Handbook of Nanophysics: Nanotubes and Nanowires

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Handbook of Nanophysics

Handbook of Nanophysics: Principles and Methods Handbook of Nanophysics: Clusters and Fullerenes Handbook of Nanophysics: Nanoparticles and Quantum Dots Handbook of Nanophysics: Nanotubes and Nanowires Handbook of Nanophysics: Functional Nanomaterials Handbook of Nanophysics: Nanoelectronics and Nanophotonics Handbook of Nanophysics: Nanomedicine and Nanorobotics

Nanotubes and Nanowires

Edited by

Klaus D. Sattler

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2011 by Taylor and Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number: 978-1-4200-7542-7 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Handbook of nanophysics. Nanotubes and nanowires / editor, Klaus D. Sattler. p. cm. Includes bibliographical references and index. ISBN 978-1-4200-7542-7 (alk. paper) 1. Nanotubes. 2. Nanowires. I. Sattler, Klaus D. II. Title: Nanotubes and nanowires. TA418.9.N35N35777 2009 620’.5--dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

2009038058

Contents Preface........................................................................................................................................................... ix Acknowledgments ........................................................................................................................................ xi Editor .......................................................................................................................................................... xiii Contributors .................................................................................................................................................xv

PART I Carbon Nanotubes

1

Pristine and Filled Double-Walled Carbon Nanotubes .....................................................................1-1 Zujin Shi, Zhiyong Wang, and Zhennan Gu

2

Quantum Transport in Carbon Nanotubes ....................................................................................... 2-1 Kálmán Varga

3

Electron Transport in Carbon Nanotubes......................................................................................... 3-1 Na Young Kim

4

Thermal Conductance of Carbon Nanotubes ................................................................................... 4-1 Li Shi

5

Terahertz Radiation from Carbon Nanotubes .................................................................................. 5-1 Andrei M. Nemilentsau, Gregory Ya. Slepyan, Sergey A. Maksimenko, Oleg V. Kibis, and Mikhail E. Portnoi

6

Isotope Engineering in Nanotube Research ...................................................................................... 6-1 Ferenc Simon

7

Raman Spectroscopy of sp 2 Nano-Carbons ........................................................................................7-1 Mildred S. Dresselhaus, Gene Dresselhaus, and Ado Jorio

8

Dispersions and Aggregation of Carbon Nanotubes ........................................................................ 8-1 Jeffery R. Alston, Harsh Chaturvedi, Michael W. Forney, Natalie Herring, and Jordan C. Poler

9

Functionalization of Carbon Nanotubes for Assembly .................................................................... 9-1 Igor Vasiliev

10

Carbon Nanotube Y-Junctions ......................................................................................................... 10-1 Prabhakar R. Bandaru

11

Fluid Flow in Carbon Nanotubes ..................................................................................................... 11-1 Max Whitby and Nick Quirke

v

vi

Contents

PART I I

12

Inorganic Nanotubes

Inorganic Fullerenes and Nanotubes ............................................................................................... 12-1 Andrey Enyashin and Gotthard Seifert

13

Spinel Oxide Nanotubes and Nanowires ......................................................................................... 13-1 Hong Jin Fan

14

Magnetic Nanotubes .........................................................................................................................14-1 Eugenio E. Vogel, Patricio Vargas, Dora Altbir, and Juan Escrig

15

Self-Assembled Peptide Nanostructures ......................................................................................... 15-1 Lihi Adler-Abramovich and Ehud Gazit

PART II I

16

Types of Nanowires

Germanium Nanowires .................................................................................................................... 16-1 Sanjay V. Khare, Sunil Kumar R. Patil, and Suneel Kodambaka

17

One-Dimensional Metal Oxide Nanostructures ..............................................................................17-1 Binni Varghese, Chorng Haur Sow, and Chwee Teck Lim

18

Gallium Nitride Nanowires ............................................................................................................. 18-1 Catherine Stampfl and Damien J. Carter

19

Gold Nanowires ................................................................................................................................ 19-1 Edison Z. da Silva, Antônio J. R. da Silva, and Adalberto Fazzio

20

Polymer Nanowires .......................................................................................................................... 20-1 Atikur Rahman and Milan K. Sanyal

21

Organic Nanowires ...........................................................................................................................21-1 Frank Balzer, Morten Madsen, Jakob Kjelstrup-Hansen, Manuela Schiek, and Horst-Günter Rubahn

PART IV

22

Nanowire Arrays

Magnetic Nanowire Arrays .............................................................................................................. 22-1 Adekunle O. Adeyeye and Sarjoosing Goolaup

23

Networks of Nanorods ..................................................................................................................... 23-1 Tanja Schilling, Swetlana Jungblut, and Mark A. Miller

PART V

24

Nanowire Properties

Mechanical Properties of GaN Nanowires ...................................................................................... 24-1 Zhiguo Wang, Fei Gao, Xiaotao Zu, Jingbo Li, and William J. Weber

25

Optical Properties of Anisotropic Metamaterial Nanowires .......................................................... 25-1 Wentao Trent Lu and Srinivas Sridhar

26

Thermal Transport in Semiconductor Nanowires .......................................................................... 26-1 Padraig Murphy and Joel E. Moore

27

The Wigner Transition in Nanowires ...............................................................................................27-1 David Hughes, Robinson Cortes-Huerto, and Pietro Ballone

28

Spin Relaxation in Quantum Wires ................................................................................................ 28-1 Paul Wenk and Stefan Kettemann

Contents

29

vii

Quantum Magnetic Oscillations in Nanowires .............................................................................. 29-1 A. Sasha Alexandrov, Victor V. Kabanov, and Iorwerth O. Thomas

30

Spin-Density Wave in a Quantum Wire .......................................................................................... 30-1 Oleg A. Starykh

31

Spin Waves in Ferromagnetic Nanowires and Nanotubes ...............................................................31-1 Hock Siah Lim and Meng Hau Kuok

32

Optical Antenna Effects in Semiconductor Nanowires .................................................................. 32-1 Jian Wu and Peter C. Eklund

33

Theory of Quantum Ballistic Transport in Nanowire Cross-Junctions ......................................... 33-1 Kwok Sum Chan

PART V I

34

Atomic Wires and Point Contact

Atomic Wires .................................................................................................................................... 34-1 Nicolás Agraït

35

Monatomic Chains ........................................................................................................................... 35-1 Roel H. M. Smit and Jan M. van Ruitenbeek

36

Ultrathin Gold Nanowires ............................................................................................................... 36-1 Takeo Hoshi, Yusuke Iguchi, and Takeo Fujiwara

37

Electronic Transport through Atomic-Size Point Contacts ............................................................37-1 Elke Scheer

38

Quantum Point Contact in Two-Dimensional Electron Gas .......................................................... 38-1 Igor V. Zozoulenko and Siarhei Ihnatsenka

PART VI I

39

Nanoscale Rings

Nanorings ......................................................................................................................................... 39-1 Katla Sai Krishna and Muthusamy Eswaramoorthy

40

Superconducting Nanowires and Nanorings................................................................................... 40-1 Andrei D. Zaikin

41

Switching Mechanism in Ferromagnetic Nanorings .......................................................................41-1 Wen Zhang and Stephan Haas

42

Quantum Dot Nanorings ................................................................................................................. 42-1 Ioan Bâldea and Lorenz S. Cederbaum

Index .................................................................................................................................................... Index-1

Preface The Handbook of Nanophysics is the fi rst comprehensive reference to consider both fundamental and applied aspects of nanophysics. As a unique feature of this work, we requested contributions to be submitted in a tutorial style, which means that state-of-the-art scientific content is enriched with fundamental equations and illustrations in order to facilitate wider access to the material. In this way, the handbook should be of value to a broad readership, from scientifically interested general readers to students and professionals in materials science, solid-state physics, electrical engineering, mechanical engineering, computer science, chemistry, pharmaceutical science, biotechnology, molecular biology, biomedicine, metallurgy, and environmental engineering.

What Is Nanophysics? Modern physical methods whose fundamentals are developed in physics laboratories have become critically important in nanoscience. Nanophysics brings together multiple disciplines, using theoretical and experimental methods to determine the physical properties of materials in the nanoscale size range (measured by millionths of a millimeter). Interesting properties include the structural, electronic, optical, and thermal behavior of nanomaterials; electrical and thermal conductivity; the forces between nanoscale objects; and the transition between classical and quantum behavior. Nanophysics has now become an independent branch of physics, simultaneously expanding into many new areas and playing a vital role in fields that were once the domain of engineering, chemical, or life sciences. This handbook was initiated based on the idea that breakthroughs in nanotechnology require a firm grounding in the principles of nanophysics. It is intended to fulfill a dual purpose. On the one hand, it is designed to give an introduction to established fundamentals in the field of nanophysics. On the other hand, it leads the reader to the most significant recent developments in research. It provides a broad and in-depth coverage of the physics of nanoscale materials and applications. In each chapter, the aim is to offer a didactic treatment of the physics underlying the applications alongside detailed experimental results, rather than focusing on particular applications themselves. The handbook also encourages communication across borders, aiming to connect scientists with disparate interests to begin

interdisciplinary projects and incorporate the theory and methodology of other fields into their work. It is intended for readers from diverse backgrounds, from math and physics to chemistry, biology, and engineering. The introduction to each chapter should be comprehensible to general readers. However, further reading may require familiarity with basic classical, atomic, and quantum physics. For students, there is no getting around the mathematical background necessary to learn nanophysics. You should know calculus, how to solve ordinary and partial differential equations, and have some exposure to matrices/linear algebra, complex variables, and vectors.

External Review All chapters were extensively peer reviewed by senior scientists working in nanophysics and related areas of nanoscience. Specialists reviewed the scientific content and nonspecialists ensured that the contributions were at an appropriate technical level. For example, a physicist may have been asked to review a chapter on a biological application and a biochemist to review one on nanoelectronics.

Organization The Handbook of Nanophysics consists of seven books. Chapters in the first four books (Principles and Methods, Clusters and Fullerenes, Nanoparticles and Quantum Dots, and Nanotubes and Nanowires) describe theory and methods as well as the fundamental physics of nanoscale materials and structures. Although some topics may appear somewhat specialized, they have been included given their potential to lead to better technologies. The last three books (Functional Nanomaterials, Nanoelectronics and Nanophotonics, and Nanomedicine and Nanorobotics) deal with the technological applications of nanophysics. The chapters are written by authors from various fields of nanoscience in order to encourage new ideas for future fundamental research. After the first book, which covers the general principles of theory and measurements of nanoscale systems, the organization roughly follows the historical development of nanoscience. Cluster scientists pioneered the field in the 1980s, followed by extensive ix

x

work on fullerenes, nanoparticles, and quantum dots in the 1990s. Research on nanotubes and nanowires intensified in subsequent years. After much basic research, the interest in applications such as the functions of nanomaterials has grown. Many bottom-up

Preface

and top-down techniques for nanomaterial and nanostructure generation were developed and made possible the development of nanoelectronics and nanophotonics. In recent years, real applications for nanomedicine and nanorobotics have been discovered.

Acknowledgments Many people have contributed to this book. I would like to thank the authors whose research results and ideas are presented here. I am indebted to them for many fruitful and stimulating discussions. I would also like to thank individuals and publishers who have allowed the reproduction of their figures. For their critical reading, suggestions, and constructive criticism, I thank the referees. Many people have shared their expertise and have commented on the manuscript at various

stages. I consider myself very fortunate to have been supported by Luna Han, senior editor of the Taylor & Francis Group, in the setup and progress of this work. I am also grateful to Jessica Vakili, Jill Jurgensen, Joette Lynch, and Glenon Butler for their patience and skill with handling technical issues related to publication. Finally, I would like to thank the many unnamed editorial and production staff members of Taylor & Francis for their expert work. Klaus D. Sattler Honolulu, Hawaii

xi

Editor Klaus D. Sattler pursued his undergraduate and master’s courses at the University of Karlsruhe in Germany. He received his PhD under the guidance of Professors G. Busch and H.C. Siegmann at the Swiss Federal Institute of Technology (ETH) in Zurich, where he was among the first to study spin-polarized photoelectron emission. In 1976, he began a group for atomic cluster research at the University of Konstanz in Germany, where he built the first source for atomic clusters and led his team to pioneering discoveries such as “magic numbers” and “Coulomb explosion.” He was at the University of California, Berkeley, for three years as a Heisenberg Fellow, where he initiated the fi rst studies of atomic clusters on surfaces with a scanning tunneling microscope. Dr. Sattler accepted a position as professor of physics at the University of Hawaii, Honolulu, in 1988. There, he initiated a research group for nanophysics, which, using scanning probe microscopy, obtained the first atomic-scale images of carbon nanotubes directly confirming the graphene network. In 1994,

his group produced the first carbon nanocones. He has also studied the formation of polycyclic aromatic hydrocarbons (PAHs) and nanoparticles in hydrocarbon flames in collaboration with ETH Zurich. Other research has involved the nanopatterning of nanoparticle fi lms, charge density waves on rotated graphene sheets, band gap studies of quantum dots, and graphene foldings. His current work focuses on novel nanomaterials and solar photocatalysis with nanoparticles for the purification of water. Among his many accomplishments, Dr. Sattler was awarded the prestigious Walter Schottky Prize from the German Physical Society in 1983. At the University of Hawaii, he teaches courses in general physics, solid-state physics, and quantum mechanics. In his private time, he has worked as a musical director at an avant-garde theater in Zurich, composed music for theatrical plays, and conducted several critically acclaimed musicals. He has also studied the philosophy of Vedanta. He loves to play the piano (classical, rock, and jazz) and enjoys spending time at the ocean, and with his family.

xiii

Contributors Adekunle O. Adeyeye Information Storage Materials Laboratory Department of Electrical and Computer Engineering National University of Singapore Singapore, Singapore Lihi Adler-Abramovich Department of Molecular Microbiology and Biotechnology George S. Wise Faculty of Life Sciences Tel Aviv University Tel Aviv, Israel Nicolás Agraït Departamento de Física de la Materia Condensada Universidad Autónoma de Madrid

Ioan Bâldea Theoretische Chemie Physikalisch-Chemisches Institut Universität Heidelberg Heidelberg, Germany Pietro Ballone Atomistic Simulation Centre Queen’s University Belfast Belfast, United Kingdom Frank Balzer Mads Clausen Institute Syddansk Universitet Sønderborg, Denmark Prabhakar R. Bandaru Department of Mechanical and Aerospace Engineering University of California, San Diego La Jolla, California

and Instituto Madrileño de Estudios Avanzados en Nanociencia Madrid, Spain

Damien J. Carter School of Physics The University of Sydney and

A. Sasha Alexandrov Department of Physics Loughborough University Loughborough, United Kingdom Jeffery R. Alston Nanoscale Science Program University of North Carolina at Charlotte Charlotte, North Carolina Dora Altbir Department of Physics Universidad de Santiago de Chile Santiago, Chile

Curtin University of Technology Sydney, New South Wales, Australia Lorenz S. Cederbaum Theoretische Chemie Physikalisch-Chemisches Institut Universität Heidelberg Heidelberg, Germany Kwok Sum Chan Department of Physics and Materials Science City University of Hong Kong Hong Kong, People’s Republic of China

Harsh Chaturvedi Optical Science and Engineering Program University of North Carolina at Charlotte Charlotte, North Carolina Robinson Cortes-Huerto Atomistic Simulation Centre Queen’s University Belfast Belfast, United Kingdom Gene Dresselhaus Francis Bitter Magnet Laboratory Massachusetts Institute of Technology Cambridge, Massachusetts Mildred S. Dresselhaus Departments of Electrical Engineering and Computer Science and Physics Massachusetts Institute of Technology Cambridge, Massachusetts Peter C. Eklund Departments of Physics and Materials Science and Engineering The Pennsylvania State University University Park, Pennsylvania Andrey Enyashin Physikalische Chemie Technische Universität Dresden Dresden, Germany and Institute of Solid State Chemistry Ural Branch of Russian Academy of Science Ekaterinburg, Russia xv

xvi

Juan Escrig Department of Physics Universidad de Santiago de Chile Santiago, Chile Muthusamy Eswaramoorthy Nanomaterials and Catalysis Laboratory Chemistry and Physics of Materials Unit and DST Unit on Nanoscience Jawaharlal Nehru Centre for Advanced Scientific Research Bangalore, India Hong Jin Fan Division of Physics and Applied Physics School of Physical and Mathematical Sciences Nanyang Technological University Singapore, Singapore Adalberto Fazzio Instituto de Física Universidade de São Paulo and Centro de Ciências Naturais e Humanas Universidade Federal do ABC São Paulo, Brazil Michael W. Forney Nanoscale Science Program University of North Carolina at Charlotte Charlotte, North Carolina

Contributors

Ehud Gazit Department of Molecular Microbiology and Biotechnology George S. Wise Faculty of Life Sciences Tel Aviv University Tel Aviv, Israel Sarjoosing Goolaup Information Storage Materials Laboratory Department of Electrical and Computer Engineering National University of Singapore Singapore, Singapore

Siarhei Ihnatsenka Department of Physics Simon Fraser University Burnaby, British Columbia, Canada Ado Jorio Departamento de Física Universidade Federal de Minas Gerais Belo Horizonte, Brazil Swetlana Jungblut Fakultät für Physik Universität Wien Vienna, Austria

Zhennan Gu College of Chemistry and Molecular Engineering Peking University Beijing, People’s Republic of China

Victor V. Kabanov Department of Complex Matter Josef Stefan Institute Ljubljana, Slovenia

Stephan Haas Department of Physics and Astronomy University of Southern California Los Angeles, California

Stefan Kettemann School of Engineering and Science Jacobs University Bremen Bremen, Germany

Natalie Herring Department of Chemistry University of North Carolina at Charlotte Charlotte, North Carolina

and Division of Advanced Materials Science Pohang University of Science and Technology Pohang, South Korea

Takeo Hoshi Department of Applied Mathematics and Physics Tottori University Tottori, Japan

Sanjay V. Khare Department of Physics and Astronomy The University of Toledo Toledo, Ohio

and Core Research for Evolutional Science and Technology Japan Science and Technology Agency Kawaguchi, Japan

Oleg V. Kibis Department of Applied and Theoretical Physics Novosibirsk State Technical University Novosibirsk, Russia

Core Research for Evolutional Science and Technology Japan Science and Technology Agency Kawaguchi, Japan

David Hughes Atomistic Simulation Centre Queen’s University Belfast Belfast, United Kingdom

Na Young Kim Edward L. Ginzton Laboratory Stanford University Stanford, California

Fei Gao Pacific Northwest National Laboratory Richland, Washington

Yusuke Iguchi Department of Applied Physics The University of Tokyo Tokyo, Japan

Jakob Kjelstrup-Hansen Mads Clausen Institute Syddansk Universitet Sønderborg, Denmark

Takeo Fujiwara Center for Research and Development of Higher Education The University of Tokyo Tokyo, Japan and

Contributors

Suneel Kodambaka Department of Materials Science and Engineering University of California Los Angeles, California Katla Sai Krishna Nanomaterials and Catalysis Laboratory Chemistry and Physics of Materials Unit and DST Unit on Nanoscience Jawaharlal Nehru Centre for Advanced Scientific Research Bangalore, India Meng Hau Kuok Department of Physics National University of Singapore Singapore, Singapore Jingbo Li State Key Laboratory for Superlattices and Microstructures Institute of Semiconductors Chinese Academy of Sciences Beijing, People’s Republic of China Chwee Teck Lim Department of Mechanical Engineering National University of Singapore Singapore, Singapore Hock Siah Lim Department of Physics National University of Singapore Singapore, Singapore Wentao Trent Lu Department of Physics Electronic Materials Research Institute Northeastern University Boston, Massachusetts Morten Madsen Mads Clausen Institute Syddansk Universitet Sønderborg, Denmark Sergey A. Maksimenko Institute for Nuclear Problems Belarus State University Minsk, Belarus Mark A. Miller University Chemical Laboratory Cambridge, United Kingdom

xvii

Joel E. Moore Department of Physics University of California and Materials Sciences Division Lawrence Berkeley National Laboratory Berkeley, California Padraig Murphy Department of Critical Studies California College of the Arts San Francisco, California and Department of Physics University of California Berkeley, California Andrei M. Nemilentsau Institute for Nuclear Problems Belarus State University Minsk, Belarus Sunil Kumar R. Patil Department of Mechanical Engineering The University of Toledo Toledo, Ohio Jordan C. Poler Department of Chemistry University of North Carolina at Charlotte Charlotte, North Carolina Mikhail E. Portnoi School of Physics University of Exeter Exeter, United Kingdom Nick Quirke School of Physics University College Dublin Dublin, Ireland Atikur Rahman Surface Physics Division Saha Institute of Nuclear Physics Kolkata, India

Horst-Günter Rubahn Mads Clausen Institute Syddansk Universitet Sønderborg, Denmark

Jan M. van Ruitenbeek Kamerlingh Onnes Laboratorium Universiteit Leiden Leiden, the Netherlands

Milan K. Sanyal Surface Physics Division Saha Institute of Nuclear Physics Kolkata, India

Elke Scheer Department of Physics University of Konstanz Konstanz, Germany

Manuela Schiek Mads Clausen Institute Syddansk Universitet Sønderborg, Denmark

Tanja Schilling Bâtiment des Sciences Université du Luxembourg Luxembourg

Gotthard Seifert Physikalische Chemie Technische Universität Dresden Dresden, Germany

Li Shi Department of Mechanical Engineering Texas Materials Institute The University of Texas at Austin Austin, Texas

Zujin Shi College of Chemistry and Molecular Engineering Peking University Beijing, People’s Republic of China

xviii

Antônio J. R. da Silva Instituto de Física Universidade de São Paulo São Paulo, Brazil and Laboratório Nacional de Luz Síncrotron Campinas São Paulo, Brazil Edison Z. da Silva Gleb Wataghin Institute of Physics University of Campinas São Paulo, Brazil Ferenc Simon Institute of Physics Budapest University of Technology and Economics Budapest, Hungary and Fakultät für Physik Universität Wien Wien, Austria Gregory Ya. Slepyan Institute for Nuclear Problems Belarus State University Minsk, Belarus Roel H. M. Smit Kamerlingh Onnes Laboratorium Universiteit Leiden Leiden, the Netherlands Chorng Haur Sow Department of Physics Faculty of Science National University of Singapore Singapore, Singapore Srinivas Sridhar Department of Physics Electronic Materials Research Institute Northeastern University Boston, Massachusetts Catherine Stampfl School of Physics The University of Sydney Sydney, New South Wales, Australia

Contributors

Oleg A. Starykh Department of Physics University of Utah Salt Lake City, Utah Iorwerth O. Thomas Department of Physics Loughborough University Loughborough, United Kingdom Kálmán Varga Department of Physics and Astronomy Vanderbilt University Nashville, Tennessee Patricio Vargas Department of Physics Universidad Técnica Federico Santa María Valparaíso, Chile Binni Varghese Department of Physics National University of Singapore Singapore, Singapore Igor Vasiliev Department of Physics New Mexico State University Las Cruces, New Mexico Eugenio E. Vogel Department of Physics Universidad de La Frontera Temuco, Chile Zhiguo Wang Department of Applied Physics University of Electronic Science and Technology of China Chengdu, People’s Republic of China

William J. Weber Pacific Northwest National Laboratory Richland, Washington

Paul Wenk School of Engineering and Science Jacobs University Bremen Bremen, Germany

Max Whitby Department of Chemistry Imperial College London and RGB Research Ltd London, United Kingdom Jian Wu Departments of Physics The Pennsylvania State University University Park, Pennsylvania Andrei D. Zaikin Institute for Nanotechnology Karlsruhe Institute of Technology Karlsruhe, Germany and I.E. Tamm Department of Theoretical Physics P.N. Lebedev Physics Institute Moscow, Russia Wen Zhang Department of Physics and Astronomy University of Southern California Los Angeles, California

and State Key Laboratory for Superlattices and Microstructures Institute of Semiconductors Chinese Academy of Sciences Beijing, People’s Republic of China

Igor V. Zozoulenko Department of Science and Technology Linköping University Linköping, Sweden

Zhiyong Wang College of Chemistry and Molecular Engineering Peking University Beijing, People’s Republic of China

Xiaotao Zu Department of Applied Physics University of Electronic Science and Technology of China Chengdu, People’s Republic of China

I Carbon Nanotubes 1 Pristine and Filled Double-Walled Carbon Nanotubes Zujin Shi, Zhiyong Wang, and Zhennan Gu .................. 1-1 Introduction • Pristine Double-Walled Carbon Nanotubes • Filled Double-Walled Carbon Nanotubes • Summary • Acknowledgments • References

2 Quantum Transport in Carbon Nanotubes

Kálmán Varga .................................................................................. 2-1

Introduction • Electronic Structure of Carbon Nanotubes • Transport in Nanotubes • Summary • References

3 Electron Transport in Carbon Nanotubes Na Young Kim .................................................................................... 3-1 Introduction • Fundamental Concepts • Electron Transport in Carbon Nanotubes • Summary • Future Perspective • References

4 Thermal Conductance of Carbon Nanotubes

Li Shi ............................................................................................. 4-1

Introduction • Theory of Thermal Conduction in Carbon Nanotubes • Measurements of Thermal Conductance of Carbon Nanotubes • Conclusion and Outlook • Acknowledgments • References

5 Terahertz Radiation from Carbon Nanotubes Andrei M. Nemilentsau, Gregory Ya. Slepyan, Sergey A. Maksimenko, Oleg V. Kibis, and Mikhail E. Portnoi .......................................................................... 5-1 Introduction • Electronic Properties of SWNTs • Thermal Radiation from SWNTs • Quasi-Metallic Carbon Nanotubes as Terahertz Emitters • Chiral Carbon Nanotubes as Frequency Multipliers • Armchair Nanotubes in a Magnetic Field as Tunable THz Detectors and Emitters • Conclusion • Acknowledgments • References

6 Isotope Engineering in Nanotube Research Ferenc Simon ................................................................................... 6-1 Introduction • State of the Art of SWCNT Research • Isotope Engineering of SWCNTs • Summary • Acknowledgments • References

7 Raman Spectroscopy of sp2 Nano-Carbons Mildred S. Dresselhaus, Gene Dresselhaus, and Ado Jorio .................7-1 Introduction • Background • Presentation of State-of-the-Art Raman Spectroscopy of sp2 Nano-Carbons • Critical Discussions • Summary • Future Perspectives • Acknowledgments • References

8 Dispersions and Aggregation of Carbon Nanotubes Jeffery R. Alston, Harsh Chaturvedi, Michael W. Forney, Natalie Herring, and Jordan C. Poler ........................................................................................... 8-1 Introduction • Background • State-of-the-Art Procedures and Techniques • Summary • References

9 Functionalization of Carbon Nanotubes for Assembly Igor Vasiliev ................................................................... 9-1 Introduction • Functionalization of CNTs • Theoretical Modeling of CNTs • Carboxylated CNTs • Thiolated CNTs • Assembly of Phenosafranin to CNTs • Summary • Acknowledgments • References

10 Carbon Nanotube Y-Junctions

Prabhakar R. Bandaru ....................................................................................... 10-1

Introduction • Controlled Carbon Nanotube Y-Junction Synthesis • Electrical Characterization of Y-Junction Morphologies • Carrier Transport in Y-Junction–Electron Momentum Engineering • Applications of Y-CNTs to Novel Electronic Functionality • Experimental Work on Electrical Characterization • Topics for Further Investigation • Conclusions • References

11 Fluid Flow in Carbon Nanotubes Max Whitby and Nick Quirke ........................................................................ 11-1 Introduction • Materials • Theory of Nanoscale Flow • Experimental Investigation of Nanoscale Fluid Flow • Controlling Fluid Flow through Carbon Nanopipes • Pumping • Interfacing, Interconnections, and Nanofluidic Device Fabrication • Applications for Fluid Flow through Nanopipes • Conclusions and Future Directions • Acknowledgments • References

I-1

1 Pristine and Filled Double-Walled Carbon Nanotubes 1.1 1.2

Introduction ............................................................................................................................. 1-1 Pristine Double-Walled Carbon Nanotubes ....................................................................... 1-2 Synthesis of Double-Walled Carbon Nanotubes • Electronic Properties of Double-Walled Carbon Nanotubes • Structure Determination of Double-Walled Carbon Nanotubes • Raman Spectra of Double-Walled Carbon Nanotubes

Zujin Shi Peking University

Zhiyong Wang Peking University

Zhennan Gu Peking University

1.3

Filled Double-Walled Carbon Nanotubes ........................................................................... 1-9 Fullerene-Filled Double-Walled Carbon Nanotubes • Inorganic-Material-Filled Double-Walled Carbon Nanotubes • Organic-Material-Filled Double-Walled Carbon Nanotubes • Chemical Reactions inside Double-Walled Carbon Nanotubes • Doping of Double-Walled Carbon Nanotubes

1.4 Summary ................................................................................................................................. 1-17 Acknowledgments ............................................................................................................................. 1-17 References...........................................................................................................................................1-17

1.1 Introduction As one of the most important materials in the nano area, carbon nanotubes have generated broad and interdisciplinary attention in the last two decades (Dresselhaus and Dai 2004). Their outstanding properties have been studied extensively and much effort has been devoted to their applications in areas of energy storage, electronics, sensors, and more (Baughman et al. 2002; Pengfei et al. 2003; Avouris and Chen 2006; Ajayan and Tour 2007). Research on carbon nanotubes has been primarily focused on multi-walled carbon nanotubes (MWNTs) (Iijima 1991) and single-walled carbon nanotubes (SWNTs) (Bethune et al. 1993; Iijima and Ichihashi 1993). Ever since the breakthrough of the macroscale selective synthesis of double-walled carbon nanotubes (DWNTs) (Hutchison et al. 2001), they have increasingly drawn scientific interest due to their attractive structures and properties. Strictly speaking, DWNTs are one kind of MWNTs. However, DWNTs’ properties are remarkably different from those of MWNTs with three or more graphitic shells. As the intermedium between SWNTs and MWNTs, DWNTs possess the advantages of both MWNTs and SWNTs, i.e., excellent mechanical and electrical properties. More importantly, DWNTs offer lots of unique characteristics over SWNTs and MWNTs. For example, DWNTs provide simple models for the investigation of intertube interaction (Saito et al. 1993a; Zolyomi et al. 2006; Tison et al. 2008). Impressive results have been attained regarding the effects of intertube interaction on the properties of DWNTs. Furthermore, the outer tube of a DWNT can serve as a protector for the inner tube (Iakoubovskii et al. 2008).

When the outer tubes are covalently functionalized, the inner tubes still retain their electronic and optical properties (Hayashi et al. 2008). Thus, the inner and outer tubes can play different roles simultaneously in electronic and optical devices. Due to the hollow structure of carbon nanotubes, they can be filled with molecules in their interior nanometer-sized space, thus providing a new class of hybrid materials with novel structures and properties (Monthioux 2002; Kitaura and Shinohara 2006). The filling of carbon nanotubes is an effective way to modify the properties of carbon nanotubes. Interactions between carbon nanotubes and the encapsulated materials including van der Waals force, electron transfer, and orbital mixing have been shown to alter the properties of carbon nanotubes markedly. On the other hand, the spatial confinement of carbon nanotubes is expected to impart novel and distinct physical and chemical properties to the encapsulated species from their corresponding bulk samples. Carbon nanotubes are transparent to light and electron beams so that they can be used as nano test tubes or nano vessels, which also serve as a protector for the molecules that are otherwise unstable in air or could be used to study the physical and chemical properties in situ, e.g., the structure, phase transition, or chemical reactions in the nanospace, using high-resolution transmission electron microscopy (HRTEM) and optical techniques. The filling of carbon nanotubes was primarily performed on MWNTs with the aim of fabricating nanowires using MWNTs as templates. The encapsulated materials include metals, oxides, halides, and carbides (Monthioux et al. 2006). After the macroscale synthesis of SWNTs was achieved, they attracted much 1-1

1-2

more attention because of their smaller diameters and more uniform and defect-free structures compared to MWNTs. The encapsulation of RuCl3 into SWNTs by Sloan et al. (1998) and the discovery of C60@SWNTs in the SWNT sample synthesized by a laser-ablation method (Smith et al. 1998) stimulated the research subject of filling of SWNTs. A large variety of molecules have been encapsulated into SWNTs, ranging from fullerenes, metallofullerenes, metal and nonmetal elements, inorganic compounds, and organic molecules (Monthioux et al. 2006). The encapsulated molecules exhibit unique properties in terms of structure, phase transition, motion behavior, and chemical properties. Meanwhile, it has been found that the electronic structure of SWNTs can be modified by dopant insertion, which makes it possible to tune the electronic properties and mechanical properties of the SWNTs. For example, arrays of C60 molecules nested inside SWNTs can change the local electronic structure of the SWNTs to give it a hybrid electronic band (Hornbaker et al. 2002); integrating organic molecules of electron donors or accepters into SWNTs provides stable and controllable doped SWNTs for fabricating molecular electronic devices (Takenobu et al. 2003). Doped endohedral metallofullerenes Gd@C82 can divide a semiconducting SWNT into multiple quantum dots, where the band gap is narrowed from 0.5 eV down to 0.1 eV (Lee et al. 2002). With regard to the encapsulation of guest molecules, DWNTs possess unique characteristics compared to SWNTs and MWNTs. On one hand, larger inner diameters of DWNTs compared to SWNTs impart the ability of accommodating large-size molecules to DWNTs. On the other hand, inner diameters of DWNTs are smaller than those of MWNTs; thus the quantum effects of the encapsulated materials are expected to be more notable in the former case. Nevertheless, there are fewer reports on the fi lling of DWNTs than both SWNTs and MWNTs, because the synthesis of pure DWNTs is more difficult than the synthesis of SWNTs and MWNTs. Since the successful macroscale synthesis of DWNTs (Hutchison et al. 2001), the fi lling of DWNTs has been drawing increasing attention. This chapter focuses on the synthesis and properties of pristine and fi lled DWNTs. Section 1.2 introduces the synthesis, and structural and electronic properties of pristine DWNTs. As an important method for characterizing carbon nanotubes, features of the Raman spectra of DWNTs are also given. Section 1.3 describes DWNTs fi lled with fullerenes, and inorganic and organic materials. The structure, phase transition, and chemical reactions of the encapsulated species as well as the doping effects on DWNTs are discussed in detail.

1.2 Pristine Double-Walled Carbon Nanotubes 1.2.1 Synthesis of Double-Walled Carbon Nanotubes The selective synthesis of DWNTs was first achieved by a hydrogen arc discharge method (Hutchison et al. 2001). The critical factor of this method is the use of sulfur and metals as a catalyst

Handbook of Nanophysics: Nanotubes and Nanowires

in the hydrogen atmosphere. From then on, large numbers of reports have emerged on the synthesis of DWNTs, which can be divided into three categories, i.e., the arc discharge method (Huang et al. 2003; Saito et al. 2003; Sugai et al. 2003; Chen et al. 2006; Qiu et al. 2006a, 2007b), chemical vapor deposition (CVD) (Ci et al. 2002, 2007; Bacsa et al. 2003; Flahaut et al. 2003; Hiraoka et al. 2003; Lyu et al. 2003; Wei et al. 2003; Zhou et al. 2003; Zhu et al. 2003; Wei et al. 2004; Li et al. 2005; Liu et al. 2005, 2007; Ramesh et al. 2005; Yamada et al. 2006; Bachmatiuk et al. 2007; Gunjishima et al. 2007; Qi et al. 2007), and the SWNTtemplate method (Bandow et al. 2001, 2004; Fujita et al. 2005; Guan et al. 2005b, 2008; Kalbac et al. 2005; Pfeiffer et al. 2007; Kuzmany et al. 2008; Shiozawa et al. 2008). The arc discharge method is an effective way of producing high structural quality carbon nanotubes. MWNTs observed by Iijima using HRTEM in 1991 were by-products in an arc discharge process toward the synthesis of fullerenes. In the following years, the synthesis of carbon nanotubes using the arc discharge method was developed (Ebbesen and Ajayan 1992; Bethune et al. 1993; Iijima and Ichihashi 1993; Saito et al. 1993b; Seraphin and Zhou 1994; Journet et al. 1997). The commonly used conditions were inert atmosphere and metals (Fe, Co, Ni, etc.) used as a catalyst. In the arc discharge process, the evaporation of graphite and metals at high temperature leads to the formation of carbon nanotubes. As for the synthesis of the DWNTs, the atmosphere of hydrogen was frequently used (Saito et al. 2003; Chen et al. 2006; Qiu et al. 2006a). Another important aspect of the synthesis condition for DWNTs is the adding of sulfur to the metal catalysts or using sulfide (Huang et al. 2003; Saito et al. 2003; Chen et al. 2006; Qiu et al. 2006a). In addition, KCl was reported to behave as a promoter for the growing of DWNTs (Qiu et al. 2006a); however, the mechanism is not clear yet. The production of carbon nanotubes by CVD is a process from gaseous carbon sources to nanotube structures catalyzed by nanoparticles. Generally, the growing conditions of SWNTs, DWNTs, and MWNTs are similar in the CVD process. In many cases, the products consist of all these kinds of carbon nanotubes. Fine control of the conditions is necessary for selectively obtaining one kind of carbon nanotubes. For the synthesis of DWNTs, catalysts reported in the literature include Fe, Co, Ni, Mo, ferrocene, sulfide, etc. Carbon sources include hydrocarbon (methane, ethene, acetylene, benzene), ethanol, etc. (Ci et al. 2002, 2007; Bacsa et al. 2003; Flahaut et al. 2003; Hiraoka et al. 2003; Lyu et al. 2003; Wei et al. 2003; Zhou et al. 2003; Zhu et al. 2003; Wei et al. 2004; Li et al. 2005; Liu et al. 2005, 2007; Ramesh et al. 2005; Yamada et al. 2006; Bachmatiuk et al. 2007; Gunjishima et al. 2007; Qi et al. 2007). One advantage of the CVD method is the facility of preparing a special assembly of DWNTs. For example, a synthesis of vertical arrays of DWNTs on flat substrates has been achieved by using catalysts with a high density (Yamada et al. 2006). Besides the above two traditional methods, another ingenious route is taking advantage of chemical reactions of guest molecules encapsulated inside SWNTs. Bandow et al. (2001) discovered that C60 molecules inside SWNTs coalesce to dimers, trimers,

1-3

Pristine and Filled Double-Walled Carbon Nanotubes

(b) (a)

5 nm

1.2.2 Electronic Properties of Double-Walled Carbon Nanotubes Generally, DWNTs are composed of two coaxial tubes interacting through van der Waals forces (DWNTs with an uncoaxial structure were occasionally observed [Hashimoto et al. 2005]). The electronic properties of DWNTs are determined basically by the electronic properties of the two constituent tubes. However, the interaction between the two tubes has significant effects on the electronic properties of DWNTs, which has been demonstrated by theoretical calculations (Saito et al. 1993a; Liang 2004; Song et al. 2005; Zolyomi et al. 2006, 2008; Lu and Wang 2007). Saito et al. (1993a) investigated the effect of intertube interaction on the energy dispersion relations of DWNTs, which consist of metal–metal, metal–insulator, and insulator–metal constituents. The splitting of some energy bands induced by intertube interactions was observed (Figure 1.2). Moreover, it was predicted that metallic@metallic (M@M)-type DWNTs are still metallic and the metallic tube retains its metallic state in the case of metallic@semiconducting (M@S) and semiconducting@metallic (S@M)-type DWNTs. Liang (2004) performed calculations considering the intertube coupling strength of commensurate DWNTs as a function of the radius difference of the inner and

3.0

3.0

2.0

2.0

E/γ0

and so on, and transform into tubes eventually upon heating at temperatures higher than ∼800°C (Figure 1.1). Detailed Raman scattering analysis revealed that the diameters of the inner tubes derived from encapsulated C60 molecules are close to that of C60 molecules at an early stage, whereas the inner tubes turn into wider tubes that match the size of the parent SWNTs upon longtime heat treatment (Bandow et al. 2004). Kalbac et al. (2005) found that the irradiation of C60@SWNTs and C70@SWNTs by ultraviolet light also yields DWNTs. Molecules that can transform into carbon nanotubes inside SWNTs are not limited to fullerenes. As the research on the fi lling of SWNTs develops rapidly, some other kinds of molecules were found to behave similarly, including ferrocene (Guan et al. 2005b, 2008), PTCDA (Fujita et al. 2005), and GdCp3 (Shiozawa et al. 2008). It was found that the diameter distributions of the inner tubes derived from C60 and ferrocene are different even for the same set of parent SWNTs (Pfeiffer et al. 2007).

E/γ0

FIGURE 1.1 HRTEM images of C60@SWNTs after heat treatment in vacuo (0)

φ( −20 V. In the case of

1000

2000

1000

500

0 –40 (a)

Ferrocene@DWNTs VDS = 1 V

800

IDS (nA)

IDS (nA)

1500

DWNTs VDS = 1 V

600 400 200

–30

–20

–10

0 10 VGS (V)

20

30

40

0 –40

–30

–20

–10

10

20

30

40

(b)

0 10 VGS (V)

20

30

40

1600 Fe@DWNTs VDS = 1 V

IDS (nA)

1200

800

400

0 –40 (c)

–30

–20

–10

0 VGS (V)

FIGURE 1.19 Drain current versus gate voltage (IDS –VGS) curves of FETs based on (a) DWNT, (b) ferrocene@DWNTs, and (c) Fe@DWNTs measured with VDS = 0–1 V. (Reproduced from Li, Y.F. et al., Nanotechnology, 17(16), 4143, 2006b. With permission.)

1-14


ferrocene@DWNTs, the conductance in the n-channel increases and the conductance in the p-channel is significantly suppressed (Figure 1.19b). This change arises from the transfer of electrons from the encapsulated ferrocene molecules to the DWNTs. Decomposition of ferrocene inside DWNTs leads to the formation of Fe@DWNTs. Fe@DWNTs exhibit a unipolar n-type semiconducting behavior (Figure 1.19c). The different properties between ferrocene@DWNTs and Fe@DWNTs are attributed to their different electron donating abilities. A higher concentration of electrons is injected into the DWNTs in the case of Fe@DWNTs. These results demonstrate that FET devices with controllable characteristics can be fabricated using DWNTs fi lled with appropriate molecules.

nanotube consists of three concentric graphene sheets. On average, the TWNTs have outer diameters ranging from 3 to 5 nm. For the individual TWNT shown in the inset of Figure 1.20, its outer diameter is 3 nm with an interlayer spacing of ∼0.4 nm. The HRTEM examinations reveal the purity or the ratio of TWNTs to all nanotubes is over 50%, which is strongly dependent on the fi lling rate and the inner diameters of DWNTs. In general, for DWNTs with outer diameters larger than 3 nm, high quality TWNTs are favored and formed more easily inside the tubes. Otherwise, few TWNTs are obtained. Figure 1.21 shows the room temperature Raman spectra of parent DWNTs and TWNTs, respectively. Several new RBM

RBM

1.3.4 Chemical Reactions inside Double-Walled Carbon Nanotubes

156

192 208 220 254

Intensity (a.u.)

Carbon nanotubes can serve as nano test tubes for reactions in their interior space. Due to the spatial confinement, the reaction process and the products exhibit distinct features compared with reactions free of confinement. In general, the polymerization of encapsulated molecules inside carbon nanotubes generates products with one-dimensional structures. Examples include a transformation from C60 into nanotubes (Bandow et al. 2001, 2004; Kalbac et al. 2005) and from C60O into linear polymers (Britz et al. 2005), etc. Qiu et al. (2007a) studied the pyrolysis of ferrocene inside DWNTs. They found that triple-walled carbon nanotubes (TWNTs) formed after high temperature annealing of ferrocene@ DWNTs, and the carbon nanotubes’ base-growth mechanism was proved. Figure 1.20 shows HRTEM images of TWNT bundles and an individual TWNT (inset), revealing clearly that each

146

50

100

(a)

150 200 250 Raman shift (cm–1)

1581

300

G band

1576

D band

5 nm

1200

10 nm (b)

FIGURE 1.20 An HRTEM image of TWNT bundles derived from ferrocene@DWNTs. Top-right inset: an individual TWNT with clearly resolved graphitic layers. (Reproduced from Qiu, H.X. et al., Chem. Commun., (10), 1092, 2007a. With permission.)

1400 1600 Raman shift (cm–1)

1800

FIGURE 1.21 Raman spectra (633 nm excitation) for empty DWNTs (bottom) and TWNTs (upper) showing (a) RBM and (b) the D band and G band. (Reproduced from Qiu, H.X. et al., Chem. Commun., (10), 1092, 2007a. With permission.)


(c)

Carbon source and catalyst

The tertiary nanotube

(b)

Growth direction

(a)

The closed cap

peaks are clearly seen in the case of TWNTs (top curve in Figure 1.21a), which is evidence that novel carbon nanotubes with smaller diameters have been formed after the annealing of ferrocene inside DWNTs. From these new Raman peaks, it can be calculated that the smallest tertiary tube has a diameter of 0.9 nm, implying that the template DWNTs should have an inner diameter of about 1.6 nm. In other words, DWNTs with inner diameters larger than 1.6 nm would favor the growth of the tertiary carbon nanotubes. Although the largest inner tube of TWNTs is 1.6 nm, this is unlikely the largest tertiary tube because the Raman peaks lower than 146 cm−1, corresponding to tubes with a diameter of 1.6 nm, are hard to detect. In comparison with the Raman spectra from DWNTs, the IG/ID ratio of TWNTs sample is lower than that of DWNTs, implying that the new nanotubes formed from ferrocene have a higher defect density than the original template DWNTs prepared by the arc discharge method. Structure details of TWNTs may render some information on their growth mechanism. It has been found that both capped and open-ended nanotubes are found inside the inner cavities of DWNTs. There are no catalyst particles at the closed-tip, see Figure 1.22a, but something black is packed right at the open-end of the tube (indicated by the lower arrow, in Figure 1.22b). Energy Dispersive x-ray (EDX) analysis reveals that black substances are small iron particles formed during the decomposition of encapsulated ferrocene when they are heated to 900°C inside the central cavity of DWNTs. Th is information

FIGURE 1.22 (a,b) HRTEM images of an individual TWNT derived from ferrocene@DWNT, showing how the tertiary carbon layer is formed. (c) A schematic illustration of the growth mechanism of the TWNTs. The scale bars are 5 nm. (Reproduced from Qiu, H.X. et al., Chem. Commun., (10), 1092, 2007a. With permission.)

1-15

leads one to believe that the formation of inner carbon nanotubes follows the base-growth mechanism, as schematically illustrated in Figure 1.22c. The growth of CNTs conforms to the vapor–liquid–solid (VLS) mechanism (Gavillet et al. 2001). Firstly, ferrocene undergoes a decomposition to release a series of hydrocarbons and Fe species at temperatures over 500°C. Theoretical calculations and molecular dynamics simulations revealed that the hydrogen atoms left the ferrocene first due to the weaker C–H bond (492 kJ/mol), and then the cyclopentadienyl ring with higher C–C bond energy (602 kJ/mol) started to break up, followed by the breakage of the C–Fe bonds (1480 kJ/mol) in combination with the remaining C–C bonds (Elihn and Larsson 2004). Hence, the fragments from the decomposition of ferrocene are sequent C–H, C, C2 and C3, and fi nally Fe. During the pyrolysis process, the Fe species serves as a catalyst and those hydrocarbons work as carbon sources. By increasing the temperature to 900°C, these hydrocarbon species and C clusters dissolve into the ion nanoparticles, diff use through them and precipitate and thus react with each other, promoting the growth of carbon tubular structures under the confi nement of DWNTs. The diameter of the newly formed tube is strictly controlled by the diameter of the template, i.e., by the inner tube diameter of DWNTs, and usually smaller by 0.7–0.8 nm. Moreover, the formation of new nanotubes is strongly dependent on the amount of ferrocene encapsulated inside DWNTs and on the pyrolysis conditions. DWNTs with large diameters contain a large amount of ferrocene, accordingly favoring the growth of nanotubes with longer lengths and good structures, while the small template is apt to produce nanotubes with fi nite length and imperfect structures. The transformation from ferrocene-filled DWNTs to TWNTs proves that the pyrolysis of molecules inside carbon nanotubes is an effective route for increasing the wall number of carbon nanotubes. An appropriate selection of the encapsulated species enables the alteration of the composition of carbon nanotubes. For instance, nitrogen-doped TWNTs were prepared by encapsulation and pyrolysis of iron(II) phthalocyanine in the interior space of DWNTs (Wang et al. 2008). Figure 1.23 shows HRTEM images of the as-prepared TWNTs, in which three main features of the TWNTs are explicit. First, the innermost walls are corrugated. In contrast, most of the innermost walls formed from ferrocene are straight. Second, there are bamboo structures in the innermost wall. Third, fullerene structures are present inside the TWNTs. In addition, it was found that the newly formed innermost wall has more defects and is less stable under electron irradiation than the parent walls. Some of the innermost walls collapsed and transformed into an amorphous structure after a long-time electron irradiation. Theoretical calculations and experimental studies have demonstrated that the presence of nitrogen atoms in the carbon nanotubes would lead to the formation of bamboo structures and fullerene structures (Sumpter et al. 2007). In addition, pyridine-like nitrogen atoms in the skeleton of carbon nanotubes result in structural defects (Terrones et al.

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7 nm

7 nm

(a)

A

B

C

(b)

FIGURE 1.23 HRTEM images of TWNTs derived from iron(II) phthalocyanine@DWNTs. (Reproduced from Wang, Z.Y. et al., Chin. J. Inorg. Chem., 24(8), 1237, 2008. With permission.)

2002). Therefore, the structural characteristics of TWNTs are probably related to the nitrogen atoms in the innermost wall of TWNTs.

335

185

Intensity (a.u.)

The modulation of DWNTs’ properties is a crucial step toward their applications. A great deal of interest is focused on the doping of DWNTs. p-Type doping of DWNTs has been achieved with electron accepters such as Br2 (Chen et al. 2003; Souza et al. 2006), I2 (Cambedouzou et al. 2004), H2SO4 (Barros et al. 2007), etc. On the other hand, n-type doping of DWNTs has been achieved with electron donors including Cs (Li et al. 2006c), K (Rauf et al. 2006; Chun and Lee 2008; Desai et al. 2008), Fe (Li et al. 2006b; Jorge et al. 2008), etc. For DWNTs doped with Cs and Fe, as shown above, the dopants are encapsulated inside the DWNTs. Whereas for other dopants listed here, both endohedral doping and exohedral doping are possible, depending on the experimental conditions. The radial charge distribution on Br2-doped DWNTs was studied as a cylindrical molecular capacitor (Chen et al. 2003). Raman scattering measurements on Br2-doped DWNTs reveal that most of the charge resides on the outer tubes. Figure 1.24 shows that the RBM of the outer tubes disappears after Br2doping, while the RBM of the inner tubes are not affected. Meanwhile, the G band of inner tubes remains at 1581 cm−1, whereas the G band of the outer tubes vanishes. These results demonstrate that the charge transfer between Br2 and DWNTs only involves the outer tubes. Th is system was modeled as a three-layer cylindrical capacitor with the bromine anions forming a shell around the outer nanotube. The calculated number of holes on the inner tube versus the total number of holes on the double-walled tubes is consistent with the experimental results. For M@S type DWNTs, the inner tube dopes slightly at a low doping level. As the doping level increases, the charge distribution rapidly begins to favor the outer tube because of the increase in the electrostatic effects. Similar results were obtained in the case of iodine-doped DWNTs (Cambedouzou et al. 2004). However, the doping effect of H2SO4 on the inner tubes of DWNTs (Barros et al. 2007) is different from that of Br2 doping. In the case of H2SO4-doped DWNTs, only the metallic inner tubes are affected, while there is no obvious change for

161 176

Before Br2-doping

342

314 335

314

342

After Br2-doping λox = 1064 nm

(a) 100

200

300

400

DWNT λox = 1064 nm

1581 Intensity (a.u.)

1.3.5 Doping of Double-Walled Carbon Nanotubes

DWNT

Before Br2-doping

After Br2-doping

1550 (b)

1590

1581

1600 Raman shift (cm–1)

1650

FIGURE 1.24 Raman spectra of DWNTs and Br2-doped DWNTs: (a) RBM band and (b) G band. (Reproduced from Chen, G.G. et al., Phys. Rev. Lett., 90(25), 257403, 2003. With permission.)

the RBM peaks of the inner semiconducting tubes in both frequency and intensity. On the other hand, n-type doping of DWNTs by potassium was investigated in detail (Rauf et al. 2006). The behavior of the RBM response of DWNTs upon potassium-doping is similar to the p-type doping of DWNTs. For the excitation of 568 nm (Figure 1.25a), the RBM of the outer tubes and the metallic inner tubes decreases significantly at a low doping level, whereas the response of the semiconducting inner tubes response is hardly

1-17


568 nm

676 nm

Outer tubes

remains unchanged for the excitation of 488 nm and downshifts for the excitation of 568 and 676 nm. A comparison of charge concentration between outer tubes and inner tubes cannot be made from the degree of the G band shift because the relationship of the G band shift of carbon nanotubes with the doping level is complicated. A nonmonotonic behavior of the G band of SWNTs upon potassium-doping has been observed (Chen et al. 2005). The frequency shift in Figure 1.25c is explained by the Coulomb interaction in a system composed of negatively charged carbon ions and positively charged potassium ions.

Inner tubes

Inner tubes

Raman intensity (a.u.)

Outer tubes

0

I

II

1.4 Summary

×5 ×5

III 150 (a)

200 250 300 350 Raman shift (cm–1)

150 (b)

200 250 300 350 Raman shift (cm–1)

G-line shift (cm–1)

15 OT IT

10

488 nm 568 nm 676 nm

5 0 –5 0

(c)

I II Intercalation step

III

FIGURE 1.25 RBM of potassium-doped DWNTs for different doping steps (0, I, II, and III) excited by lasers of (a) 568 nm and (b) 676 nm. (c) G band shift of outer tubes and inner tubes of potassium-doped DWNTs for different doping steps excited with three different lasers. (Reproduced from Rauf, H. et al., Phys. Rev. B, 74(23), 235419, 2006. With permission.)

changed. At a higher doping level, the RBM of the semiconducting inner tubes starts to decrease. For the excitation of 676 nm (Figure 1.25b), the RBM peaks at the frequency region of 250–320 cm−1 corresponding to (7, 5) and (8, 3) inner tubes. An identical inner tube exhibits a number of RBM peaks when the outer tubes are different, so each peak in Figure 1.25b represents an inner/outer tube pair. Based on the assignment of these peaks, the influence of the diameter difference between the inner and outer tubes on the charge transfer to the inner tubes was investigated. For instance, the intensity of an RBM peak from an (8, 3) inner tube with a smaller diameter difference [(8, 3)@ (14, 1) tubes] decreases faster than that with a larger diameter difference [(8, 3)@(15, 6) tubes]. Figure 1.25c depicts the G band shift of outer tubes and inner tubes of potassium-doped DWNTs excited with three different lasers. The G band of outer tubes upshifts rapidly with an increase in doping. The maximum shift values are 8, 9, and 13 cm−1 for the excitation of 488, 568, and 676 nm, respectively. In contrast, the G band of inner tubes

DWNTs possess distinct characteristics compared with SWNTs and MWNTs. The interaction between the two constituent tubes of DWNTs not only modifies the properties of DWNTs, including electronic band splitting, the occurrence of new Van Hove singularities, and semiconducting-metal transitions, but also affects the stability of encapsulated molecules. The inner and outer tubes of DWNTs exhibit different responses to endohedral or exohedral doping, which is related to the nested structure and the metallic or semiconducting nature of the constituents. Taking advantage of the different properties of the inner and outer tubes and imparting different functions to them are research issues that need to be addressed. Materials encapsulated inside DWNTs including fullerenes, inorganic and organic molecules display novel structures and properties due to the spatial confinement and the interaction with DWNTs. The new phenomena observed in the nanospace inside carbon nanotubes have promoted the understanding of the low-dimensional system.

Acknowledgments This work was supported by the National Natural Science Foundation of China (Nos. 90206048, 20371004, and 20771010) and the Ministry of Science and Technology of China (Grant 2006CB932701 and 2007AA03Z311).

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Takenobu, T., Takano, T., Shiraishi, M., Murakami, Y., Ata, M., Kataura, H., Achiba, Y., and Iwasa, Y. (2003). Stable and controlled amphoteric doping by encapsulation of organic molecules inside carbon nanotubes. Nature Materials 2(10): 683–688. Terrones, M., Ajayan, P. M., Banhart, F., Blase, X., Carroll, D. L., Charlier, J. C., Czerw, R., Foley, B., Grobert, N., Kamalakaran, R., Kohler-Redlich, P., Ruhle, M., Seeger, T., and Terrones, H. (2002). N-doping and coalescence of carbon nanotubes: Synthesis and electronic properties. Applied Physics A—Materials Science & Processing 74(3): 355–361. Tison, Y., Giusca, C. E., Stolojan, V., Hayashi, Y., and Silva, S. R. P. (2008). The inner shell influence on the electronic structure of double-walled carbon nanotubes. Advanced Materials 20(1): 189–194. Wang, Z. Y., Zhao, K. K., Shi, Z. J., Gu, Z. N., and Jin, Z. X. (2008). Preparation of nitrogen-doped triple-walled carbon nanotubes. Chinese Journal of Inorganic Chemistry 24(8): 1237–1241. Wei, J. Q., Ci, L. J., Jiang, B., Li, Y. H., Zhang, X. F., Zhu, H. W., Xu, C. L., and Wu, D. H. (2003). Preparation of highly pure double-walled carbon nanotubes. Journal of Materials Chemistry 13(6): 1340–1344. Wei, J. Q., Jiang, B., Wu, D. H., and Wei, B. Q. (2004). Large-scale synthesis of long double-walled carbon nanotubes. Journal of Physical Chemistry B 108(26): 8844–8847. Yamada, T., Namai, T., Hata, K., Futaba, D. N., Mizuno, K., Fan, J., Yudasaka, M., Yumura, M., and Iijima, S. (2006). Sizeselective growth of double-walled carbon nanotube forests from engineered iron catalysts. Nature Nanotechnology 1(2): 131–136.


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2 Quantum Transport in Carbon Nanotubes 2.1 2.2

Introduction ............................................................................................................................. 2-1 Electronic Structure of Carbon Nanotubes ......................................................................... 2-2

2.3

Transport in Nanotubes .........................................................................................................2-5

Band Structure of Graphene • Band Structure of Carbon Nanotubes Background • Transport Calculations • Conductance of Carbon Nanotubes • Carbon Nanotube Transistors • Functionalized Carbon Nanotube Devices • Spin Transport in Carbon Nanotubes

Kálmán Varga Vanderbilt University

2.4 Summary ................................................................................................................................. 2-11 References........................................................................................................................................... 2-11

This chapter summarizes the transport properties of carbon nanotubes. Due to their unusual electronic and structural physical properties, carbon nanotubes are promising candidates for a wide range of nanoscience and nanotechnology applications. We review the most important experimental and theoretical research exploring future carbon nanotube devices.

2.1 Introduction Carbon nanotubes (CNs) are thin hollow cylinders made entirely out of carbon atoms (see Figure 2.1). There are many types of CNs and CN-like structures. The most basic ones are multiwall nanotubes (with diameters, dt, of order ∼10 nm) and single-wall nanotubes (SWNT) (dt ∼1 nm). Multiwall carbon nanotubes were discovered by the Japanese scientist, Sumio Iijima, in 1991 [1] and, 2 years later, individual single-wall carbon nanotubes (see Figure 2.2) were reported [2,3]. Immediately after their discovery, it became clear that these tiny objects would have very remarkable electronic properties [4,5]. Still, it was not until 1997 that the first electronic transport measurements on CNs were performed [6,7], thanks largely to a new growth method developed by the group of R. Smalley, which enabled the production of large amounts of CN material. Currently, the physical properties of CNs are still being discovered and disputed. These studies are interesting and challenging due to the fact that CNs have a very broad range of electronic, thermal, and structural properties that change depending on the different kinds of nanotubes (defined by their diameter, length, chirality, and twist). The most appealing feature of CNs, for nanoelectronics, is their near-ballistic transport due to a limited carrier–phonon interaction [8]. When electrons travel through a conventional

metal-based wire, they encounter resistance as they bump into atoms, defects, and impurities. CNs can conduct electricity better than metals due to the ballistic transport. Ballistic transport refers to the motion of charge carriers driven by electric fields in a conducting or semiconducting material without scattering. It is a highly desirable phenomenon for a wide range of applications needing high currents, high speeds, and low power dissipations. Several groups have observed ballistic transport in CNs. In metallic tubes, the mean-free paths for phonon scattering have been found to be about 1 μm at low field and in the range of 10–100 nm at high field. For semiconducting tubes, values close to 300–500 nm and 10–100 nm have been obtained at low and high field, respectively [9–15]. The high electron mobilities (104–105 cm 2/V s) in CNs [9,16] indicate that semiconducting nanotubes should be an excellent material for a number of semiconductor applications, especially in high-speed transistors where mobility is crucial. The high mobility in CNs is partly due to the fact that graphite itself is a good conductor of electricity (about 20,000 cm2/V s at room temperature) and also partly to its one-dimensional structure. In addition, CNs can withstand about three orders of magnitude larger current densities [17,18] than a typical metal such as copper or aluminum, making them excellent nanomaterials for nanodevice applications. Moreover, due to their cylindrical shape, CNs may be an ideal element of a coaxial field-effect transistor, which is expected to be the ultimate geometry due to its short-channel behavior [19–21]. With the unique electrical properties of CNs, one could envision allcarbon electronics with metallic nanotube leads connected to semiconducting nanotube devices [22–24]. CNs can be synthesized by various methods [25,26]. The arc-evaporation method, which produces the best-quality 2-1

2-2


powerful laser to vaporize a metal-graphite target. This can be used to produce single-walled tubes with high yield.

(n,m) = (5,5)

2.2 Electronic Structure of Carbon Nanotubes 2.2.1 Band Structure of Graphene

(n,m) = (9,6)

(n,m) = (10,10)

FIGURE 2.1 CNs with different chiralities. See text for further explanation and the definition of the chiral indices (n,m).

Since CNs can be thought of as rolled-up ribbons of graphene sheets, one can start from the band structure of graphene to understand that of CNs. Many properties of CNs can be understood by a simple tight-binding model description [27–30]. This model is based on the assumption (besides the approximations present in the tight-binding description) that the effect of the curvature of the tube can be neglected and the band structure of the nanotube can be derived from the tight-binding band structure of a graphene sheet. This assumption is good for tubes of sufficiently large diameter, that is, ones whose diameter is much larger than the nearest-neighbor distance. A graphene sheet consists of carbon atoms arranged in a hexagonal lattice illustrated in Figure 2.3. The unit vectors in real and reciprocal space are defined as (see Figure 2.4)

B΄

y B x

θ

T A

R 100 Å

FIGURE 2.2

a1

Ch

O

a2

Transmission electron microscope image of an SWNT.

nanotubes, involves passing a current of about 50 A between two graphite electrodes in an atmosphere of helium. Th is causes the graphite to vaporize, some of it condensing on the walls of the reaction vessel and some of it on the cathode. It is the deposit on the cathode that contains the CNs. SWNT are produced when Co and Ni or some other metal is added to the anode. It has long been known that CNs can also be made by passing a carbon-containing gas, such as a hydrocarbon, over a catalyst. The catalyst consists of nano-sized particles of metal, usually Fe, Co, or Ni. These particles catalyze the breakdown of the gaseous molecules into carbon, and a tube then begins to grow with a metal particle at the tip. It was shown that SWNT can also be produced catalytically. The perfection of CNs produced in this way has generally been poorer than those made by arc-evaporation, but great improvements in this technique have been made in recent years. The big advantage of catalytic synthesis over arcevaporation is that it can be scaled up for volume production. The third important method for making CNs involves using a

FIGURE 2.3 Honeycomb lattice of graphene. See the text for the definition of the lattice vectors.

K΄ a1

y

b1 A

a2

b2 ky

x (a)

K

Γ

B

kx (b)

FIGURE 2.4 Real- and reciprocal-space lattice vectors of graphene. The shaded areas mark the unit cell and the Brillouin zone. a1 and a2 are the real-space lattice vectors, and b1 and b2 are the reciprocal-space lattice vectors.

2-3

Quantum Transport in Carbon Nanotubes

⎛ 3 a⎞ ⎛ 3 a⎞ a1 = ⎜ a, ⎟ , a 2 = ⎜ a, − ⎟ , 2 2 2 2⎠ ⎝ ⎠ ⎝

(2.1)

kya

h *⎞ . 0 ⎟⎠

+ 2e

−ikx a /2 3

q= 0

–2

1

2..............n

0

n + 1...

–4 –4

–2

0 kxa

2

4

FIGURE 2.5 (See color insert following page 20-16.) Plot of the bonding π band of graphene.

K΄

4

kya

0

K

2

0

E/t

Γ q= 0

1

2..............n

–3

n + 1...

