring as the "substrate" and analyzing the results in terms of level shifts to make contact with those readers more comfortable thinking in terms of quantum chemical models. To obtain the change in density of states A 9 needed so that the integral can be evaluated, we adopt a method used earlier in the theory of dilute alloys by Lifshitz (1964). Suppose the unperturbed (H '= H,,m= 0) and perturbed Hamiltonians, H ~ and H - H ~ + H', have eigenvalues sj and Ej, respectively. Then ~ (1 1.7) J But this can be rewritten as Ap(s) = -

'Imp/.

rc

_- llm Z ~

l

E-Ej.+i8

' /

e-13j+i8

ln(E-H+iS)-ln(E-H,,+iS)

J

_- _ l l m rc

O--~--Lndet( 1 )(e-H+iS) ~ ~- H,, + i8

1x Im ~oq In det (1 - G oAV)

I

Rememberthat the units of a delta function are the inverse of its argument.

(11.8)


589 ^

where G ~ is the unperturbed retarded Green's function (e - Ho + i8) -~ and V =- H~s was given in Eq. (11.5). (Since we choose here to follow convention by using retarded functions, the signs of all imaginary quantities will be the opposite of those appearing in ES and subsequent papers in that series, which used advanced Green, s functions, with infinitesimals of the opposite sign.) In scattering theory det (1 - G~ is familiar as the Fredholm determinant (of the 0th partial wave) (Gottfried, 1966). Furthermore, ^

- I m In det (1 - G~

= rl(e)

(11.9)

where rl(S) is the s-wave phase shift. (This identification is perhaps clarified by the observation that "Im In" is an arctangent, yielding an angle that amounts to a scattering phase shift.) This approach makes optimal use of the higher symmetry of the unperturbed system and the locality of the perturbation associated with adsorption. It is generalized in the scattering theory approach (Feibelman, 1989a). In terms of q, one writes the interaction energy simply as El:

A w- - (2/,) j" n(s) ds

(J 1.~0)

The Fredholm determinant contains a dense set of alternating poles and zeros, which turns into a branch cut in the continuum limit. Dreyss6 and Riedinger (1983) pointed out that one can circumvent numerical difficulties with this sort of integral by adopting the contour-integration approach (in the T = 0 limit) developed for temperature-dependent fermion Green's function problems. The result is basicalJ~y an integration of the real part of the analytic continuation of in det (1 - G~ from s~ + i0 § to s~ + ioo. This integration can be cast into a finite interval by making a substitution for the imaginary part of the energy integration variable (Liu and Davison, 1988). To evaluate the phase shift for the two-adatom problem, we arrange the matrix so that the adatom sites (a and b) and the substrate nearest neighbors to which they couple (o and^n) come first (o, a, n, b) and then all other substrate sites. The matrix (1 - G ~ V) then differs from a unit matrix only in the upper left hand 4• block:

- G x V,,,

'~

-o,,~,, v,,o

o

1

0

o

-G,\v,,.

-G~x, v,,o

1

-o,,\ v.~ 0

-c~x v~

(11,11)

1

The superscript X indicates that the substrate Green's functions can be easily generalized, for adsorption in B or C rather than A sites, to represent a (normalized) hybrid (cf. remarks after Eq. (11.5)) of substrate orbitals (Einstein and Schrieffer,

590

T.L. Einstein

1973). ~ T h e m a j o r result c o m i n g from the possibility o f c o u p l i n g to c o m b i n a t i o n s o f orbitals is that G B or G c is g e n e r a l l y very different from G A, so that the pair interaction will d e p e n d very strongly on the adsorption site. This feature arises naturally in L C A O models, in contrast to the other simple starting point, j e l l i u m m o d e l s (see below). (Braun (1981), h o w e v e r , argues that the effective c, b e c o m e s a d s o r p t i o n - s i t e d e p e n d e n t , possibly mitigating the variation with site s y m m e t r y . ) T h e d e t e r m i n a n t o f this matrix can be written as ^

det(l _ G " V ) = (1 - G,,,,G,,x, IVo,,I 2) (I - Ghh G,XlV~hl2) _ Go~G,,,,G~,~,G,,,,IV,,,,12V,,hl x x 2 (11.12) It is n o t e w o r t h y in this e x p r e s s i o n that the p a r e n t h e s e s e n c l o s e the c o n t r i b u t i o n to det(l - G ~ f r o m the adsorption of an isolated a d a t o m a at o (or b at n). F a c t o r i n g out these terms, and a s s u m i n g a d a t o m s a and b are identical, as are sites o and n, we find A

det(l-G

~

^

det (1 - G ~ V) -

[det ( 1 -

G~ ~')sing,e] 2

--x 2 x 2 = l - ( G o , , ) (G,,,) V ~

(11 . 13)

where G,,,x is a Green's function for a single adatom renormalized to account for its adsorption: G,,,x - ( I

- G .... G,,X,,IV .... 12)-' G,,,, -

[ c - c,,- V 2 G,X,,,(c)l-'

(11.14)

B e c a u s e of the logarithm, the phase shift (and hence the c h a n g e s in D O S and e n e r g y ) c h a r a c t e r i z i n g the "pair" interaction o_.fthe a d a t o m s can be o b t a i n e d directly from the phase shift associated with (1 V4(G,,X,, ) 2 ( G , ,x, ) ) 2 rather than from explicitly subtracting twice the s i n g l e - a d a t o m - p h a s e shift from the t w o - a d a t o m shift. For any n u m b e r of a d a t o m s , the single a d a t o m adsorption part factors out of the matrix (ES" G r i m l e y and W a l k e r , 1969). On the other hand, as s h o w n b e l o w in Eq. (11.20), for more than two a d a t o m s there is no way to factor out the pair effects from the h i g h e r order ones. (The feature that the s i n g l e - a d a t o m part factors out is a p l e a s a n t c o n v e n i e n c e , but with m o d e r n c o m p u t a t i o n a l p o w e r it does not p r o d u c e a significant i m p r o v e m e n t in numerical results, except perhaps in the a s y m p t o t i c r e g i m e . ) -

cos(~. R,) Explicitly, G,,An = ~ - - , where g,, is the vector in the surface plane from site o to site n, ej J e - ej+ i8 denotes the eigenvalues of H,,, and the notation on the wavevector reflects the fact that only crystal momentum in the surface plane is a good quantum number. If a single adatom sits in a bridge site between surface atoms 0 and 1, then G B.... = G A.... + GAl. If a second adatom sits between n and n+l (assuming all ,B A A A four sites colinear for simplicity, then G,,,, = G,,,, + (l/2)(G,,.~+t + G .... -l). To complete the description, one must make some statement about how the adatom-substrate hopping depends on the adsorption site, which will involve some at-least-implicit assumption about dependence on bond angles, bond lengths, local relaxations, etc. The parameter Vthat appears in the formalism corresponds to ~ times the hopping parameter between the adatom and one of the z members of the hybrid to which it couples.


591

In the LCAO framework, the formula for the pair interaction energy E,, between the adatoms adsorbed on sites o and n (which we identify as nth nearest neighbor sites on the surface) is, from Eqs. (11.9), (11.10), and (11.13), F-,F

2 f I m In [1 -(G,,X(~;)) 2 (GX,,,(e)) z V a] de E,, = rc

(11.15)

To gain some understanding of this interaction, we first expand the logarithm and consider the lowest order term (Kim and Nagaoka, 1963), which becomes a good approximation for weak coupling (small V) or large separation (small G,,.)" El:

E,, - - - - 2 Im f V4 (GX(~;)) 2 (Go,,(c)) x 2 de

(11.16)

/1; -oo

If G is neglected (which is generally a poor approximation at small separations), expression (11.16) is just the RKKY interaction energy (Ruderman and Kittel, 1954" Yosida, 1957), in which two localized spins (here localized defects) interact via coupling to a bulk conduction electron sea. If this sea is viewed as a free-electron gas, the propagator G,,, reduces to a continuum G(IRI;~), where R goes from one bulk spin/defect to the other. This bulk interaction is proportional to (x cos x sin x) x -a, where x = 2kFIRI. It is thus oscillatory in R and decays asymptotically as R -3, characteristic of Fermi surface domination. We shall discuss the decay on surfaces in the section on asymptotics. A physical interpretation of Eq. ( l l . 15) is that an electron in an occupied state starts at one adatom, hops back and forth to the substrate many times, then propagates to the second adatom, hops back and forth again for a while, then propagates back to the starting site. Alternatively, one can describe the process as a particle and a hole propagating from one adsorption site to the other (Zangwill, 1988). While Eq. (11.16) suggests that the interaction is proportional to V4, such behavior only obtains in the limit of weak coupling. For stronger coupling, the V-dependence in G eventually cancels the leading V4. This strong-adsorption case is the limit of the "surface molecule", in which the adatom and its substrate partner form a dimer which rebonds perturbatively (with the bulk coupling strength) to the substrate. The interaction between the adatoms then comes from the interference between the two dimers in the rebonding process, which does not depend on V. Grimley (Grimley, 1967; Grimley and Walker, 1969) was the first to apply the Anderson model to chemisorption, using as a substrate a semi-infinite single-band crystal with a phenomenological surface reactivity. This adjustment highlights a problem with tree-electron gas substrates, namely how to allow coupling with adatoms. If the adatom sits beyond an infinite barrier, e.g., there will be no coupling whatsoever. To avoid this problem, to put in site specificity in a natural way, and to reflect the belief that the d-bands were primarily responsible for the lateral interactions, ES modified Grimley's model by using as the substrate the (100) face

592

T.L. Einstein

of a single-band simple cubic crystal ("simple cubium") in the nearest-neighbor tight-binding approximation. Eq. (11.15) was then evaluated numerically. Table 1 1.1 capsulizes the results of ES. The energies are measured in units of one-sixth of the bandwidth (i.e., twice the hopping parameter). For the typical transition metal d-band being modeled, this unit is of order 1 to 2 eV. The Fermi energy and adatom level are measured relative to the center of the band. As the table illustrates, the pair interaction is highly anisotropic, oscillatory in sign, and rapidly decaying. At close separations the decay is precipitous, more exponential than inverse power like, dropping roughly by 1/5 with each lattice spacing, while asymptotically it decays as R-5. While asymptotic behavior is discussed in more detail in w 11.2.6, we note here that it is characteristic of dominance by a single k-state on the Fermi surface. The more complicated behavior at shorter range shows that many electronic states participate in the pair interaction. The pair interaction is comparatively insensitive to changes in to and V, somewhat more sensitive to shifts in the Fermi energy (especially for larger interadatom separation), and very sensitive to the adatom binding site. Typical values of the magnitude of the nearest, next nearest, and third nearest pair energies are 1xl0 -~, 2x10 -2, and 8• -3 units, although each of these can vary over a range of an order of magnitude. As presented here, this formalism implies that the substrate is essentially rigid during the adsorption process. In fact local distortions certainly do occur. Feibelman (1987, 1989, 1990) has emphasized that these distortions can play a crucial role, particularly at near-neighbor spacings. At farther separations, it does not seem unreasonable to believe that the distortions essentially renormalize G~,, while leaving G,,, relatively unaffected. Thus, over this range, the distortions might be taken into account by tuning the atomic and coupling parameters. n

X

11.2.2.3. Simpler illustration: pairs on a ring Many of the ideas in the preceding section may be couched in a (Green's functions) language unfamiliar to some readers. In an attempt to make the key ideas clearer to people more comfortable with the language of quantum chemistry, we present in this section results of an explicit calculation, done with Mathematica TM, in which the substrate is taken to be a ring of 50 atoms. For a system of such limited size, we can keep explicit track of what happens to all of the molecular orbitals. While I D models are a typical starting point in similar studies (Hoffmann, 1988; Whitten, 1993), we caution that consequently they contain some anomalous features which are not characteristic of most 3D substrates. In attempting to keep the following discussion uncluttered, we do not dwell on such unpleasantries as the inevitability of split-off states (due essentially to the divergence of the density of states at the band edge) and the anomalously slow decay of the interaction with separation. After exploring interaction energies from the perspective of shifting molecular orbitals, we show how the problem can be recast in the Green's function formalism presented above. The Hamiltonian of the ring itself (Hmc,~~of Eq. (11.3)) can be represented by a 50x50 matrix with non-zero entries (taken as -1/2) only along the two diagonals next to the main diagonal (i.e. entries {n, n+l }) and at the corners ({ 1,50 }, {50,1 }) to close the chain into a ring. By analogy, e.g., to benzene rings, it is well known

Table 11.1 Display of the pair interaction energy En suggesting the sensitivity of adatom arrays to changes in the Fermi level, the hopping potential V, the adatom energy level &a, and the binding-site symmetry A, C, B, and BP. (For bridge-site adsorption, there are two nearest-neighbor configurations: in B, the vector R between adatoms is in the plane formed by the adatom and its two substrate neighbors; in BP, R is perpendicular to it. Note that, e.g., for E2 there is no difference between B and BP.) One adatom sits at the origin "0"; the pair energy is for a second adatom at the nth nearest-neighbor site. The magnitude of the number given is 10 plus the common logarithm of the magnitude of the interaction. A plus (minus) sign indicates that the interaction is repulsive (attractive). Thus, table entries of . The energy unit is one-sixth the substrate band width, roughly 1-2 eV. +8.9, -7.7, and 6 . 6 represent interactions of +8x10-~,-5x10-~,and 4 x 1 0 ~respectively. Each chart is labeled by the symmetric adlayer pattern predicted. Adapted from Einstein and Schrieffer (1973) and Einstein (1979).

5

a

2. b

t

'a-

$ rn

C

% 2g 2 L

A

Binding site n =

3 1 0

C 4 2 1

5 4 3

3 1 0

B 4 2 1

5 4 3

6

9

13

3 1 0

BP 4 2 1

5 4 3

3 1 0

4

5

2

4

1

3

3.

F

594

T.L. Einstein

that the 50 eigenstates are traveling w a v e s with w a v e v e c t o r s k = nrt/25a, w h e r e a is the n e a r e s t - n e i g h b o r spacing. T h e r e are just 26 distinct e i g e n e n e r g i e s - c o s ( k a ) ; all but two (viz. 1 and - 1 ) are d o u b l y d e g e n e r a t e (due to the s y m m e t r y o f c l o c k w i s e and c o u n t e r c l o c k w i s e travel). T h e b a n d w i d t h in these units is, thus, 2, and the b a n d is c e n t e r e d about 0. T h e prescription in this Htickel m o d e l for adding extra atoms essentially f o l l o w s that in the t i g h t - b i n d i n g model. An a d a t o m h a v i n g e n e r g y ~,, ( i . e . - c ~ ' in Htickel l a n g u a g e ; ~ = 0) couples, via the h o p p i n g p a r a m e t e r - V (i.e. -13' in Htickel l a n g u a g e ; recall 13= 1/2) to an orbital o f the ring (as in atop bonding). C o n s e q u e n t l y , o n e adds a row and c o l u m n to the 5 0 • matrix. T h e d i a g o n a l entry { 51,51 } is ~,,; the only o t h e r n o n - z e r o entries are pairs o f - V ' s at { 1,51 } and { 51,1 }. F o r simplicity, to retain s y m m e t r y in ~ and to focus on c o v a l e n t effects, we take e,, = 0. As m i g h t be e x p e c t e d , the c o m p u t e d e i g e n v a l u e s m o v e away from E,, = 0, by an e n e r g y ,,,: V2 in the p e r t u r b a t i v e regime. (Actually, this interaction splits the d e g e n e r a c i e s o f the ring. T h e c o m b i n a t i o n o f the two eigenstates with an a n t i n o d e at the a d s o r p t i o n site gets shifted, while the other c o m b i n a t i o n with a node keeps its original value o f - c o s ( k a ) . ) This d o w n w a r d shifting o f orbitals increases the a d s o r p t i o n e n e r g y (absolute value o f the c h a n g e in total e n e r g y due to adsorption) as the Fermi e n e r g y a p p r o a c h e s ~,, = 0; thereafter, the a d s o r p t i o n e n e r g y decreases, e v e n t u a l l y r e a c h i n g zero (as particle-hole s y m m e t r y d e m a n d s ) . To assess pair interactions, we add a second a d a t o m n sites away from the first.' T h e matrix b e c o m e s 52• with ~,, = 0 at {52,52} and - V at {n+1,52} and {52,n+1 }. To c o m p u t e the pair interaction at close spacings, we c o m p a r e the e i g e n v a l u e s when the a d a t o m s are at n e i g h b o r i n g or n e x t - n e a r e s t n e i g h b o r sites with those w h e n they are at o p p o s i t e sides of the ring. To m a k e sense of these results, we first c o n s i d e r the situation of a d a t o m s at o p p o s i t e sides o f the ring. For an infinitely large ring one would e x p e c t results to be similar to the s i n g l e - a d a t o m case, but with shifts twice as large. For the finite case here, we note that this will occur only for states with an even n u m b e r of nodes, so that the ring eigenstates to which the a d a t o m s c o u p l e will have the s a m e a m p l i t u d e s on the two a d s o r p t i o n sites. T h e pair interaction arises from the shifts in the e n e r g y levels w h e n the a b o v e w i d e l y - s e p a r a t e d pair o f a d a t o m s are m o v e d to nearby sites. 2 W e e x p e c t that the

Note that on the ring, there is really a second pair interaction over separation (50-n)a. Because of the periodic boundary conditions used in the previous section, this effect exists implicitly in the formalism developed there. For a large ring, this second interaction is negligible, but this effect prevents us from using a small ring. The alternative of using a chain rather than a ring is undesirable because the "substrate" sites are inequivalent. 2

There are alternative definitions. Burdett and Ffissler (1990) start with ligands (viz. CO) attached to 1, 2, or 3 metal atoms, using the extended Htickel model, and seek to explain the structure of the ligand "pair potential" for a monolayer. Since it is impossible to move ligands far apart, they define the pair energy as the sum of the energy of the system with both ligands present and the energy with both absent, minus twice the energy with just one ligand attached. Some thought shows that this definition is equivalent to the one we use, assuming that our adatoms are far enough apart that they do not interact. While this perspective may be appealing, it is a chore to keep track of the electrons as they are added, and tricky to trace the evolution of the levels.


595

eigenenergy will decrease (become more energetically favorable) if the coupling is in-phase and increase if it is out-of-phase. More explicitly, we consider the eigenvectors of the antecedent of each energy level, from the original ring (without adatoms). If the eigenvector has the same sign on the two nearby sites, we expect the shift to be attractive (i.e. the level lowers in energy when the adatoms are brought to the nearby sites). The pair interaction comes from the shift of the occupied levels. How many levels are occupied is, of course, determined by the Fermi level. We generally expect that near the bottom of the band, the shifts will be attractive (negative) because the adatoms couple in phase. As the Fermi energy increases, more-rapidly oscillating eigenstates become involved. To plot and thereby analyze this behavior, we do the following: Along the horizontal axis we use the energy of each of the levels of the ring with the adatoms at opposite sides. Thus, as we set the Fermi energy further to the right along this axis, more levels become filled. At each of these 52 discrete energies, we plot as the vertical coordinate the shift of the level when the adatoms are brought to nearby sites. It is these shifts, due to the "interference" of the proximate adatoms, which in the weak-V limit tend to scale as V4 (cf. Eqs. (11.15) and (11.16)), i.e. as a next-order effect after taking into account the V2 shifts due to adsorption. The sum of these shifts, for the occupied levels, is the pair interaction. ! Thus, what we have plotted is the integrand in Eq. (11.15), essentially the phase shift 11. Near the bottom of the band, the integrand is, as noted above, generally negative, but with increasing energy, it begins to oscillate in sign. (The closer the adatoms, the larger is the energy between sign changes.) In performing this analysis, the shifts alternate between the expected behavior and a much weaker shift of the uninteractive ring eigenstates. Thus, in Fig. 11.2 we combine pairs of shifts, plotting their sum vs. the average of their (unshifted) energies (from the case of adatoms at opposite sides). We now seek to show that these results offer a decent finite-size approximation of the quasi-continuous behavior considered in the previous section. For an infinitely long chain, one can derive the analytic expression (cf., e.g., Economou (1979) or Davison and Steslicka (1992) for background information, or Kalkstein and Soven (1971), with no intralayer hopping): -i

[_e + i ~ ~ 5 - ] "

(1117)

c , , . ( s ) - ~41- - - -_- - 7

inside the band (1~1 < I); outside the band ~/1 - e2 ~ i.sgn(e) ~/E2 - 1 and Go, is pure real (Ueba, 1980). For the 50-atom ring, the substrate Green's functions can be computed numerically as G,,,(e) =

-•0

cos(kna) ~ e + cos(ka) + i8

(11.18)

k

1 Actually,it is half of the interaction, since we have been neglecting the factor-of-2 spin degeneracy in this section.

596

T.L. Einstein

0.03

0.06 f

. . . . . . . . .

,, . . . . .

~

.

.

0.02 u~

o

0.01

o

o.oo

:

O.02 0.00

._

-o.01

g c)

-0.02

-0.02 -0.04

-0.03

X

X

-0.06 -

.0

-0.5 0.0 0.5 e n e r g y r e l a t i v e to b a n d c e n t e r

1.0

.0

-0.5 0.0 0.5 e n e r g y r e l a t i v e to b a n d c e n t e r

Fig. 11.2. Integrand used to compute pair interaction energy for adatoms at nearest-neighbor and next-nearest-neighbor sites on a ring. Solid curve: continuum limit, as described in w 11.2.2.2. x- shifts of pairs of eigenvalues vs. average of their unshifted energy for a ring of 50 atoms. See text for discussion.

setting the infinitesimal at a value of, say, 0.1. ~While these Green's functions have many secondary oscillations, their overall behavior is rather similar to the analytic infinite-length Green's functions of Eq. (11.17). In any case, inserting the analytic form into Eq. (11.15), we produce the integrand (without the factor of 2) and coplot it in Fig. 11.3. We see that the couple-dozen pairs of levels from the ring provides a decent accounting for the results of an infinite ring in a form that may be more transparent. Our exercise further supports the idea that the pair interaction is a delicate mix of the couplings to all the occupied levels (or at least the half of them which are symmetric with respect to the inversion about the midpoint of the adsorption sites). Thus, a discussion in terms of H O M O (highest occupied molecular orbital) and L U M O (lowest unoccupied molecular orbital), i.e. frontier orbitals (Hoffmann, 1963, 1988) will not capture all the physics of the problem. On the other hand, with increasing separation there are more oscillations in sign as a function of oF. In the limit of large separations, reminiscent of stationary-phase problems, the interaction energy will be dominated by the endpoint of the integration, namely the behavior at By., making a frontier-orbital approach appropriate, if one has some grasp of the long-range behavior of wavefunctions at this energy. (In the section on asymptotics, we shall explore this problem further.) More importantly, in the limit of small V, the shifts and hence the interaction are quite small except when 8F is close to e,. In the limit of w e a k chemisorption, then, the H O M O / L U M O viewpoint may well offer a fruitful perspective on pair interactions. Burdett and Ffissler (1990), for example, in modeling CO adsorption find the interaction is strong only when 8F is near a large H O M O - L U M O gap. Before closing, we mention, for those particularly interested, some details skirted above. In Fig. 11.3 only 24 pairs of levels are included. In addition, there 1 The size of 8 in this discussion should be large enough so that the spiked distribution due to the discrete levels is smoothed but not so large that it is completed washed out. This parameter broadens the levels, a common way to represent a large system by a much smaller one with a limited set of eigenenergies.

597


1.5 1.0 0.5

0.5

,~

o:o

0:0

o

-0.5

-0.5

-1.0

,:,

1.0

1.5i -

|

. .0

.

.

.

-0.5 0.0 0.5 e n e r g y relative to band c e n t e r

1.5

-1.0

-0.5 0.0 0.5 e n e r g y relative to band c e n t e r

.0

Fig. 11.3. Off-diagonal Green's functions Gol (nearest-neighbor) and G02 (next-nearest-neighbor) for a ring of 50 atoms (setting B= 0.1) compared with the continuum form (solid curves), computed exactly. x: imaginary parts" 0" real parts. are pairs above and below the band, corresponding to what have been called "split-off" states (ES) and amount to localized levels outside the substrate band (where Im G vanishes). As one might guess from Eq. (11.12), they are the solutions e• of the equation - 13,,- V2(Re Goo(e~) +_Re Go,(~)) = 0

(11.19)

For the case of isolated adatoms at opposite ends of the chain, the + term is absent and the solutions ~;0 are doubly degenerate. The eigenenergies of the 52• matrices correspond virtually identically to the solutions of these equations, which use the quasicontinuum Green's functions. For 3D substrates, these states typically occur only for strong coupling (large V) but in 1D they are always present, formally due to divergent van Hove singularities in Re G at the band edge, physically because of the large number of states near the band edge in 1D models. From the figures in ES ~, one sees that E+ + e _ - 2~0 is positive. This initially counterintuitive result can be derived analytically or graphically from the generic form of the Green's functions. The fact that ~+ shifts down from E0 less than ~_ shifts up corresponds to the relative decrease in shifts in the levels as one gets farther from the band and ~,,. The split-off state involves fully in-phase hopping around the ring. Perhaps when the adatoms are close to each other, the electrons get somewhat concentrated in the region near the adatoms, so that they cannot take full advantage of the hopping all around the ring. In any case, this result leads to the spikey behavior with the unexpected sign near the band edge.

11.2.3. Multisite interactions 11.2.3.1. Three-adatom (trio) interactions In general there will be several adatoms in close proximity. Eq. (11.2) anticipated the possibility of multiadatom interactions. The expectation of ES is that overlayer

1 Bewaresome misleading analysis in wII.B.3 of ES.

598

T.L. Einstein

electronic energies are overwhelmingly dominated by nearest pair interactions. To evaluate multisite interactions, and thereby check this idea, it is straightforward (Einstein, 1979a,b) to enlarge the matrices needed to compute the phase shift in Eqs. (11.9) and (11.10). If the sites to which the adatoms bind are lth, mth, and nth nearest neighbors, we find EF

Er=,,,, - - 2 ~ Im ln(A)dlz- E , - E,, - E,

(11.20)

where u

A = 1 - V4 G,,2. (G~, G~m GZo.)- 2 V 6 G,,3,,Go, Go,. Go.

(11.21)

As indicated in w 11.2.1, Eq. (11.21) does not factor, making an explicit subtraction of pair energies necessary. There are two parts of this new interaction: (1) a new triangular path, represented by the G G G term; and (2) an "incompleted cubic" term, marked by the absence of V8 and V~2terms that would be present in E, + Em + E, if their logarithms were merged. Trial calculations using just the "triangle path", with moderate V appropriate to chemisorption, reproduce the full interaction at least qualitatively. Computations of trio energies using Eq. (11.20) suggest that their magnitudes are determined primarily by the two closest (strongest) pairs. In explicit comparisons (Einstein, 1979a,b) of E,m n -- E223, E225, E238, and E335, for typical V and •,,, for all possible substrate fillings, the first two have the strongest trio interaction energy. E223, which has a 3rd neighbor spacing as its third side, is somewhat the larger, and is nearly as strong as E 3. The other two are smaller by at least half an order of magnitude. With increasing adatom separation the trio energies fall off rapidly, much like the pair energies.

11.2.3.2. Complete overlayers While quartets and higher-order terms could be calculated, numerical noise problems from successive cancellations would become troublesome. Starting from the other extreme, one can easily show (Einstein, 1977) that the indirect interaction energy per adatom for a complete ( l x l ) adlayer of N, adatoms is El:

f

2 ~] f Im In I[1 - V2G..(~) G(k,,,E)/[1

/1;NIl

k,

I

-

V2Gaa(~) Goo(F_,)]IdF..,

-.~

(11.22)

2 2 ~Im In [1 - V2 G.,.,{ G(k,, ~ ) - G,,,,(I~)}] de

71;N,,

kll

-,Do

where the summation goes over the surface Brillouin zone (SBZ), containing N, (the number of adsorption sites) points. G(k,,e) can be computed analytically (Kalkstein and Soven, 1971), rather like a semi-infinite chain.


599

For a real monolayer, direct interactions between the closely-spaced adsorbates are likely to produce an interaction energy quite different from that predicted from Eq. (11.22). Not only does the direct interaction make a great difference for individual pairs (Burke, 1976), but it often leads to the formation of two-dimensional adlayer bands which overshadow any indirect effects (Liebsch, 1978). Therefore, we focus on the c(2x2) overlayer. Since the real space unit cell area is doubled, the SBZ is halved, most naturally taking the form of an inscribed "diamond" (square rotated by 45~ Points outside the new SBZ get folded back in, giving doubling of the (highly blurred) two-dimensional band-structure. The upshot is that for a c(2x2) adlayer, G(kai,E) in Eq. (11.22) is replace by (Einstein, 1977, 1979a) and Na = Nil~2 +

-

(11.23)

Based on these ideas, one can compare (Einstein, 1977, 1979a) the indirect interaction energy (per adatom) for a full c(2• overlayer with an explicit sum over the pair energies for all pair configurations arising in a c(2• pattern ~ only the five shortest contribute significantly - - weighting them according to the number per adatom existing in the pattern: two for pairs along the < 10> and < 11 > mirror axes, four otherwise. Overall, this curve does a good job of reproducing the c(2• plot. Trio interactions can also be included in the sum and help make up differences between the overlayer calculation and the explicit sum. Their contribution generally is important only near energies corresponding to the Hartree-Fock bonding and antibonding resonances in the DOS. In short, multisite interaction energies are not too important in total overlayer energies, although they may play a role in other circumstances. The other way to approach dense monolayers is to invoke results from the theory of alloys (Ehrenreich and Schwartz, 1976). Perhaps the simplest such scheme is the average T-matrix approximation (ATA) (Korringa, 1958), which assumes that adatoms are randomly distributed over the lattice sites. Urbakh and Brodskii (1984, 1985) work out the formal expression for Ap(~) and apply it to Pt(ll I)-H (cf. w 11.4.2.). The next level of sophistication is CPA, in which the self-energy of the "effective medium" of the alloy is calculated self-consistently; an application by Persson and Ryberg (1981) was noted in w 11.2.1.

11.2.4. Coulombic effects: self-consistency and correlation, and other improvements The issue of self-consistency has pervaded most subsequent efforts to apply tightbinding methods to the pair problem. The inability to resolve this problem in a satisfactory way is one of the greatest difficulties in extending this approach to quantitative investigations. In the LCAO framework, since the electron orbitals are fixed at the outset, self-consistency is discussed in terms of the Friedel (1958) sum rule - - which in this case requires charge neutrality within some finite range of an adatom w rather than Poisson's equation (Appelbaum and Hamann 1976). Typically, e,, is adjusted (making it a derived rather than a free parameter) (Allan 1970, 1994). The energies of nearest neighbor(s) on the surface may also be altered, thereby inviting new surface states (Kalkstein and Soven, 1971; Allan and Lenglart, 1972). Sometimes off-diagonal Coulomb terms are also included in various ways

600

T.L. Einstein

(Rudnick and Stern, 1973; Leynaud and Allan, 1975), meaning that changes in charge on a site affect the potential of its neighbors. Generally neutrality is required either at each site or just in the surface cluster consisting of the adatom and its nearest neighbor(s), excluding any longer-range oscillations. The quantitative results are rarely compelling. The qualitative results (Einstein 1975, 1979a) are plausible. A second approach assumes that in a strongly chemisorbed system, the essence of the pair interaction lies in a surface molecule. A small cluster is treated carefully, gaining an improved description of local Coulomb effects at the expense of any background effects from the substrate. From studies of W2H and W3H2, for example, Grimley and Torrini (1973) conclude that H atoms at nearest neighbor sites on W(100) will be unstable, the repulsive energy being of order 200 meV. This method is not extended readily to more widely separated pairs, since the distance from the adatom to the edge of the cluster should presumably be at least as large as its distance from the other adatom. Since the "substrate" w a v e f u n c t i o n s - via which the pair interacts m are sensitive to the details of the cluster, matching conditions to the background must be adjusted carefully. Moreover, in cases where adatoms bond to a common substrate atom, some anomalous structure may arise which should not be generalized (ES; Einstein et al., 1990). The best hope for cluster approaches is to embed them in well-characterized semi-infinite substrates (Grimley, 1976). Grimley and Pisani (1974) have taken this approach for clusters containing single adatoms and calculated in a S C F - L C A O - M O scheme. The embedded cluster technique has indeed flourished (cf. NATO conference proceedings in Pacchioni et al. (1992)). Most of the applications are to monomer adsorption, but the dissociation of (gas) dimers is also often considered (e.g. Cremaschi and Whitten (1981), Madhavan and Whitten (1982)). As noted, it is hard to imagine applying the method to larger pair separations. Feibelman (1989a) provides a lucid critique of this approach, questioning typical choices of bases and treatment of background effects. Also, since correlation is typically considered only in the cluster region, he wonders how much of the adsorbate binding energy actually comes from allowing substrate correlations in the bonding region. Grimley and Walker (1969) observe that while sizeable charge transfer might take place during chemisorption, little more should happen as a function of the relative placement of the adatoms. If energies in simple models could be determined in some plausible way, the pair interaction should work out satisfactorily even if the single-adatom results are somewhat inadequate. Moreover, the pair interaction is a rather insensitive function of e,,, as suggested by Table 11.1 and shown more convincingly by Fig. 11 of ES. Sch6nhammer et al. (1975) studied carefully the correlation effects in indirect pair interactions. Using a (100) cubium substrate with parameters appropriate to H on Ni, Sch6nhammer (1975) had previously shown from a variational approach that the single adatom binding energy is roughly 1/3 stronger than in Hartree-Fock (although the two curves did have the same structureless shape as a function of EF). They find that this correlation energy, 1/4 the binding energy, roughly cancels out when the pair interaction energy is computed. Although this cancellation is reported to be less complete for other parameters, the qualitative behavior holds for V's of


601

order the "critical hopping" (below which Hartree-Fock local moments arise). In addition to confirming the anisotropic, oscillatory behavior of the pair interaction, Schtinhammer et al. (1975) corroborate the roughly exponential fall-off with separation (for interadatom distances of order 1 to 4 lattice constants). The implication of this work is that correlation effects (in the form of careful treatments of the Anderson Coulomb term), while important for single adatom effects, can (to a reasonable approximation) be neglected in computing pair (and higher order) effects. Later studies discussed below (w 11.3.5) cast doubt on how well this result generalizes to models treating the d-band aspects of the substrate. Over the last decade or more, research in chemisorption theory has stressed generation of numerical results to fit quantitatively data from UV photoemission, ion neutralization spectroscopy, low-energy-electron-diffraction (LEED), and scanning tunneling microscopy (STM) experiments. The primary object has been to compute the spatial and energy distribution of the electron density near the surface region and to find exact locations of surface states. For these applications self-consistency (here in a Poisson's equation sense) is crucial. The first attempts to gauge the role of such effects considered the adsorption of single adatoms on jellium. In semiconductors it is difficult to propagate electrons from one adsorption site to another, from a physics viewpoint because the Fermi energy lies in the band gap, from a chemical perspective because electrons are relatively localized in covalent bonds. (Some implications are discussed in the next section.) Tosatti (1976) has considered the interaction between adatom pairs on Si(100)(2• assuming a short-range defect potential for the adatoms and linear response by the surface electrons. His pair interaction is always repulsive, oscillatory (in strength) with separation, but with an exponentially decaying envelope (due to trying to propagate electrons in the gap). Realistic slab calculations for transition and noble metals began appearing about a decade ago and are becoming more or less routine for flat surfaces. They are discussed at length in volume 2 of this handbook. Nonetheless, even today most total energy self-consistent calculations consider only a (1• overlayer, with the full symmetry of the substrate.

11.2.5. Lattice indirect interactions: phonons and elastic effects To check whether there were significant interactions mediated by phonon rather than electronic degrees of freedom of the substrate, Cunningham, Dobrzynski, and Maradudin (1973) studied the contribution to the free energy of the interaction between two identical adatoms via the substrate phonon field. In their model, the adatoms sit in the atop position on the (100) face of cubium. Results are computed as a function of the three dimensionless quantities: adatom mass over substrate mass, adatom-substrate coupling over substrate-substrate coupling, and inter-adatom separation R (in lattice constants). They find that the zero-point energy is invariably attractive and that it decreases monotonically in strength with R, going like R-7 for large R. The attraction is at most 10-4h (1) L (where O) L is the maximum phonon energy) or of order 10-6 eV, and thus nearly always negligible.

602

T.L. E i n s t e i n

Given these negative results, little further work was done on this problem. However, beginning half a decade later, considerable interest has been paid to elastic interactions on surfaces. When electronic interactions play a significant role, it is generally not just difficult but artificial to try to isolate elastic effects from other electronic effects. However, for non-metallic substrates, where there are no electrons near the Fermi energy, focus on elastic interactions may often be a fruitful perspective. Also, in the asymptotic region, the elastic interaction generally dominates for large enough separation. In this section we first give a chronological account of studies of this interaction between adatoms. We then discuss in more detail, via a few examples, the interplay between elastic and electronic effects on metal substrates, and why this perspective is often not fruitful at close range. Lau and Kohn (1977) investigate the long-range interaction between two adatoms due to classical elastic distortion of an isotropic semi-infinite substrate, finding: h

0,

_

_

_

, , ,

l

!

I

l

l l

l ~

/ /

#

I

x

,

.-xx\-\ I

...... ,...",,,..",,,. ",,,

I

" ".".,"./'1 I "',lil'J

\7\",,'.."'" " 77)""''

,,

z

z

z

1

1

1

1

~

L

l

~

/

\

l

\

,

\

.

\

(e) Fig. 11.4. Schematic of the origin of the elastic repulsion between like atoms on an elastically isotropic substrate. (a) Response of the substrate to a single atom. Here the displacement is taken to be away from the adsorbate, though it is more likely to be toward the adatom. (b) When two adatoms are present, substrate atoms between them cannot relax fully.

603


E~I~,(R)_ 1 - o" A~At, 4x~t R 3 ' Aa = ~-" F~j . ( R j - R a ) J

(11.24)

where F~j is the force exerted by adatom a at Rj, ~ is the Poisson ratio, and IX the shear modulus. For identical atoms, this interaction is always repulsive, due to frustrated relaxation of substrate atoms between the two adatoms, as illustrated in Fig. 11.4. Taking ~ = 1/2 and IX --- 10 -3 atomic units as typical, they estimate this repulsion to be of order 0.1 eV at R = 10 a.u. --- 5 /~. (If the a d a t o m - s u b s t r a t e coupling on a triangular surface has the form-3~/g0 6, where R0 = I R j - Ral is the spacing between the adatom and one of its 3 substrate neighbors, then the "virial" (Stoneham, 1977) A = - 6 T a 2Ros.) For different adatoms, the elastic interaction can have either sign. Its R -3 decay is reminiscent of the dipole-dipole repulsion. Inserting reasonable numbers for Xe pairs on Au, Lau and Kohn find at R = 10 a.u. that

I.

Q. .-..-

\ \ \ \ \ \ ~

\\\\\X

,,\\.X,\~

1 1 1 1 1 1 1 1 1 1 T t t t l l l l l /

t!

r r i I I I I II I [

,tZTrrl

~a~3]~

t Y,ZX l I

'~

[. 1 I / / 7

LI.Z/7/

\.~\A A

T ,,-,,,.~,.,.',,, ",,,'X.\

,_-.~.-~i~....-j,/ ~--'~"~'?//

/ 1 I / ,/ I

///11111

,,.,-....-........-......,,....-,...-....-....-/,/ X,.X,"~'%"-" ' " - " / ' / / / , ' I

\-..~--~~ X ",.,'~"'~--~,

i i i { i I I I I i I ],I ~ \ \ k,',,", I I I I I I I I I I t { ~ ~,\\\\ \\

(b} Fig. 11.4 (continued). Caption opposite.

604

T.L. Einstein

the elastic repulsion is 0.53 meV, compared to their dipole repulsion of 1.1 meV, i.e., about half as large. On the other hand, at nearest neighbor sites it is three or four orders of magnitude greater than the phonon-mediated attraction just discussed; in essence Cunningham and coworkers' calculation (Cunningham et al., 1973) gives the leading quantum correction to the classical distortive effects. Varying the vertical position of the adatoms relative to the isotropic substrate, Maradudin and Wallis (1980) also find the R -3 decay but find that the interaction is attractive if the average distance below the surface is greater than R/2~. Stoneham (1977) shows that if the substrate or the adatom-substrate coupling is anisotropic, then the elastic interaction between like adatoms can again be attractive. He estimates that the magnitude of interaction of neighboring bridge-bonded H on W is of order 0.1 eV, large enough to account for some measured interactions without recourse to electronic effects. He also considers additional elastic effects due to clusters of adsorbates. Lau (1978) in turn considers anisotropic substrates w i t h hexagonal or cubic symmetry. Using Green's functions derived by Dobrzynski and Maradudin (1976) and by Portz and Maradudin (1977), he works out explicit formulas. For Xe pairs on graphite, separated by 5 ]k, he finds a repulsion of 0.18 meV. On Au (100), with pairs of Xe again 5 A apart, he finds an attraction of 0.30 meV along the cube axis and a repulsion of 1.73 meV at 45 ~ He expects this anisotropic behavior to be fairly general. While these energies are quite small, he expects elastic effects to become stronger and play a significant role in distortive phase transitions. Kappus (1978) rederives the previous results on isotropic and cubic substrates, finding again the possibility of homonuclear pair attractions on anisotropic substrates. Between clusters a repulsive barrier arises, proportional to the product of the areas of the clusters, even in directions in which the long-range interaction is attractive. Kappus (1980) extends this work to consider an anisotropic force dipole tensor, which enters the calculation of the virials, but restricts the substrate to be elastically isotropic, a reasonable approximation for W. Again there is the possibility of elastic attractions between like adatoms. The formalism is applied to explain the ordered p ( 2 x l ) phase of O on W(110) (Engel et al., 1975; Wang et al., 1978). He obtains "reasonable qualitative agreement" with the pair interactions used by Williams et al. (1978) in a Monte Carlo simulation of this system. However, since they do not lead to the p(2xl ) superstructure, Kappus (1981) generalizes the model to include a nearest neighbor interaction, an electric dipole repulsion caused by adatom dipoles normal to the surface, and another long-range part coming from elastic dipoles of nearest-neighbor pairs of adatoms. This third energy leads to multisite interactions. Nonetheless, with an E2 interaction, he cannot stabilize the p ( 2 x l ) superstructure; such an interaction could, of course, arise from the electronic indirect mechanism, from small anisotropy in the elastic constants, or from a breakdown in the continuum approximation. ~ 1 This system has proved quite challenging. Rikvold et al. (1984) used a model with E 1< 0, E2 > 0, E3, and trio interactions, and still found that an attractive E51% of E l could introduce pronounced first-order behavior at both low and high coverages.


605

Theodorou (1979) proposes an intriguing approach to the overlayer structure of W( 110)-O. He noticed that on a rigid substrate the W - O - W angle of bridge-bonded O was 102.2 ~ rather than the ideal 90 ~ Presumably, then, these two W ' s would be drawn toward each other; there are two other W's, at the far ends of the "diamond" at the center of which the O sits, which are repelled by a lesser amount. From this perspective, he estimated energies per O of isolated atoms, the chain constituent of the p(2• the p(2xl) itself, and a full ( l x l ) to be 0.15 eV, 0.05 eV, 0.15 eV, and 0.29 eV, respectively. In terms of interactions, he essentially finds an attraction El = -0.10 eV which duly produces chains. A repulsion in a different direction keeps the chains apart. Unfortunately, some more distant (second-neighbor in some direction) interaction between chains is also repulsive, preventing the p ( 2 x l ) from forming. He speculates about what other interactions might overcome this repulsion, noting that the small work function change suggests that dipolar interactions are insignificant. Apparently no resolution of this problem was ever achieved and the paper seemingly has had little impact on research in adsorbate interactions, though perhaps it influenced thinking about strained superlattices in heterostructures (Tserbak et al., 1992). Tiersten et al. (1989) note that Kappus (1978) smoothly truncates 2D integrals over the surface Brillouin zone with a cutoff parameter of order the inverse lattice constant and that his interaction energies between adatoms separated by less than a few lattice spacings depends sensitively on this cutoff. Thus, they conclude that a lattice-dynamics analysis of the substrate is needed in the non-asymptotic range instead of the continuum elasticity approach. Working in a mixed representation (cf. just above Eq. 11.5) they find an expression for the pair interaction energy in terms of (Fourier-transformed) local force vectors associated with each adatom and a substrate propagator between the sites. This propagator they take to be essentially the inverse of the dynamical matrix. (In elasticity theory, the propagator is an angular-dependent term divided by the magnitude of the 2D wavevector; one then readily recovers the R-3 decay.) Tiersten et al. (1989) apply their formalism to As dimers on Si(100). They plot the interaction along the three principal directions, finding that it (1) can change sign with increasing R, (2) is highly anisotropic, (3) is rather small, less than 10 meV (often much less) once R >_ 8 ,~. They also look at interactions between H pairs on reconstructed W(100). Again they find that the interaction can be attractive or repulsive, that it depends on the direction, and is at most about 3 meV for the shortest R's, and becomes less than an meV quickly with increasing R. Presumably electronic effects are much larger for this case. In both cases the sign of the interaction at small separations can usually be understood in terms of the dominant forces on the substrate atoms or by simple arguments based on interference of the relaxations produced by the individual adatoms (cf. Fig. 11.4.). Later, Tiersten et al. (1991) consider Si(100)-O, finding generally similar qualitative features, but with larger magnitudes, around 50 meV at 4 ,~, but then falling quickly to less than 5 meV, then to less than a meV. In other words, when electronic interactions are present (on metals), they should dominate, but on semiconductors or ionic crystals, these could be the leading interaction. Recently Rickman and Srolovitz (1993) present a very general Green' s function

606

T.L. E i n s t e i n

formalism for finding the elastic interaction between defects of spatial dimensionality D and multipole character m on a surface. Specifically, they tabulate results for four generic defects: a point force (D = 0, m = 0), an impurity adatom or island (D = 0, m = 1), a stress domain (D = 1, m = 0), and a step (D = 1, m = 1). Since defects in general involve more than the lowest-order multipole, the results apply for large lateral separation R. For point interactions (D = 0) between an m-pole and an n-pole, the interaction E(R) o~ R -(''+'~+~), reproducing R -3 for the interaction between adatoms. For linear defects, E(R) o~ R -(re+n), o r C 1 ln(R/a) + C2 for m = n = 0. Further comments related to steps are deferred to w 11.4.3. The preceding discussion assumes that one can neatly distinguish between electronic and elastic interactions. Such a distinction is generally possible at moderate-to-large separations between adatoms, but fails in the "near" region: in computing carefully the electronic interaction between adsorbates (Feibelman, 1989a), relaxations can play an important role. There is clear experimental evidence that adsorption can distort the substrate in the vicinity of the binding site, although the precise nature of the deformation may be difficult to determine. For example, for N i ( l l l ) p ( 2 x 2 ) - O Narusawa et al. (1982) measured, with high-energy ion scattering, outward displacements of about 0.15 ]k of the three Ni's to which each adatom binds (i.e. substantial buckling and overall relaxation); from LEED analysis, Vu Grimsby et al. (1990) note, in addition, lateral "twist" displacements of about 0.07 A. However, Schmidtke et al. (1994) find in a subsequent LEED analysis no twisting, minimal relaxation, but buckling of 0.09/~,. In a painstaking LEED survey of Ru(0001)-S, Pfntir's group finds progressively greater substrate distortions with structures of increasing coverage: for the p(2x2) there is slight buckling and outward relaxation, o f - 0 . 0 3 ~, (Jtirgens et al., 1994). In the (q-3-• symmetry forbids such buckling; the relaxation is still comparably minimal (JiJrgens et al., 1994). In the 1/2 ML c(4x2) phase, there is substantial (-0.2 /~,) row buckling (Schwennicke et al., 1994), Ru atoms bonded to two S's relaxing more than those bonded to one S. (Moreover, the S atoms occupy fcc and hcp sites with equal probability, but are shifted laterally from the high-symmetry 3-fold position by --0.16/~,!) Finally, in the (q-7-xq-7-) at 0.57 ML, there is even stronger dependence of the Ru relaxation on the S coordination: surface atoms with 3 S's relax 0.39 A more than those with a single S (although the overall relaxation is minimal) (Sklarek et al., 1995). (It is also noteworthy that in all cases the local chemistry is preserved in the sense that S - R u bond lengths do not change by more than 0.05 ./k!) Since these displacements are based on fits to data, accuracy depends on the insight and ingenuity of the experimentalist. Using Tensor LEED and scanning tunneling microscopy, Barbieri et al. (1994) investigate two of the four ordered overlayers of S on Re(0001) (Ogletree et al., 1991) and find a similar increase in surface distortions with increasing coverage. 1 Substantial displacements of surface atoms will certainly affect the electronic states nearby (and so the interaction energy) and

1 Einstein(1991) points out that several distinct trio interactions would be needed to account for these ordered phases; presumably some of these are related to the local distortions.


607

evidently can depend on the separation between the adatoms. It is a futile exercise to sort out which portion of the interaction is elastic. As more specific systems are carefully documented, it will be interesting and important to look for trends in the evolution of buckling with coverage. 11.2.6. A s y m p t o t i c f o r m o f the indirect interaction b e t w e e n a t o m s a n d b e t w e e n steps

In this section we present more information than in w 11.2.2 about the nature of the indirect interaction between widely separated adsorbates. Our intention is to stress the general features and underlying physics while skirting explicit formulas, which can become quite complicated (Einstein 1973, 1978, 1979a; Lau and Kohn, 1978; Flores et al., 1979; Roelofs, 1980). From Eq. (11.16) we see that the asymptotic behavior hinges on the behavior of Go,(~) at large R, where R is the vector from site o to site n. While studying scattering in solids four decades ago, Koster (1954) recognized that with the competition of rapid oscillations, the solution required stationary-phase arguments. He uncovered much of the essence of our problem, finding that G,,,,(E) o,: R -I exp(i k(I;) 9R)

(11.25)

where k(e) is that wavevector along a constant-energy surface at which the velocity (viz. V~ke)is parallel to R, as illustrated in Fig. 11.5. Moreover, the proportionality constant varies inversely with the Gaussian curvature of the constant-energy surface at k. More generally, if Go,(e) o~ k -~ R-" e x p ( i k R ) , then integration by parts (Grimley, 1967) leads to the important result V4

--2

2

E, ----~--Re [G,,,(eF) Go,(~;F)]

(11.26)

and the interaction decays like R -(2re+l). For surfaces, one can show quite generally that m = 2, i.e. that G,,,(e) ,,,: k -~ R -2 e x p ( i k R ) (cf. the discussion in the paragraph after Eq. ( 11.28)) and E, ~ R25 cos(2kr.R, + r

(11.27)

if the interaction is isotropic. The complex quantity Go~ is independent of the separation and so leads to the phase factor ~; from Eq. (11.14), this factor is given explicitly by (Joyce et al., 1987) = arg(G,,)2 = 2 arg [l~F- e , , - V 2 Goo(eF)] -l

(11.28)

which vanishes when V2npo(eF) 0, the fluctuation method and the Kubo-Green approach give similar results. Using the transfer-matrix technique to calculate It, Myshlyavtsev and Zhdanov (1993) consider similar problems on a rectangular lattice with anisotropic interactions. Tringides and Gomer (1992) show that lateral interactions could produce anomalous behavior in diffusion constants measured by laser-induced diffusion compared with those from fluctuations around equilibrium, in contrast to their similar behavior in the absence of such interactions.

11.4.3. Surface states on vicinal and reconstructed fcc(110) surfaces The same mechanisms which underlie the interaction between atoms chemisorbed on flat surfaces should also play a role in the interactions between steps on vicinal surfaces. For most semiconductors the interaction potential between steps, U(l), is repulsive and decays as l-2, where I denotes the distance between steps, as reviewed by Bartelt and Williams (Chapter 2). This form describes energetic interactions expected from both dipole-dipole (Voronkov, 1968) and elastic effects (Marchenko and Parshin, 1980). As noted in w 11.2.5, this result can be argued from a very general Green's function perspective (Rickman and Srolovitz, 1993). Poon et al. (1990) found such behavior in a study of steps on Si(100) using the Stillinger-Weber (1985) interatomic potential. Using EAM to study vicinal Au (100) and (110), Wolf and Jaszczak (1992) assess how well Marchenko and Parshin's expression for two interacting steps (or another expression (Srolovitz and Hirth, 1991) for a periodic array of steps, which differs by just a numerical factor of order one) accounts for the computed step-step repulsion. They find first that the amplitude Gel of the 1-2 decay, which depends on Poisson's ratio, Young's modulus, and components of the linear force densities or stress factors, is nearly independent of l for large I. The Gem's for steps on the two surfaces are 50% and 70% of the value deduced directly from the orientational dependence of the surface free energy. The discrepancy is attributed to: (1) the fact that the expression for Qlassumes isotropic continuum-elasticity theory, while the environment near the step is highly anisotropic; (2) the bulk elastic moduli in the formula should be replaced by local responses near the surface, which have not been computed; (3) the treatment should be based on a fully relaxed flat surface with unrelaxed steps, i.e., doing the calculation correctly would require Gordian unraveling reminiscent of the comments near the end of w 11.2.5. Extending EAM calculations to six of the late

636

T.L. Einstein

transition/noble fcc metals, Najafabadi and Srolovitz (1994) also found 1-2 repulsions for l > 3ao; inclusion of a higher-order 1-3 term improved the ~2 of the fit by over an order of magnitude. Simple continuum elastic theory is thus deemed to fail at small I because it neglects the discrete atomistic nature of steps and surfaces and because the elastic field of a step cannot be adequately described by a surface force dipole alone. Detailed comparison shows that modeling steps as in-surface-plane dipole line forces in an isotropic elastic medium predicts elastic fields qualitatively different from those simulated. In both studies, it is important to remember that the EAM calculations are incapable, ipso facto, of including long-range electronic interactions since there is no Fermi-surface singularity. Recently, microscopic probes of surface structure, particularly the scanning tunneling microscope (STM) and reflection electron microscope, have permitted detailed measurements of the configuration of steps on single crystal surfaces. Specifically, the terrace-width distribution function P(I) provides a sensitive probe of step-step interactions. The simple/-2potential describes inadequately the terracewidth distributions which Frohn et al. (1991) have measured on vicinal Cu(100) surfaces" Although P(l) for Cu(1,1,7) has the shape expected for a simple repulsive potential, the width and asymmetry of P(l) for Cu(l,l,19) suggests attractive interactions between steps. Similarly, Pai et al. (1994) have recently reported STM measurements of vicinal Ag(110) surfaces in which steps appear noninteracting for {/) = 22 ]k, repulsive for (l) = 30 ~, and attractive for (l) = 40 ]k. While attractive interactions may result from surface stress relaxation in the vicinity of steps (Jayaprakash et al., 1984) or from dipole-dipole interactions (Wolf and Villain, 1990) (if dominated by the in-plane orientation), the most likely explanation is an indirect interaction between steps mediated by substrate electron states which can produce attractions at some step separations (Frohn et al., 1991; Redfield and Zangwill, 1992). In terms of the formalism in w 11.2.2.2, we can imagine the relaxation of each atom along the step edge as producing a localized perturbation on the substrate analogous to the chemisorption bond. In this perspective, we view the/-2 repulsion as arising from a naive integration along one of the steps of an r -3 point-point repulsion, thereby approximating the steps as lines of independent points (Redfield and Zangwill, 1992), although the result is more general. At small separations, the r-dependence of indirect interactions is usually quite complicated" however, for the nearest-neighbor tight-binding model, the asymptotic regime for indirect interactions via bulk states is reached in -4 lattice spacings (Einstein, 1978). In this asymptotic limit, we saw in w 11.2.6 (cf. Eq. (11.27)) that the indirect interaction reduces to r-Pcos(2kFr), where kF is the Fermi wavevector with velocity pointing in the ~ direction, p = 5 for mediation by bulk states near a surface, and we have assumed the phase factor ~ is negligible. The integration along the step edge is complicated by the oscillatory factor. ~ Redfield and Zangwill (1992) point out that, given site-site interactions of the form r-Pcos(~cr), the interrow

1 Redfieldand Zangwill (1992) pointout that this summation procedure is strictly valid only in the (weak) limit, when the local perturbation due to each site is independent of its neighbors.


637

interaction has the form r-"cos(K:r+5), with m = p - 1/2 and 5 = ~/4.1 F o r bulk electronic states, p = 5, so m = 9/2. As n o t e d in w 1 1.2.6, w h e n m e d i a t e d by [2D-isotropic] surface states, Lau and K o h n (1978) s h o w e d that the interaction d e c a y s like r -2 leading similarly to m = 3/2. T o d e c i d e which case is a p p r o p r i a t e for a particular substrate, o n e o b v i o u s l y m u s t k n o w s o m e t h i n g about the electronic structure o f the surface. O n the highly anisotropic (1 10) faces of noble metals, there a p p a r e n t l y are surface states that are p r o m i s i n g c a n d i d a t e s to m e d i a t e interactions in the [001 ] direction. H o w e v e r , these states 2 exist only (in a gap) near Y, (the intersection o f the [001 ] direction and the surface Brillouin zone b o u n d a r y ) , which w o u l d suggest m = 2 rather than m = 3/2. In M o n t e C a r l o ( M C ) simulations relying on a t e r r a c e - s t e p - k i n k ( T S K ) m o d e l o f surface structure, Pai et al. (1994) use a rather a d h o c potential e m b o d y i n g these ideas" it contains an oscillatory term at l > 6 lattice spacings and a r e p u l s i v e / - 2 - l i k e form at smaller step separations. While there is insufficient data to warrant c o n f i d e n c e in the specific potential, it is nonetheless noteworthy that this potential, with reasonable p a r a m e t e r s , can a c c o u n t for the distributions m e a s u r e d at three d i f f e r e n t (l). In s u m m a r y , vicinal A g ( 1 1 0 ) p r o v i d e s the first e v i d e n c e o f an indirect interaction m e d i a t e d by a surface state. It also illustrates that w h e n such effects occur, the l o n g - r a n g e interaction is by no m e a n s negligible. W e also note that Xu et al. (1996) are a p p l y i n g their m e t h o d using a m o d i f i e d fourth m o m e n t a p p r o x i m a t i o n to tight binding, discussed in the latter part o f w 1 1.3.5, to c o n s i d e r step interactions. T h e y fit their results to the form o f a m o n o t o n i c i n v e r s e - s q u a r e law repulsion plus an oscillatory term as discussed above, i n c l u d i n g ~, and obtain a r e m a r k a b l y consistent, if curious, set of results. Very recently, G u m h a l t e r and B r e n i g (1995) studied the s c r e e n i n g p r o p e r t i e s o f q u a s i - o n e - d i m e n s i o n a l states, such as may arise in the troughs o f r e c o n s t r u c t e d (110) fcc metals such as Ni and Cu (but not Ag) and c o n s i d e r e d h o w such states m i g h t m e d i a t e the indirect interaction b e t w e e n H atoms. T h e y derive analytic

The essence of the derivation is taking the leading term of ~ (x2+ y2)-p/2 COS(I(N/X2+ y2)dy to be o

i x /' cos(~(x + y2/2x))dx = x-(/'- I~!f(Ic~), wheref(~:,x) contains products of trigonometric functions and 1)

Fresnel integrals but has a simple asymptotic limit ~ cos ~ + 2 One state has been observed often for (110) late-transition/noble fcc metals, about 2 eV above E F (Bischler and Bertel, 1994). These states are probably too far from Er: to play an important role. However, Liu et al. (1984) calculated on Au(110) a second surface state just below EF, over a narrower range near Y, and there is some calculational evidence of a similar state on Ag (K.-M. Ho, private communication). Courths et al. (1984) reported such a state in an angle-resolved photoemission (ARUPS) study of Ag: it was found to be dispersionless at 0.1 eV below El: and sharply peaked in intensity at Y, seemingly vanishing by 20% of the distance to F. While the effect of steps and disorder are unclear, it is plausible that this state could be broadened or shifted to cross EF in some small region near Y.

638

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expressions for the consequent indirect interaction first from second-order perturbation theory (as in Lau and Kohn (1978)) and then with non-linear screening as in w 11.2.2, but with overlap included, both explicitly (Anderson and McMillan, 1967) and as a proportionality factor for the adatom-substrate eigenstate coupling (Gumhalter and Zlati6, 1980). The proportionality constant for the adatom-substrate coupling and the adatom energy ea are related by the Friedel sum rule (cf. w 11.2.4). The surface states in question run along the localized chain states in the closepacked troughs of (H-induced reconstructed) Ni and along the step edges of similar metals. When trying to parametrize the indirect interaction, the authors find that unless they assume a very slow decay corresponding to R -! in Eq. (11.27), the coupling parameter is unreasonably large. However, the specific fit to the potential in Brenig (1993) (cf. w 11.4.2) is unconvincing since that potential is: (1) represented as isotropic (while the interaction is manifestly highly anisotropic); (2) interpolated from interactions at three separations, all less than two lattice constants (and so far from the asymptotic regime); and (3) indicative of the total lateral interaction, including the direct contribution (which is likely non-negligible at the shortest separation). They and Bischler and Bertel (1993) (also Bertel and Bischler (1994)) suggest that this chain state is similar to the state seen in inverse photoemission by the latter pair. However, this particular state S~ is 6 eV above the Fermi ! energy, so presumably completely empty and hence inactive.

11.5. Discussion and conclusions Two decades ago at a Nobel Symposium (Lundqvist and Lundqvist, 1973) papers were presented on both the Kondo problem and the pair interaction, both cloaked in the Anderson model. In the conference summary, Anderson quotes Harry Suhl as saying "Like South America, the Kondo problem will always have a great future." Not only are such statements no longer "politically correct", in the meantime the Kondo model was instrumental in the formulation of the renormalization group (Wilson, 1975) and was solved exactly by Bethe ansatz methods (Andrei et al., 1983); even the two-impurity problem has been solved (Jones et al., 1989; Affleck and Ludwig, 1992). In contrast, consider what we have learned about the pair interaction. The only exact results relate to the asymptotic regime. Until recent evidence on vicinal Ag (110) of indirect interactions via surface states, these results proved of purely academic interest. There have also been exciting observations recently of standing

The main import of Bertel and Bischler's (1994) work is to show that a one-dimensional sp-derived state can exist on the surface. It is curious that the dispersion of the state is flat in the direction (in k-space) parallel to the chain (where one would expect considerable dispersion); Bertel (private communication) points out that this behavior arises because the state is antibonding in the top layer of Ni but bonding in the second layer (cf. the LCAO discussion in Bertel (1994)). Unfortunately,experimental complications have so far prevented measurements in the perpendicular direction (along which the dispersion should be flat if the states are in fact quasi-one-dimensional along the close-packed direction).


639

waves on surfaces (Crommie et al., 1993; Hasegawa and Avouris, 1993); there may be some relationship between them and the propagator which transmits the indirect interaction. The Anderson picture developed earlier at length does provide a convenient way to conceptualize the physics of the interaction. The earlier model work gives a general feeling for the relative size of the interactions associated with adatoms in various configurations. On the other hand, it has been difficult to improve the model or to achieve quantitative accounting of actual experiments, although there certainly have been several attempts in this direction (Sulston et al., 1986; Dai et al., 1987; Zhang et al., 1990; Cong, 1994; Sun et al., 1994). The model seems more limited in dealing with the adsorption of single adatoms than in the subsequent interactions between pairs. Perhaps if there were a compelling way to evaluate adjustable parameters for the single-adatom case, one might make progress in this direction. Models leave out many important pieces of physics which are particularly important in the "near" regime. Local distortions, changes in bond strength with coordination, and rehybridization subtleties have been seen to play a vital role in this regime. With scattering theory methods, compelling results have been obtained in a few simple cases. At present, progress seems to be computationally limited. It seems that in the foreseeable future, advances will come from improvements in the code rather than more powerful computers. In this regime, which is certainly the most important from a practical or chemical perspective, it is not necessary or perhaps even fruitful to concentrate on the Green's function carrying the disturbance produced by one adatom to the site of the second. Once the adatoms separate sufficiently so that they neither couple directly nor interact strongly with the same substrate atoms, the perspective stressing the propagation of disturbances should be the most appropriate. The pair interactions of remarkably few physical systems have been computed successfully. More strikingly, in many cases where two different methods have been applied, inconsistent results are found. The case of Pd(100)-H was discussed above. Consider now the case of Pt(11 I)-CO, not an ideal prototypical adsorbate from a theory viewpoint due to the two active orbitals of CO and the complicated adsorption mechanism. (It is also an intermediate case energetically, with a heat of adsorption of 1-11/2 eV (Toyoshima and Somorjai, 1979) so neither in the perturbative regime nor in the strong-adsorption regime of, say, H, which has an adsorption bond strength 1-2 eV greater (Christmann, 1988) and forms bonding and antibonding states during adsorption (Einstein et al., 1980). Persson (1989) (also Persson et al. (1990)) assumes that pair interactions depend only on separation R. Explicitly, his pair interaction consists of a Pauli (hard-core) (contributing 262 meV to E~ and negligible for larger R) and an indirect term which is also repulsive and decays (isotropically and rapidly) monotonically" (1.3 eV) exp[-(0.8 ,~-l)-R]. The two constants are chosen so that (1) the binding energy at half coverage is 0.25 eV less than that at zero coverage, and (2) the frequency of the frustrated translation at the atop site (preferred by 60 meV over bridge) increases from 49 to 60 cm -~. With this model potential he performs (off-lattice) Monte Carlo simulations which apparently do well at accounting for the experimental phase diagram. Joyce et al.