–4 –4

–2

0 kxa

(2.4)

where t is the nearest-neighbor transfer integral. Diagonalizing the Hamiltonian, we obtain the two-dimensional energy dispersion relation of the graphene, E2D(kx, ky), as follows:

E/t

Γ

–2

⎛ ky a ⎞ ⎞ cos ⎜ , ⎝ 2 ⎟⎠ ⎟⎠

⎛ ⎛ 3 kx a ⎞ ⎛ ky a ⎞ ⎛ ky a ⎞ ⎞ E2D (k ) = ±t ⎜ 1 + 4cos ⎜ cos ⎜ + 4cos2 ⎜ ⎟ ⎟ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎟⎠ ⎠ ⎝ 2 ⎠ ⎝

0

(2.3)

Here h is the nearest-neighbor interaction between the A and the B sublattices expressed as 3

3

(2.2)

where a = 3a C-C is the lattice constant of the graphene and aC-C = 1.41 Å is the carbon–carbon bond length. The first Brillouin zone is marked by shaded areas (see Figure 2.4). Carbon has four valence electrons per atom, three of which are used to form sp2 bonds with neighboring atoms in σ orbitals. The corresponding energy bands lie far below the Fermi level and do not contribute to the electrical conduction. The transport properties are determined by the remaining π electrons, which occupy the bonding and antibonding band resulting from the superposition of the 2pz orbitals. The valence orbital is thus the π(2pz) orbital and there is no interaction between the π and the σ(2s and 2px,y) orbitals because of their different symmetries. The mixing of π and σ orbitals due to the curvature is neglected as indicated above. Since the carbon atoms in a graphene plane can be divided into two sublattices A and B (bipartite lattice, see Figure 2.4), the π bands of the two-dimensional graphene is derived from the following 2 × 2 Hamiltonian matrix [31]:

⎛ h = t ⎜ eikx a / ⎝

K

2

⎛ 2π 2π ⎞ ⎛ 2π 2π ⎞ b1 = ⎜ , , b2 = ⎜ ,− ⎟ , a⎠ ⎝ 3a a ⎟⎠ ⎝ 3a

⎛0 H=⎜ ⎝h

K΄

4

2

4

FIGURE 2.6 (See color insert following page 20-16.) Plot of the antibonding π band of graphene.

1/2

.

2

This equation describes the two bands resulting from bonding (Figure 2.5) and antibonding (Figure 2.6) states. Figure 2.7 shows a surface plot of the energy dispersion, E2D. The bonding and antibonding bands touch at six K points at the corners of the first Brillouin zone. There are two triplets of Fermi points, K and K′, that are inequivalent under translations of the reciprocal lattice. On the other hand, the hexagonal lattice symmetry provides an equivalence under 60° rotations. Because of this symmetry, the K and K′ points are energetically degenerate and lead to the peculiar touching of the conduction and the valence band. At zero temperature, the bonding bands are completely

E/t

(2.5) 0

–2

–4

–2

0 kxa

2

4

–4

–2

0

2 a ky

4

FIGURE 2.7 (See color insert following page 20-16.) Band structure of graphene. The figure shows the various energy bands in the Brillouin zone.

2-4


fi lled and the antibonding bands are empty. Thus, undoped graphene is a zero-gap semiconductor. In practice, there is always some doping shift ing the Fermi energy and leading to a small density of states (DOS). This band structure (see Figures 2.5 through 2.7), which determines how electrons scatter from the atoms in the crystal lattice, is quite unusual. It is not like that of a metal, which has many states that freely propagate through the crystal at the Fermi energy. This is not the band structure of a semiconductor either; there is no energy gap with no electronic states near the Fermi energy. The band structure of graphene is instead somewhere in between these two extremes. In most directions, electrons moving at the Fermi energy are backscattered by atoms in the lattice, which gives the material an energy band gap like that of a semiconductor. However, in other directions, the electrons that scatter from different atoms in the lattice interfere destructively, which suppresses the backscattering and leads to metallic behavior. Graphene is therefore called a semimetal, since it is metallic in special directions and semiconducting in the others. Looking more closely at Figure 2.7, the band structure of the low-energy states appears to be a series of cones. At low energies, graphene resembles a two-dimensional world populated by massless fermions [32]. In passing we note that, although this model has become very popular due to its simplicity, it only gives good description close to the K point of the Brillouin zone. First-principles and more elaborated tight-binding calculations have to be used to obtain better dispersion relations [33].

The translation vector, T, is parallel to the axis of the tube (see Figure 2.3) T = t1a1 + t 2a 2 t1 =

2m + n 2n + m t2 = − , dr dr

(2.9)

where ⎧⎪d dr = ⎨ ⎪⎩3d

if n − m is not multiple of 3d d = gcd(n, m) otherwise

(2.10)

(gcd is the greatest common divisor). 3L . Using T and Ch, we can define The length of T is | T | = dr the unit cell of the CN with an area given, |Ch × T|, and the number of atoms in a unit cell is N=

| C h × T | 2L2 = | a1 × a 2 | a2dr

(2.11)

The reciprocal lattice vectors corresponding to Ch and T are K1 =

1 (−t2 b1 + t1b2 ) K 2 = N1 (mb1 − nb2 ). N

(2.12)

Now, by zone folding of the two-dimensional dispersion relation of the graphene, the dispersion relation of the CN in the tightbinding approximation is given by

2.2.2 Band Structure of Carbon Nanotubes The geometry of a CN is described by a wrapping vector. The wrapping vector encircles the waist of a CN so that the tip of the vector meets its own tail (see Figure 2.3). The wrapping vector can be any C h = na1 + ma 2 ,

(2.6)

where n and m are integers a1 and a2 are the unit vectors of the graphene lattice (see Figures 2.3 and 2.4) The angle between the wrapping vector and the lattice vector a1 is called the chiral angle. The pair of indexes (n,m) identifies the nanotube, and each (n,m) pair corresponds to diameter dt: dt =

L π

L = | C h | = a n2 + m2 + nm

(2.7)

and to a specific chiral angle, θ, (the angle between a1 and Ch) ⎛ 3m ⎞ θ = arctan ⎜ ⎟. ⎝ m + 2n ⎠

(2.8)

⎛ K ⎞ Enanotube = E2D ⎜ k 1 + μK 2 ⎟ , ⎝ | K1 | ⎠

(2.13)

where k is a continuous variable π π − 0 for a repulsive Coulomb interaction leads to g < 1. The stronger the interactions V0, the smaller the value of g. Note that g can also be greater than 1 if attractive Coulomb interactions are dominant among the particles. The TLL parameter g emerges in various 1D properties such as the fractional charge ge, the charge mode velocity v F/g, and the power-exponents of correlation functions. As one specific example of 1D conductors, metallic SWNTs have been predicted as the TLL system (Egger and Gogolin 1997, Kane et al. 1997). The transport properties in the tunneling regime, where tubes are isolated from metal reservoirs, exhibited the TLL features as the power-scaling conductance by means of the bias voltage and the temperatures (Bockrath et al. 1999). This nonlinear behavior certainly cannot be explained by a noninteracting single-particle picture. Recently, the spectral function from SWNT mats was obtained from angle-integrated photoemission measurements, which was claimed as the direct observation of the TLL features in SWNTs (Ishii et al. 2003). −

1

⎛ 8e 2 ⎛ R ⎞⎞ 2 The TLL parameter g for the SWNT is g = ⎜ 1 + ln ⎜ S ⎟ ⎟ , πvF ⎝ R ⎠ ⎠ ⎝ where RS is the screening length and R is the radius of the SWNT (Kane et al. 1997). The logarithmic dependence on RS/R explains that the value of g is rather insensitive to the actual value of RS (Kane et al. 1997), and the value falls between 0.2 and 0.3 with vF = 8 × 107 cm/s as RS/R > 4. The search of the TLL behavior in SWNTs is actively pursued since the strongly correlated SWNTs serve as a basic ingredient of quantum electron entanglers (Bena et al. 2002, Recher and Loss 2002, Bouchiat et al. 2003, Crépieux et al. 2003).

3.3 Electron Transport in Carbon Nanotubes 3.3.1 Synthesis and Device Fabrication 3.3.1.1 Synthesis The discovery of MWNTs seems fortuitous in a carbon arcdischarge chamber that was designed to produce fullerenes

(Iijima 1991). Two years later, SWNTs were found by the same arc-discharge method except that catalytic components had been added into the chamber (Iijima and Ichihashi 1993). For a systematic characterization of new materials toward functional device fabrication and quantum nature investigation, efficient synthesis methods have been on demand that aim to isolate individual nanotubes and grow specific types of SWNTs in a controllable way. There are three major synthetic methods for growing SWNTs using catalytic nanoparticles: electric arc-discharge, laser ablation, and chemical vapor deposition (CVD). Among them, the CVD method has been superior in producing high-quality SWNTs. Typically, the CVD chamber for growing SWNTs in the laboratory consists of a 1 in. diameter tube vessel inserted into a furnace, gas sources of CH4, H2 and Ar, a Si-substrate containing catalyst islands, and an exhaust system. Catalysts are essential for designating the location of SWNTs during growth. In 1999, Professor Dai’s group managed to synthesize a high-yield of SWNTs near the catalyst islands. Iron-based alumina-supported catalysts were under a carbon feedstock: 99.999% CH4 and H2 at the right concentration for 5–7 min at 900°C–1000°C followed by an Ar flush and a cool-down to room temperature (Kong et al. 1998). This work has advanced the SWNT research field in which ballistic transport studies could be performed and prototypes of nanotube-electronics can be built. The synthesis mechanism of SWNTs in the catalytic CVD method is associated with the details of nanoparticles. Recently, Li et al. (2001) have attempted to assess the role of catalysts and have shown that the diameter of SWNTs indeed closely links to the nanoparticle size based on statistical analysis. The report presents that the synthesis can be understood in three stages: First, nanoparticles as catalysts absorb decomposed carbon atoms from CH4 or other carbon feedstock in the CVD process. Second, the absorption of carbon atoms to nanoparticles continues until saturation. Once it reaches the saturation point, carbon atoms start to grow outward from the catalysts with a closed-end. Third, an excess carbon supply adds to the carbon precipitation on the surface, yielding finite-length nanotubes in the end. It is reasonable, therefore, that the SWNT diameter would be determined by the nanoparticle size as the initial basis. Although Li et al. (2001) provided valuable information as to the microscopic level of understanding of the synthesis in the catalytic CVD process, the complete controllability to produce tailor-made SWNTs with an expected diameter, chirality (roll-up vector direction), length, position, and orientation on demand is yet to be acquired, which is the present SWNT fabrication challenge. Once this goal is achieved, it is not difficult to imagine that SWNTs would become widely utilized in various applications as electrical, chemical, mechanical, and optical components. 3.3.1.2 Device Fabrication The configuration of SWNT devices for electron transport measurements resembles conventional semiconductor field-effect transistors, which have three terminals: source, drain, and gate. The fabrication goal is to produce three-terminal isolated SWNT nanotube devices on top of a Si-wafer. Figure 3.4 shows

3-7

Electron Transport in Carbon Nanotubes

PMMA SiO2 Si Global/chip alignment marks

PMMA SiO2 Si Ti/Au deposition

Thermal oxidation

E-beam for catalyst islands

SiO2 Si

PMMA SiO2 Si E-beam for electrodes

SWNT synthesis

SiO2 Si Lift-off 100 μm

100 μm

FIGURE 3.4 Schematics of the SWNT device fabrication processes. Two right bottom photographs show multiple devices in a single chip and an individual SWNT device. (From Bockrath, M. et al., Nature, 397, 598, 1999. With permission.)

the steps of conventional fabrication processes. Combinations of standard photolithography and electron-beam lithography (EBL) techniques enable one to produce about one hundred devices in a 1 cm × 1 cm Si substrate chip during a short period of time. The starting material is a 4 in. heavily doped Si-wafer, which serves as a backgate to control the electron density or the Fermi energy of carbon nanotubes. First, global wafer marks and chip marks should be patterned in the blank wafer. These marks are very useful primarily in that the overlapping processes can be performed within the lithographic resolution limit and also in that they draw boundaries of chips in the whole wafer along which the wafer can be cleaved into each chip for further steps. Those alignment marks are patterned by a photolithography recipe: photoresist spin-coating followed by exposure and development, etching process, and removal of the resist. The second step is the thermal oxidation on top of the marked Si-wafer. It is a very critical step to avoid any possible impurities on wafers during this process, since any dirt on the wafers would lead to a current leakage when devices are biased. Therefore, before inserting the wafer to a diff usion furnace, the wafer should be cleaned thoroughly and properly through the diff usion wet bench process. The next task is to pattern catalyst islands at intended locations using the EBL method consisting of polymethylmethacrylate EBL resist coating, exposure, and development. Nanotubes are then synthesized by the aforementioned CVD method with methane and hydrogen gas. The second EBL is processed for

patterning metal electrodes followed by thin metal deposition and a lift-off in acetone. Once devices are prepared, at room temperature several preliminary characterizations are performed. Atomic force microscopy imaging measurements give the number of SWNTs between the electrodes and differentiate SWNTs from MWNTs based on the tube height. Nanotubes, whose diameters are 1.5–3.5 nm from atomic force microscopy (AFM) images, are presumably considered to be SWNTs according to the statistical analysis at a given recipe. AFM images cannot identify a SWNT, MWNT, or a doublewalled nanotube with certainty unlike transmission electron microscopy (TEM). Since TEM requires conducting substrates, transport devices on top of Si-substrates are not quite adequate to TEM at this time. Thus, the determination of SWNTs or MWNTs relies heavily on the statistics of TEM results with synthesized nanotubes on conducting substrates by the same growth recipe. The current (I) vs. drain-source voltage (Vds) characteristics are measured by varying gate voltages (Vg). This room-temperature electrical characterization has a particular purpose to tell metallic tubes from semiconducting tubes. Metallic tubes have weak or no dependence on Vg in principle since there are always free electrons no matter where the Fermi energy lies. On the other hand, semiconducting tubes, where there is an energy gap between conduction and valence bands, exhibit strong Vg dependent I–V characteristics. When the Fermi energy lies in the energy gap regime, current values through tubes are suppressed to zero. In this way, no or very weak −Vg-dependent tubes can be selected,

3-8


and room-temperature characterization has also another purpose to identify the best and good ohmic contacted devices for low-temperature measurements. The subsequent two sections describe low-temperature electron transport properties: differential conductance and low-frequency shot noise. The main discussion focuses on three-terminal metallic SWNT devices that are well contacted to electrodes. The device dimension is fi xed between 200 and 600 nm by a distance between two electrodes. The Ti/Au, Ti-only, and Pd metal electrodes are used, which feature low-resistance contacts. Metallic SWNTs are considered to have both the elastic and the inelastic mean free path at least on the order of microns at low temperatures. Therefore, the electron transport within 200–600 nm-long SWNTs is believed to be ballistic, where quantum coherence is preserved inside (Kong et al. 2001, Liang et al. 2001).

3.3.2 Differential Conductance At low temperatures, an interference pattern in differential conductance was observed in well-contacted SWNTs to Ti/Au metals with finite reflection coefficients, as shown in Figure 3.5. This diamond interference pattern arises from quantum coherence along a finite SWNT length (longitudinal confi nement) due to the potential barriers at the interfaces with two metal electrodes. The spatial confinement quantizes energy levels and the energy spacing between maxima corresponds to ΔE = v F/L, where L is the SWNT length. The inset in Figure 3.5 shows that the diamond structure size is inversely proportional to L. Liang et al. modeled this system as an electronic analog FabryPerot (FP) cavity as a two-channel double-barrier problem using the Landauer–Büttiker formalism within the context of the FL theory. They captured the wave nature of electrons through an

isolated nanotube as an electron waveguide. Two interfaces at metal and tubes have a one-to-one correspondence with partially reflecting mirrors in the FP interferometer (Liang et al. 2001). Similar to photons in the FP cavity, electrons would experience multiple reflections between two barriers separating the metal reservoirs from the SWNT before escaping. The approach is to establish three 4 × 4 scattering matrices at the left and right interfaces and inside the tube. The scattering matrices contain energy-dependent components, satisfying unitary property by Born approximation (Liang et al. 2001). In metallic infi nite SWNTs, forward scatterings are dominant in comparison to backscattering and interbranch scattering. Backscattering and interbranch scattering require a big momentum transfer of 2kF between two K points in the Brillouin zone and satisfy the symmetry selection rules between two orthogonal π and π* bondings among pz-orbitals. Thus, in principle, such scattering processes inside tubes are prohibited. However, backscattering can occur at the interfaces between metal electrodes and the tube that increases the overall resistance. Meanwhile, inside the ballistic tube, the phase is accumulated over multiple reflections. The differential conductance is calculated by a simple expression, dI 2e 2 = dV h

2

∑ Tr(S *S) , T

i =1

taking a trace of the total scattering matrix, S. Indeed, this noninteracting theoretical model reproduces quantum interference diamond patterns within a ±10 mV bias voltage range, supporting that the observation of quantum interference is the evidence of the ballistic transport. Note that the diamond structure is persistent regardless of the bias voltage window in this FL model.

2.9 dI/dV (e2/h)

V (mV)

5 0 –5 1.6 –2 (a)

2

0 Vg (V)

3.3

0 –5 –2

(b)

0 Vg (V)

dI/dV (e2/h)

Vc (mV)

V (mV)

5 8 0

0 L–1 (μm–1) 5

2.9

2

FIGURE 3.5 Two-dimensional image plot of differential conductance (dI/dV) as a function of drain-source voltage (V) and gate voltage (Vg). dI/dV is renormalized by e2/h. There is a clear interference pattern whose feature is dependent on the SWNT length, (a) a 530 nm long SWNT device and (b) a 220 nm long SWNT device. (Reprinted from Liang, W. et al., Nature, 411, 665, 2001.)

3-9

Electron Transport in Carbon Nanotubes 40

20

20 10 Vds (mV)

0.40 0.35 0.30 0.25 0.20 0.15 0.10

0

0

–10 –20

–20 –10

–9

–8

–7

–6

Vg (V) –40 –9.0 –8.0 –7.0 Vg (V)

FIGURE 3.6 Density plot of differential conductance in units of 2G Q against two voltages, Vds and Vg. (From Liang, W., Nature, 411, 665, 2001. With permission.)

However, FP quantum interference pattern fringe contrast becomes reduced in magnitude at high drain-source voltages (Vds). Figure 3.6 clearly exhibits that the diamond structure disappears as Vds increases above 20 mV. This feature cannot be explained by the standard FL theory, which predicts constant oscillation amplitude regardless of the bias voltage values. An alternative attempt to explain this experimental data is to model the SWNT device within the TLL theory. It is motivated by the fact that SWNTs have exhibited the features of strong correlations among charge carriers in experiments (Bockrath et al. 1999) and in theories (Egger and Gogolin 1997, Kane et al. 1997, Peça et al. 2003) owing to intrinsic many-body interactions in 1D systems. The device consisting of a ballistic SWNT and two metal electrodes is theoretically simplified as one infinite 1D conductor with two different values of the TLL parameter g for the SWNT and metal electrodes. The interaction is assumed to be strong in

the SWNT (0 < g < 1) and weak in the higher dimensional metal reservoirs (g = 1) for metals (Peça et al. 2003, Recher et al. 2006). The four conducting transverse channels of the SWNTs in the FL theory are transformed to four collective excitations in the TLL theory: one interacting collective mode of the total charge and three neutral noninteracting collective modes including spin. There are two distinct propagating velocities, vc = v F/g (the total charge mode) and v F (the rest of three modes). The inter-channel and intra-channel scattering processes are assumed to be allowed to reflect channels partially only at the two barriers. The application of nonzero bias voltages is treated within Keldysh formalism, a powerful method to study nonequilibrium many-body condensed-matter systems in terms of Green’s functions (Mahan 2007). The transport properties are computed from correlation and retarded Green’s functions (Recher et al. 2006). Three noninteracting modes encounter backscattering at the physical barrier, whereas the interacting mode encounters the momentum-conserving backscattering due to g mismatch at the interfaces in addition to the physical barrier backscattering. It was found that one interacting mode exhibits the power-law behavior, whereas three noninteracting modes still show oscillatory behavior. The overall behavior combining two effects more closely resembles the real experiments. In Figure 3.7, experimental data (left) are compared with theoretical graphs (right) at three Vg voltages (blue, green, red). Note that the tendency of amplitude reduction in experimental data cannot be reproduced by the reservoir heating model (Henny et al. 1999), which asserts that the dissipated power Vds2(dI/dVds) leads to a bias-voltagedependent electron temperature (Liang et al. 2001). Figure 3.7, furthermore, presents the following pronounced features: the period of the oscillations at low Vds depends on the value of Vg, and it becomes elongated at high Vds. The TLL model suggests that the elongated period appears when there is nonzero contribution to the overall oscillation by only interacting mode (propagating at a slower group velocity vc) while those from the three noninteracting modes (propagating at Fermi velocity v F) are completely canceled by destructive interference. This Vg-dependent oscillation period in differential conductance has been interpreted as a signature of spin-charge separation 1.00

0.38

0.95 0.90 dI/dVds

dI/dVds

0.36 0.34

0.85 0.80

0.32 0.75 0.30 –20 (a)

0.70 –10

0 Vds (mV)

10

20

–20 (b)

–10

0 Vds (mV)

10

20

FIGURE 3.7 (a) Experimental and (b) theoretical differential conductance traces at Vg = −9 V (diamond in (a), line with peak at V = 0 V in (b)), −8.3 V (triangle in (a), line without oscillation in (b)), and −7.7 V (circle in (a), line with dip at V = 0 V in (b)) at 4 K.

3-10


in the SWNT (Peça et al. 2003). A comparison of the primary periods at different Vg values yields the TLL parameter g ∼ 0.22, which is consistent with the predicted g values in SWNT. Although it seems to be compelling evidence, further experiments focusing on the periodicity with Vds should be performed to be conclusive.

3.3.3 Low-Frequency Shot Noise The low-frequency shot noise probes the second-order temporal correlation of electron current in the nonequilibrium condition. It often manifests certain microscopic physical mechanisms of the conduction process. When Poisson statistics govern the emission of electrons from a reservoir electrode, in other words, the propagation of an electron has no relationship with the previous or the successive electron, the spectral density of the current fluctuations reaches its full shot noise spectral density, SI = 2eI, where I is the average current. In a mesoscopic conductor, nonequilibrium shot noise occurs due to the random partitioning of electrons by a scatterer, and it may be further modified as a consequence of the quantum statistics and interactions among charged carriers (Blanter and Büttiker 2000). A conventional measure for characterizing the shot noise level in mesoscopic conductors is the Fano factor F ≡ SI,m/2eI, the ratio of the measured noise power spectral density SI,m to the full shot noise value (2eI). In statistics, the Fano factor is related to be a variance-to-mean ratio, so a Poisson process whose variance equals to mean yields F = 1. Despite growing interest in the shot noise properties of the TLLs, current noise measurements in nanotubes have only recently been executed due to the difficulty of achieving highly transparent ohmic contacts and a high signal-to-noise ratio between the weak excess-noise signal and the prevalent background noise (Roche et al. 2002), although the shot noise properties of SWNTs in the tunneling (strong barriers between metals and SWNTs) regime with the TLL features have been reported recently (Onac et al. 2006).

In the ballistic regime, the shot noise properties reveal the TLL features in the SWNT device unambiguously. Two-terminal shot noise measurements were implemented at 4 K. It is critical to calibrate an experimental apparatus to reach an accurate data acquisition and analysis. For this purpose, two current noise sources are placed in parallel: a SWNT device and a full shot noise generator. The well-characterized full shot noise source quantifies complicated cryogenic transfer functions and filters. The standard of such sources is a weakly coupled light emitting diode (LED) and photodiode (PD) pair. At 4 K, the overall coupling efficiency from the LED input current to the PD output current was about 0.1%, which eliminated completely the shot noise squeezing effect due to the constant current operation (Kim and Yamamoto 1997). In order to recover the weak shot noise of the SWNT embedded in the background thermal noise, an AC modulation lock-in technique is implemented and a resonant tank-circuit together with a home-built cryogenic low-noise preamplifier is incorporated (Reznikov et al. 1995, Liu et al. 1998, Oliver et al. 1999). The specific shot noise measurement techniques for the SWNT devices are described in detail by Kim et al. (2007). Figure 3.8a presents a typical log–log plot (base 10) of SSWNT in Vds at a particular Vg. SSWNT (dot) is clearly suppressed to values below full shot noise SPD (triangle), and it suggests that the relevant backscattering for shot noise is indeed weak. Note that SSWNT and SPD have clearly different scaling slopes vs. Vds. The deviation from the full shot noise in the SWNT device is beyond the scope of the FL theory. Hence, the previous TLL theory for differential conductance is extended to compute the low-frequency shot noise spectral density, SSWNT = dteiωt {δIˆ(t ), δIˆ(0)} with

∫

δÎ(t) = Î(t) − Ī, the current fluctuation operator, and the anticommutator relation {δÎ(t),δÎ(0)} = δÎ(t)δÎ(0) + δÎ(0)δÎ(t) (Recher et al. 2006). The SWNT noise in the zero-frequency limit is expressed as

(

SSWNT = 2e coth eVds

2.5

2kBT

)

I B + 4kBT (dI /dVds − dI B /dVds )

–0.4

log (fano factor)

log (SPD, SSWNT)

–0.5 2.0 1.5 1.0

–0.6 –0.7 –0.8 –0.9 –1.0 –1.1

0.5 0.0 (a)

0.2

0.4

0.6 0.8 1.0 log (Vds)

1.2

1.4

0.0

1.6 (b)

0.5 log (Vds)

1.0

1.5

FIGURE 3.8 (a) Shot noise power spectral density vs. Vds for the LED/PD pair (SPD, triangle) and the SWNT (S SWNT, dot) at Vg = −7.9 V. The slopes of SPD, S SWNT are 1 and 0.64, respectively. The inferred g value for the SWNT is 0.16. (b) Fano factor vs. Vds on a log–log scale. The theoretical Fano factor curves where thermal noise 4k BT(dI/dVds) is subtracted are drawn for g = 1 (dotted line) and g = 0.25 (straight line) at T = 4 K. The power exponent α is −0.35 for the measured Fano factor (diamond) at Vg = −7.9 V., and the inferred g value is 0.18. The theoretical g = 1 (dotted line) plot gives α ∼ 0 as expected.

3-11


where IB is the backscattered current. In the limit of eVds > kBT, the SWNT noise spectral power density becomes simplified as SSWNT = 2eIB. The asymptotic behavior of IB follows the powerlaw scaling I B ~ Vds1+α with α = −(1/2) (1 − g)/(1 + g). The power exponent α is uniquely determined by the TLL parameter g. The experimental Fano factor F(Vds) is displayed on a log–log (base 10) scale in Figure 3.8b. The TLL model predicts that at low bias voltages eVds < kBT < /2gt F, experimental Fano factor Fexp is proportional to Vds if we subtract the thermal noise component. In addition, the slope between Fexp and Vds is insensitive to g-values in the region of log(Vds) < log (/2gtF) ∼ 0.47. On the other hand, if eVds > /2gt F, a power-law F ∼ Vdsα is expected by assumption that the backscattered current is smaller than the ideal current 2GQVds. A linear regression analysis of the Fano factor F with Vds in this region, therefore, is another means to obtain the g value. The Fano factors F for g = 0.25 (red) and g = 1 (yellow) are displayed on a log–log scale in Figure 3.8b. The experimental data (diamonds) agree well with the theoretical Fano factor of g = 0.25. The stiffer slope (α) corresponds to a stronger electron– electron interaction. The measured exponent α and inferred g values from the spectral density and the Fano factor from four different devices with various metal electrodes (Ti/Au, Ti-only, Pd) show similar statistics of α ∼ −0.31 ± 0.047 and g ∼ 0.26 ± 0.071 as derived from several Vg values for each sample. Manybody TLL behavior in the ballistic SWNT is clearly probed in the shot noise properties.

from the conductance period at various Vg and power-law scaling exponents from both shot noise and the Fano factor. The search of many-body collective phenomena in SWNT devices would provide fundamental physical knowledge in 1D electron transport properties.

3.4 Summary

References

The electron transport properties of SWNTs are discussed based on experimental results and theoretical models. The discussion started from the discovery of carbon nanotubes and the history of their research field with a focus on the electron transport area. The noninteracting FL and interacting TLL theories, which have been used widely for bulk and 1D systems respectively, are briefly introduced as background knowledge. The ballistic transport regime is clearly defined, in which no inelastic and elastic scattering occurs, consequently quantum coherence is preserved. A metallic SWNT is a model system to investigate ballistic transport properties in 1D owing to quantum many-body interactions. The differential conductance and the shot noise have been measured, and their experimental signatures are examined by the FL and the TLL theoretical frameworks. It is clear that quantum coherent properties manifest as an interference pattern in differential conductance, and unique power-law scaling behavior quantifies correlations among charge carriers. A noninteracting picture may describe conductance data in the low bias regime; however, it fails to explain the high-energy regions of experimental data. The TLL theory has explained the qualitative trend of conductance as a function of the drain-source voltage. In addition, it has captured the quantitative information of the strong electron–electron interactions both in conductance and the shot noise quantities. The strength of the interactions is parameterized by the TLL parameter, g, which has been obtained

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3-12

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4 Thermal Conductance of Carbon Nanotubes 4.1

Introduction .............................................................................................................................4-1 Fundamentals of Thermal Conduction in a Solid • Effects of Size Confi nement on Heat Conduction

4.2

Theory of Thermal Conduction in Carbon Nanotubes .....................................................4-3 Structure of Carbon Nanotubes • Electronic Thermal Conductance of Carbon Nanotubes in the Ballistic Regime • Phonon Transport in Carbon Nanotubes

4.3

Measurements of Thermal Conductance of Carbon Nanotubes .....................................4-5 Thermal Measurement of Carbon Nanotube Bundles • Direct Thermal Conductance Measurement of Individual Carbon Nanotubes • Self-Heating Measurement of Thermal Transport in Carbon Nanotubes • Optical Measurement of Thermal Transport in Carbon Nanotubes

Li Shi The University of Texas at Austin

4.4 Conclusion and Outlook.......................................................................................................4-12 Acknowledgments .............................................................................................................................4-12 References...........................................................................................................................................4-12

4.1 Introduction

conductivity (κ), cross-sectional area (A), and length (L) of the solid bar according to

Carbon nanotubes are a class of unique nanostructures that are being explored for applications in nanoelectronics, interconnects, sensors, biomedicine, and energy applications. They also provide an ideal low-dimensional system for investigating nanoscale thermal transport physics that often dictates the performance and reliability of functional devices made of carbon nanotubes. For example, the current-carrying capability of nanotubes depends on the temperature rise during self-heating, which is in turn limited by the thermal property of nanotubes. This chapter starts with a review of the fundamentals of thermal conduction in solids and a discussion of the theoretical descriptions of the unique features of thermal conduction in carbon nanotubes.

4.1.1 Fundamentals of Thermal Conduction in a Solid The rate of heat conduction (Q) along a solid bar can be obtained by using the phenomenological Fourier’s law Q = G ΔT

(4.1)

where G and ΔT are the thermal conductance and the temperature difference between the two ends of the solid bar, respectively. The thermal conductance can be calculated from the thermal

G=

κA L

(4.2)

The thermal conductance and thermal conductivity consist of contributions from different energy carriers including electrons and phonons, the latter of which are the energy quanta of crystal vibration waves, i.e., G = G e + G ph

(4.3a)

κ = κe + κ ph

(4.3b)

and

where the subscripts e and ph denote the electron contribution and phonon contribution, respectively.

4.1.2 Effects of Size Confi nement on Heat Conduction Both thermal conductance and thermal conductivity depend on the crystal structure and the temperature. While the thermal conductance depends on the sample dimension, the thermal conductivity is independent of the sample size until the size is 4-1

4-2


scaled down to the fundamental length scales such as the wavelengths and scattering mean free paths of electrons and phonons, which are both waves and particles. The electron wavelength can be calculated according to λe =

h 2m* E

(4.4)

The as-calculated electron wavelength varies from picometer (pm) for high energy free electrons to the order of 100 nm for electrons in some semimetals and semiconductors such as bismuth and indium antimonide that possess a very small effective mass. In comparison, the phonon wavelength (λph) can range in size from as small as two times the primitive unit cell size (a) of the lattice to as large as the size of a crystal. At low temperatures (T), the dominant wavelength (λph,max) of phonons that carries the maximum spectral energy density is given by an expression that resembles Wien’s displacement law for photons, i.e., hv g 2.8kB

(4.5)

where vg is the phonon group velocity kB is Boltzmann’s constant For a group velocity of about 3000 m/s, Equation 4.5 is reduced to λph,maxT ≈ 50 nm-K. Equation 4.5 fails at sufficiently high temperatures, where λph,max approaches the minimum allowable wavelength on the order of 2a, corresponding to the edge of the first Brillouin zone of the reciprocal lattice. As particles, both electrons and phonons are scattered by electrons and phonons, crystal defects, and boundaries. The scattering mean free path (l) is the average distance that electrons or phonons can travel before being scattered, and it ranges from atomic spacing in amorphous materials to over micrometers in high-quality crystals at low temperatures. In the diff usive transport limit where the length scale is much longer than the mean free path, the thermal conductivity contribution of each energy carrier type can be calculated using a simple result from the kinetic theory (Chen 2005) κ ≈ Cv x2 τ

−1 l −1 = lbulk + ls−1

(4.7)

where ls is the surface scattering mean free path, which can be estimated with the use of the specularity parameter (p) of the surface according to (Ziman 1962, Dames and Chen 2004, Moore et al., 2008)

where h is the Planck’s constant m* is the electron effective mass E is the electron energy

λ ph ,maxT ≈

When the cross-sectional dimension (d) of the solid bar is reduced to the order of mean free paths, the scattering mean free path can be modified from the bulk value (l bulk). According to the Matthiessen’s rule,

(4.6)

where C is the specific heat vx is the average velocity component of the energy carrier along the transport direction τ is the scattering mean free time

ls =

1+ p d 1− p

(4.8)

Depending on the relative magnitude of the surface roughness and the wavelength of the energy carrier, the specularity parameter of a surface with random surface roughness ranges between 0 and 1. If the surface is smooth and specular, p = 1 and ls approaches infinity. Specular surface scattering does not alter the mean free path or thermal conductivity. If the surface is rough and diff use, p = 0 and ls approaches d. Diff use surface scattering can greatly suppress the mean free path and the thermal conductivity. When the length of the solid bar is reduced to be shorter than the scattering mean free path, energy carriers can travel from one end to the other end in a ballistic manner without encountering scattering along the path. In the ballistic thermal transport regime, the thermal conductance is finite because the energy carriers are scattered at the two end contacts when they enter or exit the short solid bars from or into the two thermal reservoirs at the two ends. The maximum but finite thermal conductance found in the ballistic limit is the ballistic thermal conductance, which is in effect the reciprocal of the minimum thermal resistance achievable at the two contacts to the solid bar. In the ballistic transport regime, the thermal conductivity is not well defined because thermal conductivity is a property inherently associated with diff usive transport and is based on the assumption of a local thermal equilibrium among the energy carriers via scattering. Nevertheless, an effective thermal conductivity can still be calculated from the ballistic thermal conductance according to Equation 4.2. Apparently, the as-calculated thermal conductivity is length dependent in the ballistic transport regime because the ballistic thermal conductance is the contact thermal conductance that is length independent to the first order. When the lateral dimension (d) of the solid is reduced to be comparable with the carrier wavelength of the energy carriers, only few discrete transverse wavevector states can be supported in the solid that acts as a waveguide of the energy carrier. On the other hand, the longitudinal wavevector states can be quasi-continuous if the length of the solid is much longer than the carrier wavelength. Consequently, thermal energy is carried by few onedimensional (1D) waveguide modes that result in well-separated energy subbands in the solid. The energy quantization in the transverse direction modifies the electronic band structure or

4-3

Thermal Conductance of Carbon Nanotubes

phonon dispersion. Consequently, the density of states of electrons or phonons as well as the group velocity can be different from those in the bulk, leading to modified thermal property in the quantum transport regime.

4.2 Theory of Thermal Conduction in Carbon Nanotubes 4.2.1 Structure of Carbon Nanotubes Carbon nanotubes are made of graphitic cylinders (Dresselhaus et al., 2001). One distinguishes between multi-walled carbon nanotubes (MWNTs), consisting of a series of coaxial graphite cylinders, and single-walled carbon nanotubes (SWNTs) with a one atom thick wall, usually a small number (20–40) of carbon atoms along the circumference, and microns along the cylinder axis. The nanotube is specified by the chiral vector Ch C h = na1 + ma 2 ≡ (n, m)

(4.9)

which is often described by the pair of indices (n, m) that denote the number of unit vectors na1 and ma2 in the hexagonal honeycomb lattice contained in the vector Ch. The chiral vector Ch makes an angle θ, called the chiral angle, with the zigzag or a1 direction. A nanotube with m = 0 is called a zigzag tube; while one with n = m is called an armchair tube. Other nanotubes are called chiral tubes. The axis of the zigzag nanotube corresponds to θ = 0°, while the axis of an armchair nanotube corresponds to θ = 30°, and the general chiral nanotube axis corresponds to 0° ≤ θ ≤ 30°. The diameter of the SWNT depends on the indices n and m and is usually in the range of 1–2 nm. For comparison, a MWNT has a typical diameter E

E

Ef

Ef

k (a)

k (b)

FIGURE 4.1 The electron subbands near Fermi energy for a (a) metallic and (b) semiconducting SWNT. Contact 1

Contact 2

of about 10 nm and consists of a few tens of graphitic layers with an inter-layer spacing of about 0.34 nm.

4.2.2 Electronic Thermal Conductance of Carbon Nanotubes in the Ballistic Regime When a graphene sheet is rolled up into a tube, the allowed electron wavevector components perpendicular to the tube axis become quantized, resulting in 1D subbands in the electronic band structure of a SWNT (Dresselhaus et al., 2001). The energy (E) band structure in the wavevector (k) space for the subband near the Fermi energy Ef is shown in Figure 4.1 for a metallic and a semiconducting nanotube, with the dark lines representing filled electron states. The subband structures depend on the indices n and m and are different for different tubes, resulting in both metallic SWNTs with a vanishing bandgap and semiconducting SWNTs. It has been shown that the metallic bandstructure in a (n, m) SWNT is achieved when 2n + m = 3q

where q is an integer (Dresselhaus and Eklund 2000). All armchair carbon nanotubes with n = m are metallic, satisfying Equation 4.10. While SWNTs can either be metals or semiconductors, MWNTs are mostly metallic. At room temperature, electrons near the Fermi level in a metallic SWNT can have a long electron mean free path (Bockrath et al., 1997, Ando and Nakanishi 1998, Ando et al., 1998, Tans et al., 1998, McEuen et al., 1999, Javey et al., 2003, Park et al., 2004, Gao et al., 2005, Purewal et al., 2007). If the length of a SWNT is shorter than the mean free path, electrons can flow from one end of the tube to the other end without scattering with phonons, defects, and boundaries. The ballistic electron transport phenomenon is illustrated in Figure 4.2a. On the other hand, transport measurements of MWNTs (Collins et al., 2001b) and SWNTs (Yao et al., 2000, Collins et al., 2001a, Park et al., 2004, Pop et al., 2005, Foa Torres and Roche 2006, Lazzeri and Mauri 2006, Sundqvist et al., 2007) under high bias voltages have suggested the scattering of the energetic electrons by optical and zone boundary phonons, giving rise to diff usive transport, as illustrated in Figure 4.2b. The unique transport property of electrons near the Fermi level in a metallic SWNT arises from its unusual lattice structure. Contact 1

FIGURE 4.2

Contact 2 Diffusive conductor

Ballistic conductor

(a)

(4.10)

(b)

Electron trajectory in a (a) ballistic and (b) diff usive conductor between two contacts.

4-4


The SWNT has a periodic structure in the circumferential direction with an atomically smooth surface, resulting in the absence of diff use electron-boundary scattering in the bulk of the tube. Furthermore, although the metallic and semiconducting SWNTs are nearly identical in structure with the amount of disorder likely very similar, a long-length scale disorder breaks the semiconducting tubes into a series of dots with large barriers and a dramatically reduced conductance; whereas, metallic tubes are insensitive to disorder and remain near-perfect 1D conductors (McEuen et al., 1999). Ballistic electron transport in a short metallic SWNT at a low electric field gives rise to finite electrical conductance or resistance. For the perfect ballistic conductor shown in Figure 4.2a, the current is carried in the contacts by infinitely many modes, but inside the size-confined ballistic conductors, the current is carried by only a few 1D energy subbands due to quantization. This requires a redistribution of the current among the current-carrying modes at the contacts, leading to contact resistance. According to the Landauer formula, for a conductor with currents carried by 1D subbands, the electric current can be calculated as (Datta 1997) ∞

I=

∑ 2∫ 2π ev (k)[ f (E (k), μ ) − f (E (k), μ )]T dk

m

m

m

1

m

2

r ,m

(k)

After changing the integration variable in Equation 4.11 from wavevector k to energy E, Equation 4.11 becomes

∑ 2∫ dE h [ f (E , μ ) − f (E , μ )]T e

m

m

1

m

2

r ,m

(E )

∂f ∂f (μ1 − μ 2 ) = − (μ1 − μ 2 ) ∂μ ∂E

(4.13)

Hence, the low-bias electrical conductance that would be measured between two perfect contacts is g=

I 2e =− (μ1 − μ 2 )/e h

2

∞

∂f

∑ ∫ dE ∂E T

r ,m

m

0

(E )

(4.14)

1 h 12.9 kΩ = 2 ≈ g 2e MTr MTr

(4.16)

In a metallic SWNT, currents are carried by two 1D subbands, i.e., M = 2, leading to a ballistic resistance of 6.5 kΩ that is in effect the minimum contact resistance to the SWNT for perfect contacts. The electronic thermal conductance of a 1D conductor in the ballistic regime can be obtained in a similar manner based on the Landauer formulism. The heat current carried by the 1D subbands is obtained by replacing the charge e in Equation 4.12 with thermal energy (E − μ) carried by an electron (Chen 2005): ∞

∑ 2∫ d E m

0

( E − μ) [ f (Em ,T1 ) − f (Em ,T2 )]Tr , m (E) h

(4.17)

In the linear-response regime where the temperature difference between the two ends of the conductor is small, f (E m ,T1 ) − f (E m ,T2 ) =

df (T1 − T2 ) dT

(4.18)

The electronic thermal conductance is given by Q Ge = = T1 − T2

∞

∑ 2∫ d E m

0

( E − μ ) df Tr ,m (E) h dT

(4.19)

After changing the integration variable to the dimensionless form of x = (E − μ)/kBT and assuming μ >> kBT for the metallic limit and a constant transmission coefficient Tr, Equation 4.19 becomes Ge =

In the low-bias, linear-response regime f (Em , μ1 ) − f (Em , μ 2 ) =

R=

(4.12)

0

(4.15)

where g0 = e 2/h is the universal quantum of electrical conductance. The corresponding low-bias electrical resistance is

Q=

where the factor of 2 accounts for the spin degeneracy e is the elemental charge m is an 1D electron subband vm(k) = dEm(k) /ħdk is the electron group velocity f(E, μ) = (exp((E − μ)/kBT) + 1)−1 is the Fermi-Dirac distribution function μ is the electrochemical potential, the subscripts 1 and 2 denote the two contacts Tr is the transmission coefficient that represents the average probability that an electron injected at one end of the conductor will transmit to the other end. For a ballistic conductor, Tr = 1.

I=

g = 2g 0MTr

(4.11)

0

∞

If the transmission coefficient Tr is constant and the number of 1D subbands crossing the Fermi level is M, Equation 4.14 is simplified to the following for the metallic case of μ >> kBT:

2kB2 T Tr h

∞

∑∫

dx

m −∞

x 2e x (1 + e x )2

(4.20)

When the number of subbands crossing the Fermi level is M, the quantized electronic thermal conductance is G e = 2G 0MTr

(4.21)

where G 0 ≡ π2k B2T/3h is the universal quantum of thermal conductance. The ratio of the thermal conductance quantum to the electrical conductance quantum satisfies the WiedemannFranz law.

4-5


For a metallic SWNT, two linear subbands crossing the Fermi level contribute to an electronic thermal conductance Ge = 4TrG 0. For an intrinsic semiconducting SWNT with an energy gap of the order of 0.1 eV, the electronic thermal conductance is expected to vanish roughly exponentially when the temperature decreases to zero (Yamamoto et al., 2004).

4.2.3 Phonon Transport in Carbon Nanotubes In addition to the unique electronic structures, nanotubes also possess an unusual phonon dispersion relationship between frequency ω and wavevector k. Among 66 distinct phonon branches for a (10, 10) SWNT (Dresselhaus and Eklund 2000), only four acoustic modes with a linear dispersion have a zero frequency at the Γ point (k = 0). These are a longitudinal acoustic mode, doubly degenerate transverse acoustic modes, and a twisting acoustic mode. Among other higher energy phonon branches, the lowest phonon mode is an optical mode that is doubly degenerate and has a nonzero energy of ħωop = 2.1 meV at the Γ point. The phonon thermal conductance of a short SWNT in the ballistic phonon transport regime can be obtained in a similar manner as the electronic thermal conductance based on the Landauer formulism. After replacing the Fermi-Dirac distribution function in Equation 4.17 with the Bose-Einstein distribution, i.e., < n(ω,T) > =[exp(ħω/kBT) − 1]−1, the heat current carried by all the phonon branches can be written as (Yamamoto et al., 2004) ωmmax

Q=

∑∫

m ω min m

dω m ω m[< n(ω m ,T1 ) > − < n(ω m ,T2 ) >]Tr , m (ω) (4.22) 2π

where the integration is carried out between the lower and upper frequency limits of the phonon branch. A procedure similar to Equations 4.18 through 4.20 can be used to obtain the phonon thermal conductance k 2T G ph = B Tr , ph h

max xm

∑∫ m

x 2e x dx x (e − 1)2 min

(4.23)

xm

where a constant phonon transmission coefficient Tr,ph is assumed and x = ħω/kBT. At a sufficiently low temperature, only the four acoustic branches are occupied. For these four acoustic branches, x mmin = 0 and x mmax approaches infinity at low temperatures. Under this condition, the quantized phonon thermal conductance becomes ∞

G ph =

4kB2T x 2e x = 4G0Tr , ph Tr dx x h (e − 1)2

∫

(4.24)

0

Therefore, at the limit of ballistic electron and phonon transport where the transmission coefficient for electrons and phonons are unity, electrons and phonons each contribute to 4G 0 for a metallic SWNT at low temperatures. As the temperature increases,

higher energy phonon branches are occupied, making the phonon thermal conductance much higher than the electronic thermal conductance. The above discussion is focused on the ballistic transport regime relevant for a short nanotube at low temperatures. For a long nanotube in the diffusive transport regime, the thermal conductivity of the nanotube is dominated by the phonon contribution at a sufficiently high temperature. Diffusive phonon transport in a nanotube has been studied using molecular dynamics (MD) simulation (Berber et al., 2000). The calculated thermal conductivity of nanotubes has been compared with diamond and graphite. Because of the stiff SP3 bonds and the high speed of sound, monocrystalline diamond is one of the best thermal conductors, with the highest room temperature thermal conductivity of 3320 W/m-K reported for isotopically enriched 12C diamond (Anthony et al., 1990). Because nanotubes are held by even stronger SP2 bonds as in the basal plane of bulk graphite, they are expected to be very efficient thermal conductors. Because of the atomically smooth surface and the absence of interlayer phonon scattering or coupling to soft phonon modes of a medium, a free-standing SWNT may give rise to higher thermal conductivity than the room-temperature in-plane value of about 1950 W/m-K for graphite (Holland et al., 1966), which is also held by SP2 bonds. An equilibrium MD prediction states that the room-temperature thermal conductivity exceeds 6000 W/m-K for a free-standing (10, 10) nanotube (Berber et al., 2000). However, when nanotubes are entangled in a mat, the thermal conductivity could be suppressed by intertube phonon coupling, similar to the case of graphite, where interlayer phonon coupling quenches its thermal conductivity.

4.3 Measurements of Thermal Conductance of Carbon Nanotubes 4.3.1 Thermal Measurement of Carbon Nanotube Bundles The intriguing thermal transport phenomena in nanotubes have motivated intense experimental investigations. Hone et al. first measured the thermal conductivity of a nanotube rope with dimensions of 5 mm × 2 mm × 2 mm using a steady-state comparative method. Based on an estimated fi lling fraction of the nanotube rod, they obtained a thermal conductivity of only 35 W/m-K for the densely packed mat of the nanotube rod (Hone et al., 1999). Subsequently, a higher room-temperature thermal conductivity value exceeding 200 W/m-K was measured on a dense nanotube rope aligned in high magnetic fields (Hone et al., 2000). The low thermal conductivity can be attributed to contact thermal resistance between interconnected nanotubes in the mat and coupling of phonon modes in adjacent nanotubes in the mat. Interestingly, a linear temperature dependence of the thermal conductivity was observed in the nonaligned sample at temperatures below 30 K. The linear dependence was attributed to the occupation of only the acoustic phonon modes in a SWNT at low temperatures and a constant phonon mean free path dominated by boundary scattering. As the temperature increases,

4-6


the thermal conductivity was found to increase faster than the linear temperature dependence because higher energy phonon modes in the nanotube were occupied.

4.3.2 Direct Thermal Conductance Measurement of Individual Carbon Nanotubes For direct thermal measurement of individual nanotubes, a suspended micro-device was developed (Kim et al., 2001, Shi 2001, Shi et al., 2003). Figure 4.3 shows scanning electron microscopy (SEM) images of one of the various designs of the microdevice. This design consisted of two adjacent, low-stress silicon nitride (SiNx) membranes suspended with six 0.5 μm-thick, 420 μm-long, and 2 μm-wide silicon nitride beams. The length and number of the supporting beams were varied in different designs. One 50 nm-thick and 200 nm-wide platinum resistance thermometer (PRT) serpentine was patterned on each membrane. The PRT was connected to 200 μm × 200 μm Pt bonding pads on the substrate via 2 μm wide Pt leads on the long SiNx beams. Depending on the number of supporting beams for each membrane, up to two additional 2 μm wide Pt electrodes were patterned on each membrane, providing electrical contact to a nanotube or nanowire sample bridging between the two membranes. In the design shown in Figure 4.3, a through-wafer hole was etched under the suspended structure to allow transmission electron microscopy (TEM) characterization of the nanostructure sample bridging the two membranes. An individual nanotube or nanowire sample can be placed between the two suspended membranes by several methods. In one method, a sharp probe was used to pick up an individual nanotube. The probe was manipulated to place the nanostructure between the two membranes. The process requires a highresolution optical microscope or nanomanipulator in a scanning electron microscope. This method was employed for placing MWNT bundles and individual MWNTs between the two membranes (Kim et al., 2001), as shown in Figure 4.4a.

Alternatively, a suspension of the nanostructures in isopropanol (IPA) was dropped on a wafer piece containing many suspended devices. After the IPA evaporated, occasionally a nanostructure was left bridging the two suspended membranes. The yield could be improved by using a micro pipette to place a micro droplet of the suspension on the suspended device. In another approach, a chemical vapor deposition (CVD) method was employed to grow individual SWNTs bridging the two suspended membranes (Yu et al., 2005). In this approach, catalytic nanoparticles made of Fe, Mo, and Al2O3 were delivered to the suspended membranes using a sharp probe tip. Alternatively, a nanometer thick Fe film was patterned on the two suspended membranes. The suspended device was then placed in a 900°C CVD tube with flowing methane. The Fe film was annealed into nanoparticles at a high temperature. The catalytic Fe particles seeded the growth of SWNTs, which occasionally bridged the two suspended membranes. Figure 4.4b shows a SWNT grown between the two suspended membranes using the CVD method. Figure 4.5 shows the schematic diagram of the experimental setup for measuring the thermal conductance of the nanotube sample using the suspended device. During the measurement, the sample was placed in an evacuated cryostat. The two suspended membranes are denoted as the heating membrane and sensing membrane, respectively. When a dc current (I) flows to one of the two PRTs, a Joule heat Qh = I 2Rh is generated in this heating PRT that has a resistance of Rh . The PRT on each membrane is connected to the contact pads by four Pt leads, allowing a four-probe resistance measurement. The resistance of each Pt lead is R L . A Joule heat of 2QL = 2I 2R L is dissipated in the two Pt leads that supply the dc current to the heating PRT. The temperature of the heating membrane is raised to a relatively uniform temperature, Th, because the internal thermal resistance of the membrane is much smaller than the thermal resistance of the nanostructure sample or the thermal resistance of the long narrow beams thermally connecting the membrane to the silicon chip at the ambient temperature T0. The temperature uniformity has been verified by a numerical simulation (Yu et al., 2006).

200 μm Det SED

(a)

Mag Spot FWD E-Beam Scan 8.00 kX 3 4.990 12.0 kV H 45.26 s

10 μm

(b)

FIGURE 4.3 SEM images of a suspended device for measuring the thermal and thermoelectric properties of an individual nanotube or nanowire assembled between the two central membranes of the device: (a) low magnification SEM of the device and (b) high magnification SEM of the two central membranes of the device.

4-7


1 μm

NCEM SEI

1.00 kV

×2 500

10 μm

(a)

EHT = 1.00 kV WD = 6 mm

Signal A = InLens Date: 24 Nov 2008 Mag = 35,00 KX Pettes, M.T. Monday, November 24, 2008

(b)

FIGURE 4.4 SEM of (a) a MWNT assembled and (b) a SWNT and grown between the two membranes of the suspended device. Environment T0 Beam, Gb

QL

Vac

Th

I+

Qh

iac

QL

Ts

GS

iac

Q2

NT

Vac

Beam, Gb Environment T0

(a) Th (b)

1/Gs

Ts

1/Gb

T0

Q2

FIGURE 4.5 (a) Schematic diagram and (b) thermal resistance circuit of the measurement device. The left and right membranes are the heating and sensing membranes, respectively. The black arrow indicates the heating current flow direction. The white arrows indicate heat flow direction. The four-probe electrical resistances of the two serpentines were measured with a small ac current.

A fraction of the Joule heat, Q2, is conducted through the nanostructure sample from the heating membrane to the sensing one, raising the temperature of the latter to Ts. The sensing membrane temperature rise consists of background contributions from heat conduction due to the residual molecules in the evacuated cryostat, radiation and heating of the Si chip because of the finite spreading thermal resistance of the Si chip, and the packaging thermal resistance. When measured using a blank suspended

device without a sample bridging the two membranes, the background contribution was usually found to be much smaller than the heat conduction through the nanostructure sample. The heat flow in the amount of Q2 is further conducted to the environment through the beams supporting the sensing membrane. The rest of the heat, i.e., Q1 = Qh + 2QL − Q2, is conducted to the environment through the supporting beams connected to the heating membrane. The several beams supporting each membrane were designed to be identical. It was found by calculation that the radiation and residual molecular conduction heat losses from the membrane and the supporting beams to the environment are negligible compared to the conduction heat transfer through the support ing beams. Hence, the total thermal conductance of the supporting beams can be simplified as Gb = nκl A/L, where n is the number of the supporting beams for each membrane, κl , A, and L are the thermal conductivity, cross-sectional area, and the length of the supporting beams, respectively. We can obtain the following equation from the thermal resistance circuit shown in Figure 4.5 Q 2 = G b (Ts − T0 ) = G s (Th − Ts )

(4.25)

where Gs is the thermal conductance of the sample and consists of two components, i.e., Gs = (Gn−1 + Gc−1 )−1

(4.26)

where Gn = κnAn/Ln is the intrinsic thermal conductance of the nanostructure κn, An, and Ln are the effective thermal conductivity, crosssectional area, and length of the free-standing segment of the sample between the two membranes, respectively Gc is the contact thermal conductance between the tube and the two membranes

4-8


Considering 1D heat diff usion, one can obtain a temperature profi le in the supporting beams. A Joule heat of QL is generated uniformly in each of the two Pt leads supplying the heating current, yielding a parabolic temperature distribution along the two beams; while linear temperature distribution is obtained for the other beams without DC Joule heating. The heat conduction to the environment from the two Joule-heated beams can be derived as Qh,1 = 2(GbΔTh/n + QL/2); while that from the other supporting beams connected to the heating membrane is Qh,2 = (n − 2)GbΔTh/n, and that from all the beams connected to the sensing membrane is Qs = GbΔTs, where ΔTs ≡ Ts − T0. The energy conservation requirement, i.e., Qh,1 + Qh,2 + Qs = Qh + 2QL , is used to obtain Gb =

Qh + QL ΔTh + ΔTs

(4.27)

and Gs = Gb

ΔTs ΔTh − ΔTs

(4.28)

Qh and QL can be obtained from the measured dc current and the voltage drops across the heating PRT and the Pt leads. ΔTh and ΔTs are obtained from the measured resistance increase of the two PRTs and their temperature coefficient of resistance (TCR). Based on this direct measurement method, the thermal conductance of individual MWNTs and SWNTs have been measured. One reported result of a MWNT is shown in Figure 4.6 (Kim et al., 2001). For the MWNT, the measured thermal conductance shows a T 2.5 dependence at temperatures between 8 K and 1000

100

G (10–9 W/K)

10

1

0.1

0.01

0.001

1

10

100 T (K)

FIGURE 4.6 Measured thermal conductance (circles) of a 14 nm diameter MWNT as a function of temperature. The solid and dotted lines are 1-times and 0.4-times the calculated ballistic thermal conductance of a 14 nm diameter MWNT, respectively. (From Mingo, N. and Broido, D.A., Phys. Rev. Lett., 95, 096105, 2005.)

below 50 K, and a T 2 dependence at the intermediate temperature between 50 and 150 K. The measurement results can be better understood by reviewing the temperature dependence of the specific heat of graphite (DeSorbo and Tyler 1953, Kelly 1981), from which the MWNT is derived. At very low temperatures, the inter-layer phonon modes in graphite combined with the 2D in-plane phonon modes are expected to give rise to a T 3 dependence of the specific heat. At an intermediate temperature range, the soft inter-layer phonon modes of relatively low energy are mostly occupied and do not contribute to the specific heat. Consequently, a T 2 dependence of the specific heat has been observed as the signature of the 2D in-plane phone modes. As temperature increases further, the specific heat starts to saturate to the classical limit given as 3kb per atom. The observed quadratic temperature dependence of the thermal conductance of the MWNT in the intermediate temperature range can be attributed to a quadratic-specific heat in combination with a temperature independent mean free path, which is dominated by boundary and defect scattering. On the other hand, the T 2.5 dependence at the lower temperature range reveals a transition from the T 3 dependence to T 2 dependence of the specific heat. The diameter of the MWNT was determined to be about 14 nm using a high-resolution SEM. The measured thermal conductance below a temperature of 200 K was about 0.4 times the calculated phonon ballistic conductance of a 14 nm diameter MWNT (Mingo and Broido 2005). The 0.4 factor can be attributed to the thermal contact resistance between the two thermal reservoirs and the MWNT, as well as the static scattering processes of phonons that reduce the phonon transmission coefficient to be below unity. At temperatures higher than 200 K, the measured thermal conductance is lower than 0.4 times the ballistic conductance because phonon-phonon umklapp scattering processes reduce the phonon mean free path, resulting in a peak thermal conductance at about 320 K. The relatively high temperature corresponding to the thermal conductance peak suggests that the umklapp scattering mean free path is shorter than that for other static scattering process or the nanotube length of about 2.5 μm until at near room temperature. This method has also been used to measure the thermal conductance of individual SWNTs. One reported measurement result is shown in Figure 4.7 (Yu et al., 2005). The diameter of this SWNT was believed to be smaller than 3 nm based on a high-resolution SEM. However, the diameter determination was not accurate because of the limited resolution of the SEM. The thermal conductance was measured between a temperature of 100 and 300 K. At temperatures below 100 K, the signal to noise ratio becomes low. The observed thermal conductance was about 0.6 times that of the calculated ballistic thermal conductance of a (22, 0) SWNT 1.73 nm in diameter in the 100–300 K temperature range, without showing signatures of umklapp phonon scattering that would reduce the thermal conductance with increasing temperature.

4-9

Thermal Conductance of Carbon Nanotubes 0

G (10–9 W/K)

1

0.1

0.01

0.001

1

10

100 T (K)

FIGURE 4.7 Measured thermal conductance (circles) of a SWNT as a function of temperature. The solid and dotted lines are 1-times and 0.6times the calculated ballistic thermal conductance of a (22, 0) SWNT of 1.72 nm, respectively, and the dashed line is the calculated ballistic thermal conductance of a (10, 0) SWNT of 0.78 nm. (From Mingo, N. and Broido, D.A., Phys. Rev. Lett., 95, 096105, 2005.)

Because of the uncertainty in the diameter and the contact thermal resistance, the intrinsic thermal conductivity cannot be obtained for both the MWNT and SWNT. Figure 4.8 shows the effective thermal conductivity calculated using Equation 4.2 with the contact thermal resistance ignored, the diameter of the SWNT assumed to be the upper bound of 3 nm, and the MWNT assumed to be a 14 m solid cylinder. The intrinsic thermal conductivity is expected to be higher than the as-obtained values of ~3000 W/m-K for the SWNT and MWNT at room temperature. Nevertheless, the room-temperature thermal conductivity of the

4000

κ (W/m-K)

3000

2000

1000

0

0

100

200 T (K)

300

400

FIGURE 4.8 Calculated effective thermal conductivity of the 14 nm diameter MWNT (dots) and the SWNT (triangles) if the diameter of the SWNT is 3 nm.

nanotube is already about 50% higher than that of the pyrolytic graphite (Kelly 1981). Since the room-temperature thermal conductivity is limited by umklapp scattering, this finding suggests that umklapp scattering in nanotubes could be weaker than in graphite. Besides the suspended device, a T-shape junction sensor has been developed for the direct measurement of the thermal conductance of an individual MWNT (Fujii et al., 2005). The sensor contains a suspended pattered Pt nanofi lm that serves as both a heater and thermometer. A MWNT was placed between the center of the suspended nanofi lm and the Si substrate, and formed a T-shape junction with the suspended nanofi lm. The average temperature of the nanofi lm was calculated by measuring its electrical resistance at different electrical currents. The thermal conductivity of the nanofi lm was measured using a self-heating method when no NT was placed on the nanofi lm. The self-heating measurement was repeated after the NT was placed on the nanofi lm. The latter measurement was used to determine the thermal conductance and thermal conductivity of the MWNT. Using this method, the thermal conductivity of MWNTs of three different diameters was measured. It was found that the effective thermal conductivity increases as the MWNT diameter decreases, and exceeds 2000 W/m-K for a diameter of 9.8 nm. A common issue with these two direct thermal conductance measurement methods is that the contact thermal resistance was not eliminated from the measured thermal resistance or conductance. It is possible that the observed diameter dependence of the MWNTs (Fujii et al., 2005) was caused by contact thermal resistance, the effect of which is expected to be more apparent for larger diameter NTs because of a smaller surface-to-volume ratio (Prasher 2008).

4.3.3 Self-Heating Measurement of Thermal Transport in Carbon Nanotubes Besides the two direct thermal conductance measurement methods, other methods based on self electrical heating and resistance thermometry of individual nanotubes and wires have been reported. One reported method is based on the extension of the 3-ω technique to individual metallic microwires and MWNT bundles (Yi et al., 1999, Lu et al., 2001). In this method, the micro-wire or MWNT bundle was suspended across a trench and heated by a sinusoidal current at a frequency of ω. The temperature and consequently the electrical resistance of the suspended sample contained an oscillatory component at a frequency of 2ω. Thus, the voltage drop along the suspended sample contained an oscillatory component at a frequency of 3ω. The amplitude of the voltage oscillation at 3ω is given by

V3ω ≈

4 I 3LRR′ π 4κ A 1 + (2ωγ )2

(4.29)

4-10


where I is the root mean squared amplitude of the heating current R is the electrical resistance of the suspended sample, R′ = dR/dT κ and A are the thermal conductivity and cross-section area of the sample γ is the thermal time constant of the sample given by γ ≡ L2/π2α, where L is the suspended length α is the thermal diff usivity of the sample Apparently, this self-heating resistance thermometry method requires that the sample has a sufficiently large TCR or R′. In addition, the phase lag (ϕ) between the voltage oscillation and the temperature oscillation is given as tan φ ≈ ωγ

(4.30)

In this method, the κ and γ can be extracted by fitting the measured V3ω and ϕ at different ω values. The specific heat is obtained as C = π2 γ κ/ρL2

(4.31)

where ρ is the density. The κ measurement is usually conducted in the low frequency range where ωγ 0.1, which is equivalent to ω > 0.1π2α/L2. For an individual NT with α estimated to be of the order of 2 × 10−3 m2/s and L of about 5 μm, ω needs to be larger than about 80 MHz. This very high frequency and especially its third harmonic component exceeds the frequency range of a lock-in amplifier that is needed for measuring V3ω and ϕ. In addition, capacitive coupling can lead to a large leakage of current at this high frequency, further complicating the analysis of the measurement result. The self-heating 3ω method was used to measure the specific heat and thermal conductivity of a millimeter long suspended defective MWCN tube bundle (Yi et al., 1999). A linear temperature dependence was found in the measured specific heat from 10 to 300 K, and accompanied by a quadratic temperature dependence of the thermal conductivity at temperatures below 120 K. It was speculated that the linear-specific heat was caused by a dominant out-of-plane acoustic phonon mode in the MWNT bundle. The quadratic temperature dependence in the thermal conductivity was attributed to the linear-specific heat and a phonon mean free path that increases linearly with temperature. These findings are rather intriguing. In addition, the measured thermal conductivity was about 20 W/m-K at room temperature. The low value can be attributed to inter-tube phonon coupling, the thermal resistance at tube–tube junctions, and defects in the bundle. In two other reports (Choi et al., 2006, Wang et al., 2007), the 3ω method was used to obtain the lattice thermal conductivity of individually suspended MWNTs and individual SWNTs on a SiO2 substrate. In these self-heating methods, the electric resistance increase with increasing bias voltage was assumed to be caused by an increase in the lattice temperature alone. However, electron transport in nanotubes has been known

to be highly nonlinear. Electron transport is characterized by ballistic transport in short SWNTs at low electric fields and optical phonon emissions at high electric fields (Yao et al., 2000, Javey et al., 2003, Pop et al., 2005). Transport in MWNTs can also be nonlinear with increasing bias, which has been attributed to an increase in the number of current-carrying shells and/or to electrons subjected to Coulomb interaction that tunnel across the MWNT-electrode interface (Bourlon et al., 2006). If the nonlinear current–voltage (I–V) characteristic of a NT is fitted using a polynomial, the obtained V expression can contain an I3 term caused not only by simply an increased lattice temperature, but also by other nonlinear processes including optical phonon emissions. Consequently, the 3ω voltage U3ω is not entirely caused by a rise in lattice temperature, especially at a high field or in short SWNTs where nonequilibrium temperatures of electrons, acoustic phonons, and optical phonons need to be taken into account (Pop et al., 2005, Shi 2008). For addressing this issue, the electric current–voltage (I–V) characteristics of suspended SWNTs were measured and fitted to a coupled electron-phonon transport model that takes into account the nonequilibrium between different energy carriers (Pop et al., 2006). The lattice temperature rise and the thermal conductivity of the NT were extracted by adjusting several fitting parameters. The obtained thermal conductance of a 2 nm diameter SWNT shows inverse temperature (1/T) dependence at above room temperature, revealing the effect of the umklapp process. The effective thermal conductivity was determined to be about 3600 W/m-K at room temperature. In general, these self-heating methods provide a simpler approach for thermal measurement of nanotubes than the direct thermal conductance methods that require additional prefabricated suspended thermal sensors. Nevertheless, the inconvenience of the self-heating-based approaches are that the results obtained on the thermal properties depend on the models employed as well as on several parameters that are difficult to characterize, such as the coupling between the optical and acoustic phonons and the contact electrical and thermal resistances at the nanotube-electrode interfaces.