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(1987) present a strikingly different picture, but also achieve good agreement with different experimental data! They separate the interactions into direct, indirect, and site (atop vs. bridge, high-symmetry positions only) contributions. The direct part is formulated in terms of gas-phase Lennard-Jones potentials. The indirect part is assumed to come from sp electrons and expressed in terms of the asymptotic form, even at short range! They believe that the adatom-substrate coupling occurs via the 2rt* orbital (3 eV above EF) rather than the 5~ orbital (7 eV below). Their interaction is not purely repulsive, but oscillates in sign. Nonetheless, the results apparently fit desorption energies at four different fractional coverages ranging from 1/3 to 2/3. Wong and Hoffmann (1991) applied extended Htickel theory to CO on Ni, Pd, and Pt(111). Unfortunately, they only report results for two coverages (1/3 and 1/2), so it is unclear what the size and the sign of the pair interactions are. Very recently Jennison et al. (1995b), using a promising technique described below, found that the C O - C O interaction on Pt(111) is repulsive and decays monotonically (to at least 3 lattice spacings), similar to Persson's (1989) result. However, their admolecules are placed only on bridge (not atop) sites (favored by --0.1 eV); while the decay is sensibly less rapid, their repulsions are somewhat too strong" E 3 = 25 meV vs. E3 = 16 meV for Persson's experimentally-calibrated potential. Finally, both sets of interactions differ from the non-monotonic repulsive decay deduced by (Skelton et al., 1994) (cf. w 11.4.2). Until pair interactions can be computed readily and reliably, our general picture and its evolution provide a useful template with which to confront indeterminate interactions needed to begin Monte Carlo simulations. We have a good idea of which configurations should have comparable size (Einstein, 1979b). We can use phase boundaries to estimate interactions. When subtleties exist (Bartelt et al., 1989), they may provide particularly valuable insight into the size of small interactions. In some cases semiempirical methods can help in gauging interactions, but these usually only give significant interactions in the near region and certainly fail by the asymptotic limit, since they lack Fermi surfaces; they are best for late-transition and noble metals. Generalized tight-binding models, including d-band degeneracy and correlation effects, have been useful for mid-transition metals. Very recently Cohen et al. (1994) proposed a general tight-binding total-energy scheme that improves on previous similar schemes by adjusting the arbitrary zero of energy to eliminate the need for pair potentials; like EAM, it is in a sense an elaborate interpolation scheme, since parameters are fit using first-principles calculations. It has many times as many fitting parameters as the fourth-moment approximation method discussed earlier (Xu and Adams, 1994). It has done better than EAM in accounting for surface energies of late-transition and noble metals. Perhaps it or a related method will allow calculation of far more accurate Green's functions and, ultimately, interaction energies. As this chapter was in its final stage, Jennison et al. (1995) communicated noteworthy advances in computational capabilities. With a new Gaussian-based local-density-approximation code for massively parallel computers that uses Feibelman's LCAO method discussed in w 11.3.6, they can treat systems (large clusters, molecules) and elements (transition metals, oxygen) that pose difficulties


641

for plane-wave methods. With a three-layer 91-atom cluster of Pt, they reproduce well the details of ammonia adsorption on a seven-layer slab of Pt(111). The top layer is a hexagon 7 atoms across. Like the CO molecules mentioned above, pairs of ammonia repel each other at the close and intermediate separations that can be computed on this cluster, decaying roughly like R-3; the magnitude is somewhat greater. Specifically, they find for a pair of NH 3,s that E3 = 85 meV, which is much greater than the "through-space" dipolar repulsion, which they calculate to be 15 meV for two isolated ammonias. Multisite terms are relatively small: by comparing a compact cluster of seven molecules to a hexagonal ring of six, they find the effective E:~ drops to 75 meV, suggesting an attractive trio energy (for the associated equilateral triangles) o f - 1 0 meV. For the coadsorption case of CO at bridge sites and NO at hcp sites, there is a weak attraction at the largest computable separations. For both C O - C O and C O - N O (but not NH3-NH 3) the LDA is expected to overestimate the adsorption energies and so the interactions, consistent with the abovenoted difference from Persson's results; gradient corrections (Becke, 1988; Perdew et al., 1992) are expected (Jennison et al., 1995a) to temper this overestimate. All this work considers adatoms at or near stable sites in the holding potential. The effects of interactions on diffusion barriers, i.e. with one of the adatoms near a saddle point in the holding potential, has not yet been approached systematically. Typically some unconvincing assertion is made about this contribution, which in some cases may significantly affect the kinetics. The generic problem we have considered has broad ramifications. There are obvious extensions to defect interactions. Many analogous features occur in adsorption in electrochemical cells (Rikvold and Wieckowski, 1992). A more novel related situation is the oscillating interaction of magnetic sandwiches of varying thickness (Herman and Schrieffer, 1992; Stiles, 1993). Hopefully synergistic progress, lacking to date, will permit results from one of these problems to impact on others. In summary, there has been decent progress in understanding the general principles of lateral interactions but limited progress in achieving detailed quantitative understanding. Interest has been rekindled recently in looking for long-range effects mediated by surface or even quasi-one-dimensional states. After a decade's work, issues of correlations and self-consistency that seemed particularly troublesome earlier (Einstein, 1979a) can be dealt with, at least in simple systems in the near regime (Feibelman, 1989a). The major issue today is the role of local relaxations and hybridizing effects. In the near regime, we may well be on the verge of significant progress. In contrast, it seems that advances in treatment of the intermediate regime Will require some imaginative way to incorporate the results of careful calculations of the clean-surface (for the propagator) and single-adatom (for the coupling) problems into a general framework that recognizes that the interaction will perturb the single-adatom solution weakly at most. Acknowledgements

My work has been supported by NSF-MRG Grant DMR 91-03031. In the later stages I also benefitted from the hospitality of the IGV, Forschungszentrum Jtilich

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and the support of a Humboldt Foundation U.S. Senior Scientist Award. I have been truly fortunate to have had enlightening conversations with a large fraction of the authors, both theorists and experimentalists, cited in the references. I thank N.C. Bartelt and H. Pfntir for critical readings of drafts of the manuscript. J.B. Adams, L.W. Bruch, A.G. Eguiluz, M.C. Fallis, D.R. Jennison, S.D. Kevan, J.K. Nr and B.N.J. Persson provided comments on specific passages. I am particularly indebted to J.R. Schrieffer for having proposed a thesis problem of such enduring interest.

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T.L. Einstein

Part IV

Defects and Phase Transitions at Surfaces


C H A P T E R 12

Atomic Scale Defects on Surfaces

M.C. T R I N G I D E S Department of Physics and Astronomy Iowa State University Ames, IA 50011, USA



653

Contents

12.1.

Introduction

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

12.2.

Point defects

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

657

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

657

12.3.

12.2.1.

T h e r m o d y n a m i c s of point defects . . . . . . . . . . . . . . . . . . . . . . . . . . . .

658

12.2.2.

Detection of point defects with diffraction

658

12.2.3.

Role of defects in surface relaxation

L i n e a r defects

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

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

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

Overview 12.3.1.

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

12.3.2.

663 663

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

665

12.3.1.3. S t r e s s i n d u c e d c h a n g e s on s t e p p e d s u r f a c e s . . . . . . . . . . . . . . . .

667

Dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ovcrview

674

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

674

Clean surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

675

12.4. !. 1. M o s a i c s t r u c t u r c , strain, s t a c k i n g faults

675

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

12.4.1.3. F a c c t t i n g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

68 i 684

12.4.2.1. S u r f a c e o v e r l a y e r c o n f i g u r a t i o n s . . . . . . . . . . . . . . . . . . . . . .

684

12.4.2.2. Q u a n t i t a t i v e m e a s u r e s of o v e r l a y e r m o r p h o l o g y

686

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

12.4.2.3. O v e r l a y e r s w i t h ( I x 1) s y m m e t r y . . . . . . . . . . . . . . . . . . . . . .

688

12.4.2.4. O v e r l a y e r s w i t h s u p c r s t r u c t u r e p e r i o d i c i t y

693

T h e role of dcfects in phase transitions Overview

12.6.

677

T w o - d i m e n s i o n a l ovcrlayers on surfaces . . . . . . . . . . . . . . . . . . . . . . . .

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

12.4.2.5. O v e r l a y e r d o m a i n s t r u c t u r e s w i t h l o n g - r a n g e p e r i o d i c i t y 12.5.

672

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

12.4.1.2. L a t e r a l and v e r t i c a l d i s o r d e r on c l e a n s u r f a c e s

12.4.2.

662

12.3.1.1. Q u a n t i t a t i v e m e a s u r e s o f step m o r p h o l o g y

T w o - d i m e n s i o n a l defects 12.4.1.

660 662

Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.1.2. T h e r m a l r o u g h e n i n g of v i c i n a i s u r f a c e s

12.4.

655

........

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697 701

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

701

12.5.1.

T h e r m o d y n a m i c effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

701

12.5.2.

Kinetic effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

703

T h e Role of Defects in Crystal G r o w t h

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

704

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704

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708

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

709

Overview 12.7.

Epilogue

654

12.1. Introduction

The study of single crystal surfaces on the atomic scale has reached a sophisticated level of detailed characterization, with the invention of atom resolving imaging techniques, like scanning tunneling microscopy (Chen, 1993) and the use of data acquisition methods like diffraction lineshape analysis (Lagally, 1985). These techniques have revealed the presence of an unavoidable number of defects which interrupt the two-dimensional translational symmetry and chemical purity of the substrate, and can dramatically change the surface structure. If the ultimate goal is not only to describe but to control surface properties, it is necessary to understand the types of defects present, their mechanisms of formation and how they modify surface chemistry. Such defects are expected to exist at least as projections of defects already present in the bulk but, in addition, other types of defects exclusively characteristic of surfaces are found. The majority of the studies carried out so far have emphasized structural information on the real space configuration of the defects. It has also been realized, especially with probes that are sensitive to the electronic and structural characteristics (i.e., STM) that defects dramatically change non-structural properties (i.e., electronic structure, density of states, carrier transport). Any loss of surface symmetry will affect the available eigenstates, which in turn will affect both the static and dynamic response of the system to any external perturbation. Despite the importance of clarifying the relation between different types of defects and the electronic structure of the surface, the main focus of this chapter is to concentrate on the structural aspects of defects, i.e., the kinds of defects present on surfaces, their densities, techniques used to evaluate them quantitatively, and their role in controlling the equilibrium configuration of the surface. Defects can be classified according to different criteria. Most commonly, the dimensionality is used to categorize them: zero-dimensional or point defects (vacancies, interstitials, impurities), one-dimensional defects (steps, dislocations, etc.) two-dimensional defects (absorbed overlayers, facets, stacking faults, domains, etc.). Other criteria used, in connection to diffraction, are the extent to which they affect the long-range order in the crystal. If the atoms simply move by a small amount, but maintain on the average their equilibrium positions, they have the same relative phase difference, and long-range order is not affected (i.e., thermal disorder). This type of defect is usually referred to as a defect of the first kind (see w 1.6.2). If the damage of the crystal is so substantial (i.e., sputtering) that atoms have terminally moved out of their equilibrium positions and the phase difference between them is not fixed, the long-range periodicity is lost and all diffraction beams broaden. Such defects are referred to as defects of the second kind (see w 1.6.3). Finally, it is possible that atoms selectively occupy a fraction of the lattice 655

656

M.C. Tringides

sites (i.e., overlayers forming superstructures over the substrate lattice). A random phase relation exists between atoms in the overlayer and long-range order is lost but atoms maintain their lattice positions even at large distances. Diffraction beams broaden differently depending on the momentum transfer wave vector. Such defects are named defects of the third kind. Defects of the third kind are usually realized in surface overlayers as domains (see Fig. 1.25). For ordered overlayers with (m • n) unit cells where m and n are integers, a perfect "infinite" size domain is never attained and different types of domain walls separate equivalent domains. The presence of walls, their type and density, and limitations in the ultimate domain size realized are issues to be addressed. The mechanisms generating defects result, in general, from an interplay between kinetic and thermodynamic factors. The non-equilibrium nature of crystal growth processes allows a wide range of possibilities, where imperfect phases are stabilized over practically infinite time scales. Large energetic barriers separating the different phases lead to very slow kinetics towards the defect-free state and a finite number of defects still remains on the surface. Conversely, defects intentionally introduced on the surface affect both the static, equilibrium properties of the surface (reactivity, electronic structure, charge carrier transport) and time-dependent processes (ordering kinetics in overlayers, crystal growth). Two main techniques have been employed recently to study surface defects: quantitative diffraction analysis with several monochromatized probes (LEED, RHEED, ABS, X-ray scattering) and microscopy (STM, LEEM, SEM, TEM). The two techniques are complementary to each other. Microscopy provides local, direct imaging of the defects while diffraction is sensitive to indirect, statistical information ~tbout the overall defect distribution. Diffraction requires extra modeling to extract the defect configuration because the information is collected in reciprocal space and a non-trivial transformation, (sometimes non-unique because of the phase information lost in the scattering processes) is needed to recover the real space configuration of the defects. Microscopy provides clear, detailed, unambiguous information because it visualizes the defects, but if a small number of images are collected, then carrying out ensemble averages to determine collective parameters might be limited. Predictive power depends on sufficient statistical averaging over all the configurations present at the substrate. Ideally, a combination of the two techniques in the same experiment provides consistency in the results. This is especially the case when kinetic information is of interest in addition to the determination of the static defect structure. Microscopy is inherently slower because the higher S/N ratio needed to guarantee that the highest resolution is obtained at the expense of time resolution. Table 12.1 compares the relative advantages/disadvantages of microscopy vs. diffraction (Henzler et al., 1992). A + sign indicates that the method has advantages; a - sign indicates it has disadvantages but its application to collect useful information about a system is not ruled out. For statistical or time-dependent information diffraction is superior, while for extracting a direct and unambiguous image of the defects microscopy is better.

657

Atomic scale defects on su~. aces

Table 12.1 Comparison between diffraction vs. microscopy in their characterization of atomic scale surface defects. A +(-) sign denotes an advantage (disadvantage) of the technique. A (-) sign does not exclude the use of a technique, but the sensitivity to defects is lower than the sensitivity of the technique with a (+) sign

Microscopy General features Existence of defects Identification of defects superstructure steps point defects islands domains

+

Qualitative features Single defect shape of islands nucleation sites correlation of defects rare events

+

Diffraction

Quantitative features Lattice constants and strain Average values Size distributions steps terraces islands domains Kinetic features Changes during deposition Changes during heating annealing Reversible changes order disorder melting

+

12.2. Point defects Overview

Point defects are unavoidably present on surfaces because of their high configurational entropy. They can be observed directly with the STM or in the increase of the diffraction background. The highest sensitivity is obtained with the use of He-scattering because the scattering cross section is at least 10 times larger than the

M. C. Tringides

658

expected geometric cross section of the defect. Although the number of point defects can be easily identified, temperature dependent measurements to determine the energies of formation are limited.

12.2.1. Thermodynamics of point defects Point defects are produced thermodynamically on any surface. One can write the free energy H of the system at finite temperature T as

H= U- TS

(12.1)

where U is the internal energy and S the entropy. The formation of n isolated defects costs nAU energy, where AU is the energy of forming a defect, which is balanced

by the corresponding entropy increase AS =ksIn (N) (where ~ ~ is the usual combinatorial expression which gives the number of'ways of pos ing n defects in N possible lattice positions) so the change in free energy is AH(n) = nAU- ksTln "N" AH n By minimizing this expressions as a function of n, - 0 , the number of An de(ects produced spontaneously is obtained n =N e -aU/kT"

(12.2)

The number of point defects is always larger than this limit because other types of point defects like chemical impurities can be present, absorbed from the background gas or by diffusion from the bulk towards the surface.

12.2.2. Detection of point defects with diffraction Diffraction can be easily used to detect point defects. They are defects of the first kind, i.e., they do not affect the long-range order of the surface and the full-widthat-half-maximum (FWHM) of the diffraction spots is unchanged. Detects raise the background intensity. For uncorrelated defects (i.e., randomly positioned on the surface) which scatter with atomic scattering factorfD, different from the substrate scattering factor f,, it can be shown easily from Eq. (6.34) of Chapter 6

I(Q)=

N

+-Nf~ -

N

f, +-~fD

N (12.3)

+rN/(Nn n J Zij eikQ~r'-'i)]j1 L N /f" + -NfD The first term in the brackets does not depend on the wave vector Q and is usually

Atomic scale dejects on su~'aces

659

called the "background". The second term results in the usual ~5-function type summation, expected for an infinite lattice. The diffraction pattern is as sharp as for a perfect surface but with a higher background which can be used as a measure of the defect density. However, such measurements are difficult because even for large differences in the scattering factors between substrate and defect atoms (for example,fD = 2f~) the background is only a small fraction of the peak intensity (not more than 2-3%) for a defect density of a few percent. In practice, the absence of background in a diffraction pattern during routine substrate cleaning is used qualitatively to decide whether the surface is well-ordered. Inelastic scattering is not always completely eliminated from the measured background so that quantitative analysis should separate out its contribution..Increased background can also result from other types of disorder (i.e., thermal disorder which is discussed extensively in w 6.1.4) that remove intensity out of the Bragg peak. A convenient rule for the analysis of the background, based on the kinematic approximation (i.e., single scattering by top layer atoms) can be used for quick evaluation: the integral of the elastically diffracted intensity over the Brillouin zone is conserved, so any loss of intensity at the peak position should be found in the background. A higher sensitivity to point defects, especially for small adsorbate amounts, is possible with atomic beam scattering than with other diffraction techniques (Poelsema and Comsa, 1989). Extremely high single atom cross sections, comparable to the ones found in gas phase atom scattering have been measured in He-diffraction experiments and attributed to the long-range attractive part of the He atom interatomic potential. At very low coverages 0 0), Eq. (12.4) needs to be modified. If no interactions are present between the adatoms, one can show that I

--

=

e

-cON

(12.5)

li)

i.e., the intensity decays exponentially. Deviations from the exponential dependence can be used to identify the type of interactions between the adsorbate atoms.

660

M. C. Tringides

1'01

2.8

Q.6 ~

~

O.2 0

, 0 t

,-

CO"in"

~

IIio 2 ~ mostly D a steps are observed. This is clearly seen from the higher (1/2,0) superstructure intensity (than (0,1/2) intensity) which implies predominantly (2• type terraces. Double periodicity between successive out-of-phase conditions is measured along the reciprocal lattice rod, indicating double height steps (Saloner et al., 1987). Spot profile analysis of the split spots with HRLEED is used at the out-ofphase condition to confirm that there is a considerable amount of step meandering although it is not possible to resolve the detailed picture of the alternating degree of roughness, seen in Fig. 12.4. Contrary to the picture of a perfect staircase that implies split fundamental spots, finite intensity is present at the center, confirming that one of the domain types is more populated than the other, so the cancellation of intensity is incomplete at the center of the profile. As a function of temperature stepped surfaces of high vicinality, with Da steps, undergo transition to surfaces with SB single steps. The transformation depends both on temperature and vicinality with higher temperatures required at higher vicinalities to attain a given fraction of SB steps (De Miguel et al., 1991)). It is also evident that the large-scale domain morphology of stepped and singular Si(001) is correlated to energetic differences in the local atomic configurations of the reconstruction. From images like the one shown in Fig. 12.4, information about the energy cost of the different bonding configurations at steps is measured (Swartzentruber et al., 1990). It is straightforward to conclude that SA type steps must be energetically favored since they are more abundant. A quantitative description of the step roughness is shown in Fig. 12.8(a), with s denoting the separation between kinks and n the length of kinks (for "rough" steps, separation lengths s correspond to Sa-type steps and kink lengths n to SA-type steps). The size distributions P(s) and N(n) for both separation and length distributions can be measured accurately by statistical analysis of a large number of STM images. For non-interactive kinks P(s) is expected to be given simply by the product

P(s) = p( 1 - p)~-~

(12.10)

where p is the probability to form a kink and ( l - p ) to continue with a straight edge. Figure 12.8(a) shows that the experimentally determined P(s) follows exponential dependence on s as predicted by Eq. (12.10) which implies simply that the energy cost of a kink of length n is proportional to its length

E(n) - (Zs^ n + ~,~)

(12.11)

where ~'sA is the step energy per unit length for SA-type steps and Xc the kink formation energy. N(n), which is generated thermally at the annealing temperature T = 625 K should follow Boltzmann statistics

670

M.C. Tringides

0.6

I

"

0.4

P(s)

s

, \

~

SA

I -o, J l

~

SB

,

a-O C'-O G'~'~I

"

0.2

(a) 0.0

0

I

I

I

2

I

I ~I~,-4,-.-~

4

B 8 s (dirners)

L

1

I0

k

l

12

I0 ~

t

10 3

I

N(n)

I

3"

I0 a

I

-f

10

(b)

1

1

0

1

1

2

4

6

n (dimers)

8

I0

Fig. 12.8. Measurement of the energetic parameter ks^, k~ of the different types of Si(001) steps. (a) The separation distribution P(s) vs. s, follows an exponential dependence on s (Eq. 13.12) that proves the absence of kink interactions. (b) The kink length distribution N ( n ) vs. n, measured for the annealing temperature T = 625 K. Since there are no kink-kink interactions, the cost in energy of a kink of length n is simply E(n) = n ks^ + k~ and obeys Boltzmann statistics. By fitting the data to the form N ( n ) = No e --e~"vkr the step (per unit length) parameters ks^ = 0.04 eV X,~ = 0.06 eV are measured within 10% uncertainty. (Swartzentruber et al., 1990).

N(n) = No e -e~')/kr = No era'^ " +~)/kr

(12.12)

(The STM image is measured at room temperature but the kinetics are slow enough, as deduced from the diffusion activation energy E = 0.65 eV discussed in w 12.4.2.3, to identify the imaged configuration as a frozen-in high temperature state). The N ( n ) vs. n data shown in Fig. 12.8(b) in a semi-log plot follow Eq. (12.12). The slope of the linear segment with the heating temperature T = 625 K, can be used to extract the step energy (i.e., energy/length) ~,s^ = 0.04 eV/2a and the intercept can be related to the kink energy ),.c - 0.06 eV/2a, with 2a the spacing between dimers. Similar analysis on the "smooth" steps was used to measure the corresponding step energy for SB steps, ~.s, - 0.11 eV/2a. The experimental uncertainty in the measured values is 10%. These values are in qualitative agreement with theoretical calculations, using semi-empirical, tight binding energy models which predict ~sA = 0.02 eV/2a, X s , - 0.3 eV/2a (Chadi, 1987). The sum of the step energies of SA- and SB-type (both the experimentally measured and the theoretically estimated values) is greater than the calculated energy for double steps ~'D, = 0.05 eV/2a, so double


671

steps DB should be present under all conditions and single steps should be absent. As explained before, this is only true at higher vicinities and lower temperatures; otherwise single steps are observed, which implies that there must be additional energetic factors stabilizing the step configuration. Controlled strain experiments have indicated that a stress related contribution to the energy controls the equilibrium configuration of the Si(001). Compressive strain ~ext is applied by bending the free end of a sample, with an anvil assembly attached to a precision linear motion feed through. It is found that even for strains as low as 0.3%, a singular Si(001) surface with initially equally populated ( 2 x l ) and (1• domains, transforms into one with mostly (2xl) domains, which is the same preferred type as in the transition on the stepped surface. The transformation is reversible with strain and it occurs faster at higher temperatures. However, the final amount of asymmetry between the two domain types is independent of temperature. This implies that the transformation is different from the usual type of entropically driven order-disorder transition. Instead, the amount of asymmetry is simply related to the mechanical stress energy of the system which only depends weakly on configurational entropy. As explained in Chapter 2, surfaces even under equilibrium, can have a non-zero intrinsic stress tensor defined by I ~9E ~ = A-~c~3nij

(12.13)

where n 0 is the surface strain tensor, (ij) label possible directions on the surface, E the energy per unit area, and A0 the area of the surface unit cell. The energy can be easily calculated as a function of the atomic coordinates, in a given structure, which can be differentiated according to Eq. (12.13) to measure o 0. The Si(001) surface terraces with (2• and (1• types of reconstruction have different stress components, o~, o2, respectively, as a result of the anisotropy in the dimerization. Tight binding calculations have estimated the tensors to be o~ = 0.035 eV/A 2, O 2 - - - - 0 . 0 3 5 eV/A 2, with the negative sign indicating compressive and the positive sign tensile stress. At the steps separating different domain types there is a discontinuity in the stress which results in a force field F(~ = + (ol - o2)

(12.14)

with the sign changing at the two types of reconstruction of the step edge. This intrinsically generated stress is located at steps but it acts like an externally applied strain affecting the position of atoms over large distances. When the force field is integrated over the whole surface it adds a positive term to the surface energy which increases monotonically with the domain size L, and therefore, favors domains of smaller size. Since smaller size generates more boundaries, there is higher energy cost in the increasing number of missing bonds, which opposes the reduction of the domain size. The equilibrium configuration of the Si(001) surface is a result of the balance between these two competing terms. Mathematically exact expressions can be written in terms of the size of one type of the domain L

M.C. Tringides

672

(assuming straight step edges) and the asymmetry in fractional occupation of the two types of domains (p = 0 corresponds to equal populations of ( 2 x l ) and (1• domains, while p = 1 corresponds to a surface entirely covered by one domain type), E is the overall surface energy per area

E(L,p) = ~)'% + )'% + ~1 ~:extP(O, - 0"2) - 2 (ol

2rt 12.15)

w h e r e F~ext is the externally applied strain, ~t is the bulk modulus and v the Poisson ratio. Minimizing this expression as a function of the two parameters L, p can explain the different types of transitions on the Si(001) surface. For singular or surfaces of low vicinality, even for zero external strain tex t = 0 , E is minimized for p = 0 and some characteristic length L0 which explains why equal mixtures of the two domain types and single step heights are formed despite the lower energy cost of DB steps. The transition on singular surfaces with non-zero F_,ext :g: 0 results in p ~: 0 since the dependence on p of the second term far outweighs the weaker dependence of the last term. On a vicinal surface, one has the constraint L = constant, so E is simply minimized as a function of only p. It is clear for small enough L (i.e., high vicinality) that the contribution of the last term cannot compensate for the large number of domain walls (i.e., the energy cost of the first term in Eq. (12.15)). Refinements of Eq. (12.15) are needed to account for the entropy contribution with temperature increase, the dependence of the stress anisotropy on step meandering (straight steps in the calculation of Eq. (12.15) were assumed) and direct step-step dipole interactions due to the local strain caused by the rebonding of the S~ step. Such refinements can explain changes in the surface vicinality from double to single steps at higher temperatures and justify why a continuous transition is observed contrary to the predicted first-order transition, based on Eq. (12.15) (Webb, 1994).

12.3.2. Dislocations Dislocations are found in bulk materials as a result of kinetic limitations during crystal growth. Sometimes they can be eliminated after prolonged annealing, but often the times necessary are impractical. The definitions of the various types of dislocations have been given in w 1.6.3 with Fig. 1.24 providing a good visualization of screw and edge dislocations with their corresponding Burger vectors. Surface sensitive techniques provide information about the projection of dislocations onto the surface plane. If the distortion observed in the 2-d image is measured along different planar directions, the 3-d structure of the dislocation can be reconstructed. Since dislocations are aperiodic and irregular defects, microscopy is superior to diffraction in identifying their detailed configuration.

673

Atomic scale defects on su#aces

(a)

B1 \

B2 &

A---~ A'--~

CI

7'

7" C'

C2

(b)

Side View h :i.1~ w

~

v

Fig. 12.9. (a) STM image of a Cu(ll 1) surface show_ing a dislocation at the center. By viewing the picture at a glancing angle along the A,A' arrows ([ 1 I0] direction) a shift of d/3 is observed where d = 2.21 ,~ is the spacing between rows. If the picture is viewed along (C1, C') ([ 101] direction) the shift is 2d/3. These shifts provide the components of the Burger vector projected onto the surface plane. (b) A line scan along the A direction through the dislocation core, showing a single step height change 2.1 ]k. This and the information deduced from figure (a) uniquely specify a Burger vector b = I/2 [ 101 ]. A more detailed examination of the shifts show the presence of two peaks D~, D2 which suggests that the dislocation can be wr_itten as the sum of two partial dislocations b = I/6 [ 112] + 1/6 [211 ] (Samsavar et al., 1990). An atomically resolved S T M image of a Cu( 1 1 1 ) surface prepared by sputtering and annealing with a dislocation is shown in Fig. 12.9 ( S a m s a v a r et al., 1990). F r o m images on a larger length scale a dislocation density of 109 dislocations/cm 2 is measured; this is much higher than the densities found on s e m i c o n d u c t o r surface_s. In Fig. 12.9(a) the A arrow is along the [1 10] and the CI arrow is along the [101] direction of the C u ( l 11) surface. In the middle of the image the dislocation causes distortions that involve in-plane and out-of-plane d i s p l a c e m e n t s of atoms. T h e spacing b e t w e e n successive rows on the (111) surface (for the three e q u i v a l e n t directions) is d - 2.21 A; careful viewing of the picture at a glancing angle along the A and C I arrows shows that after crossing the dislocation, the rows are shifted by constant amounts d/3 (for rows along the A) and 2d/3 (for rows along the C l) directions. There is no shift along B2. T h e s e shifts are the projections of the d i s l o c a t i o n ' s B u r g e r vector onto the two i n d e p e n d e n t directions. Line scans along the A direction are shown in Fig. 12.9(b) which can be used to deduce the vertical positions of the atoms. The difference in the position of the atoms in the flat regions

674

M.C. Tringides

away from the dislocation is h = 2.1 A, which corresponds to the single step height on the Cu(111) surface. The lateral and vertical information about the dislocation fully determines the Burger vector b = 1/2[ 101 ]. The vector forms an angle c~ = 35 ~ with respect to the surface normal and since the dislocation line is expected to be normal to the surface (for well-annealed dislocations) it follows that this is a mixed dislocation ( a = 0 corresponds to a screw and c~ = 90 to an edge dislocation). The transition region around the dislocation core is fairly wide, far larger than what is expected, because of finite tip size effects. It clearly shows two distinct peaks D1,D2 which can be identified as the cores of two partial dislocations summing up to the mixed dislocation we have identified before; i.e., b = 1/61112] + 1/6121 1]. The vertical heights for the two partial dislocations are 2/3h and 1/3h so they add up to the single step height. This can be further confirmed if the in-plane shifts around the two cores in the surface plane are measured for each partial dislocation along two independent directions (as done before) although the uncertainty is higher since smaller displacements are involved. It is common for a dislocation to split into two partial dislocations to reduce abrupt structural distortions and to lower the strain energy.

12.4. Two-dimensional defects Overview

Structural information about the surface morphology, for clean and absorbate-covered substrates, can be readily obtained with both diffraction and STM. By measuring the intensity distribution as a function of Q= the step height distribution is obtained, while intensity measurements as a function of QII provide the terrace size distribution. This information can be verified directly with STM images. For well-annealed surfaces the number of layers exposed and, therefore, the interface roughness is relatively small. However, defects common in the bulk (mosaics, twins, strain), can be projected onto the surface. For epitaxially grown surfaces, one has the ability to manipulate growth and attempt a variety of combinanons for film/substrate materials, but usually at the cost of increasing interface roughness and lateral disorder. For ordered overlayers, the local arrangement of the overlayer atoms within the unit cell is easily observed with diffraction, even in cases where there is no commensurate relation with the substrate. Such arrangements can be also observed with STM whenever the overlayer atoms can be clearly distinguished from the substrate ones. However, identifying the long-range arrangement (type of domains, domain size distribution, shape of the domains) is a more difficult challenge. It can be done with diffraction by analysing the superstructure intensity when interference between different domains can be ignored and with STM by imaging the domains directly (although in large scale images atomic resolution is lost). LEEM is the ideal technique to map the domain distribution, but only for length scales larger than 100 ~,, the resolution of the technique.


675

The driving forces responsible for the morphology of the overlayer are fairly well understood. For ordered structures which involve the repeat of small superstructure unit cells (a few times the substrate one), short-range absorbate-absorbate interactions of different strength or sign (i.e., attractive or repulsive) are responsible. In several cases domains (which can be as long as 100-200 ]k) order in superperiodic arrangements over several ~tm. The ordering is driven by long-range stress mediated interactions that establish the superperiodic arrangement as the configuration of the minimum energy. The repeated domain size results from a competition between the stress energy that favors small domains and internal domain energy that favors large domains. Despite the abundance of techniques and experiments for the characterization of surface defect structure, quantitative information about the energetic parameters responsible for the defect formation is rather limited and can benefit from additional thermodynamic measurements. An additional complexity is that in several cases thermodynamic information might not be sufficient because defect structures form in kinetically limited, metastable configurations i.e. domain wall network in overlayer domains, interface roughness in epitaxially grown films, etc. In such cases, including all the microscopic processes with their individual kinetic rates is necessary to account for the observed defect distributions.

12.4. I. Clean surfaces 12.4.1.1. Mosaic structure, strain, and stacking faults The preparation of clean surfaces in vacuum is, in general, a lengthy process. Different cleaning recipes are used, depending on the chemical identity of the substrate: high temperature flashing for refractory metals, sputter and anneal cycles for soft metals, cleavage for semiconductors, etc. Clean surfaces can also be grown epitaxially by direct deposition of atoms after the surface has been partially smoothed out with the use of buffer layers. By manipulating the growth parameters (deposition rate, substrate temperature) the quality of the growing layer can be optimized. Regardless of the method of preparation, a residual amount of disorder is still present on the surface and needs to be characterized before overlayer adsorption. Several of the defect structures present are projections of 3-d defects that commonly exist in the bulk. As discussed in Chapter 6, different crystal grains terminate at the surface in a mosaic structure that can be easily described with diffraction. When electrons are used as the incident beam, all diffracted beams (hk) broaden with Qz, the normal component of the momentum transfer. Strain is another type of bulk defect that extends up to the surface and results in finite displacement of atoms from their equilibrium positions r,, = ro, + 8,, (Welkie and Lagally, 1982). For non-uniform strain, 8,, depends on the position of the nth atom and can be described in terms of a distribution of displacements P(8). Diffraction can be easily used to determine P(8). The resulting effects on the distribution of diffracted intensity are similar to the effects of thermal disorder, discussed in Chapter 6. The attenuation of a given (hk) beam is independent of Qz but it increases with Qll, for

M.C. Tringides

676

B.I~_ [I~

=8.88 0 450 K for Pt/Pt(l 11 ), but are absent in the in-between range. Interlayer diffusion is present at low temperatures because of the small island size and at high temperatures because of the increasing probability to cross the step edge barrier. The seemingly paradoxical result that at low temperatures a constantly roughened layer produces a smooth film is understood, if it is recognized that good quality layer-by-layer films should be grown with minimum vertical disorder (by reducing the interface width) even at the cost of maximizing the amount of lateral disorder at the top layer. Roughness is equally important for conventional or epitaxial growth but the underlying mechanism is

M. C. Tringides

708

3500

'

3000 I

'

'

'

'

'

" '

~

" " " '

'

'

" '

" " " '

C

'

'

t K

2500 .~ 2000 '6 t500 1000 5OO 0

-400

-200

0

200

Time [see]

(00

600

800

Fig. 12.31. Diffraction intensity drop vs. time plots, during epitaxial growth of Ag/Ag(11 I). No oscillations are observed when a clean substrate is used because the step edge barrier limits interlayer diffusion. When a small amount of Sb is dosed on the surface (0 = 0.2 ML) diffraction intensity oscillations are observed, indicating layer-by-layer growth, most likely because the step edge barrier is rcduced. Such experiments demonstrate the importance of impurities (i.e., surfactants) in promoting the growth of smooth films (van der Vegt et al., 1992). different. For conventional growth, the large number of available free sites, in a roughened substrate, accommodates atoms easily into stable low energy structures. For MBE growth, the fractal-like low temperature morphology provides alternative kinetic pathways for the non-equilibrium, smooth film structures to grow layer-bylayer growth. It is also well-known that crystal growth is improved with the addition of a small amount of impurities that act like stable nucleation sites of the deposited atoms to stick to. Similar impurity effects have been observed during crystal growth, by introducing small amounts of foreign elements, named surfactants. Deposition on systems that normally grow rough with the simultaneous occupation of many levels at the interface (3-d mode) can be manipulated with the addition of surfactants to grow layer-by-layer. A g / A g ( l l l ) grows in 3-d mode despite the rapid terrace diffusion because of the presence of a step edge barrier that limits interlayer diffusion. By introducing a small amount of Sb (0 = 0.2 ML) the growth mode is changed from 3-d to layer-by-layer, as can be seen from the strong oscillations observed in X-ray scattering, Fig. 12.31, which are present only on a substrate dosed with the surfactant (van der Vegt et al., 1992). The exact mechanism responsible for the change in the growth mode has not been fully identified. However, it is clear that most of the Sb floats to the surface, as successive layers are formed, so it plays the same role during the entire deposition time. Sb has the net effect of minimizing the difference between the step edge barrier E~ and terrace diffusion barrier Et, but it is not clear if this is realized because E, is reduced o r E t is increased.

12.7. Epilogue A description of the structure of different kinds of defects on single crystal surfaces


709

has been presented based mainly on the quantitative analysis of diffraction and STM images. The types of defects presented is by no means exhaustive and several interesting topics on defect structures have not been discussed (i.e., domain wall configuration in incommensurate structures, dislocation networks, correlated roughness in ultrathin films, etc.). The main emphasis of the chapter was on the structural characteristics of defects. Discussion of their effects on the electronic properties of the interface has been minimal. This is partially due to the limited amount of combined experimental work relating directly the surface electronic structure to the defect configurations. Such experiments require the combination of structural and spectroscopic techniques to be carried out in situ on the same substrate, which is instrumentally far more demanding. Potentially, the STM can be used to provide such information because the technique is sensitive to both the electronic structure and surface morphology. However, the measured quantities depend on both features and decoupling each contribution is not easy. In the current article there was more emphasis on the static defect structures (which can be produced either thermodynamically or kinetically in low temperature metastable states) than time-dependent effects. Topics relating the dynamics of defects and changes in defect distribution have not been included. For example, the diffusion of defects either on the surface or into the bulk can change the defect arrangement and the way surface properties are modified; on the microscopic time scale the coupling of defects to the vibrational motion of the crystal can change the phonon dynamics. Structural information about surface defects is readily available. It is clear from the information presented that defects of large variety and complexity are unavoidably present on crystal surfaces. The current experimental probes sensitive to atomic scale defects (STM, diffraction) can provide complementarily a clear picture of the defect configuration. The knowledge of the type, number, and defect distribution is essential to extend the picture of a surface, presented in the other chapters of the volume as a perfect, highly symmetric arrangement of atoms, into more realistic, non-ideal structures found in nature. A cknowledgements

I would like to thank Susan Eisner for the efficient and skilled typing of the manuscript and Kevin Cook for his careful preparation of the figures. I would like to thank Bill Unertl, the editor of the volume, for his valuable input in the form and substance of the manuscript. Ames Laboratory is operated by the U.S. Department of Energy by Iowa State University under Contract No. W-7405-Eng-82. This work was supported by the Director for Energy Research, Office of Basic Energy Sciences.

References Avouris, Ph. and I.W. Lyo, 1991, Surf. Sci. 242, 1. Barth, J.V., H. Brune, G. Ertl and R.J. Behm, 1990, Phys. Rev. B 42, 9307. Bauer, E. and W. Telieps, 1985, Surf. Sci. 162, 163. Burton W.K. and N. Cabrera, 1949, Disc. Farad. Soc. 5, 33

710

M. C. Tringides

Chadi, D.J., 1987, Phys. Rev. Lett. 59, 1691. Chambliss, D.D., R.J. Wilson and S. Chiang, 1991, Phys. Rev. Lett. 66, 1721. C.J. Chen, 1993, Introduction of Scanning Microscopy. Oxford University Press. Conrad, E.H. and T. Engel, 1994, Surf. Sci. 299, 398. De Miguel, J.J., C.E. Aumann, R. Kariotis and M.G. Lagally, 1991, Phys. Rev. Lett. 67, 2830. Ernst, H.J., F. Fabre, R. Folkets and J. Lapujoulade, 1994 Phys. Rev. Lett. 72, 112 Family, F., 1990, Physica A 168, 561. Fu, C.L., A.J. Freeman, E. Wimmer and M. Weinert, 1985, Phys. Rev. Lett. 54, 2261 Gawlinski, G.T., S. Kumar, M. Grant, J.D. Gunton and K. Kaski, 1985, Phys. Rev. B 32, 1575. Gotoh, Y. and S. lno, 1983, Thin Solid Films 109, 255 Hahn, P., J. Clabes and M. Henzler, 1980, J. Appl. Phys. 51, 2079. Harten, V., A.M. Lahee, J.P. Toennies and Ch. Wtill, 1985, Phys. Rev. Lett. 54, 2619. Henzler, M., M. Horn, V. Hoegen and U. Kohler, 1992, Advances in Solid State Physics, Vol. 32, ed. U. R/Sssler. Viewweg, Braunschweig/Wiesbaden. Hwang, R.Q., J. Schr6der, S. Gunther and R.J. Behm, 1991, Phys. Rev. Lett. 67, 3279. Horn, M., U. Gotter and M. Henzler, 1988, J. Vac. Sci. Tech. B 6, 727. lchninokawa, T., H. Ampo and S. Miura, 1985, Phys. Rev. B 31, 5183. Johnson, K.E., R.J. Wilson and S. Chiang, 1993, Phys. Rev. Lett. 71, 1055. Keller, K.W. and H. Htiche, 1987, Electron Microscopy in Solid State Physics, eds. H. Bethge and J. Heyendreich. Elsevier, Amsterdam. Kern, K., 1994, The Chemical Physics of Solid Surfaces, Vol. 7: Phase Transitions and Adsorbate Restructuring at Metal Surfaces, eds. D.A. King and D.P. Woodruff. Elsevier, Amsterdam. Kern, K., H. Niehus, A. Schart, P. Zeppenfeld, J. George and G. Comsa, 1991, Phys. Rev. Lett. 67, 855. Kleban, P. 1984, Chemistry and Physics of Solid Surfaces V, eds. R. Vanselow and R. Howe. Springer Vcrlag, Berlin. Kunkcl, R., B. Poelsema, L.K. Vcrheij and G. Comsa, 1990, Phys. Rev. Lett. 65, 733. Lagally, M.G., 1985, Methods of Experimental Physics, Vol. 22. Solid State Physics: Surfaces, eds. R.L. Park and M.G. Lagally, Academic Press, New York. Lagally, M.G., T.M. Lu and G.C. Wang, 1980, Ordering in Two-Dimensions, ed. S. Sinha. Elsevier, Amsterdam. Michely, T., M. Hohage, M. Bott and G. Comsa, 1993, Phys. Rev. Lett. 70, 3943. Mo, Y.W., J. Kleiner, M.B. Webb and M.G. Lagally, 1991, Phys. Rev. Lett. 66, 1998. Mouritsen, O.G. and P.J. Shah, 1989, Phys. Rev. B 40, 11445. Niehus, H. and C. Achete, 1993, Surf. Sci., 289, 19 Poelsema, B. and G. Comsa, 1989, Scattering of Thermal Energy Atoms from Disordered ,Surfaces. Springer Verlag, Berlin. Saloner, D., J.A. Martin, M.C. Tringides, D.E. Savage, C.E. Aumann and M.G. Lagaily, 1987, J. Appl. Phys. 61, 2884. Samsavar, A., E.S. Hirschom, T. Miller, F.M. Leibsle, J.A. Eades and T.C. Chiang, 1990, Phys. Rev. Lett. 65, 1607. Swartzentruber, B.S., Y.W. Mo, R. Kariotis, M.G. Lagally and M.B. Webb, 1990, Phys. Rev. Lett. 65, 1913. Takayanagi, K., Y. Tanishiro, S. Takahashi and M. Takahashi, 1985, Surf. Sci. 164, 367. Tringides, M.C., 1994, The Chemical Physics of Solid Surfaces, Vol. 7: Phase Transitions and Adsorbate Restructuring, eds. D.A. King and D.P. Woodruff. Elsevier, Amsterdam. Tringides, M.C., J.G. Luscombe and M.G. Lagally, 1989, Phys. Rev. B 39, 9377. Vanderbilt, D., O.L. Alerhand, R.D. Meade and J.D. Joannopoulos, 1989, J. Vac. Sci. Technol. B 7, 1013. van der Vegt, H.A., J.M.C. Thornton, H.M. van Pinxtesen, M. Lohmeir and E. Vlieg, 1992, Phys. Rev. Lett. 68, 3335. Wang, G.C. and T.M. Lu, 1982, Surf. Sci. 122, L635.


Wang, W.D., N.J. Wu, P.A. Thiel and M.C. Tringides, 1993, Surf. Sci. 282, 224. Webb, M.B., 1994, Surf. Sci. 299, 454. Weeks, J.D. and G.H. Gilmer, 1979, Adv. Chem. Phys. 40, 157. Welkie, D.G. and M.G. Lagally, 1982, Thin Film Solids 93, 219. Wolkow, R., 1992, Phys. Rev. Lett. 68, 2636. Yang, H.N., T.M. Lu and G.C. Wang, 1992, Phys. Rev. Lett. 68, 2612. You, H., R.P. Chiarrello, H.K. Kim and K.G. Vandervoort, 1993, Phys. Rev. Lett. 70, 2900. Wu, P.K., M.C. Tringides and M.G. Lagally, 1989, Phys. Rev. B 39, 7595. Zuo, J.K., G.C. Wang and T.M. Lu, 1988, Phys. Rev. Lett. 60, 1053. Zuo, J.K., R.J. Warmack, D.M. Zehner and J.F. Wendeken, 1993, Phys. Rev. B 47, 10743.

711


CHAPTER 13

Phase Transitions and Kinetics of Ordering L.D. ROELOFS Physics Department Haverford College Haverford, PA 19041, USA

Handbook o.[Su~. ace Science Volume 1, edited by W.N. Unertl


713

Contents

13.1.

Introduction 13.1.1.

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

Overview

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

13.1.2. Phase transition p h e n o m e n a at surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.2.1. C l e a n s u r f a c e s

13.2.

13.4.

717

13.1.2.2. A d s o r b a t e s y s t e m s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

717

Surface kinetic p h e n o m e n a

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

718

13.1.4.

Basic phase transition ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

718

13.1.4.1. T h e l s i n g m o d e l in 2-d

719

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

13.1.4.2. T r a n s i t i o n o r d e r and the free e n e r g y . . . . . . . . . . . . . . . . . . . .

720

13.1.4.3. T h e L a n d a u rules

723

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

A d s o r b a t e phase d i a g r a m s and the lattice-gas analogy The lattice-gas analogy

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

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

725 725

13.2.2.

Particle-vacancy s y m m e t r y and trio interactions . . . . . . . . . . . . . . . . . . . .

13.2.3.

C h e m i s o r p t i o n and physisorption . . . . . . . . . . . . . . . . . . . . . . . . . . . .

729

13.2.4.

Phases of m o r e c o m p l e x s y m m e t r y . . . . . . . . . . . . . . . . . . . . . . . . . . .

730

Universality and classification of transitions

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

728

736

13.3.1. Critical e x p o n e n t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

736

13.3.2.

Universality classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

746

13.3.2.1. E x p o n e n t v a l u e s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

747

13.3.2.2. M a g n e t i c m o d e l s

750

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

Surface reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

754

13.4. !. Reconstruction of metallic surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . .

755

13.4.1.1. R e c o n s t r u c t i o n r e s u l t i n g in i n c r e a s e d s u r f a c e p a c k i n g d e n s i t y . . . . . .

755

13.4.1.2. D i s p l a c i v e r e c o n s t r u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . .

757

13.4.1.3. M i s s i n g row r e c o n s t r u c t i o n s

759

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

13.4.1.4. A d l a y e r - i n d u c e d r e c o n s t r u c t i o n 13.4.2.

13.4.3.

13.4.4. 13.5.

717

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

13.1.3.

13.2.1.

13.3.

716 717

Reconstruction of s e m i c o n d u c t o r surfaces

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

764

! 3.4.2.1. ( 111 ) S u r f a c e s

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

765

13.4.2.2. (001) S u r f a c e s

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

766

Adsorption effects on reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . .

768

13.4.3.1. M e t a l l i c s u r f a c e s

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

768

13.4.3.2. S e m i c o n d u c t o r s u r f a c e s . . . . . . . . . . . . . . . . . . . . . . . . . . .

770

Diffusion in reconstruction systems

771

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

O r d e r i n g kinetics at surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.5.1.

760

772

Theoretical introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

773

13.5.1.1. G e n e r a l f r a m e w o r k

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

775

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

778

13.5.1.2. N u c l e a t i o n

714

13.5.1.3. Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

779

13.5.1.4. C o n s e r v a t i o n conditions

780

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

13.5.2. Results on late-time ordering from experiment and simulation 13.5.2.1. Verification of scaling in e x p e r i m e n t and theory

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

780 782

13.5.2.2. Finite-size effects and other limitations on e x p e r i m e n t and simulation

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

13.5.3. Ordering in coexistence regions

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

783 784

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

787

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

788

Appendix A: Physically allowed phase diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B: Critical point singularities and modern theory of critical phenomena

793 794

..........

B. 1.

Mean field theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

795

B.2.

Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

799

Appendix C: Finite size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

802 803

C.2.

Length scales

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

803

C.3. C.4.

Finite size effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Finite size scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

805 805

715

13.1. Introduction This chapter introduces the partially-related subjects of surface phase transitions and the kinetics of phase development and relates them to the broader context of two-dimensional (2-d)critical phenomena. In this introductory section the context of work in this subfield is discussed, the scope and organization of the chapter are presented, and references are given for related topics not covered in detail. The section concludes with a brief discussion of the 2-d Ising model whose known solution can be related via the lattice-gas analogy to surface phase transitions and which provides something of a paradigm for 2-d phase transitions and kinetic behavior more generally.

13.1.1. Overview Surface phase transitions and kinetic phenomena have applications ranging from the elegant theory of 2-d critical phenomena to the technologically important problems of surface reactions, diffusion and catalysis. As such, this area of surface science has been an active one since the origin of the field and has attracted the attention of an array of investigators of extraordinarily broad emphases. The area is thus sc~mewhat difficult to encompass in a brief treatment, and some aspects will c)f necessity be left to lengthier presentations or review articles. Closely related material is contained in Chapters 2, 3, 10, and 12 of this Handbook. The phase transitions exhibited by lattice gas models and the real systems they describe can be related to those occurring in simple models for 2-d magnetism. Lee and Yang (1952) first drew attention to the direct analogy between a simple attractive-interaction lattice gas model and the 2-d lsing model for ferromagnetism. Later the relevance of these ideas to experimental surface physics was noted and emphasized by several workers including Estrup (1969), Park (1969) and Doyen et al. (1975). These direct correspondences between magnetic and surface phase transition systems are presented in w 13.2. Beginning at about the same time and continuing into the 1980s, the general theory of phase transitions advanced tremendously with the advent of new caiculational techniques and theoretical constructs. The importance of the divergence of the length scale of the fluctuations that occur at continuous phase transitions was realized and exploited via scaling. The concept of universality, which allowed the grouping of disparate phase transition systems into a small number of universality classes, was broadly tested and confirmed. Members of a universality class display identical mathematical behavior in the vicinity of critical points. Universality and classification are the subject of w 13.3 with focus on their implications for surface phase transitions and the experimental tests that have so far been conducted. 716

Phase transitions and kinetics of ordering

717

The following subsections will briefly enumerate surface phenomena encompassed by these topics, identify where in this chapter or elsewhere in this volume detailed treatment is to be f o u n d - or, when no further coverage is attempted, review articles to be consulted, and present the organization of the chapter.

13.1.2. Phase transition phenomena at surfaces This subsection lists phase transition phenomena that have been observed in clean and adsorbate-covered metal and semiconductor surface systems.

13.1.2.1. Clean surfaces Surface reconstruction: The strong perturbation of a surface on the electronic structure of a metal or semiconductor drives rearrangements of surface layer atoms away from the simple, bulk-termination structure for many systems. Investigation of some of these systems, notably the Si(l 11) and W(100) reconstructions, has been one of the central thrusts of surface science. Reconstruction is discussed in w 13.4; see also Chapter 3. Surface roughening: In some cases, depending on both the material and the orientation of the surface, the thermodynamic equilibrium form of a surface may depart from simple planarity at a lower temperature than that at which the material melts. This phenomenon has been extensively investigated for a wide variety of metal and semiconductor systems both theoretically and experimentally. This and related phenomena such as the orderliness of steps on vicinal surfaces have been treated in Chapter 2. Surface melting: The process of bulk melting seems to initiate at particular crystalline surfaces for at least some materials at temperatures as much as 10% below the melting point, T,,,. Liquid-like ~ layers whose thickness increases as T,,, is approached have been observed using ion scattering and quasi-elastic neutron scattering, on (usually the more open) crystalline faces of several metals and semiconductors. The intriguing connections between surface melting and roughening have been noted by Lapujoulade and Salanon ( 1991 ). Since this phenomenon is fundamentally a bulk process B indeed it has also been called surface-induced melting B it will not be further discussed in this chapter. References and a detailed discussion are given by van der Veen (1991). Surface magnetism: A surface layer may exhibit magnetic properties differing in isotropy, symmetry or Curie temperature from the bulk it terminates. Bona fide 2-d magnetic phase transitions are then realizable, but since this is more directly a surface electronic effect than a consequence of surface structure, these phenomena are not treated in detail in this volume. The review by Falicov et al. (1990) may be consulted, and see also the chapter by Kirschner in Vol. 2 of this series.

13.1.2.2. Adsorbate systems Addition of a foreign species to a surface introduces other possible occurrences of phase development and transitions.

1 The phases cannot be true liquids since contact with the underlying solid induces partial order.

718

L.D. Roelof,;

Adsorbate ordering: Order of ( l x l ) or higher periodicity may occur within the adlayer, leading to the possibility of critical points and transitions between phases of differing symmetry. Phase diagrams ~may be determined in the (T,0) and/or (T, bt) planes and the character of transitions between phases may be ascertained and critical exponents determined. These matters are discussed in {} 13.2 and 13.3 and for physisorption systems by Suzanne and Gay in Chapter 10. Adsorbate-induced reconstruction: Adsorption typically results in displacements of nearby substrate atoms. In some cases these adsorption forces can couple to surface reconstruction modes leading to the possibilities that adsorption may induce or enhance or eliminate reconstructions of the clean surface.

13.1.3. Surface kinetic phenomena The possibility of phase changes occurring on surfaces clearly implies the opportunity of examining systems away from thermodynamic equilibrium and their approach there to, i.e. the kinetics of phase development. This chapter will focus primarily on this topic as it relates to the phase transition behavior of surfaces. The topics of significance include the development of order over time in systems which have been subjected to rapid variation of external conditions forcing the system across a first-order phase boundary, usually a temperature quench or an 'upquench'; and critical slowing down in the vicinity of continuous transitions. These topics are discussed in w 13.5. One might reasonably expect simple diffusion on surfaces to be relatively unimportant to questions of long-range order. There are, however, several situations where it in fact can play a significant role. These include: cases of reconstruction requiring mass motion of surface-layer atoms; and development of long-range order in adsorbate systems under conditions of fixed 0. Some recent developments in the study of this subsidiary topic are gathered in w 13.4.4. The purpose of this volume is primarily pedagogical, and its limitations include incompleteness of citation and (lack ol) depth of coverage. Section 13.7 comprises a brief bibliography that lists several important review articles and other sources pertinent to the subject. Readers interested in a particular topic in greater detail and more complete handling of the literature are urged to consult the sources given there.

13.1.4. Basic phase transition ideas In order to illustrate the phenomena typical of first- and second-order phase transitions in 2-d systems and to introduce the necessary vocabulary and notation it is useful to begin by considering the 2-d Ising model and its solution. Readers familiar with the general theory of phase transitions can skip this section and proceed to w 13.2.

1 A phase diagram is a 'map' showingthe phases that occur for a given systemand the nature and locations of the boundaries between them in a space defined by the relevant thermodynamic variables. Examples are given throughout this chapter.


719

13.1.4.1. The Ising model in 2-d The simplest 2-d model for a ferromagnetic system is the Ising model based on a lattice {i} of magnetic moments, s~, which can orient themselves in just two directions, s~ = +1. These moments are taken to interact via separation-dependent exchange constants, Jk--the subscript k denotes a neighbor relationship, Ji is the interaction energy for a nearest neighbor pair, J2 for a second-neighbor pair, etc. If these interactions favor alignment of neighboring moments we have the possibility of a ferromagnetic ground state. A Hamiltonian including the effect of an external magnetic field, h, may be written

H , - J, Z

s,s +

Z

s;s + ... + h Z s,

(ij) 2

(t.'/') i

i

where the index (ij)k on a sum denotes that the sum is to be taken over all kth-neighbor pairs on the lattice. This model was proposed by Ising and Lenz in the 1920s and solved for the case of a 1-d lattice. The 2-d case proved to be intractable and resisted exact solution until Onsager obtained the partition function Z,(T,h)- y j exp [-H,(is,i~)/kT]

(13.2a)

and the associated (magnetic) Gibbs free energy Gl(T,h) - - k T log Z i

(13.2b)

but for the case of h - 0 and nearest-neighbor interactions only. (The history of the Ising model is discussed by Brush (1967).) A bona fide phase transition is a change in symmetry of the material. This usually involves a change in the translational or magnetic order, the degree of which may be quantified by an order parameter. In the case of a ferromagnetic system the useful order parameter is the spontaneous magnetization, N

1 OG M(T,h) - ~ y j s~- Oh

(13.3a)

i=1

where N ~is the number of sites on the lattice. (The second equality is noted in most statistical mechanics texts, or it may be derived straightforwardly from Eqs.( 13. I and 13.2a,b.) If M ~ 0 for a sample of macroscopic size, then somehow the spins are 'communicating' across the entire sample. This characterizes what is meant by the concept of long-range order (LRO). Note that the presence of LRO implies an alteration of the symmetry of the system, in this case a loss of u p - d o w n symmetry in the spin orientation.

I The effect on transitions of the size of the system is discussed more fully in Appendix C. Here N is being used simply to normalize the order parameter conveniently.

L.D. Roeh?f;~

720

D e t e r m i n i n g M requires a differentiation with respect to h and O n s a g e r had d e t e r m i n e d Z~ only for h = 0. H o w e v e r Y a n g (1952) was able to e x t e n d O n s a g e r ' s t e c h n i q u e to d e t e r m i n e M(T, h=0) directly and two solutions are f o u n d

sin421~ji

-

/

1

w h e r e 13 -k-~T" M• vary from +l in the fully o r d e r e d state at low t e m p e r a t u r e to 0 a b o v e the critical t e m p e r a t u r e T c given by

sinh42~Ji = 1 ~ kBTc --- 2.2692 IJiI.

(13.4)

N e x t we discuss the global phase d i a g r a m implied by these results.

13.1.4.2. Transition order and the free energy The (T,h) and (T,M) phase d i a g r a m s of the model are given in Fig. 13. I. In panel (a) the bold dashed line along the h=0 axis denotes a first-order transition ~. It is c o n v e n t i o n a l to denote first-order phase b o u n d a r i e s via d a s h e d lines. The first-order line is t e r m i n a t e d by an asterisk d e n o t i n g the critical point at T c. Note that if one sets h = () and varies T through the critical point, the m a g n e t i z a t i o n will be o b s e r v e d to vary along one of the two curves M+(T) or M_(T). T h o u g h the b e h a v i o r is highly n o n a n a l y t i c at T = T c, see Eqs. (13.3 and 13.4), the m a g n e t i z a t i o n is c o n t i n u o u s so that this is a s e c o n d - o r d e r transition. Isolated s e c o n d - o r d e r transition points are usually d e n o t e d by asterisks; when they o c c u r along an e x t e n d e d curve a bold solid line is used. The model is in a f e r r o m a g n e t i c phase for h = 0 and T < To. For T > Tc the s y s t e m is p a r a m a g n e t i c . If h ~ 0, M :~ 0 for all T and there are no phase transitions. The k n o w n d e p e n d e n c e of the m a g n e t i z a t i o n on T and h allows the phase d i a g r a m to be presented also in terms of T and M as in the lower panel of Fig. 13.1. T h e shaded region b o u n d e d by the M• curves is inaccessible u n d e r normal e q u i l i b r i u m conditions, as e x p e c t e d for a f e r r o m a g n e t i c system. As first discussed in the influential t e x t b o o k of L a n d a u and Lifshitz (1969) the order o f a phase transition in a particular system is d e t e r m i n e d by the ' d e p e n d e n c e '

1 Concerning the order of a transition, phase transitions are always characterized by nonanalyticity of thermodynamic variables (see Appendix B), but come in two distinguishable varieties. One finds first-order transitions across which all thermodynamic densities (see Appendix A for a discussion of the distinction between thermodynamic fields and densities) are discontinuous - - in the present case the equilibrium magnetization jumps between M+ and M as one crosses the bold dashed line. (Nonequilibrium behavior, usually in the form of hysteresis is often observed.) In a second-order or continuous transition on the other hand, all densities vary continuously through the transition, but with still a singularity, a slope discontinuity, which thus manifests itself in the derivative of next higher order. The terminology is attributed to Ehrenfest.

721

Phase transitions and kinetics o.fordering

TII

(a)

2.0 -~

|

1.0 .I

TT

(b)

iiiiiiiiiiiii!iiiiiiiii!iii!i!iiiiiiiiiii!i!!ii~!iii!iiiiiii!i!i!i!iii!i!i!iii!iiii!ii!i!iiii ii!iiii!iiiii!ii!ii!iiiiiiiii.iiiiiiiiiii!ii!iiiiiiiiUiiiiii!!i iiiii!iiiiiiiiii!iiii i iiiilili iiii!!iii iiii iiiiiiiiiiiiiiiiiiiiiii i !iiiiii

-I .(

0

.0

Fig. 13.1. Phase diagram of the Ising model, Eq. (13.1). (a) (T,h) phase diagram; (b) (T,M) phase diagram. of its free e n e r g y on the o r d e r p a r a m e t e r a s s o c i a t e d with the p h a s e transition, t N e a r a c o n t i n u o u s p h a s e transition, M is small in m a g n i t u d e so that G ( T , h ) can be e x p a n d e d a b o u t M = 0. T h e t e r m s w h i c h a p p e a r in this e x p a n s i o n can be d e d u c e d f r o m the s y m m e t r y of the s y s t e m . F o r e x a m p l e the free e n e r g y e x p a n s i o n appropriate to the 2-d Ising m o d e l is G(T,h;M) - hM + a(T) M 2 + b(T)M 4

(13.5)

1 T and h are the canonical variables on which G, the magnetic Gibbs free energy in this case, depends. Considering the dependence G(M) means that for each value of M the overall partition function sum is restricted to include only those states having that value of M. Strictly speaking this theory should apply only close to the critical point since we want to be able to describe G adequately as a low order expansion in terms of M.

L.D. Roelofv

722

2nd - order

1st - o r d e r

F

M

M

F

F

F

M

Fig. 13.2. Free energy variation associated with first-order (left panels) and second-order (fight panels) phase transitions. The absence, for h = 0, of a term linear in M can be understood by noting that if all spins in the system are overturned G should not be affected. Analysis of other terms requires use of group theory and is treated in Landau and Lifshitz. Persson (1991) displays the determination of the Landau expansion for several models with relevance to surfaces. Equation (13.5) encompasses both types of transitions occurring in the 2-d Ising model. Figure 13.2 displays characteristic variations of G(T,h;M) for the two cases. The panels on the right show G(T,h=O;M) for the continuous case. For a < 0 (top panel) there are two equilibrium solutions with nonzero order p a r a m e t e r - this represents the situation for T < Tc. For a > 0 (bottom panel) there is only a single solution with M = 0 thus corresponding to T > Tc. As a passes through zero (middle) panel, M moves continuously between these two possibilities; and one sees in addition that the nature of the variation is such that we should expect large fluctuation of M in the vicinity of Tc - - t h a t is to say, M can vary


723

significantly with little cost in free energy. These large fluctuations, even more than the continuous variation of M, are the hallmark of a second-order phase transition. The panels on the left depict a first-order transition for T < Tc so that a has been taken to be negative. Now as h varies through zero at fixed temperature, the equilibrium value of M must move discontinuously from one minimum to the other. Note as well that this picture provides an explanation for the hysteresis often observed accompanying first-order transitions. In the lower panel, the ball describing the system has been left in the right-hand minimum, having been prevented from moving to the minimum on the left, obviously now the equilibrium situation, by the free energy barrier that will only be surmounted when h is large enough to overwhelm the effect of the term aM 2.

13.1.4.3. The Landau rules Landau and Lifshitz (1969) j also deduce from the free energy formalism several rules governing whether a given phase transition may be continuous. The rules are based only on the symmetry of the order parameters. It is important to emphasize at the beginning of this discussion that any transition can turn out to be first order. This is because a phase of virtually any s y m m e t r y can be stabilized by appropriate attractive interactions, and in that case disordering it should require a latent heat. The Landau rules only identify those transitions which have the possibility of being, on account of their symmetry, continuous. It is equally important to emphasize that the Landau free energy formulation does not account for fluctuations, i.e. the fact that important configurations of a system near its transition may be more aptly described via an order parameter which varies spatially. Therefore the 'rules' are approximate 2, and in fact some counterexamples have been found among 2-d phase transition systems to Landau's second rule, The rules, nonetheless, are still valuable in that the first and third rules seem to be valid even in 2-d and for the conceptualization of phase transitions from which they arise. The latter leads to the important notion of Landau classification, discussed in w 13.3.2. Unfortunately the Landau rules were originally formulated and are usually still presented in the rather esoteric language of group representation theory 3. Their translations into the language of surface science are, however, quite accessible and this section motivates and states them in that language and gives some examples. We will suppose that the variable being changed to drive this system through its transition is the temperature, T, but these rules apply equally well to transitions driven by varying say the chemical potential.

! This material is given in Chapter XIV, w 145 of the third edition of that text. 2 Unfortunately,the degree to which they are likely to break down increases as spatial dimensionality decreases, since fluctuations become more significant in lower dimension systems. 3 For example the second rule is stated as follows in Landau and Lifshitz (1969): "...the symmetric cube [r 3] of the representation F in question must not contain the unit representation...". Their explanation, is however, more approachable than that of most others.

724

L.D. Roelofr

The first Landau rule states that continuous transitions happen only between states whose symmetry groups lie in a group/subgroup relationship. (By group we mean the set of symmetry operations, translations, rotations and worse, of an ordered phase.) To understand this rule consider an example based on a made-up system. Suppose we have an adsorption system of rectangular symmetry, with lattice constants say, a and b, ~n the x- and y-directions respectively, and that 3 phases occur thereon: two ordered, a ( I x 3 ) and a ( l x 2 ) ; and one disordered and hence of ( l x l ) symmetry ~. Next consider the set of s y m m e t r y operations of each. The ( l x l ) phase has the largest group of operations that bring it into itself, including in particular translations by distance b in the y-direction. The (1• and (1• phases have smaller groups, since they do not contain the above-mentioned translation, for example. (They do, however, include translations by 2b and 3b respectively in the y-direction.) The groups of symmetry operations for the ( I x 2 ) and (1• phases are both obviously subgroups of that of the (1• According to the first Landau rule then there can be continuous transitions between the ( 1• 1) and either the ( 2 x l ) or the ( 3 x l ) . But it is clear that the (1• is not a subgroup of the (1• 2 and vice versa; and the first Landau rule prohibits continuous transitions directly between them. The second Landau rule, though properly stated in a more complicated manner, essentially states that continuous transitions are impossible if there are third-order terms in the free energy expansion. This is the rule that is known to be violated in some 2-d systems, most notably the 3-state Potts model (to be specified in w ! 3.3.2.2), so we will say relatively little except to motivate it. Suppose one included a third-order term c(T)M 3 in Eq. (13.5), and now imagine again the phase transition sequence presented in the right-hand panels of Fig. 13.2. In this case as a(T) passes through 0 and becomes positive, marking the disordering in the absence of an M :~ term, there will now be a new well on either the positive or negative side of M = 0. The system will remain in this minimum until a(T) becomes sufficiently positive to raise it above the one at M = 0. At that point the system will 'fall' (discontinuously) to M = 0 or exhibit hysteresis. Thus the transition must be first-order, unless by some odd coincidence b(T) happens to vanish exactly when a(T) does. The third Landau rule, sometimes called the Lifshitz criterion, has to do with higher-order periodicities and incommensurate phases. It can be phrased as follows. If the Q-vector of an ordered phase is not at a high symmetry point in the surface Brillouin zone (see below for more specificity) then there are only two possibilities for the disordering transition of the phase: either the phase becomes incommensurate, meaning that the Q-vector begins to move continuously with T before it 9

A

A

/k

,

.

A

I For a discussion of the nomenclature of surface phases, see Chapter 1. Some readers might argue with the assignment of p(lxl) symmetry to a disordered phase. This is in fact the correct assignment, since from the phase transition point of view, there has been no breaking of the substrate symmetry. The average occupancy of any regular sublattice of the lattice is the same as that of any equivalent one. 2 Thatof the (1 x2) has the translation by 2b and the other does not, so the group for the (Ix2) phase cannot be a subgroup of that of the (lx3); and likewise in reverse for the translation by 3b.

725

P h a s e transitions a n d k i n e t i c s o . f o r d e r i n g

broadens and diminishes in intensity to signal the disordering; or the transition must be first order. An example of an application of this rule is the disordering transition of the famed (7x7) reconstruction (see Fig. 6.5) of the Si(l 11) surface, whose beautiful diffraction pattern is one of the must sees of surface science. (McRae (1983) has nice photographs.) The Q-vectors of the ordered phase occur at the seventh-order positions, not high symmetry points in the hexagonal surface Brillouin zone of this (111) surface. Study of this transition has an interesting history (see w 13.4.2.1), but it is now known that the phase does not become incommensurate and that the transition is first-order, as expected from the third rule. We next discuss surface realizations of these general phase transition ideas.

13.2. Adsorbate phase diagrams and the lattice-gas analogy In accounting for the phase behavior of adsorbate systems, the principle degrees of freedom describe the occupation of a regular lattice of adsorption sites. (See Chapter 9 for a discussion of how these sites are identified experimentally.) When potential complications like surface inhomogeneities (steps, defects, impurities, etc.), reconstruction, dissociation (in the case of molecular adsorption), etc. may be ignored, it is natural to describe the cohesive energy ~ of the adlayer in terms of occupation variables, n, e {0,1 }, where {i} indexes the lattice of (assumed equivalent) binding sites for the adspecies in question. (A specific energy expression in terms of ni's will be given in Eq. (13.6).) The resulting approach, known as the lattice gas model 2, is uniquely useful for surface systems and has been widely applied. (Obviously, however, the model cannot and does not encompass some of the degrees of freedom possessed by atoms bound to surfaces, e.g. the vibrations of adatoms within their binding wells or their relaxation within the wells in response to interactions with other adatoms, and so must be applied with caution. Moreover the substrate may become more substantially involved. Some obvious limitations are discussed near the end of w 13.2.2.)

13.2.1. The lattice-gas analogy The lattice gas Hamiltonian is written in terms of the n; defined above as

H, o-e, Z , , , j (ij) ,

Z",", + ... (ij),.

Z",

(13.6)

i

1 By cohesive energy one means the change in energy, at T= 0 (i.e. no kinetic energy or zero point energy), that occurs when the originally distant and noninteracting adparticles are brought to and adsorbed on the surface. 2 'Gas' does not imply anything about the density of the phases to be described. The model can describe systems ranging from fully condensed ( Ix 1) layers to extremely dilute low-coverage systems.

L.D. Roeh?]:~

726

where Ei is an interaction energy for an ith neighbor pair of occupied sites and ijk was defined following Eq. ( 13.1 ). (E0 is a site binding energy which is often dropped from the treatment by shifting the zero of the chemical potential appropriately.) The similarities between HLG and HI (Eq. (13.1)) are obvious and the exact analogy first noted by Lee and Yang (1952) is seen through the following transformation

si ~ 2 n i - 1 J~ ~

(13.7)

l/4 Ei

h ----)- (p + ~j) where

la,, y__, giEi

(13.8)

i

with g, representing the number of ith neighbor pairs that occur per site on the lattice in question. Thus the phase behavior of a simple lattice gas with an attractive nearest neighbor interaction can be deduced from that of the Ising model; the (T,p) phase diagram is depicted in Fig. 13.3a. The dependence of the coverage 0 on T and la can be obtained from the transcription of the magnetization, N

M~~

1

~(2ni-

1)-20-

1

(13.9)

i= I

and in turn allows deduction of the (T,0) phase diagram of the attractive-nearestneighbor lattice gas shown in Fig. 13.3b. In surface nomenclature, one is dealing with an adsorbed p( 1• 1) phase ~. Some minor confusion is occasionally generated by the fact that the lattice gas analogy implicitly carries one between the canonical and grand canonical partition functions. In the case of magnetic systems one calculates the canonical partition function, Z, and from that the magnetic Gibbs free energy G(T,h) = E - T S - hM; while for simple lattice gas systems one wants the grand partition function, ~ and the associated grand potentiafl, f2((,p) = E - T S - p N = -kBT log ~. From experience with 3-d phase transitions, one is accustomed to the terms gas, liquid, f l u i d and solid for phases of differing density and character. These terms, though occasionally used in the surface phase transition literature, are not particularly meaningful since the distinguishing features of 2-d lattice gas phases are rather

1 Adsorbedp(lxl) phases seem rather uncommon in the literature, but this may be due to the fact that LEED (see Chapter 7) is not conveniently sensitive to overlayers with periodicity similar to the bulk on which they are adsorbed. Several cases of metal on metal epitaxy do occur and Kolaczkiewicz and Bauer (1985) have developed a novel technique based on work function variation to detect the boundaries of the coexistence region in this case. 2 See, for example, Chapter 23 of Morse (1969).

727


T

(a)

0.50

dilute

i

phase

0.25

dense phase

0.00

-2.0

"1

T

J (b)

dilute gas , ~1 ~

lattice'liquid' ,

ii••••!i•i•!i••!i!i•i!i••ii•i••ii!i!i•!i!iiiiii!iii !•i!i•i!i•i••iiii••iii!c i•iiiiii i ii!!i!iiiiiiiiiiiii!iiiiiii iiiiiiiiiiiiiiiiiiiii i ii!•i•i•iiii•iiiiiii!iiiiiii i.i•ivu::n sso o 2s I ~ . . " . : . i i i

0.00

0.0

0.5

1.0

o

Fig. 13.3. Transcription of lsing model phase diagram into the corresponding attractive nearest-neighbor-lattice-gas phase diagram. (a) (T,I.t) phase diagram showing first-order boundary terminated by

a critical point indicated by the asterisk. (b) (7",| phase diagram showing coexistence region occurring lor T < T,:. dissimilar to those of the 3-d situation. This subject is dealt with more completely in w 13.2.4. In the meantime, it is worthwhile noting that most ~ lattice-gas phases are 'solid' from the point of view that continuous motion of particles is hindered by the lattice potential provided by the underlying solid surface. Secondly, even in 3-d systems the only real distinction between 'gas' and 'liquid' phase is a matter of density, the two phases becoming indistinguishable above the liquid-gas critical point. The umbrella term 'fluid' therefore is sometimes used to cover both cases. The situation in the simple lattice gas phase diagram of Fig. 13.3 is similar in that 'fluid' phases of differing density are separated by a first-order discontinuity below

1 Incommensuratephases are an important exception. See Chapter 10 and w 13.2.3.

L.D. Roelq[:s

728 1.0

0.8

Au/W(110) To= 1130 K

I-

0.6

0

0.2

O.t, O (ML}

0.6

O.B

1.0

Fig. 13.4. (T,0) phase diagram of W(110)-Au as obtained via work function studies, after Kolaczkiewicz and Bauer (1985). The boundary encloses ap(l • coexistence region. The phase diagram is incomplete for experimental reasons for coverages just above the critical coverage.

a critical point, but become indistinguishable above i t - hence the occasional use of the terminology 'lattice gas' and 'lattice liquid' for phases of this sort. An example of a surface system exhibiting a phase diagram topologically similar to that of Fig. 13.3 is W(110)-Au, a p ( l • phase, whose phase diagram, obtained by Kolaczkiewicz and Bauer (1985), is shown as Fig, 13.4.

13.2.2. Particle-vacancy symmetry and trio interactions HLC gives rise to a phase diagram symmetric about 0 = l/z, as can be seen by applying the particle-vacancy transcription n; --) ( 1 - n;) in Eq. (13.6). One expects, however, at least for chemisorption systems (see Chapter 9 and w 13.2.3), that the full energy of an adlayer system will not be describable just with pairwise interactions as in Eq. (13.6). Trio interaction terms, adding to the Hamiltonian terms of the form

H, = E(~I ~.~ n;njnk (ijk)•

where (ijk)p represents a trio of sites of some particular arrangement. In some cases even higher-order terms may also be required. Trio and higher order terms are not


729

invariant under particle-vacancy interchange and therefore constitute one possible explanation for lack of symmetry about half-monolayer coverage, as seen for example for W(110)-Au in Fig. 13.4. Anomalously large trio and higher order interactions are needed to explain the pronounced asymmetry for W(110)-Au and have been attributed by Roelofs and Bellon (1989) to strong relaxation effects of the adsorbate atoms in their binding hollows as a function of local environment. This is obviously a non-lattice gas effect and illustrates a limitation of the model as applied to surface systems. See also Persson (1991) for extensions of this non-lattice gas effect to other systems. The lattice gas model cannot in simple form deal with many other real-world situations including: quantum effects as proposed for H on various substrates by Christmann et al. (1979) and Hsu et al. ( 1991 ); the effect of surface steps and other defects, particularly when these degrees of freedom are not fixed; nonregistered binding as often occurs in physisorption systems (see w 13.2.3); etc. Before proceeding to phase diagrams containing phases of more complicated symmetry a digression on two particular limiting cases of possible experimental regimes will be helpful.

13.2.3. Chemisorption and physisorption In discussing surface phase transition phenomena, it is important to bear in mind that the phases on a surface are, at least in principle, in thermodynamic contact with a 3-d gas phase of the adsorbate species. The effect of this contact is determined mostly by the strength of bonding of the adspecies to the substrate. In the case of a physically-adsorbed (or physisorbed) species (see Chapter 15) the bonding is weak ( 10 meV order of magnitude) relative to the energies of mutual interactions between the adsorbates, i.e., the Ei's of Eq. (13.6). In the physisorption limit, the adlayer and 3-d gas rapidly come to equilibrium with respect to particle exchange such that the chemical potentials of the two phases become equal. The experimentalist therefore can control the chemical potential of the adsorbed layer by controlling the pressure of the surrounding gas. (The chemical potential is simply related to the pressure for ideal gases. See Reif (1965), for example.) The (T,I.t) phase diagram can thus be directly measured, and, given a method for determining coverage on the surface, the (T,0) phase diagram may also be obtained. See for example the careful and complete X-ray diffraction study of Xe physisorbed on graphite by Hong et al. (1989)'. In the chemisorption limit on the other hand, the binding of the adsorbate particle to the surface is strong relative to the Ei's (2-5 eV order of magnitude), so that at the energy scale at which the surface phase transitions occur, the rate of desorption from the surface is so low as to be negligible. If the experimentalist has achieved ultra-high vacuum, the rate of adsorption is also slow relative to experimental time scales so that the coverage is effectively fixed. In this limit only the

1 Otherexamples are given in Chapter 10.

730

L.D. Roeh?]:v

(T,0) phase diagram can be conveniently measured; the corresponding (T,~) diagram must be inferred from the former using the known properties of phase diagrams, as will be discussed in w 13.2.4. It is noteworthy that the ability to fix and control 0 provides convenient access to the coexistence regions, e.g. Fig. 13.3. Diffusion limitations lead to interesting kinetic behavior in coexistence regions. The equilibrium state within such a region is a fully segregated coexistence of the two phases on either side of the relevant first-order line, in the present case a dilute 'gas' and a dense p ( l x l ) phase (containing a dilute gas of vacancies). For most experimental initial conditions full equilibrium is difficult to attain due to the fact that as clusters of adparticles grow they become increasingly immobile. This immobility is exacerbated by surface defects in many cases. Eventually most systems stop short of full segregation, so that the often-encountered characterization, island phase, is an accurate description of the character of the phase. The development of order in coexistence regions will be discussed further in w 13.5. An important structural concept related to the physisorption/chemisorption distinction is registry, which can be defined to be the degree to which adatoms reside at the minima of the binding potential provided for an isolated adatom by a clean surface. Obviously, if adatom-adatom interactions are significant, one would expect the adatoms to deviate from these minima. A useful generalization is that, in the absence of reconstruction, chemisorbed species tend to be well-registered, while in the case of physisorption this is less likely to be the case, though there are exceptions. In any event, lack of full registry does not necessarily rule out the use of the lattice gas approach. However, the interaction energies used in this case must reflect the (spacial) relaxation of adatoms within their binding sites (and also that of nearby substrate atoms as well!). Furthermore, these relaxations depend strongly on local coordination, so that higher-order interactions are induced (see Roelofs and Bellon, 1989).

13.2.4. Phases of more complex symmetry Many adlayer phases whose symmetry differs from that of the clean substrate occur on surfaces. The simplest may be obtained by applying the lattice gas transformation, Eq. (13.7) to the Ising antiferromagnet, whose Hamiltonian may be written exactly as in Eq. (13.1), but with the sign of Jl positive so as to encourage neighboring spins to point in opposite directions. The solution to the antiferromagnetic Ising model (on a square lattice) is straightforward because the following transcription carries one between the two models s, ~ - s , s,--) si J!---) - J!

h ~---~h~

for i ~ Sl for i e

S2

(13.10)


731

Ca) T c

~ ~ ~

. . . . . .

/

t;t;t; .

"h c

.

0

§ § § § § §

2 2 2 2 2 2

4 4 4 4 4 4

7 7 7 7 7 7

2 2 2 2 2 2

4 4 4 4 4 4

7 7 7 7 7 7

2 2 2 2 2 2

4 4 4 4 4 4

7 7 7 7 7 7

2 2 2 2 2 2

4 4 4 4 4 4

7 7 7 7 7 7

\ .

he

h

Fig. 13.5. (a) (T,h) phase diagram of the anti-ferromagnetic (AFM) Ising model. The ordering in each region is indicated by the arrays of + and -. The solid line is a second-order phase boundary separating a region of c(2• symmetry from the paramagnetic p(l• phase. (b) Phase diagram of the AFM ising model in (T,h,h~) space. Viewed in the (T,h~,0) plane the phase diagram is identical to that of the ferromagnetic Ising model, Fig. 13.1a. where S~ and S 2 denote the two interpenetrating sublattices of sites defined by c(2x2) order and where h, is the staggered field, i.e. an external antiferromagnetic field ~. The transcription, Eq. (13.10), preserves the Hamiltonian so that Onsager's solution also covers this case. The (T,h) phase diagram of the Ising antiferromagnet with J2 (and all higher-order interactions) set to 0 is given in Fig. 13.5. Because of the applicability of transformation Eq. (13.10), we know that Tc is again given by Eq. (13.4). The phase diagrams are, however, quite different because of the interchange of h and h~. h no longer forces the system into a paramagnetic phase, but only decreases the transition temperature symmetrically in both directions and does not alter the fundamental

1 Obviouslyone of those theoretical artifices somewhat difficult to realize experimentally.

732

L.D. Roelofs

nature of the transition. Thus, along the entire solid curve in Fig. 13.5, one has a continuous phase transition at which the order parameter appropriate to this transition, M s = -~

(13.11)

si i

~

i~

S2

goes to zero with temperature variation similar to that of Eq. (13.3). (Note that Eq. (13.11) differs from (13.2) because of the different symmetry of the phase being described. M~ is the appropriate order parameter since it varies from +1 in the two possible fully ordered states to 0 above the transition.) The relation between ferro- and antiferromagnetic Ising model phase diagrams is clarified in Fig. 13.5b, which shows the dramatic effect of turning on the staggered field, h~ (analogous to a magnetic field in the ferromagnetic model) in the antiferromagnetic model. For h~ ~: 0 the system has c(2x2) symmetry for all T and there are no phase transitions. It is worth emphasizing what precisely is meant by s y m m e t r y and its connection to long r a n g e o r d e r (LRO) in this context. Genuine phase transitions, marked either by discontinuities or nonanalytic variation of measurable quantities occur only in the thermodynamic limit, i.e. in the limit of infinite system size', c(2x2) symmetry then means that the M~ order parameter defined in Eq. (13.1 1) is non-zero when the sums are carried out over the entire infinite system. There are two common misapprehensions. One is that the order must be perfect. It need not be; in an infinite system an extremely small imbalance in the terms in Eq. (13.11) constitutes long-range c(2• order 2. Another error is to assume that local c(2x2) coordination constitutes c(2x2) symmetry. That condition is neither sufficient, since a part of the system not seen may lead to cancellation of the order, nor even necessary (!!!). Consider for example the low-coverage (0.14 < 0< 0.18) phase discussed by Bartelt et al. (1989), which exhibits long-range c(2x2) order, but whose local coordination is p(2x2). The widespread use of LEED (see Chapter 7) in surface physics is a two-edged sword in this respect. On the positive side, LEED allows convenient detection of the symmetry of metallic and semiconducting surface systems. On the negative side, the limited resolution,--100/~, of commercial instruments hinders distinguishing between long- and short-range order. The lattice gas analog of the Ising ferromagnet has a (T,t.t) phase diagram identical with the (T,h) phase diagram of Fig. 13.5a, but with the factor of 4 reduction in Tc resulting from Eq. 13.7b). The phase that occurs inside the continuous transition boundary is of c(2x2) character, of course, with saturation coverage 0 = 0.5, existing over the range 0.37 < 0 < 0.63 at low temperature.

! The variationwith system size N as N approaches infinity has been investigatedand employedto improve the accuracy of calculations based on noninfinite systems. This area of investigation is termed.#nite size scaling and is discussed briefly in Appendix C. 2 One might say, however, that such a phase is 'not well-ordered' despite the fact that it does exhibit long-range order.


733

The lattice-gas order parameter analogous to M~ of Eq. (13.11) can be defined as the staggered coverage

0s -- N

ni i

~

ni

(13.12)

i

0~ is the density conjugate to the field g~ in a thermodynamic framework. 0~ takes on values of +1 in the two fully-ordered degenerate c(2x2) ground states, and goes to zero in disordered or other p ( l x l ) symmetry states. Lattice-gas order parameters are usually not mentioned in the literature on surface phase transitions because, unlike the magnetic order parameter of Eq. (13.2), there is no convenient way to measure them directly. Instead, since 0, is pl'oportional to the Q{, = ( a ~, ) component of the kinematic diffraction amplitude t, 1

A(Q) - ~ ~_~ nj e i~

(13.13)

J

where {rj} are the lattice positions, one can measure the square of 0, by measuring the c(2x2) component of the kinematic diffraction intensity,

l~atQ) = A'(Q)A(Q)

(13.14)

That is, N2

I(Q") - - 4 0~

(13.15)

I(Q) can be measured via X-ray or electron diffraction (see Chapter 7): in the former case one typically needs a high intensity source and a high-Z adsorbate to achieve adequate signal; in the latter a simple LEED apparatus suffices, but coherence limitations and multiple scattering limit the resolution and complicate the analysis respectively. This exhausts phases possible in models including only nearest-neighbor interactions on a square lattice, but further complexity is possible if longer-range interactions are present. Consider the addition of an attractive second-neighbor interaction, E2, to the above picture. At low temperature, such an interaction obviously tends to stabilize clusters of c(2x2) coordination, even at low overall coverage. The result is addition of low- and high-coverage coexistence regions as shown in Fig. 13.6a. One may accordingly deduce the (T,g) phase diagram given in

1 The value of Q0 assumes a square lattice of lattice constant a.

734

L.D. Roelo.l':v (=)

Dilute Gas

Dense Gas

Coexistence ~ DiluteGas+c(2x2) 0.0

Coexistence DenseGas+c(2x2) 0.5

1.0

(b)

TT

dilute

~

(lxl)

,,,,

~

I/ /

~ -1.0

dense (lxl)

I 0

Fig. 13.6. Phase diagram of a lattice gas with E~ repulsive and E2 = -!/2 El. (a) (T,O) plane. (b) (T,l.t,l.t.~) space. The dots represent tricritical points, so named because three critical lines meet there as indicated by the addition of the la.~axis on the low coverage side. Fig. 13.6b by noting that coexistence regions indicate coverage discontinuities, i.e. first-order transitions as in the p ( l • lattice gas (recall Fig. 13.3.) These are, as usual denoted by the dashed lines in Fig. 13.6b. Above the coexistence regions the nature of the transitions is not changed and these are denoted as solid lines in the (T, la) phase diagram. The point at which a line of continuous transitions becomes first-order is called a tricritical point, because in the expanded space including a staggered chemical potential, l.t~ (which favors one c(2• sublattice over the other in analogy with the staggered magnetic field, h0, one sees in Fig. 13.6b that the point is actually the junction of three critical lines. Figure 13.6 provides a good starting point for discussing the rules that physically allowable phase diagrams must obey. This matter is discussed in Appendix A.

Phase transitions and kinetics t~ordering

735

TIK]T

25i I[10-11A1

3~176 I

20- .._._ -'-""...~.~//0.t.0 ~0.37 ~0.32

I F'*., z~176 F '4. ,so ~ ~. F R,=,rz~i lOOj--

25

0

J

J I

J

0 (o)

t

0.S

i

i

_8=0.S0

is ~ / ~ ~ o . z s ,o 5 !

I

I___

G

0

1.0

J

100 (b)

I ~ ~

1~,..__

200 T [K] 300

Fig. 13.7. Phase diagram of H/Pd(001) as presented in Fig. 15 of Binder and Landau (1981). Panel (a) shows the experimental data (the crosses) of Behm et al (1980) and the calculated phase boundary (dashed curve) which best fits the data. Panel (b) shows the raw LEED intensity data of Behm et al (1980) from which the phase boundary points were deduced. Binder and Landau note that the discrepancy between the experimental points and the calculated boundaries is due to a mingling of long- and short-range order in the experiment by a combination of finite size effects and instrumental limitations. Binder and Landau (1976) first discussed phase diagrams of this sort and later made application (Binder and Landau (1981)) to the system Pd(100)-H, whose phase diagram had been measured by Behm et al. (1980) (see Fig. 13.7). The comparison between theory and experiment is somewhat problematic because of the limitations of LEED as previously noted. Structures of p(2x2) character are also a common occurrence in square lattice surface systems; see for example the study of Ni(001)-Se by Bak et al. (1985). One might naively expect to be able to force the occurrence of this phase with a repulsive E 2 (Binder and Landau, 1976), but in fact this is not sufficient (Binder and Landau, 1980). The p(2x2) phase can be considered to be made up of double-spaced rows of adatoms running in the x- and y-d~rect~ons. Clearly, however, as shown in Fig. 13.8, with only E, and E 2 repulsions, individual rows can slide either in the x-direction or in the y-d~rectson (but not both s~multaneously) without changing the energy. Random slipping of that sort yields a phase with LRO of either ( l x 2 ) or (2x I ) symmetry. This mechanism may be the explanation of the streaky diffraction pattern observed for Ni(100)-S by Oed et al. (1990). One may continue by considering adlayers on surfaces of different symmetry: the centered-rectangular lattices often found on bcc(110) surfaces; the rectangular systems on fcc(110) surfaces and the triangular and hexagonal lattices on the f c c ( l l l ) and the basal plane of the hcp crystal structure. Phases of periodicity different from that of the substrate abound also on these surfaces; some will be noted in subsequent sections. A

A

,

A

,

.

.

736

L.D. Roeloj':v

I 0

0 0

I0

t~:

0 I

0 0

I 0

0 0

0

0 0

0 I

0 0

I

0

0

0

~I~OOQO

0 I

0 0

~O~l

0

0

0

I

0%~

0

0

I

0

0

0

I

0

0

0

0

Fig. 13.8. The p(2x2) phase formed by El and E2 repulsions only, is unstable with respect to slipping of rows in either direction. The interactions do not couple adjacent rows.

13.3. Universality and classification of transitions All continuous 2-d phase transitions can be grouped into a small number of so-called universality classes. The members of each class display identical behavior near their respective critical points. These classes, the simple magnetic models which name them, and the means by which such identifications are made are described in this section. The similarities of behavior are usually described through critical exponents which characterize the singular behavior (see Appendix B) of various thermodynamic and ordering parameters. These exponents are defined and their known values given in this section. The experimental determination of critical exponents is illustrated via discussion of the investigation of O/Ru(001) by Pfntir and Piercy (1989). The section concludes with a discussion of the real-world hindrances that complicate critical exponent measurements in surface systems. 13.3. I. Critical exponents In the vicinity of a second-order phase transition the free energy of an infinitely large system is mathematically singular at the critical point, (T c, h = 0) for a magnetic system, or (To, l.t0) for a lattice gas system. (The origin of the singularity is discussed in Appendix B.) One then expects all thermodynamic and ordering properties to vary singularly in the vicinity of the critical point, the nonanalyticities being characterized by the critical exponents. Measurements of these exponents are of interest as tests of theory of critical phenomena and to determine the classification of a given transition. It is conventional and convenient to use variables centered on the critical point to characterize these singularities. The reduced temperature, t=

T-L L

(13.16)


737

and h itself which vanishes at the critical point are the standard choices for m a g n e t i c systems. In the case of lattice gas systems, we use t as a b o v e and a reduced c h e m i c a l potential ,-..,

la - ~t - ~t0

(13.17)

It is n e c e s s a r y to pause at this point for a brief notational aside. Section 13.5 of this chapter is c o n c e r n e d with the time d e v e l o p m e n t of order, so that one also needs a s y m b o l to denote time. "t" will be used to represent time in that section, so that reduced t e m p e r a t u r e and time will be distinguished only via use of a script font for the latter. (This should not lead to intolerable confusion; "t" occurs only in this section and "t" only in w 13.5. The c o n v e n t i o n a l critical e x p o n e n t definitions for a m a g n e t i c system are as follows. M e a s u r a b l e quantity Specific heat* O r d e r p a r a m e t e r vs. temp. Susceptibility O r d e r p a r a m e t e r vs. conj. field Correlation function Correlation length (~ defined via

E x p o n e n t definition C h -- Iti-'x m -- (-t) 13 Z " Itl-V m• ,-- +_lhl~/~' F(r) - r -n ~ -- Itl-~ Fr(r) -" e -r/~

Conditions h = 0 t < 0, h = 0 h - 0 t= 0 t= h = 0 h = 0 h = 0, t ~ 0)

(13.18) (13.19) (13.20) ( 13.21 ) (13.22) (13.23) (13.24)

*The quantity is denoted Ch to denote the specific heat measured at constant field strength in a magnetic system. The lattice gas analog is C measured under conditions of constant ordering field (usually 0, of course), not order parameter. (Fr(r), the r e d u c e d correlation f u n c t i o n , is the correlation function with contributions due to L R O subtracted out. It is defined for spin and lattice gas s y s t e m s below.) In the case of Eqs. (13.18, 13.20, 13.23) and (13.24) the functional form of the variation is similar above and below To, but with different amplitudes, or leading constants. In lattice gas systems one m e a s u r e s analogous quantities to d e t e r m i n e the c o r r e s p o n d i n g exponents. The specific heat is u n c h a n g e d so that the e x p o n e n t c~ can be d e t e r m i n e d via c a l o r i m e t r i c m e a s u r e m e n t s of C , (fixed c h e m i c a l potential, i.e., pressure) for adsorption on high-surface area materials ~ - see for e x a m p l e Bretz (1977), T e j w a n i et al. (1980) or Z h a n g et al. (1986). H o w e v e r , it is more useful to focus on the correlation function 1

F(r) - ~ ~ n(r~) n(r + r,)

(13.25)

i

1 Calorimetry is not feasible for standard crystalline surfaces because of the difficulty of extracting the signal due to the surface from that of the bulk of the substrate.

L.D. Roelof~

738

for three reasons: 1-" m o r e directly m a n i f e s t s the physics of critical p h e n o m e n a (see A p p e n d i x B for a discussion of this c o n n e c t i o n and the related subjects of scaling and r e n o r m a l i z a t i o n ) ; because all the above critical e x p o n e n t s can be e x t r a c t e d f r o m m e a s u r e m e n t s of I"; and b e c a u s e F(r), or at least its F o u r i e r t r a n s f o r m , 1

I(Q) = ~ ~_~ I-'(rj) e ; ~ r,

(13.26)

J can be c o n v e n i e n t l y m e a s u r e d for m a n y adsorption s y s t e m s using electron or X-ray diffraction. I(Q) is called the structure f a c t o r and can be shown to be identical to the k i n e m a t i c diffraction intensity ~ defined in Eqs. (13.13 and 13.14). The detailed form of F differs according to the s y s t e m in which the p h a s e transition occurs, but two key aspects are g e n e r a l i z a b l e to all critical points and allow d e t e r m i n a t i o n of the critical exponents. T h e s e are the signature of L R O c o n t a i n e d in the l a r g e - r limit of F(r); and shorter range variation (for t ~ 0) that describes the fluctuations. T h e s e two aspects are transparently d i s t i n g u i s h a b l e in the m a g n e t i c (Ising) case where the analogous s p i n - s p i n correlation function 1 1-'")(r) - N y-' s(ri) s(r + ri)

(13.27)

i

varies as sketched in Fig. 13.9 for t < 0, t = 0 and t > 0. Below Tc, F~r~(r) tends for large r to a constant value of M 2 due to the L R O that occurs in that regime. Even in the context of LRO, however, the system exhibits fluctuating regions of spins o r i e n t e d in the direction opposite to that of the overall order. T h e s e regions range in size up to a value of ~(T) and therefore at shorter distances ~t)(r) displays an e x p o n e n t i a l decay 2 so that, for T < T c, the overall functional form is s o m e t h i n g like G~t~(r) =

m 2+

D_e -r/~ + (shorter range corrections)

(13.28)

with D_ being a T - d e p e n d e n t constant. To focus on the fluctuations one defines a reduced correlation function by subtracting out the contribution due to L R O

1 Multiplescattering in LEED (see Chapter 6) hinders to some extent the extraction of the simple kinematic intensity. However, the phenomena of interest with respect to phase transitions typically occur over a rather limited range of Q-vectors (in the natural units of ~c~) since they concern relatively long-range correlations. The multiple-scattering-induced variations in the diffracted intensity, on the other hand, typically vary more slowly with Q since they arise from interference between beams that have interacted with several surface atoms in a fairly small region. 2 The decay is exponential because starting from within one of the "out-of-phase" regions, the probability that a "mistake" occurs, taking the phase back to that in which the LRO is occurring, is a (temperaturedependent, of course) constant if the interactions are short-ranged so that adjacent bonds do not behave in correlated fashion. As in the case of nuclear physics, constant decay probability results in exponential decay (there in time, here in space) of a population, in this case, the members of the fluctuating, out-of-phase domain.

739


1.0

(a)

0.8"

H

s._ v

0.6

\.\ ~

0.4

0.2

0.0

-9. . , 0

,

-

20

I(0)

,

.

|

40

.

60

|

.

80

, 100

(b)

~ _ _ ~ __~

Trrl2 t,~0 !

0

Q

Fig. 13.9. Schematic Ising model" (a) correlation function for t < O, t > 0 and t = O; and (b) its Fourier transform in the t < 0 case. I(Q) is the kinematic diffraction intensity as measured in a scattering experiment, for example. From I(Q)one can determine m2, ~ and Z; the temperature dependence of these quantities give the critical exponents [3, v and 7 respectively.

l-'l/~(r) = l-'Ct)(r) - M 2

(13.29)

The (2d) Fourier transform of Eq. (13.28) gives the beam profile in reciprocal space - - the L R O contribution b e c o m e s a 8-function at Q = 0 (and the other 2d reciprocal lattice points) and the decaying exponential b e c o m e s a Lorentzian of half width at half m a x i m u m w = 1/~ centered at Q = 0, /U)(Q,t) = M2(t) 8(Q) +

Z(t)

1 + QZ ~2

(13.30)

as sketched schematically in Fig. 13.9b. (The amplitude of the Lorentzian term has been denoted by Z(t), the susceptibility as in Eq. (13.20); this identification is established below.) The second term in Eq. (13.30) is often termed the diffuse intensity contribution or the critical scattering. By m e a s u r i n g l~t)(Q) for temperatures in the vicinity of T c and resolving the T - d e p e n d e n c e of m 2 and ~, the exponents [3 and v can be determined.

740

L.D. Roelo.[2~

For t > 0 LRO is absent but short range correlations persist, giving again (to leading order) an exponentially decaying F (t), G(t)(r) = D+ e -re" + (shorter range corrections)

(13.31)

The hallmark of a critical point is the divergence of the correlation length ~ that occurs upon approaching it from either direction in T. At criticality the system exhibits domains or fluctuations of order on all length scales and F can no longer exhibit exponential decay. Instead the correlations decay more slowly, via the power law of Eq. (13.22). (For the 2-d Ising model rl takes the value 1/4.) One also finds in the large-r limit at criticality that 1-" becomes isotropic, i.e. having circular symmetry, and thus the corresponding dependence in reciprocal space becomes I(Q) ... I r-n eiQr rdr -- f rZ-neiQ(," dr

(13.32)

1 Q2-q

(Methods for determining the asymptotic dependence of Fourier transforms of singular functions are given in Lighthill (1958).) Equation (13.32) offers, at least in principle, the prospect of determining the exponent 11. Tracy and McCoy (1975) have demonstrated that in practice there are great difficulties originating from the sensitivity to variations of t away from 0. Before placing these ideas in the lattice-gas context we return to the appearance of the susceptibility as the amplitude of the diffuse intensity contribution in Eq. OM (13.30). To measure X;(t) - --~-, one does not actually need to measure the magnetization or even to apply a field. ~Z is available from the correlation function via the fluctuation-dissipation theorem (see, e.g. Reif (1965) or by twice differdhtiating the free energy, since the definition of )(; and Eq. (13.3a) imply that 1

= k ~ ~ F,.(R)

(l 3.33a)

R

This identification is not hard to prove since the definition of X and Eq. (13.3a) imply that 1 32G - N ~)2h

(13.33b)

Equation (13.33a) then follows via use of Eqs. (13.2a,b). Thus )(; is obtained from the Q = 0 value of the structure factor.


1

Z = , _ ~ (I(Q = O) -

741

(13.34)

m 2)

i.e. the intensity at the center of the diffuse part of the diffraction beam as also shown in Fig. 13.9. Finally, as first noted by Fisher and Langer (1968), and as applied to lattice gas systems by Bartelt et al. (1985), even the specific heat exponent, o~, can be extracted from the correlation function. Integrating I(Q) over an extended range about the point at which critical scattering is occurring is a way of probing the short-range correlations. Like the energy, these must vary like Q,,,~x

I~(Q) = f I(Q) d2a -- C• l-~ +fit)

(13.35)

o

where C• are constants pertaining to the variation above and below the transition temperature and fit) is a smooth analytic function. The choice of QmaxiS not critical and quality of the data may be optimized by adjusting it between the limits of the resolution of the instrument, AQ, and the entire surface Brillouin zone (see Chapter I). A small value for Qmaxreduces signal magnitude (decreasing the signal-to-noise ratio). Increasing Qm~,xincreases signal magnitude, but as the limit of the full surface Brillouin zone is approached the amplitudes, C• of the singularity in Eq. (13.35) must tend to zero since the kinematic intensity integrated over a full Brillouin zone is a conserved quantity. A typical choice is to integrate over 2-5% of the surface Brillouin zone. This method of critical exponent measurement has become popular in the literature. It has been used, for example by Clark et al. (1986) for the reconstruction transition of A u ( l l 0 ) (see w 13.4.1.3), and by Pfntir and Piercy (1989) for the adsorbate disordering transition of Ru(001)-O (see below). All these ideas are readily transposed to the cases of multiple-spaced phases seen in lattice-gas systems. Shifting from the p( l xl ) order of the 2-d Ising model and the attractive lattice gas simply shifts the location of the wavevector at which critical scattering is manifested within the surface Brillouin zone of the substrate. In the case of the c(2x2) lattice gas, the LRO 8-function and accompanying critical scattering occur at Q0 = -a'a -

as previously noted. Since measurements proceed in

diffraction space this sin~ple'shift is no obstacle, and is in fact beneficial, since in most cases there is no substrate contribution to the diffraction signal in the vicinity of the critical wavevector, whereas at the integer-order positions, that contribution would typically dominate. (The overlayer also contributes a 8-function intensity proportional to 0 at the Ghk' S, where h and k are integers.) The real-space correlation function, Eq. (13.25), for such systems manifests the long-range, multiply-spaced order, if it exists, by settling down to a regular oscillation in the limit of large R. Figure 13.10a shows this situation schematically for t < 0. The large-R limit tends to a non-zero average proportional to 0. At shorter ranges one sees the decay of

L.D. Roelo.l:~

742

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

................I ................

..... ~... ....... ........

zr' . . . . . . .

,..~

(a)

[(o)

4es2

.I.

!

~/a

Q

(b)

Fig. 13.10. Schematic correlation function (panel a) and diffracted intensity (panel b) for a lattice gas phase of double-spaced order for t < O. Solid lines denote 8-functions. correlations due to fluctuations, again in the context of the oscillations due to the multiple-spaced order. Analysis, however, proceeds in diffraction space where the total kinematic scattering intensity for t < 0 resembles the schematic sketch of Fig. 13.10b, so that abstracting away the trivial oscillations and eliminating the noncritical averaged contribution proportional to 0, requires no special effort, only the shift to the appropriate location, Q0, in diffraction space. The first diffraction-based measurements of the critical exponents of a surface phase transition were accomplished by Horn et al. (1978) in the physisorption system Kr/Graphite using X-ray diffraction. This early work was done using a high surface area form of graphite as a substrate to obtain sufficient signal strength, at the cost of rather significant finite-size limitations (see Appendix C). Later work using synchrotron X-ray sources has achieved sub-monolayer sensitivity; the reader is referred to Chapter l0 for a more complete treatment. The extraction of the critical exponents for a typical chemisorption system via LEED was first attempted by Roelofs et al. (1981) for Ni(111)-O and has since been

Phase transitionsand kineticsof ordering

743

accomplished for several other systems. The most recent example is the analysis of the order-disorder transition of a p(2• oxygen overlayer on Ru(001) by Pfniir and Piercy (1989) and this case is more instructive of the state-of-the-art. The approach used is to fit measured diffraction beam profiles about Q = Q0 to a function of the form

l(Q,t) = (m2(t) 5(Q _ Qo) + 1 + ( Q~(t) - Q,,)2 ~2 ] o T(Q)

(13.36)

where ~ denotes a 2-d convolution and T(Q) is the instrument response function (see w 7.3). T(Q) can be measured by scanning the beam in question under conditions of nearly perfect order (usually a carefully annealed, low-temperature configuration). The fit determines the functions M2(t), ~(t) and ~2(t). lin t may be obtained by directly integrating the measured profile j, or that reconstructed from the fitting procedure the latter approach giving somewhat better control over the parameter Omax in Eq. (13.35) since otherwise T(Q) contributes to the integration. Obtaining critical exponents from t-dependent data is an underdetermined problem due to the effect of the limitations of finite-size rounding (Appendix C), the lack of prior knowledge of Tc, and the fact that the power law forms of Eqs. (13.18-23) only give the lowest-order singular dependence. This last implies that one would want to use data only quite close to (the as yet unknown) To, while the first concern prevents closer approach to Tc than the point at which ~ becomes comparable to L, the linear dimension of defect-free regions on the surface 2. (This highlights the crucial importance of surface quality for exponent measurements. Step-free - - and mostly defect-free J regions of size on the order of L -~ 500 A are necessary to approach Tc within Itl < 0.01.) The approach used by Pfniar and Piercy (1989), which seems to have been quite successful, is as follows. Determine J using measurements of the T-dependence of the diffracted intensity well below the transition ~ and divide out the D e b y e - W a l l e r factor 3, assuming typically that the latter is independent of Q in the immediate vicinity of Qo; Resolve the beam profiles from the instrument response function via parametrized fits based on Eq. (13.36); Integrate the profiles over 2 - 5 % of the surface Brillouin zone to obtain lint(T); Determine Tc as the inflection point of the variation of lint(T) and use that value for all subsequent analysis; - E x a m i n e log-log plots of the variation of lint, M 2, etc. versus t to ascertain the temperature range in which the data is not seriously influenced either by finite-size -

-

-

-

1 Or equivalently, in the case of electron diffraction, by using a Faradaycup with an aperture of the requisite size.

2 Finite-sizescaling can be used to partially overcome this limitation if L can be determined, varied and controlled at least crudely. See Appendix C. 3 The Debye-Wallerfactor characterizes a non-lattice-gas effect, the loss of coherent diffraction intensity due to the uncorrelated, thermally-induced vibrations of the adatoms within their binding hollows. See Webb and Lagally (1973) for a definitive treatment.

744

L.D. Roeh~;~

effects (see Appendix C) close to Tc or by corrections to scaling away from Tc, this being the interval through which linearity is maintained; Perform least squares fits to determine the exponents using those ranges of temperature. In performing the fits it is useful to use all known constraints (and to test the effect of relaxing them!). For example, in the fit of lin t o n e expects the amplitudes C+ and C_ of Eq. (13.35) to be equal for 2-d Ising and Potts models (see following subsections for definition of Potts models). In the case studied by Pfn~r and Piercy (1989), this procedure yielded exponent values within 10% of the expected results; the most impressive verification of universality to date. (The specific values will be discussed and compared to those of the corresponding magnetic model in the following subsection.) An alternative approach to structure factor data analysis that determines Tc and the exponents y and v simultaneously is based on the notion of scaling. The origin of scaling is discussed in more detail in Appendix B and its implication, among others, is that in the vicinity of the critical point the diffuse intensity, defined in Eq. (13.30), can actually be written as a function of a single variable -

x - q~

(13.37)

where ~ is defined in Eq. (13.28) and Figs. 13.9 and 13.10 and q = Iql = IQ -Qol. The specific form of the diffuse intensity is Id(q,T) = Itl-g S•

(13.38)

The functions X§ and X_ are appropriate respectively for T > Tc and T < Tc respectively and are called scaled structure factors. An analysis based on Eq. (13.38) proceeds as follows. Begin by eliminating from the total diffracted intensity the Debye-Waller factor and the fi-function part due to LRO, and taking account of the instrument response function as in Eq. (13.36). Then one 'guesses' values for T~, y and v, and, based on those values and the variation of ~ with t in Eq. (13.23), makes plots of the function -

-

ItF Id (q,T) vs. x

(13.39)

as in Eq. (13.37) for all measured profiles either above or below the assumed value o f T c. Then one adjusts the values of T~, y and v until the plots are well superimposed. Assuming success has been achieved in superimposing the measured structure factors for a substantial temperature range above (or below) the critical point, one has demonstrated scaling and simultaneously determined the values of the exponents y and v as well as the form of the interesting scaling function X§ (or X_(x)). From these it is possible to extract the exponent 1] since X,(x) goes asymptotically to 0 like -

X•

---x 'a-2

(13.40)

for large x (see Tracy and McCoy, 1975). (As T --~ T~, the range over which the

745


, i,

3.0

I-

i , i , I ; I ;I

' I i I;

I'

I ; I '

2.0

CO

o

1.0

i I

0

o.s89

~._

~~___ o.sTs

I

I

i-

o

[-,

-

~o.ss,

~---~o.s55 i

,

0

I

2

,

i

i~

4

,

I

,

I

,

6

~

I

,

I

8

~

.

o.sqos ~

I0

| I~

-t-" k

(a)

5"0 L

4.5 .-.

1

I

I

' I

"

I'

I

-

!

4.0

-

t~

_o o

3.53.02.52o -I.75 .

-

.

.

-I.50

.

.

.

-I.25

.

.

.

-I.00

-0.75

-0,50

-0.25

3xlO - 7

Ioql o ( k )

(b) Fig. 13.11. Demonstration of scaling of the structure factor as calculated via Monte Carlo simulation of a lattice gas model for the order-disorder transition of a (ffx~J3-)R30 ~ phase by Bartelt et al. (1987). A lattice of linear dimension 60 sites and periodic boundary conditions was used. Panel (a) displays the superposition of structure factors for various temperatures above T,:, each plot being labeled via its temperature in units of E~. The value of T,: used in this plot was 0.338 Ej. Panel (b) shows an attempt to extract the value of the exponent q from the diffuse intensity for the profile with T= 0.3485 E~, the profile closest to T,: not strongly affected by the finite size of the lattice used. See text for further discussion.

746

L.D. Roeh?]:v

variation is controlled by the leading dependence specified in Eq. (13.40) becomes larger and larger.) This method has not yet been applied to experimental data, but has been tested via simulations in which the structure factor is obtained via Monte Carlo calculations (Bartelt et al., 1987). The results of this analysis of the order-disorder transition of a (f3-x'~-)R30 phase on a triangular lattice ~is reproduced in Fig. 13.11. This mode of analysis obviously cannot eliminate the influences of finite size (see Appendix C). In the case of the analysis displayed in Fig. 13.11, profiles at temperatures within 2% of Tc could not be made to scale. A second breakdown occurs at large q (and therefore for smaller x's as t increases in magnitude) and reflects the lattice structure of the substrate. The curves diverge from the common scaling function when q has reached halfway from Q0 to Q = (0,0) where a strong intensity contribution due to the substrate is centered. (The point Q0 at which ordering and critical scattering is that which characterizes ('4"3-• order.) Figure 13.11 b shows an attempt to extract the value of 1"1 using the profile closest to Tc that scales - - thus avoiding the complication of finite-size effects (see Appendix C) - - from the variation of ld VS. q which should go as indicated by Eq. (13.32). The expected value ofrl is 4/15 =0.2666... (see w 13.3.2). In Fig. 13.1 lb a trend toward larger 1"1 and thus better consistency with the expected value is found for data ranges restricted to larger values of q. However, a considerable discrepancy still remains at t = 0.02, this for a lattice size of linear dimension L = 60, which indicates the difficulty of reliably extracting a value for rl from systems of modest size.

13.3.2. Universality classes As noted above, continuous phase transitions can be classified into a rather smaller number of universality classes within which the critical exponents and certain amplitude ratios agree. This categorization (often termed Landau classification after the originator m see Schick ( 1981 ) and the textbook treatment in Landau and Lifshitz (1969)) depends on the spatial dimensionality of the lattice on which the transition occurs and the symmetry of the order-parameter whose variation defines the transition. The analysis of the symmetry of the order parameter is most conveniently done using group theory as generally applied in condensed matter physics. The discussion given here will of necessity be rather sketchy; the treatments by Schick (1981) and Persson (1992) can be consulted for further detail. Briefly, group theoretical analysis of the symmetry of the order parameter allows one to write the free energy near a continuous phase transition as an expansion in terms of the order parameter, which may have more than one component. Such an expansion, Eq. (13.5), has already been presented for the single-component-order-parameter Ising model in w 13.1.4.2. The key idea of universality is that in the vicinity of a critical point, the physics is

1 A triangular lattice can be realized in surface systems when the adsorbate binds to the hollow on a hexagonal surface such as the basal plane of graphite, or in atop binding on fcc(l I 1) and basal plane of hop crystals.

Phase transitions and kinetics ~'ordering

747

controlled by the very long-range correlations described by the divergence of ~; that therefore the local details of the system become irrelevant; and thus that the nature of the transition is controlled by more global considerations like the number of components in the order parameter and how they interact, as indicated by the free energy expansion. Stated in the language of Landau theory then, systems whose free energy expansion in terms of the order parameter are similar, have asymptotically identical critical behavior. Thus, for example, although one cannot make a direct analogy between the 2-d, square-lattice Ising model and say an attractive-interaction, lattice gas on a triangular lattice, the two systems are expected to have identical critical exponents because they agree as to dimensionality of space and symmetry of order parameter. A rather interesting illustration of classification is provided by Tejwani et al. (1980) who measured, via calorimetry, the specific heat singularity of a monolayer of He adsorbed on clean graphite and on Kr-plated graphite. Although the He layer is structurally identical in the two cases, its array of binding sites is altered from triangular (on the clean graphite) to honeycomb on the Kr-coated graphite with the result that symmetry group of the order parameter changes from that associated with the 3-state Potts model (see w 13.3.2.2) to that of the Ising model. The measurements confirmed the expected shift in value of or.

13.3.2.1. Exponent values Table 13.1 presents the exponent values of the known 2-d universality classes along with relevant surface realizations. The latter are discussed in w 13.3.2.2 where the magnetic models that name the classes are described. Table 13.1 requires some comment. The table presents the values of the exponents defined in w 13.3.1 as well as those of Yh and Yt, the leading renormalization eigenvalues, which characterize the behavior of the system near its critical point when examined on varying length scales. These eigenvalues determine the asymptotic decay of all correlation functions at T,.; Appendix B gives the details including the derivations of expressions for the critical exponents in terms of Yh and Yr. The case of the XY model with cubic anisotropy is clearly a bit different than the others in that for some of the exponents a range of values is given. This indicates that these exponents are expected to vary continuously (bUt still satisfy the scaling relations given in Appendix B) with the strength of the anisotropy, a parameter involved in the definition of the model. One therefore says that this model displays nonuniversal behavior. As demonstrated by Jos6 et al. (1977), the nature of the transition varies between the limiting cases of the Ising model and that of a Kosterlitz-Thouless transition (see w 13.3.2.2) whose critical behavior was analyzed by Kosterlitz and Thouless (1973) and Kosterlitz (1974). A calculation by Hu and Ying (1987) suggests the exponents have values rather close to those of the 2-d lsing model except in the vicinity of vanishing anisotropy where pure KosterlitzThouless behavior should occur. One should therefore expect that measured critical exponents of systems classified as XY models will often have values close to those of the Ising model. The critical exponents deriving from classical mean field theory (see any statistical mechanics textbook m e.g. Reif (1965) where the theory is called

748

L.D. Roelof~ Table 13.1 Two-dimensional universality classes, exponent values and realization

Exponent

f~ 8 ll v Yt .Vh Example of realization: Disordering of...

Ising

3-State Potts

4-State Potts

XY with cubic anisotropy

l st-order

Mean field

0* 1/8 7/14 15 1/4

1/3 1/9 13/9 14 4/15 5/6 6/5 28/15

2/3 1/12 7/6 15 1/4 2/3 3/2 15/8

[0,-co**] [1/8, oo] [7/4,00] 15 1/4

1 0 1 oo 0 1/2

0 1/2

[1,0] 15/8

2

(~3x73-3) R30 on triangular lattice

p(2x2) on triangular lattice

(Ix2) on square lattice; or (2x2) on centered-rectan gular lattice

(Ix2) on triangular lattice

I 1

15/8 c(2x2) on square or rectangular lattice; or p( I xl ) on any lattice

[1,oo]

1

3 0 1/2

2 Long-range attractive interactions on any lattice.

*Whcn cx -->0 the remaining singularity is logarithmic since for small x, Itl-x goes like I - x In Itl +... **A negative specific heat exponent implies a cusp rather than a divergence. When the exponent goes to - ,,,, the cusp becomes an essential singularity. (For a discussion of the classification of singularities see any text covering complex analysis or mathematical methods more generally; Wong (1991) is a good choice.) The essential singularity in the specific heat of the XY model is, tor all practical purposes, invisible, since the singular function and all of its derivatives vanish at the phase transition. (The function exp[-l/Itl], for example, has an essential singularity at t = 0.) See, for example the simulation study of Tobochnik and ('hester (1979). Fortunately the transition is apparent in other observables.

C u r i e - W e i s s t h e o r y , or B r o u t ( 1 9 6 5 ) h a v e b e e n i n c l u d e d in T a b l e 13. I. M e a n field t h e o r y ( M F T ) is not c o n s i s t e n t with the s c a l i n g h y p o t h e s i s and t h e r e f o r e v a l u e s o f yj, and y, h a v e not b e e n g i v e n . M F T can be and o f t e n is a p p l i e d to lattice gas m o d e l s w i t h s h o r t - r a n g e i n t e r a c t i o n s as a first t h e o r e t i c a l a t t e m p t (see S c h i c k ( 1 9 8 1 ) for a c o n v e n i e n t p r e s c r i p t i o n for c a r r y i n g o u t the c a l c u l a t i o n s ) ; but as c a n be seen f r o m T a b l e 13.1, the t h e o r y is not s u c c e s s f u l in d e s c r i b i n g p h a s e t r a n s i t i o n s in unodels with s h o r t r a n g e i n t e r a c t i o n s . M F T not o n l y fails to g i v e an a c c u r a t e a c c o u n t o f critical e x p o n e n t s , but also in s o m e c a s e s , n o t a b l y that o f the 3 - s t a t e Potts m o d e l , fails to p r e d i c t the o r d e r o f a t r a n s i t i o n p r o p e r l y . M o r e o v e r , as s h o w n by B i n d e r and L a n d a u ( 1 9 8 0 ) , p h a s e b o u n d a r y l o c a t i o n s for 2-d lattice gas m o d e l s c a l c u l a t e d via M F T m a y be i n a c c u r a t e by f a c t o r s o f 2 or 3. T h e f a i l u r e s o f M F T are d u e to its n e g l e c t o f f l u c t u a t i o n s ; the v a l u e o f e a c h d e g r e e o f f r e e d o m is set e q u a l to its s e l f - c o n s i s t e n t l y c a l c u l a t e d t h e r m a l a v e r a g e . In the c a s e o f l o n g - r a n g e a t t r a c tions, this is a less s e r i o u s a p p r o x i m a t i o n , s i n c e the s y s t e m is less a f f e c t e d by its o w n s h o r t r a n g e f l u c t u a t i o n s . I n d e e d F i s h e r et al. ( 1 9 7 2 ) f o u n d that the c r i t i c a l e x p o n e n t s for a m a g n e t i c s y s t e m with i n t e r a c t i o n v a r y i n g like s i -r3 s t- are i d e n t i c a l to

749


the mean field values given above. Translating to the lattice-gas equivalent we 1

would expect that a system exhibiting long-range ~ attractions should have mean field critical exponents. There is one important universality class not directly represented in Table 13.1, that of the Heisenberg model which can support two different forms of anisotropy. (This model and its surface realizations will be described in more detail in w 13.3.2.2.) The model is not included in the table since the best currently available theoretical evidence suggests that the model displays only either Ising-type transitions or first-order transitions. (Recent experimental work has, however, raised some important questions concerning this conclusion, as will also be discussed in w 13.3.2.2.) One therefore expects to find surface realizations of first-order phase boundaries not terminated by a critical point as in Fig. 13.3. Given this expectation, the exponents associated with a scaling treatment of first-order transitions due to Fisher and Berker (1982) have been included in Table 13.1. Note that both renorrealization eigenvalues have assumed values equal to the dimensionality of the lattice, d = 2 in this case; this being the diagnostic characteristic of the renormalization fixed point associated with a first-order transition. The exponent values are as might be e x p e c t e d only o~ and v require comment. With o~- 1 the energy, which varies as in Eq. (13.35) has no singularity upon approach to the transition. (The discontinuity in E at the first-order transition is not manifested by the critical dE exponent values.) With C-~---~ the variations in Eqs. (13.18) and (13.35) are ordinarily consistent. Obviously, however, this relation breaks down when c~ --~ 1, since the derivative of a constant vanishes and the value of ~ = 1 does not therefore imply a power-law specific heat divergence as in Eq. ( 1 3 . 1 8 ) f o r first order transitions. Rather the latent heat of the transition is exhibited as a ~i-function in C. (For systems of finite size the ~5-function is broadened thus allowing confusion with power-law growth near a critical transition. In that case one might extract an effective value of o~ from data, but there is no theoretical reason to expect that value to be related to the oc - 1 of Table 13.1. Finite size effects on first-order transitions are discussed further in Appendix C.) The value v - I/2 suggests a divergence of the correlation length, something one might not expect at a first-order transition. Fisher and Berker attribute the apparent divergence of ~ to the LRO possessed by both phases in the vicinity of a first-order transition. This interpretation, however, may be inadequate as indicated by the simulation study of the order-disorder transition of a p(2x2) phase on a honeycomb lattice of Bartelt et al. (1987). In this work apparent critical scattering and divergence of the correlation length with v~ff = 0.55+0.15 were found to be associated with an apparently first-order transition.

1 Dipolar repulsions go like r--3, but cause non (lxl) ordering (see Roelofs and Kriebel, 1987). Any order-disorder transitions that occur in that context would display the critical behavior appropriate to the relevant order parameter symmetry.

L.D. Roelo.f~

750

Some elaboration beyond this simple scheme is necessary to account for the effect of other r e l e v a n t operators. See Appendix B. 13.3.2.2. M a g n e t i c m o d e l s In this subsection the magnetic models which name and characterize the universality classes included in Table 13.1 are defined and described. All the models are for ferromagnetic behavior and can be defined on a 2-d square lattice using nearest neighbor interactions only. The description will in each case only specify the nature of the degree of freedom associated with each site and the form of the interaction, J, between neighboring 'spins'. The Hamiltonian of the 2-d Ising model was given in w 13.1.4 and therefore need not be repeated here.

(i) The Potts m o d e l s : The q-state Potts models are generalizations of the 2-d Ising model, which can be taken to be the case of q = 2. In the q-state Potts model each spin may be found to be oriented in q possible directions and interacts with its neighbors via a Kroneker ~5-function interaction.

Jq_po,t.,(ni, rt]) - - J 8,,,.,,j

( 13.41 )

The n;'s represent the degrees of freedom associated with each site and take on integer values {1, 2, ... q} to denote the q possible orientations; J is a positive constant for the ferromagnetic case. It is important to distinguish the standard Potts model defined above from the p l a n a r Ports or clock model in which the basic degree of freedom is also taken to be a vector of q possible distinct orientiations, but whose basic interaction is in the form of a dot product Jq cl,,ck(si, si) = - J si " Si

(13.42)

(The 3-state Potts and 3-state clock models are identical up to a shift of the energy zero, which has no physical significance, but for higher q the differences are non-trivial. Betts (1964) noted that the 4-state clock model can be reduced to two superimposed lsing models, and so has the critical behavior of that universality class.) Fc)r q > 4, the q-state Potts models all are known to exhibit first-order transitions. Wu (1982) in a review devoted just to the Potts model gives references detailing this work and the relations between the Potts and other models which were exploited to determine the critical exponents given in Table 13.1. He does not, however, note that the exponents for the q = 3 case are known from the solution by Baxter (1980) of the hard hexagon lattice gas model. See also Huse (1984). (ii) X Y model: In the XY model the basic degree of freedom is taken to be a spin that can be oriented in any direction in the plane. Hence the alternative name of p l a n a r model. The interaction is taken to be of dot product form Jxv(Si, SJ) = - J si " Si

where J again denotes a positive constant.

(13.43)


751

The XY model as usually discussed includes anisotropies that favor certain orientations, relative to crystalline axes, over others. One expects to encounter 2-, 3-, 4- and 6-fold anisotropies in real materials, so that the XY Hamiltonian is usually written including a single-site term of the form, Hp_..,m., - - h , ~ cos(p0,)

(13.44)

for p-fold anisotropy, where 0i represents the angle spin sg makes with say the x-axis. hp is a positive constant representing the strength of the anisotropy. Jose et al (1977) give a complete account of the behavior expected for the cases p = 2,3,4,6. In the case of p = 2 and 3 the perturbation represented by the anisotropy is relevant so that the critical behavior is governed by anisotropy. One then finds Ising behavior for the p = 2 case and 3-state Potts behavior for the p = 3 case. For p = 4 or 6 one has more interesting situations. We will focus here on the p = 4 case, that of 'cubic' anisotropy, since that is the only case that appears in Table 13.1. (It should be noted, however, that the 6-fold case has relevance to the theory of 2-d melting, see Halperin and Nelson (1978).) No exact solution for the XY model with cubic anisotropy is available, but various approximate calculations have determined the essence of phase diagram and the critical behavior of the model. The phase diagram is given in Fig. 13.12a. We begin with the situation for h4 = 0. In the absence of anisotropy there exist spin wave excitations whose energy vanishes in the limit of long wavelength. Under these circumstances the model cannot develop rigorous long-range ferromagnetic order at any finite temperature. Rather the correlation function, the analog of Eq. (13.27), displays power-law decay F(r) ~ r -n with I"1 increasing from 0 at T = 0, linearly with temperature. (See Kosterlitz and Thouless, 1973.) A power-law decaying correlation function is associated with critical behavior ~ see Eq. (13.22) and thus we must regard the h4-axis as a line of critical points with varying critical exponents, since 1"1is varying. Such a l-d locus of critical points is called a critical line and is often denoted by a line of asterisks as in Fig. 13.12a. According to spin wave theory the line would continue to arbitrarily high temperatures. Kosterlitz and Thouless (1973), however, pointed out that the h4 model supports another sort of excitation, the vortex, at sufficiently high temperature. Their analysis established that below a particular temperature, vortices and anti vortices ~ swirls in the opposite direction ~ occur in bound pairs leaving the correlation function still in the form of Eq. (13.27). However, at the so-called Kosterlitz-Thouless point (labeled 'K-T' in Fig. 13.12a) at which point 1"1has reached the value of 1/4, the vortices and anti vortices unbind resulting in a more rapid (exponential) drop off of correlations and thus constituting a transition to a paramagnetic phase. This occurs at TK_T----0.8 J. F o r h 4 :g: 0 a gap occurs in the spin-wave spectrum so that long-range magnetic order is possible at low temperature. The regions above and below the h 4 axis below TK_Tare thus ferromagnetic. (The phase diagram is symmetric about h 4 = 0 since a sign change in h4 is equivalent to a 45 ~ rotation, leading to order in one of the 45 ~ 135 ~.... directions rather than along the axes.) In the limit of large anisotropy the spins are constrained to point in just 4 directions. We thus have the 4-state clock

752

L.D. Roeh?f~

XY Model w/cubic anisotropy Y

x

(a)

+oo

,

~u

4 clock~ (Ising) h4

+

. K-T T/J

N

9 9,

~

Heisenberg Models

corner-cube anisotropy

face-centered anisotropy (b)

Fig. 13.12. Other spin models. (a) The XY model with 4-fold or cubic anisotropy. The upper panel shows the basic spin degree of freedom associated with each site as a moment of fixed magnitude able to point in any direction in the x-y plane, but with the anisotropy of 4-fold symmetry favoring certain directions over others. The lower panel shows the phase diagram of the model. See text for explanation. (b) The Heisenberg models. Preferred orientations of spins in the Heisenberg Model with corner-cube anisotropy are shown on the left; and on the right, similarly for face-centered anisotropy.


753

model in this limit, which is known to have Ising exponents. The disordering transition for large h a thus differs markedly from that at h4 = 0 and the character of the disordering transition is expected to vary continuously along the curve connecting those limits. Thus we have another critical 'line' (not straight in this case). Hu and Ying (1987) have performed a careful simulation study of this 'non-universal' behavior along this critical line, verifying the approximate character of the phase diagram shown in Fig. 13.12a and proposing forms for the variation of the various exponents with h a. For example the correlation length exponent defined in Eq. (13.23) is found to vary like v = [I - exp(-ch4)] -j

(13.45)

where c is found to have the approximate value of 3.6 when h 4 is expressed in units of J. Experimental verification of this fascinating theoretical panoply has been elusive. There has thus been some attention paid to the surface realizations the XY model, whose disordering transitions are expected to be located at varying positions along the curving critical lines. Successful measurements of critical exponents of such transitions would be useful checks of these predictions of non-universal critical behavior. The varying surface atom displacements that occur in the surface reconstruction of W(001 ) are subject to a Hamiltonian similar in symmetry to that of the XY-model so that this system, particularly when the addition of adsorbates is used to vary the Hamiltonian parameters, constitutes another possible experimental test. This matter will be explored in greater depth in w 13.4. (iii) T h e H e i s e n b e r g m o d e l s : The Heisenberg model is the extension of the Ising and XY cases to classical spins that can point in any direction in 3-d space ~. We consider the model here only with classical spin degrees of freedom, located on 2-d lattices and with ferromagnetic interactions. (A 2-d, quantum, antiferromagnetic version of the model is of some possible interest in the theory of high temperature superconductivity.) The interaction between neighboring spins is again taken to be in the form of a dot product, Jttei~(si , Si) = - J si " si

(13.46)

where J is again a positive exchange constant tending to produce ferromagnetic behavior. Again, because of long-wavelength, low-energy excitations, long-range order is possible only when anisotropies are present to create gaps in the spin-wave spectrum. Crystal fields give anisotropies of various symmetries. Uniaxial anisotropies

1 Ironically, Heisenberg proposed this model because of the failure of the 1-d Ising model to develop long-range order as determined from Ising's solution of this case. The Heisenberg model supports an even greater multiplicity of spin waves at long wave-length even than the XY-model and so is even less able than the XY-model to develop long-range order on either 1- or 2-d lattices.

754

L.D. Roelofs

lead either to Ising or XY behavior depending on whether the spin orientations along a particular axis are enhanced or suppressed. We focus here on the cases of corner-cube- and face-centered-anisotropy depicted in Fig. 13.12b, which result in models with new surface realizations. These Heisenberg models have not been solved exactly either, except that in the cornercubic case (CCH) and in the limit of large anistropy (where one has just the 8 allowed orientations) the model is identical to 3 decoupled, superimposed Ising models (one each for the x-, y- and z-directions) and so should have that critical behavior. Landau classification (see Schick, 1981) places the order-disorder transition of a p(2x2) state on a honeycomb lattice into the CCH universality class, and so there is the possibility of surface realizations. However, Landau classification is based only on the symmetry of the order parameter and interactions, and is not a quantitatively exact Hamiltonian transcription. The surface realizations might have effective Hamiltonians with s m a l l - but symmetry r e s p e c t i n g - corrections to Eq. (13.46). Grest and Widom (1981) studied the effect of small alterations of the dot product interaction of Eq. (13.46) and found that the transition becomes first-order for one type of variation and remains continuous and Ising-like for its converse. Their study did not, however, encompass all possible symmetry preserving variations of the interaction and therefore should not be considered exhaustive. Thus the most one can say at this point is that, for systems classified into the CCH universality class, either Ising critical behavior ~or first-order transitions could be expected. The face-centered case (FCH) also has surface realizations, most particularly the disordering of a p ( 2 x l ) structure on a triangular lattice as occurs for example for oxygen adsorbed on Ru(001) at 0 = 1/2. Renormalization group studies by Nienhuis et al. (1983) suggest that the transitions of the model will always be first-order in 2-d. The experimental study of the disordering of Ru(001 )p(2• ) - O by Pfntir and Piercy (1990), however, suggests that the transition is continuous and has critical exponents in rather good agreement with those of the 3-state Potts model. This apparent conflict has not been resolved at the time of this writing, but further theoretical work on the nature of the transitions of the FCH case appears to be warranted.

13.4. S u r f a c e r e c o n s t r u c t i o n

A surface is said to have reconstructed when the atoms in the surface layer and possibly those in nearby underlying layers, spontaneously, or in response to the presence of an adsorbate, shift from their expected (bulk-termination) positions in such a way as to alter the 2-d periodicity of the surface 2. Because of this symmetry

l Schick (1981 b) proposed this as explanation for the observation of Ising critical exponents by Roelofs

et al. ( 1981) for the disordering of p(2x2) oxygen on Ni(111). 2 Theoretical aspects of the driving forces for surface reconstruction are discussed in Chapter 3; experimental aspects are considered in Chapters 7, 8 and 9.


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change, reconstruction phenomena are conveniently studied with diffractive methods, especially LEED (see Chapter 7) and X-ray diffraction. In many cases reconstruction is found to be influenced by temperature and one of the guiding themes of the field has been the elucidation of the nature of the observed transitions and the detailed character of the phases involved. New experimental techniques were often developed or were first applied to address long-standing controversies, and the subtle driving forces continually challenged the accuracy of theoretical treatments of the electronic energy in the surface region. The presence of a surface usually causes alteration of equilibrium atomic positions in the perpendicular direction as well. This phenomenon is termed surface relaxation, and is observed to extend several atomic layers into the crystal. Since no symmetry changes or phase transitions are involved in surface relaxation, it is not considered further in this chapter. It would be very satisfying to offer a unified explanation for reconstruction. However, the mechanisms involved are as diverse as the types of bonding that occur in condensed matter physics, and the best one can do is to offer a categorization, perhaps along the lines of the organization of this section.

13.4. I. Reconstruction of metallic surfaces Reconstruction has been observed on many metallic surfaces. Estrup (1984) gives a useful review. Observed reconstructions can be classified into a few categories, based on the character of the atomic movement involved, and the apparent driving force. The discussion in the following subsections is similarly organized.