4.3.4 Optical Measurement of Thermal Transport in Carbon Nanotubes In addition to the aforementioned electrical thermometry methods, an optical heating and Raman thermometry technique has been developed for measuring the thermal conductance of nanotubes (Hsu et al., 2009). In this method, a laser beam was focused on a suspended nanotube and the frequency of the Raman G band was measured. Based on the calibrated Raman G band frequency shifts as a function of the nanotube temperature, the nanotube temperature at the laser spot was obtained. If the nanotube was suspended between two suspended membranes with integrated resistance thermometers, as illustrated in Figure 4.9a, the heat absorption by the NT from the laser can be obtained as Q = QL + QR

(4.32)

4-11

Thermal Conductance of Carbon Nanotubes QLaser

Environment T0

QL

QR

Beam, Gb QLaser TL Vac

iac

QL

GS,L

GS,R NT

Si

TR Vac

iac

QR

Si X

FIGURE 4.10 Schematic diagram of the contact thermal resistance measurement using laser heating and Raman thermometry. Beam, Gb Environment T0 (a) T0

1/Gb

TL

1/GS,L

Th

QL

1/GS,R

TR

1/Gb

T0

QR QLaser

(b)

FIGURE 4.9 (a) Schematic diagram and (b) thermal resistance circuit for measuring the optical absorption and thermal conductance of a nanotube suspended between two suspended membranes with integrated resistance thermometers. The white dashed lines indicate heat flow direction. The electrical resistances of the two serpentines were measured with a small ac current.

Compared to the method shown in Figure 4.5 based solely on resistance thermometry, the potential advantage of this method employing Raman thermometry and resistance thermometry is that the contact thermal resistance may be obtained by scanning the laser spot along the length of the NT. A similar contact resistance measurement approach has been demonstrated on a nanotube suspended across a trench etched in a Si substrate (Hsu et al., 2008), as illustrated in Figure 4.10. When the laser spot was focused at different locations of the suspended nanotube, the nanotube temperature rise (ΔT) at the laser spot was obtained using Raman thermometry. If the phonon transport in a nanotube is diff usive, one can solve the Fourier heat equation for 1D heat conduction. For this experiment, the total heat generation rate (Q) by the incident laser is equal to the sum of the heat flow rates to the right and left hand sides of the nanotube, which is Q = QL + QR =

where QL and QR are the heat conduction rates to the left and right membranes and subsequently through the supporting beams of the membranes to the environment. Based on the thermal resistance circuit of Figure 4.9b, Q L = G S , L (Th − TL ) = G b (TL − T0 )

(4.33)

Q R = G S , R (Th − TR ) = G b (TR − T0 )

(4.34)

and

where GS, L and GS, R are the thermal conductance of the nanotube segment to the left and to the right of the laser spot Gb is the total thermal conductance of the supporting beams for each membrane and can be measured using Equation 4.27 Th is the nanotube temperature at the laser spot measured using Raman thermometry TL and TR are the temperatures of the two membranes measured by using resistance thermometry The thermal conductance of the nanotube is obtained as

(

G S = G S , L −1 + G S , R −1

)

−1

⎛ T −T T −T ⎞ = Gb ⎜ h L + h R ⎟ ⎝ TL − T0 TR − T0 ⎠

ΔT ΔT + x L−x + Rc ,left + Rc ,right κA κA

(4.36)

where κ is the nanotube thermal conductivity A and L are the geometrical cross section and length of the suspended nanotube x is the distance of the laser spot from the left edge of the trench Rc,left and Rc,right are the contact thermal resistances at the two ends of the nanotube The measured temperature at the laser spot moved along the nanotube is given by ⎡ ⎤ ⎛ L Rc ,right − Rc ,left ⎞ x2 L Q ⎢− + x⎜ + Rc ,left ⎥ ⎟ + Rc ,right Rc ,left + 2 2 κA κA ⎢ (κ A) ⎥⎦ ⎝ (κ A) ⎠ ΔT (x ) = ⎣ L + Rc ,left + Rc ,right κA

(4.37) When the two contact thermal resistances are negligible compared to the intrinsic thermal resistance of the nanotube R NT = L/κA, Equation 4.38 is reduced to

−1

(4.35)

ΔT (x ) =

Q (− x 2 + Lx ) κ AL

(4.38)

4-12


While the amount of heat generated in the nanotubes by the laser (Q) was not determined in the experiment based on the configuration of Figure 4.10, one can still determine the ratios between the contact and nanotube thermal resistances, rl ≡ Rc,left/R NT and rr ≡ Rc,right/R NT. Equation 4.38 can be expressed in the general form of ΔT(x) = −ax2 + bx + c. One can define the following coefficient ratios α = b/a = L(1 − rl + rr) and β = c/a = L2rl(1 + rr). The ratios between the contact and nanotube thermal resistances are then given by rl ≡

Rc ,left −α + α 2 + 4β = 2L RNT

(4.39)

nanotubes. In addition to the optical method (Hsu et al., 2008), recent demonstration of a new method for measuring the contact thermal resistance to nanowires provides some promise in solving this difficult problem (Mavrokefalos et al., 2007, Zhou et al., 2007). Moreover, the crystal structure including the diameter, chirality, and defects of the nanotubes being measured are often not well characterized. Hence, there is a need to establish the thermal property-structure relationship of nanotubes. Given the increased interest and the importance of the thermal property of nanotubes on their wide range of possible applications, many of these challenges will likely be overcome in the coming years.

Acknowledgments

and Rc ,right α + α 2 + 4β rr ≡ = −1 + RNT 2L

(4.40)

Using this method, the ratios of thermal contact resistance to the thermal resistance of the nanotube were found to span the range from smaller than 0.02 to larger than 17 for four 2.6–5.0 μm long suspended SWNTs. In addition, by spatially resolving the temperature rise profile along the length of an optically heated nanotube, this technique can distinguish between diffusive and ballistic phonon transport, the latter of which would result in a constant temperature rise in the nanotube independent of the location of the heating laser spot. The results obtained on four nanotubes indicate that phonon transport in these nanotubes is diffusive or the phonon mean free path is shorter than the 2.6–5 μm suspended segment of the nanotubes. As such, thermal transport in the four nanotubes can be explained with the simple Fourier heat transport model.

4.4 Conclusion and Outlook This chapter provided an introduction to thermal conductance in carbon nanotubes and reviewed recent research progress in this topic, which is still being actively investigated. As such, it is not possible for this chapter to cover all the interesting works that have been reported on thermal transport in nanotubes. While breakthroughs have been made on the theoretical understanding and experimental characterization of thermal transport in nanotubes, the intrinsic thermal conductance and conductivity of nanotubes are yet to be found. On the theoretical front, different approaches, including molecular dynamics simulations, have been employed to calculate the thermal property of nanotubes. A common problem is that the computation capability is currently limited to short nanotubes. Consequently, long wavelength phonon modes were cut off in the simulation, leading to length-dependent thermal conductivity values obtained by these numerical methods. On the experimental side, the reported thermal conductance values all contain errors caused by contact thermal resistance, which is difficult to measure for

The author’s research on thermal transport in carbon nanotubes has benefited from collaborations with Philip Kim, Arun Majumdar, Paul L. McEuen, Choongho Yu, Deyu Li, Michael Pettes, Stephen B. Cronin, and I-Kai Hsu, and has been supported by the Department of Energy award DE-FG02-07ER46377, the National Science Foundation Thermal Transport Processes Program, and the University of Texas System.

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5 Terahertz Radiation from Carbon Nanotubes Andrei M. Nemilentsau Belarus State University

Gregory Ya. Slepyan Belarus State University

Sergey A. Maksimenko Belarus State University

Oleg V. Kibis Novosibirsk State Technical University

Mikhail E. Portnoi University of Exeter

5.1 5.2 5.3

Introduction ............................................................................................................................. 5-1 Electronic Properties of SWNTs ...........................................................................................5-2 Thermal Radiation from SWNTs ..........................................................................................5-3 Fluctuation-Dissipative Theorem • Free-Space Green Tensor • Green Tensor in the Vicinity of SWNT • Thermal Radiation Calculation • Numerical Results

5.4 5.5 5.6

Quasi-Metallic Carbon Nanotubes as Terahertz Emitters................................................5-9 Chiral Carbon Nanotubes as Frequency Multipliers ....................................................... 5-11 Armchair Nanotubes in a Magnetic Field as Tunable THz Detectors and Emitters ........................................................................................................................... 5-11 5.7 Conclusion ..............................................................................................................................5-13 Acknowledgments ............................................................................................................................. 5-14 References........................................................................................................................................... 5-14

5.1 Introduction Creating a compact, reliable source of terahertz (THz) radiation is one of the most challenging problems in contemporary applied physics (Lee and Wanke, 2007). Despite the fact that THz technology is at the boundaries of microwave and photonic technologies, it is quite underdeveloped compared to the achievements in the microwave or the photonic technology. There are very few commercially available instruments for the THz frequency region, and most of them lack the precision required to perform accurate measurements. There are also no miniaturized and low-cost THz sources. One of the latest trends in THz technology (Dragoman and Dragoman, 2004a) is to use single-walled carbon nanotubes (SWNTs) as building blocks of novel high-frequency devices. An SWNT is a hollow cylindrical molecule made up of carbon atoms (Saito et al., 1998). We can formally consider the SWNT as a graphene sheet rolled up into a cylinder along the vector R h connecting to crystallographically equivalent sites of the graphene lattice (see Figure 5.1). Th is vector is called the chiral vector and is usually defined in terms of the basic vectors, a1 and a 2 , of the graphene lattice: R h = ma1 + na 2, where m, n are integers. The dual index (m, n) is usually used to characterize SWNT type. Th ree different SWNT types are defi ned: (m, 0) zigzag SWNTs, (m, m) armchair SWNTs, and (m, n) (0 < n ≠ m) chiral SWNTs. The SWNT radius, Rcn , and chiral angle, θ (the angle between the R h and a1), are defi ned as follows:

Rcn =

cos θ =

| Rh | 3b = m2 + mn + n2 , 2π 2π

(5.1)

R h ⋅ a1 2n + m = , | R h || a1 | 2 n2 + nm + m2

(5.2)

where b = 0.142 nm is the C–C bond length. Typically, SWNTs are 0.1–10 μm in length; their cross-sectional radius varies within the range 1–10 nm, while their chiral angle is 0 ≤ θcn ≤ 30°. There are several promising proposals of using carbon nanotubes for THz applications including a nanoklystron using extremely efficient high-field electron emission from nanotubes (Dragoman and Dragoman, 2004a; Manohara et al., 2005; Di Carlo et al., 2006); devices based on negative differential conductivity (NDC) in large-diameter semiconducting SWNTs (Maksimenko and Slepyan, 2000; Pennington and Goldsman, 2003); high-frequency resonant-tunneling diodes (Dragoman and Dragoman, 2004b) and Schottky diodes (Léonard and Tersoff, 2000; Odintsov, 2000; Yang et al., 2005; Lu et al., 2006); as well as electric-field-controlled carbon nanotube superlattices (Kibis et al., 2005a,b), frequency multipliers (Slepyan et al., 1999, 2001), THz amplifiers (Dragoman and Dragoman, 2005), and switches (Dragoman et al., 2006). Among others, the idea of SWNT-based optical devices enabling the control and enhancement of radiation efficiency on the nanoscale, i.e., nanoscale antennas for THz, infrared, and visible light, is actively

5-1

5-2


a1 θ

a2

B Rh

4

A

3 1

points in the Brillouin zone of graphene (Wallace, 1947). Thus, we restrict our consideration to the π electrons, assuming that their movement can be described in the framework of the tightbinding approximation (Saito et al., 1998); the overlapping of wave functions of only the nearest atoms is taken into account. In the beginning, we apply this approach to the plane monoatomic graphite layer, and then show how the model must be modified to analyze an SWNT. To describe graphene π bands we use the 2 × 2 Hamiltonian matrix (Wallace, 1947):

2 2

0 ⎛ ˆ = H 0 ⎜ * ⎝ H12 ( px , p y )

1

FIGURE 5.1 Graphene crystalline lattice. Each lattice node contains a carbon atom.

discussed (Dresselhaus, 2004; Hanson, 2005; Burke et al., 2006; Slepyan et al., 2006). Noise properties and operational limits of such antennas are substantially determined by the thermal fluctuations of the electromagnetic field. In this chapter, several novel schemes are discussed (Kibis and Portnoi, 2005; Portnoi et al., 2006; Kibis et al., 2007, 2008; Nemilentsau et al., 2007; Portnoi et al., 2008) to utilize the physical properties of SWNTs for the generation and detection of THz radiation.

5.2 Electronic Properties of SWNTs Electrodynamic processes in any medium are dictated by its electronic properties, although they may be missing in an explicit form of the macroscopic electrodynamics equations. In that sense, SWNTs are not an exception. Many researches (Charlier et al., 2007) have been devoted to the development of the theory of electronic properties of the SWNT. Both the sophisticated methods of modern solid-state physics and fi rstprinciples simulations are among them. In this section, we give only an elementary introduction for later use in the analysis of the THz radiation from SWNTs. Each carbon atom in graphene and SWNT has four valence orbitals (2s, 2px, 2py, and 2pz). Three orbitals (s, px, and py) combine to form in-plane σ orbitals. The σ bonds are strong covalent bonds responsible for most of the binding energy and elastic properties of the graphene sheet and SWNT. The remaining pz orbital, pointing out of the graphene sheet, cannot couple with σ orbitals. The lateral interaction with the neighboring pz orbitals creates delocalized π orbitals. The π bonds are perpendicular to the surface of the SWNT and are responsible for the weak interaction between SWNTs in a bundle, similar to the weak interaction between graphene layers in pure graphite. The energy levels associated with the in-plane bonds are known to be far away from the Fermi energy in graphene, and thus do not play a key role in its electronic properties. In contrast, the bonding and antibonding π bands cross the Fermi level at high-symmetry

H12 ( px , p y )⎞ ⎟, 0 ⎠

(5.3)

where ⎧ b⎛1 ⎧ b ⎫ ⎞⎫ H12 ( px , p y ) = −γ 1 exp ⎨i px ⎬ − γ 2 exp ⎨i ⎜ px − 3 p y ⎟ ⎬ ⎝ ⎠⎭ 2 ⎩ ⎭ ⎩ ⎧ b⎛1 ⎞⎫ −γ 3 exp ⎨ −i ⎜ px − 3 p y ⎟ ⎬ . ⎠⎭ ⎩ ⎝ 2

(5.4)

Here, γ1,2,3 are the overlapping integrals; px,y are the projections of the quasi momentum of electrons, p, on the corresponding axes; and ћ is the Planck constant. As the electronic properties of graphene are isotropic in the in-plane, we set γ1 = γ2 = γ3 = γ0 in Equation 5.4, where γ0 ≃ 3 eV is the phenomenological parameter, which can be determined experimentally (see, e.g., Saito et al., 1998). The electron energy values are found as the eigenvalues of the matrix on the right side of (5.3) as ⎛ 3bp y ⎞ ⎛ 3bp y ⎞ ⎛ 3bpx ⎞ ε c ,v (p) = ±γ 0 1 + 4cos ⎜ cos ⎜ + 4cos2 ⎜ ⎟ ⎟. ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ⎝ 2 ⎠ (5.5) The plus and minus signs in this equation correspond to conduction (c) and valence (υ) bands, respectively. The range of quasi momentum variation (the first Brillouin zone) is the hexagons shown in Figure 5.2. The vertices are the Fermi points where ε = 0, which is indicative of the absence of the forbidden zone for π electrons in graphene. The dispersion properties of electrons in SWNTs are quite different from those in graphene, as a plane monolayer is transformed into a cylinder. In a cylindrical structure, an electron located at the origin and an electron located at the position defi ned by the vector R h = ma1 + na2 are identical. Hence, we should impose the periodic boundary conditions along the tube circumference on the wave functions of π electrons in SWNTs: Ψ(r + R h ) = e ipRh / Ψ(r) = Ψ(r).

(5.6)

5-3

Terahertz Radiation from Carbon Nanotubes pz

pz

2πћ/3b pφ

(a)

pφ

⎛ 3bpz ⎞ ⎛ πs ⎞ ⎛ πs ⎞ ε c ,v ( pz , s) = ± γ 02 + 4 γ 0 γ 2 cos ⎜ cos ⎜ ⎟ + 4 γ 22 cos 2 ⎜ ⎟ , ⎟ ⎝ 2 ⎠ ⎝ m⎠ ⎝ m⎠

(b)

FIGURE 5.2 SWNTs.

Strictly speaking, the curvature of the SWNT surface breaks the isotropic symmetry of electronic properties so that the overlapping integrals, γ1,2,3, in Equation 5.4 turn out to be different from one another. For the zigzag SWNTs, these integrals are as follows (Kane and Mele, 1997; Lin and Chuu, 1998): 2 γ 1 = γ 0 , γ 2 = γ 3 = (1 − 3b2/32Rcn )γ 0 . Then, instead of Equation 5.8, we have the dispersion equation:

(5.10)

First Brillouin zone for (a) zigzag and (b) armchair

The second equality here is due to the Bloch theorem. This leads to the quantization of the transverse quasi momentum of electrons: pφ = s/Rcn ,

(5.7)

where s is an integer. The cylindrical coordinate system with the z-axis oriented along the SWNT axis is used here. The axial projection, pz, of the quasi momentum is continuous. In order to derive the dispersion equation for zigzag SWNTs from Equation 5.5, one must perform the substitutions {px → p z , p y → p ϕ}, which yields ⎛ 3bpz ⎞ ⎛ πs ⎞ ⎛ πs ⎞ ε c ,v ( pz , s) = ±γ 0 1 + 4cos ⎜ cos ⎜ ⎟ + 4cos2 ⎜ ⎟ , ⎝ 2 ⎟⎠ ⎝ m⎠ ⎝ m⎠ s = 1,2,…, m.

(5.8)

For armchair SWNTs, the dispersion law is obtained from Equation 5.5 by means of the substitutions {px → pϕ , py → pz}. For chiral SWNTs, the analogous procedure is specified by {px → pz cos θ + pϕ sin θ, px → pz sin θ − pϕ cos θ}. It follows from Equation 5.7 that the first Brillouin zone in SWNTs is transformed from a hexagon to a family of onedimensional zones defi ned by segments of straight lines confined to the interior of the hexagon. Depending on the dual index (m,n), these segments can be oriented differently either by bypassing or crossing the Fermi points, as shown in Figure 5.2. Correspondingly, the forbidden zone either appears or disappears in the electron spectrum of an SWNT. In the absence of the forbidden zone, a material is a metal; otherwise, it is a semiconductor. The condition for the forbidden zone to appear is (Saito et al., 1998) m − n ≠ 3q,

(5.9)

where q is an integer. For armchair SWNTs, this condition is not valid at any m, and the forbidden zone is always absent, thus proving that the armchair SWNTs are always metallic. For zigzag SWNTs, the zone appears when m ≠ 3q, and, thus, zigzag SWNTs can be either metallic or semiconducting, depending on Rcn.

which shows the presence of the forbidden zone even for m = 3q. However, this zone is much narrower compared to that for m ≠ 3q. The nontrivial electronic structure of SWNTs dictates their response to the electromagnetic field. Due to the quasi one-dimensional nature of SWNTs, their optical response is strongly anisotropic. The optical response to the axially polarized incident electric field significantly exceeds the optical response to the electric field polarized transversely to the CNT (carbon nanotube) axis (Tasaki et al., 1998; Milošević et al., 2003; Murakami et al., 2005). Due to the quantization of the transverse quasi momentum of electrons (Equation 5.7), divergences arise in the electronic density of states (DOS) of SWNTs (Saito et al., 1998). These divergences, which are known as Van Hove singularities, produce discrete energy levels or “subbands,” the energy of which is determined solely by the chirality of SWNTs (Saito et al., 1998). As the inter-subband gap corresponds to the energy of infrared to visible light, the spectra of optical conductivity of an SWNT demonstrate the number of resonant lines in the region. In the spectral range of 1–100 THz, the nonmonotonic frequency dependence of the reflectance and transmittance of CNT-based composite media that does not follow from the standard Drude theory has been observed (Ugawa et al., 1999; Ruzicka et al., 2000). Ugawa et al. (1999) found empirically that the effective permittivity of a CNT-based composite medium can be represented as a superposition of Drudian and Lorentzian functions. The spectral width of the resonance is of the order of the resonant frequency. The origin of this resonance could be attributed the inhomogeneously broadened geometric resonance in an isolated CNT (Slepyan et al., 2006).

5.3 Thermal Radiation from SWNTs In this section, we investigate the thermal electromagnetic field radiated by an SWNT at temperature T placed in cold environment and show that the thermal radiation from metallic SWNTs can serve as an efficient source of the THz radiation. Our consideration is based on the method developed by Rytov (1958), which is known as fluctuational electrodynamics (see details in Lifshitz and Pitaevskii, 1980; Rytov et al., 1989; Joulain et al., 2005). The key idea of this method is that the thermal radiation sources in

5-4

a material are the fluctuation currents, which are due to the random thermal motion of charged carriers the material consists. To determine the statistical properties of the electromagnetic field, we have to know the statistical properties of random currents and the radiation of the elemental volume of the material. The first information is given by the fluctuation-dissipative theorem while the second information is given by the Green tensor of the system. It should be noted that the application of the equilibrium laws, such as the fluctuation-dissipative theorem, is not very rigorous in this case, but it is justified when the role of the heat transport phenomena (such as thermal conductivity) is negligible. Hence, we will not consider them further. The thermal radiation from SWNTs is of interest not only because of possible applications of THz device. Fundamental interest to the thermal radiation is dictated by the ability of nanostructures to change the photonic local density of states (LDOS), i.e., the electromagnetic vacuum energy (Agarwal, 1975; Joulain et al., 2003; Novotny and Hecht, 2006). The effect has been observed in microcavities, photonic crystals, and nanoparticles in the vicinity of surface-plasmon resonances (Novotny and Hecht, 2006). Thus, as the electromagnetic fluctuations are defined by photonic LDOS, the investigation of the thermal radiation is expected to bring new opportunities for the reconstruction of photonic LDOS in the presence of nanostructures. The apertureless scanning near-field optical microscopy provides a possibility for the experimental detection of LDOS (Joulain et al., 2003). In turn, the photonic LDOS is a key physical factor defining a set of well-known quantum electrodynamic effects: the Purcell effect and the electromagnetic friction (Novotny and Hecht, 2006), the Casimir–Lifshitz forces (Lifshitz and Pitaevskii, 1980), etc. Thermal radiation in systems with surface plasmons is known to be considerably different from blackbody radiation (Carminati and Greffet, 1999; Henkel et al., 2000; Schegrov et al., 2000). Earlier theoretical studies of SWNTs showed the existence of low-frequency plasmon branches (Lin and Shung, 1993) and the formation of strongly slowed-down electromagnetic surface waves in SWNTs (Slepyan et al., 1999). Such waves define a pronounced Purcell effect in SWNTs (Bondarev et al., 2002) and the potentiality of SWNTs in the development of Cherenkov-type nano-emitters (Batrakov et al., 2006). Geometrical resonances—standing surface waves excited due to the strong reflection from the SWNT tips— qualitatively distinguish SWNTs from the planar structures investigated in Carminati and Greffet (1999), Henkel et al. (2000), and Schegrov et al. (2000). One can expect an essential role of these resonances in the formation of SWNTs’ thermal radiation.

5.3.1 Fluctuation-Dissipative Theorem


gauge, which implies the scalar potential to be equal to zero for the electromagnetic field. For a stationary field, the correlation function depends on the time difference, t − t′, only. The Fourier transform of the correlation function is called the cross-spectral density (Joulain et al., 2005): ∞

An (r)Am* (r ′)

∫

An (r, t )Am (r ′, t ′) e iω (t − t ′ )d(t − t ′). (5.11)

−∞

Then the fluctuation-dissipative theorem for the electromagnetic field vector potential is formulated as follows (Lifshitz and Pitaevskii, 1980): An (r1 )Am* (r2)

ω

2Θ(ω, T ) ⎤ ⎡ = ⎢ + ⎥ Im ⎡⎣Gnm (r1 , r2 , ω)⎤⎦ , ω ⎣ ⎦

(5.12)

where G _(r1,r2,ω) is the retarded Green tensor n, m = x, y, z designates the Cartesian coordinate system axis Θ(ω,T) = ħω/[exp(ħω/kBT)−1], ħ and kB are the Planck and Boltzmann constants, respectively The first term in square brackets is due to the zero vacuum fluctuations, and will be omitted further. Thus, to calculate the intensity of thermal radiation emitted by an SWNT, we elaborate the method of calculation of the electromagnetic field Green tensor in the vicinity of a CNT.

5.3.2 Free-Space Green Tensor The electromagnetic field Green tensor is defined by the equation (∇r1 × ∇r1 × −k 2 )G(r1 , r2 , ω) = 4π I δ(r1 − r2 ),

(5.13)

where ∇r1 indicates that operator ∇ acts only on the variable r1 of the Green tensor _I is the unit tensor k = ω/c, ω is the electromagnetic field frequency c is the speed of light in vacuum In general, this equation should be supplemented by boundary conditions. In the Cartesian coordinate system, Equation 5.13 takes the following index form: ⎛ ⎞ ∂2 2 ⎜⎝ εiln εnkj ∂x ∂x − k δ ij ⎟⎠ G jm (r1, r2, ω) = 4πδ imδ(r1 − r 2), (5.14) 1l

The fluctuation-dissipative theorem relates the fluctuations of physical quantities to the dissipative properties of the system when it is subjected to an external action. We are interested in the space–time correlation function of the electromagnetic field fluctuations 〈An(r,t)Am(r′,t′)〉, where A(r,t) is the vector potential of the electromagnetic field. We use the Hamiltonian

ω

=

1k

where x1l,k = x1, y1, z1 and summation over the repeated indices is assumed. For each index m, Equation 5.14 gives us an independent equation that describes the evolution of the mth column of the Green tensor. Thus, for a given m, the column Gnm(r1,r2,ω) can formally be considered as a field vector, G(m)(r1;r2), induced

5-5

Terahertz Radiation from Carbon Nanotubes

at point r1 by a delta source located at point r2; here m and r2 are parameters. Let us introduce the Hertz vector, ∏(m)(r1;r2):

G(r1, r2, ω) = G(0) (r1, r2, ω) + G(SC) (r1, r2, ω),

G(m) (r1; r 2) = (k 2 + ∇∇⋅)P (m) (r1; r2)

where G _ (0)(r1,r2,ω) is the solution of the inhomogeneous Equation 5.13, for the free-space case and G _ (SC) satisfies the equation

(5.15)

Then we obtain three independent equations instead of Equation 5.14: (Δ + k 2 )P (m) (r1 ; r2 ) = −

4π e mδ(r1 − r2 ), k2

(5.16)

where em = (δxm,δym,δzm) is the basis vector of the Cartesian coordinate system. In the free-space case, the solution of Equation 5.16 is straightforward, (Jackson, 1999) and the free-space Hertz vector has the following form: P (0m) (r1; r 2) =

1 (0) 1 e ik|r1− r2 | G ( r , r , ω ) = . 1 2 k2 k 2 | r1 − r 2 |

(5.17)

e ik|r 1− r 2| | r1 − r2 |

(5.18)

where G (0) (r1, r2, ω) =

is the free-space Green function. Thus, we obtain the standard expression for the free-space Green tensor (Lifshitz and Pitaevskii, 1980): G(0) (r1, r2, ω) = ( I + k −2∇r1 ⊗ ∇r1 )G (0) (r1, r2, ω)

(5.19)

with ∇r1 ⊗ ∇r1 as the operator dyadic acting on variables r1.

(∇r1 × ∇r1 × −k 2 )G(SC) (r1, r2, ω) = 0

(5.21)

and boundary conditions on the SWNT surface. From a formal point of view, G _ (SC) can be considered as the free-space Green tensor scattered by the SWNT (see Figure 5.3). To calculate the scattered Green tensor, we use the method developed in Lifshitz and Pitaevskii (1980, see problem 1 after paragraph 77). Each column of the free-space Green tensor induces current density jz(m) on the SWNT surface, which generates the mth column of the scattered Green tensor, m = x, y, z. We take into account only the axial component of the induced current due to the fact that the SWNT length is much greater than the SWNT radius. By analogy with Section 5.3.2, we introduce three independent scattered Hertz vectors, ∏SC(m)(r1;r2) = ez∏SC(m)(r1;r2). Each Hertz vector has only z nonzero component. Further, we omit parameter r2 in the notation of the Hertz vector to simplify the designations. These Hertz vectors satisfy scalar equations, (Δ + k 2 )Π SC(m) (r) = 0,

(5.22)

and effective boundary conditions on the CNT surface. Thus, we have to solve three equations (Equation 5.22) to calculate scalar quantities ΠSC(m). We could do this in the arbitrary coordinate system. We use the cylindrical coordinate system (ρ,ϕ,z) in which the effective boundary conditions (Slepyan et al., 1999, 2006) have the simplest form:

5.3.3 Green Tensor in the Vicinity of SWNT Consider an isolated single-walled CNT of cross-sectional radius Rcn and length L, aligned along the z axis of the Cartesian coordinate basis (x, y, z) with the origin in the geometrical center of the CNT (see Figure 5.3). We restrict our consideration to the case Rcn L/2, (5.24)

cn − 0

where ⎡ ∂2 Π SC(m) (Rcn , z ) ⎤ (0) jz(m) = σ zz ⎢ + k 2 Π SC(m) (Rcn , z ) + Gzm (R, r2, ω)⎥ . 2 ∂ z ⎣ ⎦

ρ2

r2

G (0) zm

jz(m)

(5.25)

G (SC) nm

Rcn z z2

O L

FIGURE 5.3

Free-space Green tensor scattering by an SWNT.

Vector R designates the point on the CNT surface, and in the cylindrical coordinate system, has the following form: {Rcn,ϕ,z}. Due to the cylindrical symmetry of the system, the scattered Hertz potential does not depend on the azimuthal variable, ϕ. Here, σzz (ω) = −

2e 2 3 πmb(ν − iω)

m

∑ ∫ ∂ε ∂(pp , s) ∂f (∂pp , s) dp c

s =1

z

z

z

z

z

(5.26)

5-6


is the axial conductivity of the zigzag SWNT (Slepyan et al., 1999), f(pz,s) is the equilibrium Fermi distribution, and v = (1/3) × 1012 s−1 is the relaxation frequency. Applying the Green theorem (Jackson, 1999) to Equations 5.22 through 5.24, we obtain the integral equations for the normalized axial current density, jz(m) (z ; r2 ), induced on the CNT surface by the incident electric (0) field, Gzm (R, r2 , ω) (Nemilentsau et al., 2007): L /2

∫

jz(m) (z ′; r2)K (z − z ′)dz ′ + C1e −ikz + C2eikz

− L /2

=

1 2π

2π

L /2

∫

− L /2

eik|z − z ′| (0) Gzm (R ′, r2, ω)dφ′dz ′, 2ik

∫

(5.27)

0

where C1,2 are constants determined by the edge conditions, jz(m) (± L /2; r2 ) = 0 (Slepyan et al., 2006), K (z ) =

exp(ik | z |) 2iRcn − ω 2ikσ zz (ω)

π

∫ 0

eikr dφ, r

(5.28)

2 and r = z 2 + 4 Rcn sin2 (φ/2). After three independent integral equations have been solved (Equation 5.27) (for three different values of m = x, y, z) and three current density values have been calculated, we again return to the Cartesian coordinate system. Finally, we can present the solution of the scattering problem in the form of the simple layer potential (Colton and Kress, 1983; Nemilentsau et al., 2007):

(0) Gnm (r1, r2, ω) = Gnm (r1, r2, ω) +

∫

− L /2

j

(m) z

∫

(z ; r2) G (r1, R, ω)dφdz . (0) nz

5.3.4 Thermal Radiation Calculation Let us calculate the thermal radiation of a hot SWNT placed into an optically transparent cold environment. As the fluctuationdissipative theorem (Equation 5.12) is applicable only at thermal equilibrium, we cannot directly apply it in this case. To solve the problem, let us consider in more detail the case when the SWNT is in thermal equilibrium with the environment. At thermal equilibrium, the thermal fluctuation field in the system is the superposition of three different fields: thermal electromagnetic field radiated by the SWNT itself; blackbody radiation of the surrounding medium in the absence of the CNT, A(0); and the field A(s) resulting from the scattering of radiation of the medium by the SWNT. In the case of the cold medium, only the electromagnetic field radiated by the SWNT remains. Thus, to calculate the thermal radiation of the hot SWNT placed in the cold medium, we should calculate the total thermal electromagnetic field radiated in the equilibrium and separate the blackbody contribution. The thermal radiation intensity in equilibrium is easily calculated by substituting Equation 5.29 for the electromagnetic field Green tensor to Equation 5.12 for the fluctuation-dissipative theorem. To separate the blackbody radiation contribution, we use the method developed in Lifshitz and Pitaevskii (1980, see problems after Sect. 77). We introduce the blackbody radiation vector potential, A( B ) (r) = A(0) (r) + A(s) (r),

(5.30)

and calculate the correlator: An( B ) (r1)Am( B )* (r2)

2π

L /2

×

iωRcn c2

numerical integration of Equation 5.27 has been performed with integral operators approximated by a quadrature formula and subsequent transition to a matrix equation.

(5.29)

0

While deriving Equation 5.29, we assumed the incident field source distance from the CNT farther than its radius; there(m) fore, we can neglect the dependence of the current jz on the azimuthal variable, ϕ. Equation 5.29 with an arbitrary jz(m) satisfies the aforementioned equation for the retarded Green tensor and the radiation condition at |r1 − r2| → ∞. Peculiar electronic properties of CNTs (Dresselhaus et al., 2000) influence the Green tensor through the axial conductivity σzz(ω) (for details, see Slepyan et al. (1999)). The index m and the variable r2 appear in Equations 5.29 and 5.27 only as parameters. Note that these equations, as they couple the Green tensor of the system considered and the free-space Green tensor, play the role of the Dyson equation for CNTs. It is important that the role of scattering by CNTs is not reduced to a small correction to the free-space Green tensor. This means that the Born approximation conventionally used for solving the Dyson equation (Lifshitz and Pitaevskii, 1980) becomes inapplicable to our case. Because of this, the direct

(B) ≡ Dnm (r1, r2, ω).

ω

(5.31)

Then, the electric field intensity of the SWNT thermal radiation in the case when the SWNT temperature is much higher that the temperature of the surrounding medium, Iω (r0) = |E(r0)|2 is given, in view of the relation En = −ikAn, by 3

I ω (r0) = k 2

∑ ⎡⎣ | A (r ) |

2

n

0

n =1

ω

(B) (r0, r 0, ω)⎤ . − Dnn ⎦

(5.32)

To calculate the blackbody radiation correlator, we should calculate the scattered vector potential, A(s). To do this, we should solve Equations 5.22 through 5.25 with A(0) instead of the free-space (0) Green tensor, Gzm , in Equation 5.25. By analogy with Equation 5.29, the vector An( B ) (r1 ), potential, is written as An( B ) (r1) = An(0) (r1) +

Rcn c

L /2

∫

− L /2

2π

∫

(0) (r1, R, ω)dφdz , j(z ) Gnz

(5.33)

0

where the current density, j(z), induced on the SWNT surface by the free-space fluctuation electromagnetic field, A(0), is the solution of the integral equation

∫

1 2

K (z − z ′) j(z ′)dz ′ + C1e −ikz + C2e ikz =

− L /2

L /2

∫

Az(0) (R ′)e ik|z − z ′|dz ′.

− L /2

(5.34) The second term in Equation 5.33 describes scattering of the (B) free-space blackbody radiation by the SWNT. To calculate Dnm , we utilize Equation 5.33 and take into account that the correlator, (0)* , is defined by Equation 5.12 with the freeA(0) n (r1 ) Am (r2 )

Iω , 10–31 (ergs c/cm2)

L /2

ω

log (Iω)

In this section, we present the results of the numerical calculations of the thermal radiation emitted by metallic (15,0) (see Figure 5.4) and semiconducting (23,0) SWNTs (see Figure 5.5). The following parametrization of the radius vector is used throughout this section: r = {ρ, φ, z } = e x ρ cos φ + e y ρ sin φ + e z z .

(5.35)

Iω , 10–25 (ergs c/cm3)

8 6 20

30

40 50 ω (THz)

60

70

4

4 ρ0 = 100L ρ0 = 0.5L

2

2

0 (a)

Iω , 10–15 (ergs c/cm3)

Im (α), 10–13 cm3

The spectra of the thermal radiation from the SWNT (15,0) at different distances from the SWNT axis are presented in Figure 5.4a and b presents one of the spectra in the logarithmic scale. The spectrum depicted in Figure 5.4a demonstrates a number of equidistant discrete spectral lines with decreasing intensities superimposed by the continuous background. Such a structure

6

0 10

20

30

40

50

60

70

CNT, ρ0 = 0.5L

104

Blackbody log (Iω)

102 100 10–2 10

(b)

20

30

40 ω (THz)

50

60

24

0.8

16

ρ0 = 100L ρ0 = 0.5L

8

0.4 0.0

10

20

30

40

50

60

70

101

5.3.5 Numerical Results

2.5 2.0 1.5 1.0 0.5 0.0 10

1.2

0 (a)

space Green tensor, G(0) nm (r1 , r2 , ω), on the right-hand side.

8

32

Iω , 10–19 (ergs c/cm2)

5-7

Terahertz Radiation from Carbon Nanotubes

70

FIGURE 5.4 (a) Thermal radiation spectra of a metallic (15,0) SWNT. r0 = {100 L,ϕ0,0} (dashed line, left ordinate axis) and r0 = {0.5 L,ϕ0,0} (solid line, right ordinate axis); ϕ0 is arbitrary; T = 300 K; and L = 1 μm. The inset presents the CNT’s polarizability. (b) Thermal radiation from CNT in the near-field zone (solid line) compared to blackbody radiation, I ω( B ) (r0 ) = 4ω2Θ(ω, T ) /c 3 (dashed line). (From Nemilentsau, A.M. et al., Phys. Rev. Lett., 99, 147403, 2007.)

100

CNT, ρ0 = 0.5L

10–1

Blackbody

10–2 10–3 10

(b)

20

30

40 ω (THz)

50

60

70

FIGURE 5.5 Same as Figure 5.4, but for the (23,0) semiconducting SWNT. (From Nemilentsau, A.M. et al., Phys. Rev. Lett., 99, 147403, 2007.)

is inherent to spectra both in the far-field (dashed line) and nearfield (solid line) zones. The peculiarity of the near-field zone is the presence of additional spectral lines absent in the far-field zone. Thus, the thermal radiation spectra presented in the figure qualitatively differ from both blackbody radiation (Lifshitz and Pitaevskii, 1980) and radiation of semi-infinite SiC samples (Schegrov et al., 2000). In the latter case, the discrete spectrum is observed only in the near-field zone (Schegrov et al., 2000). The comparison of the thermal radiation and the SWNT’s polarizability spectra (Slepyan et al., 2006) depicted in Figure 5.4a reveals the coincidence in the far-field zone of the thermal radiation resonances and the polarizability resonances. The latest are the dipole geometrical resonances of surface plasmons (Slepyan et al., 2006) defined by the condition Re[κ(ω)] L ≅ π(2s − 1), with κ(ω) as the plasmon wavenumber; s is a positive integer. It should be noted that the polarizability (and the thermal radiation) resonances are found to be significantly shifted to the red as compared to the perfectly conducting wire of the same length, because of the strong slowing-down of surface plasmons in SWNTs: Re[κ(ω)]/k ≈ 100 (Slepyan et al., 1999). In particular, for L = 1 μm, the geometrical resonances fall into the THz frequency range. The attenuation is small in a wide frequency range below the interband transitions. Additional spectral lines in the near-field zone are described by the condition Re[κ(ω)]L ≅ 2πs. We refer to these resonances as quadrupole geometric resonances because the current density distribution for these modes is antisymmetrical with respect to z = 0 and, consequently, the dipole component of their field is identically zero. Thus, the resonant structure of the thermal radiation spectra is determined by the fi nite-length effects and also depends on the peculiar conductivity of SWNTs. Note that a similar structure of the thermal radiation spectra is predicted for the two-dimensional electron gas (Richter et al., 2007).

Resonances in the article by Richter et al. (2007) are due to the excitations of other physical nature—optical phonon modes of the barrier material. The presence of singled out resonances illustrated in Figure 5.4a allows us to propose metallic SWNTs as far-field and near-field thermal antennas for the THz range (optical thermal antennas based on photonic crystals have recently been considered in the article by Laroche et al. (2006) and Florescu et al. (2007). Taking into account the high temperature stability of SWNTs, the SWNT thermal antennas can be excited by Joule heating from the direct electric current. Low-frequency alteration of the current allows the amplitude modulation of thermal emission and, consequently, allows the use of the thermal emission for information transmission (similar to modulated RF fields in present-day radioengineering). The scattering pattern of the thermal antenna can be calculated using the approaches developed in Hanson (2005) and Slepyan et al. (2006) and is found to be partially polarized and directional. A polarization of the thermal radiation from bundles of multi-walled CNTs has been observed experimentally in Li et al. (2003). The blackbody spectrum reported in Li et al. (2003) is due to the inhomogeneous broadening originated from the SWNT length and radius dispersion and multi-walled effects. Moreover, the observation was made above the frequency range of surface plasmons. According to the article by Hanson (2005), Slepyan et al. (2006), and Burke et al. (2006) the maximal efficiency of vibrator SWNT antennas is reached at frequencies of the surface-plasmon dipole resonances. Figure 5.4a shows that the intensities of spectral lines of the thermal radiation go down with the resonance number much slowly than the polarizability peaks. Th is means that the signal–noise ratio for the SWNT-based antennas is maximal for the first resonance and decreases fast with the resonance number. As different from metallic SWNTs, semiconducting ones do not reveal isolated resonances in both far-field and near-field zones (see Figure 5.5 as an illustration). Such a peculiarity can easily be understood by accounting for the strong attenuation of surface plasmons in semiconducting SWNTs, whereas the slowing down remains of the same order. That is why in this case the Q factor of geometrical resonances turns out to be substantially smaller, and the resonances do not manifest themselves as separated spectral lines. In the same way, the thermal radiation intensity of semiconducting SWNTs is substantially smaller than that of metallic ones and displays qualitatively different spectral properties in the near-field zone: monotonous growth of the intensity with frequency inherent to the far-field zone changes into monotonous declining (see Figure 5.5a). As the thermal spectra are strongly dependent on the SWNT conductivity type and length, the near-field thermal radiation spectroscopy proposed in Schegrov et al. (2000) for testing the surface-plasmon structures can be expanded to SWNTs. Figure 5.5b demonstrates that in the frequency range considered, the blackbody radiation intensity considerably exceeds the thermal radiation of semiconducting SWNTs: I ω I ω( B ) . In the regions between geometrical resonances, the same property is


inherent to metallic SWNTs (see Figure 5.4b). This means that CNTs as building blocks for nanoelectronics and nanosensorics possess uniquely low thermal noise and, thus, provide high electromagnetic compatibility on the nanoscale: Their contribution to the electromagnetic fluctuations in nanocircuits is negligibly small as compared to the contribution of dielectric substrate. More generally, the latter example illustrates the peculiarity of the electromagnetic compatibility problem on the nanoscale, motivating future research investments into the problem. Next, we have studied the spatial structure of the electromagnetic fluctuations near SWNTs, characterized by the normalized first-order correlation tensor: An (r1) A*m (r2)

(1) g nm (r1, r2, ω) =

2

| An (r1) |

ω

ω

| Am (r2) |2

.

(5.36)

ω

The axial–axial component of this tensor is depicted in Figure 5.6. The figure clearly displays the distinctive behavior of the correlation in the far- and near-field zones. In the vicinity of geometrical resonances, where Re(κ)L ∼ 1, the near-field zone is defi ned by the condition ρ 0) are arbitrary signal amplitudes, and M (t ) = exp ⎡⎣ −(t /T1e )β ⎤⎦

(6.6)

is the reduced magnetization recovery of the 13C nuclear spins. Figure 6.14. shows the results of M(t) for the inner tubes as a function of the scaled delay time, t T1e , under various experimental conditions listed in the figure. M(t) does not follow the single exponential form with β = 1 (dashed line), but instead fits well to a stretched exponential form with β ≃ 0.65(5), implying a distribution in the relaxation times, T1. The data in Figure 6.14. is displayed on a semilog scale for the time axis in order to emphasize the data for earlier decay times and to illustrate the collapse of the data set for the upper 90% of the NMR signal. For a broad range of experimental conditions, the upper 90% of the M(t) data is consistent with constant β ≃ 0.65(5) (see inset), implying a field- and temperature-independent underlying distribution in T1. The lower 10% of the M(t) data, corresponding to longer delay times, comes from the nonenriched outer walls which have much longer relaxation times under similar experimental conditions (Tang et al. 2000, Goze-Bac et al. 2002). The collapse of the data set in Figure 6.14 to Equation 6.6 with constant β = 0.65(5) is a remarkable experimental observation. From an experimental point of view, it implies that all one needs to characterize the T and H dependence of the underlying T1 distribution is the bulk (or average) value, T1e (Equation 6.6). From an interpretational point of view, it implies that each inner tube in the powder sample has a different value of T1, yet all the T1 components and therefore all the inner tubes follow the same T and H dependence within experimental uncertainty. This finding is in contrast to earlier reports in SWCNTs where M(t) fits well to a biexponential distribution, 1/3 of which had

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Isotope Engineering in Nanotube Research

0.8 16 K 6K 238 K 360 K

M(t)

0.6

0.4

10

3.6 T 9.3 T 3.6 T 9.3 T T (K )

100

0.8

0.2

0.6 0.4

0.0 0.01

0.1 t/T 1e

1

10

FIGURE 6.14 Reduced nuclear magnetization recovery, M(t), as a function of the scaled delay time, t/T1e , for various temperature and magnetic field values. Both axes are dimensionless. Solid curve shows stretched exponential fit with β = 0.65 and dashed curve shows single exponential with β = 1. Inset shows temperature dependence of the best fit values of β at 3.6 Tesla (•) and 9.3 Tesla (∘), and average value of the data set β = 0.65 (solid line). (Reprinted from Singer, P.M. et al., Phys. Rev. Lett., 95, 236403-1, 2005. With permission.)

a short T1 value characteristic of fast relaxation from metallic tubes, and the remaining 2/3 had long T1 corresponding to the semiconducting tubes (Tang et al. 2000, Goze-Bac et al. 2002, Shimoda et al. 2002, Kleinhammes et al. 2003), as expected from a macroscopic sample of SWCNTs with random chiralities. The data for the inner tubes in DWCNTs differ in that a similar biexponential fit to M(t) is inconsistent with the shape of the recovery in Figure 6.14. Furthermore, if there were 1/3 metallic and 2/3 truly semiconducting inner tubes in the DWCNT sample, one would expect the ratio of T1 between semiconducting and metallic tubes to increase exponentially with decreasing T below the semiconducting gap (~5000 K). As a consequence, one expects an increasingly large change in the underlying distribution in T1 with decreasing T. This change would manifest itself as a large change in the shape of the M(t); however, this is not the case as shown in Figure 6.14. The possibility of two components in T1 with different T dependence can therefore be ruled out, and instead it could be concluded that all T1 components (corresponding to distinct inner tubes) exhibit the same T and H dependence within experimental scattering. The experimentally observed uniform metallicity of inner tubes is a surprising observation. Th is is suggested to be caused by the shift ing of the inner tube Fermi levels due to charge transfer between the two tube walls. Indeed, ab initio calculations found that charge transfer and hybridization can render an otherwise semiconducting tube metallic (Okada and Oshiyama 2003, Zólyomi et al. 2008).

With these arguments, the bulk average, T1e, defined in Equation 6.6 is considered and its uniform T and H dependence can be followed. The M(t) data can be fitted with the constant exponent β = 0.65(5), which reduces unnecessary experimental scattering in T1e. In Figure 6.15, we show the temperature dependence of 1/T1eT for two different values of the external magnetic field, H. The data can be separated into two temperature regimes; the high-temperature regime > ∼150 K, and the low T regime Δ,

(6.7)

otherwise,

here, E is taken with respect to the Fermi energy. Equation 6.7 is used to calculate 1/T1eT (Moriya 1963) as such

5 9.3 T 3.6 T 4 (T 1e T )–1 (10–3 s–1K–1)

1.0

3

2

1

0 0

50

100

150 T (K)

200

250

300

FIGURE 6.15 Temperature dependence of spin-lattice relaxation rate divided by temperature, 1/T1eT, in units of (103 × s−1 K−1). Solid curves are best fits to Equation 6.8 with 2Δ = 46.8(40.2) K for H = 3.6(9.3) Tesla, respectively. (Reprinted from Singer, P.M. et al., Phys. Rev. Lett., 95, 236403-1, 2005. With permission.)

6-12

Handbook of Nanophysics: Nanotubes and Nanowires ∞

1 δf = α(ω) n(E)n(E + ω) ⎛⎜ − ⎞⎟ dE, e T1 T ⎝ δE ⎠

∫

×10–3

(6.8)

5

−∞

4.5

The results of the best fit of the data to Equation 6.8 are presented in Figure 6.15, where 2Δ = 43(3) K (≡3.7 meV) is H independent within experimental scattering between 9.3 and 3.6 Tesla. The explanation in the noninteracting electron picture has two shortcomings: (1) calculations show that SG is of the order of a few 100 meV, which is two orders of magnitude larger than the experimental value; (2) the SG is expected to be strongly chirality dependent. In fact, a gap induced by electron– electron or electron–phonon correlations could be of the right order of magnitude (Bohnen et al. 2004, Connétable et al. 2005) and it could be uniform, i.e., chirality independent. The problem with such a correlation-induced gap is its magnitude: experimentally, the gap is open above 300 K, thus the critical transition of the correlation is Tc > 300 K. However, a meanfield expression between the gap and the critical temperature usually satisfies that 2Δ/TC > 3.52 (Bardeen et al. 1957) but the current value is 0.13 or smaller. To avoid this contradiction, the NMR data was reinterpreted in the framework of the TLL theory by B. Dora and coworkers (Dóra et al. 2007), which we outline here. The TLL state occurs in one-dimensional systems with strong electron–electron correlation. Formally, the interacting electrons can be treated with a noninteracting bosonic Hamiltonian that contains two TLL parameters Kc ≈ 0.2 for the charge and Ks ≈ 1 for the spin degree of freedom. The physically relevant correlation functions, such as, e.g., the spin–spin or current–current correlation functions follow power-law dependencies with exponents related to the TLL parameters. This leads to power-law behavior in the experimental measurables as, e.g., (T1T)−1 ∝ T −1. The latter result can be qualitatively explained by the localization of electrons in the TLL state, which leads to a paramagnetic-like −1 fluctuating fields, giving the increase of (T1T) with decreasing temperature (Abragam 1961). The apparent presence of the gap at low temperatures was explained by the formation of the socalled Luther–Emery liquid (Dóra et al. 2007), which is a ground state that contains a gap in the excitation spectrum and competes with the TLL state. We show the calculated (T1T)−1 values in Figure 6.16. Although, the calculation relies on essentially three parameters only (the two TLL and a vertical scaling parameter, the gap being fi xed to the temperature where the gap opens) we observe a much better agreement between the data and the calculation as compared to the gapped Fermi-liquid explanation shown in Figure 6.15. This shows that the TLL description is indeed relevant in describing the NMR data.

4 3.5 (T1T )–1 (sK)–1

where E and ω are in temperature units for clarity f is the Fermi function, f = [exp(E/T) + 1]−1 the amplitude factor α(ω) is the high temperature value for 1/T1eT .

3 2.5 2 1.5 1 0.5 0

0

50

100

150

200

250

300

T (K)

FIGURE 6.16 Calculated (T1T)−1 assuming a TLL ground state of the electrons in the inner tube SWCNTs. The magnetic field dependence is associated to the slight field dependence of the Ks TLL parameter. (Reprinted from Dóra, B. et al., Phys. Rev. Lett., 99, 166402-1, 2007. With permission.)

Summarizing the NMR studies on DWCNTs, it was shown that T1 has a similar T and H dependence for all the inner tubes with no indication of a metallic/semiconducting separation due to chirality distributions. Below ~150 K, 1 / T1eT increases dramatically with decreasing T and a gap in the spin excitation spectrum is found below Δ ~ – 20 K. The result can be understood if electrons on the inner tubes are in the TLL state above 20 K and in the Luther–Emery liquid state below. Th is makes SWCNTs the only known example of materials where the TLL state is observed using NMR.

6.4 Summary In summary, we reviewed recent advances in the isotope engineering of SWCNTs. We showed how nanotube-specific 13C isotope enrichment can be achieved by encapsulating 13C enriched fullerenes inside host SWCNTs and by transforming them into a smaller inner nanotube. The process produces highly 13C enriched inner tubes while the host outer tube and other carbonaceous side-products in the SWCNT sample consists of natural carbon. This material allows to identify Raman modes of the DWCNTs. The use of mixtures of natural and 13C enriched fullerenes allows to prove that no diff usion of carbon happens along the nanotube axis during inner nanotube synthesis, which supports the fullerene fusion model for their growth. The isotope-enriched inner tubes are excellent for NMR studies. Measurement of the 13C NMR T1 relaxation time allows to identify a non-Fermi-liquid behavior above 20 K and a low-energy,


correlation-related gap providing direct experimental evidence for the TLL state in SWCNTs.

Acknowledgments This work was supported by the Hungarian State Grants No. F61733 and NK60984, and by the Bolyai postdoctoral fellowship of the Hungarian Academy of Sciences. H. Kuzmany, R. Pfeiffer, T. Pichler, M. Rümmeli, C. Kramberger, H. Alloul, P. M. Singer, P. Wzietek, B. Dóra, M. Gulácsi, J. Bernardi, F. Fülöp, A. Rockenbauer, A. Jánossy, L. Forró, V. N. Popov, J. Koltai, V. Zólyomi, and J. Kürti are acknowledged for their participation in the reviewed works. Dario Quintavalle is acknowledged for preparing some of the figures.

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Rauf, H., Pichler, T., Knupfer, M., Fink, J., and Kataura, H. (2004). Transition from a tomonaga-luttinger liquid to a fermi liquid in potassium-intercalated bundles of single-wall carbon nanotubes, Phys. Rev. Lett. 93: 096805-1–096805-4. Reich, S., Thomsen, C., and Maultzsch, J. (2004). Carbon Nanotubes, Wiley-VCH, Weinheim, Germany. Ren, W., Li, F., Chen, J., Bai, S., and Cheng, H.-M. (2002). Morphology, diameter distribution and Raman scattering measurements of double-walled carbon nanotubes synthesized by catalytic decomposition of methane, Chem. Phys. Lett. 359: 196–202. Rochefort, A. (2003). Electronic and transport properties of carbon nanotube peapods, Appl. Magn. Reson. 67: 11540117. Rümmeli, M. H., Löffler, M., Kramberger, C., Simon, F., Fülöp, F., Jost, O., Schönfelder, R., Grüneis, R., Gemming, T., Pompe, W., Büchner, B., and Pichler, T. (2007). Isotope-engineered single-wall carbon nanotubes; A key material for magnetic studies, J. Phys. Chem. C 111: 4094. Saito, R., Dresselhaus, G., and Dresselhaus, M. (1998). Physical Properties of Carbon Nanotubes, Imperial College Press, London, U.K. Shimoda, H., Gao, B., Tang, X. P., Kleinhammes, A., Fleming, L., Wu, Y., and Zhou, O. (2002). Lithium intercalation into opened single-wall carbon nanotubes: Storage capacity and electronic properties, Phys. Rev. Lett. 88: 15502. Shlimak, I. (2004). Isotopically engineered silicon nanostructures in quantum computation and communication, HAIT J. Sci. Eng. 1: 196–206. Simon, F., Kramberger, C., Pfeiffer, R., Kuzmany, H., Zólyomi, V., Kürti, J., Singer, P. M., and Alloul, H. (2005a). Isotope engineering of carbon nanotube systems, Phys. Rev. Lett. 95: 017401. Simon, F., Kukovecz, A., Kramberger, C., Pfeiffer, R., Hasi, F., Kuzmany, H., and Kataura, H. (2005b). Diameter selective characterization of single-wall carbon nanotubes, Phys. Rev. B 71: 100–122. Simon, F., Pfeiffer, R., Kramberger, C., Holzweber, M., and Kuzmany, H. (2005c). The Raman response of double wall carbon nanotubes, in Applied Physics of Carbon Nanotubes, S. V. Rotkin and S. Subramoney (eds.), Springer, New York, pp. 203–224. Simon, F., Peterlik, H., Pfeiffer, R., and Kuzmany, H. (2007). Fullerene release from the inside of carbon nanotubes: A possible route toward drug delivery, Chem. Phys. Lett. 445: 288–292. Singer, P. M., Wzietek, P., Alloul, H., Simon, F., and Kuzmany, H. (2005). NMR evidence for gapped spin excitations in metallic carbon nanotubes, Phys. Rev. Lett. 95: 236403-1–236403-4. Slichter, C. P. (1989). Principles of Magnetic Resonance, 3rd edn. 1996 edn, Spinger-Verlag, New York. Smith, B. W. and Luzzi, D. (2000). Formation mechanism of fullerene peapods and coaxial tubes: A path to large scale synthesis, Chem. Phys. Lett. 321: 169–174. Smith, B. W., Monthioux, M., and Luzzi, D. E. (1998). Encapsulated C60 in carbon nanotubes, Nature 396: 323–324. Smith, B. W., Monthioux, M., and Luzzi, D. (1999). Carbon nanotube encapsulated fullerenes: A unique class of hybrid materials, Chem. Phys. Lett. 315: 31–36.

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7 Raman Spectroscopy of sp2 Nano-Carbons 7.1

Introduction ............................................................................................................................. 7-1 Nanoscience, Nanotechnology, and sp2 Carbon • Importance of Graphite, Carbon Nanotubes, and Graphene

7.2

Background...............................................................................................................................7-4

7.3

Presentation of State-of-the-Art Raman Spectroscopy of sp2 Nano-Carbons ...............7-6

Light–Matter Interaction and Raman Spectroscopy • Basic Concepts of Raman Spectroscopy

Mildred S. Dresselhaus Massachusetts Institute of Technology

Gene Dresselhaus Massachusetts Institute of Technology

Ado Jorio Universidade Federal de Minas Gerais

The sp2 Model System: Graphene • Adding Graphene Layers: From a Single-Layer Graphene to Graphite • Rolling Up One Graphene Layer: The Single-Wall Carbon Nanotube • Adding Tube Layers: Double- and Multi-Wall Carbon Nanotubes • Disorder in sp2 Systems • Other Raman Modes and Other sp2 Carbon Structures

7.4 Critical Discussions ............................................................................................................... 7-15 7.5 Summary ................................................................................................................................. 7-15 7.6 Future Perspectives................................................................................................................ 7-17 Acknowledgments ............................................................................................................................. 7-17 References........................................................................................................................................... 7-17

7.1 Introduction Different from scanning probe and electron microscopy-related techniques, optics is one of the oldest characterization techniques for materials science, being largely used long before nanoscience could even be imagined. In the age of nano, optics still sustains its kingdom. The advantages of optics for nanoscience relate to both experimental and fundamental aspects. Experimentally, the techniques are readily available, relatively simple to perform, possible at room temperature and under ambient pressure, and require relatively simple or no sample preparation. Fundamentally, the optical techniques (normally using infrared and visible wavelengths) are nondestructive and noninvasive because they use the photon, a massless and chargeless particle, as a probe. The sp2 carbon materials and Raman spectroscopy have a special place in the nanoworld. It is possible to observe Raman scattering from one single sheet of sp2–hybridized carbon atoms, the two-dimensional (2D) graphene sheet, and from a narrow strip of graphene sheet rolled up into a 1 nm-diameter cylinder—the one-dimensional (1D) single-wall carbon nanotube (SWNT). These observations are possible just by shining a light on the nanostructure focused through a regular microscope. Th is chapter focuses on both the basic concepts of sp2 carbon nanomaterials and Raman spectroscopy, together with their interaction.

7.1.1 Nanoscience, Nanotechnology, and sp2 Carbon To fully understand the importance of Raman spectroscopy of sp2 carbons in the nano-carbon context, it is important to understand that nano-carbons are part of the nanoworld and address structures with sizes between the molecular and the macroscopic. The Technical Committee (TC-229) for nanotechnologies standardization of the International Organization for Standardization (ISO) defines the field of nanotechnologies as the application of scientific knowledge to control and utilize matter at the nanoscale, where size-related properties and phenomena can emerge. The nanoscale is the size range from approximately 1 to 100 nm. It is not possible to clearly envisage the future results of nanotechnology, or even the limit for the potential of nanomaterials, but clearly, serious fundamental challenges have to be overcome, such as • Constructing nanoscale building blocks precisely and reproducibly • Finding and controlling the rules for assembling these objects into complex systems • Predicting and probing the emergent properties of these systems These challenges are not only technological, but also conceptual: how do you treat a system that is too big to be solved by 7-1

7-2


present-day fi rst principles calculations, and too small for statistical methods? Although these challenges punctuate nanoscience and nanotechnology, the success here will represent a revolution in larger-scale scientific challenges in the fields of emergent phenomena and information technology. The answers to questions like “How do complex phenomena emerge from simple ingredients?” and “How will the information technology revolution be extended?” will probably come from nanotechnology. It is exactly in this context that nano-carbon is playing a very important role. From one side, nature shows that it is possible to manipulate matter and energy the way integrated circuits manipulate electrons, by assembling complex self-replicating carbon-based structures able to sustain life. From another side, carbon is the upstairs neighbor of silicon in the periodic table, with more flexible bonding and unique physical, chemical, and biological properties, which nevertheless hold promise of a revolution in electronics at some future time. Three important aspects make sp2 carbon materials special for facing the nano-challenges listed in the previous paragraph: first, the unusually strong covalent bonding between neighboring atoms; second, the extended π-electron clouds; and third, the simplicity of the system. We elaborate shortly on these aspects in the following paragraphs. Carbon has six electrons: two are core 1s states, and four occupy the 2s and 2p orbitals. In the sp2 configuration, the 2s, px, and py orbitals mix to form three covalent bonds, 120° from each other in the xy plane (see Figure 7.1). Each carbon atom has three neighbors, forming a hexagonal network. These nearestneighbor sp2 bonds are stronger than the nearest-neighbor sp3 bonds in diamond, making graphene (a single sheet of sp2 atoms) stronger than diamond in tensile strength. This added strength is advantageous for sp2 carbons as a prototype material for the development of nanoscience and nanotechnology, since different interesting nanostructures (sheets, ribbons, tubes, horns, fullerenes, etc.) are stable and strong enough for exposure to many different types of characterization and processing steps. The pz electrons that remain perpendicular to the hexagonal network (see Figure 7.1) form the delocalized π electron states. For this reason, sp2 carbons, which include graphene, graphite, carbon nanotubes, fullerenes, and other carbonaceous materials, are also called π-electron materials. These delocalized

electronic states are highly unusual, because they behave like relativistic Dirac Fermions, i.e., they exhibit a massless-like linear momentum–energy relation (like a photon), and are responsible for unique optical and transport (both thermal and electronic) properties. These two properties accompany a very important aspect of sp2 carbons—the simplicity of a system formed by only one type of atom in a periodic hexagonal structure. Therefore, different from most materials, sp2 nano-carbons allow us to access their special properties from both experimental and theoretical approaches. Being able to model the structure is crucial for the development of our methodologies and knowledge.