13.4. I. I. Reconstruction resulting in increased surface packing density The more 'open' surfaces of highly coordinated metals, including AI and the noble metals, tend to reconstruct in such a way as to increase the packing density of the surface layer. On the (100) surfaces of Au, Pt and Ir a pseudo-hexagonal layer, whose spacings are not too dissimilar from that of the close-packed (111) planes, forms over the second layer which largely retains its square symmetry. (See also Chapter 3.) The case of Au(100) is typical. Recent definitive studies include: high resolution LEED by Liew and Wang (1990); scanning tunneling microscopy by Binnig et al. (1984); transmission electron microscopy by Yamazaki et al. (1988); and X-ray scattering by Zehner et al. (1991). An approximate (lx5) periodicity, as shown in the sketch in Fig. 13.13, is found. The atomic spacing in a close-packed (111 ) plane of Au is 2.88 ~. This is also the nearest neighbor distance in the unreconstructed (100) plane, but note that a reconstruction of (Ix5) periodicity - - the periodicity in this direction is evident in Fig. 1 3 . 1 3 b - gives a nearly hexagonal layer with nearest-neighbor spacings of 2.88 ~ and 2.804/~. This reconstruction is evidently driven by the improved coordination offered by the pseudo-hexagonal structure. The precise measurements of Liew and Wang (1990) indicate that there also occurs a slight compression in the other direction as well, such that 29 Au atoms occur in 28 spacings of the crystal in the (110) direction. This reduction of spacing below

756

L.D. Roel~?[;~

Fig. 13.13. Pseudohexagonal reconstruction of Au(001) and similar surfaces, after Yamazaki, et al. (1988). White circles are top-layer Au atoms, successively darker shading indicates layers further into the crystal. (a) depicts the reconstruction on a defect free surface. The overlayer is a distorted hexagon rotated by an angle from the surface crystalline axes [ 10] and [01 ]. (b) shows the effect of step edges in the [ 10] direction on the reconstruction, which is aligned to the step direction to achieve hexagonal coordination at the step edge (see dashed outline in lower portion of figure). The repeat distance along the step edge is about 28a; the sketch shows the registry of the reconstructed top layer at positions separated by half the repeat distance in that direction.

what occurs in the undistorted c l o s e - p a c k e d plane of Au probably results f r o m the fact that e v e n in the p s e u d o - c l o s e - p a c k e d a r r a n g e m e n t the Au atoms are not so fully c o o r d i n a t e d as they ' w a n t ' to be, and therefore respond by s h o r t e n i n g s o m e o f the bond distances. Finally, when the surface is relatively free of steps, the p s e u d o - h e x agonal top layer can further lower its e n e r g y via a slight rotation, as shown in Fig. 13.13a. T h e rotation angle o~ is d e t e r m i n e d by Y a m a z a k i et al. (1988) to be about 0.7 ~.

Phase transitions and kinetics ~" ordering

757

Both this rotation and the tendency of the reconstructed first plane to lock in to high-order integer multiples of the spacings established by the unreconstructed bulk are common features of overlayers that would prefer incommensurate spacings. This subject is covered in more detail in Chapter 10. Finally one should note that the areal density of atoms in the reconstructed first plane is 24% greater than that of a (100) surface. Thus the development of the reconstruction must involve large-scale motion on the surface (see w 13.4.4) and surface steps and other extended defects doubtless play a significant role. This is consistent with the observation that good order does not develop unless the material is heated above 500 K. Further detail has emerged from the X-ray diffraction studies of Zehner et al. (1991), who report a slight rotation of the pseudo-hexagonal structure for T < 970 K, an unrotated reconstruction in the range 970 K < T < 1170 K, and a disordered surface layer for all T > 1170 K up to the melting point.

13.4.1.2. Displacive reconstruction Displacive reconstruction refers to a situation in which the stable surface phase is of different symmetry than the bulk termination, but in a manner that does not involve atomic migration. A prototypical case is that of W(001) which undergoes a reconstruction to a phase characterized by c(2x2) symmetry in which zig-zag rows of atoms develop as shown in Fig. 13.14. This reconstruction, originally thought to be an adlayer phase, was first recognized as such by Debe and King (1977) and Felter, Barker and Estrup (1977). Its detailed structural character was first determined by Debe and King (1978). In this case there is no gross change in surface coordination; rather the effect appears to be driven by covalent bonding considerations primarily involving atoms in the top two planes, Singh and Krakauer (1988) display bending-charge-density plots in their ab initio electronic total energy calculations which clearly show

Fig. 13.14. Displacive reconstruction of the W(001) surface. The small arrows on two atoms in t_he top row emphasizes that the direction of motion is along the (or equivalently the direction) primarily in the surface plane. The scale of the motion is exaggerated by about a factor of 2 to improve visibility.

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L.D. Roeh~.#;

Fig. 13.15. The c(q7--• reconstruction of Mo(001) seen at glancing incidence looking along the zig-zag rows. These rows are similar to those which occur in the W(001) reconstruction (see Fig. 13.14), but here after each triple of zig-zag rows, there is a row of undisplaced atoms. additional interactions, particularly between a surface atom and the four nearest atoms in the layer beneath. The origin of this extra charge is mostly the 'dangling bonds' associated with presence of the surface. In the case of the W(001) surface, the reconstruction 'disappears' at a Tc of roughly 220 K'. The nature of the phase above Tc was initially controversial, with some workers suggesting that the ( l x l ) periodicity represented an ordered surface of bulk-termination character and others arguing for an interpretation in which the loss of periodicity was accounted for through a loss of long-range order in the displacements, whose magnitude and local character remain similar to that below To. This latter view has recently prevailed according to evidence from: LEED (Pendry et al., 1988); ion scattering (Stensgaard et al., 1989); X-ray diffraction (Robinson et al., 1989); and core-level-shift spectroscopy (Jupille et al., 1989). This evidence along with the observation that the phase transition appears to be continuous allows its classification into the universality class of the x - y model with cubic anisotropy (see w 13.3.2). Displacement magnitudes are small, on the order of 0.2 ]k in the case of W(001 ), and reconstruction energies are in the I0 meV range. These situations therefore represent stringent tests of electronic structure/total energy calculations. State-ofthe-art approaches (see Singh and Krakauer, 1988; Wang and Weber, 1987; Fu and Freeman, 1988) successfully account for the structure of the phase, and produce reasonable (order of magnitude) values for the reconstruction energy. However, detailed simulations of the disordering of the surface layer (see Roelofs et al., 1989) based on such calculations, reveal quantitative discrepancies as large as a factor of two in transition temperature. The (001) surface of molybdenum undergoes a related reconstruction, though in this case the wavevector of the reconstruction is found to be shifted slightly from the c(2x2) positions. This reconstruction was first reported by Felter et al. (1977)

1 The'unreconstructiontemperature' may appeal"to be substantiallyhigheron surfaces with closely spaced steps. (See Wendelken and Wang, 1985.)

Phase transitions and kinetics o.l'ordering

759

and was originally thought to be incommensurate. More recent studies (see Daley et al., 1993; Smilgies et al., 1993) establish that the phase actually is commensurate with c(~--x~-7--) periodicity, as shown in Fig. 13.15. The mechanism underlying this reconstruction has not been definitively settled. The theoretical accounts of C.Z. Wang et al. (1988) and X.W. Wang et al. (1988), which articulated a mechanism based on Fermi surface nesting successfully accounted for incommensurate ordering. However, a picture based on short-range interactions (see Roelofs and Foiles, 1993), seems promising for explaining a commensurate phase. Unidirectional displacive reconstruction is also possible; Itchkawitz e t a l . (1992) have reported that the top layer of K(110) shifts laterally along the [110] direction by about 0.25 ,/k. (See w 13.4.1.4 for a similar reconstruction induced by hydrogen adsorption.) 13.4.1.3. Missing row reconstructions

The (110) surfaces of several fcc noble metals are observed to reconstruct to the missing row phase shown in Fig. 13.16. This reconstruction occurs spontaneously for Ir, Pt and Au and can be induced by low-coverage alkali adsorption for Ag, Pd and Rh. (See Chapter 3 and Ho and Bohnen (1987) for experimental references.) This reconstruction is not driven simply by coordination, since there is no net change in the total number of missing first neighbors when it occurs. Thus more subtle many-body effects must be examined, and in the author's study, see Roelofs et al. (1990), of the energetics of this transition in the case of the Au(l 10) surface,

Fig. 13.16.The missing row reconstruction that occurs on fcc(110) noble metal surfaces. (a) top view; (b) side view.

760

L.D. Roelo.f~

the dominant driving force is found to be a reduction in the many-body or glue ~part of the interaction, which depends in a highly nonlinear way on coordination. These reconstructed phases appear to undergo disordering transitions at sufficiently high temperatures, but there has been some controversy over whether the disordered phase remains f l a t - but with the 1/2 layer of 'extra' rows disordered or whether the surface roughens. In the former case the transition should occur in the universality class of the 2-d Ising model as pointed out by Per Bak (1979). The first experimental study, carried out via LEED at relatively low resolution by Campuzano et al. (1985) seemed to confirm the 2-d Ising classification. Villain and Vilfan (1988) first raised the roughening possibility on theoretical grounds and noted that in that case one would not expect simple Ising behavior. Recent higherresolution studies of the Au(110) transition at 735 K, via X-ray diffraction - - see Keane et al. (1991), and the corresponding transition on Pt(110) which occurs at 960 K and has been investigated by Zuo et al. (1990) using high-resolution LEED indicate that the transition is of simple order-disorder (and thus) Ising character. The X-ray diffraction study of Keane et al. (1991) also reveals that the Au surface at least does eventually roughen as well, some 50 K above the disordering phase transition. The missing-row character of these phases was initially considered to be a problematic explanation of their apparent double-spaced periodicities, because it was thought that the extensive mass motion required for their ordering would be prevented by diffusion limitations. Many structural probes were therefore applied in attempts to shed light on this question, but the debate was only definitively settled by scanning tunneling microscopy in one of the earliest uses of that technique for metal surfaces (see Binnig et al., 1983). It was also pointed out by Campuzano et al. (1985) that the occurrence of steps initially on the surface as a result of sample preparation processes would, to some degree, mitigate the difficulty of ordering via diffusion. The significance of diffusion in non-displacive reconstruction is treated in more detail in w 13.4.4.

13.4.1.4. Adlayer-induced reconstruction The reconstruction modes considered in w 13.4.1-3 and others can also be induced by adsorption on surfaces which, when clean, do not reconstruct. There is a vast and growing set of known instances of this behavior; here we give some examples of cases that are relatively well understood at this time. It is worth noting parenthetically that in some cases these induced reconstructions are kineticaily hindered, and are not reversible. The induction by alkali adsorption of the missing-row reconstruction of w 13.4.3 has already been mentioned. The fact that coverages as low as 0.1 monolayers of K, for example, can induce the missing-row reconstruction on Ag(110) and the high electropositivity of the alkalis encouraged the hypothesis that charge donation into surface states drives the reconstruction. This long-range mechanism was supported

1 The term is due to Tosatti; see Garofalo et al. (1987).

Phase transitions and kinetics ~tf ordering

761

A

0.7 appears to cause a row-pairing reconstruction, in which adjacent, close-packed rows approach one another in pairs as shown in

762

L.D. Roelo.]:~

Fig. 13.18. Structure of the O-induced (2xl) reconstruction of Cu(110). The O atoms are denoted by the smaller solid circles; larger circles denote Cu atoms, successive darker shading indicates deeper layers of Cu. (a) top view. (b) side view, along rows. Positions of the O atoms are sketched roughly as determined by Robinson et al. (1990). Fig. 13.19. For Ni, which is immediately above Pd in the periodic table the situation is less clear. The Ni(110) surface also develops ( l x 2 ) periodicity under H-exposure, but the available experimental evidence, at this writing is insufficient to decide between the missing-row and pairing row candidate structures, although the weight of the evidence perhaps favors the latter. See Baumberger et al. (1986), for example, and Kellogg (1988) for the contrary view. A very different reconstruction category is exemplified by W ( 1 1 0 ) - H . Since tungsten is bcc, the latter surface features, long, flat-bottomed adsorption sites for H binding I as indicated schematically in Fig. 13.20a. Chung et al. (1986) suggest that if the binding well is truly flat-bottomed, then there should be an increase in H-binding energy if the entire top substrate_layer shifts uniformly in the [110] d i r e c t i o n - or the symmetrically-equivalent [110] direction, of c o u r s e - as shown in Fig. 13.20b. The driving force for the shift is that in the asymmetric well the energy of the lowest quantum state for a H-atom will be lower than in the flat-bottomed well where the first two levels are approximately degenerate. Thus we have a (H-coverage-dependent) ( l x l ) reconstruction, which manifests itself in LEED, as

1 The same hourglass adsorption hollows, but for metallic adsorbates figured in the discussion of asymmetrical phase diagrams in w 13.2.2.


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Fig. 13.19. The H-induced, row-pairing reconstruction of Pd(110). (a) The clean surface, the large circles denoting Pd atoms, with second layer atoms shaded. (b) The reconstruction whose structure has been obtained by Rieder et al. (1983) and Demuth (1977). The smaller circles denote the binding sites for hydrogen. The inequivalent binding sites atop second layer atoms denoted by less darkly shaded small circles are opened up by the reconstruction, doubtless contributing thereby to the driving force of the reconstruction. observed by Chung et al. (1986), as an alteration in the point group symmetry of the diffraction pattern with onset at a critical coverage. Substitutional adsorption is also considered, technically at least, to be adsorbateinduced reconstruction, although one could perhaps also file it under surface alloy formation. For examp)_e K adsorption on AI(I 11) at room temperature results in a LEED pattern (,~-x~3)R30 symmetry and has recently found by Stampfl et al. (1992) to arise from substitution of the surface layer AI atoms with K. Other curious adsorbate-driven reconstructions abound. A rather noteworthy example occurs on the (001) surface of Ni and possibly Cu as well. This reconstruction has a pinwheel character and was first described by Onuferko et al. (1979) for the case of Ni( 100)-C. The structure was identified principally through its symmetry which is p4g, and is depicted in Fig. 13.21. This system, as well as N i ( 0 0 1 ) - O and Ni(001)-S, have been investigated by means of lattice dynamics by Rahman and Ibach (1985), who find that the observed reconstruction can be attributed, not to forces exerted directly by the C adlayer, but to an indirect effect, the weakening of the interactions between Ni atoms in the first and second layer of the crystal. Ying (1986) subsequently pointed out that C-Ni interactions may contribute significantly to the reconstruction - - thus allowing the assumption of a smaller reduction of the interlayer Ni force constants - - if one includes the interactions beyond first neighbor and if the adsorbate layer has (the observed) c(2x2) symme-

764

L.D. Roeh?]:~

Fig. 13.20. The H-induced top-layer-shift reconstruction of W(110) as determined by Chung et al. (1986). W atoms are denoted by the large circles, top-layer atoms are shaded and second-layer atoms are white. The H adatoms are the smaller black circles. (a) At 1/2 monolayer the adlayer order is (2xl) and the substrate unreconstructed. The plot on the right shows the binding potential for the H adatoms as a function of position within the long hollows. (b) At 3/4 monolayer the adlayer displays non-primitive (2• order, the top layer of the substrate is shifted relative to underlying layers and the adsorption well for the hydrogen is no longer symmetric and flat-bottomed. try. Although O and S do not induce reconstruction of Ni(001), they similarly alter the force constants of the surface and subsurface layers so as to soften vibrational modes of the same character (see Lehwald et al., 1985). Klink et al (1993) have recently been imaged this phase of N i ( 0 0 1 ) - C via scanning tunneling microscopy, confirmed the structure of Fig. 13.21, and found that the displacement magnitudes are of order 0.55 ]k. Chorkendorff and Rasmussen (1991) have suggested a similar interpretation of an observed H-induced reconstruction of Cu(001 ). The disordering transitions of these adsorbate-driven reconstruction systems have not yet been studied in detail or theoretically modeled.

13.4.2. Reconstruction of semiconductor surfaces Because of the directionality of covalent bonding, the surface reconstructions of semiconducting materials differ considerably in structural character from those typical in metallic systems. One finds large unit cells and often the involvement of several layers, so that the structural complexity can be considerable. The (7x7) reconstruction of the Si(111) surface is the most famous example with its unit cell consisting of 49 atomic sites in the surface layer and extending 4 layers into the surface, so that one has of order 200 atoms nontrivially displaced from bulk


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Fig. 13.21. The C-induced reconstruction of Ni(001). Clean-surface positions of the first layer Ni atoms are the dashed circles and the reconstructed positions are the large solid, but unshaded circles. The C adatoms are denoted by the smaller shaded circles. termination positions. In fact, just solving this surface structure can be said to have preoccupied surface science for the 26 years from the discovery of the periodicity by Farnsworth et al. (1959), shortly after it became possible to prepare and maintain the clean surface, to the final solution of the structure by Takayanagi et al. (1985) using transmission electron diffraction. (Haneman (1987) provides a review of the study of silicon surfaces.) The structure of semiconductor surfaces is covered by Duke in Chapter 6 so that the present section will focus more briefly on what is known concerning the phase transitions of these surfaces. 13.4.2.1. ( I I I ) Surfaces

As noted above, the (111) surface of silicon exhibits a (7x7) reconstruction. Ge(111 ) reconstructs with c(2x8) periodicity; the local structure bears some resemblance to that of S i ( l l l ) 1. Both surfaces return to ( l x l ) periodicity upon sufficient heating, that of Si( 111 ) at Tc = 1130 K (see Bennett and Webb, 1981), and that of Ge(l I 1) at 573 K (see Phaneuf and Webb, 1985). In both cases the third Landau rule (see w 13.1.4.3) predicts a first-order transition unless the Q-vector of the phase moves continuously in the surface Brillouin zone while broadening. (No such m o v e m e n t has been reported; the beams are simply seen to decrease in intensity. See Bennett and Webb (1981), for example.) Nonetheless, the transition in the case of S i ( l l l ) seemed in L E E D measurements to be continuous, so much so that Bennett and

1 Althoughthe surfaces of the compound semiconductors, including GaAs for example, also reconstruct, they will not be covered in this chapter. Duke in Chapter 6 discusses their room temperature equilibrium structures in some detail and their transitions share many features with the elemental surfaces.

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Webb (1981) measured effective critical exponents, while noting that their failure to observe beam broadening and/or critical scattering was inconsistent with the assumption of a continuous transition. The resolution of this apparent paradox came shortly after Bennett and Webb (1981) conducted their LEED study ~ and involved the effect of surface steps. Beautiful experiments by Osakabe et al. (1980) (grazing incidence electron microscopy) and Telieps and Bauer (1985) (low-energy electron microscopy or LEEM), established that step edges induce, or couple favorably to, the reconstruction, so that some (7• order remains on the surface above the Tc for a flat surface and then gradually decreases in extent as T is increased further. Thus the loss of order appears to be continuous unless there are no steps at all. The transition is also found to be reversible with nucleation of (7• order occurring at step edges and then growing across the surface as the temperature is decreased. Thus the transition is first-order, and Landau's third rule has apparently survived the test. The disordering of the G e ( l l l ) reconstruction is also first-order for similar reasons, but in this case, perhaps because of differences in the coupling between the ordered phase and surface steps, that was more immediately apparent in LEED. See Phaneuf and Webb (1985).

13.4.2.2. (001) Surfaces The (001) surfaces of the common semiconductors also reconstruct, most of them by forming rows of dimers on the surface. See Chapter 6 for a picture of the structures. The most significant contribution to the reconstruction energetics comes from the formation of the dimer rows. Because Si has the diamond crystal structure, the (001) surface is 2-fold, not 4-fold symmetric, and the dimer rows form only in one direction for a given surface plane. (For the next plane down or up, however, which occur if there are steps on the surface, the dimerized rows will run in the perpendicular directions so that most experimental studies see both directions.) Chapter 6 gives more of the structural details. Based on the fact that on a given terrace this (2• phase has two possible ground states, a single component order parameter analogous to the magnetization of the Ising model suffices to describe it in the infinite system. Therefore, if the transition is continuous, we expect it to proceed via an Ising-type transition with the associated critical exponents. (Of course any transition can also turn out to be first order according to the Landau rules; see w 13.1.4.3.) This system, however, contained several further surprises, which illustrate nicely how the complexities of real systems can undermine the usefulness of simple phase transition models. First, lower temperature studies have revealed that further ordering can occur from the (2• The literature on this point is complicated, Haneman (1987) offers a summary, but low temperature scanning-tunneling-microscopy studies (see Wolkow, 1992), establish that the lowest energy phase of the surface actually has c(4• symmetry. It seems that the dimers buckle, i.e. one atom of the dimer moves

i In fact, the observations of Osakabe et al. (1980), to be discussed following, are mentioned by Bennett and Webb in a 'Note added in proof'.


767

in the +z-direction and other in the-z-direction, and they interact with one another to order into a structure in which the dimer angle alternates. Saxena et al. (1985) analyzed the implications for the phase transitions of this reconstruction. Despite the seeming complexity of the unit cell of this reconstruction, there are again actually only two ground states, for a given choice of (2• ground state. Thus from the symmetry point of view, we can again expect an Ising transition between the c(4x2) and (2xl) phases. The (2xl) phase should then be considered as a disordered version of the c(4• phase with some of the dimer buckled in one orientation and the rest in the other, but with no long-range order in those orientations. The further disordering of the (already partially disordered) (2x 1) to a (1 x 1) surface would still be classified as an Ising transition and so one might wish to measure the critical exponents. However, there is still another complication. Recent discoveries indicate that flat Si(001) surfaces should not be taken for granted. It was noted above that on successive planes of the (001) surface the ( 2 x l ) reconstruction forms in perpendicular directions. Alerhand et al. (1988) first pointed out on theoretical grounds that steps on the surface, by allowing the reconstruction to occur in the two different directions, would work to relieve the anisotropic surface stress produced by the uniaxial reconstruction. Thus the surface if prepared without miscut would be expected to rearrange itself into a regular array of up and down steps with reconstruction in alternating directions on adjacent terraces. An elegant experimental study of the effect of uniaxial strain applied to surfaces of single crystal Si on the population of reconstruction directions by Men et al. (1988) confirmed the basic idea of the coupling between surface stress and direction of reconstruction. However, no study has succeeded in observing the generation of steps on previously flat surfaces. This failure has recently been overcome by simultaneous theoretical, Tersoff and Pehlke (1992), and experimental, Tromp and Reuter (1992), work showing that the stress may also be relieved, and with lesser kinetic hindrance than imposed via creation of new steps, on nearly flat surfaces through the development of long wavelength undulations in the few existing steps on the surface. What effect does this introduction of the third dimension have on reconstruction phase transitions? The 2-d models that lack the new degrees of freedom associated with 3-d become inadequate to the full description of equilibrium. However, it should be noted that the widths of terraces between the wavy steps seen by Tromp and Reuter are in the 1000-2000 A range. Thus one could still find ordering of the (2x l) over regions that large and perhaps one could still have rather sharp disordering transitions limited only by rather insignificant finite size effects. It seems quite certain that this reasoning would apply to the disordering of the c(4x2) phase into the (2x l) and temperatures well below room temperature. However, that reasoning may not apply for the disordering transition of the (2xl) phase, since that occurs at higher temperature where the character of the undulations may have changed, but more particularly because of the coupling between the step excitation and the direction of the (2x l) ordering. This last point deserves further experimental and theoretical investigation.

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13.4.3. Adsorption effects on reconstruction Combining reconstruction and adsorption leads to interesting possible behaviors that go beyond those encompassed by the simple models that work well for these phenomena separately. One might then reasonably ask why one wants to study these more complicated situations. The answer is probably that adsorption represents one way of controllably perturbing a reconstruction system, an entree that can help determine the nature of complicated reconstructions. In other cases, one is more interested in the binding and reactive properties of adsorption on a given surface, and that surface just happens to reconstruct. (This seems to be commonly the case for adsorption on semiconducting surfaces.) From either viewpoint, many cases have now been investigated and we consider briefly in the following the effect of adsorption on reconstruction of some metallic and semiconducting surfaces. 13.4.3.1. Metallic surfaces The effect of adsorption on metallic reconstructions is highly dependent on the nature of the reconstruction. A few examples will serve to illustrate. A fairly complete theory for the effect of adsorption on the displacive systems of w 13.4.1.2 has been given by Roelofs et al. (1986). In these cases, the effect of adsorption upon the reconstruction is determined in large part by the binding site of the adsorbate, irrespective of the details of the interaction between the adsorbate and the nearby displaced substrate atoms. (The theory does not cover the case of long-range mechanisms as in the charge donation model for adsorbate-induced missing-row reconstructions, as described in w 13.4.1.4.) If the adsorption occurs into the atop site (labelled 'A' in Fig. 13.22) there is little effect on reconstruction in this case the principle interaction is with a single substrate atom. If the adatom binds in the centered site ('C' in Fig. 13.22), as for O or N on W(001), adsorption tends to oppose the reconstruction, since whatever the interaction between adsor-

Fig. 13.22. The effect of adsorption on the W(001) reconstruction. 'A' and 'C' denote sites for atop and centered adsorption respectively. 'B~' and 'B2' denote bridge sites rendered inequivalent by the reconstruction. Adsorption into either results in a rotation of the direction of displacements to lie along the axes of the surface as indicated by the two top-layer atoms adjacent to site B 1.


769

Fig. 13.23. Structure of 0.5 monolayer of H on W(001 ) as determined by Griffiths et al. (1981 ). The clean surface displacements (surface-layer W atoms are denoted by large unshaded circles) rotate around to be parallel to thex-axis, because of their attraction to the adsorbed H atoms (small black circles). The corresponding structure with displacements in the y-direction is degenerate.

bates is, it will in most cases favor equal displacements of the four nearest surfacelayer atoms, either toward, or away from the binding position. Since the reconstruction puts these four atoms at different spacing from the centered site, the two forces are bound to oppose one another. Thus, both O and N are observed to lift the W(001 ) reconstruction at relatively low coverages. For bridge-site binding, one has the opposite situation; reconstruction is enhanced by adsorption (references given in Roelofs et al. (1986)). Hydrogen is prototypical of this case and has been observed to have two important effects on the reconstruction: the reconstruction is enhanced, i.e. the disordering transition temperature is increased with increasing H c o v e r a g e - the full phase diagram is given by Barker and Estrup (1981); and the direction of the displacements is altered to be along the axes as shown in Fig. 13.23 (see Griffiths et al., 1981 ). This occurs because in bridge site adsorption the adatom interacts most significantly with two substrate atoms. Whether that interaction results in a net attractive force component in the surface plane (adsorption site B~ in Fig. 13.22) or a net repulsive force ~ (site B2) the reconstruction is favored by adsorption. Since alkali adsorption can induce the missing row reconstruction on some otherwise unreconstructed fcc(110) noble metals, it is interesting to consider their effect on those surfaces that spontaneously reconstruction. H~iberle et al. (1989) have studied the effect of K or Cs adsorption on the (1• reconstructed Au(110) surface. They find that adsorption at low coverage alters the reconstruction to (1• with 3 rows missing, one in the second layer, between successive remaining first-layer rows. At higher coverages alkali adsorption lifts the multilayer reconstruction. H~iberle, and Gustafsson (1988), for example, have investigated 1/2 monolayer of K on

1 A net repulsive force component in the surface plane acting on the neighboring surface layer atoms is possible if there is significant interaction between the adatom and the second layer substrate atoms beneath the bridge site.

770

L.D. Roeh?]:~

Au(110) using medium energy ion scattering (MEIS) and find that the equilibrium structure, which displays c(2x2) symmetry, consists of a flat (110) surface whose top layer is a K/Au alloy of c(2x2) periodicity I. The explanation for the change in behavior is that because they have donated charge to the surface, the K adatoms, which adsorb in the missing-row troughs at low coverage, repel one another strongly. When they are forced by increasing coverage into close proximity, it is advantageous for the structure to be rearranged such that Au atoms screen the K adatoms from one another; hence the alloy top-layer. The structure seen in MEIS is consistent with a first principles total energy calculation by Ho et al. (1989). Effective medium theory (see Jacobsen and NCrskov, 1988) appears to incorrectly favor a differing structure, a c(2x2) K layer atop a flat Au(110) surface. 13.4.3.2. Semiconductor surfaces Semiconducting surfaces are more reactive than most metals because of the localized and dangling orbitals (broken bonds) found in their surfaces. Therefore in some cases adsorption of say 02 on Si surfaces will lead not to subtle influences on the reconstructions, but rather to the growth of reacted areas on the surfaces, in this case forming a thin film of SIO2. Rather large doses of H2, on the other hand, are required to affect the reconstructions of the (001) and (111) surfaces, the former being eventually undone at a coverage of 0 = 2, while the (7x7) reconstruction retains its periodicity up to saturation coverage. (See Haneman (1988) for references.) The adsorption of metals on semiconducting surfaces often leads to rather complicated phase diagrams in which further study is needed to determine whether the observed order is due to adsorption or reconstruction degrees of freedom or both. See for example the rather complete study of Si(l I l ) - A u by Feidenhans'! et al. (1990) and the more recent report of further complexities of Shibata et al. (1992). For several other metallic adsorbates on the (7x7), e.g. Pb (see Ganz et al. (1991) and Ag (Ding et al. (1991) list experimental references and give a theoretical account ), one finds ('4-5x4-5-)R30 order at coverages around 1/3 monolayer. This phase may coexist with the (7x7) reconstruction elsewhere on the surface as occurs for example in the system Si(ll l ) - G a (see Zegenhagen et al., 1989). Likewise adsorption of Ag on G e ( l l l ) , at least at elevated temperatures, gives _ p h a s e diagram with coexistence between its reconstructed phase and a (',/-3-x'43)R30 overlayer. See Busch and Henzler (1990). In these cases the (7x7) reconstruction is lifted by the adlayer and the q3- order is primarily in the adlayer. In other cases metallic adsorption, rather than undoing the reconstruction, induces one of completely different character from that of the clean surface. In the system S(111)-Na, for example, a 3• pattern is seen to coexist with the (7x7) reconstruction and was found via tunneling microscopy by Jeon et al. (1992) to be of missing row character. The only generalization to make is that generalization is difficult, although there are similarities (noted by Dev et al., 1988) between metallic adsorbates in the same families on the periodic table.

1 The K atoms being larger protrude further from the surface than the Au atoms in the topmost plane.


771

13.4.4. Diffusion in reconstruction systems

The necessity of long-range mass motion in nondisplacive reconstruction has resulted in continuing interest in the problem of self-diffusion on reconstruction surfaces. Consider, for example, the (2xl) "missing-row" phase induced on Cu(110) by a half monolayer of oxygen (see w 13.4.1.4), whose reconstruction mechanism was elucidated by Coulman et al. (1990) using scanning tunneling microscopy. When oxygen is added to clean C u ( l l 0 ) the phase actually forms by means of 'evaporation' of Cu atoms from terrace edges. These atoms then diffuse across the surface until they, together with the oxygen adatoms, find a growing "island" of the (2x l) phase and form the rows of which the structure consists. (Thus Coulman et al. (1990) correctly rename the reconstruction to be of the "added-row" type.) Other cases where diffusion plays a critical role in surface reconstruction will be discussed below. The study of diffusion on surfaces has a long history ~ and full coverage would be out-of-place in this context. However, the reader interested in exploring the fundamental bases of the field is referred to the classical treatment of stochastic problems by Chandrasekhar (1943), to Bonzel (1975) who introduces the application to surfaces and to Banavar et al. (1981), who present a full theoretical account. Atomic-scale investigations of diffusion have historically been most readily conducted via Field Ion Microscopy (see Chapter 8); Ehrlich (1994) gives a useful history and summary. For the clean missing row reconstruction systems, gradual ordering is observed from a variety of starting situations at or slightly above room temperature; see, for example, Ferrer and Bonzel (1982). To investigate this perhaps surprising degree of mobility, the same total cohesive energy methods which were used by, for example Garofalo et al. (1987) and Roelofs et al. (1990), to elucidate the driving force of the transition and reproduce the phase transition behavior, can be extended to investigate the energy barriers for simple diffusive moves of Au atoms on otherwise perfect Au(1 10) surfaces, both flat and reconstructed. Figure 13.24 shows examples of atomic moves that might be significant on a flat surface. Note that some of the 'moves' are c o n c e r t e d 2 and involve both the Au 'adatom' and an atom within the surface layer. To investigate diffusion barriers one calculates the total cohesive energy as the atoms are moved in small steps along a lowest-energy pathway (determined by minimizing the energy, allowing all other atomic positions to relax in response to the moving adatom). Not surprisingly, one finds, using the Embedded Atom Method (see Daw, 1989) to model the total cohesive energy, that

I Properlygoing back all the way to Einstein's theory of Brownian motion. 2 Concertedmovementsare significant in other surface diffusion situations. Feibelman(1990) determined, via total electronic energy calculations, that for AI(001)-AI concerted movements involve an energetic barrier about 1/3 as high as simple single atom moves. Experimental support for the significance of concerted movements is also available in some systems. Tsong and Chen (1991) present evidence for Ir(001)-Ir and Ir(ll0)-lr; Kellogg (1992) shows that concerted moves are also important in the non-homogeneous situation of Pt adatoms on various surfaces of Ni.

772

L.D. Roelo.[:~

Fig. 13.24. Some basic diffusion moves on an fcc(l 10) surface. The moves on the right side of the figure are direct; those on the left are concerted.

single atom diffusion along the rows is relatively free; the barrier height or activation energy is Eact = 0.27 eV. (These results are reported in Roelofs et al. (1991).) For single atom diffusion across the row, on the other hand, one finds Eact = 1.16 eV, a barrier that would imply very slow diffusion in that direction around room temperature (kTroom---0.025 eV). Consideration of the concerted mode of movement across the rows lowers the barrier to Eac t - - 0 . 3 5 eW providing a pathway which circumvents the direct limitation and allows for rapid motion in that direction as well. In light of these calculations, the spontaneous development of good reconstructed order at temperatures around 400 K is not surprising. One expects that similar situations obtain in the other systems that reconstruct in this fashion. These results may also provide some insight into homogeneous diffusion in nonreconstructing surfaces as well, where the process must play an important role in the annealing process, etc. Diffusion also plays a key role in other surface kinetic phenomena which are the subject of the next section.

13.5. Ordering kinetics at surfaces This chapter has presented a summary of equilibrium phase transition behavior at surfaces, but has not yet considered the time development of order, a nonequilibrium phenomena. In this section we present the theoretical picture that has been developed using approximate analytic approaches and the confirmation of those ideas via simulation studies of lattice gas models and experimental investigation of chemisorption systems. It would also be well to note that kinetic phenomena pervade surface science and in their breadth deserve eventually more complete treatment, perhaps a full volume


773

later in this series. Because they are not intrinsically related to phase ordering, I have chosen not to cover some important aspects of surface kinetics including: adsorption and desorption kinetics, diffusion (but see w 13.4.4), reaction kinetics and oscillation, and the kinetics of epitaxial growth.

13.5.1. Theoretical introduction We consider the following situation. A system I possessing at least two distinct phases is allowed to come to equilibrium in one of its two phases. Then we abruptly alter the external conditions to be consistent with the other (or another) phase and observe the time development of order in the new phase. We expect the system, initially far from the new equilibrium to move toward it, but how do we characterize this development, how is it initiated, at what rate does it proceed, what universal patterns of development can be identified and what key variables determine which pattern is followed? We will pursue these questions in the following subsections, illustrating with some recent simulation results 2 for a particular model system and verifying the expected results with experimental and other simulation-based studies in w 13.5.2. Our illustrative system is a 2-d square lattice gas model whose interactions are suitable for the formation of a p(2x2) phase: El --->+oo (nearest-neighbor exclusion); E2 = 100 K; E~ = - 3 0 0 K; and E4 = 50 K. (All energies are given in the form of temperature equivalents. These interaction energies were not chosen to accurately model any particular system, but the resulting phase diagram is similar to that of Ni(001)-Se. See Bak et al. (1985).) The (T,~t) and (T,0) phase diagrams of this model are shown in Fig. 13.25. In later subsections we will discuss the development of p(2x2) order in this model under two different sets of conditions meant to simulate the typical experimental situation in, respectively, physisorption systems and chemisorption systems (see w 13.2.3). In both cases one begins with the system in equilibrium under one set of conditions, and then effectively instantaneously alters one system parameter or another in order to establish an initial out-of-equilibrium state. The experiment commences at that time, and one monitors the approach of the system to equilibrium at the new conditions. Specifically, for a system of physisorption character, imagine beginning with a clean substrate being held at a temperature at which p(2• order can be stable, and introducing to the chamber at time t = 0 the adsorbate, in gaseous form, at constant pressure which serves to fix the chemical potential ~t, at a value conducive to p(2x2) order. Thus one is instantaneously changing the experimental condition as shown via the bold arrow in Fig. 13.25a and one can follow the subsequent development

1 We assume the system is very large and discuss in w 13.5.2 the effects of size limitations. 2 The illustrative calculations -- both equilibrium (the phase diagram) and kinetic - - included in this section are original research doneby Jennifer Blue and the author, and have not been published elsewhere. Support via a CUR (Council on Undergraduate Research) Undergraduate Fellowship is gratefully acknowledged.

L.D. Roelof,;

774

400 disorder

C(2X2)

300 O0

9

[-200 p(2x2)

0

100

dv

_

0

Orv

-800

-600

-400

-200

0

l.t [K]

(a) 400

300

200-~

j ~v i ,,I ,2x,,i i i.. i i !

100

0

I

.~

0.0

0'.1

0.2

0.3

0.4

0.5

(b) Fig. 13.25. Phase diagrams for the lattice gas model used to simulate the kinetics of p(2• phase development, as determined by Monte Carlo simulation on a (48: Tc respectively.

795


where S is the entropy, M is the magnetization and Ms is the staggered magnetization. E, the internal energy (i.e. the energy not including the contribution due to the external field h in Eq. (13.1)), is a function of S, M and Ms. The phase diagram imparts the information that for h > 0 we expect the system to display long-range order with positive (spin-up) M, and for h < 0, negative M. The interest centers on the plane at h = 0, in which neither direction is favored. For large values of T or Ihsl the spins are disordered, but within the region bounded by the solid curve in Fig. 13.B. 1, symmetry is broken and the system will choose to display long-range order with either positive or negative magnetization. The boundary of this region is a "seam" where the two phases merge to become one and this is where interesting behavior might be expected. One might imagine investigating this system via scans A, B and C, in which T is held constant while the field h is set initially to a negative value and then increased through 0, stopping on the positive side. If one measured M vs. h along these scans a discontinuity at h = 0 should be seen for scan A and a smooth and gradual increase of M with h for scan C. Precisely what happens for scan B which passes directly through the seam is a more subtle matter and must be settled by calculation or measurement. It is instructive to consider the application of mean field theory. Since, formally at least, the experimentalist has control not of the extensive densities, S, M and M,, but rather of the fields, T, h, hs we must base our considerations on the (magnetic) Gibbs free energy (hereafter simply "free energy") (B.2)

G( T, h, h~) = E - T S - h M - h,M~

(For the relation of G to the partition function see Eq. (13.2).) The equilibrium phase of the system at a particular point in (T, h, hs) space is the one with the lowest value of G at that point, so one can determine the phase diagram from G. The field h, has been included in the treatment in order to characterize the effect of so-called irrelevant fields. That matter will be dealt with below; for the moment we will set h, = 0 for convenience and focus on G(T, h).

B. 1. Mean field theory Mean field theory applied to the nearest neighbor Ising model, see Brout (1965) for example, gives M = tanh

I

4JM + h ]

(B.3)

kBT

an equation that has one, two or three solutions for kBT < 4J, and just one for kBT > 4J. In the former case we need to evaluate G for each of the solutions to determine which represents the equilibrium phase. Using Eq. (B.2) one can express G in terms of M, T and h as

N

--2JM 2

hM + kBT

2

log

2

+

2

log

2

(B.4)

L.D. Roeh?]:~

796

M Curve A B C

T / Tc 0.875 1.000 1.125

! "

9

I

0.25

-0

0.25

-0.5

-

.

4

Fig. 13.B.2. Mean field results for the Ising ferromagnet. The upper panel displays the equilibrium magnetization as a function of h for the isotherms A, B, and C of Fig. 13.B.1. The lower panel display the free energy per spin g = G/N, normalized by kT vs. h for the same three isotherms. See text for further discussion.

Equation (B.3) cannot be solved analytically, but it is straightforward to do so numerically and Fig. 13.B.2 shows the solutions plotted vs. h for three values of T corresponding to the three isotherms in Fig. B. 1. The values of M can be substituted into Eq. (B.4) and the lower panel of Fig. B.2 shows the variation of G for the same three scans as well. One sees that the multiple values of M for T < 4J k---B-'which we shall shortly identify as the Tc for this transition, result in G having a triangular-shaped loop (curve A). At any given ( T , h ) the lowest value of G identifies the equilibrium phase and these values have been indicated with the bold curves. (The continuations of those curves beyond the point at which they represent the equilibrium phase describe metastable phases, leading to the possibility of hysteresis along scan A. The top segment of the loop comprises fully unstable solutions, which are completely unrealizable given their upward curvature which indicates negative susceptibility, see Eq. (13.33a).) The values of M corresponding to the equilibrium phases of G have also been drawn in bold style, and one sees that the slope discontinuity in G at h = 0 corresponds to

Phasetransitionsandkineticsof ordering

797

a magnetization discontinuity, M = M• as expected on the basis of Eq. (13.3a), as the system makes the transition from spin-down order to spin-up order. For T > Tc there is a single solution for all h and no discontinuity in M. Finally, right at Tc we must obviously have singular behavior, in both the h and T directions, as the loop in G shrinks to nothing and disappears. We can display aspects of this symmetry by considering the ~,arious derivatives of G near the critical point. First, one can establish how the M• approach 0 at Tc for h = 0. For this we examine Eq. (B.3) for h = 0 and small M, expanding the hyperbolic tangent to obtain

l (4JM13, M - 4JMkBT--3[k.T

+""

(B.5)

4J whose solutions to lowest order for T < kB are 0 (B.6)

The M - 0 solution is unstable as previously noted and the other two are the equilibrium possibilities previously represented as M• We see that M• go to 0 at Tc 4J = - - , so that our identification of the critical temperature above was correct. kB Moreover, using the standard definition, Eq. (13.16), of t, the reduced temperature we can reexpress the M• as M•

+t)3~43/ll+t-l) (B.7) = + ~f31tl (1 + t + . . )

Thus, the leading singular ~behavior as T---) Tc from below corresponds to a critical exponent 13 of 1/2. See Eq. (13.19) for the definition of 13. One can also readily examine the dependence of M on h at the critical temperature. If we set kBT = 4J in Eq. (B.3) we obtain M - tanh(M + ~)

(B.8a)

where

1 One properly considers this behavior to be singular because although M+do not diverge anywhere, their derivatives with respect to T diverge as Tc is approached from below.

L.D. Roeh?]:~

798

-

h

h =4J

(B.8b)

is a dimensionless field variable. For small M and h, which should apply near the critical point, we can expand the tanh in Eq. (B.8a) to obtain 1

M = (M + T/+ -;- (M

+

~)3 +

9 04

(B.9)

J

so that M = (3 h)I/3

_

_

h + less singular terms

(B.IO)

For this we deduce that mean field theory predicts a value of 3 for the critical exponent 5 (see Eq. (13.21 ). It is worth stopping at this point to emphasize that we these mean field results are being presented because they display the necessity of singular behavior at a phase transition, not because they have any quantitative significance. Note that in this treatment of the Ising model Tc is off (in comparison to Onsager' s exact results) by about 80% (see Eq. (13.4)); and (see Table 13.1 for the exact values) that mean field gives a value for 13that is 4 times too large and for 8 a value that is 5 times too small. Mean field theory also in some cases, including for example the 3-state Potts model, predicts the wrong transition order. Note further that identical predictions for the critical exponents can be derived from Landau theory (see w 13.1.4.2) so that we can identify Landau theory as another sort of mean field theory. The failure of mean field is due to the fact that it ignores correlations and fluctuations. These related concepts can be explained as follows. Correlation refers to the influence of neighboring spins on one another, i.e. if a given spin points upward, its neighbors are more likely than not to do so due the ferromagnetic nearest neighbor interaction. In the mean field treatment each spin interacts with the average of all other spins rather than with its actual neighbors which are more likely than average to be similarly oriented. One can improve the treatment of correlations by basing the mean field calculation on a cluster of sites, rather than a single site. This result in better accuracy for To, but does not change the calculated critical exponents, because of the role played by fluctuations in second order phase transitions. To understand the role played by fluctuations we recall that the correlation length, ~, characterizes the length over which order persists (see w 13.3.1 and Eq. (13.24) for the definition of ~). ~ therefore determines the size of the cluster that one would have to treat accurately in order to correctly predict system behavior. Unfortunately ~ is found to diverge near T~ (see Eq. (13.23)), and this has two important ramifications. First, mean field theory cannot be successful close to Tc, since no finite cluster can encompass the divergence of ~. Secondly, one realizes that the nature of the equilibrium state near Tc must be quite complex. Consider the situation for T close to but slightly less than Tc. We have a small, but nonzero value of M and a nearly divergent correlation length. The fact that the value of M is small means that there are nearly as many spins in the minority direction as in the majority


799

direction, even though the large value of ~ implies that a given spin influences a vast number of other spins. These apparently contradictory considerations can be synthesized via a conception of the equilibrium state consisting of large, coherent regions of the minority spin direction imbedded within the system which has overall more of the majority oriented spins. Thus if one did a local average ML (i.e. the average value of M within a region of linear size L), the value of M t would be found to fluctuate as one moved about the sample. This observation gives insight into the failure of the Landau version mean field theory. The variation of ML with position suggests that near Tc at least, we should regard the free energy as a function not only of M, but also of VM, the gradient of the order parameter. Treating correlations and fluctuations together presented a theoretical impasse that was not overcome until the development in the '60s and '70s of the idea of scaling and the practical calculation technique of renormalization which grew out of it.

B.2. Scaling The basic idea of scaling arises from the observation that if ~ is the physically important length scale and if it diverges near a critical point, then all other length scales become unimportant. Thus the free energy per site, which formally is a function of several variables G g = N g(t,h,h,)

(B. 1 1)

really somehow depends only on one variable, the correlation length, ~, which itself will be a function of the variables, t, h, etc., which determine how far the system is from its critical point ~. (Note that we express the temperature in terms of the variable t in this analysis to indicate that what matters is the distance from the critical point.) A function which has this property (imposed by nature in this case) is known to mathematicians as a homogeneous function 2. and may only depend on its arguments in the particular fashion flXPx, Xqy, ~,rz.... ) = ~,f(x,y, z .... )

(B.12)

The sense of Eq. (B. 12) applied to the critical free energy is that many combinations of t, h and h, result in the same value of ~ and therefore for those situations the free energy can differ only by the scale factor X. The terminology scaling is meant to imply that a rescaling of the variables implies a particular rescaling of the free energy. The seemingly innocuous Eq. (B.12) has an impressive array of consequences, some of which we detail following.

1 It is conventional to resolve g into regular (analytic) and singular contributions g = gr + gs. The following discussion applies only to the singular part of g. 2 For a good discussion of homogeneous functions and their connection to critical phenomena and a demonstration of Eq. (A. 12) see Chapter 10 of Reichl (1980).

800

L.D. Roeh?]:~

Kadanoff (1966) introduced the notion of b l o c k r e n o r m a l i z a t i o n . Suppose one takes the d-dimensional t lattice and divided it into blocks of linear dimension L, so that each block contains L a sites or spins. A new variable that describes the magnetization of the Ith block can be defined 1

s, = F 2 s,

(B.

iel

St will take on a range of values between +1. Moreover, we could think of determining a new effective Hamiltonian written in terms of the S/'s that would replace Eq. (13.1), but give the same physics, since it describes the same system. This Hamiltonian can be argued to have the same form as Eq. (13.1 ), except that we would have to use rescaled temperatures and field strengths, tL, hL, etc., to get the same degree of order. Given that it arises from a Hamiltonian of the same form, the free energy for a lattice of S/'s, g(tL, he .... ) will have the same functional form as does g(t, h .... ) and since there are fewer blocks than sites by a factor of L a the per block and per site free energies we have g(t/, hL .... ) = L a g(t, h .... )

(B. 14)

Thus we have argued that the free energy does satisfy the defining relationship for a homogeneous function and we can expect variable relationship as given in Eq. (B. 12), which we choose to write using conventional definitions of the exponents tL = T L:'

(B. 15a)

hr. = h Ly'

(B. 15b)

h,.L = h, Ly,

(B. 15c)

The exponents y, and Yh, whose values for several models are given in Table 13.1, thus characterize the rescaling of t and h necessary to preserve the free energy if the length on which the system is investigated is changed by a factor of L. Their values are related to those of the conventional critical exponents defined in Eqs. ( 1 3 . 1 8 24) and that connection will be made shortly. However, let us first deal with the third field h., and its exponent Yi. Note that if the exponent associated with a particular field is negative, as turns out to happen for h, in the case of the Ising model, then as one considers the system on ever larger length scales, the rescaled value, hs.L, grows ever smaller. That particular field must then be irrelevant to the behavior of the system near its critical point where we can go to very large L. Thus the critical behavior of the Ising model

1 For the case of surfaces d = 2, of course, but scaling theory applies to systems of other spatial dimension as well, so we will present this analysis for general d.


801

is therefore independent of the value of hs, and we will find the same critical exponents whereever we pass through the phase transition line in Fig. 13.B. 1. We term any field whose exponent is negative to be an i r r e l e v a n t field or variable. Changing the value of an irrelevant field experimentally or choosing systems in the same class, but having different values of the various definable irrelevant fields does not change the critical behavior. This is one of the aspects of the idea of u n i v e r s a l i t y defined in w 13.3.2. Irrelevant variables abound in surface phase transitions. The most common example is the coverage in the case of the order-disorder transition of s phase of greater than ( l x l ) periodicity as shown in Fig. 13.6. Having dealt with h, we will no longer include it in our formalism, but before returning to Yt and y;,, note that there is an interesting borderline case to worry about. Suppose the exponent Yl, associated with some v a r i a b l e f i s 0. Then the analogue of Eqs. (B. 15) would give a field that neither grows or diminishes as L increases. Such a field is termed m a r g i n a l and the presence of marginal fields leads to considerable complexity of behavior, including continuous variation of critical exponent values as is seen for example in the XY model (see Table 13.1). Let us return to thermal and magnetic exponents, Yt and Yh. The analysis which lead to Eq. (B. 14) can be repeated for the correlation length, which must rescale by a factor of L under the introduction of blocks CL(tL, hr.) = L-' r

h)

(B. 16)

If h = 0 we expect, according to Eq. (13.23) that ~ ~ Itl-v. Thus Eq. (B. 16) implies itl-~

= (It L:',I)-"

(B. 17)

which can only hold for general L if y , - v -!

(B. 18)

Next let us consider the magnetization, M = - ~)---~gWe rewrite Eq. (B. 14) using Eqs. (B. 15) in the form Oh g(t L",, h L~',)_ = L 't g(t, h)

(B.19)

and differentiate on both sides with respect to h to find M ( t L y,, h L:",)L:", = L a M(t, h)

(B.20)

Then, using the expected form from Eq. ( 1 3 . 1 9 ) , m ~ ( - t ) 13 we have for t < 0 and h =0 ( - t ) ~ L ~:',+:',, = L';(-t) ~

which only holds for arbitrary L if

(B.21 )

802

L.D. Roelofs

d

13= ~

- Yh

(B.22)

Y,

One can similarly derive expressions for other critical exponents in terms of y, and Yh. They are:

d ct = -Y,

(B.23)

2y h - d

7=~

(B.24)

Y, 6=

(B.25)

Yh d-Yh

1"1 = 2 + d - Yh

(B.26)

These equations are consistent with the values for all exactly solved models given in Table 13.1. Finally we conclude by noting that since the 6 measurable critical exponents can all be expressed in terms of just two scaling exponents, Yt and Yh, it should be possible to find relations between the exponents. Thus one can show: ], = 13(8 - 1)

(B.27)

tx + 213 + 7 = 2

(B.28)

v-

2--0~

(B.29)

d

rl=2-d

8+ 1

(B.30)

Equations (B.27-30) are known as scaling laws and they are also consistent with the exponent values given in Table 13.1, of course.

Appendix C" Finite size effects The genuine nonanalyticities associated with phase transitions are possible only in systems of infinite size. Real systems are finite, often distressingly so due to sample defects, and since methods have not been devised in most cases to solve for the equilibrium properties of infinite systems, theoretical results are often also available only for systems of limited size. Therefore the topic of the effect of finite size


803

considered as a "perturbation" of infinite-system expectations is of importance; this Appendix summarizes that subject briefly and introduces the related topic of finite-size scaling. C. 1. Introduction

Theorists and simulationists are often forced by the inadequacies of their techniques or computational equipment to limit the size of systems considered. One sees, for example, simulations of L• lattice gas models with periodic boundary conditions (pbcs, hereafter) applied, or in the case of transfer matrix studies exact treatment of strips that are infinite in one dimension, but very finite, say 6 or 8 sites wide in the perpendicular direction, again with the application of a periodic condition in that direction. Periodic boundary conditions are a popular choice, not because they are experimentally relevant in most cases, but because they eliminate "special" edge or corner sites, i.e. each site has the same coordination as all the others. Free or fixed boundary conditions and antiperiodic type boundary conditions can also be easily defined and incorporated into calculations, but seldom are because they are in general found to produce larger perturbations of infinite system behavior. These alternatives might, however, be more experimentally relevant, and so they may deserve greater attention that they have received. See, however, Kleban and Flagg (1981) for a simulation study of a lattice gas system with experimentally relevant boundary conditions. The effects of finite size can take quite different forms depending on the details, from rounding the transition and shifting its apparent position, to causing it to be missed entirely ~. This will be discussed in greater detail in w C.3 below. The experimental sources of finite size effects are principally surface defects and impurities if adsorbed layers are under study. These possibilities are probably too numerous and devious to be listed but the principle ones are: surface steps; point lattice defects; surface segregated substrate impurities, and immobilely bound adsorbate impurities or contaminants. Steps obviously interrupt the orderly lattice of sites and interactions, thus producing linear-type defects into the system. Sites adjacent to this defect, because their environment differs from the others contribute differently to the thermodynamic properties of the system. The effect of point defects, either of the substrate or in the form of adsorbed impurities, depends on their mobility and the nature of their interaction with the basic degrees of freedom in terms of which the order parameter is defined as discussed in w C.3. C.2. Length scales

To discuss the effect of these defects on the transitions quantitatively one must identify the associated length scale L. In the case of a surface with steps L would 1 Thisis meant in the sense that an application of a small magnetic field to a ferromagnetic system renders its behavior paramagnetic. The system is magnetized, though in varyingdegrees, in the direction of the applied field at all temperatures and thus cannot display the symmetry breaking associated with a real transition. Also at high temperature the magnetization can diminish, but never vanish entirely.

L.D. Roelo.[i~

804

naturally be the average distance between step edges; in the case of immobile point defects, it would be the typical distance between them, i.e. L --- 1/~t-n--nwhere n is their areal density. Whatever their nature, then the defects can only have an effect when the length scales characterizing the phase transition behavior becomes comparable to L. In the case of a second order transition the important length is ~, the correlation length of the fluctuations (see Eq. (13.24)). The identification of this length scale and its significance leads naturally to the idea of finite-size scaling first proposed by Fisher and Barber (1972). A fuller treatment is given by Barber (1983). The essence of the idea, similar to all scaling ideas, is that if both L and ~ are much larger than the microscopic lattice scale in a phase transition system, that the behavior can be understood in terms of the ratio L/~. Thermodynamic or ordering functions that in general depend on both L and T, can near the transitions be accurately approximated by scaling expressions. For example the free energy could be written G(L,t) = L d g(L/Itl -v)

(C. 1)

where v is the usual correlation length exponent of Eq. (13.23). This important idea can be theoretically exploited to determine critical exponent values from calculations based on finite systems and also provides much insight into the effect of finite size in experimental systems. For first-order transitions, which do not have divergent fluctuations, the above discussion needs some modification. Fisher and Berker (1982) have developed a scaling theory appropriate to this case and their treatment is summarized here. First, it is more conventional, in line with the Ising model paradigm, to think of the field variable, h, rather than temperature as being responsible for taking a system through a first-order transition. Thus the scaling form replacing Eq. (C. 1) will couple L and h, G ( L , h ) = L d g(L/Ihl -v',)

(C.2)

(For surface phase transitions, the variable analogous to h is the one that carries the system across a first-order phase boundary, often the chemical potential.) ,~ Secondly, the significant length scale to which system size is compared is not the correlation length, ~, because that does not become large near a first-order transition. Rather, one looks for the length, ~, at which the free energy cost of a fluctuation to the wrong side of the phase boundary equals some (possibly temperature-dependent) constant. One can argue in most cases that the v h of Eq. (C.2) has the value of 1/d as follows. We suppose that at the temperature of interest the order parameter value on the negative h side of the boundary has value M_. If a small positive field h is being applied, then the free energy cost of a region of dimension fluctuating to the negative order parameter side of the phase boundary is A G = hlM_l~ ~

(C.3)

At fixed temperature the probability of such a fluctuation is proportional to AG and therefore the length scale of such fluctuations is

Phase transitions and kinetics orordering

= Ch -1/'1

805

(C.4)

where C is independent of h (but not t). We can therefore expect rounding of a sharp first-order transition when this ~ is comparable to the size L of a finite system. Comparing Eqs. (C.2) and (C.4) suggests that the value of Vh is 1/d, or 1/2 for the systems of interest in surface science. C.3. Finite size effects

In this section the specific effects of finite size on equilibrium phase transition behavior are discussed in more detail. The effects of finite size limitations depend on what specifically is imposing the limit. We consider surface steps first. The least significant effect size limitations can impose is a rounding of the expected singularity. Surface steps are observed to prevent the passage of adatoms from one terrace to another in many systems. In this case the correlation lengths or ~ cannot exceed the terrace size, L. Thus rounding of the transition will set in when the one appropriate to the transition becomes comparable to L. A stronger effect may be seen when the step edges somehow couple to the order parameter. (An example would be the nucleation of the Si( 111 )(7x7) reconstruction at step edges discussed in w 13.4.2.1.) In this case the step edges act like an applied field whose (average) strength is proportional to the step spacing I/L. This applied field will shift the transition from its infinite system location, the shift being proportional to I/L. Rounding will also be observed in these cases. As noted above, point defects may also constitute a size limitation if they are quenched or immobile and if they couple to the order parameter. One might imagine impurities or defects controlling the development of adlayer order in their vicinity. This amounts to a size limitation, since if the coupling is strong, each defect will control the order in its immediate vicinity. This amounts to the application of randomly directed local fields. One expects then to find the transition shifted in various directions depending on where one is on the sample, and therefore a greater degree of rounding of the transition than would occur as expected on the basis of Eqs. (C. 1) or (C.2). C.4. Finite size scaling

One might assume from the previous subsection that finite size effects are always problematic. This is not the case, as especially Eq. (C.I) can be powerfully exploited in calculations. This primarily theoretical endeavor, known as finite size scaling, is briefly sketched here because it is frequently used in modeling surface phase transitions, where one faces the task of determining transition temperatures and critical exponents of proposed models for surface transitions in order to determine their consistency with experiment. The typical uses are as follows. One popular calculational techniques is based on the transfer matrix approach which allows calculation of all thermodynamic and ordering properties for a system of infinite length, but finite width L. In particular

806

L.D. Roelof~

the correlation length ~ is straightforwardly extracted in the transfer matrix approach. The scaling law embodied in Eq. (C. l) suggested that at the transition point, the ratio of L to ~ will be the same for systems of differing size. Hence looking for the temperature T of intersection L

L' E

(C.5)

for strips of different width, L and L', is an efficient way of estimating the infinite system transition temperature. Another popular approach for solving theoretical models is Monte Carlo simulation which produces distributions of system states with the correct equilibrium probabilities, thus allowing extraction of measurable quantities in the form of averages. Monte Carlo simulation can only be applied to finite lattices, say of size L x L , but one wants to accurately determine infinite system results, including the critical exponents and transition temperatures. Let the scaling ratio of Eq. (C. 1) be defined as the variable x x =- L/Itl -v

(C.6)

Then one can show that the dependence of the order parameter, susceptibility and heat capacity of a system of size L, near its critical point are mr. =L-IVvf,,(x) I

(C.7a)

)~" - T Lv/~fx(x)

(C.7b)

CL = L ' ~ f c ( x )

(C.7c)

where f,,(x), f x ( x ) and f c ( x ) are unknown scaling functions. Simulation calculations of me(T), xL(T) and CL(T) ~ are done for a variety of lattices sizes. One e~an then determine T~, v and 13 from Eq. (C.7a) by plotting on the same graph mLL ~/~ vs. x, adjusting Tc (which changes x since the latter depends on the r e d u c e d temperature) and the exponent values, until the runs at different L superimpose. (For an example of this technique in action see Binder and Landau (1980)). Equations (C.7b,c) can be similarly exploited to determine 7 and a (giving at the same time consistency checks for Tc and v). Note that the forms of Eqs. (C.7) also indicate that the maxima in gL(T) and CL(T) associated with the transition depend on L like L "t/vand L '~/" respectively. This allows determination of the exponent ratios y/v and a/v, which together with the scaling relations (Eqs. (B.27-30)) suffice to determine all the exponents.

I Ct~is the heat capacity per site.

Phase transitions and kinetics q]ordering

807

For first-order transitions finite size scaling can also be useful. For example (see Fisher and Berker (1982) and references therein) the maximum values of the susceptibility and the heat capacity in passing through a first-order transition are both expected to vary with system size like L d, thus allowing determination of transition order from finite size data.