7.1.2 Importance of Graphite, Carbon Nanotubes, and Graphene The ideal concept of sp2 nano-carbons starts with the graphene sheet (see Figure 7.2). Adding one or two layers produces the bilayer and trilayer graphene. Roll up a narrow strip of graphene into a seamless cylinder and you have a SWNT. Add one-layer or two-layer concentric cylinders and you have double-wall and triple-wall carbon nanotubes. Many rolled-up cylinders would make a multi-wall carbon nanotube (MWNT), and many flat layers give graphite. This ideal concept is didactic, but historically these materials came into human knowledge in the opposite order. Three-dimensional (3D) graphite is one of the longest-known forms of pure carbon, formed by graphene planes usually in an ABAB stacking (Wyckoff 1981). Of all materials, graphite has the highest melting point (4,200 K), the highest thermal conductivity (3,000 W/mK), and a high room temperature electron

(a)

(b)

π σ σ

σ π

(c)

FIGURE 7.1 The carbon atomic orbitals in the sp2 honeycomb lattice. (From Pfeiffer, R. et al., in Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties and Applications, Springer Series in Topics in Applied Physics, Vol. 111, Jorio, A. et al. (eds.), Springer-Verlag, Berlin, Germany, 2008, 495–530. With permission.)

(d)

FIGURE 7.2 The sp2 carbon materials, including (a) single-layer graphene, (b) triple-layer graphene, (c) SWNT, and (d) a C60 fullerene, which includes pentagons in the structure. (From Castro Neto, A.H. et al., Rev. Mod. Phys., 81, 109, 2008. With permission.)

7-3

Raman Spectroscopy of sp2 Nano-Carbons

mobility (30,000 cm 2/Vs; Dresselhaus et al. 1988). Graphite and its related carbon fibers (Dresselhaus et al. 1988) have been used commercially for decades. Their applications range from use as conductive fi llers and mechanical structural reinforcements in composites (e.g., in the aerospace industry) to their use in electrode materials utilizing their resiliency (e.g., in batteries; Endo et al. 2008). In 1985, a unique discovery in another sp2 carbon system took place: the C60 fullerene molecule (see Figure 7.2; Kroto et al. 1985). The fullerenes stimulated and motivated a large scientific community from the time of its discovery up to the end of the century, but their applications remain sparse to date. Carbon nanotubes arrived on the scene following in the footsteps of the C60 fullerene molecule. Carbon nanotubes have evolved into one of the most intensively studied materials, and are held responsible for co-triggering the nanotechnology revolution. The big rush on carbon nanotube science started after the observation of MWNTs on the cathode of a carbon arc used to produce fullerenes (Iijima 1991), even though they were identified in the core structure of vapor grown carbon fibers as very small carbon fibers in the 1970s (Oberlin et al. 1976) and in the 1950s in Russian literature (Radushkevich and Lukyanovich 1952; see Figure 7.3). SWNTs were first synthesized in 1993 (Bethune et al. 1993, Iijima and Ichihashi 1993). The interest in the fundamental properties of carbon nanotubes and their exploitation through a wide range of applications is due to their unique structural, chemical, mechanical, thermal, optical, optoelectronic, and electronic properties (Saito et al. 1998, Reich et al. 2003). The growth of a single SWNT at a specific location and pointing in a given direction (Zhang et al. 2001, Huang et al. 2003), and the growth of a huge amount of millimeter-long tubes with nearly 100% purity (Hata et al. 2004) have been achieved (Joselevich et al. 2008). Substantial success with the separation of nanotubes (Arnold

et al. 2006) by metallicity and length has been achieved and advances have been made with doping nanotubes for the modification of their properties (Terrones et al. 2008). Studies on nanotube optics, magnetic properties, transport, and electrochemistry have exploded, revealing many rich and complex fundamental excitonic and other collective phenomena (Jorio et al. 2008a). Quantum transport phenomena, including quantum information, spintronics, and superconducting effects have also been explored (Biercuk et al. 2008). After a decade and a half of intense activity in carbon-nanotube research, more and more attention is now focusing on the practical applications of the many unique and special properties of carbon nanotubes (Endo et al. 2008). In the meantime, the study of nano-graphite was under development, and in 2004, Novoselov et al. discovered a simple method to transfer a single atomic layer of carbon from the c-face of graphite to a substrate suitable for the measurement of its electrical and optical properties (Novoselov et al. 2004). This finding led to a renewed interest in what was considered to be a prototypical, yet theoretical, 2D system, providing a basis for the structure of graphite, fullerenes, carbon nanotubes, and other nano-carbons. Surprisingly, this very basic graphene system first prepared by Boehm (Boehm 1962) in monolayer form, which had been studied for many decades, suddenly appeared with many novel physical properties that were not even imagined previously (Geim and Novoselov 2007, Castro Neto et al. 2008). In one or two years, the rush on graphene science started. The scientific interest was stimulated by the report of the relativistic properties of the conduction electrons (and holes) in a single graphene layer less than 1 nm thick, which is responsible for the unusual properties in this system (see Figure 7.4) (Novoselov et al. 2005, Zhang et al. 2005). Many groups are now making devices using graphene and also graphene ribbons, which have a long length and a small width, and where the ribbon edges play an important role. – 120

a

010 α 000

110 100

b 100 Å (a)

(b)

D 1.37 nm (c)

FIGURE 7.3 Transmission electron microscopy images of carbon nanotubes. The early reported observations (a) in 1952 (From Radushkevich, L.V. and Lukyanovich, V.M., Zum. Fisc. Chim., 26, 88, 1952.), (b) in 1976 (From Oberlin, A. et al., J. Cryst. Growth, 32, 335, 1976. With permission.), and (c) in 1993 the observation of SWNTs that launched the field (From Iijima, S. and Ichihashi, T. et al., Nature, 363, 603, 1993. With permission.)

7-4


Rxx (kΩ)

4

2

(a)

0

80

40 30 0 20

nS (1011 cm–2)

μ (103 cm2/Vs)

40

–40

10

–80

0 –80

–40

(b)

0

40

80

Vg (V)

FIGURE 7.4 Resistivity, mobility, and carrier density as a function of gate voltage Vg (Vg in the figure) in a single-layer graphene field effect transistor device (Charlier et al. 2008). (a) Vg-dependent R xx showing a finite value at the Dirac point, where the valence and conduction bands are degenerate and the carriers are massless. The resistivity ρxx can be calculated from R xx using the geometry of the device. The inset is an image of a graphene device on a Si:SiO2 substrate. The Si is the bottom gate; five top electrodes that formed via e-beam lithography are shown in the inset. The scale bar is 5 μm. (b) Mobility μ and carrier density nS as a function of Vg (for holes Vg < 0 and for electrons Vg > 0). The mobility (dotted curve) diverges at the Dirac point due to the fi nite resistivity. (From Pfeiffer, R. et al., in Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties and Applications, Springer Series in Topics in Applied Physics, Vol. 111, Jorio, A. et al. (eds.), Springer-Verlag, Berlin, Germany, 2008, 495–530. With permission.)

Having introduced nano-carbons in the nanoworld, we now focus on the Raman spectroscopy of nano-carbons. Section 7.2 discusses the basic concepts of light–matter interaction and Raman spectroscopy. A more detailed presentation of stateof-the-art Raman spectroscopy of sp2 nano-carbons is given in Section 7.3, followed by a critical discussion in Section 7.4. Section 7.5 summarizes the main aspects of this paper, and Section 7.6 presents future perspectives.

7.2 Background 7.2.1 Light–Matter Interaction and Raman Spectroscopy When shining light into a material, part of the energy can just pass through (transmission), while the remaining energy interacts with the system through several different mechanisms.

From the light that interacts with the system, many different effects might occur: (1) a photon, which is the quantum unit of light, can be absorbed and transformed into atomic vibrations, i.e., heat, which can be represented by phonons, the quantum units of lattice vibrations; (2) a photon can be absorbed and transformed into a photon with a lower energy that is emitted. Th is process is called photoluminescence and it happens in semiconducting materials when the energy of the incident photon exceeds the energy gap between the valence and conduction bands; (3) a photon may not be absorbed, but it just shakes the electrons, which will scatter that energy back into another photon with the same energy as the incident one. Th is is an elastic scattering process named Rayleigh scattering; or (4) the photon may shake the electrons causing oscillations of the atoms according to their natural vibrational frequencies, thereby changing the electronic configuration of the atoms. In this case, when the electrons scatter the energy back into another photon, this photon will have lost or gained energy to or from the atoms. This is an inelastic scattering process called Raman scattering. Many other processes may occur but they are usually less important for the energetic balance of the light– matter interaction. The amount of light that will be transmitted, as well as the details for all the light–matter interactions will be determined by the electronic and vibrational properties of the material, thus making light a very powerful characterization tool for materials science studies, while gently perturbing the material.

7.2.2 Basic Concepts of Raman Spectroscopy This section gives the basic definitions for the terms that are used in describing Raman spectroscopy. 7.2.2.1 Raman Scattering The Raman effect refers to the inelastic scattering of light. An incident photon with energy Ei = Elaser and momentum ki = k laser reaches the sample and is scattered, resulting in a photon with a different energy E S and momentum k S. For energy and momentum conservation, ES = Ei ± Eq ,

(7.1)

kS = ki ± kq ,

(7.2)

where Eq and k q are the energy and momentum change during the scattering event, mediated by an excitation of the medium. Although electronic excitations can result from Raman scattering, the most usual scattering outcome is the excitation of atomic normal mode vibrations. These vibrational modes are related to the chemical and structural properties of materials, and since every material has a unique set of such normal modes, Raman spectroscopy can be used to probe materials properties in detail and to provide an accurate characterization of specific materials. Raman spectroscopy in particular

7-5


provides a rich variety of characterization information regarding carbon nanostructures. 7.2.2.2 Stokes and Anti-Stokes Raman Processes In the inelastic scattering process, the incident photon can decrease or increase its energy by destroying (anti-Stokes) or creating (Stokes) a quantum of normal mode vibration (i.e., a phonon) in the medium. The plus and minus signs in Equations 7.1 and 7.2 apply when energy has been received from or transferred to the medium, respectively. The probability for the two types of events depends on the excitation photon energy Ei, and this dependence can be explored for an accurate determination of electronic transition energies. Furthermore, the probability to destroy a phonon depends on the phonon population given by the Bose–Einstein distribution and, consequently, the anti-Stokes event also depends on the temperature, according to I S / I AS = exp(Eq / kBT ),

(7.3)

where IS and IAS denote the measured intensity for the Stokes and anti-Stokes peaks, respectively kB is the Boltzmann constant T is the temperature Because the anti-Stokes process is usually less probable than the Stokes process, it is usual that people only care about the Stokes spectra. In this work, when not referring explicitly to the type of process, it is the Stokes process that is being addressed. 7.2.2.3 Energy Conservation A Raman spectrum is a plot of the scattering intensity as a function of ES – Elaser. Therefore, the energy conservation relation given by Equation 7.1 is the most important aspect of Raman spectroscopy. Although the anti-Stokes process has a positive net energy, it is the Stokes spectra that are most usually measured, and for simplicity, the Stokes process is designated as being positive. The Raman spectra will show peaks at ±Eq, where Eq is the energy of the excitation associated with the Raman effect. The quantum of excitation denoting the normal vibrational modes is named the phonon, and is regularly used to describe the lattice vibrations in crystals. We will use the term “phonon” frequently in this chapter. The phonon excitation energies are found by decomposing the atomic vibrations into the vibrational normal modes of the material. 7.2.2.4 Energy Units The energy axis in the Raman spectra is usually displayed in units of cm−1. Lasers are usually described by the wavelength of the light, i.e., in nanometers, but the phonon energies are usually too small a number when displayed in nm, which is not a comfortable unit for denoting Raman shifts. The accuracy of a common Raman spectrometer is on the order of 1 cm−1, which is

equivalent to 10−7 nm. The energy conversion factors are: 1 eV = 8065.5 cm−1 = 2.418 × 1014 Hz = 11,600 K. Also 1 eV corresponds to a wavelength of 1.2398 μm. 7.2.2.5 Shape of the Raman Peak The response of a forced damped harmonic oscillator, in the limit that the peak frequency ωq is much larger than the peak width Γq, is a Lorentzian curve. Therefore, the Raman peaks usually have a Lorentzian shape. The center of the Lorentzian gives the natural vibration frequency, and the full width at half maximum (FWHM) is related to the damping, which gives the phonon lifetime. However, in specific cases, the Raman feature can deviate from the simple Lorentzian shape. One obvious case is when the feature is actually composed of many phonon contributions. Then the Raman peak will be a convolution of several Lorentzian peaks, giving rise to Gaussian, Voigt, or more complex lineshapes. Another case is when the lattice vibration couples with electrons. In this case, additional line broadening and even an asymmetric lineshape can result. These effects are observed in certain metallic sp2 carbon materials, including SWNTs (Dresselhaus et al. 2005). 7.2.2.6 Resonance Raman Effect The laser energies are usually much higher than the phonon energies. Therefore, although the exchange in energy between the light and the medium is transferred to the atomic vibrations, the light–matter interaction is mediated by electrons. Usually, the photon energy is not large enough to achieve a real electronic transition, and the electron that absorbs the light is said to be excited to a “virtual state,” from where it couples to the lattice, generating the Raman scattering. However, when the excitation laser energy matches the energy gap between the valence and conduction bands in a semiconducting medium (or between an occupied initial state and an unoccupied fi nal state more generally), the probability for the scattering event to occur increases by many orders of magnitude, and the process is then called a resonance Raman process (non-resonant otherwise). The same happens if the scattered light matches such an electronic transition. The resonance effect is extremely important in systems where electronic transitions are in the visible range, which includes sp2 carbon systems. 7.2.2.7 First and Higher-Order Raman Processes The order of the Raman process is given by the number of scattering events involved in the scattering process. The most usual case is the fi rst-order Stokes Raman scattering process, where the photon energy exchange creates one phonon in the medium with a very small momentum (q ≈ 0). If 2, 3, or more phonons are involved in the same scattering event, the process is of a second, third, or higher-order, respectively. The fi rst-order Raman process gives the basic quanta of vibration, while higher-order processes give very interesting information about harmonics and combination modes (Dresselhaus et al. 2005).

7-6

7.2.2.8 Vibrational Structure Describing the vibrations of small molecules is simple. The number of vibrational modes is given by the number of degrees of freedom for atomic motion (i.e., the number of atoms N multiplied by three dimensions, minus six; the six coming from translations along x, y, z, and rotations around these axes). The number of vibrational energy levels may be smaller than 3N − 6 since some levels can be energy degenerate. Finding the normal modes of large and complex molecules, such as proteins, is not an easy task because of the large number of degrees of freedom. In cases like that, it is common to fi nd the spectral features identified with local bondings (e.g., C–C, C=C, C=O, etc.) rather than the actual molecular normal modes. Crystals have a large number of atoms (ideally infi nite), but periodic systems are, again, quite simple to describe, although such descriptions require an understanding of the phonon dispersion relations. Since ideal crystalline structures have an infinite number of atoms, the number of vibrational levels is infinite. The vibrational structures of these systems are displayed in a plot of the phonon energy (ωq) vs. phonon wave vector q, i.e., the phonon dispersion relation for each distinct normal mode phonon. The ωq vs. q plots are composed of continuous curves called phonon branches. Being continuous, they account for the infinity of vibrational levels. The number of phonon branches depends on the number of degrees of freedom for the atomic motion for a unit cell for this material. The wave vector q is defined by the magnitude and phase for a normal mode vibration that can involve more than one consecutive unit cell.


This discussion gives an explanation for why the first-order Raman process can only access phonons at q → 0, i.e., at the Γ point. The momenta associated with the first-order light scattering process are on the order of ki = 2π/λlight, where λlight is in the visible range (800–400 nm). Therefore, ki is a very small number when compared to the dimensions of the first Brillouin zone, which is limited to vectors no longer than q = 2π/a, where the unit cell vector a in real space is on the order of tenths of nm. 7.2.2.10 Coherence It is not trivial to define whether a real system is big enough to be considered as effectively infinite and to exhibit a quasi-continuous phonon (or electron) energy dispersion relation. Whether or not a dispersion relation can be defined indeed depends on the process that is under evaluation. In the Raman process, how long does it take for an electron excited by the incident photon to decay? Considering this scattering time, what is the distance felt by an electron? These problems are described in solid–state physics textbooks by the concept of coherence. The coherence time is the time the electron takes to suffer an event such as scattering that changes its state. Thus, the coherence length is the size over which the electron maintains its integrity, its coherence, and it is defined by the electron speed and the coherence time, which can be measured experimentally. The Raman process is an extremely fast process, in the range of femtoseconds (10−15 s). Considering the speed of electrons in graphite (106 m/s), this gives a coherence length on the order of nm. Interestingly, this number is much smaller than the wavelength of visible light. Actually, this is a particle picture for the scattering process and playing with these concepts is actually quite interesting.

7.2.2.9 Momentum Conservation and Back-Scattering The q vector carries information about the wavelength of the vibration (q = 2π/λ) and the direction along which the oscillation occurs. For λ → ∞, we have q → 0, and we usually denote the center of the phonon dispersion relation as the Γ point. The phonon wavelength λ cannot be smaller than the unit cell vector a, which defines an upper limit for the relevant values of q, namely, the unit cell of the crystal in the reciprocal space, which is called the first Brillouin zone. This zone provides a bounding polygon that confines the phonon dispersion. For q > |±π/a|, the phonon structure will repeat itself. For the momentum conservation given by Equation 7.2, different scattering geometries are possible, and specific choices may be used to select different phonons due to the symmetry selection rules (Dresselhaus et al. 2005). The back-scattering configuration, for which ki and kS have the same direction and opposite signs, is the most common when working with nanomaterials, because a microscope is usually needed to focus the light onto small samples. Furthermore, in the first-order Raman process, the momentum transfer is usually neglected, i.e., kS − ki ~ 0. Phonon momentum k S ≠ 0 becomes important in defect-induced or higher-order Raman scattering processes, and the theoretical background for such processes can be found in the article by Saito et al. (2003).

7.3 Presentation of State-of-the-Art Raman Spectroscopy of sp2 Nano-Carbons Figure 7.5 shows the Raman spectra from different crystalline and disordered sp2 carbon nanostructures. The first-order Raman bands go up to 1620 cm−1, and the spectra above this value are composed of overtone and combination modes. We focus here on the Raman spectra from crystalline structures (Sections 7.3.1 through 7.3.4) and on the disorder caused by defects in the sp2 structure (Section 7.3.5). The bottom spectrum in Figure 7.5 is from amorphous carbon that exhibits a considerable amount of sp3 bonds and some hydrogen satisfying dangling bonds. This is a rich field with important applications for industry, but outside the scope of this chapter. Discussions on amorphous carbon can be found in the article by Ferrari and Robertson (2004).

7.3.1 The sp2 Model System: Graphene Among the sp2 carbon systems, monolayer graphene is the simplest and has, consequently, the simplest Raman spectra (see Figure 7.6). The ideal graphene is a 2D crystalline sheet, one atom thick, with two C atoms in a unit cell that repeats itself to infinity.

7-7


2D

G΄

514 nm

G

Intensity

Graphene

HOPG

Intensity

RBM

G Graphene

SWNT Graphite

D G D΄

D + D΄ G΄ 2G

1400

Damaged graphene

SWNH

Amorphous carbon 0

2D΄

1000

2000

3000

4000

Raman shift (cm–1)

FIGURE 7.5 Raman spectra from several sp2 nano-carbons. From top to bottom: crystalline graphene, highly oriented pyrolytic graphite (HOPG), SWNT bundles, damaged graphene, single-wall carbon nanohorns (SWNH), and hydrogenated amorphous carbon. The most intense Raman peaks are labeled in a few of the spectra.

Its Raman spectrum is marked by two strong features, named the G and G′ bands (G from graphite), as discussed below. 7.3.1.1 The G Band The most usual Raman peak observed in the Raman spectra of any sp2 carbon system has historically been named the G band. The G mode is related to the stretching of the bonds between the nearest neighbor A and B carbon atoms in the unit cell. This feature appears around 1585 cm−1 (see Figure 7.6), but exhibits some specificities according to the particular nano-carbon sp2 system under discussion and according to ambient conditions of temperature, pressure, doping, and disorder. Because of the peculiar electronic dispersion of graphene, being a zero gap semiconductor with a linear E(k) dispersion relation, the G band phonons (energy of 0.2 eV) can promote electrons from the valence to the conduction band. For this reason, the electron-phonon coupling in this system is quite strong, and gives rise to a renormalization of the electronic and phonon energies, including a sensitive dependence on electron or hole doping (Das et al. 2008). The FWHM intensity observed for graphene on top of a Si/SiO2 substrate usually ranges from 6 to 16 cm−1, depending on the graphene to substrate interactions.


2800

FIGURE 7.6 Raman spectrum of single-layer graphene and graphite. The two most intense features are the Raman allowed fi rst order G band and the second-order G′ band (labeled 2D in the figure making reference to its origin as a second-order process related to the disorder-induced D peak). The spectrum of pristine single-layer graphene is unique in sp2 carbons for exhibiting a very intense G′ band as compared to the G band. (From Ferrari, A.C. et al., Phys. Rev. Lett., 97, 187401, 2006; Pfeiffer, R. et al., in Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties and Applications, Springer Series in Topics in Applied Physics, Vol. 111, Jorio, A. et al. (eds.), Springer-Verlag, Berlin, Germany, 2008, 495–530. With permission.)

Furthermore, interesting confinement and polarization effects can be observed in the G band of a graphene nanoribbon, as shown by Cancado et al. (2004). The G1 band from the nanoribbon on top of a highly oriented pyrolytic graphite (HOPG) substrate can be separated from the substrate G2 band by use of laser heating (see Figure 7.7). The temperature rise of the ribbon due to laser heating is greater than that of the substrate and ωG1 for the ribbon decreases more than for the substrate because of the higher thermal conductivity of the substrate relative to the graphene ribbon. The ribbon G1 band shows a clear antenna effect, where the Raman signal disappears when crossing the light polarization direction with respect to the ribbon axis, in accordance with theoretical predictions (Dresselhaus et al. 2005). 7.3.1.2 The G′ Band The second most usual Raman peak observed in the Raman spectra of any sp2 carbon system has been historically named the G′ band. It is a second-order peak involving two phonons with opposite wave vectors q and −q; the atomic motion related to the phonon looks like the hexagon rings are breathing. This feature appears at around 2700 cm−1 (see Figure 7.6) for laser excitation energy of 2.41 eV (514 nm), while the highest phonon frequency in graphene is around 1620 cm−1. The scattering by the related one-phonon process is not allowed by symmetry and has been named the D-band, denoting the dominant disorder-induced band (see Section 7.3.5.1). Besides a rich dependence of the G′ band on the ambient conditions (temperature, pressure, doping), this band exhibits a

7-8


1.0 P

0.8 G1 band intensity

G1

Θ

Θ = 0°

0.6 0.4 0.2

Intensity

0.0 0

Θ = 30° (b)

30 60 (Degrees)

90

Frequency (cm–1)

1580

Θ = 60°

1576

1572

G2 G1

Θ = 90° 1568 1520 (a)

1540

1560

1580

1600 –1

Raman shift (cm )

1620 (c)

1 2 3 Laser power density (105 W/cm2)

FIGURE 7.7 (a) The G band Raman spectra from the graphene nanoribbon (G1, dashed line) and from the graphite substrate (G2, solid line). (b) The G1 dependence on the light polarization direction with respect to the ribbon axis including experimental points and theoretical predictions in the dashed curve. (c) Frequency of the G peaks as a function of incident laser power for the graphene ribbon (G2) and the HOPG substrate (G1). (From Cancado, L.G. et al., Phys. Rev. Lett., 93, 047403, 2004. With permission.)

very interesting resonance phenomena related to the laser excitation energy. By increasing (decreasing) Elaser, the G′ peak frequency ωG′ will increase (decrease), as is also the case of the D band at ωD ≈ ωG′/2 ≈ 1350 cm−1 for Elaser = 2.41 eV (see Section 7.3.5.3).

7.3.2 Adding Graphene Layers: From a Single-Layer Graphene to Graphite When increasing the number of graphene layers from one to two, two atoms are added to the unit cell (see Figure 7.8), thus increasing the number of phonon branches. Here we consider trilayer graphene as having an ABA Bernal stacking, like HOPG. Rhombohedral graphite has ABC stacking, but it will not be considered here (Wyckoff 1981). We now discuss the main changes in the Raman spectra when adding layers to the graphene system, i.e., bilayer, trilayer, many-layer (graphite) (Malard et al. 2009a). 7.3.2.1 The G′ Band While the first-order G band spectrum is approximately independent of the number of layers, the G′ spectrum shows the most characteristic changes and for this reason the G′ band can

be used to determine the number of layers (Ferrari et al. 2006) and the stacking order (Pimenta et al. 2007, Ni et al. 2008) of few-layer graphene. The G′ spectrum from single-layer graphene has one peak, while from the bilayer graphene with an AB stacking order, four distinct peaks can be observed (see Figure 7.8d). If the double layer has no stacking order, there is a small upshift in frequency, but only one peak is observed (Ni et al. 2008). The number of peaks increases with the increasing number of layers. However, it is not possible to clearly distinguish all of these peaks since the splitting among them is not larger than their linewidths. Thus, for graphite (the limit of a semi-infinite number of layers), the G′ band is found to contain only two well-defined peaks when the out-of-plane AB stacking is perfect. Turbostratic graphite (no stacking order) has, again, a one peak G′ band lineshape (Pimenta et al. 2007). 7.3.2.2 Other Raman Features Other features are usually observed in graphite, and are identified as combination modes and overtones (other than the G′ band). Some examples are the features observed above 2000 cm−1 in Figure 7.5. The mode assignment upon

7-9


514 nm A

B a1

a2

Graphite

ˆy xˆ

F E D xˆ yˆ

xˆ

5 layers

D

C

B

10 layers

Intensity (a.u.)

(a)

C

2 layers

ˆy

B 1 layers

A (b)

A

2600

(c)

2700

2800


(d)

FIGURE 7.8 Schematics for (a) mono-, (b) bi-, and (c) trilayer graphene (From Malard, L.M. et al., Phys. Rev. B, 79, 125426, 2009b. With permission.). (d) Evolution of the G′ band spectra at 514 nm (Elaser = 2.41 eV) with the number of graphene layers. (From Ferrari, A.C. et al., Phys. Rev. Lett., 97, 187401, 2006; Pfeiffer, R. et al., in Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties and Applications, Springer Series in Topics in Applied Physics, Vol. 111, Jorio, A. et al. (eds.), Springer-Verlag, Berlin, Germany, 2008, 495–530. With permission.)

which these features are based is discussed in the article by Dresselhaus et al. (2005). Furthermore, many other Raman features are observed when disorder is present in nano-carbons, as discussed in Section 7.3.5.

(5,3) a1 a2

7.3.3 Rolling Up One Graphene Layer: The Single-Wall Carbon Nanotube The SWNT can be obtained by rolling up a strip of graphene into a cylinder (see Figure 7.9). This procedure can generate tubes with different diameters (dt) by changing the width of the graphene strip, by using different chiral angles (θ), and by changing the angle between the carbon bonds and the tube axis. These two characteristic parameters (dt, θ) define the SWNT structure and are usually represented by the (n, m) indices, which describe the number of a1 and a2 graphene lattice vectors needed to build the so called chiral vector Ch = na1 + ma2. The chiral vector Ch spans the circumference of the tube cylinder (see Figure 7.9). Of course, rolling up a graphene strip is an idealized picture. The carbon nanotubes are actually formed by carbon atoms in the vapor phase self-organizing themselves to form a tube structure when growing from a nanometer size catalyst particle. Different growth methods generate samples with very different aspects, like isolated tubes on different substrates or suspended tubes over trenches, tubes in bundles, or tubes (isolated or in bundles) that are aligned along a specific direction. Furthermore, tube

Θ Ch A

1

2

3

1

2

3

4

A 5

FIGURE 7.9 Schematic diagram showing a possible rolling up of a strip of 2D graphene sheet into a tubular form. In this example, a (5, 3) nanotube is under construction and the resulting tube is illustrated on the right. (From Saito, R. et al., Physical Properties of Carbon Nanotubes, Imperial College Press, London, U.K., 1998.)

processing can generate samples in different environments, such as liquids or wrapped by DNA molecules (Joselevich et al. 2008). The most spectacular differences when comparing SWNTs and graphene are observed due to quantum confinement effects, which are also completely different from the effects on graphite ribbons due to satisfying the carbon bond requirements at the edges of the ribbon. As a very simple example, the zero-energy

7-10


translation of the graphene generates, when rolling it up into a tube, the so-called radial breathing mode (RBM), which is discussed in Section 7.3.3.1. Furthermore, due to the spatial confinement in this 1D system, their optical properties are discrete, similar to molecules. The physics behind it is related to the formation of spikes, named van Hove singularities, in the density of electronic states (Saito et al. 1998, Dresselhaus et al. 2005) and the formation of excitons (bounded electronhole pairs) for the optical transitions, as is discussed in the article by Dresselhaus et al. (2006). These van Hove singularities and the exciton formation generate an absorption picture for SWNTs that, although very rich in different phenomena (Dresselhaus et al. 2006), is mostly dominated by strong discrete optical transition levels usually denoted by Eii, where i = 1, 2, 3… enumerates the optical transition energy levels (Saito et al. 1998, Dresselhaus et al. 2005). The Eii optical transition energies depend on (dt, θ), or alternatively on (n, m), and many of them lie in the visible range, thus generating strong resonance Raman effects, as discussed in the following sections.

on the RBM measurement, as discussed in the next section. The natural linewidth (FWHM) for isolated SWNTs on a SiO2 substrate is Γ = 3 cm−1 (Dresselhaus et al. 2005), but much narrower linewidths (down to ~0.25 cm−1) have been observed for the inner tubes of double-wall carbon nanotubes (DWNTs) at low temperatures (Pfeiffer et al. 2008). The RBM gives the nanotube diameter through the use of the relation ωRBM = (A/dt)(1 + Ce/dt2)1/2 . Therefore, from the ωRBM measurement of an individual isolated SWNT (see left panel in Figure 7.10), it is possible to obtain its d t value. The parameter A = 227 cm−1 nm has been obtained experimentally for a single type of sample (Araujo et al. 2008), and it is in agreement with theoretical predictions based on the elastic constant of graphite (Mahan 2002). Most of the samples, however, exhibit an upshift in ωRBM (see Figure 7.10b) due to the tube-environment vander Waals type interactions, and this interaction can be taken into account by making the parameter Ce ≠ 0 for each specific sample (Araujo et al. 2008). The RBM spectra for SWNT bundles contain RBM contributions from different SWNTs in resonance with the laser excitation line (see Figure 7.10), and a careful spectral analysis gives the tube diameter distribution in the bundles.

7.3.3.1 The Radial Breathing Mode The RBM Raman feature corresponds to the coherent vibration of all the C atoms of the SWNT in the radial direction, as if the tube were “breathing.” This feature is unique to carbon nanotubes and occurs with frequencies ωRBM typically between 55 and 350 cm−1 for SWNTs with diameters in the range 0.7 nm < d t < 4 nm. These RBM frequencies are therefore very useful for identifying whether a given carbon material contains SWNTs, through the presence of the Raman-active RBM modes (see Figure 7.5). When the excitation laser energy is in resonance with the optical transition energy Eii from one isolated SWNT, its RBM can be seen, as shown in the left panel of Figure 7.10. The (n, m) assignment can be made based

7.3.3.2 The Resonance Raman Effect in the RBM and the (n, m) Assignment When the excitation laser energy matches one of the discrete optical transition energies Eii for a given SWNT, there is large enhancement in the Raman intensity (resonance Raman effect). Therefore, for interpreting the Raman spectra of SWNTs, a very useful guide is the so-called Kataura plot, where the transition energies Eii are plotted as a function of nanotube diameter dt (see the left panel in Figure 7.11). Such a plot can be directly related to the RBM spectra (see the right panel in Figure 7.11). (a)

1000 164(8) 148(9) (20,2)

100 (a)

Normalized intensity

Intensity

1.0

150

237(5) (10,5)

200 250 Frequency (cm–1)

300

0.5 0.0 1.0 0.5 0.0 100

350 (b)

(b)

150

200 ωRBM (cm–1)

250

300

FIGURE 7.10 (a) Raman spectra of a Si/SiO2 substrate containing isolated SWNTs grown by the CVD method. (From Jorio, A. et al., Phys. Rev. Lett., 86, 1118, 2001. With permission.) The spectra are taken at three different spots on the substrate where the RBM Raman signals from resonant SWNTs are found. The RBM frequencies (linewidths) are displayed in cm−1. Also shown are the (n, m) indices assigned from the Raman spectra for each resonant tube. The step at 225 cm−1 and the peak at 303 cm−1 come from the Si/SiO2 substrate. (b) The RBM Raman spectra for “super-growth” SWNTs (gray) and for “alcohol CVD” SWNTs (black). The samples are different and exhibit different Eii resonance values. The four spectra are obtained using different excitation laser lines: (a) 590 nm (gray) and 600 nm (black); (b) 636 nm (gray) and 650 nm (black), so that in (a) and (b) the same (n, m) SWNTs are in resonance. The gray and black spectra are shifted from each other due to the different environments seen by the SWNTs in the two types of samples. (From Araujo, P.T. et al., Phys. Rev. B, 77, 241403(R), 2008. With permission.)

7-11

Raman Spectroscopy of sp2 Nano-Carbons 2.7

2.7

2.4

2.4 Elaser (eV)

Elaser (eV)

ES44 2.1 1.8 1.5

ES11 0.9

(a)

ES22 1.2 1.5 1.8 Diameter (nm)

EM 11

ES33

1.00

ES44

0.750 EM 11

2.1

0.500 S 1.8 E22

0.250

ES33

1.5 ES11

2.1

3 (b)

0 4

5 6 7 8 1/Raman shift (10–3 cm)

9

FIGURE 7.11 (See color insert following page 20-16.) (a) The optical transition energies (dots) of SWNTs as a function of tube diameter. (From Jiang, J. et al., Phys. Rev. B, 75, 035407, 2007a. With permission.) The superscripts S and M for the Eii labels indicate the optical transitions for semiconducting and metallic SWNTs, respectively. (b) 2D color map showing the SWNT RBM spectral evolution as a function of laser excitation energy. The intensity of each spectrum is normalized to the strongest peak, and we plot the RBM results are plotted in terms of the inverse Raman shift. (a) and (b) show coincident axes and can be directly correlated. (From Araujo, P.T. et al., Phys. Rev. Lett., 98, 067401, 2007. With permission.)

If resonance with a single tube is achieved (Elaser = Eii) and its RBM is observed, it is possible to assign its specific (n, m), since (Eii, ωRBM) can be related to (dt, θ) (Jorio et al. 2001). For a more general diameter characterization of a bundled SWNT sample based on the RBMs, it is necessary to work with the Kataura plot (Dresselhaus et al. 2005). A single Raman measurement gives the RBM for the specific tubes that are in resonance with that laser line, which may not give a complete characterization of the diameter distribution of the sample. By taking the Raman spectra with many excitation laser lines, a good characterization of the diameter distribution in the sample can be obtained (Dresselhaus et al. 2005, Araujo et al. 2007). Since semiconducting (S) and metallic (M) tubes of similar diameters do not occur at similar Eii values, ωRBM measurements using several laser energies (Elaser) can also be used to characterize the ratio of metallic to semiconducting SWNTs in a given sample (Samsonidze et al. 2004, Miyata et al. 2008). A careful analysis of the resonance Raman intensities in the right panel of Figure 7.11 shows that the RBM intensity has a strong (n, m) dependence, as explained in the article by Jiang et al. (2007b). This effect is mostly due to a chiral angle dependence of the electron-phonon coupling, plus a diameter dependence of the electron-radiation interaction due to excitonic effects (Dresselhaus et al. 2006, Jiang et al. 2007b). 7.3.3.3 The G Band In contrast to the graphite Raman G band, which exhibits one single Lorentzian peak at 1584 cm−1 related to the tangential mode vibrations of the C atoms, the SWNT G-band is composed of multi-peaks due to the phonon wave vector confinement along the SWNT circumferential direction and due to symmetry-breaking effects associated with the SWNT curvature (see Figure 7.12). The G-band frequency in SWNTs can be used for (1) diameter characterization, the lower frequency G− peak exhibiting a frequency dependence on diameter (see Figure 7.12b and c); (2) distinguishing between metallic and semiconducting SWNTs, through major differences in their Raman lineshapes

(see Figure 7.12a); (3) probing the charge transfer arising from doping a SWNT (Terrones et al. 2008); and (4) studying the selection rules in the various Raman scattering processes and scattering geometries (Dresselhaus et al. 2005). Elaborating a bit more on item (2) above, the difference between the G band spectra from metal and semconducting tubes is the strong coupling between electrons and G band phonons in metals. Although up to six modes of Raman are allowed in the G band of SWNTs (Dresselhaus et al. 2005), the G band is dominated by two strong peaks that can be represented by the C–C stretching along the circumferential direction or along the axis, usually named the transverse optical (TO) and longitudinal optical (LO) modes, respectively. The LO mode strongly couples with electrons in metals, being largely downshifted in frequency and broadened (see Figure 7.12a). The physics related to this coupling have been discussed in the articles by Piscanec et al. (2004) and Ando (2006). 7.3.3.4 The G′ Band Like for the other sp2 carbons, the G′ Raman spectra provide unique information about the electronic structure of both semiconducting and metallic SWNTs. The G′-band sometimes appears (at the individual nanotube level) in the form of unusual two-peak structures for both semiconducting and metallic nanotubes (Samsonidze et al. 2003, Dresselhaus et al. 2005), even if there is no interlayer coupling, like in bilayer graphene and graphite. The two-peak G′-band Raman features observed from semiconducting and metallic isolated nanotubes are shown in Figure 7.13a and b, respectively. The presence of two peaks in the G′-band Raman feature from semiconducting SWNTs indicates resonance with two different van Hove singularities of the same nanotube, occurring independently for both the incident Elaser and scattered E laser – EG′ photons. 7.3.3.5 Other Features Carbon nanotubes also exhibit several combination modes and overtones. Basically, all the Raman features observed in graphite can also be seen in carbon nanotubes. Furthermore,

7-12

Handbook of Nanophysics: Nanotubes and Nanowires RBM

1589 (9)

(15,8)

1581(13)

HOPG

G band

1571 (9)

Intensity

Semiconducting SWNT

Raman intensity

153(10)

1592(10)

1568(8)

1588(21)

Metallic SWNT

1591 (8)

(17,3) 169(8)

1568 (8)

(15,2) 195(5) 1560 (8)

1554(60)

1590 (8)

1/q = 0.1 1450

1550

1650

120

Frequency (cm–1)

(a)

(b)

170

220

Frequency (cm–1)

1530

1570

1610

Frequency (cm–1)

ωG+

1600

Frequency (cm–1)

1580

1560 ωG–

1540 Metallic 1520

Semiconducting 1500

0

0.2

0.4

(c)

0.6 1/dt (1/nm)

0.8

1

FIGURE 7.12 (a) The G-band for highly oriented pyrolytic graphite (HOPG), one semiconducting SWNT, and one metallic SWNT. For C60 fullerenes, a peak is observed at 1469 cm−1, but it is not considered a G band. (b) The RBM and G-band Raman spectra for three semiconducting isolated SWNTs of the indicated (n, m) values. (c) Frequency vs. 1/dt for the two most intense G-band features (ωG– and ωG+) from isolated SWNTs. (From Dresselhaus, M.S. et al., Phys. Rep., 409, 47, 2005. With permission.)

carbon nanotubes exhibit several Raman features in the spectral region between 400 and 1200 cm−1, which are called intermediate frequency modes (IFMs), since their frequencies lie between the common ωRBM and ωG modes (Dresselhaus et al. 2005). Some of the IFMs are fi xed in energy and some of the IFMs are “dispersive” (Raman shift changes when changing the excitation energy) and are attributed to combination modes. It is not yet clear whether these modes are related to disorder or not. Theory relates their observation with confi nement along the tube length (Saito et al. 1998), and some supporting experimental evidence has been found for such an effect (Chou et al. 2007). However, the IFM picture is not yet fully understood.

7.3.4 Adding Tube Layers: Double- and Multi-Wall Carbon Nanotubes The differences between single-, double-, and many-wall carbon nanotubes are often classified by the diameter distribution in the tubes. MWNTs, for example, usually have tubes with very large diameters dt, with inner diameters over 10 nm, and outer diameters rising up to about 100 nm. Their spectra approach that of graphite, since no confinement effects can be observed. Only general line broadening differentiates large MWNTs from graphite (see Figure 7.14). For DWNTs, however, very interesting outer vs. inner tube effects have been observed. The most striking is the observation of many

7-13


M1 (27,3)

2695 (26)

2660 (54)

2600

2700

(a)

Raman intensity

Raman intensity

S2 (15,7)

Raman shift

2800

(cm–1)

2688 (36)

2658 (59)

2600 (b)

2700 Raman shift

2800

(cm–1)

7.3.5.1 The D Band

1000

1200

1400

1600

2400

×2 2600

2800

3000

3233

2942

×2

–3222

ASR

–2413

–1087

–1361

SR

–2946

2437

1355

1086

–2728

–1574

εL = 2.54 eV

2707

1575

FIGURE 7.13 The G′-band Raman features for (a) semiconducting (15, 7) and (b) metallic (27, 3) nanotubes showing unusual two-peak structures. (From Samsonidze, Ge. G. et al., Appl. Phys. Lett., 85, 1006, 2004. With permission.)

3200

–1

Raman shift (cm )

FIGURE 7.14 Stokes (SR) and anti-Stokes (ASR) Raman spectra from MWNTs. The small difference in frequencies when comparing the same peak in the SR and ASR spectra are due to the double resonance effect, as discussed in detail in Cancado et al. (2002) and Tan et al. (2002). (From Tan, P. et al., Phys. Rev. B, 66, 245410, 2002. With permission.)

different ωRBM for the same (n, m) inner SWNT, due to different possible outer tubes (see Figure 7.15). Furthermore, charge transfer effects depending on the metal vs. semiconducting outer vs. inner configuration have been observed (Souza Filho et al. 2007). While most of these results have been obtained from DWNTs in bundles, measurements of isolated DWNTs are expected to be very informative (Villalpando-Paez et al. 2008).

7.3.5 Disorder in sp2 Systems When the size of a graphite system is reduced, disorder effects start to be seen. Disorders can also be seen at defect locations or at graphene/graphite edges.

The so-called D band (D comes from disorder) is observed in any sp2 carbon system when the lattice periodicity is broken by a disorder mechanism, such as defects. This feature appears around 1350 cm−1 (for E laser = 2.41 eV) and the phonon-related step of the D band double resonance process corresponds to one of the steps in the G′ band process. The D band itself is a secondorder process in which the elastic scattering of the defect allows for the conservation of momentum in the phonon creation, and the defect breaks the Raman scattering selection rules. The D band has been largely used to characterize disorder in carbon materials (see Figure 7.16). The D band is dispersive, like the G′ band, and shows interesting phenomena in SWNTs due to the electron and phonon confi nement (Dresselhaus et al. 2005, Pimenta et al. 2007). 7.3.5.2 The D′ Band The so-called D′ band, appearing around 1620 cm−1, is another feature commonly observed in many of the sp2 disordered carbon systems. This feature is usually much weaker than the D band, and for this reason the D band is more often used for disorder characterization. Whereas the D band is connected with an intervalley scattering process from the K point to the K′ point, in the Brillouin zone, the D′ band is connected with an intravalley scattering process around the K point or the K′ point. The D and D′ bands tend to be sensitive to defects of a different physical origin, but these differences require further study. 7.3.5.3 The G′ Band The G′ band is not disorder induced but it can be used to study changes in the electronic and vibrational structure related to disorder. The 2D vs. 3D stacking order of graphene layers is one example. Highly crystalline 3D graphite shows two G′ peaks (see the top spectra of Figure 7.8d). When the interlayer stacking order is lost, a one-peak feature starts to develop, identified with 2D graphite, and the peak is centered near the middle of the two peaks in the G′ lineshape from ordered graphite (Pimenta et al. 2007).

7-14

Handbook of Nanophysics: Nanotubes and Nanowires 2.6

14

2.5

2

21 2.4

1.5

Laser energy (eV)

2.3

1

24

17 (6,5)

2.2

0.5 (6,4)

27

2.1

0 20

2.0

–0.5

30 –1

16

23

1.9 1.8

19 22

26

1.7

25

1.6 1.5 150

200

250

300

350

400

RBM frequency (cm–1)

FIGURE 7.15 (See color insert following page 20-16.) The 2D RBM Raman map for DWNTs. The Eii points from the Kataura plot are superimposed (green bullets). It is clear there are many more RBM features than (n, m) related Eii values. (From Pfeiffer, R. et al., in Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties and Applications, Springer Series in Topics in Applied Physics, Vol. 111, Jorio, A. et al. (eds.), Springer-Verlag, Berlin, Germany, 2008, 495–530. With permission.)

1800

2000 2.0

ID/IG

1.5 1.0

1.92 eV 2.18 eV 2.41 eV 2.54 eV 2.71 eV

0.5 2200

2300 0.0 0.00

0.01

0.02 0.03 1/La (nm–1)

0.04

0.05

0.01

0.02

0.04

0.05

(b) 30

2400

(ID/IG)*E 4L (eV4)

25 2700

20 15 10 5

500 nm (a)

0 (c)

0.00

0.03

1/La (nm–1)

FIGURE 7.16 (a) (left panels) STM images of graphite crystallites. The crystallite size L a varies with the annealing temperature in degree Celsius (displayed in the images). (right panels) The evolution of the D/G intensity ratio (ID/IG) as a function of crystallite size (a) for different laser excitation energies. All the curves in (b) collapse into one curve (c) when considering the dependence of the G band intensity on the fourth power of the 4 incident laser excitation energy, E laser . (From Pimenta, M.A. et al., Phys. Chem. Chem. Phys., 9, 1276, 2007. With permission.)

7-15


Furthermore, localized emission of a red-shifted G′ band was observed and related to the local distortion of the nanotube lattice by a negatively charged defect. The opposite occurs for p doping and this effect can be used to study SWNT doping (see Figure 7.17; Maciel et al. 2008).

7.3.6 Other Raman Modes and Other sp2 Carbon Structures In general, the observation of overtones and combination modes in condensed matter systems is rare because of dispersion effects that make these features too weak and too broad to pick out from the noisy background. The double resonance process (Saito et al. 2003), however, allows such overtones and combination modes to be quite clearly observed (Dresselhaus et al. 2005), thereby providing new information about SWNT properties. Nanowiskers, nanobuds, nanorods, and nanohorns exhibit a large number of

Undoped SWNTs

Normalized intensity

n-doped SWNTs

p-doped SWNTs

Graphene

HOPG

Amorphous carbon 2600

2700

peaks (see Figure 7.18), always related to and assignable from the phonon structure of graphene. Although we do not cover fullerenes here, when talking about overtones, it is especially interesting to mention their Raman spectra. As a molecular carbon structure, a fullerene has an especially rich overtone spectra that can be seen in both Raman and infrared spectra. A detailed discussion on this topic can be found in the book by Dresselhaus et al. (1996).

7.4 Critical Discussions As discussed in Section 7.3, Resonance Raman spectroscopy (RRS) was shown to provide a powerful metrological tool for distinguishing among the different sp2 nano-carbons, the number of layers of a graphene sample, the AB stacking order in many-layers graphene of graphite and the metallic (M) from semiconducting (S) tubes; and for determining the diameter distribution of SWNTs in a given sample, the (n, m) values for specific tubes, the doping, and many other important properties. Recent RRS measurements S S S M S S done on the E 22 , E11M , E 33 , E 44 , E 22 , E 55 , and E 66 transitions on waterassisted chemical vapor deposition (CVD)-grown SWNTs (Araujo et al. 2008) suggest that tubes in the interior of the forest of aligned SWNTs seem to be well-shielded from environmental effects and may provide a standard reference material for SWNTs that show minimal environmental effects. This suggestion needs further experimental confirmation, but if this interpretation is correct, such a reference material might represent the first nano-carbon standard reference material, which could be useful for quantitative determinations of environmental effects in nanotubes. Such effects are important for understanding the current nanotube photo-physics studies where samples routinely experience environmental effects due to substrates, wrapping agents, functionalization, strain or molecules adsorbed in suspended nanotubes, and similar problems also arise for graphene and the other nanocarbons. Controlling environmental effects are vital for a variety of uses of nano-carbons for sensors and biomedical applications. Calculations of Raman frequencies and RRS matrix elements are now at an advanced stage for SWNTs, but for understanding lifetime effects, dark singlet and triplet states are still incomplete. The characterization of MWNTs as well as many layers of graphene by RRS is at an early stage, though a good start has been made on the characterization of DWNTs and bilayer graphene that are, respectively, the simplest examples of a MWNT or of graphite. The effect of edge states in graphene is expected to be a rich field and has been, basically until now, weakly explored by spectroscopy.

2800


7.5 Summary 2

FIGURE 7.17 The G′ Raman band in different sp carbon materials measured at room temperature with Elaser = 2.41 eV (514 nm). The arrows point to defect-induced peaks in the G′ band for doped SWNTs. The p/n doping comes from substitutional boron/nitrogen atoms (Terrones et al. 2008), the nearest neighbors of carbon in the periodic table. The spectra of graphene, HOPG, and amorphous carbon are shown for comparison. (From Maciel, I.O. et al., Nat. Mat., 7, 878, 2008. With permission.)

In summary, we presented here the basic concepts of Raman spectroscopy and the Raman signatures of sp2 carbon materials. The G band at ~1585 cm−1 has a single-Loretzian peak structure for 2D and 3D materials, and shows a special lineshape for their 1D counterparts, carbon nanotubes, with a doublet rather than a single Lorentzian feature. Raman spectroscopy

7-16

1000

2000

7478

6953

6443

6923

6421

6121

5606 4848

4537

×200


3237

2462

L2 400

L1

300

200

×10 3000

4241

1.8

2927

1833 1951 ×10

×50

1130 1333 1582, 1618

228 355

(c)

5861

5363 4287 4045

×10

2662

×50

2947

288 453

1086

2447

3244

(b)

6150

5638 4853

4549

4052

2955 1914 2081 1895 2050

×10

×100

2704

1360 1581, 1623

×50

1354 1581, 1622

1074

303 481

2442

3247

5392

4300

(a)

5879

2717


4000 5000 Raman shift (cm–1)

2.0 2.2 2.4 2.6 Excitation energy (eV)

6000

7000

FIGURE 7.18 Raman spectra of graphite whiskers obtained at three different laser excitation energies: (a) 488.0 nm, (b) 514.5 nm, and (c) 632.8 nm. Note that some phonon frequencies vary with Elaser, some do not. The inset to (c) shows details of some peaks that are dispersive, and are explained theoretically by the double-resonance process. (From Tan, P. et al., Phys. Rev. B, 64, 214301, 2001. With permission.)

can also be used to differentiate between metallic and semiconducting carbon nanotubes, as well as to obtain information on tube diameter and the tube diameter distribution of a bundled sample. Another important feature is the G′ band, which appears at ~2700 cm−1 for 2.41 eV (514 nm) excitation. Th is feature is highly sensitive to the electronic structure, changing shape when comparing single-layer graphene, bilayer graphene, trilayer graphene, stack-ordered (3D) graphite, and

pyrolytic (2D) graphite and when nano-carbons are doped. The cylindrical carbon nanotubes are the only nano-carbon material showing the low frequency radial breathing modes that are strongly dependent on diameter, and usually appear in the range 50–350 cm−1. Finally, at ~1350 cm−1 (≈ωG′/2), the D peak can be observed when there is disorder in the material, i.e., any perturbation that breaks the sp2 periodic structure, and the D-band can be used for characterizing the structural quality of


carbon-based materials. Finally, many other features related to disorder, overtone, and combination modes are observed, most of them being weak, but sometimes they are strong. Although the assignment and frequency behavior has been quite well understood, the variation in intensity of these features still remains an open issue.

7.6 Future Perspectives It is clear that Raman spectroscopy of sp2 carbons remains a fruitful field of research even though many advances have already been achieved. A big step forward in the field will occur when Raman spectroscopy starts to be combined in a single instrument with microscopy techniques, either high resolution electron transmission microscopy (HRTEM) or scanning probe microscopies, like atomic force microscopy (AFM) or scanning tunneling spectroscopy (STM). Combining the information from real space (microscopy) with information from momentum space (spectroscopy) should bring in new insights, including a new understanding of different types of disorders. In this context, near-field studies should also bring in an important new understanding, since Raman spectroscopy with spatial resolution down to 10 nm can be achieved, which will provide important information on the science of defects (Hartschuh et al. 2003, Maciel et al. 2008). Furthermore, a major focus is now devoted to directing the nano-carbon field to applications, which urges studies on metrology, standardization, and industrial quality control (Jorio and Dresselhaus 2007, Jorio et al. 2008b). The development of protocols for the definition of sample parameters like structural metrics (carbon–carbon distance, surface area, tube diameter, chiral angle, ribbon width), physical properties (optical, thermal, mechanical), composition (impurity content, spatial homogeneity), and stability (dispersability, bio-compatibility and health effects) are important for both research and applications of nano-carbons. These metrological protocols are expected to be applicable not only to nanocarbon materials, but also to the exploding field of nanomaterials, where metrology issues will drive technological growth and innovation.

Acknowledgments M.S.D. and G.D. acknowledge NSF-DMR 07-04197. A.J. acknowledges financial support from CNPq, CAPES, and FAPEMIG. The authors thank Mario Hofmann for help with the preparation of the manuscript.

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8 Dispersions and Aggregation of Carbon Nanotubes Jeffery R. Alston University of North Carolina, Charlotte

Harsh Chaturvedi University of North Carolina, Charlotte

Michael W. Forney University of North Carolina, Charlotte

8.1 8.2

Structure and Properties of CNTs • Interparticle Forces in Nanotube Bundles • Thermodynamics of Nanotube Dispersions • Theory of Colloid Stability • Theory of Particle Aggregation

Natalie Herring University of North Carolina, Charlotte

Jordan C. Poler University of North Carolina, Charlotte

Introduction .............................................................................................................................8-1 Background...............................................................................................................................8-2

8.3

State-of-the-Art Procedures and Techniques ......................................................................8-8 How to Make a Dispersion of CNTs • How to Make CNT Th in Films and Devices

8.4 Summary .................................................................................................................................8-19 References...........................................................................................................................................8-19

8.1 Introduction Nanostructured carbon has shown great utility in modifying and enhancing the physiochemical properties of composite materials (Lau et al., 2006). Specifically, single-walled carbon nanotubes (SWNTs) are unique in their ability to enhance the electrical and mechanical properties of materials. The science and technology of formulating stable dispersions of SWNTs in various media has been well reviewed in the literature. This chapter is intended to be an encapsulated tutorial on the formation, stability, and properties of carbon nanotube (CNT) dispersions. That said, the reader is advised to also access several well-cited reviews in this important area. Bundles of tubes aligned parallel with each other and interacting through extensive van der Waals (vdW) forces is the thermodynamically stable state of SWNTs. If we can provide enough energy to overcome this binding energy, it is possible to disperse the SWNTs as individual tubes into various solvents and polymeric matrices. Fu and Sun (2003) review some of the earlier work on forming SWNT dispersions in various solvents and for various surface modifications of the tubes. This is a good place to get started for those unfamiliar with this field. Regardless of the solvent one wants to disperse nanotubes into, one must use some form of mechanical mixing to separate the individual tubes from the stable bundle. Mechanically grinding nanotube solids into a dispersant is effective, but often damages

the tubes too much (Chen et al., 2001a). In order to intentionally shorten and damage SWNTs, high-impact mixing techniques such as Ball–Milling are used (Pierard et al., 2004). When one’s intentions are to put as much nanostructured carbon into the matrix regardless of its final properties, these first two mechanical methods are straightforward to apply. Another mechanical method that has shown good utility in dispersing and separating SWNTs is high shear mixing. This technique forces the nanotube bundles through small pores or plates that pull individual tubes from the bundles through extrusion. The article by Hilding provides an excellent review of the mechanical mixing methods and also includes an introduction to the structure and properties of CNTs (Hilding et al., 2003). Since we want to take advantage of the superior properties of nanotubes for various applications, we must minimize damage as they are brought into dispersion. For most nanotube applications, the dispersion medium is aqueous, organic, or polymeric. The kinetic stability of nanotube dispersions depends on several factors. The nanotube–solvent interaction dominates the kinetics. One cannot achieve a kinetically stable SWNT dispersion in water without first functionalizing the nanotube surface or adding surfactants. Either way, aqueous nanotubes are no longer pristine. There are many organic solvents that can support kinetically stable dispersions of pristine SWNTs. Most of these solvents are good Lewis bases, which do not have significant 8-1

8-2

hydrogen-bonding capabilities, such as N,N-dimethylformamide (DMF). Dispersions in these solvents will be detailed below. For composite materials, nanotubes must be dispersed into polymer precursors or directly into the polymer matrix. For these systems, the higher viscosity of the matrix makes mechanical mixing more difficult. Moreover, if the polymer matrix does not interact strongly with the nanotube walls, then the nanotubes will aggregate back to their bundled form and behave like defects within the condensed polymer matrix. Many of the challenges have been overcome and are illustrated in the wellcited review by Xie et al. (2005). To effectively stabilize SWNTs in a polymer, the tubes are chemically modified, then added to polymer precursors and then the matrix is polymerized while it is being extruded, cast, or spun. To further enhance the properties of these polymer composites, the nanotubes are aligned. Nanotube orientation can be established in situ by extrusion (Fischer, 2002), by magnetic force (Kimura et al., 2002), or dielectrophoresis (Wang et al., 2008a). Aligned nanotube composites can also be formed ex situ by chemical vapor deposition (CVD) growth or self-assembly, functionalized, and fi nally set into polymer (Feng et al., 2003). Regardless of how the composite is formulated, the stability and utility of the material depends on the nanostructured carbon staying dispersed. Therefore, one must also design a system that inhibits the aggregation of the nanoscale materials. This chapter addresses the fundamental processes involved in the dispersion and aggregation of SWNTs. We address the structure and properties of CNTs and the intermolecular and interparticle attractive forces that result in nanotube bundles. The thermodynamics of solute-particle interactions leads to a dispersion limit or loading of SWNTs into a matrix. Once dispersed, the nanotubes are kinetically stabilized and this stability is discussed in terms of classical colloidal stability arguments. While a stable dispersion is typically desired, the process of aggregation under various chemical, physical, and optical stimuli is discussed. After an extended background section, we explain in detail some common methods for producing dispersions and measuring their properties. The end result of this work is to make something useful. From these stable dispersions, we aim to move the nanotubes into a more ordered and functional form. Procedures to make SWNT mats, thin fi lms, and devices for electronic, optical, and electrochemical sensing applications will be detailed.

8.2 Background 8.2.1 Structure and Properties of CNTs Carbon is the sixth element of the periodic table and is the element with the lowest atomic number in column IV. Each carbon atom has six electrons, which occupy 1s, 2s, and 2p atomic orbitals. The 1s orbital contains two strongly bound core electrons. Four more weakly bound electrons occupy the 2s and 2p valence orbitals. In the crystalline phase, the valence electrons give rise to 2s, 2px, 2py, and 2pz orbitals, which are important in forming covalent bonds in carbon materials. Since the energy difference


between the upper 2p energy levels and the lower 2s level in carbon is small compared with the binding energy of the chemical bonds, the electronic wave functions for these four electrons can readily mix with each other. Consequently, this changes the occupation of the 2s and three 2p atomic orbitals so as to enhance the binding energy of the carbon atom with its neighboring atoms. Carbon can form stable morphologies in various dimensions due to several possible hybridizations of the carbon 2s and 2p orbitals, e.g., diamond (three-dimensional), graphite (two-dimensional), SWNT (one-dimensional graphene tubules), and fullerenes (C60 zero-dimensional). The various bonding states correspond to certain structural arrangements; e.g., sp bonding gives rise to chain structures, sp2 bonding to planar structures, and sp3 bonding to tetrahedral structures. CNTs are theoretically considered as a graphene sheet appropriately rolled into a cylinder with essentially sp2 atomic bonding between the nearest neighbors. Electronic properties of SWNTs vary from semiconducting to metallic depending on the tubular structure; i.e., the diameter and chirality of the SWNTs. From the real space lattice of the two-dimensional graphene, several different tubular arrangements differing in diameter and helical arrangement of the carbon hexagons are possible. As SWNTs can be considered a rolled-up graphene sheet, the tubular arrangement of the carbon atoms can be explained in terms of the two-dimensional hexagonal lattice structure of graphene. From the origin of the lattice structure, subsequent atomic arrangements can be referred to using the coordinates of the lattice points, (n, m). As the sheet is rolled into a tube, the origin is superimposed onto itself to get the tubular structure. Due to periodic boundary conditions, only certain wave vectors are allowed. Thus, momentum vectors are allowed only for certain energies as represented by lines in the Brillouin zones (BZ). The properties of electrons in a periodic potential are calculated using band structure analyses. The collection of energy eigenstates in the first BZ is called the band structure. The momentum of an electron in the infinite periodic potential is defined by the crystalline lattice point. Since lattice points are periodic in a crystalline material, only the first BZ is considered. For a given wave vector and potential, there are a number of distinct solutions for Schrodinger’s equation of Bloch electrons in the first BZ. These solutions are denoted as different bands. In a simple one-dimensional infinite square well, energy levels are labeled by a single quantum number n and the energies are given by En = ( 2 π2/2mL2 )n2. Now consider N electrons instead of just one, with either spin in the box. Due to Pauli’s Exclusion Principle, there are two states for each energy level. The Fermi level is determined at E = 0 where states of lower energy are fully occupied, while higher energy states are completely empty such that the Fermi energy is EF = EN /2 = ( 2 π 2/ 2mL2 )(N /2)2 . The unique electronic properties of CNTs are due to the quantum confinement of the electrons with a wave vector perpendicular

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Dispersions and Aggregation of Carbon Nanotubes

to the nanotube’s axis. In the radial direction, electrons are confined by the monolayer thickness of the graphene sheet. Around the circumference of the nanotube, periodic boundary conditions come into play. For example, if a zigzag or armchair nanotube has 10 hexagons around its circumference, the 11th hexagon will coincide with the first. Because of this quantum confinement, electrons can only propagate along the nanotube axis, and so their wave vectors point in this axial direction. The resulting number of one-dimensional conduction and valence bands effectively depends on the standing waves that are set up around the circumference of the nanotube. These simple ideas can be used to calculate the dispersion relations of the one-dimensional bands, which relate momentum wave vector to energy, using a well-known dispersion relation in graphene. Dispersion relations show how the kinetic energy in different SWNTs varies with the wave vector. Each curve corresponds to a single semiconductor. Bands lower and higher than the Fermi level are denoted as bonding and antibonding π molecular orbitals. An armchair (5, 5) nanotube and a zigzag (9, 0) nanotube exhibit metallic properties. An electron in the highest occupied molecular orbital (HOMO) requires only an infinitesimally small amount of energy to excite it into the lowest unoccupied molecular orbital (LUMO). For a zigzag (10, 0) nanotube, there is a finite band gap between the HOMO and LUMO states (An et al., 2003), so this type of nanotube exhibits semiconductor properties. Significant variations in the absorption and Raman spectra are observed for slight changes in the diameter of the tubes. Thus, a small change in diameter has a major impact on the electronic and vibrational properties of SWNTs. The electronic and chiral structure of the SWNTs has been atomically resolved using various scanning tunneling technologies. Structural information like chirality and diameter, along with the band structure have been experimentally verified. In general, an (n, m) CNT will be metallic when n − m = 3q, where q is an integer. All armchair nanotubes are metallic, as are one-third of zigzag nanotubes. Plotting the density of states (DOS) with respect to the Fermi energy (see Figure 8.9) shows a number of prominent peaks called van Hove singularities (vHs). We expect the nanotube to absorb at energies corresponding to the inter-peak energy gap separations. We observe these singularities in the absorption spectra of SWNTs. This model considers independent electron–hole pairs and provides us with a basic phenomenological understanding. Experimental observations correlate better with the excitonic model. In this model, the electron in the conduction band and the hole in the valance band are found to act as quasi particles. The difference between the calculated energy and the observed energy in the absorption spectra of SWNTs is better explained by the binding energy of the exciton.

8.2.2 Interparticle Forces in Nanotube Bundles SWNTs are typically found as aggregated bundles, as illustrated in Figure 8.1. The force of adhesion between individual SWNTs in a bundle is a result of the π–π stacking and dispersion vdW forces present throughout the length of SWNT walls. Using a “universal

FIGURE 8.1 Model of (6, 6) SWNTs assembled into a closest packed bundle. The nanotubes on the right are shown with their van der Waals contact surfaces.

2Δ

2θ

l x

FIGURE 8.2 Model of two SWNTs bound to catalyst particles at the right (particles not shown) and subsequently adhered together from the left due to π–π stacking and other van der Waals forces. (From Chen, B. et al., Appl. Phys. Lett., 83, 3570, 2003. With permission.)

graphitic potential,” Girifalco calculated a Lennard–Jones type potential energy well depth of 95.16 meV/Å for a (10, 10) SWNT bound to another (10, 10) SWNT (Girifalco et al., 2000). As nanotubes are growing near each other, they bind together and sometimes twist about each other, forming a rope-like structure. A transmission electron microscopy (TEM) image of two such nanotubes was modeled by Chen et al. (2003) as illustrated in Figure 8.2. Based on the experimentally measured mechanical properties of SWNTs, and the equilibrium geometry shown here, they calculated a 0.36 nN/(unit length) binding energy between the nanotubes. Dispersing SWNT bundles and ropes into a solvent requires a significant amount of energy. Moreover, these dispersed nanotubes will re-form bundles unless they are stabilized by the solvent or other molecular and ionic species destablize in the solution.

8.2.3 Thermodynamics of Nanotube Dispersions There is no evidence that SWNTs are thermodynamically stable in a solvent, therefore we refer to dispersions rather than solutions of nanotubes. Using ultrasonication or other mechanical

8-4

mixing techniques, SWNTs can be dispersed into solvent up to some dispersion limit, DL . Keep in mind that the D L depends on many factors that are often not well controlled. Many of the amide organic solvents such as DMF or N-methyl-2-pyrrolidone (NMP) used to disperse SWNTs rapidly absorb water from the air. The amount of water in these dispersions affects the DL. Moreover, since the dispersions are only kinetically stable, the DL is time-dependent and also depends on sample manipulation. Simple mixing, vortexing, or centrifugation can initiate nanotube aggregation from a seemingly stable dispersion. To maintain the integrity of pristine SWNTs, they are dispersed into organic solvents. Properties of solvents that enable high D L (>1 mg/L) are good Lewis-based electron pair donation and low hydrogen bond donating character (Ausman et al., 2000; Bahr et al., 2001a). The best solvents also have large solvatochromic parameters such that the molecular properties of the solvent have a high polarity-polarizability (Kolling, 1981). As discussed below, SWNTs can also be dispersed into aqueous solutions. Acid oxidized SWNTs disperse into water with a high D L value; however, the ends and sidewalls of the nanotubes are highly defected. Addition of surfactants, such as sodium dodecyl sulfate, to water increase the surface energy of the nanotubes. The non-polar tails of the surfactant wrap around the nanotubes to maximize their attractive dispersion interactions. The polar head group then interacts more effectively with the water. Another successful strategy is to use single-strand DNA as a surfactant. The electrostatic energy of the DNA strand can be minimized by wrapping around the nanotube. Careful selection of the DNA strand length and sequence can effectively select specific nanotube chiralities from the dispersion (Zheng et al., 2003b). Essentially, the thermodynamics of SWNT dispersions is driven by the free energy interactions of the solvent with the nanotube. Both intermolecular interactions and entropic considerations are important.

8.2.4 Theory of Colloid Stability Molecular aggregation and self-assembly is a well-studied field and will not be reviewed here (Hill, 1964; Israelachvili, 1992). The study of the aggregation of colloidal systems is also a mature field that is well described by the DLVO theory named after Derjaguin, Landau, Verwey, and Overbeek (Verwey and Overbeek, 1948). Often, the stability of a dispersion is characterized by the Schulze–Hardy (SH) rule, where the critical coagulation concentration (CCC) is related to the valence of the charged coagulant: (CCC)αZ +−6. This behavior is predicted by the DLVO theory for certain particle geometries under a limited set of experimental conditions. Sano et al. (2001) have studied the rapid coagulation of CNTs in aqueous media as a function of valence on several inorganic coagulants. Their analysis shows good agreement with the SH rule. This result is surprising, given the assumptions made in the DLVO theory and the extraordinary properties of SWNTs (Syue et al., 2006; Tao, 2006; Yanagi et al., 2006). The kinetics of coagulation depend on the surface functionality of the pristine or modified CNTs and on the nature of the


solvent. CNT dispersions can collapse in minutes or be “stable” for months. The physical state of dispersed CNTs is still not clear. Several solution phase scattering experiments indicate that dispersions of nanotubes consist of intertwined particles forming fractal geometries (Chen et al., 2004; Saltiel et al., 2005). Only under intense ultrasonication do these light scattering results imply a more rod-like morphology consistent with dispersed and isolated SWNTs (Schaefer et al., 2003). We have shown that deposited CNTs are found in mats of bundles and ropes, and as isolated and aligned SWNTs (Chaturvedi and Poler, 2006). It is unlikely that CNTs with fractal geometry in solution would separate into isolated CNTs upon deposition. We believe an accurate description of dispersed CNTs is still needed. It is likely that a dispersion of rod-like isolated CNTs will eventually aggregate into more complex morphologies, form a floc, and segregate from the liquid phase. While interacting plates and spheres are accurately described by the DLVO theory, the geometry and surface properties of CNTs are not. The wellaccepted method of treating interacting particles in solution is to sum the attractive dispersion forces between the particles and the repulsive electrical double-layer forces. By linearizing the Poisson–Boltzmann equation and integrating, the net potential energy of interacting spheres is (Evans and Wennerstrom, 1999; Verwey and Overbeek, 1948) ⎡ −H ⎤ 64kBTN 0Γ 20 V (x )sphere − sphere = πr ⎢ 121 + exp[−κ x]⎥ 2 π 12 x κ ⎣ ⎦ where r is the radius of the sphere H is the Hamaker constant for the system Γ is a result of linearization and depends on the zeta-potential of the particles and the valence charge, Z, of the solvated ions 1/κ is the Debye length, which depends on the number density of ions in the solution, N0 and the dielectric strength of the solvent, ⎡ 1/κ = ⎢ ⎢⎣

ε r ε0kBT

∑

1/2

⎤ ⎥ 2 ( Zi e ) N i 0 ⎥ i ⎦

The sphere–sphere and slab–slab V(x)slab–slab potential energies are both results of several approximations. The Debye–Huckle approximation, used in linearizing the electrostatics, requires the zeta-potential on the particles to be less than 25 mV (k BT). CNTs in aqueous solvent have zeta potentials in the range of −15 to −65 mV. In order to integrate these equations in closed form, the second assumption is that the radius of the particle be much larger than the Debye length, κ d > 5. However, the diameter of a CNT is much smaller than the Debye length in the dispersions we are using. This requires a numerical integration to model the system properly. Obviously, the geometry of interacting CNTs is neither spherical nor planar. Recent work has derived the electrical double-layer repulsion for a sphere interacting with a cylinder (Gu, 2000). Much progress has been made on the vdW attraction

8-5


between CNTs using a universal graphite potential (Girifalco et al., 2000), resulting in a general attractive vdW potential between SWNTs (Sun et al., 2005, 2006). While progress on theoretical descriptions of CNT aggregation is being made, further experimental work is required. Recent molecular dynamics simulations of CNTs in water show an interesting solvent-induced nanotube–nanotube repulsive interaction that is not taken into account in other models (Li et al., 2006). Th is interaction should be found for other nonaqueous solvents (Giordano et al., 2007).

molecules, a depletion of solvent is created around the solute molecules. This leads to a strong attraction between solute molecules. An example of this destabilization is exemplified by the formation of micelles when molecules are dispersed in incompatible solvents. Since the attractive forces between solute particles are dependent upon the solvent, it is critical to incorporate all of these forces to determine the stability of a solution. Dissolving a solute in a solvent will change the behavior from that predicted by placing the molecules in free space.

8.2.5 Theory of Particle Aggregation 8.2.5.1 Effects of Solvent and Solute on Particle Interactions

8.2.5.2 Effects of Ionic Strength and Charge Transfer Reagents on Dispersion Stability

Particles dispersed in a medium behave differently than in free space. Solvent may change the properties of the solute molecules, thereby changing the solution’s stability. Solute molecules are in constant motion and move by displacing the solvent. This means that if the work required to displace the solvent is greater than the energy gained by the approaching solute molecules, then the molecules will repel one another. Therefore, the interactions between the solvent and solute molecules play a vital role in the stability of a solution and a dispersion. The polarizability (α) of solvent molecules is affected by the medium if the solvent and solute have different dielectric constants. A continuum approach models the solute molecules, i, as dielectric spheres with a radius, ai, and dielectric constants, εi (Israelachvili, 1995; Landau and Lifshitz, 1960). The spheres gain an excess dipole moment and the solvent, with a dielectric constant ε, feels this as a polarizability defined by

Interactions between particles are strongly dependent on the electrolyte concentration of the surrounding medium. Charged surfaces in weak electrolyte solutions will experience a strong, long-range repulsion, shown by line 1 in Figure 8.3. Particles are unable to overcome this high energy barrier. Increasing the electrolyte concentration will create a secondary minimum that appears before the primary minimum. Line 2 in Figure 8.3 shows a secondary minimum. At low electrolyte concentrations, particles are unable to overcome the energy barrier and will either aggregate in the secondary minimum or remain dispersed in the solution. These suspensions are considered kinetically stable; adhesion in the secondary minimum is weak and easily reversible. Further increasing the electrolyte concentration will lead to slow aggregation until the electrolyte concentration reaches the CCC. At the CCC, the peak of the energy barrier crosses zero and particles can fall into the primary minimum and flocculate rapidly. This potential energy is shown as line 2 in Figure 8.3. The condition required for rapid aggregation is

The values of the dielectric constants are important in determining the attractive/repulsive force between dissolved particles. These changes imply that molecules with high dielectric constants attract ions and molecules with low dielectric constants repel ions. The importance of a solvent’s ionic strength is discussed in greater detail in the following section. Solvent–solute attraction is a stabilizing interaction (Belloni, 2000). Solvent molecules will form an adsorbed layer around each solute molecule. Therefore, as solute particles approach one another, the adsorbed solvent layer must first overlap; thus, preventing the two molecules from coming into true contact. Unless the solvent molecules can stick to two surfaces at the same time, a bridging attraction, it is more difficult for the solute molecules to stick together. Furthermore, the additional layer provides a steric repulsion at short distances. An example of solute–solvent attraction is hydrophilic interactions. Water molecules bind strongly to hydrophilic surfaces and hydration repulsion stabilizes the solute molecules. Overall, dispersions with solvent– solute attraction are more stable. Conversely, solvent–solute repulsion is a destabilizing interaction (Belloni, 2000). When solute molecules repel solvent

2

Interaction energy (10–19 J)

⎛ ε −ε ⎞ 3 α i = 4 πε 0 ε ⎜ i ai . ⎝ εi + 2ε ⎟⎠

(1)

1

0

(2) –1

(3)

–2

–3 1.E-10

1.E-09

1.E-08

1.E-07

1.E-06

Distance between spheres (m)

FIGURE 8.3 Graph showing the DLVO potential energy as a function of distance for various electrolyte concentrations. Line 1 shows an potential when the electrolyte concentration value is below the CCC. Line 2 is when the electrolyte concentration is at the CCC. Line 3 is the potential when the concentration is above the CCC.

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V (x ) = 0;

∂V ( x ) = 0. ∂x x =1/ κ

This aggregation is irreversible and the suspension is considered kinetically unstable. The energy barrier will continue to fall below zero, as shown by line 3 in Figure 8.3, as the electrolyte concentration is increased above the CCC; particles will irreversibly coagulate. Electronic, vibrational, and structural properties of the nanotubes are characterized using absorption or Raman spectroscopy. Characteristic and significant changes in the UV-Vis-NIR absorption spectrum and in the Raman active vibrational modes are observed due to charge transfer, selective functionalization, or induced stress on SWNTs (Alvarez et al., 2000; Bahr and Tour, 2002; Lin et al., 2003; O’Connell et al., 2001; Paiva et al., 2004; Pompeo and Resasco, 2002; Rao et al., 1997; Yudasaka et al., 2002). Studies of the covalent and noncovalent interactions of porphyrins with CNTs are directed toward the design of novel hybrid nanomaterials combining unique electronic and optical properties of the two components. CNTs are metallic or semiconducting, based upon delocalized electrons occupying a one-dimensional density of states. However, any covalent bond on SWNT sidewalls causes localization of these electrons. Water-soluble diazonium salts (Bahr et al., 2001b) react with CNTs forming a stable covalent aryl bond. These salts are found to preferentially interact with metallic SWNTs, forming covalent bonds. It is believed this is due to greater electron density in metallic SWNTs. The reactant first forms a charge-transfer complex at the nanotube surface by noncovalent physisorption of salt onto the nanotube surface, forming a charge-transfer complex. Th is is considered to be a rapid, selective noncovalent adsorption of salts on the sidewall of SWNTs. During the second step, the charge transfer complex decomposes to form a covalent bond with the nanotube surface, thus forming more defect sites. Upon thermal treatment, this process is found to be reversible, where the salts can be washed away from the extracted metallic SWNTs. Characteristic features observed in the absorption and Raman spectra of extracted SWNTs are similar to pristine SWNTs. The noncovalent functionalization of SWNTs with molecules like porphyrins is based on π–π interactions between the two components, and thus does not disrupt the intrinsic electronic structure of CNTs, which is important for electronic applications. Variations in the characteristic absorption and Raman spectra are observed in functionalized SWNTs, due to noncovalent interaction and charge transfer. 8.2.5.3 Kinetics of Aggregation The process of colloid aggregation is well understood for monodisperse samples, where all the particles have the same size, shape, and mass. In a solution, particles move randomly due to the Brownian motion. When this motion causes two particles to collide and irreversible stick to one another, clusters begin to

form; clusters collide and stick together forming a polydisperse solution of aggregates. Th is is known as cluster–cluster aggregation. Recall, for particles to irreversibly aggregate, the energy of the interactions must be greater than the thermal energy k BT. This type of aggregation exhibits universal aggregation kinetics; therefore, the kinetics are independent of the chemical properties of the colloids. There exist two limiting regimes for irreversible aggregation: reaction limited colloid aggregation (RLCA) and diff usion limited colloid aggregation (DLCA) (Ball et al., 1987; Lin et al., 1990b; Lin et al., 1989; Sandkuhler et al., 2005). Each regime is characterized by unique aggregation kinetics and aggregate structures (Weitz and Oliveria, 1984). Analytical approaches for describing kinetics use dynamic light scattering (DLS) to follow changes in the effective hydrodynamic radius of the particles. These data are modeled using the Smoluchowski rate equation (Lin, Lindsay, Weitz, Ball et al., 1990; Lin, Lindsay, Weitz, Klein et al., 1990) to describe the number of clusters of mass, M: N ( M ) = Mn−2ψ( M/Mn ). Here, Mn is defined as the nth moment of distribution Mn (t a ) =

∑ N (M) ⋅ M ∑ N (M) ⋅ M

n

M

n −1

M

where ta is the time that has elapsed since the initiation of the aggregation. The Smoluchowski rate equation exhibits dynamic scaling where ψ(x) is a scaling function that reflects the shape of the cluster mass. Furthermore, the mass of a cluster is related to its radius of gyration, Rg, fractal dimension, df, and radius of individual particles, a, by M = (Rg /a)df . Together, these basic equations create a foundation to describe the aggregation kinetics for both limiting regimes. When the repulsive energy barrier (EB) is slightly greater than or equal to kBT, the particles must overcome the barrier to stick together. In this case, several collisions must occur before particles stick, which is defined as RLCA or slow aggregation. The rate of aggregation is limited by the probability of overcoming EB (Lin et al., 1990a), written as P ~ exp(− EB/kBT ). The probability of two clusters sticking to one another is also proportional to the number of binding sites. Therefore, as particles stick together and form clusters, the probability changes. Since larger clusters have more binding sites, they grow more rapidly than smaller clusters and individual particles. This leads to an exponential rate increase in the average cluster size over time resulting in a highly polydisperse sample. Most samples formed in the slow regime contain large aggregates and many individual, non-aggregated particles.

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Cluster growth has been studied and analytical models have been developed to describe N(M) (Lin et al., 1990a). Experimental results for the shape of the cluster mass distribution are well described by a power law equation with an exponential cutoff:

Assuming the conservation of total mass, normalization is determined by N0 =

Gold Silica Polystyrene 2000

Rh (nm)

⎛ −M ⎞ N ( M ) = AM −τ exp ⎜ ⎟ ⎝ MC ⎠

5000

1000

500

∑ N (M) ⋅ M M

200

where N0 is the initial number of colloid particles. Here, A=

100 0.1

N 0 M Cτ− 2 Γ(2 − τ)

where Γ(υ) is the gamma function, which is determined by DLS and is inversely proportional to the effective diameter. The mass of the cluster increases as MC ~ exp(t a /t 0 ) where t0 is a sample-dependent time constant. Computer simulations and experimental results suggest a value of τ ∼ 1.5 for these systems. DLS studies following the change in the average hydrodynamic radius as a function of aggregation time show linear behavior on a semi-logarithmic plot, shown in Figure 8.4 (Lin et al., 1990a; Sandkuhler et al., 2005). This confirms that the absolute rate of RLCA is exponential. 5000

2000

– Rh (nm)

1000 500

200 100 Gold Silica Polystyrene

50

20

0

1

2

3

4

5

6

7

8

9

0.2

0.5

1

2

5 10 ta (min)

20

50

100 200

FIGURE 8.5 Graph showing the change in the average hydrodynamic radius, R h, as a function of aggregation time, ta, during DLCA. Both axes are logarithmic to show power-law kinetics. (From Lin, M.Y. et al., J. Phys.-Condens. Matter, 2, 3093, 1990b. With permission.)

Reducing the repulsive energy barrier to much less than k BT changes the regime to fast aggregation or DLCA. In this regime, every collision results in particles sticking to one another and the rate is limited by the time it takes for the particles to encounter one another. For DLCA, the average cluster mass (Lin et al., 1990b), M, is defi ned by M = M0/N t . M0 is the total mass of the system and Nt is the time-dependent number N ( M ). A dynamic of clusters at time, t, given by N t (t a ) = exponent, z, describes the time dependence of M by M ~ t az . Studies following the change in an average hydrodynamic radius as a function of aggregation time show linear behavior on a log– log plot, shown in Figure 8.5, which is indicative of the power law dependence (Lin et al., 1990b; Sandkuhler et al., 2005). Additionally, most colloid aggregates form highly disordered structures that are described by fractals. The fractal dimension of each aggregate describes how completely it fills space and depends on the kinetics of its creation. Aggregates formed during RLCA have fractal dimensions greater than 2 (Lin et al., 1989; Weitz and Oliveria, 1984). This is because the particles and clusters are able to sample all possible binding sites before choosing the site with the most contact. Thus, the structures created are denser. DLCA aggregates are characterized by fractal dimensions less than 2 (Lin et al., 1989; Weitz and Oliveria, 1984). In this case, every collision results in sticking and less dense aggregates are formed. Therefore, studying the resultant aggregate structure reveals information about the kinetics of its formation.

∑

ta (h)

FIGURE 8.4 Graph showing the average hydrodynamic radius, R h, as a function of aggregation time, ta during RLCA. Rh is logarithmic to show exponential growth kinetics. (From Lin, M.Y. et al., Phys. Rev. A, 41, 2005, 1990a.)

8.2.5.4 Photon-Induced Aggregation One important concern when studying colloidal dispersions is photon-induced aggregation. When metal clusters are illuminated at their plasmon wavelength, light-induced aggregation

8-8

may occur. Studies show that when slow-aggregating metal colloids are irradiated by light, the aggregation rate is drastically accelerated (Eckstein and Kreibig, 1993; Karpov et al., 2002; Kimura, 1994; Kreibig and Vollmer, 1995; Satoh et al., 1994). There are several explanations that account for the increase in aggregation kinetics. One explanation for this acceleration is photoelectron emission changing the electric charge of either the double layers or the clusters thereby changing the Coulomb forces between the clusters. Another explanation is that light induces forces similar to vdW forces. Due to the acceleration of aggregation rates when clusters are irradiated by light, care must be taken when using optical methods to collect aggregation data. The first explanation assumes the diff use region of the EDL experiences light-induced compression; and the DLVO forces are altered (Karpov et al., 2002; Kreibig and Vollmer, 1995; Satoh et al., 1994). The negative charge on the particles decreases due to the emission of photoelectrons. This decrease leads to a smaller surface potential, and consequently a decrease in the Coulombic repulsion between the two particles. These changes are accompanied by an accumulation of positive charge at the metallic core, which leads to an increase in adsorption potential at the surface. Counter-ions will increase at the dense region of EDL, leading to the compression of the diffuse region. Reducing the EDL surrounding the colloids accelerates the aggregation rate. The second explanation suggests the photon-induced enhancement of vdW forces (Eckstein and Kreibig, 1993; Karpov et al., 2002; Kimura, 1994). Recall, the Hamaker constant describes the attractive interaction between particles and is a result of zero point fluctuations of electronic polarizations. These fluctuations are predominately determined by conduction band electrons, which exist as surface plasmons for metal colloids. Excitation of the surface plasmons by light irradiation induces electromagnetic multipolar interactions. Therefore, the vdW attraction between particles increases, which leads to an increase in the aggregation kinetics. The interaction of charge transfer reagents and SWNTs in solution is essentially driven by dispersion forces and other vdW forces. These forces between molecules are essentially electromagnetic in nature and, along with the surface charges in the solution, drive the molecules into supramolecular assemblies. The aggregation of nanoscale particles from the solution depends on the type and charge of the coagulant and also on the surface charge or zeta-potential on the nanotubes themselves. The zeta-potential of the CNTs in the aqueous solvent is in the range of ζ = –15 to −65 mV, depending on the solution pH and nanotube preparation. Along with using the charge and concentration of coagulants, photons can be used to reduce the repulsive potential barrier between the SWNTs in stable dispersion causing enhanced rapid flocculation. The double-layer repulsion term essentially depends on the counter-ion concentration and Debye length. The counter-ion concentration interacting with the SWNT surface can be changed by charge donation into the SWNT. Optically active molecules can absorb light and donate charges into the SWNT, thereby affecting the surface potential


and counter-ion concentration. It is shown that when SWNT solutions include photoactive metallodendrimers and are optically illuminated at the metal to ligand charge transfer (MLCT) absorption band, photon enhanced aggregation occurs. This rate of photon enhanced aggregation rate depends linearly on the illumination power (Chaturvedi and Poler, 2007).

8.3 State-of-the-Art Procedures and Techniques 8.3.1 How to Make a Dispersion of CNTs Dispersing CNTs in liquids is important for many reasons. CNTs are typically synthesized by laser ablation, electric arc discharge, CVD, electrolysis, or sonochemical methods (Hilding et al., 2003). Regardless of the method and growth conditions, CNTs in a condensed phase are bundled together due to π–π stacking along their length. Whether one purchases CNTs in powder form or grows them, it is necessary to disperse them in a liquid for processing. CNTs do not easily disperse into solution due to their strong tube–tube interactions and their length (hundreds of nanometers to microns). Sonication is the most common method used to de-bundle CNTs, allowing them to be dispersed in a solvent. Depending on one’s needs, three different types of sonicators are used: bath, tip, and cup-horn sonicators. Bath sonication is the gentlest form of sonication and does not cause as many sidewall defects, whereas tip or cup-horn sonicators can be far more forceful at de-bundling the CNTs. Sonication can shorten CNTs and damage the sidewalls, so sonication power and time must be chosen carefully. Furthermore, the choice of solvent also affects the sonication power and time, which can influence the dispersion quality and stability. 8.3.1.1 Aqueous versus Nonaqueous Dispersions 8.3.1.1.1 Aqueous Dispersions CNTs are completely insoluble in water when pristine. CNTs must be modified through acid treatment, a surfactant coating, or covalent functionalization. Surfactants such as sodium dodecyl sulfate (SDS) (Cardenas and Glerup, 2006; Islam et al., 2003; Moore et al., 2003; O’Connell et al., 2002; Strano et al., 2003), Triton X-100 (Hilding et al., 2003; Islam et al., 2003; Liu et al., 1998; Moore et al., 2003; Tan and Resasco, 2005; Wang et al., 2004), or sodium dodecylbenzene sulfonate (SDBS) (Cardenas and Glerup, 2006; Islam et al., 2003; Matarredona et al., 2003; Moore et al., 2003; Yamamoto et al., 2008) are used most frequently. However, other surfactants have also been explored, such as Gum Arabic (Bandyopadhyaya et al., 2002), 4-(10-hydroxy)decyl benzoate (Mitchell et al., 2002), trimethyl-(2-oxo-2-pyren-1-yl-ethyl)-ammonium bromide (Nakashima et al., 2002), PmPV-based polymer (Star and Stoddart, 2002), or even single-stranded DNA (Zheng et al., 2003a). Typical preparation methods use deionized water or D2O with 1% surfactant by weight (O’Connell et al., 2002), as this solution is above the critical micelle concentration for SDS as shown

8-9


8.3.1.2 Techniques to Determine SWNT Dispersion Properties 8.3.1.2.1 UV-Vis-NIR Spectroscopy

in Figure 8.6. However, lower surfactant concentrations can still exfoliate and de-bundle CNTs sufficiently. An alternative method to the 1% by weight method is a gradual addition of surfactant at intervals during sonication (Yamamoto et al., 2008). There is evidence that the gradual addition method may do a better job of dispersing individual tubes without a large excess of surfactant left in the solution. Once a surfactant solution has been prepared, powdered CNTs are added and one of the sonication methods is used to accelerate the de-bundling process. An alternative to using surfactants is to covalently functionalize the CNT sidewalls. Many types of covalent functionalization have been employed, including amidation, thiolation, halogenation, bromination, esterification, chlorination, fluorination, and hydrogenation (Hirsch and Vostrowsky, 2005). Covalent sidewall functionalization will significantly affect the electronic and mechanical properties, because the change from sp2 to sp3 hybridization of the sidewall carbon atoms causes irregularities in the hexagonal lattice (Hirsch and Vostrowsky, 2005). Consequently, this approach to CNT dispersion is not acceptable when studying properties of pristine CNTs. The last major method for rendering CNTs dispersible in water is to use an acid treatment. Acid treatments severely damage the integrity of the CNT sidewalls by causing many defect sites. Moreover, when the acid/CNT mixture is sonicated, CNTs will be cut at those defect sites because of the collapse of sonicationinduced cavitation bubbles (Liu et al., 1998). Acid treatment is an effective way to make CNTs dispersible or rapidly reduce the mean length of CNTs in a dispersion. This method should be avoided if features of pristine CNTs are to be studied. 8.3.1.1.2 Nonaqueous Dispersions To study the properties of pristine CNTs, nonaqueous solvents must be used since they do not require any type of surfactant, defect creation, or covalent functionalization, all of which modify CNT properties (Ausman et al., 2000). As with aqueous dispersions, sonication is used to disperse CNTs in nonaqueous solvents. A wide variety of nonaqueous solvents have been tested and the concentration of CNTs that can be dispersed in each solvent varies significantly (Ausman et al., 2000; Bahr et al., 2001a; Landi et al., 2004).

1.8

1.5

Absorbance

FIGURE 8.6 SWNT dispersions of varying concentrations 54, 27, 13.5, 6.75, and 3.38 mg/L in fi ltered, 1% w/w aqueous SDS.

SWNT samples prepared according to the methods described above can be analyzed by several techniques. UV-Vis-NIR spectrometry is a low-cost, convenient, and accurate method for ascertaining the quality and concentration of a SWNT dispersion. SWNTs are strong optical absorbers and have a distinctive optical absorption spectrum with several salient features. Figure 8.7 illustrates a typical UV-Vis-NIR absorption spectrum of a SWNT dispersion. The shape of a SWNT absorption spectrum is dominated by the background π–plasmon resonance peak at 260 nm, which comes from the sp2 hybridized bonds. If low quality, highly defected SWNTs have been dispersed, the π–plasmon resonance is the only visible feature. When high-quality, pristine SWNTs are dispersed, vHs can be easily detected in the spectrum. vHs are a consequence of the onedimensional electronic density of states in SWNTs (Dresselhaus et al., 2004). vHs manifest themselves as small peaks along the π–plasmon background and the magnitude of those peaks, with respect to the background π–plasmon resonance, can be used as a qualitative way to determine whether or not a good dispersion has been produced. In general, the ratio of the absorption from the vHs to the π–plasmon resonance should be greater than 1 for a good dispersion. As shown in Figure 8.8, there are three distinct regions of vHs in a SWNT optical absorption spectrum (Kataura et al., 1999). In the S11 and S22 regions, 1–1 and 2–2 transitions for semiconducting SWNTs are observed. The M11 region is where the vHs are due to 1–1 transition in metallic SWNTs. To quantify the concentration of SWNT dispersions, Smalley’s group developed a Beer’s Law approach to determine the

1.2

0.9

0.6

0.3 400

600

800

1000

1200

1400

1600

Wavelength (nm)

FIGURE 8.7 UV-Vis-NIR spectrum for SWNTs has the right tail of the π-plasmon resonance, with vHs peaks clearly visible. (Reproduced from Bahr et al., Chem. Commun., 193, 2001. With permission.)

8-10


M11

–1

S11

1 0

S11

–1 DOS

DOS HiPCO

S22

8.3.1.2.2 Fluorescence Spectroscopy

M11

S22

Laser M11

S11 0.5

S22

S22 1.0

Arc

M11 1.5 2.0 Energy (eV)

2.5

3.0

FIGURE 8.8 UV-Vis-NIR absorption spectrum, indicating the S11, S22, and M22 ranges. In this figure, the π-plasmon background has been removed. The insets schematically show the transitions in energy vs. density of states plots. (Reprinted from Niyogi, S. et al., Acc. Chem. Res., 35, 1105, 2002. With permission.)

concentration of SWNT dispersions based on optical absorption at 500 nm (Bahr et al., 2001a). Samples of pristine SWNTs in 16 organic solvents with varying concentrations were prepared by sonication and the optical absorption at 500 nm was measured. The solvent was then removed by oven drying and the SWNT residue was weighed, allowing Smalley et al. to determine an absorption coefficient. This approach allows the rapid quantification of the concentration of a SWNT dispersion based on optical absorption. Based on the simple tight-binding theory, the semiconducting SWNTs give rise to a series of electronic transitions between the principal mirror spikes in the electronic density of states (DOS). Electronic transition energy between corresponding levels in the valance and conduction band can be generally formulated as Sn = (2naC − C β /dt ), where n is an integer value corresponding to the transition, i.e., n = 1, 2,… for S11, S22 respectively, aC–C is the carbon–carbon bond length (0.142 nm), β is defined as the transfer integral between π–orbitals (β ≈ 2.9 eV), and d is the SWNT diameter (nm), e.g., S11 = 2na C–Cβ/d and S22 = 4aC–Cβ/d, while the metallic SWNTs show their first transition at M11 = 6aC–Cβ/d. These vHs in the absorption spectra are observed over a broad monotonic background absorption due to π–plasmon resonance. In a bulk characterization technique such as absorption spectroscopy S11, S22 transitions from different diameter semiconductor tubes and M11 transitions from different metallic tubes are simultaneously observed. π–plasmon absorption from both the SWNTs and carbonaceous impurities are observed in UV. These spectral characteristics provide an opportunity to distinguish between the SWNTs and impurities present in the sample using absorption spectroscopy. While the S11 transition is the most prominent, the second semiconducting transition (S22) is used as a reference for evaluation purity because the S22 transition is less susceptible to doping

Band-gap fluorescence provides another method by which SWNT dispersions may be characterized. Fluorescence is especially useful when analyzing SWNT dispersions because it can reveal the chirality of SWNTs present in the dispersion as well as the relative abundance of each chirality. SWNTs have characteristic excitation and emission energies depending on the chirality of the SWNT, as shown in Figure 8.9. Sample preparation for measuring SWNT fluorescence is more challenging than for UV-Vis-NIR absorption. Bundled SWNTs disturb each other’s electronic properties and quench the already weak fluorescence signal. The predominant method for preparing SWNT samples for fluorescence measurements was developed by Smalley’s group (O’Connell et al., 2002). HiPCO grown SWNTs are first dispersed into D2O/SDS by sonication. The surfactantbased method was chosen to minimize the re-aggregation of SWNTs once the dispersion has been made. Next, ultracentrifugation is employed to remove as many of the small (or large) bundles that remain partially dispersed. Ultracentrifugation fields on 5 4 Conduction 3 c2

2 1 Energy

S11

1 0

as compared to S11. The ratio A(S22)/A(T) can be used to ascertain SWNT purity, where A(S22) is the area of the S22 interband transition after linear baseline subtraction and A(T) is the total area under the spectral curve. Th is ratio was then normalized by dividing by 0.141, which is experimentally determined using the ratio of A(S22)/A(T) in the reference sample of SWNTs. This procedure is used to calculate the relative purity of SWNTs (Haddon et al., 2004). An absolute determination of the purity of SWNT is not yet possible due to the lack of reference SWNTs.

Semiconducting Energy (eV)

Energy (eV)

Absorbance (a.u.)

Metallic

c1 E11 fluorescence

0

E22 absorption

–1

v1

–2

v2

–3 Valence –4 –5

0

2

4

6

8

10

Density of electronic states

FIGURE 8.9 Schematic representation of SWNT fluorescence, where a photon of energy E22 is absorbed and a photon of energy E11 is emitted. (Reprinted from Bachilo, S.M. et al., Science, 298, 2361, 2002. With permission.)

8-11


Excitation wavelength (nm) [vn

cn transition]

900 0.3000 0.2323 0.1798 0.1392 0.1078 0.08348 0.06463 0.05004 0.03875 0.03000 0.02323 0.01798 0.01392 0.01078 0.008348 0.006463 0.005004 0.003875 0.003000

800

700

600

500

400

300 900

1000 1100 1200 Emission wavelength (nm) [c1

1300 1400 v1 transition]

1500

FIGURE 8.10 (See color insert following page 20-16.) Th ree-dimensional fluorescence scan of HiPCO grown SWNTs. Circled in white are some of the well defined fluorescence peaks that are characteristic of the chirality distribution found in SWNTs produced by the HiPCO method. (Reprinted from Bachilo, S.M. et al., Science, 298, 2361, 2002. With permission.)

the order of 100,000 g for several hours ensure that only the welldispersed, individual SWNTs remain in the supernatant, which is then used for fluorescence spectroscopy. SWNT excitation/emission peaks in the near infrared are used to uniquely identify SWNT chirality. Consequently, threedimensional fluorescence scans are commonly used to observe the emission over a range of excitation wavelengths. From these scans, a contour plot is generated, as shown in Figure 8.10. In order to correctly interpret the fluorescence spectra from SWNT dispersions, it was necessary to use both fluorescence and Raman spectroscopy to assign specific chirality SWNTs to the appropriate peaks. Over 30 assignments were made by Weisman et al. (Bachilo et al., 2002) and can be used to rapidly identify some of the SWNTs that are typically found in SWNT dispersions. A shortcoming of fluorescence is that some SWNT chirality do not fluoresce strongly enough to be detected and metallic SWNTs, of course, do not fluoresce at all. 8.3.1.2.3 Raman Spectroscopy Raman spectroscopy is a widely used tool to study the vibrational properties of materials. It is an important tool for analyzing nanomaterials, especially SWNTs. It provides ease in sample preparation and is a nondestructive, efficient tool for sample characterization. Incident photons interact with an electron that makes a transition to a higher energy virtual state where the electron interacts with phonons or Raman active vibrational modes before making a transition back to the electronic ground state. In this instantaneous process, both the energy and momenta are conserved, hence we can write that E scattered = Elaser ± hΩ for

Stokes (−) and anti-Stokes (+) scattering respectively, with Ω being the frequency of the particular phonon mode. The energy of the inelastically scattered light is measured with respect to the applied laser energy, and by convention, the Stokes shift is typically plotted as a negative shift. Typically, laser energy in the visible or near infrared is used. Laser energy does not affect the Raman shift, but if the laser energy is resonant with the electronic transition of SWNTs, it results in an exceptional increase in the observed Raman intensity. For CNTs that have a one-dimensional density of states, resonant Raman scattering dominates over nonresonant contributions. Raman scattering from nanotubes gives information about the vibrational modes of SWNTs. Raman scattering can be used for quantitative measurements to provide details about the diameter and chirality distribution of SWNTs, functionalization, stress, and charge transfer properties in SWNTs. However, it is commonly used in conjunction with other tools as Raman modes of SWNTs are found to be exceptionally sensitive to changes in the SWNT environment and aggregation states. Raman scattering is caused by the interaction of light with matter. The vibrations (phonons) of the nanotubes exhibit unique properties due to their one-dimensional tubular nature. Novel radial breathing modes (RBMs) are observed in SWNT that are not observed in any other carbon allotrope. In Raman scattering (inelastic scattering of light), photons excite or absorb phonons in the nanotubes and their frequency change is observed in the spectrum of the scattered light. Raman scattering thus enables us to understand the vibrational and electronic structure of the SWNTs.

8-12

Intensity (a.u.)


100

600

1100


2100

2600

FIGURE 8.11 Typical Raman spectra of pristine SWNT. Peaks in range (150–300 cm−1) are denoted as RBM. Peaks between the wavenumber 1550–1600 cm−1 are the G-band. And peaks about 1250 and 2570 cm−1 are the D-band and G′-band. Weak bands in the range 650–1000 cm−1 are identified as the intermediate frequency band.

Raman spectroscopy of SWNTs has been well studied and is broadly disseminated (Dresselhaus et al., 2004). It provides us with detailed information about the electronic, vibrational, and structural properties of SWNTs. Raman spectra of SWNTs have been found to be remarkably distinct from graphite due to spatial confinement at the nanoscale and their one-dimensional behavior. The most important features in typical Raman spectra of SWNTs are shown in Figure 8.11. Typical SWNT spectra have two distinct bands of transitions. At small Raman shifts ∼150–300 cm−1, RBMs are observed and at higher Raman shifts ∼1590 cm−1, the G-band is observed. There is a less intense but significant band at ∼1250 cm−1 called the D-band. The origin of other bands, including the weak intermediate frequency bands and a fairly intense peak at 2570 cm−1 called the G′-band are explained by the processes such as overtones, double resonance, or combinational modes.

Intensity (a.u.)

8.3.1.2.3.1 Radial Breathing Modes RBMs are due to the radial motion of the atoms perpendicular to the tube axis. These modes are found due to isotropic radial vibrations of the tube, which are a manifestation of the one-dimensional tubular structure of SWNTs. The RBM frequency is inversely proportional to the diameter of the tube, making it an important feature for determining the diameter distribution in a sample, as shown in Figure 8.12. The frequency of these transitions have strong diameter dependence, ωRBM = αRBM/dt. RBMs are also strongly

185

205

225

245

265

285


FIGURE 8.12 RBM region of the Raman spectra of pristine SWNTs using a 785 nm laser.

affected by the neighboring tubes and sample environment due to vdW forces. For SWNT aggregates and bundles, a constant αbundle is added to the expression for ωRBM. The modified expression is given as ωRBM = αRBM/dt + αbundle. Values of αRBM and αbundle are found to be 227 and 14 cm−1, respectively. The peaks exhibit a Lorentzian line shape described by I(ω) = I0 + (2A/π) (τ/(4(ω – ωc)2 + τ2). I(ω) is the intensity at the frequency ω and I0, ωc, A, and τ represent the constant shift, center frequency, area under the curve, and full width half maximum, respectively. 8.3.1.2.3.2 Tangential G-Band The G-band consists of several peaks in the frequency range of 1500–1600 cm−1. These Raman active modes are theoretically explained by zone folding of the graphite modes, which makes Raman inactive graphene modes become Raman active G+ and G− modes for SWNTs (Brown et al., 2001). The G-band is characterized by a tangential shear vibrational mode of the carbon atoms in the nanotube. Two prominent peaks at higher (∼1590 cm−1) and lower (∼1560 cm−1) frequencies are denoted as G+ and G−, respectively. The frequency of the G+ mode depends on the laser excitation energy. G+ is expected to be diameter independent and G− is expected to be diameter dependent. The G-band is the most intense peak in the spectrum. The physical origin of the G-band is unclear, because it depends on the combination of several processes such as the resonance, polarization effects, and electron-phonon coupling. The G-band forms a distinctive, asymmetric line shape at a lower frequency described by a Breit–Wigner–Fano (BWF) line shape. The origin of this line shape is attributed to the metallic nature of the tubes. This line shape is assumed to emerge because of plasmon–phonon coupling in metallic tubes. The ratio between the intensity of G+ and G− can be used for quantifying the amount of semiconducting and metallic SWNTs in the dispersion or in the sample. Metallic tubes are easily recognized from the broad and asymmetric BWF line shape of the G−-band. The frequency down shift of the G− is particularly strong for metallic nanotubes, with down shifts of ∼100 cm−1 for small diameter tubes. 8.3.1.2.3.3 D-Band The D-band is a common feature found in all defected sp2 carbon hybridized carbon materials. Defects in graphene result in more sp3 type bonding. The intensity of the D-band has been found to increase with the defects introduced. The most convincing explanation for the formation of

8-13


the D-band can be explained by the double resonant process. Incoming photons create an electron–hole pair in the nanotube. The electron is then scattered resonantly to another point in the Brillouin zone by a phonon with momentum k ≠ 0. This electron is then scattered by a defect back to a virtual state having the same momentum as before it was scattered by the phonon. Consequently, it recombines with the hole and creates a photon. Since the electrons are scattered back by the defect, the D-band is an important tool for characterizing the concentration of defects in the samples. The G′-band is explained by a two phonon double resonance process. The degree of disorder in the system can be quantified using the G′-band since the intensity, IG′, does not depend on the disorder, and thus the ratio, ID/IG′, is the measure of the disorder scattering. Since, both of these bands involve the same phonons, the ratio ID/IG′ should provide a good estimate for the disorder in the tubes due to defects. Moreover, the width of the G′ Lorentzian peak is inversely correlated with the degree of the nanotube dispersion (Cardenas, 2008).

8.3.2 How to Make CNT Thin Films and Devices The synthesis and proper characterization of SWNTs in 1991 (Iijima and Ichihashi, 1993), inspired many studies and encouraged the suggestion of countless applications using these unique allotropes of carbon. The properties that make CNTs unique also could enhance the performance of well-established devices and produce new technology. Devices based on classical physics, such as textile, building, or polymer materials would gain enhanced tensile strength and durability. Moreover, electronic components benefit from both the enhanced physical and tunable electrical properties of CNTs. New devices based on quantum physics and confinement phenomena are also possible when SWNTs are used as the building blocks. As with all nanofabrication, there are two schools of thought when building a device formed from nanoscale building blocks. The two methods are typically referred to as the bottom-up approach and the top-down approach. With respect to CNTs, the bottom-up approach would translate as building a device containing one to several CNTs with nanoscale dimensions. Bottom-up nanoscale devices would exploit the properties and phenomena of individual CNTs and could then be combined to form arrays or integrated into larger devices. An example of such a device is the CNTFET, which is discussed below. Top-down device fabrication uses CNTs formed together as a quasi-bulk material then manipulated and incorporated into a device. A device made this way could utilize well-established fabrication techniques and still exhibit many of the unique phenomena afforded by the individual CNTs that are interspersed throughout the device. These devices can utilize agglomerations of CNTs such as CNT bundles, ropes, fibers, or as-grown SWNT forests (Pint et al., 2008). Or they can be deposited as films of CNTs (Behnam et al., 2007; de Andrade et al., 2007; Gonnet et al., 2006; Gupta et al., 2004; Kavan et al., 2008; Kazaoui

et al., 2005; Lima et al., 2008; Liu et al., 1999; Merchant and Markovic, 2008; Ng et al., 2008; Pint et al., 2008; Song et al., 2008; Takenobu et al., 2006; Wang et al., 2008b; Wu et al., 2004; Zhang et al., 2004b; Zhu and Wei, 2008), or as mats (Deck et al., 2007; Gupta et al., 2004; Sun et al., 2008) or buckypaper (Rinzler et al., 1998; Whitby et al., 2008). 8.3.2.1 Fabrication of CNT Films and Substrate Adhesion Regardless of the naming conventions, almost all CNT fi lm fabrication processes consist of two major steps: organization of the CNTs into a fi lm and attachment of said fi lm to a substrate. An important distinction needs to be made between CNT fi lm fabrication and mats formed during CNT growth. Some of the most common CNT synthesis methods, like arc discharge and pulsed laser vaporization, produce complex structures of CNT ropes that are intertwined and entangled into mats (Gupta et al., 2004). These mats are highly disordered and there is very little control of their final morphology and properties. CNT film fabrication techniques can employ many different methods that affect the properties of a CNT fi lm. Currently, researchers fabricating CNT fi lms use a variation of one of the following five fabrication methods depicted in Figure 8.13. 8.3.2.1.1 Vacuum Filtration Vacuum fi ltration of CNT dispersions to form a fi lm seems to be the most widely used method (de Andrade et al., 2007; Liu et al., 2006; Whitby et al., 2008; Wu et al., 2004), probably due to the ease of setup and the many adjustable parameters. Figure 8.14 clearly illustrates the wide variability of results one can produce from vacuum fi ltration, from transparent 100 nm thickness flexible fi lms to buckypaper so thick and rigid they are referred to as buckydiscs (Whitby et al., 2008). There are typically two steps in the vacuum fi ltration of CNT dispersions. First, a dispersion of CNTs is fi ltered through a porous membrane or fi lter. Since fi ltration is intrinsically self-limiting, the uniformity of the fi lm on the fi lter membrane is therefore self-leveling. The second step usually involves transferring the deposited CNT fi lm from the membrane or fi lter paper to a more favorable substrate. In the case of a buckypaper thick enough to be self-supporting, transfer to a substrate may not be necessary at all, as evidenced by Figure 8.14b. The fi ltration step is simple in concept, but the method can vary enormously depending upon the desired product. While a CNT is dispersed and unbundled, it can be modified either chemically or physically, changing the inherent properties of the dispersed tubes and ultimately modifying the properties of the deposited fi lm. Changing the amount of CNTs dispersed in a solution is the most effective way to adjust fi lm properties such as thickness, opacity, rigidity, and conductivity. Sometimes it is necessary to increase the volume of the dispersion while increasing the amount of CNTs in a solution to remain below the CCC and avoid coagulation and bundling of the dispersed CNTs before they can be deposited into a film.

8-14


Dispersed CNT/DCE

x

Spray gun y

CNT dispersion CNTs

Sonication

CNT Indium layer ITO glass

Ar

Spray (a)

(b) Pull Substrate

–

APTS

+

SWNT

(c)

(d) (ii)

(i) Roller

Foil

200 μm (e)

FIGURE 8.13 CNT fi lm preparation by: (a) Vacuum Filtration, (b) Spray-Coating (Jeong et al., 2006), (c) Dip-Coating (Ng et al., 2008), (d) Electrophoretic Deposition (Lima et al., 2008) Reprinted by permission of the Royal Society of Chemistry, and (e) i. and ii. Post-Growth Extrusion. (Reprinted from Pint, C.L. et al., ACS Nano, 2, 1871, 2008; Zhang, M. et al., Science, 309, 1215, 2005. With permission.)

CNT dispersions are also affected by the choice of solvent and the possible requirement of a surfactant to stabilize the CNTs in the dispersion. Ideally, surfactant is present only in the solution phase of the process and is removed from the CNT fi lm during fi ltration. However, it has been shown that vigorous washing of a deposited fi lm is required to completely remove a surfactant, possibly due to an intercalating process that traps surfactant molecules between the intermingled and woven CNTs within a fi lm. The most apparent result of inadequate surfactant removal is the significantly reduced integrity of the film. Intercalated surfactant will inhibit the vdW interactions between the CNTs and diminishes the intrinsic electrical properties of the fi lm due to

a reduction of π-conjugation, resulting in an apparent increase of surface resistance (de Andrade et al., 2007). To reduce the risk of having excess surfactant, the minimum amount of surfactant required to maintain a stable dispersion should be used. Many publications report a surfactant concentration of 0.1%–1% by weight (de Andrade et al., 2007; Whitby et al., 2008; Wu et al., 2004), which seems to be appropriate for maintaining dispersions on the order of a few milligrams per milliliter. However, it has been difficult to remove surfactant from films made from filtrate containing 1% by weight surfactant. An effective way to reduce the amount of surfactant in the fi ltrate is to create two separate dispersions; a stock dispersion with a large

8-15


10 cm Large area transparent conductive nanotube films

(a)

(b)

FIGURE 8.14 (a) Examples of ultrathin (From Wu, Z.C. et al., Science, 305, 1273, 2004.) and (b) ultra-thick (From Whitby, R.L.D. et al., Carbon, 46, 949, 2008.) CNT fi lms and mats.

concentration of CNTs with 0.1%–1% surfactant and a second dispersion, made from an aliquot of the first and a large amount of solvent. This allows the surfactant molecules that are interacting strongly with CNTs to remain associated and maintain stability while dispersing excess surfactant in large amounts of solvent. This seems to reduce the required washings of a CNT film to a minimum, maintaining film uniformity and integrity. A chemical treatment of the CNTs before or during their dispersion can also be implemented. The most common chemical treatments use strong acids, such as concentrated sulfuric acid mixed with concentrated nitric acid to encourage defect sites along the CNTs and at their ends. This strong acid solution can release NO2(g) and should only be done under a vented chemical hood. These kinds of treatments typically affect the electrical properties of the film, changing the conductivity and photoconductivity of the end product. In a similar process, it may also be possible to physically bind CNTs with substances that would enhance or amplify desirable properties such as absorption and conductivity. After a film is deposited, it must be removed from the filter and adhered to a substrate. Removal of the film from the fi lter is a fairly straightforward process. Once the film is intimately attached to a new substrate, the filter membrane can be removed either by peeling the film from the filter or dissolving the filter. The latter tends to be the least destructive method for thinner films. A common problem that occurs during this process is that adhesion of the film to the new substrate does not occur. This method assumes that a wetted filter attached CNT film can be pressed onto a substrate and, as the wetting agent (usually water) dries, capillary forces will pull the CNT film into intimate contact with the substrate, where vdW forces hold them together firmly. This method works well and is widely used, however little or no information is presented on the preparation of the new substrates. New substrates must be sufficiently cleaned. Oxide coated substrates are cleaned by means of piranha etchant, where the substrate is placed into concentrated sulfuric acid: 30% hydrogen peroxide at a 3:1 ratio in a bath sonicator for 30 min. and rinsed copiously with deionized water. Th is wet process is followed by an oxygen plasma scrubbing to remove any surface

contaminants. Typical plasma conditions of 200 W under 200 mTorr O2(g) for 30 min are sufficient. Polymer substrates, such as polycarbonate, are also cleaned using similar methods. Leaving the substrate in water or methanol immediately aft er plasma cleaning helps preserve the surface until one is ready to transfer the fi lm. 8.3.2.1.2 Spray-Coating Spray-coating is widely used to deposit fi lms, adhesives, and pigments. It is also capable of producing very thin and uniform CNT fi lms. Spray-coating to deposit CNT fi lms is the most precise method of deposition. Figure 8.13b illustrates the spray-coating method using an air brushing technique. Another variation of this film fabrication method is inkjet printing. Inkjet printing has been used by many groups to print patterned layers of nanoparticles on substrates just like a printer lays down ink when printing on paper. These two variations can print highly resolved images using either masks or a computer controlled printing mechanism patterning CNT fi lms onto a substrate. The initial preparation of the CNTs is similar to that of vacuum fi ltration. The dispersion is made following established techniques, with the stipulation that no surfactant is used. N-methyl2-pyrrolidone (NMP) has been observed as being a good solvent for this purpose (Beecher et al., 2007). Then, using either an air gun or printing mechanism, the CNT solution is sprayed as an aerosol onto the substrate (Artukovic et al., 2005; Beecher et al., 2007; de Andrade et al., 2007; Merchant and Markovic, 2008). Figure 8.15 shows an image of an electrode pattern in which the gaps are printed with CNTs (Beecher et al., 2007). Problems arise in the form of continuity while using this method. The CNT fi lm morphology depends on the temperature of the substrate, droplet size, and solvent type. Typically, the capillary forces induced by the drying droplets will ball up the CNTs or bundle and separate CNTs within the droplet creating an incongruent layer. Th is effect will manifest as a higher fi lm resistivity. 8.3.2.1.3 Dip-Coating Dip coating has the distinction of being the simplest of the CNT fi lm deposition techniques. Using previously described

8-16


40 μm

FIGURE 8.15 Interdigitated CNT electrodes deposited by spray coating. (From Beecher, P. et al., J. Appl. Phys., 102, 7, 2007.)

methods, CNT dispersions are created with CNT concentrations of the order of milligram per milliliter (Ng et al., 2008; Song et al., 2008). Then, much like one would dip a candle, the substrate is immersed into the dispersion in a vertical orientation. The dipping process is repeated until a coating of CNTs of the desired thickness has formed. The thickness can usually be estimated by a calibrated optical absorption curve correlated with AFM measurements. As one would imagine, this process is highly dependent on the attraction and adhesion of CNTs to the substrate surface, so a clean substrate is always required and in many cases a coating of polyethylene terephthalate (PET) is used as the substrate coated on glass (Song et al., 2008). In other cases, when a conductive substrate is required, a binding agent can be utilized (Ng et al., 2008). Figure 8.13c illustrates dip-coating with a glass substrate utilizing 1,2-aminopropyltriethoxysilane (APTS) as an adhesion promoter. 8.3.2.1.4 Electrophoretic Deposition Electrophoretic deposition (EPD) utilizes an electric field to deposit charged particles onto a surface (Boccaccini et al., 2006; Guo et al., 2007; Lima et al., 2008; Poulin et al., 2002; Wang et al., 2007; Zhao et al., 2005). EPD is particularly attractive to those wanting to incorporate CNTs with electrodes. Typically, a conductive material acts as both an electrode and a substrate, attracting CNTs towards an oppositely charged electrode and building up a densely and uniformly packed/bonded layer. Until recently, this method produced CNT fi lms attached to opaque conductive substrates. New methods have shown that a substrate coated with a very thin layer of metal will serve as an electrode for the deposition of CNTs. The ultrathin metal layer is then oxidized to form an optically transparent metal-oxide such as Al 2O3 or TiO2 (Lima et al., 2008). 8.3.2.1.5 Post-Growth Extrusion Post-growth extrusion (PGE) is a noteworthy recent development for forming films from CNTs. PGE is a term that suggests extruding a film from an as-grown CNT material,

typically a CNT forest. Two different methods have been utilized for this technique (Pint et al., 2008; Zhang et al., 2005). The first method uses an adhesive tape to adhere to the exterior edge of a multi-walled carbon nanotube (MWNT) forest. Subsequently, the edge of the MWNT forest is extended from the substrate with a constant pulling force. This method results in an extended continuous transparent sheet of unraveled MWNTs (Zhang et al., 2005). The second technique uses a foil laid over the top of a CNT forest, followed by a roller pressing the foil to compress and bend the CNTs over each other, forming them into a highly aligned dense film (Pint et al., 2008). Figure 8.13e (i) and (ii) illustrate these techniques. Either chemical or physical means can be used to attach these films to new substrates. 8.3.2.2 Devices Made from CNT Dispersions 8.3.2.2.1 CNT-Based Field Effect Transistors The exceptional mechanical, electronic, and opto-electronic characteristics of SWNTs make them important materials for novel sensors and nanodevices. The field effect transistor (FET) response of SWNT devices has been fairly well documented (Appenzeller et al., 2002; Avouris, 2002; Guo et al., 2004; Snow et al., 2003). These carbon nanotube field effect transistors (CNT-FETs) are used to probe the electronic properties of functionalized SWNTs. 8.3.2.2.1.1 Theory As discussed above, carbon has four valence electrons. In SWNTs, these electrons in trigonal planar sp2 orbitals form strong σ covalent bonds with the neighboring carbon atoms. The robust mechanical properties of SWNTs are mainly caused by strong covalent bonds between the tightlybound σ-orbitals. The electron in a 2pz orbital forms a weaker π bond with the 2pz orbital of the neighboring C atoms. The resulting π-band structure defi nes the Fermi surface, and hence these orbitals are responsible for the electronic transport properties. 8.3.2.2.1.2 Basics of One-Dimensional Transport A detailed discussion of the conductance of CNTs as one-dimensional conductors can be found in various reviews and textbooks (e.g., Datta, 1995; Landauer, 1989). Mesoscopic systems such as SWNTs are defined by two characteristic lengths. The fi rst is the mean free path Lm, which is the average length that an electron travels before it is scattered by a scattering center. Impurities, lattice mismatch, and any potential variation can act as scattering centers. The second characteristic length is the Fermi wavelength λF, defi ned by λF = 2π/k F, which is the de Broglie wavelength for electrons at the Fermi energy. 8.3.2.2.1.3 Resistance of a Ballistic Conductor Figure 8.16 illustrates a narrow two-dimensional conductor with width W and length L (W 74* >57*

0–66 0–104

FIGURE 11.7 Plot of the equilibrium force acting on a carbon nanotube during partial immersion in three different liquids measured using an AFM as a force balance. The right-hand plot shows derived contact angles. The asterisked values for water should be treated with caution since the contact angle may be equal to or greater than 1 (implying complete wetting) within the measurement error. The authors’ derivation of internal contact angle is discussed in the text. (From Barber, A.H. et al., Phys. Rev. B, 71, 115443, 2005. With permission.)

11-12


equation to describe the wetting of both the outside and the inside of a nanotube:

18 16 14

(11.17)

where Fr = Fout + Fin (i.e., the restoring force is the sum of the wetting forces on both the inside and the outside of the nanotube). The ratio of dout to din was taken to be 2.5:1 from the TEM. There were a number of difficulties noted by the authors that make the interpretation of the imbibition experiments problematic. No internal wetting was observed for the two organic liquids. Th is may be due to tubes that are not properly open (the TEM images included in the paper are not defi nitive) or it may be due to defects in the nanotubes, which were synthesized using CVD. Notwithstanding these problems, the authors interpret the greatly increased pull-in forces observed for the open nanotubes in water as being due to the influence of imbibition. As discussed previously in Section 11.2.2 on materials, the Drexel group has published a study of the wetting of amorphous carbon fi lms by a range of polar and nonpolar liquids (Mattia et al., 2006a). These films were produced using CVD on alumina substrates and are assumed to have surface properties similar to templated carbon nanopipes. Th is assumption was supported by Raman and the infrared spectroscopy of the two sets of materials. CVD amorphous carbon surfaces were generally found to be more hydrophilic than carbon nanotubes. Contact angles, particularly for polar liquids, decreased substantially following exposure to alkali solutions, which are commonly used for etching during the manufacture of nanopipes. Conversely, annealing amorphous carbon at high temperatures in an inert atmosphere promotes graphitization and leads to a sharp increase in contact angles and reduced wettability of nanopipes by water (Mattia et al., 2006b). As also discussed earlier, Raman spectroscopy offers a unique opportunity to probe the intermolecular forces between nanotubes and fluids, especially as peak shift s are of a level that is easily detectable by modern Raman instruments. Figure 11.8 shows the predicted solvent shift of the radial breathing mode for a [22,0] nanotube in water obtained by molecular dynamics simulation and theory (based on treating the nanotube and its solvent shell as elastic membranes) for a range of possible water-carbon nanotube Lennard-Jones interaction strengths ε. The figure shows the separate contributions from the internal and external wetting of the tube and a combination of both. The solid line is the experimental upshift for diameter corresponding to a [22,0] SWNT taken from Izard et al. (2005). The intersection between this line and the triangles indicates a [22,0] CNT–water interaction of around 0.4 kJ/mol. Th is is weaker than predicted from external wetting only. Detailed predictions of water carbon interactions will require more accurate Raman data for pure solvent; however, the approach seems promising.

12 Δ f (cm–1)

Fr = γ π(dout cos θout + din cos θin )

10 8 6 4 2 0 0.0

0.2

0.4

0.6

0.8

εwater-C (kJ mol–1)

FIGURE 11.8 Upshift in the RBM; theory (lines) against simulation (symbols) for [22,0] nanotubes as a function of the nanotube-water interaction strength εwater-C. Solid symbols correspond to atomistic simulations, open symbols correspond to mean field shell approximation, for water present on the inner surface (squares), outer surface (diamonds) or both surfaces (triangles). Errors are of the order of ±1 wave number. The hatched region corresponds to intermediate values of the nanotubewater interaction strength, which result in contact angles of water on graphite between 42° and 86°. (From Longhurst, M.J. and Quirke, N., J. Chem. Phys., 125, 184705, 2006b.)

Finally, with regard to dynamic wetting, a comprehensive review of the experimental results in this field, including consideration of nanoscale phenomena, has recently appeared (Ralston et al., 2008)

11.4 Experimental Investigation of Nanoscale Fluid Flow We turn now to consider a series of recent experiments that directly investigate the flow of fluids through the central pores of carbon nanotubes and nanopipes. Two of these experiments report dramatically enhanced flow rates in small ( 100 μm), where viscosity of the bulk fluid dominates, decane flows faster than water (Cheng and Giordano, 2002). Majumder et al. attributed the relationship reversal they observed as being due to the interaction between the water molecules and the hydrophobic carbon walls, with formation of a hydrogen bond network leading to very low friction. Further theoretical support for the “super-flow” papers appeared in 2008 with the publication of a molecular dynamics simulation of water flowing through carbon SWNTs with diameters in the range of 2.1–2.5 nm, i.e., similar to the DWNTs used in the Holt et al. study (Joseph and Aluru, 2008b). These authors predicted a flow enhancement of 2052 times compared to the non-slip Poiseuille flow, which they attributed to the occurrence of a pronounced depletion layer in the water adjacent to the carbon walls. According to their model, the density of water molecules is just 5% of the bulk value in this region. They postulate that this low-density layer acts as a kind of lubricant to facilitate super flow in smooth walled carbon nanopipes. Far less enhancement (0.85 Å or half the diameter of a water molecule. The authors suggest that the gating effect is due to the perturbation of the water–wall interaction in this highly confined regime and may have similarities to the way that pores in biological membranes switch on and off. In the same year, the Kentucky group again published the results of experiments in which the flow of ionic species is controlled by voltage applied to a functionalized carbon nanotube membrane (Majumder et al., 2007). They used long diazonium-based actuator molecules tethered both near the ends of the MWNT and along the inner pore walls. The former positioning was achieved by undertaking key steps of the reaction while rapidly flowing an inert solvent through the core during electrochemical functionalization. By applying a voltage in the range of ±200 mV, Majumder et al. found a potential at which the selective transport of large and small probe molecules was achieved. They postulate, in the case of the actuator molecules tethered near the nanopipe tip, that an applied positive charge has the effect of drawing the negative tails of the actuators into the pore entrance, thus partially blocking it. In the case of actuator molecules tethered inside the pore, negative charging of the nanopipes causes the tails to be repelled from the walls into the center of the channel, again partially blocking it. Selectivity between the two isovalent but differently sized probe molecules was as high as 23 times when the nanopipe channels were switched closed. A different electrical method of controlling transport is to modify the wetting behavior of a fluid be applying a potential to the contact surface, a phenomenon known as electrowetting. Using this technique, mercury can be persuaded to enter the interior of carbon MWNTs by capillary uptake, which normally it will not (Chen et al., 2005). A subsequent MD simulation of nanoscale wetting shows that the contact angle of water nanodroplets on a


graphite surface is remarkably sensitive to an applied potential (Daub et al., 2007). The authors explain the effect as being due to the modulation of interfacial hydrogen bonding in the nanodrop, which in turn affects the interfacial tensions.

11.6 Pumping Flow switching and control is just one of several basic facilities required for the realization of practical nanofluidic devices. No less fundamental are the methods for inducing flow through nanochannels by pumping. A variety of pumping methods involving carbon nanotubes have been proposed in the literature and explored using MD simulations. These include: using two-beam coherent lasers to induce an electric field gradient along the axis of the nanotube such that the resulting electron transport in the carbon walls induces motion in intercalated ions (Kral and Tomanek, 1999); using ultrasound or pulsed laser heating to induce surface acoustic waves that travel along the carbon nanotube, propelling gas molecules inside the central pore by a form of nanoperistalsis (Insepov et al., 2006); placing a sequence of precisely positioned external charges near the pore entrances of nanotubes to induce water molecules to enter or exit by dipole coupling (Gong et al., 2007; Hinds, 2007); locally heating fluid imbibed by a SWNT from a cooler reservoir to generate high internal pressures and jetting (Longhurst and Quirke, 2007); use of thermal gradients (Shiomi and Maruyama, 2009); and the alignment of water molecules inside a carbon nanopipe by an external electric field to promote unidirectional transport due to asymmetrical coupling between rotational and translational motions (Joseph and Aluru, 2008a). A recent theoretical paper has examined whether simple mechanical propellers, as widely deployed in macroscale pumps, can be used at the nanoscale (Wang and Kral, 2007). The authors conclude that performance is highly sensitive to the surface interactions between the driven fluid and the material of the propeller surface. Micron scale motors are of course known in the form of flagella used by spermatozoa and many unicellular organisms to achieve mobility. As suggested in the introduction to this chapter, the sophisticated biological mechanisms found in specialized transport proteins embedded in cell membranes can also be considered as nanofluidic pumps. A recently published review of transport mechanisms involving the confinement of water in nanopores provides extended background (Rasaiah et al., 2008).