Author index Allan, G. 642 Allan, G. s e e Leynaud, M. 646 Allan, G. s e e Lopez, J. 646 Allen, L.R. s e e Conrad, E.H. 358 Allen, S.M. 788 Alloneau, J.M. s e e Thorel, P. 574 Alnot, M. s e e Fargues, D. 569 Als-Nielsen, J. s e e Bohr, J. 568 Als-Nielsen, J. s e e Braslau, A. 358 Als-Nielsen, J. s e e McTague, J.P. 572 Als-Nielsen, J. s e e Pluis, B. 359 Altman, M.S. s e e Robinson, I.K. 135,791 Alverez, M.M. s e e Snyder, E.J. 420 Amer, N.M. s e e Martensson, P. 135,498 Aminpirooz, S. 493 Aminpirooz, S. s e e Becker, L. 494 Aminpirooz, S. s e e Schmalz, A. 500, 648 Ampo, H. s e e Ichninokawa, T. 710 Anazawa, T. s e e Edamoto, K. 225 Ancilotto, F. 133 Andersen, J.N. 493,494 Andersen, J.N. s e e Nielsen, M.M. 499 Andersen, O.K. s e e Nowak, H.J. 135 Anderson, H.L. 418 Anderson, J.A. s e e G6pel, W. 226 Anderson, M.S. s e e Snyder, E.J. 420 Anderson, P.W. 494, 642 Andersson, S. 494 Ando, A. s e e Hirata, A. 226 Andrei, N. 642 Andres, S.R. 358 Andzelm, J. s e e Dunlap, B.I. 181 Angeraud, F. s e e Larher, Y. 572 Angot, T. 567 Angot, T. s e e Sidoumou, M. 574 Anno, K. s e e Kono, S. 134

Aberdam, D. 225 Abraham, F.F. 567 Abraham, F.F. s e e Koch, S.W. 571 Abraham, F.F. s e e Poon, T.W. 98,647 Abraham, K. s e e Dederichs, P.H. 643 Abraham, M. s e e Kroll, C. 226 Abukawa, T. 133,493 Abukawa, T. s e e Kono, S. 134 Achete, C. s e e Niehus, H. 710 Adachi, H. s e e Ellis, D.E. 496 Adachi, H. see Tsukada, M. 183 Adams, D.L. see Aminpirooz, S. 493 Adams, D.L. see Nielsen, M.M. 499 Adams, D.L. s e e Schmalz, A. 648 Adams, D.L. s e e Stampfl, C. 501,792 Adams, J.B. see Liu, C.L. 97 Adams, J.B. see Xu, W. 650 Addato, S.D. s e e Pedio, M. 500 Affleck, I. 642 Agra's't, N. 418 Ahsan, S. s e e Kahn, A. 267 Ai, R. 418 Ai, R. s e e Marks, L.D. 419 Aika, K. s e e Ozaki, A. 500 Aizawa, T. see Itoh, H. 226 Aizawa, T. s e e Souda, R. 227 Akimitsu, J. see Mochrie, S.G.J. 572 Akinci, G. s e e Bak, P. 788 Albano, A.G. s e e Chini, P. 495 Albano, E.V. s e e Miranda, R. 572 Albrecht, T.R. 418 Alder, B.J. s e e Ceperly, D.M. 133 Alerhand, O.L. 96, 265,788 Alerhand, O.L. s e e Vanderbilt, D. 268,710 Alfonso, C. 96 Allan, D.C. s e e Teter, M. 136

809

810

Anton, A.B. s e e Rahman, T.S. 500 Antonik, M.D. 225 Anz, S.J. s e e Snyder, E.J. 420 Aono, M. 225 Aono, M. s e e Katayama, M. 497 Aono, M. s e e Oshima, C. 227 Aono, M. s e e Souda, R. 420 Aono, M. s e e Takami, T. 501 Aono, M. s e e Watanabe, S. 501 Appelbaum, J.A. 642 Applebaum, J.R. 180 Armour, D.G. s e e Verheij, L.K. 501 Arthur, J.R. s e e Cho, A.Y. 266 Artioli, G. s e e Smyth, J.R. 183 Aruga, T. 494 Aspnes, D.E. 265 Astaidi, C. 494 Astaldi, C. s e e Cautero, G. 495 Astaldi, C. see Comicioli, C. 495 Atkins, P.W. 180 Atrei, A. s e e Galeotti, M. 225 Audibcrt, P. 568 Augustyniak, W.M. s e e Martinez, R.E. 97 Aumann, C.E. 265 Aumann, C.E. s e e de Miguel, J.J. 87,710 Aumann, C.E. s e e Saloner, D. 710 Averili, F.W. s e e Ellis, D.E. 496 Avery, N.R. s e e Rahman, T.S. 500 A vouris, P. 265,709 Avouris, P. s e e Hasegawa, Y. 645 A vouris, P. see Lyo, I.-W. 135,498 Avron, J.E. 96 Axiirod, B.M. 568,642 Baberschke, K. s e e D6bler, U. 495 Baca, A.G. s e e Tobin, J.G. 501 Bachelier, V. s e e Courths, R. 643 Bachrach, R.Z. 265 Bachrach, R.Z. s e e Bringans, R.D. 266, 494 Bachrach, R.Z. s e e Olmstead, M.A. 499 Bachrach, R.Z. s e e Uhrberg, R.I.G. 501 Baddorf, A.P. s e e Itchkawitz, B.S. 790 Baddorf, A.P. see Mundenar, J.M. 499 Bader, M. 494 Badcr, S.D. s e e Falicov, L.M. 789 Badt, D. s e e Wilhelmi, G. 501 Badziag, P. 225,265

Author

Baer, D.R. s e e Blanchard, D.L. 180, 225 Baer, D.R. s e e Wang, L.Q. 227 Baerends, E.J. s e e Ellis, D.E. 496 Baetzold, R.C. 180 Bagus, P.S. 494 Bagus, P.S. s e e Bauschlicher, C.W. 494 Bagus, P.S. s e e Hermann, K. 497 Bagus, P.S. s e e Pacchioni, G. 647 Bak, P. 568,788 Bak, W. s e e Corey, E.R. 495 Baker, J.A. s e e Garcia, N. 359 Bakhtizin, R.Z. s e e Park, C. 500 Balibar, S. 96 Bancel, P.A. s e e Keane, D.T. 790 Bancel, P.A. s e e Stephens, P.W. 574 Banerjea, A. s e e Smith, J.R. 98, 136, 648 Bar-Yam, Y. s e e Kaxiras, E. 267 Bfir, M. 494 Baratoff, A. s e e Salvan, F. 500 Barber, M.N. 788 Barber, M.N. s e e Fisher, M.E. 789 Barbieri, A. 225,642 Barbieri, A. s e e Batteas, J.D. 494 Bard, A.J. 418 Bardi, U. 568 Bardi, U. s e e Galeotti, M. 225 Bare, S.R. 494 Bare, S.R. s e e Hofmann, P. 497 Barker, J.A. 642 Barker, R.A. 133,788 Barker, R.A. s e e Felter, T.E. 789 Barnes, C.J. 494 Barnes, C.J. s e e Lindroos, M. 498 Barnes, R.F. 358 Barnett, M.E. s e e Klemperer, O. 359 Bar6, A.M. 642 Baroni, S. 133 Barrett, J.H. 418 Barrett, R.C. s e e Tortonese, M. 420 Barrie, A. 494 Bart, F. 225 Barteit, N.C. 96, 642, 788 Bartelt, N.C. s e e Bak, P. 788 Bartelt, N.C. s e e Hwang, R.Q. 645 Barteit, N.C. s e e .loos, B. 97 Barteit, N.C. s e e Kodiyalam, S. 97 Bartelt, N.C. s e e Ozcomert, J.S. 98

index

Author

index

Bartelt, N.C. s e e Pai, W.W. 98,647 Bartelt, N.C. s e e Phaneuf, R.J. 98 Bartelt, N.C. s e e Roelofs, L.D. 648 Bartelt, N.C. s e e Taylor, D.E. 649 Bartelt, N.C. s e e Wang, X.-S. 98 Bartelt, N.C. s e e Wei, J. 99 Bartelt, N.C. s e e Williams, E.D. 99, 268 Barth, J.V. 133,709 Barth, J.V. s e e Schuster, R. 500, 648,791 Bartos, I. s e e Van Hove, M.A. 136 Barzel, G. s e e Scheffler, M. 500, 648 Baskes, M.I. s e e Daw, M.S. 134, 643 Baskes, M.I. s e e Foiles, S.M. 644 Baskes, M.I. s e e Roelofs, L.D. 648,791 Baski, A.A. s e e Nogami, J. 499 Baski, A.A. s e e Shioda, R. 136, 500 Bassett, D.W. 494, 642 Bassett, W.A. s e e Liu, L.-G. 182 Batchelor, D.R. see Aminpirooz, S. 493 Batchelor, D.R. s e e Schmalz, A. 648 Batra, I.M. s e e Garcia, N. 359 Batra, I.P. 133,494 Batra, I.P. s e e Himpsel, F.J. 267 Batteas, J.D. 494 Baudoing-Savois, R. see Rundgren, J. 227 Bauer, E. 96,418,642, 709, 788 Bauer, E. see Engel, T. 569 Bauer, E. see Kolaczkiewicz, J. 790 Bauer, E. s e e Pinkvos, H. 419 Bauer, E. s e e Telieps, W. 792 Bauer, E. s e e Williams, E.D. 99 Bauer, H.E. s e e Ichimura, S. 226 Bauer, R.S. s e e Bachrach, R.Z. 265 Baumberger, M. 788 Baumberger, M. see Rieder, K.H. 791 B~iumer, M. 225 Bauschlicher, C.W. 494 Bauschlicher, J.C.W. s e e Cox, B.N. 495 Baxter, R.J. 788 Bayer, P. see Muschiol, U. 499 Bayer, P. s e e Wedler, H. 501 Beaume, R. 568 Becher, U. s e e Zeppenfeld, P. 575 Becke, A.D. 642 Becker, L. 494 Becker, L. s e e Aminpirooz, S. 493 Becker, L. s e e Pedio, M. 500

811

Becker, L. s e e Schmalz, A. 500, 648 Becker, R.S. 265,494 Becker, R.S. s e e Kubby, J.A. 267 Beckschulte, M. 418 Bedrossian, P. 133, 494 Bedrossian, P. s e e Zegenhagen, J. 792 BEe, M. 568 Beeby, J. 358 Behm, B.J. 133,494, 642, 788 Behm, R.J. s e e Barth, J.V. 133, 709 Behm, R.J. s e e Brune, H. 495 Behm, R.J. s e e Christmann, K. 643,789 Behm, R.J. s e e Couiman, D.J. 495,789 Behm, R.J. s e e Gritsch, T. 496 Behm, R.J. s e e Hwang, R.Q. 710 Behm, R.J. s e e Imbihl, R. 645 Behm, R.J. s e e Kleinle, G. 498 Behm, R.J. s e e Moritz, W. 646 Behm, R.J. s e e Schuster, R. 500 Behm, R.J. s e e Schuster, R. 648,791 Behm, R.J. s e e Van Hove, M.A. 649 Behm, R.J. s e e Wintterlin, J. 501 Belin, M. s e e Rousset, S. 98 Bellman, A.F. 494 Bellon, R.J. s e e Roelofs, L.D. 648, 791 Benbow, R. s e e Broden, G. 494 Bendt, P. 133 Bennemann, K.H. s e e Dreyss6, H. 643 Bennett, P.A. 788 Bennett, P.A. s e e Robinson, i.K. 43 Bentley, J. s e e Wang, Z.L. 227,421 Berker, A.N. s e e Aierhand, O.L. 265 Berker, A.N. s e e Caflisch, R.G. 568 Berker, A.N. s e e Fisher, M.E. 789 Berlincourt, D.A. s e e Jaffee, H. 419 Bermond, J.M. s e e Alfonso, C. 96 Bermond, J.M. s e e Heyraud, J.C. 97 Bermudez V.M. 180 Bernasek, S.L. s e e Langell, M.A. 226 Berndt, W. s e e Welton-Cook, M.R. 183 Berry, F.J. 180 Berry, S.D. s e e Lind, D.M. 227 Bertei, E. 642 Bertel, E. s e e Bischler, U. 642 Besenbacher, F. 494 Besenbacher, F. s e e Eierdal, L. 495 Besenbacher, F. s e e Feidenhans'l, R. 496

812

Besenbacher, F. s e e Jensen, F. 497,790 Besenbacher, F. s e e Klink, C. 790 Besenbacher, F..Tee Mortensen, K. 499 Besoid, G. 494 Besold, G. s e e Eggeling, von, C. 495 Besold, G..Tee Mtiller, K. 646 Bethge, H. 418 Betts, D.D. 788 Beveridge, D.L..Tee Pople, J.A. 183 Biberian, J.P. s e e Coulomb, J.P. 569 Biberian, J.P. s e e Perdereau, J. 135 Biberian, J.P. s e e Suzanne, J. 574 Biberian, J.P. s e e Van Hove, M.A. 136 Bickcl, N. 225 Bickel, N..Tee Rous, P.J. 360 Biegelsen, D.K. 266 Bienfait, M. 568 Bicnfait, M. s e e Bardi, U. 568 Bienfait, M. s e e Coulomb, J.P. 568,569 B ienfait, M. s e e Gay, J.M. 570 Bicnfait, M. s e e Giachant, A. 570 Bienfait, M. s e e Seguin, J.L. 574 Bicnfait, M. see Suzanne, J. 574 Bienfait, M. s e e Venables, J.A. 575 Bienfait, M. s e e Zeppenfeld, P. 575 Biersack, J.P. 418 Bilalbegovic, G. 96 Bilz, H. s e e Martin, A.J. 182 Binder, K. 642, 788,789 Binder, K. see Kehr, K.W. 790 Binder, K. see Kinzel, W. 646 Binder, K. s e e Selke, W. 648 Binnig, B. 96 Binnig, G. 266,418 Binnig, G. see Ohnesorge, F. 227 Binnig, G.K. 789 Binning, G. see Salvan, F. 500 Binning, G.K. 133 Bird, R.B..Tee Hirschfelder, J.O. 570, 645 Birgeneau, R.J. 568 Birgeneau, R.J. see Dimon, P. 569 Birgeneau, R.J. seeHong, H. 571,790 Birgeneau, R.J. seeHorn, P.M. 571,790 Birgeneau, R.J. s e e Mochrie, S.G.J. 572 Birgeneau, R.J. see Nagler, S.E. 572 Birgeneau, R.J. s e e Robinson, I.K. 135,791 Birgeneau, R.J. s e e Specht, E.D. 574

Author

Birgeneau, R.J. s e e Stephens, P.W. 574 Birkhoff, G. 642 Bischler, U. 642 Bischler, U. s e e Bertel, E. 642 Bisson, C.M. s e e Schwoebel, P.R. 648 Biswas, R. 133 Bjurstrom, M.R. s e e Jin, A.J. 571 Black, J.E. s e e Hall, B. 570 Black, J.E. s e e Rahman, T.S. 500 Blakely, J.M. 96 Blakely, J.M. s e e Keeffe, M.E. 97 B lanchard, D.L. 180, 225 Blanchard, D.L. s e e Conrad, E.H. 358 Blanchin, M.G. s e e Epicier, T. 225 Blank, H. s e e Bienfait, M. 568 Bliznakov, G. s e e Surnev, L. 501 Block, J.D..Tee Robinson, I.K. 135 Bludau, H. 494 Bludau, H. s e e Gierer, M. 496 Bludau, H. s e e Hertel, T. 497 Biudau, H. s e e Over, H. 499, 500 Blugel, S. s e e Kobayashi, K. 134 Blyholder, G. 494 Boato, G. 358 Boato, G. ,Tee Glachant, A. 570 Bobonus, M. s e e Haase, O. 496 Bockel, C. 568 Bockel, C. s e e Dupont-Pavlovsky, N. 569 Bockel, C. s e e Menaucourt, J. 572 Bockel, C. s e e Rdgnier, J. 573 Bf~gh, E. s e e Aminpirooz, S. 493 Br E. s e e Schmalz, A. 648 Bohnen, K.P. 133, 418 Bohnen, K.P. see Chan, C.T. 133 Bohnen, K.P. s e e Ho, K.-M. 134, 790 Bohr, J 266, 568 Bohr, J s e e Braslau, A. 358 Bohr, J s e e D'Amico, K.L. 569 Bohr, J s e e Feidenhans'l, R. 266, 496 Bohr, J s e e Kjaer, K. 571 Bohr, J s e e McTague, J.P. 572 Bohr, J s e e Robinson, I.K. 360 Bol'shov, L.A. 642 Born, M. 358 Bomchil, G. s e e Beaume, R. 568 Bomchil, G. s e e Meehan, P. 572 Bomchii, G. s e e Thorel, P. 574

index

Author

index

Bonevich, J.E. 418 Bonevich, J.E. s e e Marks, L.D. 419 Bonnell, D.A. 225 Bonnell, D.A. s e e Liang, Y. 182, 226 Bonnell, D.A. s e e Rohrer, G.S. 183, 227 Bonnetain, L. s e e Delachaume, J.C. 569 Bonnetain, L. s e e Khatir, Y. 571 Bonzel, H.P. 96, 494, 789 Bonzel, H.P. s e e Breuer, U. 96 Bonzel, H.P. s e e Fetter, S. 496, 789 Bonzel, H.P. s e e Pirug, G. 500 Bonzel, H.P. s e e Wesner, D.A. 501 Bootsma, G.A. 568 Borbonus, M. s e e Koch, R. 498 Bormet, J. 494 Born, M. 180 Bornemann, P. see Engel, T. 569 Bothe, F. see Heinz, K. 497 Bothe, F. s e e Oed, W. 791 Bott, M. 418 Bott, M. s e e Michely, T. 710 B6ttchcr, A. 494 Bouchdoug, M. 568 Bouchet G. see Aberdam, D. 225 Boudart, M. 180 Boudriss, A. 225 Bouldin, C. 568 Bouldin, C.E. s e e Woicik, J.C. 502 Bourdin, J.P. 642 Bouzidi, J. s e e Krim, J. 571 Bowker, M. 642 Bowker, M. s e e Leibsle, F.M. 498 Bowker, M. s e e Murray, P.W. 499 Bozzolo, G. see Rodrigucz, A.M. 135,648 Bozzolo, G. see Smith, J.R. 98,648 Bradshaw, A.M. 494 Bradshaw, A.M. see Bonzel, H.P. 494 Bradshaw, A.M. see Hayden, B.E. 496 Bradshaw. A.M. s e e Hofmann, P. 497 Bradshaw. A.M. s e e Horn, K. 497, 571 Bradshaw. A.M. s e e Persson, B.N.J. 647 Bradshaw, A.M. s e e Pfntir, H. 500 Bradshaw, A.M. s e e Robinson, A.W. 791 Bradshaw A.M. see Rose, K.C. 420 Brand, J.L. 643 Brandes, G.R. see Canter, K.F. 180 Brandes, G.R. see Horsky, T.N. 267

813

Braslau, A. 358 Braun, O.M. 643 Brener, R. 568 Brener, R. s e e Shechter, H. 574 Brener, R. s e e Wang, R. 575 Brenig, W. 643 Brenig, W. s e e Gumhalter, B. 645 Brenig, W. s e e SchOnhammer, K. 648 Brennan, D. 494 Bretz, M. 568, 789 Bretz, M. s e e Campbell, J.H. 568 Bretz, M. s e e Dutta, P. 569 Bretz, M. s e e Quateman, J.H. 573 Breuer, U. 96 Bridge, M.E. 494 Briggs, G.A.D. s e e Burnham, N.A. 418 Bringans, R.D. 266, 494 Bringans, R.D. see Biegelsen, D.K. 266 Bringans, R.D. s e e Olmstead, M.A. 499 Bringans, R.D. see Uhrberg, R.I.G. 136, 501 Brinkman, F. s e e Coppersmith, S.N. 568 Brock, J.D. s e e Robinson, I.K. 791 Brodde, A. s e e B~umer, M. 225 Brodde, A. s e e Wiihelmi, G. 501 Broden, G. 494 Brodskii, A.M. 643 Brodskii, M.I. see Urbakh, A.M. 649 Brommer, K.D. 133 Brook, R.J. 225 Brooks, R.S. s e e Lamble, G.M. 498 Broughton, J.Q. see Brundle, C.R. 494 Brout, R.H. 789 Brown, G.S. s e e Stephens, P.W. 574 Bruch, L.W. 568,643 Bruch, L.W. see Gottlieb, J.M. 645 Bruch, L.W. s e e Klein, J.R. 646 Bruch, L.W. see Unguris, J. 575 Bruckcr, C. s e e Broden, G. 494 Brugger, R.M. s e e Taub, H. 574 Brundle, C.R. 494 Brundle, C.R. s e e Bagus, P.S. 494 Brundle, C.R. s e e Hopster, H. 497 Brune, H. 495 Brune, H. s e e Barth, J.V. 133, 709 Brune, H. s e e Wintterlin, J. 501 Brush, S.G. 789 Buckett, M.I. 225

814 Buckett, M.I. s e e Ai, R. 418 Buckett, M.I. s e e Marks, L.D. 419 Buerger, M.J. 358 Bukshpan, H. s e e Shechter, H. 574 Bullock, E.L. 133 Burandt, B. s e e Claessen, R. 225 Burch, R. 643 Burchhardt, J. s e e Nielsen, M.M. 499 Burchhardt, J. s e e Stampfl, C. 501,792 Burdett, J.K. 643 Burdick, S. s e e EI-Batanouny, M. 134 Burg, B. s e e Koch, R. 498 Btirgler, D. s e e Tarrach, G. 227 Burke, N. 643 Burnham, N.A. 418 Burnham, N.A. s e e Landman, U. 419 Burns, A.R. s e e Jennison, D.R. 645 Burns, G. 43 Burns, M. see Morkoq, H. 227 Bursill, L.A. s e e Smith, D.J. 227 Burt, M.G. 495 Burton, W.K. 709 Busch, H. 789 Busek, P.R. 418 Busing, W.R. 358 Buslaps, T. see Claessen, R. 225 Bustamante, C. 418 Butler, D.M. 568 Bykov, V. see Teplov, S.V. 420 Bylander, D.M. 495 Cabrera, N. see Burton, W.K. 709 Cadoff, 1. s e e Nolder, R. 227 Caflisch, R.G. 568 Cahn, J.W. 96 Cahn, J.W. s e e Allen, S.M. 788 Calisti, S. 568 Calicn, H.B. 96 Calicnfis, A. see LindstrOm, J. 227 Cammarata, R.C. 96 Campargue, R. 358 Campbell, C.T. s e e Over, H. 499 Campbell, D.A. see Eguiluz, A.G. 644 Campbell, I.M. 180 Campbell, J.H. 568 Campuzano, J.C. 495,789 Canter, K.F. 180

Author

index

Canter, K.F. s e e Horsky, T.N. 267 Cantini, P. s e e Boato, G. 358 Cao, R. 495 Cao, Y. 358 Cappus, D. s e e Bfiumer, M. 225 Car, R. 133 Cardenas, R. s e e Gavrilenko, G.M. 644 Cardillo, M.J. s e e Lambert, W.R. 267 Carlsson, A.E. 643 Carneiro, K. 568 Carneiro, K. s e e Taub, H. 574 Carstensen, H. s e e Claessen, R. 225 Carter, E.A. s e e Weakliem, P.C. 268 Casanova, R. 643 Catlow, C.R.A. s e e Lewis, G.V. 182 Caush, M. 180 Cautero, G. 495 Cautero, G. s e e Dhanak, V.R. 495 Celli, V. s e e Hill, N.R. 359 Celli, V. s e e Rieder, K.H. 359 Ceperly, D.M. 133 Ceva, T. 568 Ceva, T. s e e Marti, C. 572 Chadi, D.J. 133, 180, 266, 710 Chadi, D.J. s e e lhm, J. 267 Chadi, D.J. s e e Mailhiot, C. 267 Chadi, D.J. see Qian, G.-X. 268 Chambers, S.A. 180, 225,266 Chambers, S.A. s e e Tran, T.T. 227 Chambliss, D.D. 133, 710 Chan, C.-M. 495 Chan, C.-M. s e e Van Hove, M.A. 183, 360, 501 Chan, C.T. 133 Chan, C.T. see Ding, Y.G. 134, 495,789 Chan, C.T. s e e Ho, K.-M. 134, 790 Chan, C.T. s e e Takeuchi, N. 136 Chan, C.T. see Tomfinek, D. 649 Chan, C.T. s e e Wang, X.W. 136, 792 Chan, C.T. s e e Xu, C.H. 136 Chan, C.T. s e e Zhang, B.L. 136 Chan, M.H.W. 568 Chan, M.H.W. s e e Jin, A.J. 571 Chan, M.H.W. s e e Kim, H.K. 571 Chan, M.H.W. s e e Migone, A.D. 572 Chan, M.H.W. s e e Pestak, M.W. 573 Chan, M.H.W. s e e Zhang, Q.M. 575,792 Chandavarkar, S. 495

Author

index

Chandavarkar, S. s e e Fisher, D. 496 Chandrasekhar, S. 789 Chang, C.C. 225 Chang, C.S. 225 Chang, H.L.M. s e e Guo, J. 226 Chang, S.C. s e e Lubinsky, A.R. 227 Chapon, C. s e e Duriez, C. 225 Chelikowsky, J.R. 133 Chelikowsky, J.R. s e e Cohen, M.L. 180 Chen, C.-L. 643 Chen, C.-L. s e e Tsong, T.T. 792 Chen, C.J. 418, 710 Chen, D.M. s e e Bedrossian, P. 133,494 Chen, D.M. s e e Zegenhagen, J. 792 Chen, J. s e e Weitering, H.H. 136 Chen, J.G. 495 Chen, S.P. s e e Voter, A.F. 649 Chen, W. 266 Chen, W. s e e Kahn, A. 267 Chen, Y.C. see Flynn, C.P. 570 Chern, G. s e e Lind, D.M. 227 Chcrnov, A.A. 96 Chernov, A.A. s e e Haneman, D. 267 Chester, G.V. s e e Tobochnik, J. 792 Chester, M. 133,495 Chestcrs, M.A. 568 Chctty, N. 133 Chetty, N. s e e Stokbro, K. 136 Chetwynd, D.G. s e e Smith, S.T. 420 Chevary, J.A. s e e Perdew, J.P. 647 Chiang, S. see Chambliss, D.D. 133,710 Chiang, S. s e e Johnson, K.E. 710 Chiang, S. s e e Wilson, R.J. 136 Chiang, S. s e e W611, Ch. 136 Chiang, T.C. see Hirschorn, E.S. 267 Chiang, T.C. see Samsavar, A. 710 Chiaradia, P. s e e Bachrach, R.Z. 265 Chiarrello, R.P. s e e You, H. 711 Ching, W.Y. 643 Chini, P. 495 Chinn, M.D. 358,568 Chinn, M.D. s e e Fain, S.C. 569 Chinn, M.D. s e e Shaw, C.G. 574 Cho, A.Y. 266 Chou, Y.C. s e e Hong, I.H. 497 Christensen, A.N. 225 Christensen, A.N. s e e Johansson, L.I. 226

815

Christensen, A.N. s e e LindstrOm, J. 227 Christman, K. s e e Imbihl, R. 645 Christmann, K. 495,643,789 Christmann, K. s e e Behm, R.J. 494, 642, 788 Christmann, K. s e e Gierer, M. 496 Christmann, K. s e e Koch, R. 498 Christmann, K. s e e Over, H. 499 Christmann, K. s e e Schwarz, E. 500 Christmann, K. s e e Van Hove, M.A. 649 Chua, F.M. s e e Kuk, Y. 646 Chubb, S.B. 495 Chung, J.W. 789 Chung, J.W. s e e Roelofs, L.D. 791 Chung, S. 568 Chung, T.T. 568 Chung, Y.W. 225 Chung, Y.W. s e e Zschack, P. 228 Citrin, P.H. s e e Riffe, D.M. 500 Clabes, J. s e e Hahn, P. 710 Claessen, R. 225 Clark, A. 97 Clark, D.E. 789 Clark, R. s e e Nagler, S.E. 572 Clarke, D.R. 225 Cochran, W. s e e Lipson, H. 359 Cocke, D.L. 225 Coddens, G. s e e Zeppenfeld, P. 575 Coenen, F.P. s e e Wesner, D.A. 501 Cohen, J.B. s e e Zschack, P. 228 Cohen, J.M. s e e Liu, C.L. 97 Cohen, M.L. 180 Cohen, M.L. s e e lhm, J. 134, 267 Cohen, M.L. s e e Louie, S.G. 135 Cohen, M.L. s e e Northrup, J.E. 267 Cohen, P.I. 568 Cohen, P.I. s e e Lent, C.S. 359 Cohen, P.I. s e e Pukite, P.R. 359 Cohen, R.E. 643 Colbourn, E.A. 180 Cole, M.W. s e e Jung, D.R. 571 Cole, M.W. s e e Klein, J.R. 646 Cole, M.W. s e e Vidali, G. 575,649 Colella, N.J. 568 Colenbrander, B.G. s e e Turkenburg, W.C. 420 Collart, E. 225 Collazo-Davila, C. 418 Collins, I.R. s e e Fisher, D. 496

816

Collins, J.B. 643 Collins, J.B. s e e Rikvold, P.A. 648 Colton, R.J. see Burnham, N.A. 418 Colton, R.J. s e e Hues, S.M. 419 Colton, R.J. s e e Landman, U. 419 Comelli, G. 495 Comelli, G. s e e Comicioli, C. 495 Comelli, G. s e e Dhanak, V.R. 495 Comicioli, C. 495 Comrie, C.M. 495 Comrie, C.M. s e e Bridge, M.E. 494 Comsa, G s e e B o t t , M . 418 Comsa, G s e e David, R. 569 Comsa, G s e e Kern, K. 571, 710 Comsa, G s e e Kunkel, R. 710 Comsa, G s e e Michely, T. 710 Comsa, G see Poelsema, B. 573,710 Comsa, G see Zeppenfeld, P. 575 Condon, N.G. s e e Murray, P.W. 227 Cong, S. 643 Conrad, E.H. 358,710 Conrad, E.H. see Cao, Y. 358 Conrad, E.H. s e e Robinson, I.K. 360 Considine, D.M. 180 Convert, P. s e e Razafitianamaharavo, A. 573 Conway, K.M. 495 Cook, M.R. s e e Himpsel, F.J. 267 Cooper, B.R. s e e Fernando, G.W. 134 Copel, M. 133,495 Coppersmith, S.N. 568 Cord, B. 180, 225 Cord, B. s e e Courths, R. 643 Corey, E.R. 495 Cornsa, G. s e e Poelsema, B. 359 Corson, D. s e e Lorrain, P. 182 Cotton, F.A. 43 Coulman, D. s e e Gritsch, T. 496 Couiman, D.J. 495,789 Coulomb, J P. 568,569 Coulomb, J P. s e e Beaume, R. 568 Coulomb, J P. s e e Bienfait, M. 568 Coulomb, J P. see Gay, J.M. 570 Coulomb, J P. s e e Glachant, A. 570 Coulomb, J.P. s e e Krim, J. 571 Coulomb, J.P. see Madih, K. 572 Coulomb, J.P. s e e Razafitianamaharavo, A. 573

Author

index

Coulomb, J.P. s e e Suzanne, J. 574 Coulomb, J.P. s e e Trabelsi, M. 575 Coulon, M. s e e Delachaume, J.C. 569 Coulon, M. s e e Khatir, Y. 571 Coulon, M. s e e Tabony, T. 574 Courths, R. 643 Courths, R. s e e Cord, B. 180, 225 Courths, R. s e e Pollak, P. 500 Cowan, P.L. 643 Cowan, P.L. s e e Woicik, J.C. 502 Cowie, B. s e e Kerkar, M. 497 Cowley, J.M. 225, 358 Cowley, J.M. s e e Busek, P.R. 418 Cowley, J.M. s e e Gajdardziska-Josifovska, M. 225 Cowley, J.M. s e e Liu, J. 419 Cowley, J.M. s e e Yao, N. 184, 228 Cowley, R.A. s e e Andres, S.R. 358 Cox, B.N. 495 Cox, D.F. 180, 225 Cox, D.F. s e e Henrich, V.E. 181 Cox, D.F. s e e Semancik, S. 183 Cox, P.A. s e e Henrich, V.E. 226 Cremaschi, P.L. 643 Crommie, M.F. 643 Croset, B. see Madih, K. 572 Croset, B. 569 Croset, B. s e e Coulomb, J.P. 569 Croset, B. s e e Marti, C. 572 Croset, B. s e e Razafitianamaharavo, A. 573 Croweil, J.E. s e e Chen, J.G. 495 Crozier, P.A. 418 Crozier, P.A. s e e Gajdardziska-Josifovska, M. 225 Cui, J. 495,569 Cui, J. s e e Jung, D.R. 571 Cunningham, S.L. 643 Cunningham, S.L. s e e Williams, E.D. 650 Curtiss, C.F. s e e Hirschfelder, J.O. 570, 645 Cvetko, D. s e e Bellman, A.F. 494 Czanderna, A.W. s e e Lu, C. 572 D'Amico, K.L. 569 D'Amico, K.L. s e e Liang, K.S. 359 D'Amico, K.L. s e e Specht, E.D. 574 Daboul, D. s e e Seehofer, L. 500 Dabrowski, J. 266

Author

index

Dahl, L.F. s e e Corey, E.R. 495 Dai, X.Q. 643 Daimon, H. 133 Daiser, S. s e e Miranda, R. 572 Daley, R.S. 133,789 Daley, R.S. see Hildner, M.L. 790 Danner, H.R. s e e Coulomb, J.P. 569 Danner, H.R. s e e Taub, H. 574 Danner, H.R. s e e Trott, G.J. 575 Darville, J. see de Fr6sart, E. 180 Das Sarma, S. s e e Kodiyalam, S. 97 Dash, J.G. 97, 569 Dash, J.G. see Bienfait, M. 568 Dash, J.G. s e e Bretz, M. 568 Dash, J.G s e e Chung, T.T. 568 Dash, J.G see Ecke, R.E. 569 Dash, J.G s e e Gay, J.M. 570 Dash, J.G s e e Glachant, A. 570 Dash, J.G s e e Goodstein, D.L. 570 Dash. J.G s e e Huff, G.B. 571 Dash. J.G s e e Hulburt, S.B. 571 Dash. J.G s e e Kjems, J.K. 571 Dash. J.G s e e Krim, J. 571 Dash. J.G s e e Migone, A.D. 572 Dash J.G s e e Motteler, F.C. 572 Dash J.G s e e Muirhead, R.J. 572 Dash J.G s e e Pengra, D.B. 573 Dash J.G see Seguin, J.L. 574 Dash, J.G s e e Shechter, H. 574 Dash, J.G s e e Zhu, D.M. 575 David, R. 569 David, R. s e e Kern, K. 571 David, R. s e e Zeppenfeld, P. 575 Davidov, A.S. 418 Davidson, E.R. 133 Davidson, E.R. s e e Feller, D. 181 Davies, J.A. s e e Jackman, T.E. 419, 497 Davies, J.A. s e e L'Ecuyer, J. 419 Davies, J.A. s e e Norton, P.R. 647 Davis, H.L. s e e Gruzalski, G.R. 226 Davis, H.L. s e e Itchkawitz, B.S. 790 Davis, R.F. s e e Chang, C.S. 225 Davis, R.F. s e e Kevan, S.D. 498 Davis, R.F. s e e Liaw, H.P. 226 Davis, R.F. s e e Tobin, J.G. 501 Davison, S.G. 643 Davison, S.G. s e e Liu, W.-K. 646

817

Davison, S.G. s e e Sulston, K.W. 649 Davisson, C.J. 359 Davydov, S.Yu. 643 Daw, M.S. 133, 134, 643,789 Daw, M.S. s e e Einstein, T.L. 644 Daw, M.S. s e e Fallis, M.C. 644 Daw, M.S. s e e Felter, T.E. 644 Daw, M.S. s e e Foiles, S.M. 644 Daw, M.S. s e e Gumbsch, P. 97 Daw, M.S. s e e Roelofs, L.D. 648, 791 Daw, M.S. s e e Wright, A.F. 650 Dayan, M. 225 De Andres, P. s e e Oed, W. 499 De Beauvais, C. s e e Marti, C. 572 De'Bell, K. s e e Piercy, P. 647 De Cheveigne, S. s e e Rousset, S. 98 De Fr6sart, E. 180 De Miguel, J.J. 87, 710 De Miguei, J.J. s e e Aumann, C.E. 265 De Paola, R.A. s e e Heskett, D. 497 De Souza, E.P. s e e Rapp, R.E. 573 De Tacconi, N.R. s e e Liu, S.H. 646 De Vita, A. s e e Manassidis, I. 227 De Wette, F.W. 180 Debe, M.K. 134, 789 Deckert, A.A. s e e Brand, J.L. 643 Dederichs, P.H. 643 Degenhardt, D. 569 Delachaume, J.C. 569 Delachaume, J.C. s e e Tabony, T. 574 Delbouille, A. s e e Boudart, M. 180 Delchar, T.A. s e e Woodruff, D.P. 183 Delley, B. s e e Spiess, L. 136 Delley, B. s e e Ye, L. 136 Delly, B. 180 Demmin, R.A. s e e Song, K.-J. 98 Demuth, J.E. 495,643,789 Demuth, J.E. s e e Alerhand, O.L. 265 Demuth, J.E. s e e Hamers, R.J. 496 Demuth, J.E. s e e Marcus, P.M. 498 Demuth, J.E. s e e van Loenen, E.J. 136 Denier, A.W. s e e Viieg, E. 136 Denier van der Gon, A.W. 266 Denier van der Gon, A.W. s e e Tromp, R.M. 420 Dennison, J.R. s e e Larese, J.Z. 572 DePristo, A.E. s e e Raeker, T.J. 647

818

DePristo, A.E. s e e Stave, M.S. 136 Derouane, E.G. s e e Boudart, M. 180 Derry, T.E. 180 Desjonqu~res, M.C. 643 Desjonqu~res, M.C. s e e Bourdin, J.P. 642 Desjonqu~res, M.C. s e e Oils, A.M. 647 Deutsch, M. s e e Braslau, A. 358 Dev, B.N. 789 DeWette, F.W. s e e Reiger, R. 183 Dhanak, V.R. 495 Dhanak, V.R. s e e Comelli, G. 495 Dhanak, V.R. s e e Comicioli, C. 495 Dhanak, V.R. s e e Murray, P.W. 499 Dick, G.B. 180 DiDio, R.A. s e e Gruzalski, G.R. 226 DiDio, R.A. s e e Mundenar, J.M. 499 Diederich, F.N. s e e Snyder, E.J. 420 Diehl, R.D. 495,569 Diehl, R.D. s e e Barnes, C.J. 494 Diehl, R.D. s e e Chandavarkar, S. 495 Diehl, R.D. s e e Cui, J. 495 Diehl, R.D. s e e Fain, S.C. 569 Diehl, R.D. s e e Fisher, D. 496 Diehl, R.D. s e e Kerkar, M. 497 Dieleman, J. s e e Dijkkamp, D. 97 Dietrich, S. 569 Dijkkamp, D. 97 Dijkkamp, D. s e e Elswijk, H.B. 496 Dillon, J.A. s e e Farnsworth, H.E. 789 Dimon, P. 569 Dimon, P. s e e Mochrie, S.G.J. 572 DiNardo, N.J. 418 DiNardo, N.J. s e e Weitering, H.H. 136 Ding, M.Q. s e e Barnes, C.J. 494 Ding, Y.G. 134, 495,789 Dtibler, U. 495 Dobrzynski, L. 643 Dobrzynski, L. s e e Cunningham, S.L. 643 Dobson, P.J. s e e Joyce, B.A. 267 Doering, D.L. 495 Dolle, P. 569 Dolle, P. s e e Fargues, D. 569 Domany, E. 569 Dong, C.-Z. s e e Madey, T.E. 97 Dong, C.-Z. s e e Song, K.-J. 98 Dornisch, D. 134, 495 Dose, V. s e e Scheid, H. 500

Author

Dtitsch, B. s e e Oed, W. 499 Dovesi, R. s e e Cause, M. 180 Dovesi, R. s e e Pisani, C. 183 Dow, J.D. s e e Vogl, P. 183 Doyen, G. 789 Drabold, D. s e e Ordej6n, P. 135 Draper, C.F. s e e Hues, S.M. 419 Drechsler, M. 97 Dresselhaus, G. s e e Henrich, V.E. 181 Dreyss6, H. 643 Dreyss6, H. s e e Stauffer, L. 648 Drir, M. 569 Drir, M. s e e Nham, H.S. 573 Drittler, B. s e e Dederichs, P.H. 643 Droste, Ch. s e e Scheffler, M. 500, 648 Dubois, L.H. 495 Ducros, P. s e e Aberdam, D. 225 Duffy, D.M. s e e Tasker, P.W. 227 Dufour, L.-C. 180, 225 Dufour, L.-C. s e e Boudriss, A. 225 Dufour, L.-C. s e e Nowotny, J. 182 Duke, C.B. 180, 181,266, 418 Duke, C.B. s e e Godin, T.J. 266 Duke, C.B. s e e Horsky, T.N. 267 Duke, C.B. s e e Kahn, A. 267 Duke, C.B. s e e LaFemina, J.P. 182, 267 Duke, C.B. s e e Lessor, D.L. 267 Duke, C.B. s e e Lubinsky, A.R. 227, 267 Duke, C.B. s e e Mailhiot, C. 267 Dumas, M. s e e Chen, W. 266 Dumas, M. s e e Kahn, A. 267 Dumas, Ph. s e e Thibaudau, F. 501 Dunham, D. s e e Stoehr, J. 420 Dunlap, B.I. 181 Dunn, D. s e e Ai, R. 418 Dunn, D. s e e Marks, L.D. 419 Dunn, D.N. 418 Dunphy, J. s e e Barbieri, A. 642 Dupont-Pavlovsky, N. 569 Dupont-Pavlovsky, N. s e e Razafitianamaharavo, A. 573 Dupont-Pavlovsky, N. s e e R6gnier, J. 573 Duraud, J.P. s e e Bart, F. 225 Duriez, C. 225 Dutheil, A. s e e Gay, J.M. 570 Dutta, P. 569 Duval, X. s e e Matecki, M. 572

index

Author

index

Duval, X. s e e Menaucourt, J. 572 Duval, X. s e e R6gnier, J. 573 Duvai, X. s e e Thomy, A. 574 Dzioba, S. 225 Eades, J.A. s e e Samsavar, A. 710 Ealet, B. 225 Ealet, B. s e e Gillet, E. 225 Eastman, D.E. s e e Himpsel, F.J. 181 Eastman, D.E. s e e Lang, N.D. 571 Eberhardt, W. 495 Eberhardt, W. s e e Heskett, D. 497 Ebina, K. 643 Ecke, R.E. 569 Eckert, J. 569 Eckert, J. s e e Grier, B.H. 570 Eckert, J. s e e Satija, S.K. 573 Economou, E.N. 644 Edamoto, K. 225 Eden, V.L. 569 Edmonds, T. 569 Edwards,J.C. see Antonik, M.D. 225 Egdell, R.G. 181 Eggeling, von, C. 495 Eggleston, C.M. see Johnsson, P.A. 226 Eguiluz, A.G. 644 Ehrenreich, H. 644 Ehrhardt, J.J. see Fargues, D. 569 Ehrlich, G. 644, 789 Ehrlich, G. s e e Fink, H.-W. 644 Ehrlich, G. s e e Wang, S.C. 501 Ehrlich, G. see Watanabe, F. 649 Eierdal, L. 495 Eigler, D.M. s e e Crommie, M.F. 643 Einstein, T.L. 97, 134, 644, 789 Einstein, T.L. s e e Bak, P. 788 Einstein, T.L. s e e Bartelt, N.C. 96,642, 788 Einstein, T.L. s e e Eisner, D.R. 97 Einstein, T.L. s e e Joos, B. 97 Einstein, T.L. see Khare, S.V. 97, 645 Einstein, T.L. s e e Nelson, R.C. 98 Einstein, T.L. see Pai, W.W. 98,647 Einstein, T.L. s e e Roelofs, L.D. 648,791 Einstein, T.L. s e e Taylor, D.E. 649 Einstein, T.L. s e e Wang, X.-S. 98 Einstein, T.L. s e e Williams, E.D. 99 Eisner, D.R. 97

819

Eiswirth, M. 496 Eiswirth, M. s e e B~ir, M. 494 Eiswirth, M. s e e Krischer, K. 498 EI-Batanouny, M. 134 El-Batanouny, M. s e e Hsu, C.-H. 790 Elgin, R.L. 569 Ellenson, W. s e e Nielsen, M. 573 Ellenson, W. s e e Satija, S.K. 573 Ellenson, W.D. s e e Eckert, J. 569 Ellis, D.E. 181,496 Ellis, D.E. s e e Guo, J. 181,226 Ell[is, T.H. 569 Elsaesser, C. s e e Ho, K.M. 134 Elswijk, H.B. 496 Elyakhloufi, M.H. s e e Ealet, B. 225 Emmett, P.H. 496 Eng, P. s e e Meyerheim, H.L. 498 Eng, P.J. s e e Smilgies, D.-M. 792 Engel, T. 359, 496, 569 Engel, T. s e e Conrad, E.H. 358,710 Engel, T. s e e James, R.W. 359 Engel, T. s e e Kaufman, R. 359 Engel, T. s e e Sander, M. 227 Engel, T. s e e Szabo, A. 227 Engel, W. 418,496 Engel, W. s e e Griffith, O.H. 419 Engel, W. see Jakubith, S. 497 Engel, W. s e e Rose, K.C. 420 Engel, W. s e e Rotermund, H.H. 500 Engelhard, M.H. s e e Wang, L.Q. 227 Engelhardt, H.A. s e e Madey, T.E. 498 Engelhardt, H.A. s e e Pfn0r, H. 500 English, C.A. s e e Venables, J.A. 575 Enta, Y. s e e Kinoshita, T. 498 Epicier, T. 225 Ercolessi, F. 134, 644 Ercolessi, F. s e e Bilalbegovic, G. 96 Ercolessi, F. s e e Garofalo, M. 790 Ercolessi, F. s e e Tosatti, E. 136, 649 Eriksen, S. s e e Egdell, R.G. 181 Erley, W. s e e Bar6, A.M. 642 Ernst, H.J. 710 Ernst, K.H. s e e Over, H. 499 Ernst, K.H. s e e Schwarz, E. 500 Ertl, G. 43, 359, 418, 496,789 Erti, G. s e e B~ir, M. 494 Ertl, G. s e e Barth, J.V. 133,709

820 Erti, G. s e e Behm, B.J. 133,494, 642, 788 Ertl, G. s e e Bludau, H. 494 Ertl, G. s e e Bonzel, H.P. 494 Ertl, G. s e e B/Sttcher, A. 494 Ertl, G. s e e Brune, H. 495 Ertl, G. s e e Christmann, K. 495,643,789 Ertl, G. s e e Coulman, D.J. 495,789 Ertl, G. s e e Doyen, G. 789 Ertl, G. s e e Eiswirth, M. 496 Erti, G. s e e Engel, T. 496 Ertl, G. s e e Gierer, M. 496 Ertl, G. s e e Gritsch, T. 496 Ertl, G. s e e Hertel, T. 497 Ertl, G. s e e Imbihl, R. 497,645 Ertl, G. s e e Jacobi, K. 497 Ertl, G. s e e Jakubith, S. 497 Ertl, G s e e Kieinle, G. 498 Ertl, G s e e Krischer, K. 498 Ertl, G s e e Miranda, R. 572 Ertl, G s e e Moritz, W. 646 Ertl, G s e e Over, H. 499, 500 Ertl, G s e e Rotermund, H.H. 420, 500 Ertl, G see Schuster, R. 500, 648,791 Ertl, G s e e Shi, H. 500 Ertl, G. s e e Trost, J. 501 Ertl, G. see Van Hove, M.A. 649 Ertl, G. s e e Woratschek, B. 502 Erwin, S.C. 134 Estrup, P.J. 496, 789 Estrup, P.J. s e e Barker, R.A. 133,788 Estrup, P.J. s e e Chung, J.W. 789 Estrup, P.J. s e e Daley, R.S. 133,789 Estrup, P.J. see Felter, T.E. 789 Estrup, P.J. s e e Hildner, M.L. 790 Estrup, P.J. see Robinson, I.K. 135,791 Estrup, P.J. see Roelofs, L.D. 791 Evans-Lutterodt, K. s e e Robinson, I.K. 135, 791 Everts, H.-U. see Sandhoff, M. 648 Ewald, P.P. 181 Eyring, L. s e e Busek, P.R. 418 Fabre, F. s e e Ernst, H.J. 710 Fadley, C.S. s e e Bullock, E.L. 133 Fahnle, M. s e e Ho, K.M. 134 Fain, S.C. 569 Fain, S.C. s e e Chinn, M.D. 358,568

Author

Fain, S.C. s e e Cui, J. 569 Fain, S.C. s e e Diehl, R.D. 569 Fain, S.C. s e e Eden, V.L. 569 Fain, S.C. s e e Osen, J.W. 573 Fain, S.C. s e e Shaw, C.G. 574 Fain, S.C. s e e Taub, H. 792 Fain, S.C. s e e Toney, M.F. 575 Fain, S.C. s e e You, H. 575 Faisal, A.Q.D. 569 Faisal, A.Q.D. s e e Hamichi, M. 570 Faisal, A.Q.D. s e e Venables, J.A. 575 Falicov, L.M. 789 Falkenberg, G. s e e Seehofer, L. 500 Fallis, M.C. 644 Family, F. 710 Fan, F.R. s e e Bard, A.J. 418 Fan, W.C. s e e Over, H. 499 Fargues, D. 569 Farnsworth, H.E. 789 Farnsworth, H.E. s e e Schlier, R.E. 135,268 Farrell, H.H. 266 Fasolino, A. s e e Wang, C.Z. 136, 792 F~issler, T.F. s e e Burdett, J.K. 643 Faul, J.W.O. 569 Fedak, D.G. 134 Fedorus, A.G. s e e Bol'shov, L.A. 642 Fedyanin, V.K. s e e Gavrilenko, G.M. 644 Feenstra, R.M. 266, 418 Feenstra, R.M. s e e Pashley, M.D. 268 Fehlner, F.P. 181 Feibelman, P.J. 134, 644, 789 Feibelman, P.J. s e e Knotek, M.L. 226 Feibelman, P.J. s e e Williams, A.R. 649 Feidenhans'l, R. 266, 359, 496, 789 Feidenhans'l, R. s e e Bohr, J. 266 Feidenhans'l, R. s e e Dornisch, D. 134, 495 Feidenhans'l, R. s e e Grey, F. 266 Feile, R. 570 Fein, A.P. s e e Feenstra, R.M. 418 Feldman, L.C. 418 Feldman, L.C. s e e Headrick, R.L. 496 Feldman, L.C. s e e Stensgaard, I. 136 Feller, D. 181 Felter, T.E. 644, 789 Felter, T.E. s e e Daley, R.S. 133, 789 Felter, T.E. s e e Hildner, M.L. 790 Felton, R.C. s e e Prutton, M. 183,227

index

Author

index

Feng, Y.P. s e e Kim, H.K. 571 Fenter, P. s e e Hfiberle, P. 790 Fernando, G.W. 134 Ferrante, J. s e e Rodriguez, A.M. 135,648 Ferrante, J. s e e Rose, J.H. 648 Ferrante, J. s e e Smith, J.R. 98, 136, 648 Ferreira, O. s e e Tejwani, M.J. 574, 792 Ferrer, S. 496, 789 Ferrer, S. s e e Bonzel, H.P. 494 Ferret, P. s e e Epicier, T. 225 Feulner, P. s e e Pfntir, H. 500 Feynman, R. 181 Feynman, R.P. 134 Fink, H.-W. 418,644 Fink, J. see Claessen, R. 225 Finney, M.S. see Howes, P.B. 134 Finnis, M.W. 134, 644 Finzel, H.-U. 359 Fiolhais, C. s e e Perdew, J.P. 647 Fiorentini, V. 134 Firment, L.E. 225 Fisher, A.J. 181 Fisher, D. 496 Fisher, D. s e e Kerkar, M. 497 Fisher, D.S. s e e Coppersmith, S.N. 568 Fisher, D.S. s e e Fisher, M.E. 97 Fisher, G.B. s e e Root, T.W. 500 Fisher, H.J. s e e Murray, P.W. 227 Fisher, M.E. 97,789 Fisher, M.E. s e e Huse, D.A. 571 Flagg, R. s e e Kieban, P. 790 Flavell, W.R. s e e Egdeil, R.G. 181 Fleming, R.M. 359 Fleszar, A. s e e Scheffler, M. 500, 648 Flipse, C.F.J. see Murray, P.W. 227 Flodstr6m, S.A. s e e Hammar, M. 226 Flores, F. 644 Flores, F. s e e Garcia-Moliner, F. 134 Flores, F. see Joyce, K. 645 Flynn, C.P. 570 Flynn, C.P. s e e Y adavalli, S. 228 Flynn, D.K. s e e Behm, B.J. 133 Flytzani-Stephanopoulos, M. 97 Fock, V.A. 181 Fogedby, H.C. 789 Foiles, S.M. 644 Foiles, S.M. s e e Daw, M.S. 134, 643

821 Foiles, S.M. s e e Einstein, T.L. 644 Foiles, S.M. s e e Felter, T.E. 644 Foiles, S.M. s e e Roelofs, L.D. 648,791 Foiles, S.M. s e e Schwoebel, P.R. 648 Folkets, R. s e e Ernst, H.J. 710 Folman, M. s e e Shechter, H. 574 Folman, M. s e e Uram, K.J. 501 Ftilsch, S. s e e Schimmelpfenning, J. 574 Fong, C.Y. s e e Fallis, M.C. 644 Fong, C.Y. s e e Wright, A.F. 650 Fontes, E. s e e Guryan, C.A. 570 Ford, W.K. s e e Blanchard, D.L. 180, 225 Ford, W.K. s e e Lessor, D.L. 267 Ford, W.K. s e e Wan, K.J. 501 Frahm, R. s e e Greiser, N. 570 Francis, S.M. s e e Leibsle, F.M. 498 Frank, F.C. 570 Frank, H. s e e Finzel, H.-U. 359 Frank, H.H. 496 Frank, K.H. s e e Ferrer, S. 496 Frankel, D. s e e G6pel, W. 226 Frankl, D.R. s e e Chung, S. 568 Frankl, D.R. s e e Jung, D.R. 571 Franz, R.U. s e e Rotermund, H.H. 420 Frauenfelder, H. 570 Frederiske, H.P.R. 97 Freeland, P.E. s e e Zegenhagen, J. 502, 792 Freeman, A.J. s e e Fu, C.L. 134, 710, 789 Freeman, A.J. s e e Spiess, L. 136 Freeman, A.J. s e e Wimmer, E. 136, 501,650 Freeman, A.J. s e e Ye, L. 136 Freeman, D.L. 644 Freimuth, H. 570 Freimuth, H. s e e Cui, J. 569 French, T.M. 225 Frenken, F.W.M. s e e Kuipers, L. 97 Frenken, J.W.M. 496 Frenken, J.W.M. s e e Denier van der Gon, A.W. 266 Frenken, J.W.M. s e e Pluis, B. 359 Frenken, J.W.M. s e e Smeenk, R.G. 500 Frenken, J.W.M. s e e Van Pinxteren, H.M. 98 Freund, H.J. s e e Btiumer, M. 225 Fricke, A. s e e Mendez, M.A. 498 Friedel, J. 644 Friedel, J. s e e Bourdin, J.P. 642 Friedel, P. s e e Lanoo, M. 182

822 Friedman, D.J. s e e Bullock, E.L. 133 Fritsche, L. s e e Noffke, J. 499 Fritzsche, V. s e e Wedler, H. 501 Frohn, J. 97,644 Frohn, J. s e e Poensgen, M. 98 Fryberger, T.B. s e e Cox, D.F. 180, 225 Fu, C.L. 134, 710, 789, 790 Fu, C.L. s e e Wimmer, E. 501 Fuchs, G. s e e Epicier, T. 225 Fuchs, H. s e e Salvan, F. 500 Fuggle, J.C. 496 Fuggle, J.C. s e e Steinkilberg, M. 501 Fujiwara, T. see Nowak, H.J. 135 Fukushi, D. see Takami, T. 501 Fuoss, P.H. 359 Fuoss, P.H. s e e Robinson, I.K. 43 Furuya, K. s e e Andrei, N. 642 Fuselier, C.R. 570 Gadzuk, J.W. 496 Gajdardziska-Josifovska, M. 225 Gajdardziska-Josifovska, M. see Crozier, P.A. 418 Galatry, L. 570 Galeotti, M. 225 Gallagher, J. 644 Galli, G. s e e larlori, S. 181 Gameson, I. 570 Ganachaud, J.P. see Bourdin, J.P. 642 Ganz, E. 496, 790 Gao, G.B. s e e Morkor H. 227 Gao, Y. s e e Chambers, S.A. 225 Garcia, A. 418 Garcia-Moliner, F. 134 Garcia, N. 359 Garcia, N. see Rieder, K.H. 359 Garfunkei, E. s e e Novak, D. 227 Garfunkel, E. s e e Song, K.-J. 98 Garibaldi, V. 359 Garofalini, S.H. 181 Garofalo, M. 790 Garrett, B.C. s e e Truong, T.N. 649 Gauthier, S. s e e Rousset, S. 98 Gauthier, Y. s e e Hammar, M. 226 Gauthier, Y. s e e Rundgren, J. 227 Gautier, M. s e e Bart, F. 225 Gavrilenko, G.M. 644

Author

index

Gawlinski, E.T. s e e Saxena, A. 791 Gawlinski, G.T. 710 Gay, J.G. s e e Ricter, R. 135 Gay, J.M. 570 Gay, J.M. s e e Bienfait, M. 568 Gay, J.M. s e e Denier van der Gon, A.W. 266 Gay, J.M. s e e Krim, J. 571 Gay, J.M. s e e Meichel, T. 572 Gay, J.M. s e e Pluis, B. 359 Gay, R.R. 225 Geisinger, K.L. 181 George, J. s e e Kern, K. 710 George, S.M. s e e Brand, J.L. 643 Gerber, C. s e e Binnig, G. 266, 418 Gerber, Ch. s e e Binnig, G.K. 789 Gerber, Ch. s e e Binning, G.K. 133 Gerlach, R.L. 496 Germer, L.H. 496 Germer, L.H. see Davisson, C.J. 359 Giannozzi, P. s e e Baroni, S. 133 Gibbs, D. s e e D'Amico, K.L. 569 Gibbs, D. s e e Zehner, D.M. 792 Gibbs, G.V. s e e Geisinger, K.L. 181 Gibbs, G.V. s e e Hill, R.J. 181 Gibbs, J.W. 97 Gibson, A. 181 Gibson, J.M. see Pohland, O. 98 Gibson, K.D. 570 Gibson, W.M. see Narusawa, T. 647 Gierer, M. 496 Gierer, M. see Bludau, H. 494 Gierer, M. s e e Hertel, T. 497 Gierer, M. see Over, H. 499, 500 Gierlotka, S. s e e Pluis, B. 359 Giesen, M. s e e Frohn, J. 97,644 Giesen, M. see Poensgen, M. 98 Giesen-Seibert, M. 644 Gijzeman, O.L.J. see Bootsma, G.A. 568 Gillan, M.J. s e e Manassidis, I. 227 Gilles, J.M. s e e de Frdsart, E. 180 Gilles, N.S. see Fuselier, C.R. 570 Gillet, E. 225 Gillet, E. s e e Ealet, B. 225 Giimer, G.H. s e e Leamy, H.J. 43 Gilmer, G.H. s e e Weeks, J.D. 711 Giiquin, B. s e e Larher, Y. 572 Gilquin, B. s e e Ser, F. 57

Author

823

index

Girard, C. 570, 644 Girard, C. s e e Galatry, L. 570 Girard, C. s e e Girardet, C. 570 Girard, C. s e e Meichel, T. 572 Girard, J.C. s e e Rousset, S. 98 Girardet, C. 570 Girardet, C. s e e Girard, C. 570, 644 Girardet, C. s e e Lakhlifi, A. 571 Girardet, C. s e e Meichei, T. 572 Gittes, F.T. 570 Gjostein, N.A. s e e Fedak, D.G. 134 Glachant, A. 570 Glachant, A. s e e Bardi, U. 568 Glachant, A. s e e Beaume, R. 568 Glander, G. see Tong, S.Y. 268 Glander, G.S. s e e Wei, C.M. 136 Glanz, G. s e e Hofmann, P. 497 Glasser, M.L. s e e Gumbs, G. 645 Gl6bl, M. s e e Scheid, H. 500 Glueckstein, J.C. see Nogami, J. 499 Gobcli, G.W. s e e Lander, J.J. 135 Godfrey, M.J. s e e Needs, R.J. 97 Godin, T.J. 181,266 Goldberg, J.L. see Wang, X.-S. 98 Goldenfeld, N. 790 Goldman, A.I. see Guryan, C.A. 570 Goldman, A.I. see Stephens, P.W. 574 Goldmann, M. s e e Ceva, T. 568 Gollisch, H. 645 Golovchenko, J. s e e Ganz, E. 496, 790 Golovchenko, J.A. s e e Bedrossian, P. 133,494 Golovchenko, J.A. s e e Hwang, I.-S. 497 Golovchenko, J.A. see Martinez, R.E. 97 Goluvchenko, J.A. s e e Zegenhagen, J. 792 Golze, M. 570 Gomer, R. 418,645 Gomer, R. s e e Tringides, M. 649 Gomer, R. s e e Uebing, C. 649 Gomer, R. s e e Wang, C. 575 Gonser-Buntrock, C. s e e Schwarz, E. 500 Goodstein, D.L. 570 Goodstein, D.L. s e e Elgin, R.L. 569 Goodstein, D.L. s e e Hamilton, J.J. 570 Goodwin, L. 134 G6pel, W. 226 Gi3pel, W. see Kroll, C. 226 Gordon, R.G. 645

Gossler, J. s e e Hofmann, P. 497 Gotoh, T. 181 Gotoh, Y. 710 Gotter, U. s e e Horn, M. 710 Gottfried, K. 645 Gottlieb, J.M. 645 Graham, W.R. s e e Copel, M. 495 Grant, M. s e e Gawlinski, G.T. 710 Gray, H.B. 266 Greber, T. s e e Btittcher, A. 494 Greene, R.L. s e e Greiser, N. 570 Greenler, R.G. s e e Campuzano, J.C. 495 Greg, F. s e e Feidenhans'l, R. 266 Grehk, T.M. 496 Greiser, N. 570 Gremaud, G. s e e Burnham, N.A. 418 Grempel, D.R. s e e Villain, J. 360 Grest, G.S. 790 Grey, F. 266 Grey, F. s e e Dev, B.N. 789 Grey, F. s e e Dornisch, D. 134, 495 Grey, F. s e e Feidenhans'l, R. 496, 789 Grier, B.H. 570 Griffith J.E. 267 Griffith, J.E. s e e Kubby, J.A. 267 Griffith, O.H. 419 Griffith, O.H. s e e Rempfer, G.F. 419 Griffith, O.H. s e e Skoczylas, W.P. 420 Griffiths, K. 790 Griffiths, K. s e e Bare, S.R. 494 Griffiths, K. s e e Jackman, T.E. 419 Griffiths, R.B. 97,790 Griffiths, R.B. s e e Butler, D.M. 568 Griffiths, R.B. s e e Domany, E. 569 Griffiths, R.B. s e e Niskanen, K.J. 573 Grimley, T.B. 645 Grimsby, D. 496 Gritsch, T. 496 Grobecker, R. s e e B6ttcher, A. 494 Gronsky, R. s e e Falicov, L.M. 789 Gronwald, K.D. 359 Grout, P.J. s e e Joyce, K. 645 Groves, G.W. s e e Kelly, A. 226 Grozea, D. s e e Collazo-Davila, C. 418 Gruber, E.E. 97 Grumbach, M.P. s e e Ordej6n, P. 135 Grunze, M. s e e Golze, M. 570

824 Gruyters, M. s e e Jacobi, K. 497 Gruzalski, G.R. 226 Guan, J. s e e Madey, T.E. 97 Guillermo, B. s e e Smith, J.R. 136 Guinea, F. s e e Rose, J.H. 648 Guinier, A. 359 Gumbs, G. 645 Gumbsch, P. 97 Gumhalter, B. 645 Gunnarsson, O. s e e Jones, R.O. 134, 181 Gunther, S. s e e Hwang, R.Q. 710 Gtintherodt, H.-J. 645 Gtintherodt, H.-J. s e e Tarrach, G. 227 Gunton, J.D. s e e Collins, J.B. 643 Gunton, J.D. s e e Gawlinski, G.T. 710 Gunton, J.D. s e e Rikvold, P.A. 648 Gunton, J.D. s e e Saxena, A. 791 Guo, J. 181,226 Guo, J. s e e Ellis, D.E. 181 Guo, Q. s e e Murray, P.W. 499 Guo, T. s e e Blanchard, D.L. 180, 225 Guo, T. s e e Wan, K.J. 501 Gurney, R.W. 496 Guryan, C.A. 570 Gustafsson, T. s e e Chester, M. 133,495 Gustafsson, T. see Copel, M. 133,495 Gustafsson, T. s e e Haberle, P. 134, 790 Gustafsson, T. see Novak, D. 227 Gustafsson, T. see Zhou, J.B. 228 Guthmann, C. s e e Balibar, S. 96 Gwanmesia, G.D. s e e Susman, S. 183 Gygi, F. see lariori, S. 181 Haas, G. s e e Rotermund, H.H. 420 Haase, J. see Aminpirooz, S. 493 Haase, J. s e e Bader, M. 494 Haase, J. see Becker, L. 494 Haase, J. s e e Pangher, N. 500 Haase, J. see Pedio, M. 500 Haase, J. s e e Schmalz, A. 500, 648 Haase, O. 496 Haase, O. s e e Koch, R. 498 Haberen, K.W. s e e Pashley, M.D. 267, 268 Haberland, H. s e e Woratschek, B. 502 Haberle, P. 134, 790 H~iberle, P. s e e Zhou, J.B. 228 Haensel, R. s e e Degenhardt, D. 569

Author

Haftel, M.I. 134, 645 Haga, Y. 419 Hfiglund, J. s e e Hammar, M. 226 Hagstrum, H.D. 419 Hahn, P. 710 Hfikansson, K.L. s e e Hammar, M. 226 Haler, J.F. 226 Hall, B. 570 Halperin, B.I. 570, 790 Halperin, B.I. s e e Coppersmith, S.N. 568 Halperin, B.I. s e e Nelson, D.R. 573 Halpin-Healy, T. 570 Hamann, D. s e e Feibelman, P.J. 644 Hamann, D.R. 134 Hamann, D.R. s e e Appelbaum, J.A. 642 Hamann, D.R. s e e Applebaum, J.R. 180 Hamann, D.R. s e e Biswas, R. 133 Hamann, D.R. s e e Lambert, W.R. 267 Hamann, D.R. s e e Tersoff, J. 136, 420 Hamers, R.J. 496 Hamers, R.J. s e e Alerhand, O.L. 265 Hamers, R.J. s e e van Loenen, E.J. 136 Hamichi, M. 570 Hamichi, M. s e e Faisal, A.Q.D. 569 Hamichi, M. s e e Venables, J.A. 575 Hamilton, J.J. 570 Hammar, M. 226 Hammer, L. see Eggeling, von, C. 495 Hammer, L. s e e Mendez, M.A. 498 Hammer, L. s e e Oed, W. 499 Hammonds, E.M. s e e Horn, P.M. 571,790 Han, W.K. 134 Hanada, T. s e e Hikita, T. 181,226 Hanayama, M. s e e Morishige, K. 572 Hanekamp, L.J. s e e Bootsma, G.A. 568 Haneman, D. 134, 181,267, 359, 790 Hanke, G. s e e Lang, E. 359 Hannaman, D.J. 496 Hansen, F.Y. 570 Hansen, F.Y. s e e Coulomb, J.P. 569 Hansen, F.Y. s e e Trott, G.J. 575 Hansen, F.Y. s e e Wang, R. 575 Hansen, G.D. s e e Rikvold, P.A. 648 Hansson, G.V. s e e Bachrach, R.Z. 265 Hansson, G.V. s e e Nichoils, J.M. 135,499 Hansson, G.V. s e e Uhrberg, R.I.G. 268 Hara, S. 226

index

Author

825

index

Harbison, J.P. s e e Farrell, H.H. 266 Hardiman, M. s e e Venables, J.A. 575 Harp, G.R. s e e Stoehr, J. 420 Harris, J. s e e Liebsch, A. 359 Harrison, W.A. 134, 181,267 Harten, U. 134 Harten, V. 710 Hartman, J.K. s e e Gadzuk, J.W. 496 Hartung, V. s e e Schtinhammer, K. 648 Hasegawa, F. s e e Kumagai, Y. 498 Hasegawa, T. 496 Hasegawa, Y. 645 Htiser, W. s e e Heidberg, J. 570 Hashizume, T. s e e Jeon, D. 134, 790 Hashizume, T. s e e Park, C. 500 Hashizume, T. s e e Taniguchi, M. 501 Hastings, J.B. see Eckert, J. 569 Hastings, J.M. s e e Larese, J.Z. 572 Hathaway, K.B. s e e Falicov, L.M. 789 Hau, U. see Courths, R. 643 Hawkes, P.W. 419 Hayami, W. s e e Souda, R. 227 Hayden, B.E. 496, 790 Haydock, R. s e e Gallagher, J. 644 Haydock, R. s e e Gibson, A. 181 Haydock, R. s e e Haneman, D.R. 359 Hayes, F.H. s e e Brennan, D. 494 Hayes, W. 226 Hazma, A.V. see Schildbach, M.A. 183 He, Y.L. see Zuo, J.K. 792 Headrick, R.L. 496 Hehre, W.J. 181 Heidberg, J. 570 Heidemann, A.D. s e e Larese, J.Z. 571 Heilman, P. s e e Lang, E. 359 Heimann, P. see Himpsel, F.J. 181 Heine, V. s e e Burt, M.G. 495 Heine, V. s e e Finnis, M.W. 134 Heiney, P. s e e Horn, P.M. 571,790 Heiney, P.A. s e e Birgeneau, R.J. 568 Heiney, P.A. s e e Guryan, C.A. 570 Hciney, P.A. see Stephens, P.W. 574 Heinz, K. 497 Heinz, K. see Besold, G. 494 Heinz, K. s e e Bickel, N. 225 Heinz, K. s e e Chubb, S.B. 495 Heinz, K. s e e Eggeling, von, C. 495

Heinz, K. s e e Lang, E. 359 Heinz, K. s e e Mendez, M.A. 498 Heinz, K. s e e MUller, K. 646 Heinz, K. s e e Muschiol, U. 499 Heinz, K. s e e Oed, W. 499, 791 Heinz, K. s e e Pendry, J.B. 791 Heinz, K. s e e Rous, P.J. 360 Heinz, K. s e e Wedler, H. 501 Held, G. s e e Barbieri, A. 642 Held, G. s e e Feenstra, R.M. 266 Held, G. s e e Greiser, N. 570 H61d, G. s e e JiJrgens, D. 645 Held, G. s e e Keane, D.T. 790 Held, G. s e e Lindroos, M. 498 Held, G. s e e Pfntir, H. 500 Held, G. s e e Schwennicke, C. 648 Heigesen, G. s e e Ocko, B.M. 135 Hellmann, H. 134 Henderson, B. 181 Henderson, M.A. 226 Hennig, D. s e e Methfessel, M. 97 Henrich, V E. 181,226 Henrich, V E. s e e Gay, R.R. 225 Henrich, V E. s e e Lad, R.J. 226 Henrich, V E. s e e Rohrer, G.S. 183,227 Henrich, V E. s e e Zhang, Z. 184 Henriot, M s e e Bart, F. 225 Henry, C.R. s e e Duriez, C. 225 Henzler, M. 359, 710 Henzler, M. s e e Busch, H. 789 Henzler, M. s e e Hahn, P. 710 Henzler, M. s e e Horn, M. 710 Henzler, M. s e e Schimmelpfenning, J. 574 Hering, S.V. 570 Herman, F. 645 Herman, G.S. s e e Bullock, E.L. 133 Hermann, K. 497, 570 Hermann, K. s e e Watson, P.R. 649 Hermanson, J.C. s e e Wan, K.J. 501 Hermsmeier, B.D. s e e Stoehr, J. 420 Herrera-Gomez, A. s e e Woicik, J.C. 502 Herring, C. 97 Hertel, T. 497 Hertel, T. s e e Bludau, H. 494 Hertel, T. s e e Gierer, M. 496 Hertel, T. s e e Over, H. 499 Hertz, J.A. s e e Einstein, T.L. 644

826

Heskett, D. 497 Heskett, D. s e e Frank, H.H. 496 Heslinga, D.R. 497 Hess, G.B. 570 Hess, G.B. s e e Drir, M. 569 Hess, G.B. s e e Nham, H.S. 573 Hess, G.B. s e e Youn, H.S. 575 Heydenreich J. s e e Bethge, H. 418 Heyraud, J.C. 97, 134 Heyraud, J.C. s e e Alfonso, C. 96 Heyraud, J.C. s e e Metois, J.E. 97 Hibma, T. s e e Heslinga, D.R. 497 Hibma, T. s e e Peacor, S.D. 227 Hickernell, D.C. see Bretz, M. 568 Higashiyama, K. 497 Higashiyama, K. s e e Kono, S. 135 Hikita, T. 181,226 Hildner, M.L. 790 Hildner, M.L. s e e Daley, R.S. 133,789 Hill, N.R. 359 Hill, R.J. 181 Hill, T.L. 97 Hillert, B. s e e Becker, L. 494 Hillert, B. see Pedio, M. 500 Himpsel, F. see Eberhardt, W. 495 Himpsel, F.J. 181,267 Himpscl, F.J. s e e Lang, N.D. 571 Himpsel, F.J. s e e McLean, A.B. 498 Hinncn, C. see Liu, S.H. 646 Hirabayashi, K. 181 Hirata, A. 226 Hirschfelder, J.O. 570, 645 Hirschom, E.S. s e e Samsavar, A. 710 Hirschorn, E.S. 267 Hirth, J.P. s e e Srolovitz, D.J. 648 Hjalmarson, H.P. s e e Vogl, P. 183 Hjclmberg, H. 497,645 Hjeimberg, H. s e e Johansson, P.K. 645 Hnace, B.K. see Poirer, G.E. 227 Ho, K.M 134, 790 Ho, K.M s e e Bohnen, K.P. 133,418 Ho, K.M s e e Chan, C.T. 133 Ho, K.M s e e Ding, Y.G. 134, 495,789 Ho, K.M s e e Fu, C.L. 134, 790 Ho, K.M s e e Louie, S.G. 135 Ho, K.M s e e Liu, S.H. 646 Ho, K.M. s e e Takeuchi, N. 136