11.7 Interfacing, Interconnections, and Nanofluidic Device Fabrication Also necessary for the fabrication of practical nanofluidic devices incorporating nanotubes and nanopipes are general architectural facilities for forming and accurately positioning channels, for interconnecting fluid conduits, for forming branches, and for interfacing to macroscale inputs and outputs. Microfabrication techniques, originally developed for use in the semiconductor

11-19

Fluid Flow in Carbon Nanotubes

industry, offer a broad range of relevant capabilities (Harnett et al., 2001; Madou, 2002; Lorenz et al., 2004; Flachsbart et al., 2006; Han et al., 2006). These use “top-down” methods such as lithography and etching to form patterns and structures such as those typically found in integrated circuits. Features as small as a few tens of nanometers can be defined in this way using the latest methods. For fabrication below this size limit, self-assembly or “bottom-up” techniques are appropriate. These take advantage of natural chemical and physical processes to rationally arrange and organize nanoscale objects, for example, through the formation of self-assembling molecular monolayers and the natural segregation of block co-polymers (Liang et al., 2004). The self-ordering of pores in AAO templates, discussed above in the contexts of nanopipe synthesis, is another instance. A combination of top-down and bottom-up methodologies is a promising strategy for fabricating practical devices with nanoscale components (Mendes and Preece, 2004; Mijatovic et al., 2005; Riegelman et al., 2006). A related suite of resources, already tailored to handling fluids, has been developed for applications in the rapidly advancing field of microfluidics, where great progress has been made in recent years (Whitesides and Stroock, 2001; deMello, 2006; Whitesides, 2006; Abgrall and Gue, 2007). Th is technology platform provides a solid foundation for nanofluidic device design. Using microfluidic technology, the “lab on a chip” concept has been realized with the launch of commercial miniaturized devices for sensing, diagnosis, and synthesis. Designers take advantage of the unique characteristics of microfluidic systems including laminar flow, very small sample volumes, rapid and highly efficient mixing schemes, and the possibility of ultra-sensitive (ultimately single molecule) detection (deMello and deMello, 2004; Dekker, 2007; Horsman et al., 2007; Kuswandi et al., 2007; Hansen and Miro, 2008; Mansur et al., 2008; Winkle et al., 2008). There are a number of reports of experimental work addressing the challenge of microfluidic to nanofluidic interfacing. Liquid samples for characterization or processing using nanoscale components are likely to start out as macroscale droplets. These must be guided into progressively smaller channels for ultimate delivery to the nanoscale device components. A number of potential problems arise. One is channel blocking due to

large macromolecules or insoluble debris (Stone et al., 2004). This can already be a limiting hazard with microscale lab-on-a-chip systems. Work is currently taking place in many laboratories to understand the dynamics of particulate transport at the nanoscale and to find low friction coatings or channel materials that help to reduce blocking (Sharp and Adrian, 2005). Another problem is more fundamental: the large size of polymers, including biologically relevant polymeric molecules such as DNA, which are often tangled and tightly folded in vivo. A typical DNA molecule from a virus has a length of 100–200 kb and will form a random coil with a radius of some 700 nm in aqueous solution at 20°C (Cao et al., 2002). Th is is several times greater than the pore diameter of even large carbon pipes and two orders of magnitude greater than the diameter of a SWNT, such as might be functionalized to detect the presence of specific base sequences (Heller et al., 2006). DNA molecules will fit into the central channel of even small nanotubes, but only when unraveled and fed into the pore opening lengthwise. The entropic barrier to achieve this from the disordered state is very high and therefore such long molecules are normally excluded. To overcome this problem, investigators at Princeton University used optical lithography to fabricate an array of microchannels forming a gradient fluidic device to interface the microscale to the nanoscale (Cao et al., 2002). They used a novel modified form of diffraction gradient lithography involving a photosensitive blocking mask resist on a silicon wafer substrate. The technique is inherently parallel and both faster and more efficient than using e-beam lithography. The end result is a massive array of microposts, with a continuous reduction in the gaps that form fluidic channels as the chip is traversed from one side to the other. To test the device, long DNA strands stained with a fluorescent dye were introduced on the microscale side. Diffusion of the molecules was then observed under a UV light and captured on video. Still frames from the recording showed individual DNA molecules straightening out and moving through the interface in an extended configuration towards the nanoscale region (see Figure 11.10). The authors reported the transport of the stretched DNA molecules with significantly greater efficiency compared

10 μm

10 μm (a)

(b)

FIGURE 11.10 In the above pair of optical micrographs (reproduced with permission of the authors) the left image shows the chip after development of the photoresist. A continuous reduction in the gaps between the posts can be seen moving in a horizontal direction from right to left . The right-hand image shows integrated video recordings of fluorescent-labeled DNA molecules entering the gradient zone and becoming elongated as they enter the nanochannels on the left side of the picture. (From Cao, H. et al., Appl. Phys. Lett., 81, 3058, 2002.)

11-20

to random diff usion through comparable nanoscale pores in the absence of any gradient interface (Cao et al., 2002). Wang et al. have created a nanoscale preconcentration device using standard photolithography and etching methods, which has achieved concentration factors in the range of 106 –108 (Wang et al., 2005). They exploit the electrokinetic trapping effect found in nanofluidic fi lters. A team from Oak Ridge National Laboratory in Tennessee reports a general method for creating patterned arrays of silica nanopipes precisely positioned over pores in a silicon nitride membrane on a silicon substrate (Melechko et al., 2003). The method requires expensive equipment and top-down fabrication techniques, but allows a high-degree of control over the device architecture. Critically, precise control of the nanopipe location is achieved by deterministic positioning of catalyst particles for CVD growth. Bau et al. describe another method for constructing a nanofluidic device comprising a single carbon nanopipe (diameter 250 nm) connecting two fluid reservoirs (Bau et al., 2005). Dielectrophoresis was used to orient and manipulate the nanopipe into position prior to depositing a dividing wall on top. Other workers have fabricated nanofluidic devices with microfluidic interconnects using interferometric lithography (O’Brien et al., 2003) and the creation of multiple polymer layers involving contact printing with a thermally cured adhesive (Flachsbart et al., 2006). Finally, there have been efforts to realize Y-junctions and other interconnect topologies involving carbon nanotubes (Ho et al., 2001; Wang et al., 2006). The motivation for such research has primarily been driven by interest in the electronic properties of junction structures, where transistor-like behavior can be expected (Kim et al., 2006; Choi et al., 2007). The integrity of the inner fluid channel is not critical for such applications and so the techniques developed are not immediately applicable to nanofluidic devices. Some preliminary investigation of mechanical stability, fluid flow dynamics, and possible applications of Y-junctions, such as for ion separation, have been made using MD simulation (Hanasaki et al., 2004; Meng et al., 2006; Park et al., 2006). The varied approaches described above currently enable prototyping of functional nanofluidic devices for research and testing. Once effective designs have been demonstrated to the proof-of-concept stage, less costly and highly repeatable fabrication methods will need to be developed to make large scale, affordable manufacturing possible. This will require considerable effort and investment, which can only be justified if there are clear applications for nanofluidic devices incorporating carbon nanotubes.

11.8 Applications for Fluid Flow through Nanopipes The size scale of nanopores discussed in this chapter is essentially the same as the dimension of many important biological entities (antibodies, enzymes, viruses, DNA molecules). Thus, carbon nanotubes and nanopipes are potential conduits, concentrators, detectors, containers, and probes for biomedical


applications. Doubtlessly, many challenges remain before such devices become practical including: ensuring the mechanical strength and biochemical compatibility of nanocomponents in proximity with living cells and tissue, developing methods for assembling huge numbers of nanoscale components, using precautions to avoid fouling of the channels and surfaces, controlling defects in the components, and managing information flow from nanoscale sensors to the outside world.

11.8.1 Filtering and Purification An obvious nanofluidics application involving carbon nanotubes is materials separation, purification, and processing; taking advantage of the small pore size to selectively transport target molecules while excluding larger contaminants. Desalination of seawater, which is likely to require nanotubes with very small pore diameters, is an exciting example with enormous global demand. Physical fi ltering can be supplemented with electrochemical and surface modification strategies (Schoch et al., 2008). Selectivity has been demonstrated for gas transport through membranes incorporating nanotubes (Hinds, 2006; Pietrass, 2006; Sholl and Johnson, 2006). Promising results have been demonstrated using EOF, specifically taking advantage of the electric double layer on electokinetic transport (Yuan et al., 2007). A number of studies were mentioned in Section 11.2.2 involving fi ltration through tangled mats of SWNTs and MWNTs with fluids passing between rather than through the nanopipes (Srivastava et al., 2004; Lu et al., 2005; Yan et al., 2006b; Brady-Estevez et al., 2008). An interesting study, which fully exploits the interior pores of carbon nanotubes, presents evidence to show that viruses can be destroyed by disassembly induced by confinement inside the pore (Fan et al., 2008). A review of the environmental applications of carbon nanomaterials, including the use of carbon nanotubes for fi ltering and purification, is available in the article by Mauter and Elimelech (2008).

11.8.2 DNA Sensing Rapid progress has been made during the past several decades in sequencing DNA and in analyzing the resulting genetic information. The field of bioinformatics accelerates our understanding of living systems. This knowledge is enabling improvements in medical care, particularly by customizing therapies to the specific needs of the individual. To fully realize this promise, it will be necessary to achieve a dramatic reduction in the time and cost associated with DNA analysis. Ultimately, the goal is make sequencing of a person’s genome in a few hours using equipment that costs just tens, rather than tens of millions, of dollars possible (Hood, 2004). Researchers in this swift ly developing field are looking to nanofluidics to provide key components of the necessary technology (Tegenfeldt et al., 2004). An MD simulation study in 2003 indicated that short DNA molecules spontaneously enter carbon nanotubes in an aqueous environment (Gao et al., 2003). Similar studies have been

11-21

Fluid Flow in Carbon Nanotubes

carried out for RNA using an aligned SWNT membrane with a pore diameter of 1.5 nm (Yeh and Hummer, 2004). In order to fit, the nucleic acid chains must stretch out into a linear configuration that may be a useful feature for subsequent sequencing. An electric field is used to drive the charged macromolecules through the carbon nanochannels. Soon after these simulation papers appeared, the first experimental results were published confirming that DNA can enter 50 nm gold nanotubes (Kohli et al., 2004) and that 5–8 μm long DNA sequences can be transported through 50 nm wide silica nanochannels (Fan et al., 2005a). Monitoring of the ionic current was used to detect the passage of the DNA molecules. More recent studies have investigated ways of slowing the transit times with the ultimate aim of identifying the individual nucleotide bases as they pass through the nanopore (Kim et al., 2007). An optical readout from parallel arrays of nanopores has also been demonstrated (Mulero and Kim, 2008).

11.8.3 Drug Delivery The application of nanoscale principles and engineering to the challenge of drug delivery is a rapidly advancing field with a fast-growing literature and high levels of commercial and government investment (Martin and Kohli, 2003; Sinha et al., 2004; Wagner et al., 2006; Emerich et al., 2007; Sahoo et al., 2007; Hervella et al., 2008; Singh et al., 2008; Venugopal et al., 2008). A growing capability to control both physical and chemical characteristics at very small scales ( 6 nm, which is in agreement with the experimental findings (Seifert et al. 2002) (Figure 12.6). In accordance with experimental data, multilayered nanosystems are more stable than the monolayered ones, owing to the attractive interlayer van der Waals’ interactions. Moreover, inorganic SWNTs have not been synthesized yet. A similar approach was used to compare the relative stability of SWNTs and the corresponding nanorolls of anatase TiO2 (Enyashin and Seifert 2005). The study has shown that the strain energy for a nanoroll with a cross-section of an Archimedian spiral depends on the distance L between the coils and the starting angle φ1:

400

800

1,200 N, atom

1,600

2,000

0 (b)

40,000

80,000

120,000 160,000 200,000

N, atom

FIGURE 12.6 Total energies of multilayered nanostructures of MoS2: (a) 1D nanostripes and nanotubes (From Seifert, G. et al., J. Phys. Chem. B, 106, 2497, 2002. With permission.) (b) zero-dimensional nanoplatelets, octahedral fullerenes and spherical fullerene-like nanoparticles depending on the number of atoms within a unit cell and number of walls k. Dashed line, energy of a MoS2 monolayer. (From Bar-Sadan, M. et al., J. Phys. Chem. B, 110, 25399, 2006a. With permission.)

12-6


can be explained by the difference in surface tensions of the two sides of the layer (Zhang et al. 2005). For instance, titanate nanotubes are delivered from the proton-containing layers of H2Ti3O7, which are able to exchange the protons by other cations like Na+. The difference in the concentration of Na+ ions deposited on both sides leads to the difference in surface tension, which can exceed bending forces and lead to the self-rolling of the layers. Evidently, a similar mechanism is involved in the formation of VOx nanorolls using the adsorption of alkylamines or other organic molecules initially on one side of the monolayer (Patzke et al. 2002). The formation of such INTs attracted much interest, since they can be formed in a natural environment, as has been shown for nanotubes and nanorolls of alumina- and magnesiasilicates, which bundle into minerals species like imogolite (Guimaraes et al. 2007) and chrysotile (Piperno et al. 2007), respectively. These nanotubes represent curved aluminum Al(OH)3 or magnesium Mg(OH)2 hydroxide sheets, where at one side the hydroxyl groups are completely substituted by silicate anions. In these cases, the incommensurabilty in the lattice vectors of hydroxide and silica sublattices results in negative values of energy for the rolling of the planar sheet into a cylinder (Figure 12.5e) i.e., in contrast to most of the nanotubes, these tubes are not metastable. The description of the stability for such nanotubes implies a second contribution in the strain energy (Equation 12.5) (Guimaraes et al. 2007), which is responsible for the difference in the surface tensions, Δσ < 0, between the outer and inner tube surfaces (internal and external surface energies): Estr Yh3 Δσh = + N R 24ρa R2

relieving the large inherent strains in quasi-spherical fullerenelike structures. It has also been established that the ratio of the wall thickness to the radius of a multi-walled particle is determined by the ratio of the surface energy to the energies of curvature and dislocation (or grain boundary). The number and the distribution of the defects within a fullerene-like particle are not known. However, one may assume that the number of defects is quite small in comparison with the total number of atoms in a nanoparticle, as each defect introduces a significant curvature. Assuming the absence of defects within the shell of a spherical nanoparticle, the strain energies per atom for this morphological type can be derived using the theory of elasticity as (Enyashin et al. 2007b) Estr Yh3 (1 + σ) = , N 12ρa R2

(12.10)

where σ is the Poisson ratio. The difference between Equations 12.5 and 12.10 by 2(1 + σ) reflects the fact of a higher curvature of a sphere compared with a cylinder and, accordingly, a lower stability of fullerenic forms at a given radius R (Figure 12.6). The nanoparticles with regular shapes have a countable number of defects. Thus, the exact expression for their strain energies can be derived. It may be assumed that the energy of a polyhedral fullerene can be subdivided into contributions from the atoms of the facets, the edges, and the corners with the corresponding energies ε∞, εe, and εc, respectively (Bar-Sadan et al. 2006a). For example, the strain energy of an octahedron, based on a layered MX 2 compound, with k-layers can be written as

(12.9)

The interaction between the walls of coaxial nanotubes may also play an important role in the formation of these nanostructures, especially if they are composed of layers of different chemical compounds. Mendelev et al. proposed a thermodynamic model for two-phase multilayered fi lms and nanotubes (Mendelev et al. 2002). They showed that the dominance of one or another nanoform is determined by the balance between the surface and interfacial energies, the energies of interactions between the interfaces and interfaces and surfaces, and the bending energies. The stability of IFs can be also evaluated using the classic theory of elasticity or an atomistic approach. Such an approach may be refined on the basis of quantum-mechanical calculations for representative smaller entities of the whole structure. As in the corresponding carbon-based nanostructures, a spherical shape can not be built from strained, but otherwise unperturbed fragments of a hexagonal crystalline sheet, but structural defects are required to provide the curvature of the fullerene-like shape. As shown in the framework of a continuum model (Srolovitz et al. 1995), line defects such as dislocations and more extended defects such as grain boundaries and stacking faults are intrinsic features of spherical fullerene particles, once the thickness exceeds a critical value. Such defects are essential for

Estr 6ε e = N

∑

k i =1

( Ni − 6) − 6ε ∞

∑

k i =1

∑

Ni

k i =1

Ni + 36kε c

+

k −1 εvdW k (12.11)

The energies as a function of size (number of atoms N) for octahedral MoS2-based fullerenes obtained using Equation 12.11 are drawn in Figure 12.6 in comparison with the corresponding energies for spherical fullerene-like particles and nanoplatelets. All single-walled fullerenes are less stable than a flat monolayer and even the nanoplatelets with their unsaturated edges. Obviously, the stability of fullerene-like particles and octahedral particles increases with the number of van der Waals bound shells. However, the most striking result is the occurrence of “crossover points” between the energy curves of the nano-octahedra and the nanospheres at the values of N of the order of a few 105 atoms. Thus, several phase transitions between the different morphologies can be predicted: first, for particles with less than 1.5 × 104 atoms, the flat platelet structure is the most stable one. Second, in the interval between 1.5 × 104 and 1.5 × 105 atoms, the most stable structures are the octahedral particles

12-7

Inorganic Fullerenes and Nanotubes

(nano-octahedra) with a small number of dangling bonds at the corners and flat facets. Third, above 105 atoms, the most stable “zero-dimensional” modifications are quasi-spherical, fullerenelike particles. It turns out, therefore, that the nano-octahedra are the smallest hollow clusters of MoS2, i.e., they can be considered as the genuine IFs of this compound. Obviously, this observation is not limited to MoS2 and can probably be extended to various other layered compounds. Equations 12.9 and 12.10 can also explain the preference for the formation of uncapped inorganic tubes with open ends, because the strain energies for spherical or octahedral fullerene-like caps are too large for the typical radii of the nanotubes. This conclusion is supported by the quantum-mechanical density-functional tight-binding (DFTB) calculations of the MoS2 octahedral fullerenes (Bar-Sadan et al. 2006a; Enyashin et al. 2007a,b). Geometry optimizations of hollow (MoS2)x particles up to x = 576 reveal that the initially assigned octahedral shape of the small stoichiometric fullerenes (with x < 100) was unstable. The hollow structures of the larger nano-octahedra were found to be stable. While the facets remain unchanged, changes occur at the corners: two sulfur atoms split off from each corner and the Mo:S composition degrades to MoxS2x−12. Additional molecular dynamics simulations showed a considerable distortion of the initial structure around the corners already at ambient conditions, but the fullerenic facets and edges preserve their integrity (Figure 12.4e). This can be attributed to the high strain energy in the corners of the nano-octahedra. The numerous observations of nanotubes based on “nonlayered” compounds like MgO (Li et al. 2003b), ZnO (Xing et al. 2003), and Mn5Si3 (Yang et al. 2004), which will be described below, evoke the question of their stability. These nanotubes have preferentially prismatic faceted morphology and, actually, can be represented as hollow monocrystals fashioned from the corresponding bulk compounds. Atomistic simulations using a force-field method on MgO nanotubes (Enyashin et al. 2006b; Enyashin and Ivanovskii 2007) and quantum-chemical calculations on TiSi 2 and TiC nanotubes (Enyashin and Ivanovskii 2005a,b; Enyashin and Gemming 2007) show that the energy of such nanotubes weakly depends on their perimeters. Though, by increasing the wall thickness, the energy of a nanotube based on a nonlayered compound will approach the energy of the bulk material.

12.4 Synthesis of Inorganic Nanotubes and Fullerenes There is a broad range of synthetic strategies that have been developed for the preparation of INTs and fullerene-like nanoparticles. While the structure and shape control of inorganic nanoparticles is not as effective as in the case of carbon fullerenes and nanotubes, some of the successful methods demonstrated partial controllability. Furthermore, heuristic arguments derived from a combination of theory and experiment allowed one to draw some far-reaching conclusions as to the propensity of a

particular technique to produce certain nanostructures. Despite some exceptional cases, unlike their carbon counterparts, INTs and fullerenes appear preferentially in multiwall structures. Quantum-mechanical calculations provide a clue to this phenomenon, but far more work is needed on both the experimental and theoretical fronts to clarify the full complexity of these inorganic nanostructures. Remarkably though, chemistry provided some neat ways to produce various fullerene-like nanoparticles and nanotubes, other than carbon, in large amounts (a few kg/ day and more) and at affordable costs ( direction. The mechanical


properties in this direction are determined by the tight Mo−S chemical bond. On the other hand, the mechanical parameters of bulk MoS2 crystals along the c-axis < 001 > exhibit appreciably inferior values. Here, the properties are determined by the weak van der Waals interactions between the layers. The shear modulus of a beam is expressed as: G = Y/2(1 + ν), with ν = the Poisson ratio. This expression shows that the three important mechanical parameters are not independent. In particular, if one assumes the value of 0.3 for the Poisson ratio of a WS2 nanotube, G is found to be 57 GPa. DFTB calculations of the intralayer shear modulus yielded the values of 53 and 81.7 GPa for zigzag and armchair single-wall MoS2 nanotubes, respectively. This value is significantly different from C44 values that were previously determined by the neutron and x-ray scattering of the linear compressibilities for bulk 2H–MoS2 (15 GPa; Feldman 1976). This data indicated that the interlayer shear of multi-wall MoS2, which is affected by the weak van der Waals interactions, is appreciably smaller than the one obtained within a layer. To prove this issue quantitatively, a bending test of an individual WS2 nanotube was carried out (Kaplan-Ashiri and Tenne 2007), which allowed to determine a shear of 2 GPa reflecting the slippage between the adjacent WS2 layers of the nanotubes and designated as a sliding modulus. The DFTB calculation of the interlayer shear of two adjacent layers in 2H–MoS2 resulted in a modulus of 4.09 GPa, which is in reasonable agreement with the experimental data for the multi-wall WS2 nanotubes (the van der Waals interaction between adjacent layers in the two materials is not likely to be very different). Due to the strong C–C bond in sp2 (graphitic) hybridization, the Young’s and the shear moduli of carbon nanotubes are appreciably higher than those of most INTs. Furthermore, being made of carbon, these nanotubes are very light. Nonetheless, carbon nanotubes, and especially the multi-wall ones, which are very important, for example, in ultra-high strength nanocomposites, suffer from a number of disadvantages making the presently available inorganic (WS2 or MoS2) nanotubes suitable for a variety of mechanical applications. In particular, the C–C bond is unstable under compression and transforms easily into the sp3 (diamond) bond. Contrarily, WS2 or MoS2 do not have a highpressure phase, and under high pressure, they eventually break down making their inorganic fullerene-like and tubular nanoparticles much more robust under compression. Furthermore, the narrow scattering in the strength and elongation data of the WS2 nanotubes (Kaplan-Ashiri et al. 2004, 2006; Kaplan-Ashiri and Tenne 2007) indicates that they are almost free of critical defects, permitting a predictable assessment of their mechanical behavior in a variety of media. The mechanical properties of WS2 nanotubes under loading were also studied by in situ TEM and SEM experiments (Sheng Wang et al. 2008). Centers of deformation in the form of kinks occurred while the nanotubes were loaded until complete or partial fracture occurred. These kinks can be correlated with the nonlinear elastic deformations of the nanotubes, which were observed before. A hint for the nanotubes’ remarkable resistance against fracture was also demonstrated in the deformation

12-13

experiments. Here a rip occurred in the nanotube but it has not propagated. The incredible capability of the tubular morphology to highly deform without failure was demonstrated as well (Figure 12.9). The deformation mechanism of the WS2 nanotubes is somehow different from the one observed for carbon nanotubes because no ripples were observed. This difference can be related to the more complex atomic structure of the WS2 layers, which makes it stiffer than carbon nanotubes. The mechanical properties of INTs of other compounds also attracted attention and they were studied by means of both theoretical and experimental methods. The Young’s moduli were found to be ∼80 GPa for MoTe2 (Wu et al. 2007), ∼290 GPa for GaS (Köhler et al. 2004), ∼240 GPa for imogolite (HO)3Al2O3SiOH (Guimaraes et al. 2007), and 159±125 GPa for chrysotile Mg3Si2O5(OH)4 (Piperno et al. 2007) nanotubes. The stretching of the mentioned INTs leads mainly to the deformation of the valence angles. These values are a bit lower than for carbon nanotubes having Y ∼ 1–2 TPa. Here, Y is determined by the strong C–C covalent bonds along a tube axis, which suffers the strain. Evidently, INTs with similar mechanisms of deformation should also be more resistant to the strain. It is nicely illustrated, for example, by BN nanotubes, which have a graphite-like structure of walls and Y ∼ 0.5 –1 TPa (Hernandez et al. 1999). Quite recently, rhenium disulfide ReS2 nanotubes with Y ∼ 0.4 TPa were claimed to be the most rigid among the INTs with a nongraphitic structure of walls (Enyashin et al. 2009). This uniqueness may be explained by the presence of intralayer covalent bonding between the metal atoms within ReS2, which is nearly absent in other dichalcogenides. For a long time, d-transition metal dichalcogenides (MX 2, M = Mo, W, Nb, Ta, X = S, Se) were known as compounds with excellent antifrictional properties (Kalikhman and Umanskii 1972). Not only did they reduce the friction coefficient, they were also found to lubricate the reciprocating metal contacts at higher loads than traditional lubricants such as, for example, grease. These results were obtained in the last few years through a long series of experiments, and a realization was found in the preparation of fluid or dry solid lubricants and self-lubricating metal coatings with nanostructured fullerene-like WS2 (MoS2) additives (Spalvins 1971; Moser and Levy 1993). The general dependency of the friction coefficient on the load, however, is that of a standard grease-based lubricant. Early on it was hypothesized that the spherical fullerene-like MS2 nanoparticles would behave like nanoball bearings thereby providing superior solid lubrication as compared to the existing technology. Further work suggested that under mechanical stress the nanoparticles would slowly deform and exfoliate, transferring MS2 nanosheets onto the underlying surfaces (third-body effect), and continue to provide effective lubrication until they are completely gone, or oxidized (Figure 12.10). The beneficial effect of the powder of fullerene-like nanoparticles as an additive to lubricating fluids has been studied in quite some detail and this phenomenon has been summarized (Hu and Zabinski 2005; Joly-Pottuz et al. 2005; Rapoport et al. 2005). This effect is particularly important when the clearance (gap) between the two mating surfaces and

12-14


(a)

20 nm (b)

FIGURE 12.10 Tribological properties of inorganic nanoparticles are the subject of both experimental and theoretical research and find industrial application. (a) Snapshots of molecular dynamics DFTB simulation are shown for the structural evolution of a double walled MoS2 nanotube under squeezing between two Mo(001) planar gripes. (From Stefanov, M. et al., J. Phys. Chem. B, 112, 17764, 2008. With permission.) (b) Damage of fullerene-like WS2 particle under friction and wear. (From Rapoport, L. et al., Wear, 229, 975, 1999. With permission.)

the surface roughness are approximately of the same order of magnitude as the nanoparticles themselves, i.e., 30–300 nm. More recently, fullerene-like WS2 nanoparticles were impregnated into metal and polymer fi lms, endowing them with a self-lubricating character (Chen et al. 2002a; Katz et al. 2006; Friedman et al. 2007) and offering them a variety of applications. Clearly, the rolling and sliding friction of the nanoparticles is not possible in this case unless they are gradually released from the metal/polymer/ceramic matrix onto the surface. Here too, some of the beneficial effects of such nanoparticles can be attributed to their gradual exfoliation and the transfer of WS2 nanosheets onto the asperities of the mating metal surface (third body effect). Furthermore, the bare metal surface is shown to oxidize during the test, leading to a gradual increase in the friction coefficient to very high values (0.3–0.6). In contrast to this, the metal surface impregnated with fullerene-like nanoparticles does not seem to oxidize during the tribological test, although the coverage of the metal surface by the nanoparticles does not appear to exceed 20%–30%. This observation suggests that the temperature of the WS2-impregnated interface is lower than that of the pure metal surface during the tribological test. It, furthermore, suggests that the fullerene-like nanoparticles may act as a kind of “cathodic protection” against the oxidation of the metal surface, which prevents the oxidation of the metal surface. This technology offers numerous applications, among them various medical devices, like improved orthodontic practice (Katz et al. 2006). In order to capitalize on these applications, NanoMaterials, Ltd. recently constructed a manufacturing pilot plant with a production capacity of about 75 kg/batch and sales of their product under the title “NanoLub” have been launched. The friction process at high loads was studied also at the atomistic level using quantum-mechanical DFTB simulations (Stefanov et al. 2008). The effect of high loads was performed on cylindrical, defect-free MoS2 nanotubes of different diameters, chirality, and number of tube walls (Figure 12.10). Two external Mo grips apply a mechanical pressure and gradually squeeze the nanotubes, while chemical interactions between the

grips and the nanotubes were deliberately excluded. The strain– stress curves of single-, double-, and triple-walled nanotubes have shown several remarkable trends: (1) the strain–stress relation is steeper for smaller nanotubes, which is in agreement with the well-known inverse proportionality of the tube energy with respect to its diameter (Seifert et al. 2000a,b, 2002); and (2) the strain–stress relation is essentially independent of the tube’s chirality. In contrast to tensile strain–stress curves, the systems deviate from linearity towards higher stress values, that is, if the nanotubes are deformed it becomes harder to deform them any further. The reason for this behavior is that the tubes form planar surface segments close to the grips that are connected with half-tubes, so further compression is essentially equivalent with the compression of tubes with smaller diameters. Further compression leads to irreversible deformation of the nanotubes, and the fi nal product of this process is, however, MoS2 sheets attached to the two grips, at least by van der Waals interactions, in a face-to-face position, which is ideal for lubrication. It might be meaningful to interpret this result as local coating of the grips. This study has also shown that the strain–stress relationship of the MWNTs is determined by the smallest, innermost tube. In all simulations, the innermost tube determines the breaking process of the whole nanostructure. The tubes are being broken from the inside to the outside: when the innermost tube bursts and unbends under the load, much of the original stress is transferred to the next-largest tube, which breaks down and so forth. The result of the process is, however, the same as for the SWNTs: at the end, MoS2 platelets are formed partially attached to the grips, which will provide good lubrication at the position of the closest contact of the grips. The results of this study, which are also supported by the existing experimental data, suggest that the ball-bearing effect does not seem to play a major effect in the lubrication process provided by the fullerene-like nanoparticles. Th is is supported by the fact that the nanostructures break easily under mechanical pressure, but also because the friction coefficient of MoS2 platelets and nanostructures is identical for smaller loads. The excellent lubrication of nanostructures is hence

12-15


interpreted as “nano-coating,” by attaching lubricating platelets to those parts of the material that are exposed most closely to each other i.e., the asperities.

12.5.2 Electronic Properties

3

2

2

1 0 –1

1 0 –1

–2

(a)

DOS, arb.units

3

Energy, eV

Energy, eV

The electronic band structure of many INTs has been studied with many semi-empirical and density-functional-based methods. In the latter, the Kohn–Sham orbital energies were used for the representation of the band structures. Simple zone-folding schemes for the estimation of the nanotube band structure from the band structure of the corresponding layered structure are less meaningful for INTs than for carbon nanotubes. This is mainly due to the much more significant structural relaxation in the case of the rolled layer for inorganic tubes as compared to carbon nanotubes. Th is conjecture was nicely shown for the nanotubes of MoS2 and WS2 . In an analogy to carbon nanotubes, the qualitative picture of the band structure of armchair and zigzag MoS2 can be derived from the band structure of the molecular MoS2 layer with its hexagonal structure (Seifert et al. 2000a,b)—see Figure 12.11. But there is a strong reduction in the gap size with the decreasing radius of the tube. Th is reduction is caused by the compression of the inner sulfur “shell” in the S–Mo–S triple layer in the tubular structure as compared with the flat undistorted triple layer in MoS2. Th is calculated bandgap reduction is consistent with experimental observations of the optical-absorption spectra of MoS2 nanotubes, inorganic fullerene-like structures, and scanning tunneling microscopy (STM) studies of WS2 nanotubes (Scheffer et al. 2002). A similar size dependence on the electronic gap was also predicted for GaS and GaSe nanotubes (Côté et al. 1998; Köhler et al. 2004). Many INTs investigated up to now are semiconductors or insulators and they show a considerable dependence on the gap size of the tube diameter. This holds for hypothetical SiH- and GeH-based nanotubes (Seifert et al. 2001a,b), hypothetical phosphorus tubes (Seifert and Hernandez 2000), Bi tubes (Su et al. 2002), and many others. The diameter dependence of the gap size is in all cases nearly independent of the chirality of the nanotube. However, in many cases, the zigzag nanotubes have

a direct bandgap, whereas for the armchair nanotubes, the gap is an indirect one. Also, the electronic properties of nanotubes based on compounds with insulating character (wide band semiconductors) remain insulating and weakly depend on their chirality, which follows from DFTB calculations of TiO2 (Enyashin and Seifert 2005), AlOOH (Enyashin et al. 2006a), Al(OH)3 (Enyashin and Ivanovskii 2008), and imogolite nanotubes (Guimaraes et al. 2007). For several tubular structures, it has been shown that the semiconducting nanotubes can be transformed into metallic ones by intercalation or substitution. Th is has been demonstrated theoretically, e.g., for Si-based nanotubes in terms of silicide nanotubes CaSi 2 (Gemming and Seifert 2003), for BC-based nanotubes by intercalation with Li (Ponomarenko et al. 2003), and by partial substitution of Mo by Nb in MoS2 nanotubes (Ivanovskaya et al. 2006). Doping of semiconducting MoS2 nanotubes by Re results in the change of the character of conductivity to n-type (Deepak et al. 2008). NbS2 nanotubes (Seifert et al. 2000a) should be metallic with the Fermi energy in the Nb electronic d-band, which is related to a rather high density of states at the Fermi energy. Boronbased nanotubes (Kunstmann and Quandt 2005; Quandt and Boustani 2005) and metal-boride nanotubes (Quandt et al. 2001; Guerini and Piquini 2003; Ivanovskaya et al. 2003) should also be metallic. An analysis of the optical properties of various kinds of INTs was undertaken in recent years. Many of the measurements were taken from an ensemble of nanoparticles with a few studies dedicated only to individual nanotubes. Raman and IR spectroscopies were used to follow the structural transformation of TiO2 powder into titanate nanotubes (Qian et al. 2005a). The conversion of the anatase/rutile nanoparticles to sodium titanate during the NaOH reflux was confi rmed by the loss of the B2g and the second order of the B1g phonon mode at 398 cm−1 of anatase that individually peaked at 516 and 784 cm−1. Instead, a new peak appears at 906 cm−1, which is typical for the short Ti–O bond stretching in the layered sodium titanate and the shoulder at 3208 cm−1, which was assigned to the Ti–OH bonds. Th is study confirmed the existence of the Ti–OH bonds in the nanotubes.

Γ

X

–2

(b)

Γ

X

3 2 1 0

(c)

–15 –10 –5 0 Energy, eV

5

FIGURE 12.11 The band structures of the MoS2 nanotubes: (a), (22,0) zigzag; (b), (14,14) armchair configurations. (From Seifert, G. et al., Phys. Rev. Lett., 85(1), 149, 2000b. With permission.); and (c), the total (solid line) and Mo4d- (painted area) density of states for octahedral fullerene (MoS2)576. (From Enyashin, A.N. et al., Angew. Chem. Int. Ed., 46, 623, 2007. With permission.)

12-16

Optical absorption, photoluminescence (PL), and luminescence excitation of titanate nanotubes were undertaken (Bavykin et al. 2005). The bandgap of the nanotubes (3.87 eV) is close to that of the layered sodium titanate (3.84 eV), but is appreciably higher than that of the anatase phase of titania (3.2 eV). Studies have shown that changing the internal diameter of TiO2 nanotubes in the tube diameter range of 2.5–5 nm does not lead to any changes in the position of absorption and emission bands, indicating small quantum size effects in this size range. It was concluded that the electronic structure of TiO2 nanotubes is very close to that of TiO2 nanosheets (Bavykin et al. 2005). In yet another study, a strong and broad sub-bandgap PL with a peak at 570 nm was observed in samples consisting of titanate nanotubes. The PL was associated with the Ti–OH complex within the tubular structure (Qian et al. 2005b). The electronic structure and optical properties of VOx – alkylamine nanotubes were investigated by absorption, photoelectron, and electron energy-loss spectroscopies (Liu et al. 2005). The photoemission and core-level electron energy-loss spectroscopies confirmed the mixed-valence character of VOx – alkylamine nanotubes. Indeed, the vanadium ion was found to have an average valency of +4.4 in these nanotubes. In another study, the temperature dependence of the optical gap of VOx – alkylamine nanotubes was determined (Cao et al. 2004). The optical gap at 0.56 eV was found to be insensitive to the tube diameter. The Raman spectrum of VOx –alkylamine nanotubes was measured and the optical transitions were assigned to the various modes (Chen et al. 2004; Souza et al. 2004). In latter work (Souza et al. 2004), Cu ions were exchanged successfully with the alkylamine moiety (dodecylamine), which was followed by both Raman and IR spectroscopies. The optical properties of MoS2 nanotubes and fullerene-like nanoparticles have been reported in some detail before. More recently, the optical–limiting (OL) properties of MoS2 nanotubes in aqueous suspensions were investigated (Loh et al. 2006). The OL performance of MoS2 nanotubes at 1064 and 532 nm was found to surpass that of the carbon-nanotube sample. The resonance Raman spectrum of individual WS2 nanotubes was recorded (Rafailov et al. 2005). The Raman spectra of agglomerated WS2 nanotubes were measured under hydrostatic pressure as well. The 2D (in-plane) Grüneisen parameter value found for the WS2 nanotubes is 0.45, compared to the value of 2 for carbon nanotubes, which shows the softness of the WS2 nanotube material. Another interesting observation is the blocking of the rigid-layer (E2g) mode (33 cm−1) in the fullerene-like nanoparticles. This observation indicates that the shear of the two layers with respect to each other in the unit cell is damped in the closed fullerene-like structure. In a separate study (Luttrell et al. 2006), the IR reflectivity at room temperature and at 10 K of bulk (2H) and fullerene-like WS2 nanoparticles was studied. It was found that the oscillator strength of the E1u transition is appreciably stronger in the former (2H) material at both room temperature and 10 K. By analyzing the two modes at both the 2H (bulk) and fullerenelike WS2 nanoparticles, it was concluded that the interlayer


charge polarization, i.e., electron transfer from the metal to the sulfur atom, is appreciably smaller in the 2H as compared to the fullerene-like nanoparticles. On the other hand, the interlayer charge polarization between the layers is slightly larger in the fullerenic nanoparticles as compared to the 2H, indicative of the somewhat larger interlayer interaction as compared to the 2H–WS2 particles. The electronic and optical properties of octahedral IFs in spite of quasispherical fullerene-like particles have still not been investigated experimentally because of the low stability of such nanostructures. However, the first theoretical studies (Enyashin et al. 2004a,b) using the semi-empirical Extended Hückel Theory (EHT) method have shown that small hollow fullerenes of dichalcogenides MX 2 (M = Mo, Nb, Ti, Zr, Sn, X = S, Se) should be metalloid irrespective of the electronic character of the bulk material. The results of an ab initio study on the same sulfide particles and chlorides, like NiCl2, FeCl2, and CdCl2 also yielded a very low HOMO–LUMO gap and also predicted a noticeable spin-polarization of d-metal atoms in the case of MoS2, NiCl2, and FeCl 2 fullerenes (Enyashin and Ivanovskii 2004, 2005a; Enyashin et al. 2005). A bond population analysis demonstrated on both semi-empirical and ab initio levels of theory that the M–S bonds on the inner surface of the particle were weaker than on the outer surface, and indirectly suggested a propensity to form nonstoichiometric structures. The fully quantum-mechanical DFTB studies (Bar-Sadan et al. 2006a; Enyashin et al. 2007a,b) support the conclusions about the metalloid character of MoS2 nano-octahedra drawn from the semi-empirical modeling (Figure 12.11). The densities-of-states are quite similar in profi le to those of semiconducting nanotubular and bulk MoS2 (Seifert et al. 2000b). The general features of the electronic spectra are the same in all cases. However, in spite of this great similarity and in sharp contrast to the nanotubular and bulk MoS2, the HOMO–LUMO gap in all investigated MoS2 fullerenes does not exceed a few 0.01 eV, nearly irrespective of the nanocluster size (Figure 12.11). The valence band is composed of mixed Mo4d–S3p-states. As in the bulk, the states around the HOMO and LUMO levels are mainly Mo4d-states. A Mulliken charge distribution analysis shows a charge transfer from the Mo to the S atoms with average charges of −0.41 e and −0.47 e for the internal and external S atoms and +0.91 e for the Mo atoms, hence there is a tendency to form a surface dipole that points along the facet normals. Referring to the activity of nanoplatelets, such an enlarged electronic density on the external sulfur atoms of a fullerenic wall can be associated with a high reactivity. This finding opens up a perspective for the MoS2 nano-octahedra as a catalyst similar to the nanoplatelets that await an experimental verification. DFTB calculations (Enyashin and Seifert 2007) reveal that in spite of the above mentioned (MoS2) IFs with metal-like character, the fullerenes of wide-band gap semiconductors like TiO2 should be also semiconducting like their bulk or nanotubular forms (Enyashin and Seifert 2005). The calculated densities of states for TiO2 fullerenes are similar to those of the anatase bulk phase and various 1D titania nanostructures, which were

12-17


discussed above, do not differ for particles of different sizes. The valence band of the fullerenes is composed of O2p states, whereas the lower part of the conduction band is formed by Ti3d states. All titania fullerenes have large HOMO–LUMO gaps of about 4.3–4.7 eV.

12.6 Chemical Modification In addition to the structure-property relationship of pure inorganic compounds in nanostructured and bulk allotropes, which was described in detail in the previous sections, the chemical properties of nanosized allotropes can also be modified in comparison to their bulk materials. The most promising approaches are doping, defect formation, intercalation, surface functionalization, and incorporation into composites. The nanotubes, fullerene-like particles, and fullerenes (nano-octahedra) are especially well-suited candidates for these purposes, because being hollow they provide additional ways for functionalization. Four possibilities may be distinguished by the placement of an admixture into the hollow nanostructures: the substitution within the walls (intra-layer doping), the intercalation in the space between the walls (interlayer doping), an adsorption on the outer surfaces of the walls (exohedral functionalization), or an injection into the inner cavities (endohedral functionalization). The first attempts in these directions have already been accomplished and were described above by the substitutional doping in wall-doped Mo1−xWxS2, Mo1−xNbxS2 and Mo1−xRexS2 nanotubes (Deepak and Tenne 2008). Ti-doped MoS2 nanostructures were produced by pyrolysing H2S over an oxidized Ti–Mo alloy powder (Hsu et al. 2001). However, such chemical modification is possible only during the synthesis because direct substitutional doping is a very energy-expensive process due to the quite strong covalent bonding within a monolayer. Crystalline Fe-doped trititanate nanotubes were synthesized via a wet-chemistry method (Han et al. 2007). These nanotubes exhibited noticeable catalytic activity in the water-gas-shift reaction. Magnetic measurements indicated that the Fe-doped trititanate nanotubes comprised a mixture of ferromagnetic and paramagnetic phases. The second alternative has more perspectives: the processes of intercalation and de-intercalation by insertion of alkali-metal atoms into MoS2 and WS2 fullerene-like particles (Zak et al. 2002; Kopnov et al. 2008). A small increase of the a-axis was observed and the XPS analysis of the rubidium intercalated material showed a rise in the Fermi energy as a result of the intercalation, endowing the originally p-type nanoparticles an n-type character. Mg2+ cations were electrochemically injected in various MoS2 nanostructures including nanotubes and fullerene-like particles (Li and Li 2004). It was shown that even larger particles, such as organic surfactant molecules or even fragments of graphite can be incorporated into MoS2 nanotubes (Mirabal et al. 2004; German et al. 2005). MoS2 nanotubes alloyed with gold or silver and with encapsulated nickel sulfide Ni17S18 have been synthesized (Remškar et al. 2000; Hofmann et al. 2002). Third, results of the covalent surface functionalization of fullerene-like MoS2 particles by organic ligands have been

reported (Tahir et al. 2006). Surface functionalization was achieved using an nitrilotriacetic ligand functionalized with a fluorescent 7-nitrobenzofurazan unit and using a polymeric ligand carrying a nitrilotriacetic group and a catechol type ligand, which has been used as the anchor groups for the functionalization of metal oxides. It provides the basis of a toolbox to construct supramolecular assemblies of organic–inorganic hybrid nanomaterials. Multi-walled BN nanotubes grown by a chemical vapor deposition (CVD) method were used for functionalization by naphthoyl chloride C10H7COCl, butyryl chloride CH3(CH2)2COCl, and stearoyl chloride CH3(CH2)16COCl to fabricate the respective covalent adducts of alkylcarbonyl groups and BN walls, which showed a dramatic change in the electronic structure of BN nanotubes (Zhi et al. 2005, 2006). These studies demonstrated that INTs and fullerene-like particles may be functionalized in the same manner as the analogous carbon nanostructures. Finally, such a functionalization (by organic molecules or by metal and semiconductor nanoparticles) may even facilitate their dispersion in composites. One recent example concerns the integration of MoS2 with carbon nanostructures: carbon nanotubes covered by the MoS2 sheets (Song et al. 2004) and coaxial MoS2@carbon nanotubes, which demonstrate improved MoS2 properties regarding lithium storage (Wang and Li 2007). Already these initial and still not numerous studies on modified nanostructures underline the necessity for more profound experimental and theoretical investigations. In this way, the great potential of these versatile nanomaterials due to their wealth of polymorphic structures may be explored and exploited in large-scale applications.

12.7 Conclusions Currently, many layered compounds have shown the ability to be prepared in nanostructured allotropes—nanotubes and fullerenes. The first known to be artificially curved into hollow nanostructures and the most explored type of such compounds are d-metal chalcogenides, especially, MoS2 and WS2. An extensive effort was especially devoted to the development of advanced synthetic routes and the characterization of the nanotubes and fullerene-like particles of tungsten and molybdenum disulfides. This is not surprising in view of their excellent tribological and mechanical properties, which was recently capitalized in a series of commercial products. In the meantime, analogous structures of BN, VOx, H2Ti3O7, and TiO2 were discovered suggesting numerous applications for these nanoparticles. Potential applications for such nanostructures in ultrastrong nanocomposites, as biosensors, catalysts for green chemistry, and renewable energy devices are being intensively explored. However, the materials science of the INTs and fullerenes is currently at an incipient stage of development. The investigation of the properties of INTs and fullerenes is still lacking in many points and needs an evaluation both from experimental and theoretical perspectives. No doubt, the results of further

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investigations of INTs and fullerenes promote not only the development of our ideas about matter but also will find their incarnation in advanced technological decisions.

Acknowledgments The authors would like to acknowledge Prof. Reshef Tenne from the Weizmann Institute of Science (Rehovot, Israel) for the helpful discussions and valuable comments made during the preparation of the manuscript.

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13 Spinel Oxide Nanotubes and Nanowires 13.1 Introduction ...........................................................................................................................13-1 General Fabrication Methods for Nanowires and Nanotubes • Basics of Spinels

13.2 MgAl2O4 ..................................................................................................................................13-4 13.3 ZnO-Based Spinels ................................................................................................................13-5 ZnAl 2O4 • ZnGa 2O4 • ZnFe2O4 • ZnCr2O4 • Zn2TiO4 • Zn2SnO4

Hong Jin Fan Nanyang Technological University

13.4 Twinning of Spinel Nanowires .......................................................................................... 13-11 13.5 CoAl2O4 .................................................................................................................................13-12 13.6 LiMn2O4 ................................................................................................................................13-12 13.7 Summary and Conclusions ................................................................................................13-13 Acknowledgments .......................................................................................................................... 13-14 References ........................................................................................................................................ 13-14

13.1 Introduction Spinel oxides have been one of the most important oxides in material science. In the form of ceramics, spinels have applications such as gas sensors, pigment materials, phosphors, catalysts, battery electrodes, infrared windows, and transparent electric conductors. However, ceramics are made up of highly compressed grain particles whose sizes range from nano- to micrometers. With nanotechnology, it is possible to make nanoparticles with more uniform sizes and fi ner grain structures. If traditional ceramic spinels are substituted with such nanostructured spinel nanoparticles, it is expected that some of the application performance can be enhanced. Driven by this, the preparation and characterization of nanoscale spinel oxides, in the form of nanoparticles, nanotubes, and nanowires (dimensions below 100 nm), have been recently receiving much attention (Song and Zhang 2004, Zeng et al. 2004a,b, Wang et al. 2005: 2928, Tirosh et al. 2006). Because of a large surface-to-volume ratio and symmetry breaking on the surface, nanoscale spinel oxides have shown physical properties different from their bulk counterparts. Th is chapter discusses only quasi one-dimensional (1D) nanostructures (e.g., tubes and wires). We will first provide an introduction to the fabrication of nanowires and nanotubes, followed by some fundamentals of spinel oxides. In the main body, a material-by-material summary will be given of the examples of spinel-type 1D nanostructures that have been demonstrated so far. As there are relatively few physical characterizations of these nanomaterials, our focus

is on the fabrication and associated structure properties. We will discuss the merits and drawbacks of different fabrication routes, analyze the common features of the structures (particularly the twinned nanowires), and also point out the potential spinel-forming interface reaction problem that is still not so well known to the metal-oxide nanowire community and hence may not have been addressed. Some well-known binary spinel-type oxides, such as Fe3O 4 and Co3O 4, will not be included here.

13.1.1 General Fabrication Methods for Nanowires and Nanotubes The fabrication of solid nanowires can be realized through various strategies, as elaborated in several review articles (Fan et al. 2006c, Kuchibhatla et al. 2007, Comini et al. 2009). For example, molecules/atoms physically fi ll up a hollow channel to form nanowires, or they self-organize in a 1D manner due to an anisotropic elongation, or they grow with the assistance of a metal catalyst, the way that carbon nanotubes grow. The most commonly applied techniques are the vapor–transport–deposition and hydrothermal growth. Spinel nanowires have been synthesized following these two methods. However, the difficulty for ternary spinel nanostructures is the control of morphology and composition. The thermal evaporation of the powder mixture of two constituent oxides (or metal plus one oxide) usually results in more than one type of nanostructure in terms of their morphology and phase. A relatively new fabrication method is the solid-state reaction of core–shell nanowire, where the core and 13-1

13-2


shell are the two constituent materials. Compared to thermal evaporation, this method allows for better control. Unlike nanowires, which can grow directly from the precursor molecules, the formation of nanotubes usually needs the usage of templates. Templates can be of a negative type (such as porous anodic alumina and polycarbonate, which possess 1D parallel channels) or of a positive type (such as solid nanowires). For negative templates of porous alumina, in most cases, the pores are a pure morphology-defining agent, which means that the template does not react with the depositing material; but in some cases, the template itself reacts with the depositing precursor. In other words, the templates can be both reactive and nonreactive. The same holds true for positive templates. In general, the template-based fabrication methods for spinel nanotubes can be classified into the following five categories (references can be found in Table 13.1):

3. Reaction at pore surfaces of a porous alumina with the other oxide material 4. Infi ltration of liquid precursors inside nanochannels followed by solidification 5. Transformation of a nanowire through a solid–state reaction directly to a nanotube through the Kirkendall effect

1. Thermal annealing of a core–shell nanowire to form a spinel nanoshell, followed by eliminating the core via etching or dissolution 2. Reaction of a nanotube of one oxide with the other oxide in a gas or liquid TABLE 13.1 Material MgAl2O4 ZnAl2O4

ZnGa2O4

ZnFe2O4 ZnCr2O4 Zn2TiO4 Zn2SnO4

ZnSb2O4 Mn2SnO4 CrAl2O4 Mg2TiO4 CoAl2O4 LiMn2O4 Fe3O4

As we can see, the common feature of these methods is that they are based on templates. Compared to the growth of solid nanowires, such template-based processes for nanotubes allow more control in the tube wall thickness, total diameter, and composition stoichiometry.

13.1.2 Basics of Spinels The AB2O4 type spinel has a cubic lattice structure, which at low temperatures consists of an fcc sublattice of O with the divalent cation (A 2+) and the trivalent cations (B3+) occupying one-eighth of the tetrahedral interstices and one half of the octahedral interstices, respectively (see Figure 13.1). One conventional unit cell comprises 8 formula units: 8 A metal cations, 16 B metal cations,

A Summary of Recently Reported One-Dimensional Spinel Oxide Nanomaterials Nano Morphology

Fabrication Method

Wires Tubes Tubes

Reaction of Mg vapor with ceramic alumina substrate Solid-state reaction of core–shell nanowires Reaction of porous alumina with Zn precursor

Tubes Wires Helical wires and springs Shells with Ga2O3 core Tubes Chainlike wires Wires Tubes Twinned zigzag wires Ribbons Rings Chainlike zigzag wires

Solid–state reaction of core–shell nanowires Thermal evaporation of ZnO/Ga powders, with Au catalyst Thermal evaporation of ZnO/Ga powder Reaction of Ga2O3 nanowires with Zn vapor Reaction of ZnO nanowires with Ga–O vapor and removal of ZnO core Infiltration of nanoparticles into porous alumina and ripening Solution infiltration in silica template Reaction of Zn nanowires with CrO2Cl2 vapor Solid-state reaction of ZnO–Ti core–shell nanowires Thermal evaporation of powder mixture and vapor–solid Thermal evaporation of powder mixture and vapor–solid Thermal evaporation of ZnO/SnO powder

Twinned zigzag wires

Thermal evaporation of ZnO/SnO powder

Thin rods Wires and belts Wires Tubes Shells with MgO core

Hydrothermal synthesis Thermal evaporation of Sb2O3/ZnO powders Thermal evaporation of MnCl2/Sn powder, with Au catalyst Reaction of porous alumina with Cr Pulsed laser deposition and in situ reaction with MgO nanowires Electrodeposition of Co within porous alumina and annealing Hydrothermal reaction Epitaxial deposition on MgO nanowires Atomic layer deposition inside porous alumina

Tubes with peapod structure Wires Tubes Tubes

Refs. Wu et al. (2003) Fan et al. (2006a: 5157) Wang and Wu (2005), Wang et al. (2006), Zhao et al. (2006) Fan et al. (2006b: 627) Bae et al. (2004) Bae et al. (2005) Chang and Wu (2006) Li et al. (2006) Jung et al. (2005) Liu et al. (2006) Raidongia and Rao (2008) Yang et al. (2007, 2009) Wang et al. (2004b: 435, 2005) Wang et al. (2004b: 435) Wang et al. (2004a: 177, 2008a,b: 707), Kim et al. (2008) Wang et al. (2004a: 177, 2008a,b: 707), Chen et al. (2005), Jeedigunta et al. (2007) Zhu et al. (2006) Zeng et al. (2004a) Na et al. (2006) Liu et al. (2008) Nagashima et al. (2008) Liu et al. (2008) Hosono et al. (2009) Liu et al. (2005) Bachmann et al. (2007), Escrig et al. (2008)

13-3

Spinel Oxide Nanotubes and Nanowires

(a)

(b) AO

(a)

(b)

FIGURE 13.1 (a) A ball-and-stick and (b) polyhedral model of the AB2O 4 spinel structure. Cations A locate at tetrahedral interstices and surrounded by four oxygen; whereas cations B at octahedral interstices and surrounded by six oxygen. The oxygen atoms have an fcc-type close pack sublattice occupying the corners of the polyhedra. (Reproduced from Fan, H.J. et al., J. Mater. Chem., 19, 885, 2009. With permission.)

and 32 O anions. The divalent, trivalent, or quadrivalent metal cations can be Mg, Zn, Fe, Mn, Al, Cr, Ti, and Si. Some crystals have an inverse spinel structure, which contains only half A cations at tetrahedral sites while the other half A and B are at the octahedral sites. A review article on spinel crystal structures is available in Sickafus et al. (1999). Solid-state reactions of the type AO + B2O3 → AB2O 4 are a common method used for the fabrication of spinel-oxide fi lms or single crystals. Traditional studies on spinel-formation reactions are usually conducted at planar interfaces or in the form of a powder mixture by bringing two solid binary oxides, or a solid oxide and a vapor or liquid phase, into contact at temperatures near or higher than 1000°C (Schmalzried 1974, Bolt et al. 1998). Figure 13.2a and b show this process schematically. In addition, this sol–gel method is also commonly used for the fabrication of spinel thin fi lms (e.g., see Lee et al. 1998, Meyer et al. 1999). The name spinel originally refers to MgAl2O4. It has been a model material for the study of spinel-forming solid-state reaction thermodynamics and kinetics, as well as cation disordering (Irifune et al.1991, Askarpour et al. 1993, Redfern et al. 1999). MgO is cubic with a lattice constant (0.421 nm) about half of most cubic spinels. Moreover it has the same fcc-type oxygen sublattice as that in cubic spinels. The lattice misfit, defined as 2(af − as)/(af + 2as), where af and as are, respectively, the lattice

B2O3

AB2O4 (c)

FIGURE 13.2 Interface solid-state reaction for the growth of spinels. (a) Conventional powder reaction where two powders are mixed thoroughly and calcinated at high temperatures (near 1000°C). Spinels form at the contact face in which the material transport methods involve volume and surface diff usion, grain boundary diff usion, and evaporation and deposition. (b) Film-substrate interface reaction to grow spinel fi lms. One of the oxide materials can also be vapors which react with the other oxide due to high substrate temperatures. (c) Nanostructures (tubes or wires) of spinels formed via solid-state reactions of a solid nanowire core with a shell or with vapors of the other oxide.

constants for the film and substrate, is around 4%. Hence, singlecrystal MgO substrates have been frequently used as one of the reactants for the formation of a variety of spinels through solidstate reactions. Besides MgAl2O4, spinels of Mg2TiO4, MgFe2O4, MgIn2O4, MgCr2O4, MgCo2O4, etc., can be formed through reactions between the corresponding oxide and a MgO(100) substrate (Hesse and Bethge 1981, Hesse 1987, Sieber et al. 1996, 1997). A cubic-to-cubic orientation of (001) [100]spinel ⎢⎢ (001) [100]MgO has been established. A detailed electron microscopy analysis of the lattice misfits between MgO and various MgObased spinels was available by Sieber et al. (1997). In other materials, when the two oxides are non-cubic, the O sublattice has to rearrange its type, e.g, from hcp in sapphire to fcc in cubic, which easily results in the grain growth of different orientations and a rough interface. The growth process of classical spinel oxides involves Wagner’s cation counterdiff usion mechanism (Carter 1961, Rigby and Cutler 1965), namely, cations migrating through the reaction interface in opposite directions and the oxygen sublattice remaining essentially fi xed (see Figure 13.3a). This mechanism applies to many types of spinels, for example, MgAl 2O 4, ZnFe2O 4, and Mg2TiO4. The reaction of ZnO (wurzite structure; lattice constants: a = 0.3250 nm, c = 0.5207 nm) with Al2O3 into ZnAl2O4 spinel (cubic structure; lattice constants: a = 0.880 nm) is, however, unique: The growth mechanism involves the diff usion of both Zn and O and an effective unilateral transfer of ZnO into the spinel (see Figure 13.3b). This was first pointed out by Bengtson and Jagitsch (1947) and later readdressed by Navias (1961), Branson (1965), and Keller et al. (1988). This means that an inert marker plane placed at the initial interface will be found at the ZnO/spinel interface for the ZnO−Al2O3 reaction, whereas in the case of the

13-4


AO

AB2O4

B2O3

13.2 MgAl2O4

3B2+ 2A3+ (a) AO or A

AB2O4

sion and grain boundary diffusion might become the dominating transport process over the volume diffusion.

B2O3

A+O (b)

FIGURE 13.3 Schematics of the two main diff usion mechanisms during the spinel-forming solid-state reactions. (a) Wagner counterdiffusion. Typical reaction is MgO−Al 2O3. (b) Uni-directional diff usion. Typical reaction is ZnO−Al 2O3. The dashed line indicates the initial interface. An inert marker plane placed at the initial interface will be found at the ZnO/spinel interface in (b), whereas within the spinel layer in (a) dividing this layer in a ratio of 1:3 as a result of charge neutrality of the overall diff usion flux.

MgO−Al2O3 reaction, the marker plane is within the spinel layer (see Figure 13.3). When the nanoscale templates are the materials of one of the oxides, the same reaction process and diff usion mechanism can occur, resulting in the formation of spinel nanowires or tubes (see Figure 13.2c). This is a straightforward and controllable fabrication route. In addition, in order to grow spinel 1D nanostructures of high crystalline quality, one of the oxides should be a single-crystal substrate. Th is is essentially the same as the fi lm-substrate reactions in conventional studies. On the other hand, there are also reports of the direct formation of spinel oxides nanowires by the thermal evaporation of powder mixture of the two oxides or two metals. Table 13.1 lists the examples that have been published so far of spinel-type ternary oxide nanotubes and nanowires. It should be mentioned that Fe3O4, in its form of (Fe3+)(Fe3+ Fe2+)O 4, is a prototype inverse spinel ferrite. There are a number of publications of Fe 3O 4 nanotubes and nanowires, but they will not be covered in this chapter. In bulk solid-state reactions, the diffusion of atoms through the growing product phase generally represents the rate-controlling step; therefore, a high annealing temperature and long time are usually needed for a complete sintering reaction. But a temperature higher than 1300°C brings a stoichoimetry problem because of the volatility of the reactants, e.g., ZnO. In a nanoscale system, the diffusion path is shortened so that the reaction steps at the interface may become rate controlling. Furthermore, during the growth of hollow nanoparticles or nanotubes, surface diffu-

Conventional interests in the magnesium aluminate spinel are the growth kinetics and mechanism order–disorder phase transitions (Irifune et al. 1991, Askarpour et al. 1993, Redfern et al. 1999). From the application point of view, MgAl 2O 4 is a material of interest for optical windows in the visible to mid-wavelength infrared ranges, and had been expected to be a low-cost substitute for sapphire or other optical ceramics. Therefore, most of the synthesis experiments of MgAl 2O 4 spinel focus on the modification of its microstructure, in terms of a more uniform grain size and fi ner grain microstructure, to improve its ceramic strength. From this content, nanostructures (wires and tubes) of MgAl 2O4 spinel might be advantageous. The fi rst report on MgAl 2O4 spinel nanowires was made by Wu et al. (2003), who obtained spinel nanowires (sometimes the product is spinel/Al 2O3 core–sheath wires) through the reaction of Mg vapors with a ceramic alumina substrate at 1350°C. The growth direction of the nanowires was along [011]. Generation of the Mg vapor by carbothermal reduction, and subsequent formation of eutectic droplets was the keypoint here to make the reaction process different from the traditional powder-mixture solid–solid reaction. In our own experiments of the vapor-phase synthesis of MgO nanowires by evaporating Mg3N2 powder at 950°C inside an alumina tube, a spinel layer and possibly nanostructures might have also formed. The implementation of the solid-state reaction using nanowires as one reactant provides a generic method for the synthesis of spinel nanotubes. Following this concept, MgAl 2O 4 nanotubes have been fabricated by using single-crystalline MgO nanowires as the template (Fan et al. 2006a: 5157). The nanowires were fi rst coated with a conformal layer of alumina via atomic layer deposition (ALD). The interfacial solid–solid reaction took place upon annealing the core–shell nanowires at 700°C−800°C under ambient conditions. During the reaction, metal cation pairs (Mg 2+ and Al3+) diff use in opposite directions while the oxygen sublattice is stationary. Th is is the well-accepted Wagner counterdiff usion mechanism for MgAl 2O 4. As the core nanowire has a larger diameter than necessary for a complete spinel reaction with the alumina shell, part of the core remains and an MgO/spinel core–shell nanowire structure was formed (see Figure 13.4a). In order to have a pure phase of spinel nanotube, the remaining MgO core was dissolved in an (NH4)2SO 4 solution. Figure 13.4b shows a TEM image of a particular web-like MgAl 2O 4 spinel nanotube, which is a perfect single crystalline with a cubic cross section (indicated by the arrow). Th is cubic structure was inherited from an excellent interface lattice matching (the lattice misfit equals to −4.1%) with an MgO core whose side faces are also {100} planes.

13-5


MgO core

MgAl2O4 shell

30 nm

(a)

040 022 004

100 nm (b)

FIGURE 13.4 Spinel MgAl2O4 nanotubes based on the reaction and Wagner cation counterdiff usion. (a) Schematics of the fabrication process. (b) TEM image of one MgO-MgAl 2O4 core–shell nanowire. (c) TEM image of one complicated nanotubes web of spinel MgAl2O4. The arrow indicates the square cross section of the nanotubes. Inset is the corresponding electron diff raction pattern revealing the single crystallinity. (Reproduced from Fan, H.J. et al., Nanotechnology, 17, 5157, 2006a. With permission.)

13.3 ZnO-Based Spinels Among all semiconductor 1D nanomaterials, ZnO is the most studied material compared to Si, Ge, and III–V. This is driven partly by the simple and cheap growth setup and high throughput of ZnO nanostructures. ZnO itself is a multifunctional material because of its intrinsic interesting physical properties. Research on the nanostructure ZnO has boomed since 2001 when the Wang group (Pan et al. 2001) and the Yang group (Huang et al. 2001) reported ZnO nanobelts and nanowires, respectively. Worldwide there are now numerous groups dedicated to the basic research and device application of ZnO in its various morphologies from thin fi lms to quantum wells to nanostructures. Most of the attention on 1D ZnO nanostructures is

focused on their structural, optical, and optoelectronic properties. Also, due to their easy availability, ZnO nanowires have been widely used as a physical (nonreactive) template for the fabrication of nanotubes like single-crystalline GaN (Goldberger et al. 2003), amorphous alumina (Shin et al. 2004) and silica (Chen et al. 2005a,b), and Pt−BaTiO3−Pt double-electroded ferroelectrics (Alexe et al. 2006). In most cases, compounds based on ZnO are technologically more important than its pure phase. For example, alloying with Cd or Mg is an effective route for tuning the optical bandgap of ZnO; doping with lithium or vanadium has been shown to enhance the piezoelectricity because the spontaneous polarization (due to the non-centrosymmetric crystal structure P63mc) is enlarged by replacing the Zn ions with smaller Li or V ions that cause lattice distortion (Joseph et al. 1999, Yang et al. 2008a: 012907); and alloying ZnO with In2O3 gives thermoelectric applications (Kaga et al. 2004). Most of the work is still on bulk materials or fi lms for their potential applications like SAW (surface acoustic waves) devices, piezoelectric actuators, and transparent conductors. ZnO can form spinel-type ternary compounds with many other oxides. They have a formula of either ZnM[32 +]O4 or Zn2M[4+] O4 (M = Al, Ga, Fe, In, Sn, Sb, Ti, Mn, V, Cr). The formation process can be either solid-state reactions by thermal annealing, or hydrothermal growth, or in-situ vapor–liquid–solid (VLS) growth by co-evaporating powders of the constituent oxides. There is now increasing interest in ZnO-based spinel 1D nanostructures within the domain of ZnO research (see Table 13.1). Such nanoscale materials show a large diversity of morphologies, such as tubes, wires, periodically zigzagged and/or twinned wires, which might reflect the uniqueness of the nanoscale solid-state reaction (versus bulk reactions) and VLS growth of ternary compound nanowires (versus binary compounds). Most of the ZnO-based spinels are wide bandgap semiconductors and phosphorous materials.

13.3.1 ZnAl2O4 Porous aluminum oxide (see Figure 13.5a) is the most widely used template for the fabrication of 1D nanostructures, and generally it is used as a morphology-defining agent. When in contact

400 nm (a)

(b)

100 nm

(c)

FIGURE 13.5 ZnAl2O4 by reacting Zn precursor with porous alumina template. (a) Schematics of the alumina template. (b) Single-crystal ZnAl 2O4 nanonet. (Reprinted from Wang, Y. and Wu, K., J. Am. Chem. Soc., 127, 9687, 2005. With permission.) (c) Polycrystalline ZnAl 2O4 nanotubes. (Reprinted from Zhao, L. et al., Angew. Chem. Int. Ed, 45, 8042, 2006. With permission.)

13-6


with ZnO, however, the alumina is also one of the reactants used to form zinc aluminate spinel (ZAO). Wang et al. (2005, 2006) and Zhao et al. (2006) obtained ZAO nanotubes by reacting a Zn-containing precursor with the porous alumina template (see Figure 13.5). In Wang’s experiment, the Zn vapor was generated by an H2 reduction of ZnO or ZnS powder, whereas no spinel was formed using the pristine ZnO solid source because of an insufficient contact. As shown in Figure 13.5b, the overall single-crystalline ZAO nanonets were obtained after annealing at 680°C for 1000 min (Wang and Wu 2005) or 650°C–660°C for 800 min (Wang et al. 2006). A variation of the tube wall thickness was realized by varying the temperature and reaction time, but was restrained below 20 nm because of the limited diff usion depth of Zn vapor into alumina. Note that a multilayer of ZAO/ Zn4Al22O37/ZnO was observed on the surfaces of the template. The thin (800 min at 660°C), a ZnO nanonet can also be formed epitaxially on the ZAO nanonet surface. Such an epitaxial growth might explain the vertical alignment of ZnO nanowires that we obtained using a ZAO covered sapphire as the substrate. In Zhao’s experiment, the Zn precursor was provided by the [Zn(TePh)2(tmeda)] solution that was infi ltrated into the alumina pores (Zhao et al. 2006). After annealing in air for 24 h at 500°C, nanowires of the single-crystalline Te core surrounded by a polycrystalline ZAO shell was formed. It was suggested that the reaction underwent an intermediate phase of ZnTe that was then oxidized to ZnO plus elemental Te. Such freshly

formed ZnO then reacted with the alumina pore wall to form the spinel. After the removal of the Te core, ZAO nanotubes with a thin (8 nm) and polycrystalline wall were obtained (see Figure 13.5c). We have demonstrated a different fabrication route to ZAO spinel nanotubes based on the Kirkendall effect (see Figure 13.6; Fan et al. 2006b: 627; Fan et al. 2007). In this strategy, pre-synthesized ZnO nanowires (10−30 nm thick, up to 20 μm long) were coated with a 10 nm thick conformal layer of alumina via ALD. Thus, the formed ZnO−Al 2O3 core–shell nanowires were annealed in air at 700°C for 3 h, causing an interfacial solid-state reaction and diff usion. Because the reaction is effectively a one-way transfer of ZnO into the alumina, it represents an extreme Kirkendall effect (Aldinger 1974). It is noted that this nano Kirkendall mechanism also accounts for the formation of other spinel nanotubes of CoAl 2O4 (Liu et al. 2008) and ZnCr2O4 (Raidongia and Rao 2008), ZnCo2O 4 hollow cubes (Tian et al. 2008), as well as the conversion of Zn wires into ZnO nanotubes (Qiu and Yang 2008). Upon a suitable matching of the thickness of the core and the shell, highly crystalline single-phase spinel nanotubes are obtained, as shown in Figure 13.6a and b. Figure 13.6b shows a TEM image of one nanotube together with the remaining gold particle atop the pristine ZnO nanowire. The spinel nanotube wall thickness can be precisely controlled through the variation of the alumina shell thickness i.e., the number of ALD cycles. In addition, a suitable annealing temperature is very important. Figure 13.6c illustrates how the morphology of the final product depends on the annealing temperature (Yang et al. 2008c: 4068): Tiny voids are generated at the ZnO/Al2O3 interface near 600°C, but do not grow to large voids due to kinetic

100 nm (a)

20 nm (b)

500°C

500 nm

ZnO

Al2O3

Al2O3

600°C 700°C 800°C 900°C

(c)

(d)

FIGURE 13.6 ZnAl 2O4 tubular nanostructures by solid-state reaction of ZnO/Al 2O3 core–shell nanowires. (a,b) TEM images of straight nanotubes. (Reprinted from Fan, H.J. et al., Nat. Mater., 5, 627, 2006b. With permission.) (c) Effect of the annealing temperature to the morphology. (Reprinted from Yang, Y. et al., J. Phys. Chem. C, 112, 4068, 2008c. With permission.) (d) Complex tubular structures using hierarchical ZnO nanowires as the starting material. (Reprinted from Yang, Y. et al., Chem. Mater., 20, 3487, 2008b. With permission.)


reasons. The reaction at 700°C results in good crystalline nanotubes as long as the thickness of the wires and shells matches well to each other. Interestingly, a higher annealing temperature near 800°C results in hollow nanotubes even if the initial ZnO nanowire is thicker than necessary for a complete spinel formation of the former shell. However, when the temperature is further increased, the tube wall collapses driven by the thermodynamic instability. Therefore, the temperature window for an optimal solid-state reaction of the core–shell nanowires is 700°C−800°C. The conformity and uniformity characteristics of ALD are essential to the formation of smooth nanotubes. Particularly, the synthesis of complex tubular ZAO nanostructures by such a shape-preserving transformation is possible if one starts with hierarchical three-dimensional (3D) ZnO nanowires (Yang et al. 2008b: 3487). Recent literature shows an abundant variety of ZnO 3D nanostructures including bridges, nails, springs, and stars. Figure 13.6d gives one example of a Chinese firecracker-like 3D hollow ZAO spinel. Such uniform hollow 3D structures cannot be obtained by other coating methods like pulsed laser deposition, sputtering, or physical vapor deposition, which have poor step coverage characteristics. Because of the above mentioned reaction, we expect that the ZnO nanowires grown inside the porous alumina template, especially those by Zn electrodeposition plus post oxidation (Li et al. 2000, Liu et al. 2003, Fan et al. 2006d: 213110) are not pure ZnO, but a mixture of ZnO with a layer of ZAO or Zn4Al22O37, since their growth or annealing temperatures were high enough for the spinel-forming solid-state reaction to occur. Unfortunately, none of these papers have shown a high-resolution TEM of the nanowires. Likewise, the growth of ZnO nanowires directly on the lattice-matched sapphire (single crystal Al2O3) might also end up with the formation of a ZAO spinel, as pointed out recently by Grabowska et al. (2008).

13.3.2 ZnGa2O4 Zinc gallate (ZGO) is most intensively studied among all spinel nanostructures, because of its interesting luminescent properties. It is a promising candidate for a blue light source, for a vacuum fluorescent display (Itoh et al. 1991), and for gas sensing (based on surface states-related electric property). ZGO has a wide bandgap of 4.4 eV, thus it is transparent from the violet to near ultraviolet region. It is proposed that ZGO can be a host for full color emitting materials when doped with various activators: Mn2+ for green (Shea et al. 1994) and Eu3+ or Cr3+ for red (Rack et al. 2001). The demonstrated fabrication routes of 1D ZGO nanostructures can be divided into two main categories. The first type of fabrication is a high-temperature (around 1000°C) co-evaporation of a mixture of ZnO–Ga powders and deposition substrates. The growth is driven by the well-known vapor–liquid–solid mechanism with gold nanoparticles acting as the catalyst. One drawback of this method is the difficulty in controlling the Ga/Zn ratio so that a ZnO phase might co-exist with the ZGO

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spinel even if the source material is mixed with a defi ned molar ratio. Nevertheless, Bae et al. (2004) fabricated pure cubic phase spinel nanowires vertically standing on an Si substrate using a gold nanoparticle catalyst for the vapor–liquid–solid growth by employing this method. All the nanowires grow along the [111] direction. Using the same technique, Feng et al. (2007) obtained ultralong nanowires on the Au covered Si substrate, which were mainly ZGO but with a small amount of ZnO. A particularly interesting discovery was the helical ZGO nanowires wrapped around the straight ZnSe nanowire support, as well as freestanding ZGO nanowires self-coiled into a spring-like structure (Bae et al. 2005). These zigzag wires elongate periodically along the four equivalent directions, with an oblique angle of 45° (so as to maintain a coherent lattice at the kinks). According to the authors (Bae et al. 2005), this unique growth behavior might be related to the ZnSe nanowires, which provide the Zn source at the beginning and simultaneously an epitaxial substrate for ZGO. Note that the evaporation temperature used by the authors was as low as 600°C. The second type of fabrication route is a two-step process: synthesis of single-crystalline nanowires followed by their reaction with the other oxide. The latter step is a typical solidstate reaction on the nano scale, which results in an outer ZGO layer surrounding the nanowire core. Pure ZGO spinel can be obtained by dissolving the remaining core material. The spinel nanoshell or nanotubes can be single crystalline as a result of the epitaxial relationship between the growing ZGO and the core material. Chang and Wu (2006) made a systematic study of the formation of the ZGO layer on top of β-Ga2O3 nanowires. Nanowires of a single-crystalline Ga 2O3 core and a chemical vapor deposition (CVD)-grown polycrystalline ZnO shell were annealed at 1000°C for 1 h. Depending on the thickness of the ZnO layer, the final product after the solid-state reaction was either Ga 2O3/ZGO core–shell nanowires, ZGO nanowires, or ZGO nanowires capped with ZnO nanocrystals. In any case, the ZGO were single crystalline. Li et al. (2006) did a similar experiment but in the opposite way: they started with ZnO nanowire array, which in situ reacted at 500°C with a Ga xO y surface layer deposited from the vapor, forming a ZGO spinel layer. Because of the epitaxial interface, ZnO[110]⎥⎥ ZGO[112]; ZnO[110]⎥⎥ ZGO[440], the ZGO layer as well as the subsequent ZGO nanotubes were single crystalline. Figure 13.7a shows the nanotubes together with an electron diff raction pattern that proves its single-crystallinity. Here, the phase was controlled by the deposition temperature, unlike Chang and Wu’s case where the phase was controlled by the shell thickness. Another special way to obtain ZGO spinel nanotubes was reported by Gautam et al. (2008). In their experiment, Ga-doped ZnS nanowires were heated up in an oxygen gas flow at 500°C−850°C for 200 min, during which the Ga:ZnS nanowires were converted into nanotubes whose walls are comprised of ZnO + ZGO composite fragments (see Figure 13.7c). While the overall 1D structure and morphological homogeneity of the initial nanowire were preserved after the reaction, the nanotube wall became

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(111) 389 nm

ZnGa2O4[121] Intensity (a.u.)

(111)

(202) (a)

380 nm

ZnGa2O4 nanotubes ZnGa2O4 nanowires

O

Zn Zn Zn Ga O Ga Zn Zn Zn O Ga Zn O Zn Zn Ga

464 nm

200 nm

200 (b)

300

400 500 600 Wavelength (nm)

O O O O

700

(c)

FIGURE 13.7 ZnGa 2O4. (a) ZnGa2O4 nanotubes by reaction of initial ZnO nanowire with Ga-O vapor and removal of the remaining ZnO core. Insets are the corresponding electron diffraction pattern and high-resolution TEM image showing the single-crystallinity of the nanotube. (From Li, Y. J. et al., Appl. Phys. Lett., 88, 143102, 2006. With permission.) (b) Room temperature cathodoluminescence spectra showing sharp a UV and a blue peak of ZnGa 2O4 nanotubes relative to the spectrum from solid nanowires. (Reprinted from Li, Y.J. et al., Appl. Phys. Lett., 88, 143102, 2006. With permission.) (c) ZnGa2O4–ZnO compound nanotubes and the proposed formation mechanism which involves a faster outward diff usion of Zn/Ga than the inward diff usion of O. (Reprinted from Gautam, U.K. et al., Adv. Mater., 20, 810, 2008. With permission.)

polycrystalline and rugged. The possible formation mechanism involves unequal diff usion rates for the outward Ga/Zn and inward oxygen during the oxidation of the ZnS nanowires at the interface plane. This is similar to the Kirkendall effect in the case of the ZnO + Al 2O3 reaction discussed in Section 13.3.1. As the Ga concentration in the initial ZnS nanowires is only on doping level, it is not surprising that the nanotube walls are not pure ZGO. Annealing of a Ga-doped ZnO nanowire will certainly not give similar nanotubes as above. All the so-far demonstrated ZGO nanowires or nanotubes are undoped, so the observed emissions in their luminescence spectra are related mainly to the self-activated Ga−O emission or composition stoichiometry (e.g., O vacancies, Zn insterstitials). Two main peaks have been commonly observed: one near 380 nm and the other near 430 nm. An example from Li et al. (2006) is shown in Figure 13.7b, but there seems to be little point in comparing data from different authors since the peak position is highly sensitive to the preparation condition and can shift by more than 200 nm depending on the Zn/Ga stoichiometry. For the application in nanoscale light source, doping with rare-earth elements into the 1D ZGO nanostructures is needed. ZGO spinels are also known to have applications as solid sensors. Like ZnO, the large surface-to-volume ratio makes the surface states play a major role in the optoelectronic properties of 1D nanostructures. The optically driven oxygen and temperature sensing behavior of ZGO nanowires was recently demonstrated (Feng et al. 2007). The current across individual nanowires was nearly zero at ambient conditions, whereas the current increases drastically under illumination by 254 nm UV light (the bandgap of ZGO is around 270 nm). Such enhancement was attributed to the increased charge carrier concentration and decreased contact resistance. The current level is also dependent on the temperature and oxygen pressure, giving the ability to construct sensing devices.

13.3.3 ZnFe2O4 Nanostructured ferrite spinels are of special interest because of their size- and morphology-related magnetic properties. While most ferrite spinels have inverse spinel structure and are ferrimagnetic, ZnFe2O4 (ZFO) have a normal spinel structure and show long-range antiferromagnetic ordering at its equilibrium state below the Néel temperature TN ≈ 9−11 K (Ho et al. 1995, Schiessl et al. 1996). At room temperature, it is paramagnetic because of the weak magnetic exchange interaction of the B site Fe3+ ions. Nanoscale ZFO, however, can have a higher TN or show ferromagnetic with the magnetization generally increasing with grain size reduction. There are a number of reasons for this feature, such as the inversion of the cation distribution (Fe3+ into tetrahedral interstices and Zn2+ into octahedral interstices), small size effect (a large surface area gives uncompensated magnetization moment), and nonstoichiometry (Oliver et al. 2000, Grasset et al. 2002, Hofmann et al. 2004, Yao et al. 2007). For example, the ZFO nanoparticles with diameters of ≈10 nm have shown a saturated magnetization of Ms = 44.9 emu/g at room temperature (Yao et al. 2007). The magnetic behavior of the nanocrystals was related to the ferromagnetic coupling of Fe ions at A–B sites in the (Zn1−xFex)[Zn xFe2−x]O4 particles and surface spin canting. While there are quite a number of reports on ZFO nanoparticles, there are strangely only a few publications on ZFO nanotube or wires. Jung et al. (2005) fabricated ZFO nanowires by annealing the densely packed ZFO nanoparticles infiltrated into porous alumina templates. The nanowires are superparamegnetic and show evidence of an increase in the coercive field and Mr/Ms value compared to the unannealed nanoparticles. Liu et al. (2006) adopted a strategy similar to the standard solution synthesis of perovskite nanotubes/wires (e.g., Hernandez et al. 2002). In their experiment, Zn(NO3) 2 · 6H2O and Fe(NO3)3 · 9H2O with 1:2 molar ratio of Zn/Fe were dissolved into mesopores of silica SBA-15 template. After drying, decomposition, and annealing,


polycrystalline ZFO nanowire bundles were obtained, which also shows enhanced paramagnetism compared to the bulk phase. Different from the above negative template methods, ZFO nanoshells using ZnO nanowires as the template were prepared. Single-crystalline ZnO nanowires were coated with a uniform amorphous layer of ALD Fe2O3 using a Fe2(OtBu)6 precursor (Bachmann et al. 2007). After annealing the core–shell nanowires at 800°C for 4 h, an outer layer of ZFO surrounding the remaining ZnO was formed. It is expected that the solid-state reaction of ZnO−Fe2O3 occurs by a Wagner counterdifussion mechanism. Unlike the ZnO−Al2O3, which is characterized by a very smooth inner surface of the nanoshells, the interface after the solid-state reaction is rugged but the ZFO layer is single-crystalline. The crystallinity can be attributed to an orientation relationship ZnFe2O4(111) [110]//ZnO(0001) [1120], as established by Zhou et al. (2007). Since these nanoshells have a large aspect ratio of up to 100 and tunable shell thickness on an atomic scale, it would be interesting to examine its anisotropy magnetic properties. Systematic characterization is still underway.

13.3.4 ZnCr2O4 There is a recent report on the conversion of elemental metal nanowires into the corresponding nanotubes, among which is the spinel ZnCr2O4 (Raidongia and Rao 2008). The reaction is an unconventional solid–vapor type: solid Zn nanowires reacted with a CrO2Cl2 vapor in an oxygen environment at 400°C for 3 h in a resistance furnace. The nanotube walls are granular and polycrystalline and possibly contain pores; nevertheless, the XRD spectrum does correspond to a cubic, normal, spinel-type structure. Note that the reaction temperature was much lower than 900°C, temperatures above which are used for the conventional solid–solid reaction of the ZnO–Cr2O3 powder mixture. As was learned from the properties of bulk or thin fi lms, it would be of great interest to investigate the antiferromagnetic ordering (Martinho et al. 2001) and gas-sensing properties (Zhuiyko et al. 2002) of the ZnCr2O4 nanotubes.

13.3.5 Zn2TiO4 Zinc titanate is a useful material for low-temperature sintering dielectrics. Usually, there are three modifications for the ZnO– TiO2 system: Zn2TiO4 (inverse spinel, cubic), ZnTiO3 (hexagonal), and Zn2Ti3O8 (cubic). ZTO has been widely used as a regenerable catalyst as well as an important pigment in the industry. It is also a good sorbent for removing sulfur-related compounds at high temperatures. Since the nanoscaled ZTO is expected to achieve low-temperature sintering, a desirable property for microwave dielectrics, increasing attention has been paid to the preparation of ZTO in a nanoscaled form in recent years. Conventional synthesis methods such as high-energy ball milling and a high-temperature solid reaction have been reported to prepare ZTO nanocrystallites (Manik et al. 2003). However, the reports on 1D ZTO nanostructure growth are

13-9

scarce. Only recently, Yang et al. (2007) reported the synthesis of twinned ZTO nanowires by using ZnO nanowires as a reactive template. In the experiments, ZnO nanowires with an axis direction along [0110] were coated with amorphous Ti by magnetron sputter deposition to form ZnO/Ti core–shell nanowires. After thermal annealing at 800°C for 8 h at low vacuum, the solid-state reaction led to a phase transformation from wurtzite ZnO to ZTO. The final product contained a large amount of (111)-twinned ZTO nanowires elongated along the [111] direction. It was discussed that two types of nonuniformities have contributed to the formation of ZnTO subcrystallites, eventually causing the zigzag morphology of the nanowires: first, the sputter-deposited Ti layer was not conformal and second, the electrons and ions bombardment causes content fluctuation in the ZnO nanowires. We synthesized 1D twinned ZTO nanowires via a solid-state reaction approach using ZnO nanowires as the template (Yang et al. 2009). Different from above, a very thin (5 nm) layer of TiO2 was deposited via ALD. The TiO2 layer was amorphous and surrounded the ZnO nanowires with a uniform thickness. The ZnO/TiO2 core–shell nanowires transformed into a zigzag structure, too, after annealing at 900°C for 4 h in an open furnace (see Figure 13.8a and b). The twinned nanowire is composed of large parallelogram-shaped subcrystallites, but does not present periodic stacking (see Figure 13.8b). The interplanar spacings of 0.49 nm measured from the high-resolution TEM image perfectly match the d111 lattice distance of ZTO crystal, demonstrating the [111]-oriented growth direction of each individual grain. The fast Fourier transform (FFT) pattern of the twin boundary (TB) (inset of Figure 13.8c) reveals clearly the (111) twin structure: two (111) mirror planes sharing a common (111) face. The twinning angle across the boundary is measured to be about 141°, and the relative rotational angle is about 70.5° (see Figure 13.8c and d). These results are in agreement with other cubic nanowires like SiC and InP (Section 13.4), and Zn2SnO4 (Section 13.3.6). No misfit dislocations were observed at the interface. It is proposed that the formation of twinned ZTO nanowires in our case includes multiple stages, which differs from the one by Yang et al. (2007) Since amorphous TiO2 easily crystallizes at a temperature of 500°C or higher, the initially continuous TiO2 shell on the surface of ZnO nanowires are expected to transform into anatase TiO2 islands first. The islands are textured as a result of the volume shrinkage of the amorphous phase. During the annealing at 900°C, the TiO2 phase could be incorporated in the ZnO lattice as a segregation at early stages because of the faster diff usion speed of Ti4+ as compared with that of Zn2+. Subsequently, ZTO spinel nanocrystallites are formed through the lattice rearrangement and were attached to the surface of the unconsumed ZnO core (note that the TiO2 layer was only 5 nm thick). No voids or tubular structures appeared at the ZTO/ZnO interface, indicating that the inner-diff usion of Ti4+ is prominent for the TiO2/ZnO couple. With prolonged annealing at 900°C, the unconsumed ZnO core is desorbed or evaporated at 900°C through the gaps of the nanocrystallites. Subsequently,

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ZnO nanowire

Twinned Zn2TiO4 nanowires

25 nm (b) [111] 111

111

0.49 nm

TiO2 thin layer 141°

141°

TB

TB

Annealing

TB TB

TB

Annealing Annealing

141°

200 nm TB

(a)

(c)

[111]

2 nm

10 nm

(d)

Spinel nanocrystallites (e)

FIGURE 13.8 Zn 2TiO4 zigzag twinned nanowires by solid-state reaction of ZnO/TiO2 core–shell nanowires. A uniform shell of TiO2 was coated using atomic layer deposition and the annealing condition is: 900°C, 4 h. (a,b) TEM images at different magnifications showing the zigzag morphology of the nanowires. Note that the twinning is non periodic. (c) A closer view at one twin boundary and the corresponding FFT pattern. (d) Several twinning boundaries with a fi xed zigzag angle of 141°. (Reproduced from Fan, H.J. et al., J. Mater. Chem., 19, 885, 2009. With permission.) (e) Schematics of the proposed formation process of the Zn2TiO4 zigzag nanowires.

the loosely interconnected nanocrystallites were [111]-orient attached, coalesced, and fi nally evolved into twinned nanowires seen in Figure 13.8b. By comparing the above two solid-state reaction experiments (Yang et al. 2007, 2009), we are able to draw the following conclusions: first, the preferred [111] growth direction of the twinned ZTO nanowires is independent on the orientation of the starting ZnO nanowires ([0110] or [0001]). During sintering, individual ZTO nanocrystallites may rotate to constitute an energetically favorable assembly along the [111] direction, since the surface energy of [111] facets is the lowest for an fcc structure. Secondly, the diff usion species in the ZnO/TiO2 couple is unambiguously Ti4+, not the metallic Ti as described in the case of the ZnO/Ti couple. Lastly, a fluctuation of the ZnO content is not a necessity for forming the subcrystallites.

13.3.6 Zn2SnO4 As in the case of Zn2TiO4, Zn2SnO4 (ZSO) crystals also have an inverse spinel structure. The interest in ZSO has been in the electrical circuit as a transparent conducting electrode as a replacement of the expensive ITO (In-doped SnO2). This is because of its high electron mobility, high electrical conductivity, and low visible absorption. Like other metal–oxide semiconductors, ZSO can also have application potential in photovoltaic devices and Li-ion batteries (Rong et al. 2006, Tan et al. 2007). While ZSO nanoparticles have been demonstrated to be useful as electrodes of dye-sensitized solar cells with a light-to-electricity efficiency of 3.8% (Rong et al. 2006), it is envisaged that 1D nanostructures could enhance the performance due to a continuous electrical conduction path. One-dimensional ZSO nanostructures are fabricated mainly by thermal evaporation of a powder mixture of ZnO + SnO2 or Zn + Sn metal (and some by hydrothermal reaction). No reports have been published so far on the solid-state reaction of

core–shell nanowires, although a controlled growth of ZnO and SnO2 nanowires has become mature in many research labs. The high-temperature evaporation and deposition method involves a complicated thermodynamic and kinetic process. The partial pressures of oxygen and ZnO/SnO vapor are very inhomogeneous, which causes disturbances of the Au–Zn–Sn ternary droplet size in the VLS growth and hence diameter oscillations (Jie et al. 2004). Moreover, such an evaporation method usually ends up with a mixture of different phases and structures in one growth set. For example, ZnO, ZSO nanowires, and chainlike ZTO nanowires were identified in a single VLS growth (Jie et al. 2004). The ZSO composition was off stoichoimetric with an element ratio of Zn:Sn:O = 1.75:1:(2.6−3.5), indicating a deficiency of zinc and oxygen. Reports on a series of ZSO nanostructures with very different morphologies have been published by the Xie group, who conducted similar evaporation experiments (Wang et al. 2004a: 177,b: 435, 2008b: 707, Chen et al. 2005). The sofar demonstrated ZSO quasi-1D nanostructures are listed in Table 13.1, including smooth belts (Wang et al. 2004b: 435, 2005: 2928), rings (Wang et al. 2004b: 435), chainlike single-crystal wires (Wang et al. 2004a: 177, 2008b: 707, Kim et al. 2008), twinned wires (Wang et al. 2004a: 177, 2008b: 707, Chen et al. 2005, Jeedigunta et al. 2007), and short rods (Zhu et al. 2006). In the following section, we selectively discuss only two types of rather extraordinary nanostructures. The first is the twinned nanowire (see Figure 13.9a). These 1D ZSO nanostructures have a twinning morphology similar to the aforementioned ZTO. Wang et al. (2004a: 177,b: 435, 2008b: 707) reported VLS-grown twinned ZSO nanowires by evaporating powder mixtures at 1000°C with two types of weight ratios. In one case, a ratio of ZnO:SnO = 2:1 gave twinned nanowires (among many other shaped wires), whereas in the other case, a ratio of ZnO:SnO = 1:4 was needed for the production of similar twined wires. All these ZSO wires have a [111] growth direction, with the twin planes being (111) and the twinning direction

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Spinel Oxide Nanotubes and Nanowires [111]

[111]

(111) (111)

[111]

(002)

100 nm 200 nm

111T

002T

111

111T 111T

111

111

111 002 111

(a)

(b)

FIGURE 13.9 Zn2SnO4. (a) Pseudoperiodic twinning nanowire. (b) Single-crystal nanowires by periodic stacking of rhombohedral nanocrystals. Top: schematics of the structural; middle: SEM images; bottom: the corresponding electron diff raction patterns. The arrows in the diff raction patterns indicate the nanowire growth directions. (Reprinted from Wang, J.X. et al., Cryst. Growth Des., 8, 707, 2008b. With permission.)

Twinning as planar defects in nanowires is becoming a very interesting topic and is widely studied for cubic crystals including not only the above ZTO and ZSO but also binary compounds like GaAs, GaP, InP, ZnSe, and SiC (Xiong et al. 2006, Davidson et al. 2007, Mattila et al. 2007, Wang et al. 2008a: 215602). These nanowires are formed through either solid-state reactions or metal-catalyzed unidirectional growth. Quasiperiodic twinnings have been observed intersecting the entire wire cross-section. It is known that the (111) twined crystals have a relative rotational angle of 70.5° and the zigzag angle between the two twinning nanounits is about 141°. This is in accordance with all the so-far demonstrated nanowires of the cubic phase. Figure 13.10 schematically illustrates the two typical growth processes of twinned nanowires. The first (Figure 13.10a) represents the spinel nanowires by solid-state reactions, for example, the aforementioned ZTO (Figure 13.8). This is mainly a selfassembling process where subcrystallites rearrange to minimize the total energy: two adjacent (111) faces meet via a shear movement along the [211] direction of subcrystallites that are driven by a dislocation-induced strain, and then the mirror planes arrange symmetrically to reduce the total energy. Alternatively, the twinned structure can also be formed when new crystallites start to grow in a limited space between two existing grains via an Ostwald ripening process during the sintering stage. φ0 Liquid

θL

Vapor or solvent

ν

[111]

}B {111} A

h Solid

{111

perpendicular to the wire axes, same as was observed for ZTO nanowires. The second is the single-crystal chainlike nanowire (see Figure 13.9b). Structurally, this type of wire is formed by a sequential stacking of rhombohedral nanocrystals along the [111] direction. The stacked nanocrystals constitute a single-crystal wire on the whole, as revealed by the electron diffraction pattern (e.g., the bottom of Figure 13.9b). This means lattice coherence at the boundary of the nanocrystals. Jie et al. (2004) observed the so-called diameter-modulated ZSO nanowires composed of linked ellipses. Wang et al. (2005: 2928) reported diamond-like wires composed of intersected rhombohedra (see Figure 13.9b). But according to the careful electron tomography study by Kim et al. (2008), who also obtained both types of chainlike ZSO nanowires, the above two structures are essentially the same, that is, the observed zigzag angles appear different (125° or 120°) under the electron microscopy simply because of a rotation of the wire around its axis. Hydrothermal synthesis as a low-cost, high-yield method for 1D nanostructures has been applied to single-crystalline ternary compound nanowires like BaTiO3 and PbTiO3, which are important ferroelectric materials (Rorvik et al. 2008), and recently to ZSO nanowires. A hydrothermal process was applied with the use of hydrazine hydrate as the alkaline mineralizer for the growth of ultrathin (2−4 nm in diameter) ZSO nanowires (Zhu et al. 2006). The optical bandgap was determined to be 3.87 eV by diffuse UV-vis reflectance measurement, which is a blueshift of 0.27 eV from bulk ZSO (3.6 eV), indicative of the quantum confinement effect.

13.4 Twinning of Spinel Nanowires

Twin

Metal

(a) Annealing

(b) Vapor–liquid–solid or solution–liquid–solid

FIGURE 13.10 Two main formation processes proposed for twinned nanowires of cubic crystals including spinels. (a) Nanocrystallites are formed first through solid-state reactions, and then transformed into twinned nanowires during annealing. Th is process is mainly for ternary alloyed nanowires like Zn2TiO4 and Zn2SnO4. (b) Twinned nanowires formed by vapor–liquid–solid or solution–liquid–solid growth processes. It is for both ternary compound spinel nanowires and binary nanowires like GaP, InP, GaAs, and ZnSe. Inset of (b) shows the threephase boundary where a fluctuation in the droplet/nanowire contact angle causes nucleation of a twinned plane. (Reproduced from Fan, H.J. et al., J. Mater. Chem., 19, 885, 2009. With permission.)

13-12

In the second route (see Figure 13.10b), the nanowires grow through a VLS or solution–liquid–solid process with a ternaryalloy droplet at the growth front as the catalyst. The ZSO twinned nanowires (see Figure 13.9) and most of the binary compound zinc blende nanowires (Xiong et al. 2006, Davidson et al. 2007, Mattila et al. 2007, Wang et al. 2008a: 215602) fall into this category. In this case, the formation of twins is strongly related to the contact angle fluctuation at the three-phase boundary (inset of Figure 13.10b), as elaborated by Davidson et al. (2007). A smooth untwined nanowire has (112) sidewall faces. But twinning eliminates the (112) sidewall surface by converting it to two lower energy (111) mirror planes. The sidewall surface alternates between the two (111) mirror planes with subsequent twins in order to maintain a straight nanowire growth along the [111] direction. In the Davidson model, fluctuations in the droplet/nanowire contact angle for Au-seeded nanowires are most likely necessary for twinning to occur. This requirement (the contact angle at the three-phase boundary must fluctuate sufficiently) indicates that even when twin planes might be expected based on their formation in bulk crystals, they may not form in nanowires. Interestingly, this model also suggests that nanowires with smaller diameters ( 0.3. These results have been corroborated by using the public OOMMF computer program package (OOMMF year unknown) (Escrig et al. 2007b). Thus, we focus the following description on core-free magnetic nanotubes. For the F configuration, the magnetization M(r) can be approximated by M0z, where M0 is the saturation magnetization and z is the unit vector parallel to the axis of the nanotube. In this configuration, the exchange contribution is nil, and the total energy is given by the dipolar contribution and the uniaxial anisotropy (Escrig et al. 2007b,c):

uniaxial anisotropy are both nil. Thus, only exchange and cubic anisotropy characterized by the constant Kc contributes to the total energy (Escrig et al. 2007b,c):

5,000

0

0

20

40

60

80

R/Lx

FIGURE 14.6 Phase diagrams for magnetic nanotubes with β = 0.9 giving the regions in the RL plane where one of the configurations has lower energy: F phase to the upper left, V phase to the lower right.

14-8


a representative range for the geometrical parameters. Regions where each phase dominates are graphically indicated. However, it is clear that most of the phase diagram for long iron thin tubes is dominated by the F phase, namely, with the magnetic moment along the axis with two possible and equivalent orientations. This property is what makes such magnetic nanotubes suitable for binary information storage.

14.3.3 Reversal Modes in Magnetic Nanotubes The polarization of the magnetic moment along one of the two possible orientations in state F can be changed by means of external magnetic fields. This is precisely what is done in the process of magnetic recording in a bunch of magnetic nanotubes. There are some studies concerning the way this process is achieved, which we summarize this section. The properties of virtually all magnetic materials are controlled by domains—extended regions where the spins of individual magnetic centers are tightly locked together and point in the same direction. A domain wall forms where two domains meet. Measurements on elongated magnetic nanostructures highlighted the importance of nucleation and the propagation of a domain wall going from the extreme of one domain to the extreme of the other domain with opposite magnetization; this is called the magnetization reversal process (Wernsdorfer et al. 1996, Atkinson et al. 2003, Thomas et al. 2006). For one isolated magnetic nanotube, magnetization reversal (e.g., the change in the magnetization from one of its energy minima M(r) = M0z to the other M(r) = −M0z) can occur by one of the following three idealized mechanisms (Landeros et al. 2007): coherent mode (C) in which all spins are supposed to reverse their spins coherently and simultaneously; vortex mode (V) with zero net magnetization in a segment of the tube; or transverse mode (T) with a net magnetization component in the (x, y) plane in a segment of the tube as depicted in Figure 14.7. Instabilities arise at any of the ends where surface magnetic moments are weekly coordinated to the rest of the material and propagates from one end to another (or from both ends to the center). In the case of any of the last two mechanisms (T or V), a domain wall appears at any end of the tube and propagates toward the other end. According to the available simulations, the actual mechanism followed to achieve the reversal is strongly dependent on the geometry of the nanotube. The energy cost associated with the reversal for each of the three modes (C, V, and T) can be calculated in a way similar to the energy calculations done to find the phase diagram above (Landeros et al. 2007, Usov et al. 2007). It turns out that the C mechanism is extremely costly in energy and can be left out of the analysis. In the case of the other two mechanisms (V and T), a domain wall is tentatively defi ned and continuity of the magnetic quantities at the borders is imposed (Chen et al. 2007, Landeros et al. 2007, Usov et al. 2007). Energy for each configuration is found. Analytic studies have been complemented by numerical simulations to decide upon which mechanism requires less energy to switch the magnetization

C

L

V

wV

T

wT

FIGURE 14.7 (See color insert following page 20-16.) Magnetization reversal modes in nanotubes. Arrows represent the orientation of magnetic moments within the tube. Left: Coherent-mode rotation, C. Center: Vortex-mode rotation, V, with a domain wall of thickness w V. Right: Transverse-mode rotation, T, with a domain wall of thickness w T.

completely from one end to the other. The reader who is interested in the details of this study is referred to the already quoted literature. Now we go directly to a discussion on the conditions for the prevalence of one mechanism over the other. The results of the simulations indicate that the preferred reversal mode depends on the actual material and on the geometry mainly through the internal and external radius (Landeros et al. 2007, Usov et al. 2007). For each β value, there exists a critical radius, Rc(β), at which the cost in energy is the same for V and T mechanisms. Thus, for R < Rc(β), the tube reverses its magnetization creating a transverse domain wall (T), while for R > Rc(β), a vortex domain wall (V) appears. The locus for the equilibrium curve Rc(β) can be found looking like the lower curve in Figure 14.9. It can be seen that the formation of the vortex domain walls is favored for the dimensions at which magnetic nanotubes are presently obtained. Actually, nanotubes with a radius smaller than a few exchange lengths are very difficult to produce currently (Nielsch et al. 2005a,b, Bachmann et al. 2007, Daub et al. 2007a,b). Thus, the propagation of vortex-like domain walls is the expected reversal process. However, for some particular nanotubes with values of R and β lying close to the boundary line separating the T and V mechanisms (see Figure 14.9), instabilities may appear giving chances to the T mechanism. MC simulations have even shown that in such cases the reversion can begin as a T process, changing to the V mechanism as the domain wall propagates to the center of the tube (Landeros et al. 2007). In any case, this is a field where experimental progress is needed as simulations depend strongly on the assumptions. Thus, for instance, ideal long nanotubes are considered to have anisotropy with the magnetization aligned along the nanotube

14-9

Magnetic Nanotubes

axis. In such models, a coherent reversal of the magnetization would be needed to invert the magnetic polarization. However, such mechanisms are extremely costly in terms of energy. Contrary to such an ideal picture (Stoner and Wohlfarth 1948), magnetic nanotubes tend to exhibit inhomogeneous reversal processes. Micromagnetic and MC simulations provide a more realistic picture to deal with the switching process for real nanotubes, which can still be put in agreement with analytic models (Escrig et al. 2007a, 2008b, Allende et al. 2008). We now proceed to study each reversion mode independently. 14.3.3.1 Transverse Mode In the simple model by Escrig and collaborators (Escrig et al. 2007a, 2008b), the coercive field H nT can be approximated by an adapted Stoner–Wohlfarth model (Stoner and Wohlfarth 1948) in which the length of the coherent rotation is replaced by the width of the transverse domain wall, w. Following this approach, H nT 2K (w) = M0 μ 0 M 02

(14.5)

qJ 0 (q) − J1(q) βqJ 0 (βq) − J1(βq) − =0 qY0 (q) − Y1(q) βqY0 (βq) − Y1(βq)

(14.9)

Here Jp(z) and Yp(z) are Bessel functions of the fi rst and second kind, respectively. Equation 14.8 has an infinite number of mathematical solutions, out of which only the one with the smallest nucleation field has to be considered (Aharoni 1996) for practical purposes. Therefore, the nucleation field depends on α(β), which is related to the internal and external radii of the tube. A crossover between the two reversal modes T and V (that have been previously presented) has been reported (Escrig et al. 2008b). It turns out that the curling mode (V) is more stable for thinner tubes, whereas thicker tube walls favor the transverse mode (T). However, the absolute values are computed for the coercivity by means of equations. Values obtained by means of Equations 14.5 and 14.7 are greater than the experimental data, a discrepancy which can be solved by considering the weak magnetostatic interactions among nanotubes that are present in the experiment but they were left out of the theoretical calculations based on a single nanotube.

14.3.4 Magnetostatic Interactions among Nanotubes

where K (l) =

1 μ 0 M 02 ⎣⎡1 − 3N z (l)⎦⎤ 4

(14.6)

The calculated switching field for an isolated nanotube based on the model of Escrig et al. (2007a, 2008b) is always lower in the T mode than in the Stoner–Wohlfarth approximation (Stoner and Wohlfarth 1948). 14.3.3.2 Curling Mode The curling mode (Aharoni 1996, 1997) is a noncoherent calculation that minimizes the total magnetic energy. When magnetic reversal occurs via curling, a magnetic vortex structure is formed at the nanotube ends and for magnetic tubes with an infinite length, an analytical solution can be calculated (Chang et al. 1994, Escrig et al. 2007a, 2008b). For an infinite tube, the nucleation field for the V mode, H nV, is given by (Chang et al. 1994, Escrig et al. 2007a) H nV L2 = α(β) x2 M0 R

A couple of isolated magnetic nanotubes can attract or repel each other upon approach depending on their relative orientations. Based on the nanotubes reported in the literature, the interacting force has been estimated to be of the order of tens to hundreds of microdynes (Suarez et al. 2009). The performance of such an experiment could provide the basis for testing the different theoretical models at the nanoscale. In the case of nanotubes produced by porous membranes, they are trapped in the array and the interaction in such a triangular lattice (with an overall hexagonal appearance) tends to produce all kinds of orientations, frustrating many local fields and tending to give a null effective magnetization in the absence of external fields (Figure 14.8). In a way, each tube interacts with

(14.7)

s

The function α(β) in previous expressions has been obtained by means of two different methods. Escrig and collaborators (Escrig et al. 2007a) obtained an analytical approach using a Ritz model, which leads to 8 (14 − 13β2 + 5β 4 ) α(β) = 3 (11 + 11β2 − 7β 4 + β6 )

(a)

(14.8)

Equation 14.7 had been previously obtained by Chang and collaborators (Chang et al. 1994) starting from Brown’s equations. They obtained α(β) = q2, where q satisfies the condition

d

(b)

FIGURE 14.8 (a) Relative position of interacting tubes: d is the interaxial distance and s is the vertical separation. (b) Hexagonal array of magnetic nanotubes on a triangular lattice where contributions from the 6 nearest neighbors, the 12 second-degree neighbors, and the tubes situated farthest away from the probe tube combine to form the so-called stray field.

14-10


the stray fields produced by the array, so an effective antiferromagnetic coupling between neighboring tubes is expected, thus reducing the coercive field. In these interacting systems, the process of magnetization reversal can be viewed as the overcoming of a single energy ΔE. In an array with all the nanotubes initially magnetized in the same direction, the magnetostatic interaction between neighboring tubes favors the magnetization reversal of some of them. A reversing field aligned opposite to the magnetization direction lowers the energy barrier, thereby increasing the probability of switching. Escrig and collaborators (Escrig et al. 2008a) have calculated this magnetostatic interaction using a simple mathematical expression. They start by calculating the magnetic field generated by a ferromagnetic tube assuming that it is a continuum material (see Section 14.3.1.1). From the expression for the field, the researchers can calculate if the field is attractive or repulsive at any point in space. Next, they put another magnetic tube near the first one and calculate the interaction energy between them. The expression for the energy they obtained is usually quite complicated and has to be numerically solved. However, assuming that the tubes investigated satisfy R/L 9 × 10−7 Torr and temperatures between 300°C and 380°C. The maximum pressure in these experiments is limited to 10−5 Torr by the design of the TEM. The gases are leaked continuously into the microscope column to ensure a constant pressure during wire growth. Under these conditions, 〈111〉-oriented Ge wires that are several hundreds of nanometers long can be grown for times between 1 and 6 h. TEM images are acquired at video rate (30 frames/s). Wires grown under continuous, as well as intermittent, electron beam irradiation exhibit similar growth rates indicating that the electron beam does not affect the wire growth kinetics. Substrate temperatures are measured before and after the deposition using an infrared pyrometer. After growth, the surface can be cleaned by heating to 1250°C, so that a series of growth experiments can be carried out on the same sample in one area. For such a series, the relative temperature can be measured to within 20 K, while for different samples the measurement uncertainties in absolute temperature are ~50 K. In these experiments, the Ge2H6 pressure range accessible for growth was between 1 × 10−7 and 1 × 10−5 Torr. Sustained epitaxial growth of the 〈111〉-oriented single-crystalline Ge wires was observed at 250°C < T < 400°C. In contrast to Si, Ge wires are bounded by smooth sidewalls. Figure 16.5 is a typical bright field TEM image of Ge NWs obtained during deposition at T = 330°C using 4.8 × 10−6 Torr Ge2H6. Note that the wire tips show smoothly curved catalyst particles indicative of the liquid phase and suggest that the Ge wires grow via the VLS process. Th is is an important observation since TE = 361°C for AuGe alloy (see Figure 16.4) while liquid droplets are observed at T = 330°C. In order to understand the Ge wire growth kinetics, a series of experiments were carried out while systematically varying the substrate temperature and Ge2H6 pressure during growth.

In one such experiment, the effect of the substrate temperature on the AuGe catalyst state was studied during wire growth at a constant Ge2H6 pressure. Solidification of the droplets occurred at temperatures far below (~100 K) TE and required significantly higher temperatures (>400°C) to re-establish the liquid phase. This hysteresis in the solid–liquid phase transformation is seen in all Ge growth experiments and for wires with a range of diameters (20–140 nm). Interestingly, the wires continue to grow even after the catalyst particle has solidified, i.e., via the vapor–solid–solid (VSS) process. Measurements made on several wires showed that VSS growth is 10–100 times slower than VLS growth at the same Ge2H6 pressure and temperature, presumably due to weaker surface reactivity and/or lower diff usivity through the solid. Both VLS and VSS growth were observed to occur simultaneously on neighboring wires in some instances. All the wires, irrespective of the growth mode, are crystalline and the only obvious difference between the growth modes is that VSS process yields more tapered wires owing to their relatively slower growth rates. Th is demonstration of dual growth modes may be relevant to the controversy regarding the role of VSS and VLS growth in other systems (Persson et al. 2004, Harmand et al. 2005). The growth experiments carried out while varying Ge2H6 pressure at a constant substrate temperature showed that a significant Ge2H6 pressure appears to be essential for stabilizing the liquid state below TE. Whenever the Ge2H6 pressure is reduced during VLS growth, the solidification of catalyst droplets was observed (see Figure 16.6). Here, the first image shows a typical VLS-grown Ge wire. In this experiment, growth was initiated using 4.6 × 10−6 Torr Ge2H6 and continued at pressures ≥1.1 × 10−6 Torr for ~78 min. The Ge2H6 pressure was then reduced to 2.8 × 10−7 Torr while maintaining a constant temperature. Within 681 s, the droplet abruptly solidifies. The fact that the droplets can solidify confirms that the temperature is definitely below the bulk TE, independent of any uncertainties in temperature calibration.

35 nm

[1 – 12 ]

[111]

– [110]

t=0s

679 s

681 s

100 nm

FIGURE 16.5 A typical bright-field TEM image obtained from a Si(111) sample during the VLS growth of Ge nanowires. Most wires grow epitaxially in the 〈111〉 direction. (Adapted from Kodambaka, S., Tersoff, J., Ross, M.C., and Ross, F.M., Proc. SPIE, 7224, 72240C, 2009.)

FIGURE 16.6 Representative bright-field TEM image series showing the solidification of AuGe catalyst on top of a Ge wire when the Ge2H6 pressure is reduced during growth at a constant temperature T = 340°C. In this experiment, Ge2H6 pressure was dropped from 1.1 × 10−6 Torr to 2.8 × 10−7 Torr at t = 0. (Adapted from Kodambaka, S., Tersoff, J., Ross, M.C., and Ross, F.M., Proc. SPIE, 7224, 72240C, 2009.)

16-6


(h/d)

0.90

h

0.85

0.80

d 0.75 0

1

2 t (103 s)

3

4

FIGURE 16.7 Plots of aspect ratio (top curve) of a AuGe eutectic droplet on top of a Ge wire (diameter = 59 nm) and Ge2H6 pressure (bottom curve) vs. t at T = 355°C. The data is acquired from a TEM video sequence as the pressure is cycled repeatedly between 1.9 × 10 −6 Torr and 8.4 × 10−6 Torr. Aspect ratio (h/d) is defined as the ratio of the height h and the base width d of the droplet as labeled on the TEM image. (Adapted from Kodambaka, S. et al., Science, 316, 729, 2007.)

This behavior is typical of all droplets observed, although the exact time delay between smaller and larger droplets depends on the growth history and in some cases can be as long as several tens of minutes. In order to understand the role of Ge2H6 pressure on the droplet state, shapes of AuGe droplets were measured during wire growth as a function of Ge2H6 pressure at a constant T. Figure 16.7 is a typical plot of the aspect ratio of a droplet as the pressure is varied repeatedly between higher and lower values. Although the changes are small, note that when the Ge2H6 pressure is decreased, the aspect ratio decreases and when the Ge2H6 pressure is increased, the aspect ratio increases. Clearly, the droplet shape is varying with Ge2H6 pressure, suggesting that there are observable changes in surface energy with pressure (Kodambaka et al. 2007). All the above observations suggest that the liquid phase may be effectively stabilized against solidification by Ge supersaturation, which arises from the growth process (Kodambaka et al. 2007). For more details, the reader is encouraged to refer to the article by Kodambaka et al. (2007). In conclusion, in situ TEM experiments enable the quantitative determination of the NW growth mechanisms. In case of Au-catalyzed growth of Ge NWs, it was found that the AuGe catalyst state may be either solid or liquid below the bulk eutectic temperature, with the state depending not just on temperature but also on the Ge2H6 pressure and history. Remarkably, both VLS and VSS processes can operate under the same conditions to grow Ge wires. Most surprisingly, a significant Ge2H6 pressure is essential for growth via the VLS process below the eutectic temperature. Clearly, in situ observations provide valuable insights into the physical processes controlling the morphological and structural evolution of NWs and are expected to be general and applicable to other material systems. Large-scale fabrication of NW-based devices for the above-mentioned applications requires precise control over NW morphology (shape, length, and size), crystalline structure, chemical composition, and interfacial abruptness. Th is is an

extremely challenging task governed by a complex interplay of the thermodynamics of the materials and the kinetics of nucleation and growth processes. For example, growth orientations of Ge wires can be varied between 〈111〉, 〈110〉, or 〈112〉 by changing the growth temperature, precursor, and the growth technique (Hanrath and Korgel 2005). However, very little is known concerning the factors affecting the wire orientation (Tan et al. 2002, Borgström et al. 2004, Wu et al. 2004a, Schmidt et al. 2005, Wang et al. 2006). Despite several years of research in this area, none of the following has been achieved, even for relatively simple elemental Si and Ge NWs: (1) 〈100〉-oriented Si or Ge wires, (2) Si/Ge heterostructures with atomically-abrupt interfaces (Clark et al. 2008), or (3) large-scale synthesis of sub-5 nm diameter wires. Success in the rational synthesis of NWs with desired architecture can only be achieved through a fundamental understanding of all the processes influencing the nucleation and growth.

16.4 Properties of Ge Nanowires Having looked at the VLS growth of Ge NWs, we turn to their properties and applications. NWs could be the building blocks for the post-CMOS era bottom-up assembly of nano devices. The primary requirements for this new technological era are compatibility with traditional silicon manufacturing processes and integration for cost reduction. For this reason, Ge is the front runner to replace silicon. Ge has a lighter effective mass for the electron and hole charge carriers than Si implying higher carrier mobilities and thus high-performance transistors. Figure 16.8, adapted from the article by Yu (Yu et al. 2006), shows that bulk Ge has direct (~0.88 eV) and indirect (~0.66 eV) band gaps making it an attractive material for electronic and photonic circuitry applications. These band gaps can be further tuned for specific electronic/photonic applications by controlling the wire

300 K

Energy

Ex

ΔE

EΓ2

EΓ1

Eg

Eg = 0.66 eV Ex = 1.2 eV EΓ1 = 0.8 eV EΓ2 = 3.22 eV ΔE = 0.85 eV E∞ = 0.29 eV

E∞

Wave vector Heavy holes Light holes Split-off band

FIGURE 16.8 Band gaps in bulk Ge. (Adapted from Yu, B. et al., J. Clust. Sci., 17, 579, 2006.)

16-7

Germanium Nanowires

dimensions thus giving rise to a myriad of design possibilities. Some of the quantum effects leading to various properties and applications for Ge NWs are discussed here.

16.4.1 Structural Properties For NWs, the surface to volume ratio is very large. Th is leads to the surface reconstruction of the atoms in the NW as seen in Figure 16.9, which shows the simulated Ge NWs with H termination of the structures adapted from Medabonia et al. (2007). After analyzing bond lengths in these structures, Medaboina et al. observed that bond lengths between Ge atoms at the surface were relaxed by ~1%, whereas no relaxation was observed for atoms in the interior region away from the surface. When the NWs were allowed to dimerize, it was observed that the atoms relaxed by ~50% as expected. Cross-sections of wires along [110] were found to have cylindrical structures for all diameters. Along the [001] and [111] axes, the smaller wires with diameters d < dc have circular cross-sections. As seen from Figure 16.9, the critical diameter dc, above which the cross-section acquires a faceted shape, lies in the range of 2.0 nm < dc < 3.0 nm for [001] and [111] axes. The value of dc is determined by the competition to lower the total energy between the energies of the different exposed surfaces of H terminated Ge under the constraint of a fi xed volume. Wires along the [001] direction appear in Figure 16.9 to have a rectangular bonding geometry rather than the expected square surface arrangement of atoms in a single [001] surface layer. This expected square arrangement of atoms of a diamond lattice is not made of the nearest [001]

neighbor atoms. The rectangular structure seen in Figure 16.9 arises because the middle atoms, along the longer side of the rectangle, are the nearest neighbors of the top layer atoms and are one layer below the top layer. For larger diameters, d > dc = 2.15 nm wires along [001], cross-sections were found to be octagonal-shaped with facets of the [001] and [110] type, which were normal to [001]. For larger diameters, d > dc = 2.11 nm wires along [111], they were hexagonal-shaped with facets of the [110] type, which were normal to [111]. Thus, we see that the arrangement of atoms of a NW cut along a particular direction depends on its diameter, its axis orientation, and the termination of its surface. We will see in the following discussion that the properties of a NW are intimately related to its crystal structure.

16.4.2 Electronic Properties and Applications As seen from Equation 16.12, the quantum confi nement of electrons (in the NW cross-section) leads to the difference between successive energy states to increase as NW diameter decreases. This leads to an increase in the band gap between the fi lled and unoccupied electronic energy states, which correspondingly increases with a decreasing diameter. For example, in InP NWs, it was found that the band gap E g varies with diameter d of the wires as Eg ~ 1/d1.45. Simple particle in box type explanations (as shown in Section 16.2) though qualitatively adequate, do not account for this scaling quantitatively (Yu et al. 2003). As expected from previous theoretical analysis, Figure 16.10 shows the dependence of the band gap of hydrogen passivated Ge NWs

[110]

[111]

(a)

(c)

(e)

(b)

(d)

(f )

(Ge-185,H-60) (Ge-69,H-32) (Ge-133,H-40) (Ge-89,H-44) FIGURE 16.9 Cross-sectional views of the relaxed Ge NWs. (a) NW[001] (2.03), (b) NW[001] (3.03), (c) NW[110] (2.12), (d) NW[110] (Ge-170,H-66) (Ge-326,H-90) (3.3), (e) NW[111] (2.11), (f) NW[111] (3.03). Larger circles represent Ge atoms and the outer smaller ones represent the H atoms used to saturate dangling bonds. Local geometrical patterns corresponding to the axis of the wires are evident: square shapes for [001] axis, hexagonal for [110], and parallelograms for [111]. (Adapted from Medaboina, D. et al., Phys. Rev. B, 76, 205327, 2007.)

16-8


value of the valence band maximum decreases in energy and the absolute value of the conduction band minimum increases in energy as the thickness of the wire decreases. This leads to an increase in the electronic band gap with a decreasing NW diameter. The band gap could be either direct or indirect, depending on the crystallographic orientation. NWs along [110] and thin (d < 1.3 nm) ones along [001] have direct band gaps occurring at the gamma point. This is different from bulk Ge band structure as shown in Figure 16.8. Such wires, due to their direct gaps, would be suitable for applications in optics. Wires along [001] were found to transit from direct to indirect band gaps as the diameter increased above 1.3 nm, while all wires along [111] had indirect band gaps. Another important factor used to control the electronic properties of NWs is doping. Dopants of n- and p-type in NWs are required for faster and low power consuming logic devices. Doping has been studied in several NWs and has been used to design inverters, LEDs, and bipolar transistors. Obtaining good electronic properties for n- and p-channel FETs is more complicated (Greytak et al. 2004). Band bending caused by doping Ge NWs has been demonstrated experimentally (Wang and Dai 2006). Figure 16.12 shows the simulation results of band structures of doped ~2 nm diameter Ge NWs for a single wire along each crystallographic direction [001], [110], and [111]. A high level of doping (0.5%–1.5%) obviously has an impact on the electronic structures of Ge NWs. As shown in Figure 16.12, adding a p-type (n-type) dopant moves the Fermi energy toward the valence band (conduction band). The maximum of the valence band and the minimum of the conduction band increase

4.5 [001]-axis [110]-axis [111]-axis

4 3.5

Eg (eV)

3 2.5 2 1.5 1 0.5 0

0.5

1

1.5

2

2.5

3

3.5

d (nm)

FIGURE 16.10 Dependence of band gap on the wire diameter and orientation. (Adapted from Medaboina, D. et al., Phys. Rev. B, 76, 205327, 2007.)

on their diameter and orientation (Medaboina et al. 2007). Band structures for different Ge NWs (shown in Figure 16.9) as calculated by Medaboina et al. are shown in Figure 16.11. As expected, the band structure varies depending on the wire orientation, diameter, and passivation material. The dispersion of the valence band for wires with approximately the same diameter is greatest for wires along [110] and least for wires along [111]. As expected from a quantum size effect, Figure 16.11 shows the absolute

0.1

0.2

0.3

0.4

0.5

(c)

[111] 2.5 2 1.5 Energy (eV)

2.5 2 1.5 1 0.5 0 –0.5 –1 –1.5 –2 –2.5

1 0.5 0 –0.5 –1

0

0.1

0.2

0.3

0.4

0.5

–1.5 (e) 0

2.5

2

2

2

1.5

1.5

1.5

1 0.5 0

1 0.5 0 –1

–1

–1.5

–1.5

0.3

0.4

0.5

(d)

0.4

0.5

0.1

0.2

0.3

0.4

0.5

0

–0.5

0.2

0.3

1

–1 0.1

0.2

0.5

–0.5 –1.5 (b) 0

0.1

2.5

Energy (eV)

2.5

Energy (eV)

Energy (eV)

Energy (eV)

[110]

Energy (eV)

[001] 4 3.5 3 2.5 2 1.5 1 0.5 0 –0.5 –1 –1.5 (a) 0

–0.5

0

0.1

0.2

0.3

0.4

0.5 (f )

0

FIGURE 16.11 Band Structure of Ge NW along [001], [110], and [111] directions. The Fermi level in each panel is set to zero and is shown by the (Ge-25,H-20) (Ge-185,H-60) (Ge-17,H-12) dotted line. (a) NW[001] (1.12) and (b) NW[001] (3.03) represent the band structure along the [001] direction. (c) NW[110] (1.12) and [Ge-326,H-90] (Ge-133,H-40) (Ge-62,H-42) (d) NW[110] (3.3) represent the band structure along the [110] direction. (e) NW[111] (1.23) and (f) NW[111] (3.03) represent the band structure along the [111] direction. (Adapted from Medaboina, D. et al., Phys. Rev. B, 76, 205327, 2007.)

16-9

(g)

2.5 2 1.5 1 0.5 0 –0.5 –1 –1.5 –2 –2.5

0.2

0.3

0.4

0.5

(b)

0

0.1

0.2

0.3

0.4

0.5

(e)

0

0.1

0.2

0.3

0.4

0.5

(h)

2.5 2 1.5 1 0.5 0 –0.5 –1 –1.5 –2 –2.5

2.5 2 1.5 1 0.5 0 –0.5 –1 –1.5 –2 –2.5

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FIGURE 16.12 Comparison of band structures of Ge NWs with and without doping. The Fermi level in each panel is set to zero and is shown by (Ge-89,H-44) (Ge-88,H-44,P-1) (Ge-68,H-32,P-1) (Ge-169,H-66,P-1) the dotted line. (a) NW[001] (2.03), (d) NW[110] (2.12), and (g) NW[111] (2.11) are n-type doped wires. (b) NW[001] (2.03), (Ge-169,H-66,B-1) (Ge-69,H-32) (Ge-170,H-66) (Ge-88,H-44,B-1) (Ge-68,H-32,B-1) (e) NW[110] (2.12), and (h) NW[111] (2.11) are undoped wires. (c) NW[001] (2.03), (f) NW[110] (2.12), and (i) NW[111] (2.11) are p-type doped wires. (Adapted from Medaboina, D. et al., Phys. Rev. B, 76, 205327, 2007.)

(decrease) in energy with the addition of p-type (n-type) dopant when measured relative to the Fermi level. Figure 16.12 also shows that the doping of wires does not have a significant effect on the dispersion of valence and conduction bands. Thus, the effect of doping on the band structure of hydrogenated Ge NWs is similar to that in doped bulk Ge. This property would make Ge NWs suitable for applications that require the tuning of the NW conductivity by doping. Ge NW-FETs are ideal for very-low-power circuit applications. The majority of the published work (Yu et al. 2006) shows that Ge NW-FETs function at nano- to micro-amperes on-state current with a large on-to-off current ratio (10 4–106). The switching energy of a single-NW-FET is 3 to 6 orders of magnitude lower than that of a typical top-down conventional FET. The standby power is almost negligible due to the pA-level off-state leakage per NW. These excellent properties make Ge NW-FETs an ideal nano electric device for micro or nano power

chips with significantly improved power performance tradeoff (Yu et al. 2006).

16.4.3 Optical Properties and Applications Germanium’s compatibility with Group III–V materials and germanium oxide’s optical properties allow for the realization of radical integrated optoelectronic circuitry designs. Photoluminescence (Canham 1990, Duan et al. 2000, Katz et al. 2002) data revealed a substantial blue shift with a decreasing size of NWs. It has also been shown recently that Ge NWs could be used in optoelectronic components fabricated within siliconbased technology (Halsall et al. 2002). Optical properties of NWs also mainly depend on the size of the wire, orientation of the wire (Bruno et al. 2007), and passivation material and doping utilized (Wu et al. 2004b). As explained earlier, the band gap and band structure change with wire size and orientation

16-10

leading to numerous ways of obtaining a particular optoelectronic functionality. The simulation of optical properties of NWs becomes expensive if the proper incorporation of many body effects like self energy, local field, and excitonic effects are taken into account correctly (Bruno et al. 2005). Ge NWs of diameters in the range of 0.8 nm have the main absorption peak in the visible range (Bruno et al. 2007), which could lead to an efficient application of Ge NWs in optoelectronic devices when compared with Si NWs. Already, Ge NW-based nano devices like solar cells, magnets (Alguno et al. 2003), and FETs (Wang et al. 2003) have been characterized.