A uthor

Ho, K.M. s e e Wang, X.W. 136, 792 Ho, K.M. s e e Xu, C.H. 136 Ho, K.M. s e e Zhang, B.L. 136 Ho, P.S. s e e Poon, T.W. 98,647 Ho, W. s e e Richter, L.J. 647 Hobson, J.P. s e e Edmonds, T. 569 H~che, H. s e e Keller, K.W. 710 Hochella, Jr., M.F. 181 Hochella, Jr., M.F. s e e Johnsson, P.A. 226 Hoegen, V. s e e Henzler, M. 710 Hoeven, A.J. s e e Dijkkamp, D. 97 HOfer, H. s e e Wintterlin, J. 501 Hoffmann, F.M. 497 Hoffmann, F.M. s e e Bradshaw, A.M. 494 Hoffmann, F.M. s e e Heskett, D. 497 Hoffmann, F.M. s e e Pfntir, H. 500 Hoffmann, R. 181,497,645 Hoffmann, R. s e e Halet, J.F. 226 Hoffmann, R. s e e Jansen, S.A. 226 Hoffmann, R. s e e Wong, Y.-T. 650 Hofmann, P. 497 Hofmann, P. s e e Bare, S.R. 494 Hofmann, P. s e e Bradshaw, A.M. 494 Hofmann, S. s e e Ichimura, S. 226 Hohage, M. s e e Michely, T. 710 Hohenberg, P. 134, 181,497 Hohlfeld, A. s e e Horn, K. 497 Hoinkes, H. 359 Hoinkes, H. s e e Finzel, H.-U. 359 Holland, B.W. s e e Onuferko, J.H. 791 Holland, B.W. s e e Zimmer, R.B. 360 Hollins, P. s e e Horn, K. 497 Holloway, P.H. 497, 645 Holloway, S. s e e N~rskov, J.K. 499, 647 Holmstr(Sm, S. s e e Nordlander, P. 647 Holub-Krappe, E. s e e Frenken, J.W.M. 496 Hong, H. 571,790 Hong, I.H. 497 Honjo, G. s e e Hopster, H. 497 Hopster, H.J. s e e Falicov, L.M. 789 Horn, K. 497,571 Horn, K. s e e Frenken, J.W.M. 496 Horn, K. s e e Hermann, K. 570 Horn, M. 710 Horn, M. s e e Henzler, M. 710 Horn, P.M. 571,790

index

Author

827

index

Horn, P.M. s e e Dimon, P. 569 Horn, P.M. s e e Greiser, N. 570 Horn, P.M. s e e Hong, H. 571,790 Horn, P.M. s e e Mochrie, S.G.J. 572 Horn, P.M. s e e Nagler, S.E. 572 Horn, P.M. s e e Specht, E.D. 574 Horn, P.M. s e e Stephens, P.W. 574 Horng, S.F. s e e Horsky, T.N. 267 Horsky, T.N. 267 Horsky, T.N. s e e Canter, K.F. 180 Horton, L.L. s e e Wang, Z.L. 227 Hosaka, S. s e e Hasegawa, T. 496 Hoshino, T. s e e Dederichs, P.H. 643 Hosokawa, Y. s e e Kirschner, J. 419 Hosoki, S. s e e Hasegawa, T. 496 Hou, Y. s e e Aono, M. 225 Houm~ller, A. s e e NCrskov, J.K. 499 Houston, J.E. s e e Park. R.L. 359 Howes, P.B. 134 Hrbek, J. 497 Hsu, C.-H. 790 Hsu, T. 226 Hsu, T. s e e Kim, Y. 226 Hu, G.Y. 790 Hu, P. see Barnes, C.J. 494 Hu, P. s e e Lindroos, M. 498 Hu, W.Y. see Tong, S.Y. 268 Huang, H. 134, 497 Huang, H. see Over, H. 499 Huang, H. s e e Tong, S.Y. 268,501 Huang, H. s e e Wei, C.M. 136 Huang, H. see Wu, H. 502 Huang, Y. s e e Tobin, J.G. 501 Huang, Z. 497 Hubbard, A.T. 419 Huber, D.L. s e e Ching, W.Y. 643 Hudson, J.B. see Holloway, P.H. 497, 645 Huerta-Garnica, M. s e e Rousset, S. 98 Hues, S.M. 419 Huff, G.B. 571 Huff, W.T. s e e Huang, Z. 497 Htifner, S. s e e Courths, R. 643 Hui, K.C. 497 Hui, K.C. s e e Wong, P.C. 502 Hulburt, S.B. 571 Humbert, A. s e e Thibaudau, F. 501 Hurst, J. s e e Pinkvos, H. 419

Hurych, Z. s e e Broden, G. 494 Hurych, Z. s e e Soukiassian, P. 136 Huse, D.A. 571,790 Hussain, M. s e e Horn, K. 497 Hussain, Z. s e e Huang, Z. 497 Hwang, I.-S. 497 Hwang, I.-S. s e e Ganz, E. 496, 790 Hwang, J. s e e Pate, B.B. 182 Hwang, R.Q. 645,710 Hwang, R.Q. s e e Ogletree, D.F. 647 Hwang, R.Q. s e e Williams, E.D. 360 Hybertsen, M.S. 134 Hybertsen, M.S. s e e Becker, R.S. 494 Hybertsen, M.S. s e e Zegenhagen, J. 502 Hyde, B.G. s e e O'Keefe, M. 182 lannotta, S. s e e Ellis, T.H. 569 larlori, S. 181 Ibach, H s e e Frohn, J. 97,644 Ibach, H s e e Giesen-Seibert, M. 644 lbach, H s e e Lehwald, S. 498,791 Ibach, H s e e Poensgen, M. 98 Ibach, H s e e Rahman, T.S. 500 Ibach, H see Sander, D. 98 lbach, H s e e Voigtltinder, B. 649 lchimura, S. 226 lchinokawa, T. s e e Itoh, H. 226 Ichinokawa, T. s e e Kirschner, J. 419 Ichinose, T. s e e Itoh, H. 226 lchninokawa, T. 710 Ignatiev, A. 571 Ignatiev, A. s e e Over, H. 499 lhm, G. s e e Jung, D.R. 571 lhm, G. s e e Vidali, G. 575, 649 lhm, J. 134, 267 lhm, J. s e e Aspnes, D.E. 265 Ikeda, N. s e e Kirschner, J. 419 ll'chenko, L.G. s e e Braun, O.M. 643 lllas, F. s e e Bagus, P.S. 494 Imbeck, R. s e e B6ttcher, A. 494 lmbihl, R. 497,645 lmbihl, R. s e e Eiswirth, M. 496 Imbihi, R. s e e Moritz, W. 646 Imbihl, R. s e e Rose, K.C. 420 Indovina, V. s e e Boudart, M. 180 Inglesfield, J.E. 790 Ino, S. s e e Daimon, H. 133

828 Ino, S. s e e Gotoh, Y. 710 Ipatova, I.P. s e e Maradudin, A.A. 359 Ishida, H. 497 lshikawa, T. s e e Takahashi, T. 136, 501 Ishikawa, Y. s e e Kirschner, J. 419 Ishimoto, K. s e e Kumagai, Y. 498 lshizawa, Y. s e e Aono, M. 225 lshizawa, Y. s e e Otani, S. 227 lshizawa, Y. s e e Souda, R. 227,420 Israelachvili, J.N. 419 Itchkawitz, B.S. 790 Itoh, H. 226 Jackman, T.E. 419, 497 Jackman, T.E. s e e Norton, P.R. 647 Jackson, A.G. 43,226, 359 Jackson, D.P. s e e Jackman, T.E. 497 Jackson, K. 134 Jackson, K.A. s e e Leamy, H.J. 43 Jackson, K.A. s e e Perdew, J.P. 647 Jacobi, K. 497 Jacobi, K. s e e Ranke, W. 268 Jacobi, K. s e e Shi, H. 500 Jacobsen, K.W. 497, 790 Jacobsen, K.W. s e e Feidenhans'i, R. 496 Jacobsen, K.W. s e e Stokbro, K. 136 Jacoby, M. s e e Marks, L.D. 419 Jaehnig, M. s e e Gt~pel, W. 226 Jaffee, H. 419 Jtiger, R. s e e St6hr, J. 501 Jahns, V. s e e Meyerheim, H.L. 498 Jakubith, S. 497 Jakubith, S. s e e Rotermund, H.H. 500 Jaloviar, S.G. s e e De Miguel, J.J. 87 James, R. s e e Kaufman, R. 359 James, R.W. 359 Jamison, K.D. s e e Behm, B.J. 133 Janak, J.L. s e e Moruzzi, V.L. 646 Janata, J. 181 Jansen, S.A. 226 Jardin, J.P. s e e Desjonqu/~res, M.C. 643 Jasnow, D. s e e Ohta, T. 791 Jasperson, S.N. 571 Jaszczak, J.A. s e e Wolf, D. 99, 650 Jaubert, M. s e e Glachant, A. 570 Jayaprakash, C. 97,645 Jayaram, G. s e e Collazo-Davila, C. 418

Author

index

Jefferson, D.A. s e e Smith, D.J. 227 Jefferson, D.A. s e e Zhou, W. 228 Jeng, S.-P. s e e Zhang, Z. 184 Jennings, G. s e e Campuzano, J.C. 789 Jennison, D.R. 645 Jensen, F. 497, 790 Jensen, F. s e e Feidenhans'l, R. 496 Jensen, F. s e e Mortensen, K. 499 Jensen, L.H. s e e Stout, G.H. 360 Jentjens, R. s e e Giesen-Seibert, M. 644 Jentz, D. s e e Barbieri, A. 642 Jeon, D. 134, 790 Jepsen, D.W. s e e Demuth, J.E. 495 Jepsen, D.W. s e e Marcus, P.M. 498 Jin, A.J. 571 Joannopoulos, J.D. s e e Alerhand, O.L. 96, 265,788 Joannopoulos, J.D. s e e Brommer, K.D. 133 Joannopoulos, J.D. s e e Kaxiras, E. 267 Joannopoulos, J.D. s e e Needels, M. 267 Joannopoulos, J.D. s e e Rappe, A.M. 135 Joannopoulos, J.D. s e e Vanderbilt, D. 268, 710 Johansson, L.I. 226 Johansson, L.I. s e e Hammar, M. 226 Johansson, L.I. s e e Lindberg, P.A.P. 227 Johansson, L.I. s e e Lindstrtim, J. 227 Johansson, L.I. s e e Rundgren, J. 227 Johansson, P.K. 645 Johansson, P.K. s e e NCrskov, J.K. 499 Johnson, D.D. 645 Johnson, E.D. s e e Cocke, D.L. 225 Johnson, K.E. 710 Johnson, K.L. 419 Johnson, R.J. s e e Feidenhans'l, R. 496 Johnson, R.L. s e e Bohr, J. 266 Johnson, R.L. s e e Dev, B.N. 789 Johnson, R.L. s e e Dornisch, D. 134, 495 Johnson, R.L. s e e Feidenhans'l, R. 266, 496, 789 Johnson, R.L. s e e Grey, F. 266 Johnson, R.L. s e e Seehofer, L. 500 Johnsson, P.A. 226 Joly, Y. s e e Rundgren, J. 227 Jona, F. s e e Himpsel, F.J. 267 Jona, F. s e e Huang, H. 497 Jona, F. s e e Kleinle, G. 498 Jona, F. s e e Over, H. 499, 500

Author

index

Jona, F. s e e Quinn, J. 135,500 Jona, F. s e e Sokolov, J. 500 Jona, F. s e e Yang, W.S. 502 Jona, F. s e e Zanazzi, E. 360 Jones, A.V. s e e Ignatiev, A. 571 Jones, B.A. 645 Jones, E.R. s e e McKinney, J.T. 359 Jones, R.G. s e e Kerkar, M. 497 Jones, R.O. 134, 181 Joos, B. 97 Jordan-Sweet, J.L. s e e Keane, D.T. 790 Jos6, J.V. 790 Josell, D. 97 Joyce, B.A. 267 Joyce, K. 645 Joyner, R.W. s e e MacLaren, J.M. 646 Jung, D.R. 571 Jupille, J. 790 JiJrgens, D. 645 J~irgens, D. see Schwennicke, C. 648 J~irgens, D. s e e Sklarek, W. 648 Kaburagi, M. 645 Kaburagi, M. s e e Ebina, K. 643 Kaburagi, M. s e e Urano, T. 183 Kadanoff, L.P. 790 Kadanoff, L.P. s e e Jos6, J.V. 790 Kahn, A. 267 Kahn, A. s e e Chen, W. 266 Kahn, A. s e e Duke, C.B. 266 Kahn, A. s e e Horsky, T.N. 267 Kahn, A. s e e Lessor, D.L. 267 Kalkstein, D. 645 Kampshoff, E. s e e Heidberg, J. 570 Kanaji, T. see Urano, T. 183 Kanamori, J. s e e Kaburagi, M. 645 Kang, H.C. 645 Kang, J.M. s e e Gay, J.M. 570 Kang, M.C. s e e Zhang, T. 650 Kang, W.M. s e e Tobin, J.G. 501 Kang, W.M. s e e Tong, S.Y. 501 Kao, C.-T. s e e Ohtani, H. 647 Kaplan, R. 226 Kappus, W. 645 Kara, A. s e e Chung, S. 568 Kardar, M. s e e Caflisch, R.G. 568 Kardar, M. s e e Halpin-Healy, T. 570

829 Kariotis, R. 97 Kariotis, R. s e e Aumann, C.E. 265 Kariotis, R. s e e de Miguel, J.J. 87, 710 Kariotis, R. s e e Hamichi, M. 570 Kariotis, R. s e e Lagally, M.G. 267 Kariotis, R. s e e Swartzentruber, B.S. 98,268, 710 Kariotis, R. s e e Venables, J.A. 575 Karlin, B.A. s e e Woicik, J.C. 502 Karlsson, C.J. s e e Northrup, J.E. 135 Kaski, K. s e e Gawlinski, G.T. 710 Kaski, K. s e e Rikvold, P.A. 648 Kasper, E. s e e Hawkes, P.W. 419 Katayama, M. 497 Kato, H. s e e Edamoto, K. 225 Kato, M. s e e Katayama, M. 497 Kaufman, R. 359 Kaufman, R. s e e James, R.W. 359 Kaukasoina, P. s e e Fisher, D. 496 Kawai, M. s e e Hikita, T. 181,226 Kawai, S. s e e Matsumoto, T. 182, 227 Kawai, S. s e e Nakamatasu, H. 182 Kawai, S. s e e Oshima, C. 227 Kawai, S. s e e Tanaka, H. 227 Kawai, T. s e e Matsumoto, T. 182, 227 Kawai, T. s e e Tanaka, H. 227 Kawasaki, K. s e e Ohta, T. 791 Kawazu, A. 497 Kawazu, A. s e e Sakama, H. 268 Kawazu, A. s e e Shioda, R. 136, 500 Kaxiras, E. 267 Kaxiras, E. s e e Lyo, I.-W. 135,498 Kaxiras, E. s e e Martensson, P. 135,498 Kaxiras, E. s e e Rappe, A.M. 135 Keane, D.T. 790 Keat, P.P. s e e Shropshire, J. 183 Keating, P.N. 134 Keeffe, M.E. 97 Kehr, K.W. 790 Keller, D. s e e Bustamante, C. 418 Keller, K.W. 710 Kellogg, G. 790 Kellogg, G. s e e Schwoebel, P.R. 648 Kelly, A. 226 Kemmochi, M. s e e Kirschner, J. 419 Kendall, K. s e e Johnson, K.L. 419 Kendelewicz, T. s e e Soukiassian, P. 136

830 Kendelewicz, T. s e e St6hr, J. 501 Kendelewicz, T. s e e Woicik, J.C. 502 Kenik, E.A. s e e Wang, Z.L. 227 Kerkar, M. 497 Kern, K. 571,710 Kern, K. s e e David, R. 569 Kern, K. s e e Zeppenfeld, P. 575 Kern, R. 97 Kern, R. s e e Quentel, G. 573 Kersten, H.H. s e e Turkenburg, W.C. 420 Kesmodel, L.L. s e e Van Hove, M.A. 136 Kevan, S.D. 267, 498 Kevan, S.D. s e e Skelton, D.C. 648 Kevan, S.D. s e e Wei, D.H. 649 Khare, S.V. 97,645 Khare, S.V. s e e Einstein, T.L. 134 Khare, S.V. s e e Nelson, R.C. 98 Khatir, Y. 571 Khokonov, K.B. s e e Kumikov, V.K. 97 Khor, K.E. s e e Kodiyalam, S. 97 Kikuta, S. s e e Takahashi, T. 136, 501 Kilcoyne, A.L.D. s e e Robinson, A.W. 791 Kim, H.-Y. s e e Vidali, G. 649 Kim, H.K. 571 Kim, H.K. s e e You, H. 711 Kim, H.K. see Zhang, Q.M. 575,792 Kim, H.Y. s e e Jung, D.R. 571 Kim, H.Y. s e e Vidali, G. 575 Kim, Y. 226 Kim, Y. s e e Hsu, T. 226 Kim, Y.S. s e e Gordon, R.G. 645 Kimura, Y. s e e Shibata, A. 791 King, D.A. s e e Bare, S.R. 494 King, D.A. s e e Barnes, C.J. 494 King, D.A. s e e Bowker, M. 642 King, D.A. s e e Debe, M.K. 134, 789 King, D.A. s e e Griffiths, K. 790 King, D.A. s e e Hofmann, P. 497 King, D.A. s e e Jupille, J. 790 King, D.A. s e e Lamble, G.M. 498 King, D.A. s e e Lindroos, M. 498 King, D.A. s e e Stensgaard, I. 792 King-Smith, R.D. s e e Stich, I. 136 King-Smith, R.D. s e e Ramamoorthy, M. 227 Kingdon, K.H. s e e Langmuir, I. 498 Kingsbury, D.L. s e e Ma, J. 572 Kinniburgh, C.G. 181,359

Author

Kinoshita, T. 498 Kinosita, K. s e e Gotoh, T. 181 Kinzel, W. 645,646 Kinzel, W. s e e Selke, W. 648 Kirkpatrick, S. s e e Jos6, J.V. 790 Kirschner, J. 419 Kiskinova, M. 498 Kiskinova, M. s e e Cautero, G. 495 Kiskinova, M. s e e Comeili, G. 495 Kiskinova, M. s e e Dhanak, V.R. 495 Kittaka, S. s e e Morishige, K. 572 Kittel, C. 134, 182, 498,571 Kittel, C. s e e Ruderman, M.A. 648 Kjaer, K. 571 Kjaer, K. s e e Bohr, J. 568 Kjems, J.K. 571 Kjems, J.K. s e e Taub, H. 574 Klapwijk, T.M. s e e Heslinga, D.R. 497 Kleban, P. 710, 790 Kleban, P. s e e Bak, P. 788 Kleban, P.H. s e e Clark, D.E. 789 Klebanoff, L.E. s e e Tobin, J.G. 501 Klein, J. s e e Rousset, S. 98 Klein, J.R. 646 Klein, M.L. s e e Peters, C. 573 Klein, M.L. s e e Ruiz-Suarez, J.C. 573 Kleiner, J. s e e Mo, Y.W. 710 Kleinle, G. 498 Kleinle, G. s e e Over, H. 499 Kleinman, L. s e e Batra, I.P. 494 Kleinman, L. s e e Bylander, D.M. 495 Klemperer, O. 359 Klier, K. 498 Klimov, A s e e Galeotti, M. 225 Kiink, C. 790 Klink, C. s e e Mortensen, K. 499 Klitsner, T. 267 Klitsner, T. s e e Becker, R.S. 265 Knorr, K. 571 Knorr, K. s e e Faul, J.W.O. 569 Knorr, K. s e e Koort, H.J. 571 Knorr, K. s e e Shirazi, A.R.B. 574 Knorr, K. s e e Volkmann, U.G. 575 Knorr, K. s e e Weimer, W. 575 Knotek, M.L. 226 Kn~Szinger, H. 182 Kobayashi, K. 134

index

Author

831

index

Koch, J. s e e Ertl, G. 496 Koch, R. 498 Koch, R. s e e Haase, O. 496 Koch, S.W. 571 Koch, S.W. s e e Abraham, F.F. 567 Kochanski, G.P. s e e Griffith J.E. 267 Kodiyalam, S. 97 Koel, B.E. s e e Parrott, L. 500 Koestner, R.J. 498 Koestner, R.J. s e e Van Hove, M.A. 136, 501 Kogut, J.B. 646 Kohler, U. s e e Henzler, M. 710 Kohn, W. 134, 182, 498,646 Kohn, W. s e e Hohenberg, P. 134, 181,497 Kohn, W. s e e Lang, N.D. 498 Kohn, W. s e e Lau, K.H. 646 Kolaczkiewicz, J. 790 Kolaczkiewicz, J. s e e Rogowska, J.M. 648 Kolasinski, K.W. s e e Kubiak, G.D. 182 Koma, A. s e e Hirata, A. 226 Komolov, S.A s e e Mr P.J. 227 Kono, S. 134, 135,498 Kono, S. s e e Abukawa, T. 133,493 Kono, S. s e e Higashiyama, K. 497 Kono, S. s e e Kinoshita, T. 498 Koort, H.J. 571 Koranda, S. s e e Stoehr, J. 420 Kordesch, M. s e e Engel, W. 496 Kordesch, M.E. 419 Kordesch, M.E. s e e Garcia, A. 418 Kordesch, M.E. s e e Rotermund, H.H. 500 Korringa, J. 646 Kortan, A.R. s e e Roelofs, L.D. 791 Korte, U. s e e Schwegmann, S. 500 Kose, R. s e e Over, H. 500 Koster, G.F. 646 Koster, G.F. s e e Slater, J.C. 183 Koster, G.F. s e e Slater, J.R. 136 Kosterlitz, J.M. 359 Kosterlitz, J.M. 790 Kosterlitz, M. 571 Kotliar, B.G. s e e Jones, B.A. 645 Kouteck, J. 646 Krakauer, H. s e e Fernando, G.W. 134 Krakauer, H. s e e Roelofs, L.D. 135 Krakauer, H. s e e Roelofs, L.D. 791 Krakauer, H. s e e Singh, D. 136

Krakauer, H. s e e Singh, D. 791 Krakauer, H. s e e Wimmer, E. 136 Krakauer, H. s e e Wimmer, E. 650 Krakauer, H. s e e Yu, R. 136 Kramer, H.M. 571 Kramer, H.M. s e e Venables, J.A. 575 Krans, R.L. s e e Frenken, J.W.M. 496 Kress, W. s e e de Wette, F.W. 180 Kress, W. s e e Reiger, R. 183 Kreuzer, H.J. s e e Payne, S.H. 98, 647 Kriebel, D.L. s e e Roelofs, L.D. 791 Krim, J. 571 Krim, J. s e e Bienfait, M. 568 Krim, J. s e e Gay, J.M. 570 Krim, J. s e e Migone, A.D. 572 Krim, J. s e e Muirhead, R.J. 572 Krischer, K. 498 Krischer, K. s e e Eiswirth, M. 496 Kroll, C. 226 Kronberg, M.L. 226 Kruger, P. see Landemark, E. 267 Kubala, S. s e e Engel, W. 496 Kubby, J.A. 267 Kubby, J.A. s e e Soukiassian, P. 136 Kubiak, G.D. 182 Kubiak, G.D. s e e Sowa, E.C. 183 Kudo, M. s e e Hikita, T. 181,226 Kuhlenbeck, H. s e e B~iumer, M. 225 Ktihnemuth, R. s e e Heidberg, J. 570 Kuipers, L. 97 Kuk, Y. 646 Kulik, A.J. s e e Burnham, N.A. 418 Kumagai, Y. 498 Kumar, S. s e e Gawlinski, G.T. 710 Kumikov, V.K. 97 Kunkel, R. 710 Kunz, A.B. 182 Kuppers, J. s e e Ertl, G. 43, 359, 496 Ktippers, J. s e e Woratschek, B. 502 Kurtz, R.L. 226 Kvick, ~. s e e Smyth, J.R. 183 L'Ecuyer, J. 419 Lackey, D. s e e Hayden, B.E. 790 Lad, R.J. 226 Lad, R.J. s e e Antonik, M.D. 225 Ladas, S. s e e Imbihl, R. 497

832 Laegsgaard, E. s e e Klink, C. 790 LaFemina, J.P. 182, 267 LaFemina, J.P. s e e Blanchard, D.L. 180, 225 LaFemina, J.P. s e e Gibson, A. 181 LaFemina, J.P. s e e Godin, T.J. 181,266 Lagally, M.G. 267, 359, 571, 710, 790 Lagally, M.G. s e e Aumann, C.E. 265 Lagally, M.G. s e e Barnes, R.F. 358 Lagally, M.G. s e e Ching, W.Y. 643 Lagally, M.G. s e e de Miguel, J.J. 87, 710 Lagally, M.G. s e e Haneman, D. 267 Lagally, M.G. s e e Martin, J.A. 359 Lagally, M.G. s e e Mo, Y.W. 710 Lagally, M.G. s e e Saloner, D. 710 Lagally, M.G. s e e Swartzentruber, B.S. 98, 268,710 Lagally, M.G. s e e Tringides, M.C. 710 Lagaily, M.G. s e e Wang, G.-C. 98,649 Lagally, M.G. s e e Webb, M.B. 268,360, 792 Lagally, M.G. s e e Welkie, D.G. 711 Lagally, M.G. see Wu, P.K. 711 Lagerof, K.P.D. s e e Lee, W.E. 226 Lahee, A.M. 498 Lahee, A.M. s e e Campuzano, J.C. 789 Lahee, A.M. see Harten, U. 134, 710 Lakhlifi, A. 571 Lam, D.J. s e e Ellis, D.E. 181 Lain, D.J. s e e Guo, J. 181,226 Lambert, R.M. 43 Lambert, R.M. s e e Bridge, M.E. 494 Lambert, R.M. s e e Comrie, C.M. 495 Lambert, W.R. 267 Lambeth, D.N. s e e Falicov, L.M. 789 Lamble, G.M. 498 Landau, D.P. s e e Binder, K. 642, 788,789 Landau, L.D. 571,790 Landemark, E. 267 Lander, J.J. 135 Landman, U. 419 Landree, E. s e e Collazo-Davila, C. 418 Lanclskron, H. s e e Pendry, J.B. 791 Lang, E. 359 Lang, N.D. 419, 498, 571,646 Lang, N.D. s e e Klink, C. 790 Lang, N.D. see N~rskov, J.K. 499, 647 Lang, N.D. s e e Williams, A.R. 649 Lang, P. s e e Dederichs, P.H. 643

Author

Langell, M.A. 226 Langmuir, I. 498 Langreth, D.C. 135 Lanoo, M. 182 Lapeyre, G.J. s e e Wu, H. 502 Lapujoulade, J. 790 Lapujoulade, J. s e e Ernst, H.J. 710 Lapujoulade, J. s e e Villain, J. 360 Larese, J.Z. 571,572 Larese, J.Z. s e e Chung, S. 568 Larese, J.Z. s e e Guryan, C.A. 570 Larher, Y. 572 Larher, Y. s e e Nardon, Y. 572 Larher, Y. s e e Ser, F. 57 Larher, Y. s e e Teissier, C. 574 Larher, Y. s e e Terlain, A. 574 Larsen, P.K. s e e Joyce, B.A. 267 Larson, B.E. s e e Brommer, K.D. 133 Larson, B.E. s e e Hsu, C.-H. 790 Lau, K.H. 646 Lau, K.H. s e e Kohn, W. 646 Lauter, H.J. s e e Croset, B. 569 Lauter, H.J. s e e Cui, J. 569 Lauter, H.J. s e e Degenhardt, D. 569 Lauter, H.J. s e e Feile, R. 570 Lauter, H.J. s e e Freimuth, H. 570 Lauter, H.J. s e e Gay, J.M. 570 Lauter, H.J. s e e Kjaer, K. 571 Lauter, H.J. s e e Madih, K. 572 Lauter, H.J. s e e Taub, H. 792 Lauter, H.J. s e e Tiby, C. 574, 575 Lauter, H.J. s e e Wiechert, H. 575 Law, D.S.L. s e e Lindberg, P.A.P. 227 Law, D.S.L. s e e Lindstr6m, J. 227 Lazneva, E.F. s e e MOiler, P.J. 227 La~gsgaard, E. s e e Eierdal, L. 495 L~egsgaard, E. s e e Feidenhans'i, R. 496 Ltegsgaard, E. s e e Jensen, F. 497, 790 Le Boss6, J.C. 646 Leadbetter, A.J. 182 Leamy, H.J. 43 Lebehot, A. s e e Campargue, R. 358 Lee, B.W. s e e Lubinsky, A.R. 227,267 Lee, K.B. s e e Guryan, C.A. 570 Lee, P.A. s e e Coppersmith, S.N. 568 Lee, T.D. 791 Lee, W.E. 226

index

Author

833

index

LeGoues, F.K. s e e Tromp, R.M. 420 Lehmann, M.S. s e e Wright, A.F. 184 Lehmpfuhl, G. 419 Lehmpfuhl, G. s e e Wang, N. 421 Lehwald, S. 498, 791 Lehwald, S. s e e Rahman, T.S. 500 Lehwald, S. s e e Voigtl~inder, B. 649 Leibsle, F.M. 498 Leibsle, F.M. s e e Hirschorn, E.S. 267 Leibsle, F.M. s e e Murray, P.W. 227, 499 Leibsle, F.M. s e e Samsavar, A. 710 Leidheiser, Jr. H. s e e Klier, K. 498 Lelay, G. 135 Lemonnier, J.C. s e e Campargue, R. 358 Lenc, M. s e e Muellerova, I. 419 Lenglart, P. s e e Allan, G. 642 Lenssinck, J.M. s e e Dijkkamp, D. 97 Lent, C.S. 359 Lent, C.S. s e e Pukite, P.R. 359 Lenz, J. see Schwarz, E. 500 Lcrner, E. s e e Bienfait, M. 568 Lerner, E. see Krim, J. 571 Lerner, E. see Rapp, R.E. 573 Lessor, D.L. 267 Lessor, D.L. s e e Blanchard, D.L. 180, 225 Lessor, D.L. see Horsky, T.N. 267 Leung, W.Y. s e e Chung, S. 568 Levi, A.C. 359 Levi, A.C. s e e Garibaldi, V. 359 Levy, H.A. see Busing, W.R. 358 Lewis, G.V. 182 Leynaud, M. 646 Li, C.H. see Kevan, S.D. 498 Li, X.P. 135 Li, Y. s e e Murray, P.W. 499 Li, Y.S. s e e Jennison, D.R. 645 Li, Z.G. s e e Smith, D.J. 420 Li, Z.R. s e e Chan, M.H.W. 568 Li, Z.R. see Migone, A.D. 572 Liang, K.S. 359 Liang, W.Y. s e e Zhou, W. 228 Liang, Y. 182, 226 Liang, Y. s e e Chambers, S.A. 225 Liao, D.K. s e e Hong, I.H. 497 Liaw, H.P. 226 Liebau, F. 182 Liebermann, R.C. s e e Susman, S. 183

Liebsch, A. 359, 646 Lieske, N.P. 182 Liew, Y.F. 791 Lifshitz, I.M. 791 Lifshitz, I.M. s e e Landau, L.D. 790 Lighthill, M.J. 646, 791 Lin, D.L. s e e Zheng, H. 650 Lin, D.S. s e e Hirschorn, E.S. 267 Lin, M.E. s e e Morkoq, H. 227 Lin, X.F. s e e Wan, K.J. 136, 501 Lind, D.M. 227 Lindberg, P.A.P. 227 Linder, B. s e e MacRury, T.B. 646 Lindgren, S.A. 498 Lindhard, J. 419 Lindner, H. s e e Oed, W. 499 Lindner, Th. s e e Horn, K. 497 Lindroos, M. 498 Lindroos, M. s e e Barnes, C.J. 494 Lindroos, M. s e e Fisher, D. 496 Lindroos, M. s e e PfniJr, H. 500 Lindsley, D.H. 227 Lindstri3m, J. 227 LindstrOm, J.B. s e e Lindberg, P.A.P. 227 Linke, U. s e e Sander, D. 98 Lippel, P.H. s e e Canter, K.F. 180 Lippel, P.H. s e e W~311,Ch. 136 Lipson, H. 359 Littleton, M.J. s e e Mott, N.F. 182 Litzinger, J.A. 572 Litzinger, J.A. s e e Butler, D.M. 568 Liu, C.L. 97 Liu, F.C. s e e Ma, J. 572 Liu, F.C. s e e Zeppenfeld, P. 575 Liu, H. s e e Himpsel, F.J. 267 Liu, J. 419 Liu, L.-G. 182 Liu, S.H. 646 Liu, W.-K. 646 Liu, W.-K. s e e Dai, X.Q. 643 Liu, W.-K. s e e Sulston, K.W. 649 Lo, W.J. s e e Chung, Y.W. 225 Lo, W.K. 419 Ltiffler, U. s e e Wedler, H. 501 Lohmeir, M. s e e van der Vegt, H.A. 710 Longoni, V. s e e Chini, P. 495 Lopez, J. 646

834 Lopez, J. s e e Le Boss6, J.C. 646 Lopez Vazquez-de-Parga, A. s e e Ogletree, D.F. 647 Lorrain, P. 182 Louie, S.G. 135 Louie, S.G. s e e Becker, R.S. 494 Louie, S.G. s e e Chelikowsky, J.R. 133 Louie, S.G. s e e Hybertsen, M.S. 134 Louie, S.G. s e e Tomfinek, D. 649 Louie, S.G. s e e Vanderbilt, D. 183 Louie, S.G. s e e Zhu, X. 268 Lovesey, S.W. s e e Marshall, W. 572 L6wdin, P.O. 646 Lowe, J.P. 182 Lowenstein, J. s e e Andrei, N. 642 Lu, C. 572 Lu, H.C. s e e Zhou, J.B. 228 Lu, Ping s e e Smith, D.J. 420 Lu, T.-M. s e e Presicci, M. 359 Lu, T.-M. s e e Wang, G.-C. 649, 792 Lu, T.M. see Lagally, M.G. 710 Lu, T.M. see Wang, G.C. 710 Lu, T.M. s e e Yang, H.N. 711 Lu, T.M. see Zuo, J.K. 711 Lu, W. see Smith, R.L. 227 Lu, W.C. s e e Dai, X.Q. 643 Lu, W.C. see Zhang, T. 650 Lu, Y.-T. s e e Zhang, Z. 99 Lubinsky, A.R. 227, 267 Lucovsky, G. s e e Pantelides, S.T. 182 Ludwig, A.W.W. s e e Affleck, I. 642 Luedtke, W.D. s e e Landman, U. 419 Lundgren, E. s e e Andersen, J.N. 493,494 Lundgren, E. s e e Nielsen, M.M. 499 Lundqvist, B. 646 Lundqvist, B.I. s e e NCrskov, J.K. 499 Lundqvist, S. 646 Lundqvist, S. s e e Lundqvist, B. 646 Lundqvist, S. s e e March, N.H. 135 Lupis, C.H.P. 572 Luschka, M. s e e Finzel, H.-U. 359 Luscombe, J.G. s e e Tringides, M.C. 710 Lutz, C.P. s e e Crommie, M.F. 643 Lutz, M.A. s e e Feenstra, R.M. 266 Lyo, I.W. 135,498 Lyo, I.W. s e e Avouris, Ph. 709 Lyo, S.-W. s e e Avouris, P. 265

Author

index

Ma, J. 572 Maca, F. s e e Scheffler, M. 500 Mfica, F. s e e Scheffler, M. 648 Macdonald, J.E. s e e Conway, K.M. 495 Macdonald, J.E. s e e Vlieg, E. 360 MacDonald, J.E. s e e Pluis, B. 359 MacDonald, J.E. s e e Van Silfhout, R.G. 268 MacDowell, A.A. s e e Robinson, I.K. 135, 791 MacGillavry, C.H. 43 MacLachlan, A.D. 572 MacLane, S. s e e Birkhoff, G. 642 MacLaren, J.M. 267,498,646 MacRae, A.U. s e e Germer, L.H. 496 MacRury, T.B. 646 Madden, H.H. s e e Park, R.L. 43, 573 Madelung, E. 182 Madey, T.E. 97, 498 Madey, T.E. s e e Maschhoff, B.L. 227 Madey, T.E. s e e Song, K.-J. 98 Madey, T.E. s e e Thiel, P.A. 183 Madhavan, P. 646 Madih, K. 572 Maglietta, M. 498 Magnanelli, S. s e e Bardi, U. 568 Mahanty, J. 646 Mailhiot, C. 267 Maiihiot, C. s e e LaFemina, J.P. 267 Mak, A. s e e Hong, H. 571,790 Mak, A. s e e Specht, E.D. 574 Maksym, P.A. 182 Malic, R.A. s e e McRae, E.G. 267 Manassidis, I. 227 Mangat, P. s e e Soukiassian, P. 136 Mansfield, M. s e e Needs, R.J. 97 Manzke, R. s e e Claessen, R. 225 Mao, D. s e e Chen, W. 266 Maradudin, A.A. 359, 646 Maradudin, A.A. s e e Cunningham, S.L. 643 Maradudin, A.A. s e e Dobrzynski, L. 643 Maradudin, A.A. s e e Eguiluz, A.G. 644 Maradudin, A.A. s e e Portz, K. 647 March, N.H. 135,646 March, N.H. s e e Flores, F. 644 March, N.H. s e e Joyce, K. 645 March, N.H. s e e Lundqvist, S. 646 March, N.H. s e e Mahanty, J. 646 Marchenko, V.I. 97,646

Author

index

Marcus, P.M. 498 Marcus, P.M. s e e Chubb, S.B. 495 Marcus, P.M. s e e Demuth, J.E. 495 Marcus, P.M. s e e Himpsel, F.J. 267 Marcus, P.M. s e e Quinn, J. 135,500 Marcus, P.M. s e e Sokolov, J. 500 Marcus, P.M. s e e Yang, W.S. 502 Maree, P.M.J. 267 Mark, P. s e e Duke, C.B. 266 Mark, P. see Lubinsky, A.R. 227, 267 Marks, L.D. 419 Marks, L.D. s e e At, R. 418 Marks, L.D. see Bonevich, J.E. 418 Marks, L.D. s e e Buckett, M.I. 225 Marks, L.D. see Collazo-Davila, C. 418 Marks, L.D. s e e Dunn, D.N. 418 Marshall, W. 572 Mfirtensson, P. 135,498 Mfirtensson, P. see Grehk, T.M. 496 Mfirtensson, P. see Nicholls, J.M. 135,499 Marti, C. 572 Marti, C. s e e Ceva, T. 568 Marti, C. s e e Coulomb, J.P. 569 Marti, C. see Croset, B. 569 Marti, C. s e e Thorel, P. 574 Marti, O. s e e Spatz, J.P. 420 Martin,, R.M. s e e Ordej6n, P. 135 Martin, A.J. 182 Martin, J.A. 359 Martin, J.A. s e e Lagally, M.G. 359 Martin, J.A. s e e Saloner, D. 710 Martin, J.l. s e e Roelofs, L.D. 791 Martin, R.M. s e e Chetty, N. 133 Martin-Rodero, A. s e e Joyce, K. 645 Martinez, R.E. 97 Martinez, V. s e e Soria, F. 500 Martini, K.M. s e e EI-Batanouny, M. 134 Martins, J.L. s e e Troullier, N. 136 Martir, E.I. s e e Roelofs, L.D. 648 Maschhoff, B.L. 227 Mashkova, E.S. 419 Mason, B.F. 572 Mason, M.G. s e e Kevan, S.D. 498 Mason, R. 498 Masri, P. s e e Coulomb, J.P. 568 Matecki, M. 572 Matecki, M. s e e Coulomb, J.P. 569

835 Matecki, M. s e e Dolle, P. 569 Materer, N. s e e Barbieri, A. 642 Materlik, G. s e e Dev, B.N. 789 Mathias, H. s e e Lind, D.M. 227 Mathiez, Ph. s e e Thibaudau, F. 501 Matsumoto, T. 182, 227 Matsumoto, T. s e e Tanaka, H. 227 Matsunami, N. s e e L'Ecuyer, J. 419 Matsushima, T. s e e Imbihl, R. 645 Matsushima, T. s e e Moritz, W. 646 Mattera, L. s e e Boato, G. 358 Mauck, M.S. s e e Rempfer, G.F. 419 Mayer, J.E. s e e Born, M. 180 McAlpine, N. s e e Haneman, D. 267 McCartney, M.R. s e e Gajdardziska-Josifovska, M. 225 McCartney, M.R. s e e Smith, D.J. 420 McColm, l.J. 227 McCormick, W.D. s e e Goodstein, D.L. 570 McCoy, B.M. s e e Tracy, C.A. 792 McGrath, R. s e e Diehl, R.D. 495 McKay, S.R. s e e Unertl, W.N. 360 McKee, R.A. s e e Wang, Z.L. 227 McKinney, J.T. 359 McLachlan, A.D. 646 McLean, A.B. 498 McLean, E.O. s e e Bretz, M. 568 McMillan, W.L. s e e Anderson, P.W. 642 McMurry, H.L. s e e Taub, H. 574 McRae, E.G. 267,791 McTague, J.P. 572 McTague, J.P. s e e Kjaer, K. 571 McTague, J.P. s e e Nielsen, M. 573 McTague, J.P. s e e Novaco, A.D. 499, 573 McTague, J.P. s e e Taub, H. 574 Meade, R.D. 97,498 Meade, R.D. s e e Alerhand, O.L. 96, 788 Meade, R.D. s e e Bedrossian, P. 133, 494 Meade, R.D. s e e V anderbilt, D. 268,710 Mednick, K. s e e Bylander, D.M. 495 Medvedev, V.K. s e e Braun, O.M. 643 Meehan, P. 572 Mehl, M.J. s e e Cohen, R.E. 643 Mehl, M.J. s e e Langreth, D.C. 135 Met, M.N. s e e Tong, S.Y. 268 Meichel, T. 572 Meichel, T. s e e Gay, J.M. 570

836 Meier, F. s e e Pierce, D.T. 500 Melle, H. 135 Melmed, A. 419 Men, F.K. 791 Men, F.K. s e e Tong, S.Y. 268 Men, F.K. s e e Webb, M.B. 268 Menaucourt, J. 572 Menaucourt, J. s e e Bockel, C. 568 Menaucourt, J. s e e Bouchdoug, M. 568 Menaucourt, J. s e e R6gnier, J. 573 Mendez, M.A. 498 Mendez, M.A. s e e Wedler, H. 501 Menzel, and E. s e e Melle, H. 135 Menzel, D. 498 Menzel, D. s e e Fuggle, J.C. 496 Menzel, D. s e e Hofmann, P. 497 Menzel, D. s e e Lindroos, M. 498 Menzel, D. see Madey, T.E. 498 Menzel, D. s e e Michalk, G. 498 Menzel, D. see Pfntir, H. 500 Menzel, D. s e e Steinkilberg, M. 501 Mcrrill, R.P. s e e Cocke, D.L. 225 Mcrzbacher, E. 419 Methfessel, M. 97, 135 Methfessel, M. s e e Fiorentini, V. 134 Metiu, H. see Zhang, Z. 99 Mctois, J.E. 97 Metois, J.J. see Alfonso, C. 96 Mctois, J.J. see Heyraud, J.C. 97, 134 Meyer, G. see Mfirtensson, P. 135,498 Meyer, J.A. 646 Meyer, J.A. s e e Kuk, Y. 646 Meyer, R.J. s e e Duke, C.B. 266 Meyerheim, H.L. 498 Michalk, G. 498 Michely, T. 710 Michely, T. s e e Bott, M. 418 Miedema, A.R. 498 Migone, A.D. 572 Migone, A.D. s e e Chan, M.H.W. 568 Migone, A.D. s e e Zhang, S. 575 Miller, J.S. s e e Plummer, E.W. 500 Miller, M. s e e Vlieg, E. 360 Miller, T . s e e Samsavar, A. 710 Miller, W.A. s e e Tyson, W.R. 98 Millis, A.J. s e e Jones, B.A. 645 Mills, D.L. s e e Hall, B. 570

Author

index

Mills, D.L. s e e Rahman, T.S. 500 Mills, Jr., A.P. s e e Canter, K.F. 180 Mills, Jr., A.P. s e e Horsky, T.N. 267 Milne, R.H. s e e Hui, K.C. 497 Mimata, K. s e e Morishige, K. 572 Miner, K.D. s e e Chan, M.H.W. 568 Mintmire, J.W. s e e Dunlap, B.I. 181 Miranda, R. 499, 572 Misawa, S. s e e Hara, S. 226 Mitchell, K.A.R. s e e Grimsby, D. 496 Mitchell, K.A.R. s e e Hui, K.C. 497 Mitchell, K.A.R. s e e Parkin, S.R. 500 Mitchell, K.A.R. s e e Vu Grimsby, D.T. 649 Mitchell, K.A.R. s e e Wong, P.C. 502 Miura, S. s e e Ichninokawa, T. 710 Miyano, K.E. s e e Woicik, J.C. 502 Miyazaki, E. s e e Edamoto, K. 225 Mo, Y.W. 710 Mo, Y.W. see Lagally, M.G. 267 Mo, Y.W. s e e Swartzentruber, B.S. 98,268, 710 Mochida, A. s e e Edamoto, K. 225 Mochrie, S.G.J. 359, 572 Mochrie, S.G.J. s e e Feidenhans'l, R. 496 Mochrie, S.G.J. s e e Nagler, S.E. 572 Mochrie, S.G.J. s e e Song, S. 98 Mochrie, S.G.J. s e e Yoon, M. 99 Mochrie, S.G.J. s e e Zehner, D.M. 792 Modesti, S. see Astaldi, C. 494 Molchanov, V.A. s e e Mashkova, E.S. 419 Moler, E.J. s e e Huang, Z. 497 Moliere, G. 419 Moiler, M. s e e Spatz, J.P. 420 Moiler, M.A. s e e Ruiz-Suarez, J.C. 573 M611er, P. s e e Eiswirth, M. 496 Mr P.J. 227 M6nch, W. 499 Moncton, D.E. s e e D'Amico, K.L. 569 Moncton, D.E. s e e Dimon, P. 569 Moncton, D.E. s e e Mochrie, S.G.J. 572 Moncton, D.E. s e e Nagler, S.E. 572 Moncton, D.E. s e e Stephens, P.W. 574 Moncton, D.E. s e e Specht, E.D. 574 Monkenbusch, M. 572 Montrol, E.W. s e e Maradudin, A.A. 359 Monty, C. s e e Dufour, L.C. 225 Moog, E.R. 572

Author

837

index

Moog, E.R. s e e Unguris, J. 575 Moore, A.J.W. 97 Moore, I.D. 646 Moore, I.D. s e e Flores, F. 644 Moore, W.T. s e e Hui, K.C. 497 Morgante, A. s e e Bellman, A.F. 494 Morgante, A. s e e Btittcher, A. 494 Mori, R. s e e Kumagai, Y. 498 Morikawa, Y. 135 Morikawa, Y. s e e Kobayashi, K. 134 Morishige, K. 572 Morita, S. s e e Ueyama, H. 420 Moritz, W. 135,646 Moritz, W. s e e Dornisch, D. 134, 495 Moritz, W. s e e Gierer, M. 496 Moritz, W. s e e Kleinle, G. 498 Moritz, W. s e e Meyerheim, H.L. 498 Moritz, W. s e e Michalk, G. 498 Moritz, W. s e e Over, H. 499 Moritz, W. s e e Stampfl, C. 501,792 Moritz, W. see Zuschke, R. 502 Morkoq, H. 227 Morkot/, H. see Strite, S. 227 Morrison, J. s e e Estrup, P.J. 496 Morrison, J. s e e Lander, J.J. 135 Morrison, J.A. s e e Peters, C. 573 Morse, P.M. 791 Mortensen, K. 499 Mortensen, K. s e e Bedrossian, P. 133,494 Mortensen, K. s e e Zegenhagen, J. 792 Moruzzi, V.L. 646 Moser, H.R. s e e Heskett, D. 497 Mott, N.F. 182 Motteler, F.C. 572 Mouritsen, O.G. 710 Mouritsen, O.G. s e e Fogedby, H.C. 789 Mowforth, C. s e e Morishige, K. 572 Mowforth, C.W. 572 Mross, W. 499 Mueilerova, I. 419 Muirhead, R.J. 572 Muller, K. s e e Bickel, N. 225 Muller, K. s e e Lang, E. 359 Muller, K. s e e Rous, P.J. 360 MOiler, E.W. 419 Mtiller, J. 499 Mtiller, J.E. 646

Mtiller, K. 646 Mtiller, K. s e e Besold, G. 494 Miiller, K. s e e Chubb, S.B. 495 Mtiller, K. s e e Eggeling, von, C. 495 MOiler, K. s e e Gruzalski, G.R. 226 Miiller, K. s e e Mendez, M.A. 498 Mtiller, K. s e e Oed, W. 499 MiJller, K. s e e Pendry, J.B. 791 Mullins, W.W. 97 Mullins, W.W. s e e Gruber, E.E. 97 Mundenar, J.M. 499 Munoz, M.C. s e e Soria, F. 500 Murakami, S. s e e Gotoh, T. 181 Murata, Y. s e e Aruga, T. 494 Murata, Y. s e e Gotoh, T. 181 Murray, P.W. 227,499 Murray, P.W. s e e Leibsle, F.M. 498 Muryn, C.A. s e e Murray, P.W. 227 Muscat, J.-P. 647 Muscat, J.P. 499 Muschiol, U. 499 Muto, Y. 572, 647 Mykura, H. s e e Blakely, J.M. 96 Myshlyavtsev, A.V. 647 Nagayoshi, H. 499 Nagayoshi, H. s e e Kono, S. 135 Nagler, S.E. 572 Nahm, H.S. s e e Drir, M. 569 Nahr, H. s e e Finzel, H.-U. 359 Najafabadi, R. 647 Nakagawa, K. s e e Maree, P.M.J. 267 Nakamatasu, H. 182 Nakamura, N. s e e Kono, S. 134 Nakatani, S. s e e Takahashi, T. 136, 501 Nakayama, T. s e e Takami, T. 501 Napartovich, A.P. s e e Bol'shov, L.A. 642 Narasimhan, S. 135 Nardon, Y. 572 Narusawa, T. 647 Nastasi, M. s e e Tesmer, J.R. 420 Naumovets, A.G. 499, 647 Naumovets, A.G. s e e Bol'shov, L.A. 642 Neddermeyer, H. s e e Btiumer, M. 225 Neddermeyer, H. s e e Wilhelmi, G. 501 Needels, M. 267 Needels, M. s e e Brommer, K.D. 133

838 Needs, R.,I. 97 Neilsen, M. s e e Dornisch, D. 495 Nelson, D.R. 573 Nelson, D.R. s e e Halperin, B.I. 570, 790 Nelson, D.R. s e e Jos6, J.V. 790 Nelson, J.S. s e e Klitsner, T. 267 Nelson, R.C. 98 Nenow, D. 573 Neubert, M. s e e Schwarz, E. 500 Neugebauer, J. 499 Neugebauer, J. s e e Bormet, J. 494 Neugebauer, ,i. see Schmalz, A. 500, 648 Neve, J. s e e Lindgren, S.A. 498 Newns, D.M. s e e Muscat, J.P. 499 Ncwns, D.M. see Nc~rskov, ,I.K. 499 Newton, J.C. s e e Wang, R. 575 Newton, M.D. s e e Hill, R.J. 181 Ng, Lily s e e Uram, K.J. 501 Nham, H.S. 573 Nicholas, J.F. 98, 135,499 Nicholls, J.M. s e e Grehk, T.M. 496 Niehus, H. 710 Niehus, H. s e e Kern, K. 710 Nielsen, M. 573 Nielsen, M. s e e Bohr, J. 266, 568 Nielsen, M. s e e Dornisch, D. 134 Nielsen, M. see Dutta, P. 569 Nielsen, M. s e e Feidenhans'l, R. 496, 789 Nielsen, M. s e e Grey, F. 266 Nielsen, M. s e e Kjaer, K. 571 Nielsen, M. s e e McTague, J.P. 572 Nielsen, M. s e e Stampfl, C. 792 Nielsen, M.M. 499 Nielsen, M.M. s e e Aminpirooz, S. 493 Nielsen, M.M. s e e Stampfl, C. 501 Nielson, M. s e e Feidenhans'l, R. 266 Nienhuis, B. 791 Niessen, A.K. s e e Miedema, A.R. 498 Nightingale, M.P. 573,647 Niskanen, K.,I. 573 Nodine, M.H. s e e Gay, R.R. 225 Noffke, J. 499 Nofke, J. s e e Hermann, K. 570 Nogami, J. 499 Nogami, J. s e e Kono, S. 135 Nogami, J. s e e Shioda, R. 136, 500 Nogami, ,i. s e e Wan, K.J. 136, 501

Author

Nogler, S.E. s e e Specht, E.D. 574 Nolden, I. s e e Van Beijeren, H. 98 Nolder, R. 227 Nomura, E. s e e Katayama, M. 497 Noonan, J.R. s e e Gruzalski, G.R. 226 Nordlander, P. 647 Norman, D. s e e Lamble, G.M. 498 Norris, C. s e e Conway, K.M. 495 Norris, C. s e e Howes, P.B. 134 Norris, C. s e e Van Silfhout, R.G. 268 Nc~rskov, J. s e e Stoltze, P. 648 N~rskov, J.K. 135,499, 647 Nr J.K. s e e Besenbacher, F. 494 N~rskov, J.K. s e e Feidenhans'l, R. 496 Nc~rskov, J.K. s e e Jacobsen, K.W. 497, 790 NCrskov, .I.K. s e e Stokbro, K. 136 Northrup, J.E. 135,267, 499 Northrup, J.E. s e e Biegelsen, D.K. 266 Northrup, J.E. s e e Bringans, R.D. 494 Northrup, J.E. s e e Nicholls, J.M. 135,499 Northrup, J.E. s e e Uhrberg, R.I.G. 501 Northrup, J.E. s e e Zegenhagen, ,i. 792 Northrup, Jr., C.J.M. 182 Norton, P.R. 647 Norton, P.R. s e e Jackman, T.E. 419, 497 Novaco, A.D. 499, 573 Novaco, A.D. s e e Kjems, J.K. 571 Novaco, A.D. s e e McTague, J.P. 572 Novak, D. 227 Nowak, H.,I. 135 Nowotny, J. 182 Nowotny, J. s e e Dufour, L.-C. 180 Nozieres, P. 98 Nunes, R.W. s e e Li, X.P. 135 Nussbaum, R.H. 573 Nyberg, G.L. s e e Bare, S.R. 494 Nyholm, R. s e e Andersen, J.N. 493, 494 O'Keefe, M. 182 Ocai, C. s e e Bader, M. 494 Ochab, J. s e e Bak, P. 788 Ocko, B.M. 135 Ocko, B.M. s e e Zehner, D.M. 792 Oed, W. 499, 791 Oed, W. s e e Mendez, M.A. 498 Oed, W. s e e Muschiol, U. 499 Oed, W. s e e Pendry, J.B. 791

index

839

A uthor index

Oen, O.S. 419 Ogletree, D.F. 499, 647 Ogletree, D.F. s e e Barbieri, A. 642 Ogletree, D.F. s e e Van Hove, M.A. 268, 649 Ohdomari, I. s e e Hara, S. 226 Ohmura, Y. s e e Flores, F. 644 Ohnesorge, F. 227 Ohta, H. s e e Kinoshita, T. 498 Ohta, M. s e e Ueyama, H. 420 Ohta, T. 791 Ohtani, H. 499, 647 Okamoto, N. s e e Takahashi, T. 136, 501 Oils, A.M. 647 Olesen, L. s e e Klink, C. 790 Olmstead, M.A. 267,499 Oimstead, M.A. s e e Uhrberg, R.I.G. 136, 501 Ong, P.J. s e e Smilgies, D.M. 136 Onuferko, J.H. 791 Oppenheimer, J.R. s e e Born, M. 180 Ordej6n, P. 135 Ortega, A. s e e Pfntir, H. 500 Osakabe, N. 98,791 Osen, J.W. 573 Oshima, C. 227 Oshima, C. see Aono, M. 225 Oshima, C. s e e ltoh, H. 226 Oshima, C. s e e Souda, R. 227,420 Osuch, K. 227 Otani, S. 227 Otani, S. s e e Aono, M. 225 Otani, S . s e e Souda, R. 227,420 Outka, D.A. s e e D6bler, U. 495 Over, H 499, 500 Over, H s e e Bludau, H. 494 Over, H s e e Gierer, M. 496 Over, H s e e Hertel, T. 497 Over, H s e e Huang, H. 134, 497 Over, H s e e Stampfl, C. 501,792 Overhauser, A.W. s e e Dick, G.B. 180 Overney, R.M. 419 Ozaki, A. 500 Ozcomert, J.S. 98 Ozcomert, J.S. s e e Pal, W.W. 98,647 Pacchioni, G. 647 Packard, W.E. s e e Men, F.K. 791 Packard, W.E. s e e Tong, S.Y. 268

Pai, W.W. 98,647 Pai, W.W. s e e Ozcomert, J.S. 98 Palmari, J.P. s e e Bienfait, M. 568 Palmberg, P.W. 573 Palmer, R.L. s e e Kern, K. 571 Pan, J.M. s e e Maschhoff, B.L. 227 Pandey, K.C. 135 Pandey, K.C. s e e Kaxiras, E. 267 Pandey, K.C. s e e Mfirtensson, P. 135,498 Pandit, R. 573 Pandy, K.C. 182, 267 Pangher, N. 500 Pangher, N. s e e Aminpirooz, S. 493 Pangher, N. s e e Comelli, G. 495 Pantelides, S.T. 182 Paolucci, G. s e e Comelli, G. 495 Paolucci, G. s e e Dhanak, V.R. 495 Papaconstantopoulos, D.A. 135 Papaconstantopoulos, D.A. s e e Cohen, R.E. 643 Papanikolaou, N. s e e Dederichs, P.H. 643 Park, C. 500 Park, C.Y. s e e Abukawa, T. 493 Park, R.L. 43,359, 573,791 Park, R.L. s e e Hwang, R.Q. 645 Park, R.L. s e e Roelofs, L.D. 791 Park, R.L. s e e Taylor, D.E. 649 Park, R.L. s e e Williams, E.D. 360 Park, S.I. s e e Nogami, J. 499 Parkin, S.R. 500 Parkin, S.S.P. s e e Falicov, L.M. 789 Parmigiani, F. s e e Bagus, P.S. 494 Parmigiani, F. s e e Pacchioni, G. 647 Parr, R.G. 182 Parrinello, M. s e e Car, R. 133 Parrinello, M. s e e Ercolessi, F. 134, 644 Parrinello, M. s e e larlori, S. 181 Parrott, L. 500 Parry, D.E. 182 Parshin, A.Y. s e e Marchenko, V.I. 97,646 Pashitskii, E.A. s e e Braun, O.M. 643 Pashley, M.D. 267, 268,419 Passell, L. 573 Passell, L. s e e Carneiro, K. 568 Passeli, L. s e e Dutta, P. 569 Passell, L. s e e Eckert, J. 569 Passell, L. s e e Grier, B.H. 570

840 Passell, L. s e e Kjems, J.K. 571 Passell, L. s e e Larese, J.Z. 571,572 Passell, L. s e e Satija, S.K. 573 Passell, L. s e e Taub, H. 574 Passell, L. s e e You, H. 575 Passler, M.A. s e e Hannaman, D.J. 496 Pate, B.B. 182 Patel, J.R. s e e Zegenhagen, J. 502, 792 Paton, A. s e e Duke, C.B. 266 Paton, A. s e e Horsky, T.N. 267 Paton, A. s e e Kahn, A. 267 Patterson, H. s e e Grier, B.H. 570 Patterson, H. s e e Satija, S.K. 573 Paul, A. 182 Pauling, L. 227, 268, 573,647 Payne, M.C. s e e Needels, M. 267 Payne, M.C. s e e Stich, I. 136 Payne, M.C. s e e Teter, M. 136 Payne, S.H. 98,647 Peacor, S.D. 227 Pcdersen, J.S. 359 Pcderson, J.S. see Feidenhans'l, R. 266 Pederson, J.S. s e e Grey, F. 266 Pcderson, M.R. see Jackson, K. 134 Pcderson, M.R. s e e Perdew, J.P. 647 Pedio, M. 500 Pedio, M. s e e Becker, L. 494 Pchike, E. s e e Tersoff, J. 792 Peierls, R. 182, 573 Pelz, J.P. s e e B irgeneau, R.J. 568 Pendry, J.B. 359, 791 Pendry, J B s e e Andersson, S. 494 Pendry, J B s e e MacLaren, J.M. 267,498,646 Pendry, J.B s e e Muschiol, U. 499 Pendry, J B s e e Oed, W. 499 Pendry, J B see Rous, P.J. 360 Pendry, J.B s e e Wedler, H. 501 Pengra, D.B. 573 Pbpe, G. s e e Gay, J.M. 570 Perdereau, J. 135 Perdew, J.P. 135,647 Perez-Sandoz, R. s e e Weitering, H.H. 136 Perry, T. s e e Smith, J.R. 98, 136, 648 Pershan, P.S. s e e Braslau, A. 358 Persson, B.N.J. 647, 791 Pestak, M.W. 573 Peter, J. s e e Harten, U. 134

Author

Peters, C. 573 Peters, C. s e e Specht, E.D. 574 Peters, C.J. s e e Hong, H. 571,790 Peterson, L.D. s e e Farrell, H.H. 266 Petot-Ervas, G. s e e Dufour, L.C. 225 Pettifor, D.G. s e e Goodwin, L. 134 Pflanz, S. s e e Over, H. 499 Pfntir, H. 500, 791 Pfntir, H. s e e Jtirgens, D. 645 Pfntir, H. s e e Lindroos, M. 498 Pfn~ir, H. s e e Michalk, G. 498 Pfntir, H. s e e Piercy, P. 647 Pfniar, H. s e e Sandhoff, M. 648 Pfn~ir, H. s e e Schmidtke, E. 648 Pfntir, H. s e e Schwennicke, C. 648 Pfntir, H. s e e Sklarek, W. 648 Pfntir, H. s e e Sokolowski, M. 648 Phaneuf, R.J. 98,268, 791 Phaneuf, R.J. s e e Williams, E.D. 99 Phillips, J.C. 268 Phillips, J.M. 227 Phillips, J.M. s e e Bruch, L.W. 643 Phillips, K. s e e G6pel, W. 226 Pianetta, P. s e e Cao, R. 495 Pianetta, P. s e e Nogami, J. 499 Pianetta, P. s e e Woicik, J.C. 502 Pickett, W.E. 135 Pierce, D.T. 500 Pierce, D.T. s e e Falicov, L.M. 789 Piercy, P. 647 Piercy, P. s e e Pfntir, H. 500, 791 Piggins, N. s e e Pluis, B. 359 Pignet, T. see Christmann, K. 643 Pinkvos, H. 419 Pirug, G. 500 Pisani, C. 183 Pisani, C. s e e Causer, M. 180 Pisani, C. s e e Grimley, T.B. 645 Pitzer, K.S. s e e Sinanoglu, O. 574, 648 Plancher, M. s e e Doyen, G. 789 Plass, R. s e e Collazo-Davila, C. 418 Plass, R. s e e Marks, L.D. 419 Pluis, B. 359 Plummer, E.W. 500 Plummer, E.W. s e e Heskett, D. 497 Plummer, E.W. s e e Itchkawitz, B.S. 790 Plummer, E.W. s e e Mundenar, J.M. 499

index

Author

841

index

Poeisema, B. 359, 573,710 Poelsema, B. s e e Kunkel, R. 710 Poensgen, M. 98 Poensgen, M. s e e Frohn, J. 97, 644 Poensgen, M. s e e Giesen-Seibert, M. 644 Pohland, O. 98 Poirer, G.E. 227 Polanyi, M. 500 Polatoglou, H.M. s e e Tserbak, C. 649 Pollak, P. 500 Polli, M. s e e Bellman, A.F. 494 Pollmann, J. s e e Landemark, E. 267 Polzonetti, G. s e e Bagus, P.S. 494 Poon, T.W. 98,647 Pople, J.A. 183 Pople, J.A. s e e Hehre, W.J. 181 Popova, S.V. s e e Stishov, S.M. 183 Poppa, H. s e e Pinkvos, H. 419 Portz, K. 647 Prade, J. see Reiger, R. 183 Praline, G. s e e Parrott, L. 500 Preikszas, D. s e e Rose, H. 420 Presicci, M. 359 Press, W. 573 Preuss, E. s e e Bonzel, H.P. 96 Price, G.L. 573 Price, G.L. s e e Venables, J.A. 575 Prigge, D. s e e Ertl, G. 496 Prince, K.C. s e e Bellman, A.F. 494 Prince, K.C. s e e Comelli, G. 495 Prince, K.C. s e e Comicioli, C. 495 Prince, K.C. s e e Dhanak, V.R. 495 Prince, K.C. s e e Murray, P.W. 499 Prinz, G. s e e Falicov, L.M. 789 Pritchard, J. see Chesters, M.A. 568 Pritchard, J. s e e Horn, K. 497 Pritchard, J. s e e Roberts, R.H. 573 Prokrovskii, V.L. 573 Prutton, M. 183,227 Prutton, M. see Welton-Cook, M.R. 183 Pu Hu, Zi s e e Zuschke, R. 502 Puga, M.W. s e e Tong, S.Y. 268 Pukite, P.R. 359 Pulay, P. 135 Purcell, K.G. s e e Jupille, J. 790 Purcell, K.G. s e e Stensgaard, I. 792 Puschmann, A. s e e Bader, M. 494

Qian, G.-X. 268 Quate, C.F. 419 Quate, C.F. s e e Albrecht, T.R. 418 Quate, C.F. s e e Binnig, G. 418 Quate, C.F. s e e Nogami, J. 499 Quate, C.F. s e e Shioda, R. 136, 500 Quate, C.F. s e e Tortonese, M. 420 Quateman, J.H. 573 Quentel, G. 573 Quinn, J. 135,359, 500 Quinn, J. s e e Huang, H. 497 Quinn, J. s e e Over, H. 500 Qvarford, M. s e e Andersen, J.N. 493,494 Rabe, K.M. s e e Rappe, A.M. 135 Rablais, J.W. s e e Teplov, S.V. 420 Radom, L. s e e Hehre, W.J. 181 Rae, A.I.M. s e e Mason, R. 498 Raeker, T.J. 647 Raeker, T.J. s e e Stave, M.S. 136 Rahman, T.S. 500 Rahman, T.S. s e e Lehwald, S. 791 Raich, J.C. s e e Fuselier, C.R. 570 Rakova, E.V. 227 Ramamoorthy, M. 227 Ramana, M.V. s e e Fernando, G.W. 134 Ramsey, J.A. s e e Prutton, M. 183, 227 Ramseyer, T. s e e Roelofs, L.D. 135,791 Rangelov, G. 500 Rangelov, G. s e e Kiskinova, M. 498 Rangelov, G. s e e Surnev, L. 501 Ranke, W. 268 Rapp, R.E. 573 Rappe, A.M. 135 Rastomjee, C.S. s e e Rose, K.C. 420 Rausenberger, B. 419 Rausenberger, B. s e e Rose, K.C. 420 Ravikumar, V., D. Wolf, V.P. Dravid, Rayment, T. s e e Gameson, I. 570 Rayment, T. s e e Meehan, P. 572 Rayment, T. s e e Mowforth, C.W. 572 Raynerd, G. s e e Venables, J.A. 575 Razafitianamaharavo, A. 573 Redfield, A.C. 98,647 Reed, D.S. s e e Robinson, I.K. 360 Reeder, R.J. 183 R6gnier, J. 573

842 R6gnier, J. s e e Thomy, A. 574 Reichl, L.E. 791 Reider, K.H. 183 Reif, F. 791 Reiger, R. 183 Reihl, B. s e e Ferrer, S. 496 Reihl, B. s e e Lang, N.D. 571 Reihl, B. see Nicholls, J.M. 135 Reinecke, T.L. s e e Tiersten, S.C. 649 Reineker, P. s e e Spatz, J.P. 420 Rempfer, G.F. 419 Rempfer, G.F. see Skoczylas, W.P. 420 Rettner, C.T. s e e Barker, J.A. 642 Reuter, M.C. s e e Tromp, R.M. 420, 792 Reutt-Robey, J.E. s e e Ozcomert, J.S. 98 Reutt-Robey, J.E. s e e Pai, W.W. 98,647 Rhead, G.E. s e e Perdereau, J. 135 Rhodin, T.N. s e e Broden, G. 494 Rhodin, T.N. s e e Demuth, J.E. 643 Rhodin, T.N. see Gadzuk, J.W. 496 Rhodin, T.N. s e e Gerlach, R.L. 496 Rhodin, T.N. s e e lgnatiev, A. 571 Ricci, M. s e e Ealet, B. 225 Richardson, N.V. s e e Bare, S.R. 494 Richter, D. see Grier, B.H. 570 Richter, D. s e e Larese, J.Z. 571 Richter, L.J. 647 Rickard, J.M. see Duriez, C. 225 Rickard, J.M. s e e Quentel, G. 573 Ricken, D.E. s e e Robinson, A.W. 791 Rickman, J.M. 647 Rioter, R. 135 Ricdel, E.K. s e e Nienhuis, B. 791 Rieder, K.H 359, 360, 791 Rieder, K.H s e e Baumberger, M. 788 Ricder, K.H s e e Engel, T. 359, 496 Rieder, K.H s e e Haase, O. 496 Ricder, K.H s e e Koch, R. 498 Rieder, K.H s e e Swendsen, R.H. 360 Riedinger, R. s e e Dreyss6, H. 643 Riedinger, R. s e e Stauffer, L. 648 Rifle, D.M. 500 Rikvoid, P.A. 647, 648 Rikvold, P.A. s e e Collins, J.B. 643 Riley, F.L. 227 Roberts, A.D. s e e Johnson, K.L. 419 Roberts, R.H. 573

Author

index

Robinson, A.W. 791 Robinson, I.K. 43, 135, 360, 791 Robinson, I.K. s e e Bohr, J. 266 Robinson, I.K. s e e Fuoss, P.H. 359 Robinson, I.K. s e e Headrick, R.L. 496 Robinson, I.K. s e e Meyerheim, H.L. 498 Robinson, I.K. s e e Smilgies, D.M. 136, 792 Robinson, M.T. 420 Rocca, M. s e e Lehwald, S. 791 Rocker, G. s e e G6pel, W. 226 Rodge, W.E. s e e Abraham, F.F. 567 Rodriguez, A.M. 135 Rodriguez, A.M. 648 Roelofs, L.D. 135,648, 791 Roelofs, L.D. s e e Bartelt, N.C. 642, 788 Roelofs, L.D. s e e Payne, S.H. 98,647 Roetti, C. s e e Caus,~, M. 180 Roetti, C. s e e Pisani, C. 183 Rogowska, J.M. 648 Rohrer, G. 420 Rohrer, G.S. 183, 227 Rohrer, G.S. s e e Smith, R.L. 227 Rohrer, H. s e e Binnig, B. 96, 266, 418 Rohrer, H. s e e Binnig, G.K. 789 Rohrer, H. s e e Binning, G.K. 133 Rojo, J.M. s e e Miranda, R. 499 Rollefson, R.J. s e e Grier, B.H. 570 Rolley, E. s e e Balibar, S. 96 Root, T.W. 500 Rose, H. 420 Rose, J.H. 648 Rose, K.C. 420 Rosei, R. s e e Bellman, A.F. 494 Rosei, R. s e e Cautero, G. 495 Rosei, R. s e e Comelli, G. 495 Rosei, R. s e e Comicioli, C. 495 Rosei, R. s e e Dhanak, V.R. 495 Rosei, R. s e e Murray, P.W. 499 Rosenbaum, T.F. s e e Nagler, S.E. 572 Rosenblatt, D.H. s e e Kevan, S.D. 498 Rosenblatt, D.H. s e e Tobin, J.G. 501 Rosenblatt, D.H. s e e Tong, S.Y. 501 Rotermund, H.H. 420, 500 Rotermund, H.H. s e e B~ir, M. 494 Rotermund, H.H. s e e Engel, W. 496 Rotermund, H.H. s e e Jakubith, S. 497 Rottman, C. 98

Author

843

index

Rottman, C. s e e Bartelt, N.C. 96 Rottman, C. s e e Jayaprakash, C. 97,645 Rouquerol, J. 573 Rouquerol, J. s e e R6gnier, J. 573 Rous, P.J. 98, 360 Rous, P.J. s e e MacLaren, J.M. 267,498 Rous, P.J. s e e Nelson, R.C. 98 Rousina, R. s e e Dzioba, S. 225 Rousseau-Violet, J. s e e Le Boss6, J.C. 646 Rousset, S. 98 Rovida, G. s e e Bardi, U. 568 Rovida, G. s e e Galeotti, M. 225 Rownd, J.J. s e e Haneman, D. 267 Rowntree, P. 573 Rowntree, P.A. s e e Ruiz-Suarez, J.C. 573 Rubin, Y. s e e Snyder, E.J. 420 Rubio, G. s e e Agra'ft, N. 418 Rudberg, E. 360 Ruderman, M.A. 648 Rudge, W.E. s e e Koch, S.W. 571 Rudnick, J. 648 Rudolf, P. see Astaldi, C. 494 Rudolf, P. s e e Cautero, G. 495 Ruiz-Suarez, J.C. 573 Ruland, W. 573 Rundgren, J. 227 Rundgren, J. s e e Hammar, M. 226 Rundgren, J. s e e Lindgren, S.A. 498 Ruska, E. 420 Russo, J. s e e Liang, K.S. 359 Ryberg, R. s e e Persson, B.N.J. 647 Saam, W.F. s e e Jayaprakash, C. 97, 645 Saam, W.F. see Nightingale, M.P. 573 Sacedon, J.L. s e e Soria, F. 500 Sacks, W. s e e Rousset, S. 98 Sacramento, P. s e e Collins, J.B. 643 Sagawa, S. s e e Higashiyama, K. 497 Sagawa, T. s e e Kono, S. 135 Sagncr, H.-J. s e e Frank, H.H. 496 Saiki, K. see Hirata, A. 226 Saito, Y. 573 Sakai, A. s e e Lambert, W.R. 267 Sakama, H. 268 Sakawa, H. s e e Kawazu, A. 497 Sakuma, E. s e e Hara, S. 226 Sakurai, T. s e e Jeon, D. 134, 790

Sakurai, T. s e e Park, C. 500 Sakurai, T. s e e Taniguchi, M. 501 Salahub, D.R. 183 Salaneck, W.R. s e e Plummer, E.W. 500 Salanon, B. s e e Lapujoulade, J. 790 Salanon, M. s e e Falicov, L.M. 789 Saldin, D.K. s e e MacLaren, J.M. 267, 498 Saldin, D.K. s e e Oed, W. 499 Saldin, D.K. s e e Rous, P.J. 360 Salmeron, M. s e e Barbieri, A. 642 Salmeron, M. s e e Miranda, R. 499 Salmeron, M. s e e Ogletree, D.F. 647 Saloner, D. 710 Salvan, F. 500 Salvan, F. s e e Nicholls, J.M. 135 Salvan, F. s e e Thibaudau, F. 501 Samant, M.G. s e e Stoehr, J. 420 Samsavar, A. 710 Samsavar, A. s e e Hirschorn, E.S. 267 Sander, D. 98 Sander, M. 227 Sanders, D.E. s e e Stave, M.S. 136 Sandhoff, M. 648 Santucci, A. s e e Galeotti, M. 225 Sarid, D. 420 Sarikaya, M. 420 Saris, F.W. see Smeenk, R.G. 500 Saris, F.W. s e e Turkenburg, W.C. 420 Saris, F.W. see Van der Veen, J.F. 501 Satija, S. s e e You, H. 575 Satija, S.K. 573 Satija, S.K. s e e Passell, L. 573 Sato, H. 98 Satoko, C. s e e Tsukada, M. 183 Satti, D. s e e Thibaudau, F. 501 Sautet, P. s e e Barbieri, A. 642 Savage, D.E. s e e Saloner, D. 710 Savage, T.S. s e e Ai, R. 418 Saxena, A. 791 Schabel, M.C. s e e Northrup, J.E. 135 Schabes-Retchkiman, P.S. 573 Schabes-Retchkiman, P.S. s e e Venables, J.A. 575 Schtifer, J.A. s e e G6pel, W. 226 Schaffroth, Th. s e e Besold, G. 494 Schardt, B. s e e Ocko, B.M. 135 Schart, A. s e e Kern, K. 710

844 Schaub, T. s e e Tarrach, G. 227 Scheffler, M. 500, 648 Scheffler, M. s e e Bormet, J. 494 Scheffler, M. s e e Dabrowski, J. 266 Scheffler, M s e e Fiorentini, V. 134 Scheffler, M s e e Horn, K. 571 Scheffler, M s e e Methfessel, M. 97 Scheffler, M s e e Neugebauer, J. 499 Scheffler, M s e e Schmalz, A. 500, 648 Scheffler, M s e e Stampfl, C. 501,792 Scheffler, M. s e e Stumpf, R. 649 Scheid, H. 500 Schick, M. 573, 791 Schick, M. s e e Dietrich, S. 569 Schick, M. see Domany, E. 569 Schick, M s e e Gittes, F.T. 570 Schick, M s e e Kinzel, W. 645 Schick, M s e e Nienhuis, B. 791 Schick, M s e e Nightingale, M.P. 573 Schick, M s e e Pandit, R. 573 Schiff, L.I 360 Schildbach, M.A. 183 Schildberg, H.P. s e e Cui, J. 569 Schildberg, H.P. s e e Freimuth, H. 570 Schinlmelpfenning, J. 574 Schleyer, P.V.R. s e e Hehre, W.J. 181 Schlier, R.E. 135,268 Schlicr, R.E. s e e Farnsworth, H.E. 789 Schl6gi, R. see Erti, G. 496 Schluter, M. 268 Schmalz, A. 500, 648 Schmalz, A. s e e Aminpirooz, S. 493 Schmalz, A. s e e Becker, L. 494 Schmicker, D. 574 Schmidt, G. see Besold, G. 494 Schmidt, G. s e e Bickel, N. 225 Schmidt, G. see Eggeling, von, C. 495 Schmidt, K. s e e Koch, R. 498 Schmidt, L.D. see Flytzani-Stephanopoulos, M. 97 Schmidt, L.D. s e e Root, T.W. 500 Schmidtke, E. 648 Schmidtlein, G. s e e Pendry, J.B. 791 Schnatterly, S.E. s e e Jasperson, S.N. 571 Schnell-Sorokin, A.J. 98 Schober, O. s e e Christmann, K. 495 SchOnektis, O. s e e Heidberg, J. 570

Author

Schtinhammer, K. 648 Schott, J. s e e Hayden, B.E. 790 Schreiner, D.G. s e e Park. R.L. 359 Schrieffer, J.R. 648 Schrieffer, J.R. s e e Einstein, T.L. 644 Schrieffer, R. s e e Herman, F. 645 Schrtider, J. s e e Hwang, R.Q. 710 Schr6der, U. s e e de Wette, F.W. 180 Schr6der, U. s e e Reiger, R. 183 Schr6dinger, E. 183 Schuller, I.K. s e e Falicov, L.M. 789 Schulman, L.S. s e e Avron, J.E. 96 Schultz, H. s e e Dornisch, D. 495 Schultz, P.A. s e e Jennison, D.R. 645 Schulz, H. s e e Dornisch, D. 134 Schuster, R. 500, 648, 791 Schwartz, C. 574 Schwartz, L.M. s e e Ehrenreich, H. 644 Schwarz, E. 500 Schwarz, E. s e e Gierer, M. 496 Schwarz, E. s e e Koch, R. 498 Schwarz, E. s e e Over, H. 499 Schwegmann, S. 500 Schwennicke, C. 648 Schwennicke, C. s e e Schmidtke, E. 648 Schwennicke, C. s e e Sklarek, W. 648 Schwoebel, P.R. 648 Schwoebei, R.L. s e e Blakely, J.M. 96 Scire, R. s e e Young, R. 421 Scoles, G. s e e Ellis, T.H. 569 Scoles, G. s e e Rowntree, P. 573 Scoles, G. s e e Ruiz-Suarez, J.C. 573 Sears, M.P. s e e Jennison, D.R. 645 Seehofer, L. 500 Seguin, J.L. 574 Seguin, J.L. see Bienfait, M. 568 Seguin, J.L. s e e Suzanne, J. 574 Seguin, J.L. s e e Venables, J.A. 575 Seifert, R.L. s e e Yang, Y.-N. 99 Seiler, H. s e e Ichimura, S. 226 Seitz, F. 183 Selke, W. 648 Selke, W. s e e Kinzel, W. 646 Selloni, A. s e e Ancilotto, F. 133 Selloni, A. s e e Takeuchi, N. 268 Semancik, S. 183 Semancik, S. s e e Cox, D.F. 180, 225

index

Author

845

index

Semancik, S. s e e Doering, D.L. 495 Septier, A. 420 Ser, F. 57 Sesselmann, W. s e e Woratschek, B. 502 Shah, P.J. s e e Mouritsen, O.G. 710 Sham, L.J. s e e Kohn, W. 134, 182, 498,646 Sharma, Y.P. s e e Taub, H. 574 Shaw, C.G. 574 Shechter, H. 574 Shechter, H. s e e Brener, R. 568 Shechter, H. s e e Krim, J. 571 Shechter, H. s e e Wang, R. 575 Sheiko, S. s e e Spatz, J.P. 420 Shek, M.L. s e e Johansson, L.I. 226 Shelton, J.C. s e e Blakely, J.M. 96 Sheth, R. s e e Roelofs, L.D. 791 Shi, H. 500 Shi, H. s e e Jacobi, K. 497 Shiba, H. 574 Shibata, A. 791 Shibata, Y. s e e Oshima, C. 227 Shih, H.D. s e e Huang, H. 497 Shioda, R. 136, 500 Shirazi, A.R.B. 574 Shirley, D.A. s e e Huang, Z. 497 Shirley, D.A. s e e Kevan, S.D. 498 Shirley, D.A. s e e Tobin, J.G. 501 Shirley, D.A. s e e Tong, S.Y. 501 Shivaprasad, S.M. s e e Madey, T.E. 97 Shkrcbtii, A.I. s e e Takeuchi, N. 268 Shore, J.D. s e e Roelofs, L.D. 648 Shropshirc, J. 183 Shu, Q.S. s e e Ecke, R.E. 569 Shuttleworth, R. 98 Sibcncr, S.J. s e e Gibson, K.D. 570 Siboulet, O. s e e Rousset, S. 98 Sidoumou, M. 574 Sidoumou, M. s e e Audibert, P. 568 Sigmund, P. 420 Silverman,, P.J. s e e Stensgaard, 1. 136 Siiverman, P.J. see Kuk, Y. 646 Siivi, B. 183 Simon, A. 500 Sinanoglu, O. 574, 648 Sinclair, J.E. s e e Finnis, M.W. 134, 644 Singh, D. 136,791 Singh, D. s e e Roelofs, L.D. 135,791

Singh, D. s e e Yu, R. 136 Singh, D.J. s e e Perdew, J.P. 647 Sinha, S.K. s e e Dutta, P. 569 Sinha, Sunil K. 791 Skelton, D.C. 648 Skelton, D.C. s e e Wei, D.H. 649 Skibowski, M. s e e Claessen, R. 225 Skinner, A.J. s e e Goodwin, L. 134 Sklarek, W. 648 Skoczylas, W.P. 420 Skottke-Klein, M. s e e Over, H. 499 Slater, J.C. 183 Slater, J.R. 136 Siavin, A.J. s e e Feenstra, R.M. 266 Slijkerman, W.F.J. s e e Hara, S. 226 Slyozov, V.V. s e e Lifshitz, I.M. 791 Smeenk, R.G. 500 Smeenk, R.G. s e e Van der Veen, J.F. 501 Smilgies, D.M. 136, 792 Smit, L. s e e Derry, T.E. 180 Smith, D.J. 227,420 Smith, G.W. s e e Weakliem, P.C. 268 Smith, J.R. 98, 136, 648 Smith, J.R. s e e Ricter, R. 135 Smith, J.R. s e e Rose, J.H. 648 Smith. J.V. s e e Smyth, J.R. 183 Smith, P.V. s e e Zheng, X.M. 184 Smith, R.L. 227 Smith. S.T. 420 Smouluchowski, R. 136 Smyth, J.R. 183 Sneddon, L.G. s e e Mundenar, J.M. 499 Snyder, E.J. 420 Sokolov, J. 500 Sokolowski, M. 648 Solomon, E.I. s e e Gay, R.R. 225 Somers, J. s e e Horn, K. 497 Somers, J.S. s e e Robinson, A.W. 791 Somorjai, G.A. 43, 98, 183 Somorjai, G.A. s e e Barbieri, A. 225,642 Somorjai, G.A. s e e Batteas, J.D. 494 Somorjai, G.A. s e e Chung, Y.W. 225 Somorjai, G.A. s e e Dubois, L.H. 495 Somorjai, G.A. s e e French, T.M. 225 Somorjai, G.A. s e e Koestner, R.J. 498 Somorjai, G.A. s e e MacLaren, J.M. 267,498 Somorjai, G.A. s e e Ogletree, D.F. 499, 647

846 Somorjai, G.A. s e e Ohtani, H. 499,647 Somorjai, G.A. s e e Toyoshima, I. 649 Somorjai, G.A. s e e Van Hove, M.A. 98, 136, 268, 501,649 Song, K.-J. 98 Song, K.-J. s e e Madey, T.E. 97 Song, S. 98 SCrensen, O.T. 227 Soria, F. 500 Soszka, W. s e e Turkenburg, W.C. 420 Souda, R. 227,420 Souda, R. s e e Aono, M. 225 Soukiassian, P. 136 Soven, P. 648 Soven, P. s e e Kalkstein, D. 645 Sowa, E.C. 183 Spackman, M.A. s e e Geisinger, K.L. 181 Spadicini, R. s e e Garibaldi, V. 359 Spaepen, F. s e e Josell, D. 97 Spanjaard, D. see Bourdin, J.P. 642 Spanjaard, D. see Desjonqu6res, M.C. 643 Spanj,'lard, D. see Oils, A.M. 647 Spatz, J.P. 420 Spccht, E.D. 574 Spcnce, J.C.tt. 420 Spcnce, J.C.H. see Lo, W.K. 419 Spcnce, J.C.H. see Zuo, J.M. 421 Spicer, W.E. 501 Spicer, W.E. s e e Woicik, J.C. 502 Spiess, L. 136 Srolovitz, D.J. 648 Srolovitz, D.J. s e e Najafabadi, R. 647 Srolovitz, D.J. s e e Rickman, J.M. 647 Stair, P.C. 360 Stair, P.C. see Collazo-Davila, C. 418 St:lit, P.C. see Marks, L.D. 419 Stair, P.C. s e e Van Hove, M.A. 136 Stampfl, C. 501,792 Stancioff, P. s e e EI-Batanouny, M. 134 Stanlcy, H.E. 574 Stark, J.B. s e e Robinson, I.K. 43 Starke, U. see Heinz, K. 497 Starke, U. see Oed, W. 499,791 Starkey, E.K. s e e Batteas, J.D. 494 Stauffer, L. 648 Stave, M.S. 136 Stechel, E.B. s e e Jennison, D.R. 645

Author

Steele, W.A. 574 Stefan, P.M. s e e Johansson, L.I. 226 Stefanou, N. s e e Dederichs, P.H. 643 Steffen, B. s e e Bonzel, H.P. 96 Steinkilberg, M. 501 Steinkilberg, M. s e e Fuggle, J.C. 496 Stensgaard, I. 136, 792 Stensgaard, I. s e e Eierdal, L. 495 Stensgaard, I. s e e Feidenhans'l, R. 496 Stensgaard, I. s e e Jensen, F. 497, 790 Stensgaard, I. s e e Klink, C. 790 Stensgaard, I. see Mortensen, K. 499 Stephens, P.W. 574 Stephens, P.W. s e e Guryan, C.A. 570 Stern, E.A. s e e Bouldin, C. 568 Stern, E.A. s e e Rudnick, J. 648 Steslicka, M. s e e Davison, S.G. 643 Stevens, K. s e e Horsky, T.N. 267 Stewart, G.A. s e e Butler, D.M. 568 Stewart, G.A. s e e Litzinger, J.A. 572 Stich, I. 136 Stiles, K. see Duke, C.B. 266 Stiles, K. see Horsky, T.N. 267 Stiles, M.D. 648 Stillingcr, F. 136, 648 Stishov, S.M. 183 Stocker, W. see Baumberger, M. 788 Stocker, W. s e e Rieder, K.H. 791 Stockmeyer, R. see Monkenbusch, M. 572 St6hr, J. 420, 501 Stt~hr, J. s e e Di3bler, U. 495 Stokbro, K. 136 Stoll, E. s e e Binnig, G.K. 133, 789 Stoltenberg, J. 574 Stoltze, P. 648 Stoneham, A.M. 648 Stoneham, A.M. s e e Flores, F. 644 Stoneham, A.M. see Hayes, W. 226 Stoner, N. s e e Van Hove, M.A. 501 Stott, M. 648 Stout, G.H. 360 Strite, S. 227 Strite, S. s e e Morkog:, H. 227 Stroscio, J.A. s e e Feenstra, R.M. 418 Sttihn, B. see Wiechert, H. 575 Stulen, R.H. s e e Felter, T.E. 644 Stulen, R.H. s e e Sowa, E.C. 183

index

A uthor

847

index

Stumm, W. 183 Stumpf, R. 648,649 Sudo, A. s e e Nakamatasu, H. 182 Suematsu, H. s e e Hong, H. 571,790 Sugawara, Y. s e e Ueyama, H. 420 Suhl, H. s e e Levi, A.C. 359 Suhren, M. s e e Heidberg, J. 570 Sullivan, T.S. s e e Coulomb, J.P. 569 Sullivan, T.S. s e e Ecke, R.E. 569 Sulston, K.W. 649 Sun, Q. 649 Surnev, L. 501 Surnev, L. see Kiskinova, M. 498 Surnev, L. s e e Rangelov, G. 500 Susman, S. 183 Suter, R.M. s e e Greiser, N. 570 Sutter, R.M. s e e Colella, N.J. 568 Sutton, L.E. 501 Sutton, M. see Dimon, P. 569 Sutton, M. s e e Hong, H. 571 Sutton, M. s e e Mochrie, S.G.J. 572 Sutton, M. see Nagler, S.E. 572 Sutton, M. see Passell, L. 573 Sutton, M. see Specht, E.D. 574 Suzanne, J 574 Suzanne, J see Angot, T. 567 Suzanne, J see Audibert, P. 568 Suzanne, J s e e Beaume, R. 568 Suzanne, J s e e Bienfait, M. 568 Suzanne, J s e e Brener, R. 568 Suzannc, J. s e e Calisti, S. 568 Suzanne, J. see Coulomb, J.P. 568,569 Suzannc, J. s e e Gay, J.M. 570 Suzanne, J s e e Kramer, H.M. 571 Suzanne, J see Krim, J. 571 Suzanne, J see Meichei, T. 572 Suzanne, J s e e Migone, A.D. 572 Suzanne, J s e e Passell, L. 573 Suzanne, J s e e Seguin, J.L. 574 Suzanne, J s e e Shechter, H. 574 Suzanne, J s e e Sidoumou, M. 574 Suzanne, J s e e Venables, J.A. 575 Suzanne, J see Wang, R. 575 Suzuki, S. see Kinoshita, T. 498 Sverdlov, B. see Morkoq, H. 227 Swartz, L.E. see Biegelsen, D.K. 266 Swartzentruber, B.S. 98, 268, 7 l0

Swartzentruber, B.S. s e e Becker, R.S. 265,494 Swartzentruber, B.S. s e e Lagally, M.G. 267 Swartzentruber, B.S. s e e Webb, M.B. 268 Swendsen, R.H. 360, 574 Swiech, W. s e e Williams, E.D. 99 Szabo, A. 227 Szasz, L. 183 Szeftel, J.M. s e e Rahman, T.S. 500 Tabony, T. 574 Taglauer, E. s e e Beckschulte, M. 418 Takahashi, M. s e e Takayanagi, K. 136, 268, 710, 792 Takahashi, T. 136, 501 Takami, T. 501 Takata, K. s e e Hasegawa, T. 496 Takayanagi, K. 136, 268, 7 I0, 792 Takayanagi, K. s e e Haga, Y. 419 Takayanagi, K. s e e Shibata, A. 791 Takayanagi, K. s e e Yamazaki, K. 792 Takeuchi, N. 136, 268 Talapov, A.L. see Prokrovskii, V.L. 573 Tan, Y.T. see Baetzold, R.C. 180 Tanaka, H. 227 Tanaka, H. see Matsumoto, T. 182, 227 Tanaka, K. s e e Taniguchi, M. 501 Tanaka, T. s e e Otani, S. 227 Tang, S.P. s e e Spiess, L. 136 Taniguchi, M. 501 Tanishiro, Y. s e e Osakabe, N. 98 Tanishiro, Y. s e e Sato, H. 98 Tanishiro, Y. s e e Takayanagi, K. 136, 268, 710, 792 Tanishiro, Y. s e e Yamazaki, K. 792 Tarrach, G. 227 Tasker, P.W. 183,227 Tasker, P.W. s e e Baetzold, R.C. 180 Taub, H 574, 792 Taub, H s e e Carneiro, K. 568 Taub, H s e e Coulomb, J.P. 569 Taub, H s e e Hansen, F.Y. 570 Taub, H see Kjems, J.K. 571 Taub, H s e e Krim, J. 571 Taub, H s e e Larese, J.Z. 572 Taub, H s e e Suzanne, J. 574 Taub, H s e e Trott, G.J. 575 Taub, H. s e e Wang, R. 575

848 Tayler, L.I. s e e Roelofs, L.D. 135 Taylor, D.E. 649 Taylor, L.L. s e e Roelofs, L.D. 791 Taylor, T.N. s e e Parrott, L. 500 Teissier, C. 574 Teitel, S. s e e Jayaprakash, C. 97 Tejwani, M.J. 574 Tejwani, M.J. 792 Telieps, W. 792 Telieps, W. s e e Bauer, E. 418,709 Teller, E. s e e Axilrod, B.M. 568, 642 Teplov, S.V. 420 Terakura, K. s e e Ishida, H. 497 Terakura, K. s e e Kobayashi, K. 134 Terlain, A. 574 Terminello, L.J. s e e McLean, A.B. 498 Tersoff, J. 136, 420, 792 Tersoff, J. s e e Feenstra, R.M. 418 Tesmer, J.R. 420 Testardi, L.R. s e e Lind, D.M. 227 Teter, M. 136 Thciss, S.K. see Ganz, E. 496, 790 Thcodorou, G. 649 Thcodorou, G. s e e Tserbak, C. 649 Thibaudau, F. 501 Thicl, P.A. 183 Thici, P.A. see Behm, B.J. 133 Thiel, P.A. s e e Wang, W.D. 711 Thomas, G. see Griffiths, K. 790 Thomas, G.E. 501 Thomas, R.K. s e e Meehan, P. 572 Thomas, R.K. s e e Morishige, K. 572 Thomas, R.K. s e e Mowforth, C.W. 572 Thomlinson, W. s e e Carneiro, K. 568 Thomson, R.M. see Wert, C.A. 43 Thorny, A 574 Thorny, A see Bockel, C. 568 Thorny, A see Bouchdoug, M. 568 Thorny, A see Coulomb, J.P. 569 Thorny, A s e e Dolle, P. 569 Thorny, A s e e Dupont-Pavlovsky, N. 569 Thorny, A s e e Marti, C. 572 Thorny, A s e e Matecki, M. 572 Thorny, A s e e Menaucourt, J. 572 Thorny, A s e e Razafitianamaharavo, A. 573 Thomy, A s e e R6gnier, J. 573 Thorel, P. 574

Author

index

Thorel, P. s e e Coulomb, J.P. 568,569 Thorel, P. s e e Croset, B. 569 Thornton, G. s e e Leibsle, F.M. 498 Thornton, G. s e e Murray, P.W. 227,499 Thornton, J.M.C. s e e van der Vegt, H.A. 710 Thouless, D.J. s e e Kosterlitz, J.M. 359, 571, 790 Tiby, C. 574, 575 Tiersten, S.C. 649 Tobin, J.G. 501 Tobin, J.G. s e e Kevan, S.D. 498 Tobin, J.G. s e e Tong, S.Y. 501 Tobochnik, J. 792 Toennies, J.P. 360, 575 Toennies, J.P. s e e Harten, V. 710 Toennies, J.P. s e e Lahee, A.M. 498 Toennis, J.P. s e e Schmicker, D. 574 Tom~inek, D. 649 Tom~inek, D. s e e Dreyss6, H. 643 Tommasini, F. s e e Bellman, A.F. 494 Tommei, G.E. s e e Garibaldi, V. 359 Tompa, H. s e e Ruland, W. 573 Toney, M. see Bohr, J. 266 Toney, M.F. 575 Toney, M.F. s e e Shaw, C.G. 574 Tong, S.Y. 268,501 Tong, S.Y. see Hong, I.H. 497 Tong, S.Y. s e e Huang, H. 134, 497 Tong, S.Y. s e e Kevan, S.D. 498 Tong, S.Y. s e e Over, H. 499, 500 Tong, S.Y. s e e Tobin, J.G. 501 Tong, S.Y. see Van Hove, M.A. 360, 501 Tong, S.Y. s e e Wei, C.M. 136 Tong, S.Y. s e e Wu, H. 502 Tong, W.M. s e e Snyder, E.J. 420 Tong, X. s e e Pohland, O. 98 Tonner, B.D. s e e Stoehr, J. 420 Tonner, B.P. 420 Toppe, W. s e e Schwegmann, S. 500 T/3rnevik, C. s e e Hammar, M. 226 TCJrnqvist, E. s e e Narusawa, T. 647 Torrini, M. s e e Galeotti, M. 225 Torrini, M. s e e Grimley, T.B. 645 Tortonese, M. 420 Torzo, G. s e e Taub, H. 792 Tosatti, E. 136, 649 Tosatti, E. s e e Ancilotto, F. 133

Author

849

index

Tosatti, E. s e e Bilalbegovic, G. 96 Tosatti, E. s e e Ercolessi, F. 134, 644 Tosatti, E. s e e Garofalo, M. 790 Tosatti, E. s e e larlori, S. 181 Tosatti, E. s e e Takeuchi, N. 268 Tosatti, E. s e e Trayanov, A. 575 Tosatti, E. s e e Wang, C.Z. 136, 792 Toth, L.E. 227 Toyoshima, I. 649 Trabelsi, M. 575 Tracy, C.A. 792 Tracy, J.C. 501 Trafas, B.M. s e e Yang, Y.-N. 99 Tran, T.T. 227 Trayanov, A. 575 Trayanov, A. s e e Nenow, D. 573 Tr6glia, G. s e e Ol/~s, A.M. 647 Trevor, P.R. see Lambert, W.R. 267 Tringides, M.C. 649, 710 Tringides, M.C. s e e Saloner, D. 710 Tringides, M.C. see Wang, W.D. 711 Tringides, M.C. see Wu, P.K. 711 Tromp, R. s e e Himpsel, F.J. 267 Tromp, R.M 420, 792 Tromp, R.M s e e Maree, P.M.J. 267 Tromp, R.M s e e Rotermund, H.H. 420 Tromp, R.M see Schnell-Sorokin, A.J. 98 Tromp, R.M s e e Smeenk, R.G. 500 Tromp, R.M s e e van Loenen, E.J. 136 Tromp, R.M. s e e van der Veen, J.F. 501 Trost, J. 501 Trost, J. s e e Brune, H. 495 Trott, G.J. 575 Trott, G.J. s e e Coulomb, J.P. 569 Troullier, N. 136 Truhlar, D.G. s e e Truong, T.N. 649 Truong, T.N. 649 Tsen, S.C.Y. s e e Smith, D.J. 420 Tserbak, C. 649 Tsong, I.S.T. s e e Chang, C.S. 225 Tsong, T.T. 420, 649, 792 Tsong, T.T. see Casanova, R. 643 Tsong, T.T. s e e Chen, C.-L. 643 Tsong, T.T. s e e Cowan, P.L. 643 Tsong, T.T. s e e Mtiller, E.W. 419 Tsukada, M. 183 Tsukada, M. s e e Watanabe, S. 501

Tsuno, K. 420 Tucker, Jr., C.W. 501 Tung, R.T. s e e Robinson, I.K. 360 Tung, R.T. s e e Wei, J. 99 Turkenburg, W.C. 420 Ttishaus, M. s e e Persson, B.N.J. 647 Tyson, W.R. 98 Uchida, Y. s e e Lehmpfuhl, G. 419 Uchida, Y. s e e Wang, N. 421 Ude, M. s e e Takami, T. 501 Ueba, H. 649 Uebing, C. 649 Ueda, K. s e e Sakama, H. 268 Ueyama, H. 420 Uhrberg, R.I.G. 136, 268,501 Uhrberg, R.I.G. s e e Bringans, R.D. 494 Uhrberg, R.I.G. s e e Landemark, E. 267 Uhrberg, R.I.G. s e e Northrup, J.E. 135 Uhrberg, R.I.G. s e e Olmstead, M.A. 499 Umbach, C.C. s e e Keeffe, M.E. 97 Umbach, E. see Tobin, J.G. 501 Unertl, W. s e e Golze, M. 570 Unertl, W.N. 360, 420 Unertl, W.N. s e e Bak, P. 788 Unertl, W.N. s e e Clark, D.E. 789 Unertl, W.N. s e e Jackman, T.E. 497 Unguris, J. 575 Unguris, J. s e e Cohen, P.I. 568 Uram, K.J. 501 Urano, T. 183 Urbakh, A.M. 649 Urbakh, M.I. s e e Brodskii, A.M. 643 Uzunov, D.I. 792 Vainshtein, B.K. 43 Valbusa, U. s e e Ellis, T.H. 569 Van Acker, J.F. s e e Andersen, J.N. 493 Van Beijeren, H. 98 Van Beijeren, H. s e e A vron, J.E. 96 Van den Berg, J.A. s e e Verheij, L.K. 501 Van der Merwe, J.H. s e e Frank, F.C. 570 Van der Veen, J.F. 183,420, 501,792 Van der Veen, J.F. s e e Conway, K.M. 495 Van der Veen, J.F. s e e Denier van der Gon, A.W. 266 Van der Veen, J.F. s e e Derry, T.E. 180

850

Van der Veen, J.F. s e e Frenken, J.W.M. 496 Van der Veen, J.F. s e e Hara, S. 226 Van der Veen, J.F. s e e Himpsel, F.J. 181 Van der Veen, J.F. s e e Maree, P.M.J. 267 Van der Veen, J.F. s e e Pluis, B. 359 Van der Veen, J.F. s e e Smeenk, R.G. 500 Van der Veen, J.F. s e e Tromp, R.M. 420 Van der Veen, J.F. s e e Van Silfhout, R.G. 268 Van der Veen, J.F. s e e Vlieg, E. 360 Van der Vegt, H.A. 710 Van der Werf, D.P. s e e Heslinga, D.R. 497 Van Hove, M.A. 98, 136, 183, 268,360, 501, 649 Van Hove, M.A. s e e Barbieri, A. 225,642 Van Hove, M.A. s e e Batteas, J.D. 494 Van Hove, M.A. s e e Christmann, K. 643, 789 Van Hove, M.A. s e e Koestner, R.J. 498 Van Hove, M.A. s e e MacLaren, J.M. 267,498 Van Hove, M.A. s e e Ogletree, D.F. 499 Van Hove, M.A. see Ohtani, H. 499, 647 Van Hove, M.A. s e e Somorjai, G.A. 98 Van Hove, M.A. see Sowa, E.C. 183 Van Hove, M.A. see Van der Veen, J.F. 792 Van Hove, M.A. see Watson, P.R. 649 Van Huong, C.N. s e e Liu, S.H. 646 Van Loenen, E.J. 136 Van Loenen, E.J. s e e Dijkkamp, D. 97 Van Loenen, E.J. s e e Elswijk, H.B. 496 Van Pinxteren, H.M. 98 Van Pinxtesen, H.M. s e e van der Vegt, H.A. 710 Van Schilfgaarde, M. s e e Methfessel, M. 135 Van Sciver, S.W. 575 Van Sciver, S.W. s e e Hering, S.V. 570 Van Silfhout, R.G. 268 Vanderbilt, D. 98, 136, 183,268,710 Vanderbilt, D. s e e Alerhand, O.L. 96, 265,788 Vanderbilt, D. s e e Bedrossian, P. 133,494 Vanderbilt, D. s e e Li, X.P. 135 V anderbiit, D. s e e Meade, R.D. 97,498 Vanderbilt, D. s e e Narasimhan, S. 135 V anderbilt, D. see Ramamoorthy, M. 227 Vandervoort, K.G. s e e You, H. 711 Vannerberg, N.-G. 501 V aughn, P.A. s e e Shropshire, J. 183 V edula, Yu.S. s e e Naumovets, A.G. 499, 647 Venables, J.A. 575

Author

index

Venables, J.A. s e e Calisti, S. 568 Venables, J.A. s e e Faisal, A.Q.D. 569 Venables, J.A. s e e Hamichi, M. 570 Venables, J.A. s e e Price, G.L. 573 Venables, J.A. s e e Schabes-Retchkiman, P.S. 573 Venables, J.A. s e e Seguin, J.L. 574 Veneklasen, L. 420 Verhei ~, L.K. 501 Verheij, L.K. s e e Kunkel, R. 710 Verheij, L.K. s e e Poeisema, B. 573 Vernitron Corp., 421 Verway, E.J.W. 183 V erwoerd, W.S. s e e Badziag, P. 265 Verwoerd, W.S. s e e Osuch, K. 227 V ickers, J.S. s e e Becker, R.S. 265,494 Vickers, J.S. s e e Kubby, J.A. 267 Victora, R.H. s e e Falicov, L.M. 789 Vidali, G. 575,649 Vieira, S. s e e Agra'it, N. 418 Vilches O.E. s e e Bretz, M. 568 Vilches O.E. s e e Coulomb, J.P. 568, 569 Vilches O.E. s e e Ecke, R.E. 569 Vilches O.E. s e e Hering, S.V. 570 Vilches O.E. see Ma, J. 572 Viiches O.E. s e e Stoltenberg, J. 574 Vilches O.E. s e e Tejwani, M.J. 574, 792 Vilches O.E. s e e Van Sciver, S.W. 575 Vilches. O.E. s e e Zeppenfeid, P. 575 Vilfan, . s e e Villain, J. 792 Villain, J. 360, 792 Villain, J. s e e Wolf, D.E. 650 V issar, R.J. s e e Collart, E. 225 Vlieg, E. 136, 360 Vlieg, E. see Conway, K.M. 495 Viieg, E. s e e Headrick, R.L. 496 Vlieg, E. s e e van der Vegt, H.A. 710 Vogl, P. 183 Voigtlfinder, B. 649 Volin, K.J. see Susman, S. 183 Volkmann, U.G. 575 Volkmann, U.G. s e e Faul, J.W.O. 569 Volimer, R. s e e Schmicker, D. 574 Vollmer, R. see Toennies, J.P. 575 Volokitin, A.I. 649 V on Oertzen, A. s e e Engel, W. 496 Von Oertzen, A. s e e Jakubith, S. 497

Author

index

Von Oertzen, A. s e e Rotermund, H.H. 500 V ora, P. s e e Dutta, P. 569 Voronkov, V.V. 98,649 Vosko, S.H. s e e Perdew, J.P. 647 Voter, A.F. 649 Voter, A.F. s e e Liu, C.L. 97 Vu Grimsby, D.T. 496, 649 V vedensky, D.D. s e e MacLaren, J.M. 267, 498,646 Wachutka, G. s e e Scheffler, M. 500, 648 Wagner, R. 421 Walker, J.A. s e e Prutton, M. 183,227 Walker, J.S. s e e Domany, E. 569 Walker, S.M. s e e Grimley, T.B. 645 Wallden, L. s e e Lindgren, S.A. 498 Wailis, R.F. s e e Eguiluz, A.G. 644 Wailis, R.F. s e e Maradudin, A.A. 646 Waiters, A.B. s e e Boudart, M. 180 Wan, K.J. 136, 501 Wandclt, K. s e e Miranda, R. 572 Wang, C. 575 Wang, C.P. s e e Over, H. 499 Wang, C.Z. 136, 792 Wang, C.Z. s e e Xu, C.H. 136 Wang, G.C. 98,649, 710, 792 Wang, G.C. s e e Ching, W.Y. 643 Wang, G.C. see Lagally, M.G. 710 Wang, G.C. s e e Liew, Y.F. 791 Wang, G.C. s e e Wendelken, J.F. 792 Wang, G.C. see Yang, H.N. 711 Wang, G.C. s e e Zuo, J.K. 711,792 Wang, J. s e e Ocko, B.M. 135 Wang, L.Q. 227 Wang, M. s e e Dai, X.Q. 643 Wang, N. 421 Wang, R. 575 Wang, R. s e e Gay, J.M. 570 Wang, R. s e e Krim, J. 571 Wang, S.C. 501 Wang, S.K. s e e Wang, R. 575 Wang, S.W. s e e Van Hove, M.A. 268,649 Wang, W.D. 711 Wang, X.-S. 98 Wang, X.-S. see Wei, J. 99 Wang, X.W. 136, 792 Wang, Y. s e e Perdew, J.P. 135

851 Wang, Y. s e e Susman, S. 183 Wang, Y.C. s e e Chang, C.S. 225 Wang, Y.R. s e e Duke, C.B. 266 Wang, Y.R. s e e Kahn, A. 267 Wang, Z.L. 227, 421 Wang, Z.L. s e e Yao, N. 184 Ward, J. s e e Young, R. 421 Warmack, R.J. s e e Zuo, J.K. 228, 711 Warren, B.E. 360, 575 Washkiewicz, W.K. s e e Robinson, I.K. 43, 360 Wasterbeck, E. s e e Simon, A. 500 Watanabe, F. 649 Watanabe, F. s e e Ehrlich, G. 644 Watanabe, S. 501 Watson, P.R. 649 Watt, I.M. 420 Weakliem, P.C. 268 Weaver, J.H. s e e Yang, Y.-N. 99 Webb, M.B. 268, 360, 711,792 Webb, M.B. s e e Barnes, R.F. 358 Webb, M.B s e e Bennett, P.A. 788 Webb, M.B s e e Cohen, P.I. 568 Webb, M.B s e e Lagally, M.G. 267 Webb, M.B s e e McKinney, J.T. 359 Webb, M.B s e e Men, F.K. 791 Webb, M.B s e e Mo, Y.W. 710 Webb, M.B s e e Moog, E.R. 572 Webb, M.B s e e Phaneuf, R.J. 268, 791 Webb, M.B s e e Swartzentruber, B.S. 98,268, 710 Webb, M.B. s e e Tong, S.Y. 268 Webb, M.B. s e e Unguris, J. 575 Webber, P.R. s e e Bassett, D.W. 494 Weber, T. s e e Stillinger, F. 136, 648 Weber, W. s e e Wang, X.W. 136, 792 Wedler, H. 501 Weeks, J.D. 98, 711 Wegner, F.J. 792 Wei, C.M. 136 Wei, C.M. s e e Hong, I.H. 497 Wei, C.M. s e e Tong, S.Y. 268,501 Wei, D.H. 649 Wei, D.H. s e e Skelton, D.C. 648 Wei, J. 99 Wei, J. s e e Williams, E.D. 99 Wei, S.H. s e e Singh, D. 136 Weibel, E. s e e Binnig, G. 266, 789

852

Weimer, W. 575 Weinberg, W H. s e e Chan, C.-M. 495 Weinberg, W H. s e e Christmann, K. 643, 789 Weinberg, W H. s e e Comrie, C.M. 495 Weinberg, W H. s e e Kang, H.C. 645 Weinberg, W H. s e e Rahman, T.S. 500 Weinberg, W H. s e e Thomas, G.E. 501 Weinberg, W.H. s e e Van Hove, M.A. 183, 501,360, 649 Weinberg, W.H. s e e Williams, E.D. 501,650 Weinert, B. s e e Noffke, J. 499 Weinert, M. s e e Fu, C.L. 134, 710 Weinert, M. s e e Wimmer, E. 136, 650 Weiss, A.H. s e e Braslau, A. 358 Weiss, G.H. s e e Maradudin, A.A. 359 Weiss, H. see Ertl, G. 496 Weiss, H. see Schmicker, D. 574 Weiss, W. see Barbieri, A. 225 Weitcring, H.H. 136 Wcitcrling, H.H. s e e Heslinga, D.R. 497 Welkie, D.G. 711 Wclton-Cook, M.R. 183 Wclton-Cook, M.R. s e e Prutton, M. 183,227 Wcndeken, J.F. s e e Zuo, J.K. 711 Wendclken, J.F. 792 Wcndclken, J.F. s e e Zuo, J.K. 228 Wengelnik, H. see Wilhelmi, G. 501 Wern, H. see Courths, R. 643 Wcrt, C.A. 43 Wcrtheim, G.K. see Rifle, D.M. 500 Wesner, D.A. 501 W estrin, P. s e e Lindgren, S.A. 498 Wetzl, K. see Eiswirth, M. 496 Wheeler, J.C. see GriMths, R.B. 790 Whetten, R.L. see Snyder, E.J. 420 White, A.F. s e e Hochella, Jr., M.F. 181 White, G.K. 575 White, J.D. s e e Cui, J. 495 White, J.M. s e e Parrott, L. 500 White, J.M. s e e Poirer, G.E. 227 White, J.W. 575 White, J.W. see Meehan, P. 572 White, J.W. s e e Tabony, T. 574 Whitten, J.L. 649 Whitten, J.L. s e e Cremaschi, P.L. 643 Whitten, J.L. s e e Madhavan, P. 646 Wicksted, J.P. see Larese, J.Z. 571

Author

index

Widom, M. s e e Grest, G.S. 790 Wiechers, J. s e e Brune, H. 495 Wiechert, H. 575 Wiechert, H. s e e Cui, J. 569 Wiechert, H. s e e Feile, R. 570 Wiechert, H. s e e Freimuth, H. 570 Wiechert, H. s e e Knorr, K. 571 Wiechert, H. s e e Koort, H.J. 571 Wiechert, H. s e e Tiby, C. 575 Wiechert, H. s e e Weimer, W. 575 Wieckowski, A. s e e Rikvold, P.A. 647 Wiesendanger, R. s e e Gtintherodt, H.-J. 645 Wiesendanger, R. s e e Tarrach, G. 227 Wilhelmi, G. 501 Wilhelmi, G. s e e Btiumer, M. 225 Willenborg, K. s e e Dederichs, P.H. 643 Williams, A.A. s e e Pluis, B. 359 Williams, A.R. 649 Williams, A.R. s e e Lang, N.D. 4 9 8 , 5 7 1 , 6 4 6 Williams, A.R. s e e Moruzzi, V.L. 646 Williams, B.R. s e e Mason, B.F. 572 Williams, E.D. 99, 268,360, 501,650 Williams, E.D. s e e Barteit, N.C. 96 Williams, E.D. s e e Hwang, R.Q. 645 Williams, E.D. s e e Kodiyalam, S. 97 Williams, E.D. s e e Phaneuf, R.J. 98 Williams, E.D. s e e Taylor, D.E. 649 Williams, E.D. s e e Wang, X.-S. 98 Williams, E.D. s e e Wei, J. 99 Williams, R.S. s e e Katayama, M. 497 Williams, R.S. s e e Snyder, E.J. 420 Williams, W.S. 228 Willis, C.R. s e e Hsu, C.-H. 790 Willis, R.F. s e e Campuzano, J.C. 789 Willis, R.F. s e e Jeon, D. 134, 790 Wilsch, H. s e e Finzel, H.-U. 359 Wilson, E.B. s e e Pauling, L. 573 Wilson, K.G. 650, 792 Wilson, R.J. 136 Wilson, R.J. s e e Chambliss, D.D. 133, 710 Wilson, R.J. s e e Johnson, K.E. 710 Wilson, R.J. s e e WOlI, Ch. 136 Wimmer, E. 136, 501,650 Wimmer, E. ,see Fu, C.L. 134, 710 Winkelmann, K. s e e Toennies, J.P. 360 Winkler, R.G. s e e Spatz, J.P. 420 Wintterlin, J. 501

Author

853

index

Wintterlin, J. s e e Behm, R.J. 494 Wintterlin, J. s e e Brune, H. 495 Wintterlin, J. s e e Coulman, D.J. 495,789 Wintterlin, J. s e e Kleinle, G. 498 Wintterlin, J. s e e Trost, J. 501 Witzel, St. s e e Pollak, P. 500 Wohlgemuth, H. 502 Wohlgemuth, H. s e e Gierer, M. 496 Wohigemuth, H. s e e Over, H. 499 Wohlgemuth, H. s e e Schwarz, E. 500 Woicik, J. s e e Pate, B.B. 182 Woicik, J.C. 502 Wojciechowski, K.F. s e e Rogowska, J.M. 648 Wold, A. 183 Wolf, D. 99, 650 Wolf, D. s e e Moritz, W. 135 Wolf, D. s e e Zuschke, R. 502 Wolf, D.E. 650 Wolf, E. s e e Bom, M. 358 Wolf, J.F. s e e Frohn, J. 97, 644 Wolf, J.F. s e e Poensgen, M. 98 Wolff, P.A. s e e Schrieffer, J.R. 648 Wolken, G., Jr. 360 Wolkow, R. 711,792 W611, Ch. 136 W611, Ch. s e e Harten, V. 134, 710 W611, Ch. s e e Lahee, A.M. 498 Wong, C.W. 792 Wong, P.C. 502 Wong, Y.-T. 650 Wonka, U. s e e Finzel, H.-U. 359 Wood, E.A. 43 Woodruff, D.P. 183,421 Woodruff, D.P. s e e Kerkar, M. 497 Woodruff, D.P. s e e Onuferko, J.H. 791 Woodruff, D.P. s e e Robinson, A.W. 791 Woratschek, B. 502 Wortis, M. 99 Wortis, M. s e e Pandit, R. 573 Wortis, M. s e e Rottman, C. 98 Wright, A.F. 184, 650 Wright, A.F. s e e Leadbetter, A.J. 182 Wright, C.J. s e e Flores, F. 644 Wu, F.Y. 792 Wu, H. 502 Wu, J. s e e Pate, B.B. 182 Wu, N.J. s e e Wang, W.D. 711

Wu, P.K. 711 Wu, Y. s e e Stoehr, J. 420 Wu, Y.K. s e e Vu Grimsby, D.T. 496, 649 Wyckoff, R.G.W. 228 Wyckoff, W.G. 268 Wycoff, R.W.G. 184 Wyrobisch, W. s e e Bradshaw, A.M. 494 Wyrobisch, W. s e e Hofmann, P. 497 Xie, J. s e e Sun, Q. 649 Xiong, F. s e e Ganz, E. 790 Xdng, F. s e e Ganz, E. 496 Xu, C.H. 136 Xu, G. s e e Tong, S.Y. 268 Xu, J. s e e Rowntree, P. 573 Xu, J. s e e Ruiz-Suarez, J.C. 573 Xu, P. s e e Dunn, D.N. 418 Xu, W. 650 Yadavalli, S. 228 Y aegashi, Y. s e e Kinoshita, T. 498 Yagi, K. 421 Y agi, K. s e e Osakabe, N. 98,791 Y agi, K. s e e Sato, H. 98 Yagi, K. s e e Takayanagi, K. 136 Y agi, K. s e e Y amazaki, K. 792 Y alabik, M.C. s e e Rikvoid, P.A. 648 Yalisove, S. s e e Copel, M. 495 Yamada, M. s e e Bullock, E.L. 133 Yamazaki, K. 792 Y ang, C.N. s e e Lee, T.D. 791 Yang, H.N. 711 Yang, H.N. s e e Zuo, J.K. 228 Y ang, M.H. s e e Y adavalli, S. 228 Yang, W. s e e Parr, R.G. 182 Yang, W.S. 502 Y ang, W.S. s e e Huang, H. 497 Y ang, X. s e e Cao, R. 495 Yang, Y.-N. 99 Yaniv, A. 650 Yao, N. 184, 228 Yates, Jr., J.T. s e e Chen, J.G. 495 Y ates, Jr., J.T. s e e Uram, K.J. 501 Ye, L. 136 Ying, S.C. 792 Ying, S.C. s e e Chung, J.W. 789 Ying, S.C. s e e Han, W.K. 134

854 Ying, S.C. s e e Hu, G.Y. 790 Ying, S.C. s e e Roelofs, L.D. 791 Ying, S.C. s e e Tiersten, S.C. 649 Yip, S. s e e Poon, T.W. 98, 647 Yoon, M. 99 Yoshida, S. s e e Hara, S. 226 Yoshikawa, S. s e e Nogami, J. 499 You, H. 575,711 Youn, H.S. 575 Young, R. 421 Yu, R. 136

Zaima, S. s e e Oshima, C. 227 Zanazzi, E. 360 Zanazzi, E. s e e Galeotti, M. 225 Zangwili, A. 184, 650, 792 Zangwill, A. s e e Redfield, A.C. 98,647 Zaremba, E. s e e Stott, M. 648 Zartner, A. s e e Hofmann, P. 497 Zastavnjuk, V.V. s e e Teplov, S.V. 420 Zegenhagen, J. 502,792 Zeglinski, D M. s e e Ogletree, D.F. 647 Zehner, D.M 792 Zchner, D.M s e e Gruzalski, G.R. 226 Zehner, D.M s e e Mundenar, J.M. 499 Zehner, D.M s e e Yoon, M. 99 Zehner, D.M s e e Zuo, J.K. 228, 711 Zeiger, H.J. s e e Gay, R.R. 225 Zcller, R. s e e Dederichs, P.H. 643 Zemansky, M.W. 99 Zeng, H.C. s e e Parkin, S.R. 500 Zcppenfeld, P. 575 Zeppcnfeld, P. s e e Bienfait, M. 568 Zeppenfeld, P. s e e David, R. 569 Zeppenfeld, P. s e e Kern, K. 571, 710 Zerner, M.C. s e e Salahub, D.R. 183

Author

Zettlemeyer, A.C. s e e Klier, K. 498 Zhang, B.L. 136 Zhang, J.P. s e e Marks, L.D. 419 Zhang, Q.M. 575,792 Zhang, Q.M. s e e Kim, H.K. 571 Zhang, Q.M. s e e Larese, J.Z. 572 Zhang, S. 575 Zhang, T. 650 Zhang, T. s e e Dai, X.Q. 643 Zhang, T. s e e Sun, Q. 649 Zhang, Z. 99, 184 Zhang, Z.P. s e e Ai, R. 418 Zhdanov, V.P. s e e Myshlyavtsev, A.V. 647 Zheng, H. 650 Zheng, X.M. 184 Zhou, J.B. 228 Zhou, M.Y. s e e Hui, K.C. 497 Zhou, M.Y. s e e Parkin, S.R. 500 Zhou, M.Y. s e e Wong, P.C. 502 Zhou, W. 228 Zhu, D.M. 575 Zhu, D.M. s e e Pengra, D.B. 573 Zhu, X. 268 Zia, R.K.P. s e e Avron, J.E. 96 Zieger, H.J. s e e Henrich, V.E. 181 Ziegler, J.F. s e e Biersack, J.P. 418 Ziman, J.M. 136 Zimmer, R.B. 360 Zlati, V. s e e Gumhalter, B. 645 Zschack, P. 228 Zunger, A. 184 Zunger, A. see Bendt, P. 133 Zunger, A. s e e ihm, J. 134 Zuo, J.-K. s e e Wang, G.-C. 792 Zuo, J.K. 228,711,792 Zuo, J.M. 421 Zuschke, R. 502

index

Subject index adsorbed surfaces 159 adsorption 64, 68, 142, 322, 453,465,476, 53, 579, 592, 665,773 adsorption/desorption rate 633 adsorption effects on reconstruction 767 adsorption energies 594, 641 adsorption, heat and entropy of 533 adsorption, isosteric heat of 533, 534, 535 adsorption isotherms 509, 511,533,549, 554, 555, 561-563,566 adsorption sites 317,434, 460, 472, 474, 615 adsorption-induced dipoles 584 advanced Green's function 589 AES (Auger electron spectroscopy) 166, 217, 219, 456, 507,554, 661 affinity energy 587 AFM 195, 197, 223, 363, 373 Ag 67, 91, 105, 121, 127,676 Ag adsorption 124 Ag/Ag(l 11) 708 Ag(100) 83 Ag(110) 67, 116, 122, 449, 449, 637, 760 A g ( l l l ) 83,558 Ag( 11 l)-Cs 469, 470 ",/-3-x'~-f3Ag/Si( 111 ) and Au/Si( 111 ) 125 AIP 249 AI 111, 116, 128 AI adsorption 124 A! and Sn on GaAs(llO) 124 AI(OOI)-Na 464 AI(II0) 116 AI(I 11) 460, 474, 476, 476, 763 AI(II I)-K 469, 473,475,476 AI( 11 l)-Na 476 AI(I 11)-O 448,456 AI(I 1 l)-Rb 469

ab initio calculations 153 ab initio computations 156 ab initio methods 141, 144 ab-initio molecular dynamics simulations 110 ab initio psuedopotentials 148 ab initio SCF-LCAO method 147

aberrations in emission microscopes 417 accommodation 552 accommodation coefficient 689 activated process 438,450, 453 activation barrier 456, 582 activation energy 247, 450, 689, 772 adaptive grids 111 adatom interactions 659, 787 adatom model 487 adatom-substrate hopping 590 adatom-substrate coupling 601,604 adatoms 239, 289, 451,471,579, 665 adatom-adatom interactions 515 added-row 451 adhesion images 376 adiabatic calorimetry 535 adiabatic cleavage 58 adiabatic demagnetization 531 adiabatic process 57 adparticles 471 adsorbate-defect complex 177, adsorbate ordering 718 adsorbate-induced reconstructions 132, 718 adsorbate-adsorbate interactions 616, 697, 473,518, 517,564, 567 adsorbates 176, 325 adsorbate-substrate interactions 515, 517, 518, 564, 567 adsorbed layers 427 adsorbed overlayers 248 855

856 o~-A1203171, 194, 198, 207, 209 ct-A1203 (0012) 208 ot-Al203 (0001) 199, 200, 208,224 a-alumina 171, 172 y-alumina 207 AlAs 249 alkali adatoms 582 alkali adsorption 760, 761,769 alkali halides 175 alkali metal/Si(111) 127 alkali metals 116, 120 alkali metals on semiconductors 130 alkali-metal atoms 428,459 allotropes 174 alloy formation 476 AIN 219,220 ammonia 459 ammonia synthesis 477 ammonia synthesis reaction 459 amplitude ratios 746 amplitudes 737 Anderson model 586, 591,609, 638 angle-resolved photoemission 129 angle resolved photoemission spectroscopy (ARPES) 233 angular dependence of the surface free energy 77 angular-resolved UPS 440 anharmonic effects 666 anion terminated 258 anionic vacancies 175 anions 248,249, 253,261,264 an~sotropic temperature effects 437 antsotropy in the dimerization 671 anJsotropy of the surface tension 68 annealing 244 anti-NiAs structure 218 anti-Bragg condition 295 antibonding 430, 452, 484, 638,639, 697 antibonding electronic states 246 antibonding molecular orbital 479, 582 antiferromagnetic field 731 antiferromagnetic Ising model 730 antiferromagnetic Ising model phase diagrams 732 antiphase walls 684 Ar 511,518,540, 564

Subject index

Arrhenius plot 621 As 129 As capping 262 As dimers 262 As trimer 259 asymptotic interaction 607, 608 asymptotic regime 638 atom-atom correlation function 273 atom basis 194 atom diffraction 37 atom probe 387 atom scattering 273,555 atomic beam scattering 659 atomic connectivity 234 atomic coordinates 113 atomic force microscope s e e AFM atomic forces 144 atomic form factor 274, 327 atomic height steps 199 atomic packing 189 atomic polarizabilities 505 atomic relaxations 246 atomic resolution 380 atomic scale resolution 376 atomic scattering factor 289, 290, 658 atomic sizes 233 atomic steps 289 atomic vacancies and substitutions 174 atomic vibrations 79 atoms, rows of 389 atoms, strings of 389 atop sites 21,580, 587,639, 640, 768 attenuation coefficient 279, 284, 325 attenuation length 281 Au 91, 105, 127, 755 Au adsorption 124 Au(100) 120, 122, 755 Au(110) 120, 449, 619, 760, 771 Au(110)(Ix2) 116 A u ( l l l ) 120, 122,405,700 Au(111)(22xq-33) reconstruction 699 Auger de-excitation 478 Auger electron spectroscopy see AES Auger electrons 307 auto-correlation function 37,299 autocatalytic process 441 autocatalytic reaction process 483

Subject index

autocompensation 142, 164, 167, 17 I, 172, 174, 177. 247,250, 256, 259, 265 autocompensation principle 248 autocorrelation function 508,686, 687 average T-matrix approximation (ATA) 599 Axilrod and Teller 507 azimuthal ordering 468, 482 B. AI, Ga, In on Si(100) and Si(l I 1) 128 B 5 site 488 back focal plane 400 back-bonding 431,432, 483 back-donation 430, 431,468,485 background intensity 658 backscattering spectra 393 ballistic surface erosion 199 band gap 115 band structure 106, 154, 626 band structure effects 105 bandwidth 609 bare interaction 634 basal planes 532 basis functions 109 basisscts 110, 148 basis vectors 22 basis vectors of the reciprocal lattice 22 BaTiO 3 (100) 211 bcc 8.604, 611. 615. 620. 621. 623-625 bcc substrates 624 Bc 111 Beeby approximation 349 Beeby matrix inversion 323 bending-vibrational modes 435 bent bonding 435 benzoate 365 BeO 202 Bi 233 Bi atoms on Si 403 Bi/Si(l I 1) 129 Bi2Sr2CaCu208_ x 211 binary collisions 387,388 binding energy 6 5 , 5 4 7 , 5 4 8 , 5 8 0 binding sites 684, 695, 768 binding states 613 block renormalization 800 blocking 195 blocking dips 392

857 Blyholder model 429, 431 BN 219, 532 body-centered-cubic see bcc Boltzmann factor 35 o-bond 241,246 bond angles 104, 105, 142, 590 bond distortions 490 bond lengths 104, 107, 113, 142, 162, 169, 171, 172, 175, 178, 253, 447, 460, 469-471,474, 481,590, 606, 626, 630 bond-length-conserving rotations 255, 263,265 bond orientational order 521 bond-saturation model (BSM) 618 bond strength 456 bonded atoms 668 bonding 245,430, 587,639, 690, 697 bonding in ceramics 189 bonding orbital 582 Born approximation 274, 344 Born-Mayer repulsive term 155 Born-Oppenheimer approximation 145 bouncing mode 512 bound state 584 bound state resonances 348 Bragg angle 338 Bragg peak 659 Bragg points 293. 295 Bravais lattices 5. 19, 193, 233 points 5 unit cell vectors 5 bridge bonds 604 bridge sites 434, 587,593,640, 769 bright and dark field imaging 401,405 Brillouin zone 23, 288,667, 676, 686 Brillouin zone boundary 23 broken symmetry 793 brownian mobility 542 brownian motion 513 Brush 719 Buckingham potential 155 buckled dimers 662 buckled surface 475 buckling 178,485,606 buffer layers 675 bulk crystallography 5 bulk density of states 132 bulk elastic constants 155

858 bulk lattice constant 254 bulk modulus 672 bulk phonons 285 bulk states 115 Burger vector 41,663,672, 673,700 buried interfaces 344, 406 buried oxygen 459 butadiene-iron-tricarbonyl (BIT) 546 butane 512, 512, 512 c-BN 220 c-BN (001) 220 C ( I I I ) 161 C(I 1 l)-(2xl) 162 c(2x2) 581,598,627, 631 C2H 4 513 C2H 6 513,515,521,542 CaCO 3 196 cadmium chloride 511 calorimetry 533,535,566, 747 adiabatic 535 isothermal 535 canonical partition functions 726 canonical ensemble 65 cantilevers 378,379 CaO 202 capillary condensation 531,532, 566 carbides 143,212,222 carbon monoxide, adsorption of 428 carbonyl chemistry 427 carrier transport 655 Car-Parrinello method 109, 110 catalysis 139, 166, 176, 443,716 catalysts 171 catalytic CO oxidation reaction 440 catalytic poisoning 632 catalytic processes 117 catalytic reactions 174, 427 cation terminated surface 258 cations 248,249, 253, 261,264 CD 4 539 CdS 231,249, 263 CdSe 231,249 CdTe 249,403 centered meshes 13 centered sites 21,587,626 ceramic materials 187

Subject index ceramic synthesis 139 ceramics, bonding in 188 cesium peroxide (Cs20 2) 478 cesium suboxide (Cs! iO3) 478 cesium superoxide (CsO2) 478 Cf3CI 517 CF3CI 517 CH 4 509,513,514,559 chains 130, 237 chalcogens 446, 630 channeling 195,392, 393, 397 channeling/blocking 392 charge corrugation 349 charge density 107, 108, 116 charge density distribution 104 charge gradient 619 charge neutral surface 167 charge neutrality 178,200 charge state 171 charge transfer 247,449, 457,459, 464, 483, 486, 582, 600, 630, 633 charge transfer effects 105 charge-induced reconstruction 465 charged neutral surface 142 chemical adsorption 65 chemical bonding 159, 235,427,507 chemical forces 245 chemical potential 53, 57, 64, 65,580, 588, 633,723, 726, 729, 737, 773,804 chemical properties 273 chemical sensor applications 166 chemical sensors 176 chemical turbulence 443 chemical vapor deposition (CVD) 326 chemical waves 442 chemisorption 115, 174, 176, 427,443,448, 459, 505,580, 582, 584, 585,587,600, 636, 729, 787,788 chemisorption bond 582 chemisorption energy 431,450 chemisorption state 445 chemisorption, weak 596 classical force fields 174 classical force models 122 classical models 104 classical potential models 140, 154 classical potentials 104, 117, 145, 155, 175, 176

Subject index classical turning point 345,348 classification 747 classification of overlayer meshes 30 Clausius-Clapeyron line 533, 551,552 clean solid surfaces 55 cleavage 56, 58, 160, 172, 188, 233,235, 243, 247-249, 679 cleavage faces 236, 258,265 cleavage process 142 cleavage surfaces 142, 256 cleaved surfaces 234 cluster binding energy 627 cluster calculations 144, 448,582 cluster density 688 cluster methods 112 cluster models 156, 158, 159 cluster variation 631 clusters 112, 473,600, 613 Co on transition metals 427 CO covered Pt(ll0) 364 CO on Ni(100) 431 CO on Pt(100) 413,417 CO on Pt(ll 1) 659, 660 CO/Ru(0001) 430 Co(1010) 453 coadsorbate phase 480 coadsorption 428, 461,483,485,586 coesite ! 74 coexistence 509, 510, 563,567, 615 coexistence regions 533,730, 780 coherence length 532, 553, 686 coherence limitations 733 coherent domains 277 coherent lattices 30 coherent potential approximation 583 cohesive energy 104, 189, 507, 618,629, 725, 771 coincidence lattices 30, 686 collective excitations 541 collective motion 513 Coloumb and exchange interactions 150 combined space method 323 commensurability 483, 517 commensurate adlayer 515 commensurate lattices 30 commensurate phase 555,561 commensurate solid phase 559

859 commensurate solids 510, 513,514, 522, 545, 567 commensurate structure 515, 560 commensurate-incommensurate transition 516, 519, 554, 567 complete wetting 562 completely ionic material 163 complex meshes 15 composition 53 compound formation 456, 477 compound semiconductors 247, 248 cohapressibility 701 concentration 53 concerted movements 771 condensation 534, 550, 551,564, 688 configurational entropy 65, 81,633 conjugate honeycomb-chained-trimer 126 constituents 508 contamination 549 continuous phase transitions 567 continuous transition 525,561,720, 766 continuum approximation 604 continuum elasticity 605 contractions 448 contractive reconstructions 119, 132 contrast transfer function 402 convergence 108, ! 12 convergent beam-REM 405 convexity 69, 72 convolution theorem 304 CoO 201,202 coordination 618,639 coordination number 470, 474, 617, 624 coordination polyhedra 189, 190 core-electrons 108 core-level-shift spectroscopy 758 corner-cube- and face-centered-anisotropy 754 corrected effective medium theory 616 correlated collisions 392 correlated roughness 709 correlation effects 112, 600 correlation energy 148 correlation function 580, 632, 633, 665,737, 738,741,751 correlation length 701,740, 749, 753, 798, 799, 804, 806 correlation length exponent 804

860 correlations 38,545, 581,600, 625,678,747, 798,799 corrosion 139, 176, 443 corrugation 348, 349, 468, 469, 471--473, 481, 483 corrugation function 345 corundum 171, 179, 193, 198,207 Coulomb forces 163,374 Coulomb potential 154, 155 Coulomb repulsion 120, 609, 616, 623 Coulombic interactions 144, 148, 154, 155 coupled channel calculation 348 coupled channel techniques 275,344, 348 covalent bonding 106, 132, 188, 231,764 covalent radius 471 coverage 64, 510, 550, 561,665,678,680, 726, 774, 785 CPA 599 Cr20 3 207, 209 Cr3C 2 213 or- and 13-cristobalite 174 critical angle 335 for X-rays 330 critical behavior 448 critical exponents 508,522, 535, 718, 736, 737,742, 743,746-748,766, 788,798, 800, 801,804-806 critical hopping 601 critical line 751 critical mismatch 516 critical phenomena 508 critical point 511,531,567, 716, 718,720, 728,736, 797,794, 799 critical scattering 739, 741,749, 766 critical temperature 509, 533, 797 critical wetting temperature 563 crossover point 445 crystal growth 691,704, 705 crystal lattices 5 crystal shape 53 crystal structure factor 276, 289 crystal symmetry 189 crystal truncation rods 280, 299, 305 crystalline metal oxides 142 crystallographic definition of the step 71 crystallographic orientation 68 crystallography of a plane 10, 11

Subject index Cu 91,105 Cu on Cu(100) 684 Cu(100) 684 Cu(110) 116, 365,364, 449, 464, 761 Cu(110)(2• reconstruction 364, 697,698 Cu(110)-K 761 Cu(110)p(2• 450 Cu(110)-O 451 Cu(111) 673, 674 Cu(l I l)-Cs 469 Cu(113) 667 Cu(115) 667 Cu3Au(001) 679 cubic anisotropy 751 cubic sphalerite structure 215 cubium 600, 601 CuCI 249, 251 d-bands 591 d-shells 105 dangling bond electrons 141 dangling bonds 127, 129, 132, 141, 142, 163, 167, 170, 177, 217, 231,234, 236, 242, 246-248,250, 488,668,669, 758 rehybridization of 234, 246 saturation of 234 dangling orbitals 770 dark field imaging 401 Darwin width 335,339 Davidson scheme 109 deBroglie relation 309, 356 deBroglie wavelength 304, 382 Debye temperature 286 Debye-Waller 352 Debye-Waller effect 666 Debye-Waller exponent 286 Debye-Waller factor 326, 511,548,743,744 decomensuration 231 defect interactions 223,641 defect microstructure 197 defect sites 174 defect states 131 defect structures 157 defects 35, 142, 159, 194, 195, 221,224, 231, 245,277, 289, 290, 298,343, 456, 458, 655,656, 674, 699, 704, 708, 725,784, 803, 805

Subject index

of the first kind 38, 39, 655,658 of the second kind 38, 39, 655 of the third kind 656 defocus 401 degree of ionicity 189 delocalized rr bonding 243 dendritic islands 691 densities 793 density function theory 242 density functional calculations 474 density functional formalism 105, 107, 464 density functional method 144, 145, 147, 149, 151 density functional pseudopotential 173 density functional theories 108, 115 density matrix methods 107, 111 density of states 581,588,616, 655 depolarization 467, 468,470 of dipoles 462 depth of field 412 desorption 461,467, 549,630, 633,773 desorption temperature 479 devil's staircase 517 diagonalization procedures 109 diamond 179,231 diamond cleavage surface 160 diamond lattice 236 diamond surfaces 142, 160 dielectric 139 dielectric constant 139 dielectric permittivity 139, 165 difference map 340 diffraction 29, 273,537,538,655-666, 674, 678,683,688,697, 706, 708 Laue condition for 279 diffraction intensity oscillations 707,708 diffraction lineshape 288, 319 diffraction methods 33 diffraction oscillations 680 diffraction pattern 400 diffraction peaks 327 diffraction profiles 325,538 diffraction rod 305 diffuse intensity 739 diffusion 53, 124, 513,515,634, 658,686, 691,707,716, 718,730, 760, 761, 771-773,776

861 diffusion anisotropy 689, 690, 691 diffusion barriers 690, 693,771 diffusion coefficients 358,530, 547,635,688 diffusion constant 787 diffusion mechanisms 122 diffusion of defects 709 diffusion time 787 diffusivity 93 dimensionality 511 dimer pair energy 624 dimer rows 130, 668,688, 689, 697 dimerization 178 dimers 237, 239, 246, 247,249, 260, 261,582, 630, 661,662, 677 dimer-adatom-stacking fault model 123, 237 diperiodic groups 18 dipolar repulsions 85 dipole moment 459, 464, 466, 467, 470, 472, 507,582 dipole-dipole coupling 436, 583 dipole-dipole interactions 88,469, 582, 583, 635,636 dipole-dipole repulsion 467,471,473,603 dipole-quadrupole force 583 dipoles 505 dipoles, screening between the 471 direct imaging 398 direct interactions 579, 581,580, 586, 587, 626 direct lattice 21 direct methods 363 direct space 273 directional bonding 105 directional d-bonding 119 directions 23 directions of a form 6 disclinations 523,567 discrete Gaussian model 86 disk of confusion 402 dislocation, edge orientation 41 dislocation line 39 dislocation, mixed 41 dislocation networks 709 dislocation, screw orientation 41 dislocations 199, 219, 221,223, 523,523,545, 554, 567, 655,672, 673,705,707 disorder 35,242, 680, 701,703, 788 disorder effects 703

862 disordered structures 289 disordering transitions 760 dispersion 609 dispersion curves 155, 513, 558 dispersion forces 506 dispersion relations 513 displacement sensors 371 displacive reconstructions 118, 119, 757, 759 dissociation 431,445,549, 725 dissociation energy 445,621 dissociative adsorption 580 dissociative chemisorption 448 distorted rhenium oxide 193 dividing surface 60, 61 domain boundary pinning 704 domain boundary 42 domain boundary pinning 704 domain degeneracy 695 domain growth kinetics 693 domain size 671,675,698,701,703,704 domain size distribution 687,703,704, 775 domain walls 42, 520, 523, 545,554, 567,656, 672,688,693,695,696, 701,709 domains 281,317,655,656 domains 664, 669, 674, 684-687,693, 694, 696, 697, 702, 703 domains on stepped surfaces 696 o donation 485 donor-acceptor mechanism 429 double alignment technique 391,392 double-layer missing-row reconstruction 454 dynamic LEED 322 dynamic linear dielectric function 507 dynamic polarizability 507 dynamical coupling 513 dynamical matrix 605 dynamical scattering 326 EAM (embedded atom method) 105, 122, 614, 616, 617,619, 625,629, 636, 771 edge dislocation 39 EDIM (embedded diatomics-in-molecules) 618 EELS (electron energy loss spectroscopy) 143, 211,615,634 effective medium 599 effective medium theory s e e EMT effective radius 469

Subject index

Eikonal approximation 345,346 Einstein mode 513 elastic constants 104, 604 elastic deformation 375 elastic distortion 602 elastic effects 601,604, 635 elastic incoherent structure factor (EISF) 541 elastic interactions 88, 90, 606, 698 elastic mean free path 279 elastic medium 636 elastic moduli 635 elastic repulsions 85 elastic strain, release of 490 elastic stress in the surface region 486 elastic yield 307 elasticity theory 93 electric dipole moments 582 electric multipole 508 electrical insulator 139 electrochemical cells 641 electrochemical methods 117, 369 electrochemistry 176 electron bombardment 175 electron charge density 351 electron correlation 141, 149, 151, 162 electron correlation effects 149 electron correlation energy 148 electron counting 142 electron density 151 electron density gradient 342 electron diffraction 733 electron double counting terms 154 electron-electron interactions 146, 148, 151, 152, 153 electron-energy loss spectroscopy see EELS electron exchange coupling 610 electron gas 109 electron-gas model 156 electron-ion energy 151 electron microscopy 91,363, 398, 766 electron spin 146 electron-stimulated desorption (ESD) 194, 205,310, 555 electron-stimulated reactions 199 electron tunneling 364 electronegative adsorbates 582 electroneutrality 430

863

Subject index

electronic charge density 108 electronic correlations 623 electronic friction 448 electronic hopping 579 electronic properties 21,273 electronic structure 106, 235, 367,655 electronic wavefunctions 104, 108 electronic work function 461 electropositive adsorbates 582 electrostatic dipole layer 461 electrostatic interaction 146, 486, 582 elemental semiconductors 236 elementary excitations 112 ellipsometry 560, 566 embedded atom method s e e EAM embedded cluster model 613-616 embedded cluster technique 600 embedded diatomics-in-molecules s e e EDIM embedding functions 617-619 emission electron microscopy 408 empirical classical potential models 144, 159 empirical quantum-mechanical methods 141 empirical schemes 619 empirical techniques 107 cmpiricai tight-binding methods 153, 159 EMT 105,613,614, 615,629, 770 corrected 616 energetics 234 energetics of surface defects 122 energy barrier 684 energy barriers 771 energy expansion 721 energy potential relief 471 enthalpy 534, 535 entropic repulsion 91 entropic step repulsion 81 entropy 60, 79, 81, 82, 85,534, 535,657,658, 665 epitaxial growth 210, 224, 242, 684, 699, 702, 773 equilibrium 53,767 equilibrium crystal shape 53 equilibrium faceted surfaces 68 equilibrium surface atomic geometry 144 equivalent-crystal-theory 105 ESDIAD 436 ethane 510, 514, 521,524, 525,525,542, 543,

550, 553 evaporation 451 Ewald construction 278, 279, 281,309 Ewald sphere 279, 315, 320 Ewald term 108 exchange constants 719 exchange-correlation energy 107, 109, 152 exchange-correlation functionals 107 exchange-correlation potentials 153 exchange energies 151 exchange interaction 150 exchange mechanisms 629 exchange-overlap forces 505 exfoliated graphite 532 extended defects 175, 363 extended Htickel model 594, 640 extended systems 112 extensive variables 70 extrinsic stacking fault 676 face-centered cubic s e e fcc facet planes 197 faceted TaC( 11O) 216 faceting 55, 68, 69, 74, 122, 199, 224, 242, 681,682 faceting transitions 73,581 facets 30, 67, 197, 201,221,655 family of bulk lattice planes 35 far asymptotic region 609 Faraday Cup 313 fcc 8,606, 619 fcc(100) 446 f c c ( l l l ) 116,446 Fe(211) 449 Fej_xO 190, 201,202 Fel_xO(l 11) 206 ot-Fe203(O001) (10]-2) 194 ~-Fe203(O001) 194 ot-Fe203 171, 196, 207 Fe304 206 Fe304(001) 207 Fe304(111) 206 FeAI20 4 206 feedback systems 371 Fermi energy 382, 440, 588,595,601,607, 638 Fermi level 115, 132, 464, 488 Fermi surface 106, 580, 592, 610, 636, 640

864 Fermi surface domination 591 Fermi surface effects 105 ferrimagnetic 206 ferroelectrics 210 ferromagnetic phase 720 ferromagnetic system 719 FeTiO 3 209 field desorption 386 field emission 315 field evaporation 386 field ion microscopy s e e FIM film growth 406 FIM 363, 380, 381,385, 611,620, 623,626, 627,632, 663, 771 FIM, resolution of 384 FIM tip 621,623 finite-size effects 343, 701,702, 743,749, 767, 783 finite-size limitations 702 finite-size rounding 743 finite-size scaling 803,804, 805 finite-size scaling analysis 702 Finnis-Sinclair models 105 first ionization potential 462 first-order condensation 563 first-order discontinuity 727 first-order melting 525 first-order phase transition 243, 718 first-order transition 550, 561,672, 720, 723, 724, 749, 750, 765,766 first-principles calculations 618 first-principles methods 104, 107, 141, 144 Fischer-Tropsch reaction 428 five-dimensional transition and noble metal surfaces 120 FLAPW (full potential linear augmented plane wave) 108 flood electron gun 307 fluctuation-dissipation theorem 740 fluctuations 681,722, 723,738,742, 748, 778, 784, 798, 799, 804 fluid phase 726, 727 fluorite 193, 202 focal length 402 Fock operator 150 force sensor 372 force sensor, cantilever beam 373

Subject index

forces 110 forces on each atom 158 foreshortening 405 form factor 274 four-circle geometry 331 fourfold hollow site 684 fourfold sites 455 fourfold-coordinated hollow sites 448 Fourier transform 738-740 fractal 691,708 fractal dimensionality 691 fracture 178 Frank and Van der Merwe theory 516, 519 Fraunhofer diffraction 400 Fredholm determinant 589 free-electron models 116 free energy 53, 76, 658,675,723, 736, 740, 746, 776, 777,799, 800, 804 free energy barrier 723 free energy expansion 747 free-electron gas 591,614 free-fermion approximation 92 freezing 565 Fresnel reflectivity 342 Fresnel theory 561 Friedel (1958) sum rule 599 Friedel oscillations 608, 617 Friedel sum rule 638 frontier orbitals 581,596 full potential linear augmented plane wave s e e FLAPW fullerene 375 Ga 128 Ga adsorption 124 Ga trimer 259 Ga vacancy 258 GaAs 124, 130, 231,249, 254, 367 G a A s ( - l - l - l ) 259 GaAs(100) 260 GaAs(100)-c(2• 235 GaAs(110) 123 GaAs(l I 1)-(2• 258 G a A s ( l l l ) and (-1-1-1) 258 GaN 218-220 GaP 249 GaP(I 11)-(2x2) 258

Subject index

gas phase 726 GaSb 249 Gaussian approximation 277, 278 Gaussian functions 147 Ge 111, 160, 231,233, 236, 246 Ge(100) 244 Ge( 111 )-(2xl) 247 Ge(l 11) 487, 765 Ge( 111 )-c(2x8) 247 Ge(l 11)(x/3-• 127 Ge(lll)-(2• 162, 163 germanium 243 Gibbs 55, 60, 68 Gibbs adsorption equation 64, 66 Gibbs dividing surface 60, 64 Gibbs free energy 719, 726, 795 glass formation 139 glassy phases 224 glide lines 16, 18 glide plane 455 glide-plane symmetry 481 global minimum ! 14 glue model 105, 122,617 gold 338 gradient corrections 107 gradient corrections 619 grain boundaries 35, 159, 174, 219, 291,363, 387 Gram-Schmidt 587 grand canonical partition functions 726 grand partition function 726 grand potentials 63, 69, 726 grand thermodynamic potential 57 graphite 377, 509-514, 516-518, 520, 521, 524, 525,532, 540, 542, 543,545,546, 550, 560, 563,564, 566 graphitic (2x2) 615 graphitic (2x2) or (2x2)-2H 618 Green's function 111, 112, 156, 157, 158, 588, 589, 592, 596, 604, 605,609, 610, 611, 612, 624, 628,635,639, 640 advanced 589 matching techniques 112 techniques 144 Grimley 584 ground state 114, 581 ground state energy 109

865 groundwater transport of contaminants 139 Group III elements AI, Ga, In on Si(100) 128 groups 16, 724 growth exponent 703 growth kinetics 703 growth mode 708 growth rate 705 GW approximation 115 H 3 site 487 halides 143 Halperin-Nelson model 525 Hamaker constant 374 Hamiltonian 108, 146 Hamiltonian matrix 106, 107, 146, 153 hard hexagon lattice gas model 750 hard wall approximation 345 hard wall model 345 hard-sphere radii 469, 471,473 hard-square model 631 harmonic approximation 518 harmonic-oscillator approximation 471 harpooning 462 Hartree (or Coulomb) interaction 150 Hartree and exchange-correlation potential 108 Hartree and exchange interactions 153 Hartree energy 152, 153 Hartree-Fock 162, 587, 600 Hartree-Fock approximation 505 Hartree-Fock bonding and antibonding resonances 599 Hartree-Fock calculations 107 Hartree-Fock computations 173 Hartree-Fock level 149 Hartree interaction 148 Hartree-type calculation 107 hcp 8, 606 hcp crystals, lattice planes 8 hcp sites 447,448,480 hcp(0001) 446 He 116 3He 511 4He 511 He diffraction 165,456, 659, 707 He scattering 116, 566, 657, 660, 666, 667,684 heat and entropy of adsorption 533 heat capacity 511, 59, 522, 535, 806, 807

866 heats of adsorption 431,549, 551,552 heats of cleavage and adsorption 64 heavy domain walls 42 height-height distribution functions 37 Heisenberg model 749, 753,754 Heisenberg uncertainty principle 384 helium atom scattering 275 helium atoms, wavelength of the 356 Hellmann-Feynman theorem 110 Heimholtz coils 308 Helmholtz free energy 56, 63, 69 Hermann-Mauguin notation 18 Hermitian matrix eigenvalue equation 109 Herring 55, 68 herringbone packing 521 herringbone pattern 122, 521 herringbone reconstructions 700 herringbone solid 521,525 herringbone structures 699 heteroepitaxial growth 124, 248 heteroepitaxial systems 687 hetcrogeneous catalysis 428,459 heterolytic dissociation 177 heterostructures 605 hexagonal aligned incommensurate phase 555 hexagonal-close-packed s e e hcp hexagonal rotated incommensurate phase 555 hexagonal rotated phase 557 hexagonal wurtzite structure 215 hexatic liquid crystal 523 hexatic orientational order 525 hexatic phases 559 hexto interaction 619 HfC 213,214 HfN 218 high-density surfaces 446 high energy electron diffraction (HEED) 273 high-order commensurate adlayer 515 high resolution electron energy loss spectroscopy s e e HREELS high resolution electron microscopy s e e HREM high resolution LEED 313 high-temperature annealing 175 high-symmetry sites 448,465,471 higher order commensurabilities 517 highest occupied molecular orbitals s e e HOMO hill- and valley structure 68

Subject index

[hkl] zone 33 holding potential 507, 641 hollow sites 21,434, 446, 465,471,699 HOMO (highest occupied molecular orbital) 429, 596 homoepitaxial growth 684, 688,690, 692, 697 homoepitaxy 689 homogeneous function 799 homologous classes 235 honeycomb-chained-trimers 125, 126, 492 honeycomb structure 447 honeycomb symmetry 83 hopping 384 hopping translations 542 hot adatoms 448 HREELS (high resolution electron energy loss spectroscopy) 440, 447, 455,464, 471, 480, 484 HREM (high resolution electron microscopy) 197, 198 Hi.ickel model 594 m extended 594 hybridization 429,464, 465,486, 490, 612, 641 hydrogen 160 hyperoxide ions 0 2 478 hysteresis 370, 723,724, 796

ICISS 165 ideal crystal 8 ideal gas 64 ~deal substrate lattice 25 deal surface 8 dentity transformation 16 iluminated area 332 iimenite 193,209 mage charge 465 image formation 400 image intensifiers 386 image plane 584 image shift 462 imperfections 35 improper rotations 16 impurities 42, 202, 586, 655,704, 708,725, 784, 803, 805 impurity atoms 39 In 128 In adsorption 124

Subject index

in-phase condition 302, 667,679, 692, 693 in-phase scattering 295 lnAs 249 incoherent lattices 30 incoherent scattering 540 incommensurability 516, 517, 557 incommensurate adlayer 515 ncommensurate chains 517 ncommensurate close-packed surface atomic layer 120 incommensurate-commensurate transition 547 incommensurate lattices 30 incommensurate layer 490 incommensurate overlayers 358 incommensurate phase 490, 521,545,559, 560, 561,724 with a hexagonal network of walls 520 incommensurate reconstruction 759 ncommensurate solids 510, 513, 523,545,567 ncommensurate structures 515, 518,686 ncommensurate superstructure 479 ncomplete wetting 562 ndependent particle partition function 84 ndex of refraction 330 ndirect interactions 579, 585, 607,611,612, 637 inelastic mean free path 279 inelastic mean free path length 306 inelastic neutron scattering 540 inelastic scattering 306, 659 InN 219,220 inner potential 319, 320, potential 322 inP 249 lnSb 249 InSb(l 11)-(2• 258 instantaneous atomic positions 37 instrument resolution 303 instrument response function 304, 743 insulators 115 integrated intensities 335 intensive variables 70 interaction energies 82, 640 interaction potentials 505,507 lnteratomic interactions 104 interface thickness 680 Interface width 683 interfaces 55, 159

867 interference fringes 405 ~nterference function 276, 290 ~nterferometry 380 nterlayer diffusion 707 nternal energy 55, 56, 63, 658 nternal polarization 464 international notation 18 International Union of Pure and Applied Physics 5 interplanar distances 113 interplanar spacings 62 interstitial atoms 189 interstitials 39, 655 ~ntrinsic stacking fault 676 ~ntrinsic step anisotropy 665 nverse photoemission 115, 129, 440, 465,638 nverse spinel structure 191 nversions 16 ion backscattering spectra 390 ion backscattering techniques 363,387 Ion beam induced damage 197 ion bombardment 175,239, 241,244, 249, 258 ion neutralization spectroscopy 601 ion scattering 113, 125, 143,717,758 ion scattering spectrometry see ISS 213, 233, 396 ionic bonding 188, 231,464, 470 Ionic bonding framework 163 ionic insulator surface 154 ionic insulators 144 ~onlc interaction 428 ionic materials 169, 178 ionic potential 108 iomc radius 189 ionically bonded insulating systems 153 ionlcity 233,254, 471 iomzation 587 ~omzation cross section 551 ionization gauge 551 ionization level 383 ionization probability 381 IPES 457 Ir 122, 755 Ir(001) 403 Ir(l 11) 447 iron oxide 206 irrelevant fields 795,801

868

Subject index

Ising antiferromagnet 730 Ising behavior 522 Ising model 83, 84, 632, 704, 705,716, 718-722, 726, 747,750, 760, 766, 794, 798,800 lsing transition 524 sland formation 448,476 island growth 486 sland nucleation 706 sland phase 730 sland separation distribution 687 island shape 690 islands 440, 458,627,687,688,689, 705 :sosteric heat of adsorption 507, 533,534, 535, 633 isothermal calorimetry 535 isothermal process 57 isotherms 509, 550, 553,555,564, 566 ISS (ion scattering spectrometry) 203,213, 233,396

kink density 73, 79, 85, 88, 89 kink energy 74, 82, 83, 84, 90, 91,670 kink formation energy 242, 665,669 kink-kink interactions 83, 670 kink-kink separation 93 kink sites 223 kinks 33, 71, 79, 87, 122, 174, 176, 239, 565, 663,665,666, 669, 670 kinks, thermal excitation of 85 KMnF 3 169, 170 knife edge singularity 85 Kohn-Sham self-consistent approach 609 Kohn-Sham equation 108, 109 Kohn-Sham method 151, 152 Kondo problem 638 Kosterlitz-Thouless point 751 Kosterlitz-Thouless transition 747 krypton 516, 520, 545,558,563 krypton/graphite 522 KZnF 3 170

Jahn-Teller transition 697 jcllium 119, 507,582, 583,585,586, 590, 601, 607,608,609,611,612,613,616, 631 jeilium model 461 jump to contact 375

La203 198, 199 lamellar halides 532 Landau classification 746 Landau rules 723,766 Landau theory 747, 798 Landau's first rule 724 Landau's second rule 723, 724 Landau's third rule 724, 765,766 Langdau expansion 722 Langmuir adsorption model 65 Langmuir-Gurney picture 476 Langmuir-Gurney model 461,464, 466, 483 laser-induced diffusion 635 late-time growth 777 latent heat 534 lateral disorder 674 lateral force mode 376 lateral interaction energy 555 lateral interactions 580, 584, 617, 626, 627, 631,634, 635,638, 641 lateral length distribution 680 lateral resolution of the STM 367 lattice 655 lattice constant 257 lattice, direction in a 6 lattice dynamics 557,634

K 111, 120 K(110) 759 keatite 174 kinematic analysis 273 kinematic approximation 287, 322, 659, 678, 687 kinematic approximation in 3-d 274 kinematic diffraction amplitude 733 kinematic diffraction intensity 733, 738 kinematic model 326 kinematic scattering 336, 345 kinematic scattering intensity 742 kinematic scattering theory 557 kinetic accessibility 261 kinetic energy 151 kinetic limitations 187 kinetic oscillations 428,435,441 kinetic phenomena 772 kinetics 141, 162, 163,656, 704, 788 kink corner energies 83

Subject index

lattice gas 581,585,620, 631,632, 695 lattice gas analogy 716 lattice gas Hamiltonian 725 lattice gas models 615,703,716, 725,773 lattice gas phase 728 lattice gas system 803 lattice gas transformation 730 lattice line 16 lattice liquid 620 lattice liquid phase 728 lattice mismatch 41,290, 516, 567,697 lattice planes 6 family of 6 lattice sites 79, 581,678 lattice-gas 580, 633 lattice-gas order parameters 733 Laue conditions 537 for diffraction 279 layer groups 18 layer spacings 326 layer-by-layer growth 680, 705-708 layered perovskite 211 LCAO (linear combination of local orbital) 108,591,599 LDA (local density approximation) 107, 109, 112, 115, 116, 117, 122, 628,640 lead zirconate titinate ceramic 369 ledges 33 LEED (low energy electron diffraction) 37, 54, 113, 121, 125-127, 143, 160, 162, 165, 166, 172, 194, 203-205,207,208, 210, 211,214, 215,217-219, 223,233, 234, 241,254, 258,273, 305,358,436-438, 447,448,450, 453,455-458,468, 471-476,480, 482, 485,507, 511,514, 518,548,566, 601,606,615,631,661, 694, 697,732, 738,742,755,758,760, 762, 763, 765 m resolution 320 LEEM (low energy electron microscopy) 408, 766 m contrast mechanisms in 408,409 resolution in 409 samples for 411 left-handed coordinate system 28 Legendre transformations 57 LEIS (low energy ion scattering) 194, 396, 399

869 Lennard-Jones 505 Lennard-Jones one-dimensional diagram 445 Lennard-Jones potentials 104, 105,506, 548, 640 Lennard-Jones repulsion 583 Lennard-Jones interaction 93 Lenz 719 LEPD (low energy positron diffraction) 143 level mixing 462 Li 111 Li and Na on Ru(0001) 469 LiF(001) 352 lifetime-broadening 462 Lifshitz criterion 724 light domain walls 42 limitations to resolution 363 LiNbO 3 171,209 line tension 84 linear bridge sites 21 linear chains 482 linear-combination of atomic orbitals s e e SCF-LCAO linear combination of local orbital s e e LCAO linear sites 580 linear transformation 27 liquid phase 726 liquid-gas critical point 727 liquid-like structures 469 LiTaO 3 171 LMTO (linear muffin-tin orbitais) 108 local coordination 189 local density approximation s e e LDA local density functional method 431 local density of states 132, 623 local distortions 615 local equilibration 53 local functional 109 local minimum 113 local orbital based methods 111, 113 local relaxations 590 local roughness 295 localization of a state 115 London dispersion forces 505 long-bridge sites 450, 452, 615 long-range correlations 699 long-range forces 565 long-range interactions 637, 697

870 long-range order 289, 325,343,469, 479, 508, 632,655,697, 701,718,719, 732 long-range translational order 523 longitudinal mode 513 Lorentz factor 303, 338 Lorentzian function 542, 739 low coverage 112 low energy electron diffraction see LEED low energy electron microscopy ,tee LEEM low energy ion scattering see LEIS low energy positron diffraction see LEPD low index directions 31 low index surfaces 8 lowest unoccupied molecular orbitals s e e LUMO LUMO (lowest unoccupied molecular orbital) 429, 596 M203 207 Madclung sum 155 magnctlc fields 325 magnetlc forces 374 magnetic ordering 587 magnetxc sandwiches 641 magnetic shielding 308 magnettte 206 magnetization 719, 720, 726, 795,797, 800, 801 magnification 381 marginal fields 801 mass flow 53 mass transport 53,450 matrix method 317 matrix notation 25, 27 Maxwell relationship 58, 64 MBE (molecular beam epitaxy) 235,249, 258, 259, 260, 261,262, 706 template 239 McLachlan modification 583 mean field 107,631 mean field theory 587,747, 795,796 mean square displacement 511 mean square displacements of steps 91 mean square fluctuation 663 mean squaref height fluctuations 299, 300, 678 mean squared vibrational amplitude 286 medium energy ion scattering s e e MEIS

Subject index

MEIS (medium energy ion scattering) 126, 162, 194, 392, 457, 770 melting 231,514, 523,524, 525,546 melting point 565 melting transition 342, 469 meso-scale structures 376 mesoscopic ordered domains 122 metal carbonyls 428,429,430, 434, 435 metal dimers 128 metal overlayers on semiconductors 123 metal-semiconductor transition 242 metal surfaces 104, 116, 427, 660 metal trimer overlayer 489 metal-carbonyl complexes, vibrational spectra of 430 metal-semiconductor bonds 489 metallic bonding 188,464, 469 metallic contacts 124 metallization 473 metals 115 metal-metal bonds 127, 489, 491 metal-semiconductor bonds 127,491 metal-semiconductor interfaces 104, 123,486 metastable (1• structure 455 metastable phases 113, 124, 476, 796 methane 511,542 Mg 111 MgO 163, 177, 201,406, 507,513, 533, 539, 559, 566 MgO(001) 164, 165, 176 MgO(100) 187, 1 9 5 , 2 0 1 , 2 2 3 , 5 1 3 , 5 1 4 , 5 4 2 MgO(l 11) 224 mica 676 microchannel plate 386 microelectronics 174, 239 microfacets 205 microscopy 656 Miller indices 6, 22, 33, 35 mirror electron microscopy 408 mirror plane 17 mirror-reflection planes 17 miscut angle 242 mismatch 516 critical 516 lattice 516 missing row 441 missing row model 454

871

Subject index

missing row phase 759 missing row reconstructions 119, 132, 449, 456, 465,759, 761,768,769, 771 missing row structure 451 mixed-basis pseudopotential approach 110 mixed-basis technique 111 mixed dislocation 674 mixed phases 42 mixed representation 587,605 MnO 201,202 Mo 132 Mo(001) 106, 118 Mo(100) 116 Mo(310) 416, 416 Mo2C 213 ]'- MoC 192 mobility 468,546 molecular adsorbate systems 427 molecular adsorption 431 molecular beam epitaxy see MBE molecular coordination chemistry 255 molecular dynamics 105, 119, 159, 162, 508 molecular dynamics simulations 110 molecular orbitals of CO 429 molecular orientational order 521 molecular vibrations 512 Moliere potential 394 momentum transfer 300, 347 momentum transfer vector 274 Monte Carlo 631 Monte Carlo calculations 745 Monte Carlo simulations 79, 119, 6 ! 3, 614, 615, 618, 619,627,632, 633,634, 639, 703,704, 705,706, 806 Monte Carlo techniques 394 mosaic 41,289, 290, 291,674, 676 mosaic structure 675 MOssbauer spectroscopy 511,545 Mott-Littleton approach 156, 159, 175, 176 mound structures 684 muffin tin 631 muffin-tin orbitals 106 muffin-tin spheres 613, 614 Mulliken (1934) electronegativity 587 Mullins 81 multicomponent systems 62 multigrids 111

multiple height steps 679 multiple scattering 284, 322, 326, 346, 347, 678,733,738 multisite interactions 604 Na 111 NaCI 507, 708 NaF(001) 354 NaxWO 3 (100) 211 NbC 213 NbN 218 Ne 518 nearest neighbors 189 nearest-neighbor repulsion 581 negative surface excesses 62 neon 511 neutralization 397 neutron scattering 535,566, 717 neutron scattering experiments 533 next-nearest-neighbor attraction 581 Ni 105 Ni island nucleation 699 Ni(001) 316, 317 Ni(00 I)-C 764 Ni(001)c(2• 457 Ni(100) 457,686 Ni( 100)-C 763 Ni( 100)-K 470 Ni(100)-O 448,456 Ni(l 10) 449, 762 Ni(110)(2• 440 Ni(110)(2• 453,455 Ni(l 10)-CO 440 Ni(110)p(2• 450, 453 Ni(110)p(3• 453 Ni(l 11)-O 742 Ni(l 11)-K 468,469 Ni(l 13) 667 Ni(115) 666, 667 Ni(771) 453 NiO 201,202 NiO(100) 202, 222, 457,686 N i O ( l l l ) 457 niobium pentoxide 193 nitrides 143, 218,222 nitrogen 511,521 Ni-CO 430

872 no load point 375 noble-metal adsorption on silicon and germanium surfaces 125 noble metals 116, 117,637, 640 noble metals on Si or Ge 127 non-rotated hexagonal phase 520 non-stoichiometric surfaces 142 non-bonded atoms 668 non-bonding electronic states 246 non-channeling 393 non-contact imaging 376 non-directional bonding 581 non-equilibrium structure factors 7903 non-interacting steps 302 non-registered binding 729 non-uniform strain 675 non-universal behavior 747,753 non-wetting 562 normal mode 286 normal vibrational modes 285 notation 35 Novaco-McTague rotation 518,523, 524 Novaco-McTague epitaxial rotation 549 nucleated islands 688 nucleation 473,478,688, 766, 775,776,778, 788 nucleation and growth 775,785 nucleation processes 689 number of the space group 19 O/Ru(001) 736 O/W(100) 693,694 O/W(110) 685,694 O/W(211) 704 02-anion sublattice 199 object wavefunction 400 occupied density of states 364 octahedral coordination 188 octahedral interstices 207 octahedral interstitial sites 192 octahedrai sites 191 on-top position 484, 485 on-top sites 20, 434, 469, 471,475,476, 580 one-electron density 151 one-electron eigenfunctions 152 one-electron eigenvalues 151 one-electron electronic energy 148

Subject index one-electron functions 151 one-electron kinetic energy and electron-ion (nuclear) attraction 149 one-electron wavefunctions 147, 149, 150 one-dimensional defects 655 one-electron energies 613, 616 one-phonon scattering 287 Onsager 719 optical interferometry 378 optical lever 378, 380 optical transfer 400 order parameters 719, 721,723, 732, 733, 746, 747, 804-806 order-disorder transition 119, 448,521-523, 535,567, 671,697 order(N) methods 111 ordered phase 42, 613 ordering kinetics 661 orientational order 521 orientationai order-disorder transition 525 orientational phase diagram 68 orientationai phase separation 55, 68 orientational phase separation of vicinal Au(l 11) 79 orientational phase separation of vicinai Pb(l 11) 79 orientational phase separation of vicinal Si(l 11) 79 out-of-phase condition 297, 302, 328,663, 666, 667,669, 679, 692, 693 out-of-phase scattering 295 overhang 81 overlap matrix 106, 147 overlayer lattice 42 overlayer mesh 25 overlayer meshes, classification of 30 overlayer structures 457 overlayer unit mesh vectors 25 overlayers 655 oxidation 427,457 oxide structures 191,456 oxides 143,222, 427 oxygen adsorption 443 oxygen deficient surface 166 oxygen-induced relaxations 447 p(2x2) oxygen on Ni(l 11) 754 oxygen penetration 457

Subject index

oxygen vacancies 175 p 3 conformation 264 pa~r correlation function 300, 523 pa~r distribution function 37, 39, 298 pair energy 586, 598 pa~r interaction energy 593 pa~r interactions 579, 581,586, 594, 638,639 pa~r potential 154, 625 pair-wise interactions 79 parallel imaging 398 paramagnetism 720 partial dislocations 674, 676 particle-hole symmetry 594 particle-vacancy interchange 729 particle-vacancy transcription 728 partition function 719 Patterson function 37, 299 Pauli exclusion principle 149, 505 Pauling law of electronegativity 189 Pauling radius 469, 471-474 Pb(110) 342, 343 Pd 105, 122 Pd(100)-H EAM 626 Pd(110) 449 Pd(110)-CO 440 Peierls distortion 162 perfect absorption 280 periodic boundary conditions 803 Periodic Table 431 perovskite (100) surfaces 169 perovskite 175, 178, 193,210 peroxide ions 02- 478 phase boundaries 221,511,533,535,628, 631,640 phase coexistence 784 phase contrast 401 phase diagrams 128, 468,509, 510, 521,523, 533,535,552, 582, 613,615,617,618, 626, 631,639, 686, 693, 702, 718, 720, 726, 728, 729, 735, 751,788, 793,794 phase equilibria 74 phase separation 69, 77, 78 phase shifts 275,590, 613,614 phase transitions 53,242, 473,508,516, 521, 522,535,549, 551,552, 567, 604, 681, 693,701,704, 788

873 phase transition (7x7) ~ ( l x l ) 242 phases 581 phases of a form 7 phenacite structure 193, 220 phonon dispersion 557, 634 phonon dynamics 709 phonon energies 155 phonon excitation 448 phonons 330, 511, 513, 601 photocatalysis 210 photocathodes 459, 478 photoemission 114, 115, 162, 448,601 photoemission data 159 photoemission electron microscope (PEEM) 442 photoemission spectroscopy 476 photon stimulated desorption (PSD) 194 phthalocyanine 241 physisorbed films 565 physisorption 505,536, 548,580, 583,583, 729, 742, 773,778 physisorption state 445 physisorption studies 531 7t and 7t* orbitals 246 rt and n:* states 247 rt-bonded chain model 123, 131, 132 7t-bonded chain S i ( l l l ) ( 2 x l ) system 114 rt-bonded chains 160, 161, 162, 163, 236, 240, 243,247 ~-bonds 160, 161, 241,246 piezoelectric materials 369 piezoelectric scanning device 369, 370 pinwheel reconstruction 763 planar model 750 planar sp 2 bonded chains 240 Planck's constant 146 plane wave based methods 108 plane wave pseudopotential 130 plane waves 147 plastic deformation 375 point and space group symmetry 16 point defects 174, 175,202, 221,223,289, 290, 358,363, 386, 655,657, 658, 805 point groups 16, 19 point lattice defects 803 point of general position 18 point of special position 18

Subject index

874 point operation 16 point-group symmetry 579 point-to-point resolution 368,402 poisoning 616 Poisson's equation 599 Poisson's ratio 94, 635,672 polar surfaces 207, 210, 236, 248,249, 258 polarizability 154, 464, 583 polarization 155,610 polarization factor 326 polarization potential 155 polycrystalline surfaces 416 polymorphism 521,567 polytypes 215 polytypes of SiC 217 porosity 224 positional correlation 525 positional disordering 522 positional order 521 positive surface excesses 62 potential barrier 445 Ports models 750 3-state Potts model 522, 724, 747, 748, 754, 798 4-state clock model 750, 751 4-state Potts model 448, 724 clock model 750 planar Potts 750 q-state Ports models 750 power law 680, 703 power-law decay exponents 611 power-law line shape 301 preconditioned conjugate gradient scheme 109 premelting 547 premeiting transition 243 pressure 70 pressure for ideal gases 729 pressure measurement 551 primitive cells 579 primitive lattice vectors 317 primitive meshes 13 primitive overlayer unit mesh 29 principles of semiconductor surface reconstruction 246 projected band structure 115 projected electron density 340 Prokrovskii and Talapov 519

promoters in catalytic reactions 477 promoting 616 pseudo-potential plane wave methods 110 pseudo-potential plane waves 108 pseudo-hexagonal layer 755 pseudo-hexagonal structure 755,757 pseudo-potential methods 108, 113 pseudo-potentials 148 Pt 53, 54, 105, 122, 755 Pt(ll0) (1• 397 Pt Co 382 P t o n P t ( l l l ) 689 Pt/Pt(l 11) 680 Pt(001) 396 Pt( 100)c(4• 413 Pt(110) 414, 415,449, 760 Pt(110) (1• 398,440, 441 Pt(110) (2• 440 Pt(110)-CO 435,438 P t ( l l l ) 520, 557 Pt(I 1 I)-CO 639 Pt(l 1 I)-K 468,469 Pulay forces 110 pull-off force 376 Q-resolution 312 quantum chemistry 592 quantum chemistry codes 112 quantum gases 531 quantum mechanical calculations 233 quantum mechanical methods 144 quantum mechanical models 179 quantum mechanical potentials 144 quantum size effect 478 quarto interaction energy 586 quartos 619 quartz 194 or- and I]-quartz 174, 193 quartz microbalance 566 quasi-atom approach 616 quasi-chemical methods 631,634 quasi-one-dimensional states 610, 637, 641 quasielastic incoherent neutron scattering (QENS) 541 quench 780, 805 radial scan mode 339

Subject index

rainbow angles 351 rainbow scattering 351 RAIRS 440 random direction 393 random phase approximation 687,693 random steps 300 random field Ising model (RFIM) 704 rare gas adsorption 583 rare gases 509 Rayleigh ansatz 347 reaction kinetics 773 real and reciprocal space lattices, transformations between 29 real space images 363 real space lattice 21,555 rebonding 127, 132,591 reciprocal lattice 21 reciprocal lattice, basis vectors of the 22 reciprocal lattice rods 537, 553,669, 682, 683, 692 reciprocal lattice vectors 22, 278, 315,347 reciprocal mesh vectors 22 reciprocal space 273,291 Recknagel formula 409 reconstructed (110) fcc metals 637 reconstructed Au( I 11) 405,406 reconstructed phases 209 reconstructed Si(100) 364 reconstructed surfaces 141, 281 reconstruction 68, 112, 122, 131, 132, 194, 208, 214, 220, 233,241,243, 247,248, 253, 2 8 1 , 4 4 7 , 4 5 3 , 4 5 5 , 6 1 9 , 661,669, 718,725,730, 741,787 (7• 764, 765, 770 contractive type 120 effects on adsorption on 768 recursion method 107 reduced correlation function 737 reduced free energy 71 rcduced surface free energy 72 reduced surface tension 74, 76, 78, 81 reduced temperature 736, 737,776, 806 reentrant 563 reentrant aligned incommensurate solid phase 559 reentrant fluid 523 reflection anisotropy microscopy 416

875 reflection electron microscopy see REM reflection high-energy electron diffraction s e e RHEED reflections 16 refractory metals 675 registry 730 regular point system 18 rehybridization 235,243,248, 251, 616, 630, 639 rehybridization of the dangling bonds 234 relativistic effects 118 relaxation 62, 112, 116, 122, 131, 132, 200, 206, 208, 223,233, 246, 281,327,447, 569, 476, 606, 629, 641,664 relevant operators 750 reliability (R) factors 323 REM (reflection electron microscopy) 93, 198, 200, 209, 404, 636 u holography 405 m resolution 405 renormalization 738, 799 renormalization eigenvalues 747 renormalization fixed point 749 renormalization group 638 renormalized forward scattering (RFS) 323 repulsive dipole--dipole interaction 472 resistive anode 312 resolution 544 FIM 384 function 305 LEEM 409 REM 405 SFM 366, 376 TEM 402 resonance 353 resonance energies 610 resonant bond 488 resonant bonding 487 resonant scattering 352, 557 rest atoms 239, 243 retarded Green's function 589 reverse scattering perturbation (RSP) 323 Rh 122 Rh(100)-Cs 469 RHEED (reflection high energy electron diffraction) 125, 165, 170, 194, 209, 234, 456, 553, 566

876 ridge-and-valley morphology 681 ring structure 693 rippling 215 RKKY interaction energy 591 rocking surface mode 512 rocksalt 163, 175, 178, 179, 193,214 rocksalt structure 190, 192, 201, 213, 218 rocksalt structure alkali halides 142 rod scans 331 root-mean-square displacement 39 rotated hexagonal phase 520 rotated incommensurate solid phase 559 rotated structures 469 rotation 16, 511,514, 542 rotational axes 17 rotational diffusion 522, 542, 543 rotational diffusion coefficient 515 rotational epitaxy 469 rotational motions 514 rotational scattering 541 rough stcps 668 roughened surfaces 705 roughening 231 roughening temperature 81,301,565,706 roughening transition 665--678,705 roughness 219,669, 674, 679 roughness parameter 301 row-pairing reconstruction 761 Ru relaxation 606 Ru(0001) 447,460, 461,476, 477 Ru(0001)('~3• 469, 479 Ru(0001 )(~/3• 470 Ru(0001 )(2x2)-Cs 472 Ru(0001)-CO 435,436, 437 Ru(0001)p(2x2)-Cs system 472 Ru(0001)-Cs 468,470, 470, 472 Ru(0001)-K 470, 472, 473 Ru(0001 )-Li 473 Ru(0001)-Na 470, 473 Ru(0001 )-O 447 Ru(0001 )-Rb 473 Ru(001) 691,743 Ru3(CO)!2 435 rumpling 195,201, 219, 486 Rutherford backscattering spectroscopy (RBS) 392 Rutherford scattering cross section 392

Subject index

rutile 179, 193, 203 rutile (110) surface 166 rutile surface 175 sapphire 194, 207, 209 saturation of the dangling bonds 241 Sb 129, 233,708 Sb adsorption 124 scaled structure factors 744 scaling 235,716, 738,744, 780, 782, 799 scaling exponents 802 scaling function 779, 806 scaling laws 257,265, 802, 806 scanning electron microscopy s e e SEM scanning force microscopy s e e SFM scanning probe microscopies 224, 363 scanning tunneling microscopy s e e STM scanning tunneling spectroscopy (STS) 233 scattered-wave theory 628 scattering angle 275 scattenng cross section 344, 555 scattenng length 274, 630 scattenng matrix 322 scattenng plane 303,305 scattenng theory 589, 630 scattenng theory methods 628,639 SCF-LCAO (linear-combination of atomic orbitals) 144, 145, 147, 149, 151, 152 Scherzer contrast transfer function 401 Scherzer defocus 404 Sch6nflies notation 17, 18 Schottky barrier 124, 130 Schrtidinger equation 108, 146, 344 one-electron 147, 157 screened Coulomb potential 387 screening 392, 471,472, 476 screening between the dipoles 471 screening charge 465 screening length 388 screening potential 108 screw and edge dislocations 35,672 screw dislocation 39, 705,706 second moment 625 second-order phase transition 718,736, 798 second-order transition 701,702, 720 second-phase precipitates 224 secondary electrons 307

Subject index

secondary emission ratio 307 secondary ion mass spectroscopy see SIMS segregation 224 selection rules 540 selective adsorption 352 selenides 142 self-adsorption 473 self-affine scaling 681 self-consistency 599 self-consistent field 144 self-consistent Hartree-Fock 587 self-consistent matrix Green's-function (MGF) scattering theory 629 self-consistent pseudopotential calculations 129 self-similarity 704 SEM (scanning electron microscopy) 273 semiconductor surface reconstruction 234, 261,265 principles of 246 semiconductor surfaces 91, 104-106, 108, 111, 114, 123,231,427,660 semiconductor systems 131 semiconductors 115, 601,611,635,675,676 metal overlayers on 123 semiempirical methods 640 semiempiricai quantum-mechanical methods 141 semiempirical potentials 507 semiempirical techniques 107 SEXAFS 449, 474 SFM (scanning force microscopy) 363,372, 373 resolution of the 376 shadow cone 396, 388-390, 397 shape transition 691 shear 523 shell models 155, 165 short-bridge site 615 short-range correlation 632 short interaction range 104 short-range potential 155 Si !11, 123, 160, 231,233,236, 246, 338 Si dimer chains 128 Si(001) 660, 668-671,688,689 Si(0010) 663 Si(lO0) (2• 241

877 Si(100) 90, 123, 124, 130, 662, 689, 693 Si(110) 660 S i ( l l l ) (2• 123, 132, 162, 163,235,240, 247 Si(l I 1) (7• 132, 241,247, 328, 329, 513, 662, 677, 696, 725, 805 S i ( l l l ) 91, 93, 123, 124, 127-129, 161,412, 487,488, 717 Si(l I 1) surface reconstructions with ('~-•162 124 Si(l I 1) ('43-•162 129 Si(l I 1) ('43-• 696 Si(111) (5• 404 SiC, polytypes of 217 2H-SiC 215 3C-SiC 215,217 4H-SiC 215,217 6H-SiC 215,217 6H-SiC (0001) 218 or-SiC 215 [3-SIC 215,217 13-SiC(l 11) 217,218 [3-Si3N4 220 oc-SiO2 194 ot-Si3N 220 I3-Si3N4 221 silica 174,212 silicates 212 silicide layer 124 silicon 236 silicon carbide 215 silicon nitride 220 silicon wafers 123 silicon-on-sapphire 209 silver 524, 525 simple cubium 592 simple metals 108, 132 s~mple lattices 30 SIMS (secondary ion mass spectroscopy) 456 simulation techniques 105 s~ngle alignment technique 390 s~ngle-bond scission 161, 162, 236 s~ngle crystal substrates 427 s~ngle height steps 665 s~ngle scattering limit 274 s~ngularity 793 site binding energy 726

878 site switching 485 Si-Si dimers 217 skimmer 355 slab 113 slab calculations 144, 156, 158,601 slab thickness 158 Slater determinant 149, 151 Slater exchange-correlation potential 152 Siater-Koster 2-center approximation 106 Slater-type orbitals 147 Smoluchowski 117 Smoluchowski smoothing effect 132 SnO 2 166, 203 SnO 2(110) 166, 168,203 soft metals 675 softening of a surface phonon 118 solid l-solid 2 transitions 567 solid-on-solid (SOS) model 79 solid phase 726 solid state physics 5 solid-on-solid model 80, 299,665 solid-ovcrlayer interface potentials 325 solid-solid interface 342 solid-vacuum interface 112, 342 solid-vapor interface 61 solitary wave 443 solubility 474 source extension 320 s,p electrons 116 sp 2 bonding 237,490 sp 2 chains 237, 241,246, 258,263 sp 2 conformation 264 sp 2 coordinated Si atoms 237 sp 2 coordination 256 sp 2 hybridization 173 sp 2 surface chain 237 sp2-bonded surface layer 475 sp 3 bonding 237,488,489 sp 3 orbitals 487 sp3-type dangling bonds 486 SPA-LEED (spatially analyzed LEED) 202, 315 space charge 315 space groups 16 notation 19 number of 19 short form symbol for 19

Subject index

spatial coherence 314 spatial self-organization 441 spatially analyzed LEED see SPALEED specific heat 535, 633,749 specific heat exponent 741 spectromicroscopy 416 spectroscopic ionicity 257 speed ratio 355 sphalerite structure 193, 219, 220 spherical aberration 401,402 spin interactions 631 spin-polarized electrons 459 spin-polarized low energy electron microscopy 416 spinel lattice 206 spinel structure 193, 206 spiral dislocation 707 spiral reaction fronts 414 split positions concept 471 split-off states 597 spot profile analysis 669 spreading pressure 66 sputtering 397 SrO 210 SrO (Type I1) terminations 170 SrTiO 3 169,210 SrTiO 3 (100) 210 SrTiO 3 (2• 171 stability of surfaces 55 stacking faults 15, 199, 239, 386, 6 5 5 , 6 7 5 , 6 9 6 staggered chemical potential 734 staggered coverage 733 staggered field 731 staggered magnetic field 734 standing waves 638 static lattice 223 statistical distributions 90 statistical mechanics 55, 79 step bunching 78, 79, 681 step collisions 82 step crossing 81 step density 72, 73, 76, 79, 80 step diffusivity 85, 87, 88, 91 step distributions 94 step edge barrier 702, 707, 708 step edge diffusion 690 step edge roughness 663

879

Subject index

step edge stiffness 88 step edges 33, 71, 83, 223,239, 663,681 step energy 93,665,670 step fluctuations 665 step formation 82 step formation energy 82, 242 step free energy 73, 81, 83, 85 step height 663,664 step height distribution 678 step height multiplicity 665 step interaction coefficient 94, 95 step interaction energy 82 step interactions 77 step lengths 302 step meandering 665,669, 672, 697 step pressure 73, 77 step roughness 669 step-step dipole interactions 672 step-step interactions 81, 83, 91,93,636, 664, 665 step-step interaction energies 242 step-step repulsions 83 step-step separation 93 step-step repulsion 292,635 step stiffness 82, 88, 91 step-terrace array 197 step wandering 55, 82, 85, 90 stepped surfaces 80, 159, 231,236, 242, 302, 453,662, 666 steps 30, 35, 67, 79, 96, 122, 174, 176, 197-199, 221,223, 224, 291,363,565, 606,607,635,636, 655,663,665,668, 679,688,689,696, 705,725 stereographic projection 30, 31,663 stcreographic triangle 33 sticking coefficient 479, 552 sticking probability 441,457 stimulated desorption 358 stishovite 174 STM (scanning tunneling microscopy) 54, 67, 90, 91, 93, 116, 122, 125, 126, 130, 195, 196, 202-205,207, 211,214-216, 218, 223,234, 239, 241,242, 259, 273,363, 364, 371,372,438,449, 451,453,454, 456-458,476, 490, 492, 601,606, 623, 633,636, 655, 661-664, 669, 673,674, 688,689, 690, 693,697,699, 700, 709,

755,760, 761,764, 766, 771 lateral resolution 367 resolution 366 tip 369 stoichiometry 235 strain 93, 175, 178, 235, 481,626, 665, 671, 672, 674, 675,686 strain energy 699 strain gauge 380 stress 93, 104, 177,242, 245,290, 428,488, 491,635,636, 663, 664, 665,671,675, 698,699 stress anisotropy 668,672 stress relaxation 697 stress-mediated interactions 93 striped incommensurate phase 519, phase 555 striped phase 520 structure 775 structure factor 738,740, 744, 779 subgroup 724 sublimation 534 substitutional adsorption 763 substitutional atoms 289 substitutional model 490, 491 substitutional site 474, 476 substrate lattices 686 substrate relaxation 618,627, 628 substrate-mediated interaction 483 subsurface oxygen 456 subsurface sites 617 sulfides 142 superlattice 692 superlattice reflections 327 superlattice rods 284, 293, 297,339 superlattices 199, 208 superperiodicity 698 superstructures 661,685 surface, definition of 55 surface atomic geometry 235 surface band structure 114 surface barrier shift 462 surface Brillouin zone 114, 246, 598,605, 724, 741,743 surface charge 194 surface charging 140, 179 surface chemical bonding 233,235,249, 253, 256 m

Subject index

880 surface crystallography 3, 9, 339 surface dangling bonds 261 surface decomposition 179 surface defects 171, 174, 176, 281,414, 657 surface diffraction rods 280 surface diffusion 56, 438,442, 580, 681 surface dynamics 179 surface electron charge distribution 344 surface electron density 119 surface electronic properties 113 surface electronic structure 104, 112 surface energy 114, 142, 671, 681 surface excess 60, 63 surface excess quantities 60 surface faceting 55 surface force constant 512 surface free energy 242, 635 angular dependence of the 77 surface imperfections 363 surface magnetism 717 surface mechanical properties 375 surface melting 77, 122,565-567 surface mesh 194 surface i~lolccule 591 surfacc morphology 96 surface orientation 53 surface peak 393,394 surface phonon spectrums 358 surface phonons 285,286 surfacc plasmons 307 surface poisoning 630 surface rate processes 634 surface reactions 716 surface reactive site 176 surface reconstruction 110, 117, 122, 131, 14(), 141, 164, 169, 172, 177, 178, 197-200, 223,255,389, 390, 395,427, 438,441,450, 451,453,665,681,717, 753,755,771 surface relaxation 110, 117, 122, 140, 14 !, 164, 169, 170, 172, 173, 175-178, 212, 219, 249, 254, 389, 390, 395,660, 755 surface resonance scattering 406 surface resonances 114 surface roughening 55, 122,563,565,666, 717 surface roughness 195,328 surface segregated substrate impurities 803

surface segregation 175 surface stability 69 surface-state bands 248 surface states 112, 114, 115, 162, 211,234, 246, 250, 259, 261,450, 585,599, 601, 610, 611,635,637, 638,641 surface steps 803,805 surface stoichiometries 140, 188 surface strain tensor 671 surface strains 142 surface stress 77, 94, 95,767 changes in 94 surface structural properties 113 surface structure 140 properties 113 surface symmetry 325 surface tensile stress 122 surface tension 55, 56, 57, 58, 63, 65, 69, 81, 114, 301 anisotropy of the 68 as a function of orientation 83 changes in 56 values of 56 surface terminations 200 surface thermodynamics 55, 60 surface topography 366 surface topology 172, 173, 178, 179 surface V I center 176 surface X-ray diffraction 125-128,273, 326 surface X-ray scattering 113, 126 surface X-ray scattering system 331 surface-force constants 457 surfaces states 112 surfactants 705,708 survival of the largest 785 susceptibility 739, 796, 806, 807 symmetry 516 symmetry frustration 515 symmetry operations 724 synchrotron radiation 326 T 4 site 487 TaC 192, 195,213,214 TaC(I 10) 682 TaC(110) transition 683 TaN 218 ~5-TaN 192

Subject index

tantalates 212 tapping mode 376 TED (transmission electron diffraction) 172, 241,765 TEM (transmission electron microscopy) 214, 273,401,755 resolution in 402 temperature 53, 68 tensor LEED 324 terrace-ledge-kink model 33 terrace size 663,678,702 terrace size distribution 665,678 terrace-step-kink 86 terrace-step-kink model 637 terrace-width distribution 91, 93, 636 terraces 33, 79, 81, 198,239, 297,679, 689, 703 tetrahedral bond 487 tetrahedral coordination 188 tetrahedral sites 191 tetrahedral structure 264 tetrahedrally coordinated compound semiconductors 235,236, 248 tetrahedrally coordinated semiconductors 231 THEED (transmission high energy electron diffraction) 554 thermal accommodation 381 thermal average 285 thermal desorption 580 thermal desorption spectra 467 thermal diffuse scan 319 thermal diffuse scattering 330 thermal disorder 675 thermal equilibrium 53 thermal excitation of kinks 85 thermal expansion coefficients 326 thermal vibration 341, 351 thermal vibrational amplitudes 326, 393 thermal vibrations 39, 322 thermodynamic densities 793 thermodynamic faceting 53 thermodynamic fields 793 thermodynamic phase separation 68 thermodynamics 53, 55 thin film processing 567 Thomson scattering formula 326 three-body terms 105

881 three-dimensional metals 111 three-phase coexistence 563 three-way automotive exhaust catalyst 454 threefold adatom site 487 threefold hollow 488 threefold site 487 threefold-coordinated hcp site 485 threefold-coordinated site 455 threefold-fcc hollow sites 449 threefold-hollow position 476 threefold-hollow site 481 through bond 579 Ti20 3 171,207, 209 TiC 192, 195,213 TIC(100) 195,213 T i C ( I l l ) 214 tie bar 72, 76, 78 tight-binding 144, 611, 612, 618, 619, 620, 624, 634 tight-binding calculations 110, 118,623 tight-binding chain 595 tight-binding methods 104, 106, 141, 144, 145, 148, 149, 153, 154, 162, 599 tight-binding models 179, 586,609, 617,636, 640, 670 t~ght-binding substrate 610 t~ght-binding total energy calculations 258 t~lt angle 253 t~lted dimer model 244 tilted dimers 239, 240, 241,242, 246 t~me-of-flight methods 397 TiN 218,219 TiO 2 166, 194,203,210 TiO2(001) 205 TiO2(100) 197, 198 TiO2(110) 166, 168,203,204, 205 TiO2(l 11) 224 TiO 2 (Type I) terminations 170 tip 378, 379, 380, 386 tip-substrate interactions 374 topological reconstruction 178 topology 142, 171, 175, 176 torsions 512 total energy 104, 108, 111, 113, 114, 144 total energy calculations 125 total energy functional 144 transfer function 304

882 transfer matrices 633,634, 803, 805 transfer-matrix techniques 632, 635 transfer width 304, 339 transformations 29 transition metal carbides 213 transition metal carbonyl complexes 430 transition metal nitrides 218 transition metal substrate 630 transition metal surfaces 111,474 transition metals 105, 106, 111, 116, 117, 579, 592, 616, 6 1 8 , 6 2 3 , 6 2 5 , 6 4 0 transition temperature 759, 805,806 translation 511 translation operation 11 translation vector 5, 42 translational and orientational order 525 translational and rotational motions 513 translational diffusion 542 translational diffusion coefficient 514, 515, 542 translational motions 514 transmission electron diffraction s e e TED transmission electron microscopy s e e TEM transmission high energy electron diffraction see TtlEED transversc modc 513 transverse scan 339 trial wavcfunctions 109 triangulation methods 387 tricritical point 733 a- and [5-tridymite 174 trigonai prism 192 trimcrs 247,489, 491,492, 618,621,623,624, 685 trio energies 598,624, 632 trio interaction energy 586 trio interactions 597, 607,618,620, 621,624, 632,728 trtos 619 triple-bond scission 161 triple dipole 507,583 triple point 509, 531,533 triple point wetting 562, 567 triple point wetting transition 562 tripod scanner 369 true atomic resolution 372, 376 tube scanner 369

S u b j e c t index

tungsten bronzes 211 tungsten carbide 193 tungsten carbide structure 192 tunneling 380 tunneling current 363, 364 tunneling probability 366, 382 tunneling tip 378 tunneling transmission probability 366 twin boundaries 41,386, 677 twinning axis 41 twins 674 two-dimensional Bravais lattice 11 two-dimensional Cs oxides 482 two-dimensional defects 655 two-dimensional melting 751 two-dimensional melting transitions 567 two-dimensional phase transitions 358, 508 two-dimensional point groups 18 two-dimensional space groups 18, 43 two-electron exchange interaction 149 two-electron Hartree screening (or Coulomb) interaction 149 Type I termination 169 Type !1 termination 170 ultrahigh vacuum (UHV) 123,241 ultraviolet photoelectron spectroscopy see UPS uncertainty principle 478 uniaxial rotation 515 uniform stepped surface model 295 uniform strain 676 uniformly stepped surface 78 unit cells 5, 8 basis of 8 positions 8 unit matrix 28 unit mesh transformation 25 unit mesh vectors 11 unit meshes 11, 13,684, 685,696 centered rectangular 13 hexagonal 13 oblique 13 rectangular 13 square 13 unit vectors 684 universal binding curve 616 universality 716, 744, 746, 801

883

Subject index

universality classes 448, 508,522, 716, 736, 746, 747 unoccupied density of states 364 unpaired bonds 486, 489 unpaired electrons 428, 491 unpaired orbitals 487 untiited dimers 242 up-quenching 718,780 UPS (ultraviolet photoemission spectroscopy) 143,430, 436, 456, 457,464, 465 V203 171,207, 209 V205 195, 196 vacancies 38, 42, 79, 208, 213, 214, 217, 218, 219, 221,222, 247, 289, 290, 474, 475, 629, 655,661,662, 665,705 vacancy defects 211 vacancy form factor 290 vacancy-vacancy interactions 630 vacuum 139 vacuum level 115 vacuum-solid interface potentials 325 valence band transitions 307 valence electrons 106, 116 van der Waals 344, 377,507, 579 van der Waals attractive term 155 van der Waals diameters 516 van der Waals dispersion force 374 van der Waals interaction 445,505,506, 583 van der Waais potential 275,583,584 van der Waals radius 516,548 van der Waals systems 563 vanadates 193, 212 vaporization 534 variance 508 VC 195,213 VC x 214 very low energy electron diffraction see VLEED vibration 511 vibration isolation 371,378 vibration properties 21 vibrational amplitudes 286, 325 vibrational force constants 257 vibrational frequencies 583 vibrational mode 511,540 vibrational motion 39, 709

vibrational partition function 65 vibrational spectra of metal-carbonyl complexes 430 vibrational spectroscopy 436 vibrational states 634 vibrations 547 vicinal 231 vicinal Ag(110) 681 vicinal plane 663 vicinal Si(l I 1) 681 vicinal surfaces 30, 33, 74, 79, 205,239, 242, 606, 635,636, 665, 681,696, 717 vicinal ZnO surfaces 223 vicinality 669, 705 Vidicon 310 virial coefficient 507 VLEED (very low energy electron diffraction) 409 VN 192,218 void channels 220 volumetry 533 vortex 751 W 53, 132 W/W(100) 692, 693 W(001) 106, 328,329, 768 W(001), c(2• reconstruction of 118 W(001) reconstruction 697,757,758 W(100) reconstructions 717 W(110) 693 W(110)-H 762 W ( I I I ) 54 Warren approximation 277 wave vectors 274 wavelength of the helium atoms 356 wavelets 111 WC 192, 213 weak chemisorption 596 weak coupling 591 wetting 562, 564-567 complete 562 u incomplete 562 triple point 567 white graphite 219 Wigner-Seitz cells 23, 117 WKB approximation 366 WO 3 211

884 Wood notation 25 work function 115,262, 365,382, 442, 449, 462, 466, 470, 478, 633 work function change 114 work-function lowering 461 Wulff plots 69, 224 wurtzite 193,202, 231,239, 241,246 wurtzite cleavage faces 262 wurtzite structure 220 X-ray diffraction 37 X-ray photoelectron diffraction see XPD X-ray photoemission spectroscopy see XPS X-ray diffraction 204, 453,557,729, 733,742, 757,758, 760 X-ray reflectivity 330, 342 X-ray resolution function 339 X-ray scattering 321,358, 511,543, 708,755 X-rays, critical angle for 330 Xc 511, 516, 518,520, 524, 525,557,566 XPD (X-ray photoelectron diffraction) 143 XPS (X-ray photoemission spectroscopy) 143, 448,456, 457

Subject index

XY model 750, 751,753,801 XY model with cubic anisotropy 747, 751,758 YBa2Cu3OT_x 211 Young's modulus 94, 635 zero and one-phonon intensities 287 zero creep 56 zero force 113 zero-point energy 584 zigzag chains 440, 455, 481,482 zi,,zao,:, ,:, type displacement 118 zincblende 215,220, 231,239, 248,256, 260 zincblende (110) cleavage faces 250 zincite 231 zirconia 212 ZnO 202, 203, 231,249, 263 ZnS 249 ZnSe 249 ZnTe 249 ZrC 195,213 ZrN 192,218, 219 ZrO 2 212

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