Handbook of Nanophysics: Nanotubes and Nanowires Lateral deflection Photo diode

Laser

Mechanical stability: The high surface to volume ratio of NWs gives them interesting mechanical properties, as well. For example, simulations on Au NWs show that a spontaneous transition from face centered cubic (fcc) to body centered tetragonal (bct) structure occurs in 〈001〉 oriented gold (Au) NWs or cross-sections less than 4 nm2. The simulations showed no transitions when the wires were oriented along 〈111〉 or 〈110〉 directions (Diao et al. 2003). It may be speculated that similar phenomena driven by surface reconstruction may cause structural phase transitions into the interior of Ge NWs. However, there has been no such observation to date. Mechanical strength: If NWs are to be used as building blocks in nano devices, their mechanical properties need to be characterized for avoiding mechanical failure. With current NW production techniques, NWs can be grown as single crystals with the absence of any structural defects such as point or line defects. This sometimes leads to the high mechanical strength and stiffness of NWs close to their theoretical single crystal limits (Wong et al. 1997, Diao et al. 2003, Kis et al. 2003).This property makes them attractive for use in composites and nano-electromechanical devices. Many test set-ups have been used to characterize the mechanical properties of NWs. Figure 16.13 shows the most commonly used clamped beam experimental set-up for lateral loading of the Ge NW bending test using a atomic force microscope (AFM). Despite the widespread use of the clamped beam configuration, a comprehensive model that accounts for the detailed shape of these force-displacement (F-d) curves over the entire elastic region is yet to be described and validated for NW systems. Heidelberg et al. (2006) provides a method for the complete description of the elastic properties in a double-clamped beam configuration over the entire elastic regime for diverse wire systems. This method can be used to perform a comprehensive analysis of F-d curves using a single closed-form analytical description. It can be applied to extract linear material constants such as the Young’s modulus (E) and to describe the entire elastic range and hence to identify the yield points for dramatically different systems of NWs. The same set-up and theory have been used for tests on Ge NWs (Ngo et al. 2006) of sizes between 20 and 80 nm. These tests revealed that the bulk moduli of NWs is comparable with that of bulk Ge but their mechanical strength

Laser Manipulation path

(a)

16.4.4 Mechanical Properties and Applications

Normal deflection

(b)

FIGURE 16.13 Schematic of the lateral nanowire manipulation experiment setup: (a) before manipulation, (b) during manipulation. The nanowire under investigation is suspended over a trench in the substrate and fi xed by Pt deposits at the trench edges. During the manipulation the lateral and normal cantilever deflection signals are simultaneously recorded. (Adapted from Heidelberg, A. et al., Nano Lett., 6, 1101, 2006.)

is closer to the theoretical value for pure bulk Ge and is substantially greater than the mechanical strength of any whisker semiconductor. Simple bending stress analysis on the clamped– clamped beam system shows a linear variation of displacement with force (Heidelberg et al. 2006). Th is cannot explain the nonlinear behavior seen in Figure 16.13. As the NW is displaced, an axial tensile force is inherently induced due to the stretching of the NW. This force affects the total stress experienced by the NW, leading to an enhancement of its rigidity and thus non-linearity of the F-d curve. The curve fit to the elastic deformation plot using the closed-form solution (Heidelberg et al. 2006) gives a Young’s modulus of 137.6 GPa and a fracture strength of 17.1 GPa. An important observation is that the Ge NW breaks without plastic deformation, revealing its brittle nature.

16.4.5 Surface Chemistry and Applications Two commonly occurring adsorbents for Si and Ge surfaces are hydrogen and oxygen. An adsorbent layer on a surface can significantly affect the surface reconstruction and also lead to faceting or de-faceting transitions of the surface. Density functional theory (DFT) computations (Mingwei et al. 2006) show that the passivation of Ge NWs with more stable ethine reduces the quantum confi nement effects when compared with that of Ge NWs passivated with hydrogen. NWs have been thought to be the active elements in sensing devices because the large surface area is believed to cause the sensitive dependence of electronic properties on changes in surface adsorption by different chemical species (Wang and Dai 2006). Ge NWs along [111], which have facets of [110] and [100] types have been studied (Medaboina et al. 2007) by changing the surface hydrogen concentration. This orientation has the tendency of the unhydrogenated facets to dimerize. Hydrogen atoms were removed from

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Germanium Nanowires

the surface of relaxed wires. Band structures of these wires were found to have electronic states with energies lying in the middle of band gap due to the occurrence of unpaired electrons on the surface of the NW after the removal of H atoms. This change in the electronic properties with a change in hydrogen concentration for wires along [111] could be potentially exploited in sensing applications, due to the disappearance of the band gap, which would cause a significant increase in electronic conduction. Synthesized Ge NWs could be oxidized when exposed to air forming Ge/GeO2 surfaces leading to the high density of surface states causing appreciable Fermi level pinning (Kingston 1957). This oxidation phenomenon is more severe for small wires. So Ge NWs have to be chemically passivated with appropriate adsorbents to avoid the degradation of their electronic properties. Band bending could occur in Ge NWs due to the pinning of Fermi level at surface states (Wang et al. 2004, 2005). Band bending due to surface oxidation of Ge NWs has been explored by Wang and Dai (Wang and Dai 2006).

16.5 Simulation Methods The grand challenge of producing complex nanodevices by elementary nano building blocks still faces several hurdles. NWs due to the nano-length scale along one axis also show properties intermediate between bulk and single atoms. This broken symmetry means quantum mechanical computations are expensive and a multi-scale approach is necessary. A thorough theoretical understanding of the properties of nano structures involves understanding the issues of their symmetry, lower dimensionality, size, and chemical composition. Computational techniques are very useful to deal with the multiplicity of these parameters. Table 16.2 lists some of the simulation methods that have been commonly used to characterize nanostructures. All these theoretical techniques attempt to solve complex body problems arising due to the interaction between different atoms. Moving down the table increases assumptions employed in the theory thus simplifying the model and decreasing the fidelity of the result, the advantage being reduced computational cost. The simulation method that would yield the best results in a reasonable amount of time differs from case to case and is entirely at the discretion of the user. For the interested reader, Martin (2004) gives additional information of these simulation methods. Quantum Monte Carlo (QMC) simulations approximate the many electron solutions by a TABLE 16.2 Hierarchy of Computational Simulation Methods Used for Studying Nanostructures Quantum Monte Carlo (QMC) GW Time dependent density functional theory Time independent density functional theory (DFT) Tight binding (TB) Classical molecular dynamics (MD) Note: Going down the column increases the physical assumptions in the input parameters, but yields computation-time savings leading to large scale simulations.

wave function obtained from Monte Carlo methods. Many types of QMC (Hammond et al. 1992) techniques are available for various applications. These simulations are very expensive and so are limited to few tens of atoms. In DFT, the many-electron system is approximated by functionals that depend on spatial electron density. DFT (Thomas 1927, Hohenberg and Kohn 1964, Kohn and Sham 1965, Parr and Yang 1989, Fiolhais et al. 2003) has been a workhorse in condensed matter physics with comparatively less computational expense than QMC. After the exchange and correlation interaction effects were added to DFT (Kohn and Sham 1965), it has also been used in computational chemistry with improved results. Some of the disadvantages of DFT include the difficulty of accounting for the exchange and correlation effects in strongly correlated materials like high temperature superconductors. Specifically as related to NWs, DFT methods do not account correctly for electron–hole interactions that could become appreciable for small NWs (Bruno et al. 2007). In tight binding (TB) methods (Turchi et al. 1997, Slater and Koster 1954, Ashcroft and Mermin 1976, Goringe et al. 1997), the Hamiltonian of the many-body system is approximated as the Hamiltonian of an isolated center at each lattice point and the resultant atomic orbitals are assumed to become localized within a lattice constant. These assumptions make the computational solution to larger systems with many atoms less expensive. In classical molecular dynamics (MD) simulations, the many atom problem is approximated by a potential or force field between atoms or molecules. MD (McCammon and Harvey 1987, Haile 2001, Oren et al. 2001, Schlick 2002) is generally used when interaction effects between thousands of atoms or molecules are of interest. It is very commonly used in large material systems and bio-molecule (or large protein) simulations.

16.6 Conclusions We have shown a rudimentary theory connecting the physical dimensions of a NW to its properties. We have described stateof-the-art Ge NW growth techniques. TEM characterization of the growth process was reported. A brief overview of the effect of Ge NW diameter, orientation, passivation, and doping on surface reconstruction, band gap, and band structure has been provided. The resulting electrical, optical, mechanical, and surface properties have been explained and their applications have been cited. Different nanoscale simulation techniques used for the numerical analysis of NW were briefly listed. Ge NWs have a considerable potential for next generation microelectronics, photonics, and nanodevices owing to their high carrier mobilities, tenability of their band gaps, and recently discovered high mechanical strength at NW scales. This chapter should serve as a quick reference for Ge NW growth methods, properties, and applications.

Acknowledgments SVK thanks the NSF, DARPA, Wright Center for PVIC from the State of Ohio, Wright Patterson Air Force Base, and Kirtland Air Force Base for funding this work.

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17 One-Dimensional Metal Oxide Nanostructures Binni Varghese National University of Singapore

Chorng Haur Sow National University of Singapore

Chwee Teck Lim National University of Singapore

17.1 Introduction ........................................................................................................................... 17-1 17.2 Controlled Synthesis of 1D Metal Oxide Nanostructures .............................................. 17-2 Vapor Phase Growth • Liquid Phase Growth

17.3 Physical Properties of 1D Metal Oxide Nanostructures ............................................... 17-10 Electrical Properties • Mechanical Properties • Optical Properties • Field Emission Properties

17.4 Concluding Remarks........................................................................................................... 17-21 References......................................................................................................................................... 17-21

17.1 Introduction Metal oxides have numerous technological applications and provide an excellent platform to study various fundamental physical processes and phenomena existing in the material systems. Metal oxides crystallize in a multitude of crystal structures and exhibit diverse properties. For example, several metal oxides have the ability to undergo reversible surface oxidation and reduction processes due to the adsorption of certain specific gases (Henrich and Cox 1994). The adsorption of gas molecules results in band bending at the surface and effectively modifies the surface conductivity. Such a change in the conductivity due to gas adsorption is readily detectable in most of the metal oxides, including SnO2, In2O3, and ZnO. Deviations in the properties of metal oxides due to the adsorption of specific gases render them as potential gas sensors. Transition metal oxides, in particular, are attractive for their range of properties (Rao 1999). This is partly due to the partially fi lled d-orbital and the mixed valency of the constituent transition metal atoms and the defect-induced self-doping capability. Transition metal oxides are potentially useful in a variety of applications including as a catalysis in the petroleum industry, magnetic data storage in information technology, and gas sensing. Surface processes play key roles in various applications of metal oxides. Nanostructures of metal oxides with huge surface areas are therefore valuable for many potential applications. A large surface area of nanostructures could result in the improvement of material functionalities. In addition to the properties that originate from the large surface area, confinement effects in low-dimensional systems are expected to provide additional properties that can be tuned by varying the physical size or shape. The confinement effect occurs when the size of

the nanostructures are comparable to the characteristic length scale of the physical properties of interest (typically in the subten nanometer range). Nanostructures hold great promise in device applications where small size, faster operation, and high density integration is of great importance. In addition to the technological applications, studies on nanometric metal oxide structures may aid in the improvement of our understanding of various fundamental physical phenomena associated with metal oxides. High quality nanostructures of metal oxides with a tailored geometrical size and shape are needed for studying their behavior at the nanometric regime. In general, nanostructures can be fabricated either by lithography based “top down” approaches or self assembly based “bottom up” approaches. For metal oxides, “bottom up” approaches were found to be more effective in craft ing structures at the nanometric regime. Over the years, a variety of metal oxide nanostructures with varied dimensionality and morphologies including nanoparticles (Jun et al. 2006), nanowires (NWs) (Lu et al. 2006, Wang 2007b), nanobelts (Rao and Nath 2003, Wang 2004a, 2007b), nanorods (Patzke et al. 2002), nanotubes (NTs) (Patzke et al. 2002, Bae et al. 2008), core-shell, and other complex hierarchical structures (Rao and Nath 2003, Wang 2004a, Jun et al. 2006, Lu et al. 2006, Wang 2007b) were synthesized by adopting various “bottom up” methods. Once appropriate nanostructures are obtained, one can explore various properties and nanoscale phenomena using these structures. Due to the sizedependent properties, there is a need for the characterization of individual nanostructures. Techniques that allow the manipulation and investigation of properties of individual metal oxide nanostructures are in the forefront of low- dimensional material research. 17-1

17-2

In this chapter, an overview of the research activities on one-dimensional (1D) metal oxide nanostructures is presented. The chapter is organized as follows. After this short introduction, various available methods for the synthesis of 1D metal oxide nanostructures are described. The relative merits and demerits of each synthesis approach to create nanostructured metal oxides are highlighted. Following this, selected physical properties of 1D metal oxide nanostructures are discussed. The techniques adopted for the characterization of individual metal oxide nanostructures are detailed. In addition, properties of metal oxide nanostructures that led to the discovery of various prototype nanodevices are emphasized. This chapter ends with some remarks on the future perspective of 1D metal oxide nanostructures.

17.2 Controlled Synthesis of 1D Metal Oxide Nanostructures The discovery of carbon nanotubes (CNTs) in 1991 (Ijima 1991) and the realization of their amazing physical properties stimulated interest in inorganic nanomaterials as well. Over the years, efficient methods have been established to synthesize metal oxide nanostructures with fine control over their chemical composition, crystal structure, dimensionality, size, and shape. Depending on the medium in which nanostructures are formed, the growth techniques are broadly classified as (1) liquid phase growth and (2) vapor phase growth. In the following section, various synthesis techniques are described with a special emphasis on the new developments in the field.


droplet surfaces. Such adsorbed gas molecules undergo surface and bulk diff usion in the metal droplet and form a eutectic mixture (liquid). As the metal droplet supersaturates with the precursor atoms or compounds, phase segregation occurs, leading to the formation of nuclei at the droplet–substrate interface. Subsequent growth occurs as more and more atoms joined to the nuclei at the liquid–solid interface. The metal droplet functions as a virtual template by promoting crystal growth at the liquid–solid interface and restricting growth in other directions. The metal droplet remains at the tip of the resultant nanostructure and solidifies in the post-growth cooling phase to form a nanoparticle. The appearance of such nanoparticles at the tip of the nanostructures indicates the vapor–liquid–solid (VLS) growth mechanism. VLS routes often promote anisotropic growth leading to the formation of 1D nanostructures. The use of a metal catalyst and the formation of an eutectic mixture largely reduces the activation energy required for the growth of nanostructures via the VLS route compared with noncatalytic growth. Moreover, the growth conditions can be retrieved from the binary phase diagram of the metal component of the targeted nanostructure and the catalyst metal. The diameter of the as-grown structures is largely determined by the size of the metal droplet. For a sustained growth via the VLS route, the stability of the catalyst liquid droplet is essential. Using thermodynamic considerations, the minimum equilibrium size of a metal droplet can be expressed as (Tan et al. 2003, Li et al. 2007b) rm =

17.2.1 Vapor Phase Growth In vapor phase growth, nanostructures are formed from gaseous state precursor reactants. Using vapor phase techniques, highly crystalline, contamination-free nanostructures can be synthesized. The major advantage of vapor phase growth is the feasibility of manipulating and organizing nanostructures during their growth. In addition, hybrid and complicated nanostructures with multiple functionalities can be synthesized by vapor growth techniques. The much needed impurity doping, which is essential for constructing various nanodevices, can be realized using vapor phase growth methods. It is customary to further divide the vapor phase growth techniques in terms of the governing mechanisms. Different vapor phase growth strategies used for the growth of metal oxide nanostructures are elaborated in the following section. 17.2.1.1 Vapor–Liquid–Solid Growth The growth of micron sized whiskers from gas phase reactants on substrates covered with metal impurities was developed more than 40 years ago (Wagner and Treuting 1961, Wagner and Ellis 1964). When metal coated substrates are annealed above a certain temperature, the metal fi lm melts and forms droplets. Due to the high sticking coefficient of the liquid as compared with the solid substrate, the reactant gases adsorb on the metal

2Ω l σ lv , kBT ln S

(17.1)

where Ωl is the volume of an atom in the liquid σlv is the liquid–vapor surface energy kB is the Boltzmann constant T is the temperature S is the degree of supersaturation Th is sets a limit for the smallest achievable size of a nanostructure by VLS growth. In addition, the morphology, density, and size of the 1D structure formed by VLS growth are dependent on the nature of the metal catalysts (Song et al. 2005b). Studies showed that metals that have a low meting point and are oxidation resistant are found to have a better catalyzing capability. For example, Figure 17.1 shows the SEM images of SnO2 NWs grown via the VLS route by using different catalyst metals (Nguyen et al. 2005). Thermal chemical vapor deposition (CVD) technique can be employed to grow nanostructures through the VLS route. In a typical setup, a tube furnace with a vacuum-sealed ceramic tube is used. One end of the ceramic tube is connected to a vacuum pump and the other end is connected to gas cylinders through mass flow controllers. Substrates coated with a catalyst metal thin fi lm are placed inside the tube furnace and heated to form

17-3

One-Dimensional Metal Oxide Nanostructures

Ta

1 μm

W

2 μm

Ir

1 μm

(a)

(b)

(c)

Pt

Au

Al 1 μm

500 nm (d)

1 μm (e)

10 μm (f )

FIGURE 17.1 SEM images of SnO2 NWs grown on α-sapphire substrate via VLS mode using different metal catalysts (as indicated in the top left corner). (From Nguyen, P. et al., Adv. Mater., 17, 1773, 2005. With permission.)

metal droplets. The gas phase precursors are introduced at an optimal flow rate into the tube furnace. The pressure inside the ceramic tube is regulated and controlled. The availability of precursor gases is an issue in the VLS-based nanostructure synthesis. The reactant gases can be produced by evaporating respective metals or metal nitrides in the presence of oxygen (Choi et al. 2000, Guha et al. 2004, Johnson et al. 2006). Yang et al. initiated a method to create the reactant gases using carbothermal reduction of metal oxide powder and successfully synthesized ZnO NW using the VLS route by using gold as the catalyst metal (Huang et al. 2001b, Yang et al. 2002). The ZnO was first reduced by carbon into Zn and CO/CO2 in the high temperature zone of the tube furnace. The Zn metal evaporated and transported to the substrates placed at the low temperature zone. This is followed by a metal catalyst–assisted growth of ZnO NWs. The density and diameter of the as-synthesized NWs is controlled by the thickness of the gold catalyst film. NWs with diameters as small as 40 nm can be synthesized using gold as the catalyst. A number of researchers have utilized the carbothermal reduction–assisted VLS method to synthesize 1D nanostructures of SnO2, Ga 2O3, In2O3, Al2O3, ZnO, and V2O5 (Kam et al. 2004, Rao et al. 2004, Nguyen et al. 2005, Song et al. 2005, Zhang et al. 2005). The laser ablation–assisted VLS growth developed by Lieber’s group was found to be effective for the growth of many 1D semiconductor nanostructures (Morales and Lieber 1998). In laser-assisted VLS growth, a high-intensity laser beam evaporates the target containing NW material and condensates on a substrate with catalyst metal clusters. The laser ablation– assisted VLS growth was used for growing metal oxide nanostructures including In2O3 NWs (Stern et al. 2006) and ZnO NWs (Son et al. 2007). Pulsed laser deposition (PLD) was also used for vaporizing respective bulk materials and subsequent growth of nanostructures by the VLS route (Morber et al. 2006, Son et al. 2007).

In some cases, the VLS growth of nanostructures can take a different route in which the constituent metal of the targeted oxide nanostructure itself functions as the catalyst. The governing mechanism of such growth is usually denoted as a self-catalytic VLS mechanism (Mohammad 2006). In addition to its simplicity, the self catalytic growth avoids the unintentional doping of the nanostructures due to the use of a foreign metal catalyst. Many metal oxide nanostructures such as dentritic ZnO NWs (Fan et al. 2004a), SnO2 NWs (Chen et al. 2003, 2004a), CuO nanofibers (Hsieh et al. 2003a), indium doped tin oxide NWs (Chen et al. 2004c), and Al4B2O9 NWs (Liu et al. 2003b) were synthesized following the self-catalytic growth. Nanostructures of mixed metal oxides or impurity doping can be achieved by choosing a mixture of appropriate source materials or gas phase components (Wan et al. 2006). By a one step evaporation method using a mixture of In and Sn as the source for reactant vapor production, SnO2–In2O3 heterostructured NWs have been produced (Kim et al. 2007). The SnO2 NWs covered with a In2O3 shell were formed most likely due to the difference in the bulk and surface diff usion coefficients of the InOx and SnOx species in the catalyst droplet. The VLS approach has the great advantage of yielding highquality single-crystalline nanostructures. In most occasions, the VLS grown nanostructures are dislocation free. The morphology of the nanostructures formed by the VLS route depends on the selection of catalyst particles, source material, thickness of the catalyst layer, and growth duration (Yang et al. 2002, Ng et al. 2003, Nguyen et al. 2005, Zhang et al. 2007a). By precisely adjusting the catalyst layer thickness, Ng et al. demonstrated the possibility of creating 1D and two-dimensional (2D) ZnO structures on different substrates (Figure 17.2; Ng et al. 2003). The VLS route has the feasibility of manipulating and positioning the NWs during growth (Fan et al. 2006c). For many applications, the proper alignment and precise positioning of nanomaterials is necessary. In the VLS route, aligned

17-4


subsequent growth of nanostructures. Since the gaseous reactants directly condense into solid structures, the governing mechanism is known as the vapor–solid (VS) mechanism. Probability in the formation of nuclei via the vapor–solid process can be expressed as (Blakely and Jackson 1962, Dai et al. 2003)

500 nm

1 μm

(a)

⎛ −πσ2 ⎞ Pn = A exp ⎜ , ⎝ kBT 2 ln α ⎟⎠

2 μm (b)

925°C 1D nanowire VLS growth

Nanowall VLS growth

500 nm (c)

Au surface diffusion and aggregation at a node

(d)

FIGURE 17.2 ZnO nanostructures of different morphologies synthesis via VLS route by choosing appropriate catalyst layer thickness. (a) SEM image of quasi-3D ZnO nanostructures grown on a sapphire using ∼40–50 Å Au thin fi lm as the catalyst. The inset shows a SEM perspective view. (b) ZnO nanowalls on a sapphire substrate. (c) ZnO NWs on a highly ordered pyrolytic graphite substrate using ∼15 Å thick Au fi lm as the catalyst. (d) Schematic illustration showing the growth mechanism of ZnO nanowalls and NWs. (From Ng, H.T. et al., Science, 300, 1249, 2003. With permission.)

nanostructures can be produced by using lattice matching substrates. For example, vertically aligned ZnO nanostructures can be obtained with substrates like sapphire (Huang et al. 2001, Yang et al. 2002), GaN (Fan et al. 2006a), and SiC (Ng et al. 2004). Positioning of the nanostructures can be accurately achieved by the VLS route using various catalyst patterning strategies (Wang et al. 2004). The actual growth mechanism of metal oxide nanostructures using the VLS method is complicated due to the presence of oxygen. The mechanism that governs the growth of metal oxide nanostructures through the VLS route still remains controversial. Nanostructures of ZnO, for example, can form via the VLS route for a broad range of temperatures. The state of the catalyst alloy particle (solid or liquid) during the growth over this entire range of temperature is unclear. Campos et al. proposed a vapor–solid–solid (VSS) growth mechanism instead of the VLS mechanism for the growth of ZnO using gold as the catalyst at low temperatures (Campos et al. 2008). 17.2.1.2 Vapor–Solid Growth The growth of nanostructures from gas phase reactants could be possible even in the absence of any metal catalyst. Gas phase precursor reactants of the targeted nanomaterial are directly adsorbed on the substrates, followed by nucleation and the

(17.2)

where A is a constant σ is the surface energy α is the supersaturation ratio T is the temperature in Kelvin kB is the Boltzmann constant The supersaturation ratio is given by, α = p/p0, with p as the vapor pressure and p0 as the equilibrium vapor pressure of the condensed phase at the same temperature. Similar to the VLS method, the thermal CVD technique can be used for growing nanostructures via the VS route. The source material is normally placed at a high temperature zone of the furnace. The substrates to support the nanostructures were located at a lower temperature zone. The reactant gases were first formed by using techniques such as thermal evaporation (Zhang et al. 1999, Pan et al. 2001, Wang 2003, Zhou et al. 2003a,b, Lilach et al. 2005, Zhou et al. 2005a, Chueh et al. 2006, Zhao et al. 2006) of the respective source materials. Reactants were then transported by carrier gas to the substrate kept at a favorable temperature. The resultant morphology of the nanostructures largely depends on the substrate temperature, processing pressure, carrier gas flow rate, and source material (Wang 2003). Pan et al. reported a versatile approach to create metal oxides in a unique nanobelt morphology by direct evaporation of the respective metal oxide powders without using any metal catalysts (Pan et al. 2001). Despite the crystallographic structure diversity among binary oxides including ZnO, SnO2, In2O3, CdO, and Ga2O3, nanobelts are readily formed via the VS route (Figure 17.3). Using the VS route, nanostructures including ZnO nanotubes (Mensah et al. 2007), ZnO NWs (Umar et al. 2005), and nitrogen doped tungsten oxide NWs (Chang et al. 2007) were also synthesized. The versatility of creating complex hierarchical nanostructures via the VS route has been established. Such capability will facilitate our efforts to achieve high density integration of nanostructure assembly. Following the VS route, ZnO comblike structures (Wang et al. 2003) and three-dimensional (3D) WO3-x NW networks (Zhou et al. 2005b) have been reported. Lao et al. prepared hierarchical ZnO nanostructures on In 2O3 NWs by using ZnO, with In 2O3 and graphite powders as source materials (Lao et al. 2002). The InOx vapors first evaporated and formed In 2O3 NWs on the collector substrates. Then ZnOx vaporized and the secondary growth produced branches on the already existing In 2O3 NW sidewalls. The radial In 2O3–SnO2 heterostructure was also reported by

17-5


(b)

(a)

17.2.1.3 Template-Assisted Growth

[1

00

]

100 010

200 nm (c) 2 μm

200 nm (e)

(d)

200 nm (f )

200

020

0]

[10

2 μm

200 nm

FIGURE 17.3 (a) Metal oxide nanobelts via VS route. (b) SEM and TEM image of the In2O3 nanobelts, respectively. (c) A nanobelt with an abruptly reduced width. (d) SEM image of CdO nanobelts and sheets. (e, f) TEM images and a corresponding electron diff raction pattern of the CdO nanobelts. (From Pan, Z.W. et al., Science, 291, 1947, 2001. With permission.)

Vomiero et al. using the VS approach (Vomiero et al. 2007). In another example, Sun et al., produced SnO2 hierarchical nanostructures in a multi-step thermal evaporation method (Sun et al. 2007). To fabricate vertically aligned structures through the VS route, one can either choose a lattice matching foreign substrate to promote heterogeneous epitaxial growth or a seed layer for homogeneous epitaxial growth (Fang et al. 2007, Li et al. 2007). Figure 17.4 displays the SEM images of vertically well-aligned ZnO NWs on substrates with a ZnO seed layer. The structural characterizations reveal the high crystalline quality of such synthesized NWs. In addition, with modification of the surface roughness of the substrates with nonmatching lattices, one can also effectively improve the alignment of nanostructures via the VS route (Ho et al. 2007). The exact physical mechanism that governs the anisotropic growth of nanostructures via the VS route is not clear. The morphology of the resultant nanostructures is found to be largely determined by the anisotropy in the growth rates of different crystallographic surfaces. Certain crystal surfaces have relatively higher surface energy and tend to grow faster to minimize the total energy of the system resulting in anisotropic crystal growth. In addition, the presence of defects like screw dislocations also facilitates the growth in the VS process.

The use of proper templates to direct crystal growth is a versatile technique for producing monodisperse metal oxide nanostructures. By using the template-assisted approaches, various compositions of materials can be crafted at the nanometric regime. Either negative templates with nanosized pores (e.g., anodic alumina (AAO), track etched polycarbonate fi lms) or positive templates (e.g., NWs and CNTs) can be used as scaffolds to confi ne crystal growth. The use of templates to create oxide nanostructures was first reported in the early 1990s. Early efforts on the template-assisted synthesis of metal oxide nanostructures were focused on CNTs as positive templates (Ajayan et al. 1995). In the CNT templating method, the surface of the CNT is fi rst coated with desired metal oxide. Th is is followed by the removal of the CNT templates either by thermal heating or by chemical means. Recently, many other 1D nanostructures were employed as a template for creating various metal oxide nanostructures. The epitaxial deposition of technologically important mixed oxides such as superconducting YBCO, magnetic LCMO, ferroelectric PZT, and Fe3O 4 on vertically oriented MgO NWs by pulsed laser deposition has been reported (Han et al. 2004). Figure 17.5 describes the experimental procedure and the results of the morphological and structural characterizations of as-obtained MgO/YBCO core-shell structures. A similar approach was used for the creation of MgO/titanate hetrostructures (Nagashima et al. 2008). The NW templating method can be effectively used for the realization of nanostructures of structurally complicated multinary metal oxides by various solid–state reaction mechanisms (Chang and Wu 2007, Yang 2005, Fan et al. 2006b). Well-aligned β-Ga 2O3 NWs were coated with ZnO using the metal organic CVD technique and subsequent annealing at 1000°C in an O2 atmosphere produced Ga2O3/ZnGa2O4 core-shell NWs, single-crystalline ZnGa 2O4 NWs, and ZnGa2O4 NWs inlaid with ZnO nanocrystals (Chang and Wu 2005). Spinel Zn 2TiO4 NWs were synthesized by coating ZnO NWs with Ti and subsequent annealing at 800°C in a low vacuum condition. Heating causes a solid–state reaction via diff usion of Ti atoms into the ZnO leading to the phase transformation from wurtzite ZnO to spinel Zn2TiO4 (Yang et al. 2007). In another example, depositing Al2O3 on ZnO NWs using the atomic layer deposition (ALD) technique and subsequent annealing of the resultant ZnO–Al2O3 core-shell structures produced spinel ZnAl2O4 nanotubes by using the nanoscale kirkendall effect (Fan et al. 2006b). 17.2.1.4 Direct Growth by Solid–Vapor Interaction Whisker or needle-shaped metal oxide structures have drawn the attention of the scientific community as early as the 1950s (Cowley 1954, Takagi 1957). Microscopic studies on pure metal pieces thermally oxidized in air or in an oxygen environment revealed the presence of whiskers grown perpendicular to the metal surface. The anisotropic growth of oxides along a certain crystal axis is believed to be due to the presence of screw

17-6


Count number

150 100 50 0

200 nm

100 150 200 250 Diameter (nm)

Acc.V Spot Magn Det WD 25.0 kV 3.0 10000× SE 8.0


2 μm

(a)

500 nm

(b)



10 μm

(c)

2 μm

(d)

FIGURE 17.4 SEM images of vertically aligned ZnO nanorod arrays using ZnO seed layer. (a and b) Top view. (c and d) 50° tilted view. (From Li, C. et al., J. Phys. Chem. C, 111, 12566, 2007a. With permission.)

(a)

Intensity (a.u.)

MgO (200) (006)

300 nm (b)

(003)

(e)

(005)

d1 = 4.9 nm d2 = 2.75 Å YBCO

(004)

20

(110)

30

(200)

(007)

40 50 60 2θ (degree)

(009)

70

(c)

YBCO + MgO

50 nm

80 (d)

(f )

FIGURE 17.5 NWs as templates to form core-shell nanostructures. (a) Schematic illustration of vertically aligned MgO NWs and core-shell NWs consisting of MgO NWs coated with the desired material by PLD process. (b) SEM image of YBCO NWs after the PLD process. (c) XRD data of as-grown YBCO NWs on a MgO (100) substrate. (d) Low magnification TEM image showing the MgO/YBCO coreshell structure. (e) TEM image of MgO/YBCO NWs. (f) HRTEM image of the MgO/YBCO NW. (From Han, S. et al., Nano Lett., 4, 1241, 2004. With permission.)

dislocations (Cowley 1954). Another type of whiskers with pores along their axes was grown when beryllium metal was heated in a silica furnace tube in hydrogen with a trace of water vapor (Edwards and Happel 1962). In this particular growth mode, a metal ball always appeared at the tip of the whisker. By heating a

W foil that is partly covered by a SiO2 plate in the Ar atmosphere at ∼1600°C, Zhu et al. observed the formation of tungsten oxide tree-like microstructures with nanoneedle branches (Zhu et al. 1999). Gu et al. reported the formation of tungsten oxide NWs on W wires/foil by heating in an Ar atmosphere (Gu et al. 2002).

17-7


They suggested that the mechanism of formation of NWs on the clean metal surface may be governed by the VS mechanism. Recently, a number of reports presented the direct growth of metal oxides on the respective metal surfaces heated at the right conditions (Dang et al. 2003, Fu et al. 2003, Xu et al. 2004. Wen et al. 2005, Rao et al. 2006, Varghese et al. 2008b).

sub-oxides of the metal. Th is is followed by the nucleation of the most stable oxide phase. Surface diff usion of the constituent atoms towards the nucleated crystals fueled the growth of the nanostructures. The morphology of the resultant nanostructures is most likely controlled by kinetic factors. 17.2.1.4.2 Plasma-Assisted Direct Growth

17.2.1.4.1 Hotplate Method We have developed a simple yet efficient method of growing metal oxide nanostructures in large quantities by heating metal foils in ambient conditions on a thermal hotplate. Using this technique, vertically oriented α-Fe2O3 nanoflakes (Zhu et al. 2005, Yu et al. 2006), Co3O4 nanowalls (Yu et al. 2005c), CuO NWs (Yu et al. 2005b), and CuO–ZnO hybrid nanostructures (Zhu et al. 2006) have been synthesized. The coverage and size of the hotplate grown nanostructures can be controlled by varying the growth time as shown in Figure 17.6 (exemplified by the growth of α-Fe2O3 nanoflakes). Surprisingly, the hotplate method produced metal oxide nanostructures with high crystalline quality. Being at a low temperature (200°C–550°C) and a catalyst-free method, the hotplate technique is particularly attractive. The morphology and size of the nanostructures can be controlled by simply varying the growth duration (Yu et al. 2005). The hotplate method for the direct growth of nanostructures on metal foils has advantages of low cost and large scale production. The mechanism that governs the direct growth of oxide nanostructures on respective metal substrates is not well understood. Yu et al. proposed a solid–liquid–solid (SLS) mechanism for the direct growth (Yu et al. 2005c). Due to heating, surface melting occurs even at low temperatures. The adsorbed oxygen atoms from the surroundings react with the melt forming various

(a)

17.2.2 Liquid Phase Growth Metal oxide nanostructures with controlled size, shape, and structure can be synthesized by solution-based methods using

(b)

1 μm (d)

Plasma-assisted direct growth is another feasible direct growth method for producing nanostructures of low melting metal (for example, gallium) oxides in large quantities. Sharma et al. synthesized β-gallium oxide tubes, NWs, and nanopaintbrushes by heating molten gallium in microwave plasma containing mono-atomic oxygen and hydrogen mixture (Sharma and Sunkara 2002). Using the plasma-assisted direct growth technique, Varghese et al. have demonstrated the feasibility of tailoring morphology of metal oxide nanostructures (Varghese et al. 2007, 2008a). Here, the respective metal foils were heated in the presence of oxygen at low pressure conditions in a vacuum chamber. The chamber is equipped with a RF plasma generator. By varying the plasma power, vertically oriented Co3O4 NWs, nanowalls, or a mixture of these two structures were successfully synthesized (Figure 17.7). Similar techniques can be used for the creation of NiO nanowalls and nanoflake-like morphologies. Such control over the morphology of the nanostructures is believed to be due to the variation in the rate of oxide formation with plasma power.

(c)

1 μm

1 μm (e)

(f )

FIGURE 17.6 Synthesis of α-Fe2O3 nanoflakes on Fe foil by the hotplate method. (a) Optical image of the Fe foil before heating. (b and c) Optical images of the Fe foil after heating at 300°C for 10 min and 24 h, respectively. (d–f) Corresponding SEM images of the foil surfaces shown in (a–c). (From Yu, T. et al., Small, 2, 80, 2006. With permission.)

17-8


1 μm

1 μm (a)

(b)

1 μm (c)

100 nm (d)

FIGURE 17.7 (a and b) SEM images of vertically aligned cobalt oxide NWs grown on cobalt foil, by heating Co foils in the presence of oxygen. (c and d) SEM images of cobalt oxide nanowalls grown by heating the Co foils in an oxygen plasma. (From Varghese, B. et al., Adv. Funct. Mater., 17, 1932, 2007. With permission.)

relatively simple laboratory equipments. In solution-based methods, metal precursors are dissolved in appropriate solvents and the nucleation and growth of the nanostructures are controlled by the degree of supersaturation, temperature, pH value, etc. Due to the large surface energy associated with the nano entities, suitable surfactants are employed to stabilize and/or direct the growth of nanostructures. The solution-based growth can be broadly classified as (1) aqueous, (2) nonaqueous, and (3) template-assisted solution routes. In the aqueous solution process, appropriate metal salts are dissolved in water and the oxide is formed when the mixture is heated at certain optimum temperatures. On the other hand, the nonaqueous solvothermal process involves mostly organic solvents as the growth medium. The shape and morphology control is achieved by properly selecting the growth parameters or using appropriate surfactants/ligands to direct the growth. In the template-assisted method, the 1D growth control is achieved by confi ning the growth inside nanopores or channels. A brief note on these techniques is provided in this chapter. 17.2.2.1 Aqueous Solution Route The most common solution-based synthesis of metal oxide nanostructures is the aqueous solution method in which the chemical reaction leading to the formation of nanostructures takes place in the presence of water. Nanostructured metal oxides, particularly transition metal oxides, in the form of spherical or faceted nanoparticles (Seshadri 2004) to highly anisotropic NWs or nanotubes (Vayssieres et al. 2001a,b, Vayssieres 2003) have

been synthesized via the aqueous solution method. Normally, metal alcoxides or metal halides are used as the metal precursors. The purity as well as crystal quality of nanostructures can be improved by carrying out the synthesis at elevated temperatures. Such reactions can be conducted in a closed container like autoclave at high pressure. The high pressure allows the reaction temperature to be higher than the boiling point of water. The reaction in such a closed system at high temperatures above the boiling point of solvents is known as the solvothermal process. If the solvent is water, it is called a hydrothermal reaction. For a more detailed description on the hydrothermal synthesis of metal oxide nanostructures, please refer to recent reviews by Mao et al. (2007). 17.2.2.2 Nonaqueous Solution Route Recently, the nonaqueous solution route to synthesize crystalline metal oxide nanostructures has attracted much attention (Niederberger 2007). In the nonaqueous solution route, the growth medium is usually organic solvents. Park et al. demonstrated the feasibility of using the nonaqueous method to synthesize ultra large scale metal oxide nanocrystals (Park et al. 2004). Nanostructures of mixed metal oxides, which are otherwise hard to synthesize, can be controllably produced using nonaqueous solvothermal routes. O’Brien et al. reported a generalized method for synthesizing complex oxides like BaTiO3 nanoparticles using an “injection-hydrolysis” protocol (O’Brien et al. 2001). In a typical experiment, barium titanium ethyl hexano-isopropoxide is injected into a mixture of

17-9


biphenyl ether and stabilizing agent oleic acid at 140°C under argon or nitrogen. The mixture is cooled to 100°C and a 30 wt% hydrogen peroxide solution is injected through the septum (vigorous exothermic reaction). The solution is maintained in a close system and stirred at 100°C over 48 h to promote further hydrolysis and crystallization of the product in an inverse micelle condition. Size control is achieved by varying the reagent concentration. Using this technique, ferroelectric BaTiO3 nanoparticles 6–12 nm in size were synthesized. Single crystalline perovskite nanorods of BaTiO3 and SrTiO3 were synthesized by the decomposition of bimetallic alkoxide in the presence of coordinating ligants (Jeff rey 2002). The reaction carried out at a temperature of ∼100°C in a mixture of heptadecane, H2O2, and oleic acid. The anisotropic growth is attributed to the precursor decomposition and crystallization in a structured inverse micelle medium formed by precursors and oleic acid under these reaction conditions. Size and shape control in solution routes can be achieved by employing various strategies. An efficient means of obtaining nanostructures with a uniform size is through the Ostwald ripening process (Zeng 2007). The Ostwald ripening process occurs during the aging of the nanomaterial suspension by which the growth of bigger structures is facilitated at the cost of smaller ones due to size-dependent dissolution. Considerably narrow size distribution on the nano-products can be achieved by the separation of the nucleation and growth process. A principally different approach to form anisotropic nanostructures including metal oxides is the so-called oriented attachment of nanoparticles during aging (Penn and Banfield 1998, Pacholski et al. 2002). The oriented attachment refers to the process in which adjacent particles spontaneously selforganized to share a common crystallographic orientation (Penn and Banfield 1998). Metal oxide nanostructures with complex morphologies can be produced using the oriented attachment mechanism (Zitoun et al. 2005). 17.2.2.3 Template-Assisted Liquid Phase Growth The use of nano-porous materials as a host for the synthesis of nanostructures was pioneered by Martin’s group (Martin 1994, Parthasarathy and Martin 1994). Early attempts were focused on the synthesis of metals and conducting polymer structures via the template-assisted method. The fi lling of nanopores of the negative templates by means of solution-based techniques is a feasible way to synthesize metal oxide nanostructures. The subsequent removal of the template by selective etching yields nanostructures. As mentioned earlier, the most common negative templates used for the growth of nanostructures are AAO and track-etched polycarbonate. The AAO templates can be produced by anodizing pure Al foils in various acids. AAOs have high chemical, thermal, and mechanical stability, which makes them an ideal template for nanofabrication. Porous AAO templates with high nanopore density, in various pore sizes are now available. The solution-based fi lling of the template pores is either achieved by sol–gel chemistry or electrochemical route.

17.2.2.3.1 Sol–Gel Processing The template-assisted sol–gel chemistry route is a viable method for producing nanostructures of many chemical compositions. In the sol–gel technique, a suspension of the colloidal sol of the materials was fi rst prepared by the hydrolysis and polymerization of precursor molecules. Either inorganic metal salts or organic metal alkoxides can be used as precursors. The subsequent condensation of as-prepared sol yields the gel. The pores of the templates can be fi lled by the as-prepared sol by direct infi ltration due to capillary action or the electrophorectic method (Shankar and Raychaudhuri 2005). The template-assisted sol–gel technique is particularly useful for many materials to be sculptured into 1D nanostructures (NWs or nanotubes) using the appropriate processing conditions. Lakshmi et al. fi rst extended the template-assisted sol–gel method to produce an array of 1D metal oxides (Lakshmi et al. 1997). They demonstrated the feasibility of using the AAO template to create 1D nanostructures of TiO2, ZnO, and WO3 by the direct immersion of the template in respective sols prepared using the sol–gel chemistry approach. Due to the capillary action, the sols fi ll the pores of the AAO template. Heat treatment and the subsequent removal of the template by dissolution in aqueous NaOH solution yields the respective 1D nanostructures. The end-products can be nanotubes or NWs depending on the immersion time and sol temperature. Following this pioneering work, numerous materials were engineered into nanometric structures using the template-assisted sol–gel route. One of the advantages of using the well-developed sol–gel chemistry method is the possibility to control the stoichiometry of complex multi-component oxides that are not straightforward or impossible to achieve via vapor phase techniques (Zhou and Li 2002, Jian et al. 2004, Yang et al. 2006b). Recently, Kim et al. reported the preparation and ferroelectric properties of ultra-thin walled Pb(Zr,Ti) O3 (PZT) nanotube arrays by using the AAO template-assisted sol–gel process (Kim et al. 2008). In their work, the infi ltration of AAO nanopores was facilitated by spin coating. The schematic of the processing steps, images of the template, and products are shown in Figure 17.8. 17.2.2.3.2 Electrochemical Deposition The electrochemical deposition in conjunction with templates is a viable low temperature method for the production of various metal nanostructures. The deposition is normally carried out in a conventional three electrode electrochemical bath with the template to be deposited configured as the cathode. Since most of the templates are insulating, a metal coating on one of the surfaces is essential in order to use them as electrodes. The salt solution of the metal to be deposited was used as the electrolyte. The production of metal oxide nanostructures by the electrochemical deposition route can be realized by either direct oxide deposition (Hoyer 1996, Zheng et al. 2002, Oh et al. 2004, Takahashi et al. 2004) or by adopting a post-oxidation protocol on the electrochemically deposited metal nanostructures (Yi et al. 2008).

17-10


Pb

Zr

Ti

Distance (nm) 0

100

200

300

50 nm

Spin-coating (b)

Intensity (a.u.)

PZT sol–gel drop on the PAM

PAM on the custom support Annealing 50 nm (d)

(c) PZT–NTs in the PAM Selective etching

100 nm (a)

PZT–NT arrays

(e)

1 μm (f )

FIGURE 17.8 (a) A schematic illustration of the PZT–NT arrays synthesis procedure using template-assisted sol–gel route. (b) SEM image of a typical porous alumina membrane with a corresponding cross-sectional image shown in the inset. (c) SEM image of a alumina template after spincoating with a PZT sol–gel solution. (d) The lower part of this figure is a STEM image and the upper part is the EDS line profi le along the green line in the STEM image. The dotted red line highlights the periodic intensity of Pb, Zr, and Ti in the sectioned PZT–NTs. (e and f) SEM image of PZT–NTs after wet chemical etching for 15–30 min, respectively. (From Kim, J. et al., Nano Lett., 8, 1813, 2008. With permission.)

17.3 Physical Properties of 1D Metal Oxide Nanostructures Electrical properties of low-dimensional nanostructures show deviation from their bulk form. The variation in the electronic properties of materials with dimensionality can be explained on the basis of the difference in the electronic density of states (Yoffe 2002). In general, the density of states, ρ ∝ ED/2−1, where E is the energy and D is the dimensionality (3, 2, or 1 depending on whether it is 3D, 2D, or 1D). In addition, the spatial confi nement in nanostructures causes a blueshift in their band gap with a reduction in size. The shift in band gap of nanostructures, ΔE g ∝ 1/d2 , where d is the characteristic size of the nanostructure. Due to the varied degree of confi nement, the band gap shift evolves differently with size in nanostructures of different dimensionality (Yoffe 2002, Yu et al. 2003). The evolution of band gap with size and dimension is displayed in Figure 17.9 (Yu et al. 2003). Here, the band gap energy was estimated from the energy states of electron and holes calculated using the simple particle-in-a-box model. Thus, the

ΔEg

17.3.1 Electrical Properties Dots

Wires Wells

1/d 2

FIGURE 17.9 Predictions of band gap variation with size for 2D (wells), 1D (wires), and 0D (dots) materials. (From Yu, H. et al., Nat. Mater., 2, 517, 2003. With permission.)

electric transport properties of nanostructures are expected to be dependent on the characteristic size and shape. The electrical transport properties of nanostructures can be sensitively affected by the large electron scattering at the

17-11


boundaries (Zhang et al. 2000). The large surface scattering results in an increase in the resistivity of the nanostructures compared with the bulk materials. This makes the electrical properties of the nanostructures sensitive and dependent on the surface and surrounding medium (Nanda et al. 2001). Metal oxides exhibit the whole spectrum of conductivity, which ranges from metallic through semiconductor to insulators. The electrical properties of metal oxide nanostructures are particularly interesting as one can follow how the myriad of electrical phenomena observed in bulk evolve with size and shape at the nanometric regime. Over the years, research efforts have been focused on the transport properties of metal oxide nanostructures. In particular, the transport properties of 1D metal oxides have attracted tremendous attention owing to their possible dual role as functional electronic components as well as interconnects. An overview of the electric transport properties of metal oxide 1D nanostructures is given in the following sections. 17.3.1.1 Electrical Properties of 1D Metal Oxides The electrical transport properties of 1D structures are usually determined by performing two-probe or four-probe measurements by laying them across metal electrodes with a few micron gap. The nanostructures are first dispersed on a substrate and then suitable metal electrodes are deposited using appropriate masking techniques. Alternatively, one can deposit the nanostructures on substrates that are pre-patterned with metal electrodes. In some cases, aligning strategies were employed to precisely place the nanostructures across the electrodes (Smith et al. 2000, Hu et al. 2006). Studies suggested that for most of the nanostructured metal oxides investigated, the electrical properties resemble their bulk material (semiconducting, metallic, or insulating). However, significant quantitative variations from the bulk material properties are often observed. The I–V characteristics of CdO nanoneedles (Liu et al. 2003a), RuO2 NWs (Liu et al. 2007b), and ITO NWs (Wan et al. 2006) showed metallic behavior. Whereas ZnO NWs (Kang et al. 2007), SnO2 NWs (Ramirez et al. 2007), VO2 nanobelts (Liu et al. 2004), V2O5 NWs (Muster et al. 2000), W18O49 NWs (Shi et al. 2008), and Nb2O5 NWs (Varghese et al. 2009) exhibited semiconducting I–V characteristics. A semiconductor NW with metal pads on either side can be modeled as a metal–semiconductor–metal circuit. Due to the work function difference between the NW material and the metal pads, the two contacts form Schottky-type barriers. This circuit is equivalent to two Schottky diodes connected back to back through a semiconductor (Zhang et al. 2006a). The I–V characteristics of such circuits at an intermediate bias condition are determined by the reverse biased Schottky junction. Assuming the thermionic field emission theory for the reverse biased Schottky junction, the current–voltage relationship for such circuits at intermediate bias conditions can be expressed as (Zhang et al. 2006, 2007b) ⎛ e 1 ⎞ ln I = ln(S ) + V ⎜ − ⎟ + ln J s , kT E 0 ⎠ ⎝

(17.3)

where S is an area factor e is an electronic charge k is the Boltzmann constant (1.38 × 10−23 m2 kg s−2 K−1) T is the temperature Js is a slowly varying function of the applied voltage E 0 is a function of the carrier density (n) and is given by the equation E 0 = E 00 coth(eE00/kT), where E00 = (h- /2)(n/(m* εs ε0))1/2 , with m* as the effective mass of electrons in the NW material, εs is its - is relative permittivity, ε0 is the permittivity of free space, and h the Planks constant. This implies that the slope of ln(I) versus the voltage curve can be approximated to e/kT − 1/E0. Through this analogous, we can retrieve the various electronic transport parameters of the NWs from their respective I–V curves. According to this theoretical formulation, Zhang et al. estimated the carrier density in ZnO NW to be ∼1.0 × 1017 cm−3, which is of the same order of magnitude as the bulk ZnO (Zhang et al. 2006a). 17.3.1.2 Nanowire Field Effect Transistors Quantitative information regarding the carrier type (electron or holes), carrier density, and carrier mobility could be retrieved from measurements on a NW designed in a three terminal device configuration. The working principle is analogous to the field effect transistor (FET) in the micro electronic industry. Typically, NWs are dispersed on a degenerately doped Si substrate with an SiO2 over-layer. Th is is followed by patterned electrode fabrication using lithography techniques. More advanced metal deposition techniques using electron beam lithography or focused ion beam (FIB) techniques can be used to selectively deposit metals to perform single NW level measurements. The metal contacts on either side of the NW could function as source (S) and drain (D) electrodes and the bottom Si bulk can be used as the gate electrode. At the moderate doping level, the Debye screening length (λd) of most of the metal oxides is in the range of 10–100 nm (Kolmakov and Moskovits 2004). Th is implies that one can control the current through the NW by varying the gate voltage (Vg). In such configurations, the total charge on the NW can be expressed as (Martel et al. 1998) Q = CVgT, where C is the NW capacitance with respect to the back gate and VgT is the threshold gate voltage required to completely deplete the carriers from the channel. By assuming the NW as a metallic cylinder, C=

2πεε 0 L , ln(2h / r )

where L is the NW length r is its radius h is the thickness of the SiO2 layer ε is the average dielectric constant of the NW material

(17.4)

17-12


Nanowire Source Drain SiO2 (100 nm) P++ Si back gate

1.0 μm (a)

1.0 μm

(b)

Ids (μA)

2

Ids (μA)

30

(c)

Vds= 1 V

1

Vg = 30 V

0 –20 –10 0 10 20 Vg (V)

20

25 V

20 V 10

15 V 10 V

0

0.5 V 0

5

(d)

10

15

20

Vds (V)

FIGURE 17.10 (a) FESEM image of as grown ZnO NWs. (b) Schematic of a ZnO NW FET. (c) FESEM image of a ZnO NW across metal electrodes. (d) Typical Ids–Vds curves for different gate biases from 0 to 30 V with 5 V step. The inset shows Ids–Vg curve measured at Vds = 1 V. (From Maeng, J. et al., Appl. Phys. Lett., 92, 233120, 2008. With permission.)

The carrier density is given by n = Q/eL cm−1. The carrier mobility (μ) can be determined according to dI/dVg = μ(C / L2)V, where I and V are the source–drain current and voltage, respectively. The FET performance of many semiconducting metal oxide nanostructures has been reported recently. A single ZnO NW FET, for example, was reported by many researchers (Fan et al. 2004b, Chang et al. 2006). By appropriately doping the NWs, the FET characteristics of ZnO NWs can be tuned (Yuan et al. 2008a,b). Recent studies revealed that the performance of single ZnO NWs largely depends on their surface microstructures (Hong et al. 2008). Maeng et al. fabricated a FET using ZnO NWs and its performance was evaluated under different environments (Maeng et al. 2008). The SEM image of the ZnO NW arrays, the schematic of the single NW FET design, the SEM image of a typical NW FET, and the performance of the as fabricated device are shown in Figure 17.10. Improved device performance was achieved by fabricating the vertical surrounded-gate FETs ZnO NWs (Ng et al. 2004). This design facilitates high density integration and eliminates the alignment and lithographic issues associated with the horizontal single NW-based FETs.

17.3.1.3 Conductometric Nano Sensors Most of the commercially available gas sensors contain doped or pristine metal oxides. Th is is due to the selective adsorption of specific analyte molecules on certain metal oxide surfaces (Henrich and Cox 1994). As a result, its properties modify and provide quantitative information on the presence of the analyte molecules. If the sensor functions on the basis of the electrical conductivity changes of the active material, it is called a conductometric type sensor. Presumably, the huge surface fraction of the nanostructures enhances the sensing capability of the metal oxides compared with the coarse-grained polycrystalline bulk materials (Kolmakov and Moskovits 2004). In addition, nanostructures having reduced defects and free of dislocations will improve the stability and performance in sensing applications (Comini 2006). The conductivity of the ZnO NWs was found to be highly sensitive to UV light (Kind et al. 2002, Li et al. 2004). The ultraviolet photoconductivity of ZnO was found to be significantly enhanced when the size was reduced to the nano regime (Kind et al. 2002). Such variations in the electrical conductivity are attributed to the desorption

17-13


of the adsorbed oxygen species from the surface of the NW. Th is effect can be utilized for the fabrication of ultrafast optical switches and photodetectors. 17.3.1.4 Nanowire FET Sensors The FET characteristics of metal oxide nanostructures are found to be highly dependent on the surrounding medium. ZnO NW FETs are sensitive to oxygen partial pressures (Maeng et al. 2008). Individual and multiple In2O3 NWs in the FET configuration are found to be sensitive to NO2 gases at the ppb level (Zhang et al. 2004). The sensing properties of NW FETs depend on the doping level of the NWs as well. Zhang et al. reported the sensing capability of single In2O3 NW transistors for the detection of NH3 gas (Zhang et al. 2003). Sysoev et al. demonstrated an electronic nose device based on conductivity measurement on an array of three kinds of metal oxide (Ni surface doped and pristine SnO2, TiO2, and In2O3) NWs in a single chip to selectively detect the presence of H2 and CO gases in an oxygen environment (Sysoev et al. 2006). The electrical properties of metal oxide nanostructures are dependent on various factors such as size, defects and microstructures, surface properties, and environment in which the measurements are carried out. Due to these multiple factors, electrical properties of the same kind of NWs reported by different researchers showed large inconsistencies (Schlenker et al. 2008).

17.3.2 Mechanical Properties The theoretical calculations and subsequent experimental verification of the ultrahigh strength of CNTs have stimulated intensive research on the mechanical properties of nanosized structures (Ebbesen 1994, Treacy et al. 1996, Wong et al. 1997). Different from bulk materials, the mechanical properties of many nanostructures vary as a function of their characteristic size. Such size effect has great importance from both a fundamental as well as a technological point of view. Particularly, studies on the mechanical properties of nanostructures will provide greater insight into the fundamental mechanism of material deformation and failure. Due to their small size, the experimental characterization of the mechanical properties of nanosized structures proves to be challenging. The challenges include their manipulation, application of force, and the measurement of the corresponding deformation. Accuracy in the range of nano-Newton in force and nanometer in deflection measurements are required to extract elastic constants of the nanostructures. Recently, direct bending or indentation techniques using an atomic force microscope (AFM) were developed to characterize the mechanical properties of NWs/NTs (Wong et al. 1997, Salvetat et al. 1999, Kis et al. 2003, Zhu et al. 2007). Another way to extract the elastic constants of NWs/NTs is to excite them into mechanical resonance vibration inside electron microscopes (Treacy et al. 1996, Poncharal et al. 1999, Yu et al. 2000).

17.3.2.1 AFM-Based Techniques AFM provides exceptionally high precision in force and deflection measurement at the nano-Newton and nanometer level, respectively. The nanoscale three-point bend test, lateral force microscopy, or the nanoindentation test can be performed using an AFM on a nanostructure to extract its elastic constants. A brief overview on the experimental strategies developed using an AFM for mechanical testing is discussed below and the obtained results on various metal oxide nanostructures are highlighted. 17.3.2.1.1 Nanoscale Three-Point Bend Test The nanoscale three-point bend test can be performed using an AFM on suspended NWs across the trenches fabricated on hard substrates like Si. An AFM cantilever of an accurately calibrated force constant is used for applying a normal force on the midpoint of the suspended NW. From the recorded force–distance curve (vertical deflection of the cantilever versus Z-piezo position), the force and the corresponding deflection on the NW can be estimated (Tombler et al. 2000). By assuming the suspended NW as an end clamped cantilever beam, the Young’s modulus is given by Y=

FL3 , 192δI

(17.5)

where F is the force L is the suspended length of the NW δ is the deflection I is the second moment of area of the NW For a NW with a cylindrical cross-section, I = πd4/64, with d as the diameter. Tan et al. reported the elastic properties of CuO NWs by the nanoscale three-point bend test using AFM (Tan et al. 2007). The effects of crystallinity, surface properties, and size of the CuO NW on its elastic constant were discussed. Following similar methodologies, Cheong et al. observed a size-dependent Young’s modulus of WOx NWs (Cheong et al. 2007). 17.3.2.1.2 Lateral Force Microscopy One of the early works on the mechanical characterization of 1D structures was based on quantifying the lateral force signal obtained while an AFM cantilever was used for deflecting a one-end pinned NW (Wong et al. 1997). Song et al. demonstrated the feasibility of AFM lateral force microscopy technique to characterize the mechanical properties of vertically aligned ZnO NWs avoiding the tedious manipulation and assembly steps (Song et al. 2005a). 17.3.3.1.3 Nano-Indentation In nano-indentation, a sharp tip made of a hard material and with a known geometry and elastic properties is used to make an indent on the material to be tested. From the details of

17-14


1 μm

1 μm 10 μm (a)

(b) 400

Young’s modulus, E (GPa)

350 300

Single crystal along [220] (Theoretical calculation)

250

Polycrystalline (Theoretical calculation)

200 150 100 50 0 0

(c)

50

100 150 Diameter, D (nm)

200

250

FIGURE 17.11 (a) SEM image of a SiN TEM grid with circular holes with dispersed Co3O4 NWs. Inset shows a closeup view of the highlighted region (white circle) showing a suspended NW. (b) SEM image of a suspended NW after securing the ends with Pt deposition. (c) Plot of Young’s modulus of the Co3O4 NWs against diameter. (From Varghese, B. et al., Nano Lett., 8, 3226, 2008c. With permission.)

applied load and penetration depth, the mechanical properties of materials can be extracted. For nano-indentation tests, the nanostructures are dispersed on a hard substrate. It is recommended that the nanostructures are secured by using the FIB technique or the like, to prevent sliding during indentation. The nano-indenter is often used in conjunction with an AFM, so that the testing and subsequent imaging of the indent region can be carried out (Tao et al. 2007, Tao and Li 2008). Lucas et al. investigated the size-dependent elastic modulus of the ZnO nanobelts using a modified nano-indentation technique (Lucas et al. 2007). The observed aspect ratio dependence on the elastic constant of ZnO nanobelts was attributed to the growth direction-dependent aspect ratio and variation in defects. AFM-based techniques provide a platform for acquiring force–deflection curves simultaneously during the application of force. However, AFM techniques lack the in situ structural characterization and imaging during the test. To circumvent some of these drawbacks, we have developed a combinatory approach that permits both visualization of the microscopic details as well as characterization of its mechanical properties (Varghese et al. 2008). In this approach, suspended NW configurations are constructed on a SiN TEM grid with circular

holes. The position of each suspended NW can be noted and this facilitates multiple experiments on the same NW. The elastic constants were obtained from the nanoscale three-point bend test using AFM and the microstructure of the same NW was examined using TEM. Figure 17.11a shows SEM images of the SiN TEM grid with Co3O4 NWs. The ends of the suspended NWs are secured using Pt deposition (Figure 17.11b). Figure 17.11c shows a plot of Young’s modulus versus the size of the Co3O4 NWs. 17.3.2.2 Resonance Method The resonance tests developed to obtain the mechanical properties of 1D nanostructures are normally conducted inside electron microscopes for visualization purposes. First, experimental calculations of Young’s modulus of the CNTs were performed by setting a freestanding CNT into thermal vibration inside a TEM and measuring the resonance frequency (Treacy et al. 1996). Later, mechanical vibrations induced by an electric field emerged as a versatile tool for nano-mechanical characterization (Poncharal et al. 1999). When a static potential is applied to the projected NW/NT, its end is electrically charged and attracted to the counter electrode. If the NW/NT is not perpendicular to the counter electrode, it bends towards the counter

17-15


2 μm

(a)

(b)

(c)

(d)

FIGURE 17.12 Images of ZnO NWs of different length and diameter captured during resonance vibration arising from the application of alternating electric field. (From Huang, Y. et al., J. Phys.: Condens. Matter., 18, L179, 2006. With permission.)

electrode. On the other hand, applications of alternating field to such protruding NWs/NTs results in dynamic deflections. If the frequency of the applied field is varied, the NW/NT can be excited into the resonance vibrations. The resonance frequency of such cantilevered beams is given by fi =

βi2 L2

YI , ρA

(17.6)

where βi is a constant for the ith harmonic with the values β1 = 1.875 and β2 = 4.694 L is the length of the cantilever beam Y is the elastic modulus I is moment of inertia ρ is the mass density A is the cross-sectional area of the nanostructures For a NW with a circular cross-section with diameter d, the expression can be written as fi =

βi2 d Y . 2π L2 16ρ

(17.7)

For nanotubes with an outer diameter d and inner diameter di, the expression is fi =

βi2 1 Y (d 2 + di2 ) . 8π L2 ρ

(17.8)

For a NW with an equilateral triangular cross-section, fi =

βi2 1 2π L2

Yd 2 . 24ρ

(17.9)

Chen et al. studied the size-dependent Young’s modulus of the ZnO NWs by resonance technique inside SEM (Chen et al. 2006). Young’s modulus of other metal oxide nanostructures including,

ZnO nanobelts (Bai et al. 2003), WOx NWs (Liu et al. 2006), and β-Ga2O3 NWs (Yu et al. 2005a) were also determined using the resonance method. Figure 17.12 shows the resonance oscillation of ZnO NWs of different lengths and diameters (Huang et al. 2006). Table 17.1 shows a summary of the mechanical properties of the various metal oxide nanostructures estimated using the techniques described above. Many of the metal oxide nanostructures exhibited a size-dependent elastic behavior. The origin of the size-dependent elastic modulus is a topic of fundamental research. As size shrinks to nanoscale, the surface atoms contribute to the material properties significantly. The surface atoms are in a state of strain due to the uneven bonding compared with the atoms in the interior of the nanostructures. The effects of surface stress and reduced defects are considered to be the causes of the size-dependent mechanical properties of the nanosized materials. Recently, theoretical modeling on the mechanical properties of nanostructures by considering a combined effect of surface and bulk properties was reported. Miller et al. explained the size-dependent elastic properties of nanostructures by taking into account the surface elasticity (Miller and Shenoy 2000). Their model predicts that the deviation of an elastic property D from that of conventional continuum mechanics Dc can be expressed as D − Dc S h =α =α 0, Dc Yh h

(17.10)

where α is a dimensionless constant that depends on the geometry of the structural element h is a length defining the size of the structure h0 ≡ S/Y is a material length that sets the scale at which the effect of the free surfaces become significant The quantity S is a surface elastic constant relevant to the structural element being considered and Y is the corresponding elastic modulus of the bulk material. Similarly, Chen et al. modeled the NW as composite material with a bulk core and a surface with different elastic properties

17-16

Handbook of Nanophysics: Nanotubes and Nanowires TABLE 17.1 Mechanical Properties of Metal Oxide Nanostructures Method

Material

Bending test

Lateral force microscopy Nano indentation

Resonance

Young’s Modulus

CuO nanowires WOx nanowires Co3O4 nanowires Nb2O5 nanowires ZnO nanowires

70–300 GPa 10–110 GPa 4–250 GPa 5–30 GPa 29 ± 8 GPa

ZnO nanowires ZnO nanobelts Mg2B2O5 nanowires Al4B2O9 nanowires Al18B4O33 nanowires ZnO nanowires ZnO nanobelts WOx nanowires Ga2O3 nanowires

— 31.1 ± 1.3 Gpa 125.8 ± 3.6 GPa 80–120 GPa 80–200 GPa 130–250 GPa 35–50 GPa 100–300 GPa ∼300 GPa

(Chen et al. 2006). The flexural rigidity of such composite material can be written as YI = YbI b + SI s ,

(17.11)

where Ib and Is are the moment of inertia of the cross-section of the core and the shell, respectively Yb is the Young’s modulus of the bulk core

(17.12)

where rs is the depth of the shell D is the total NW diameter The experimental data obtained from the resonance method fit well in the above equation. However, the applicability of this model is limited by the arbitrary nature of the constants S and rs. On the other hand, adding the surface stress contribution to the total energy of a bend NW, Cuenot et al. derived an expression for the Young’s modulus as (Cuenot et al. 2004) L2 8 Y = Yb + τ(1 − ν) 3 , 5 d

Size Dependent

Reference

— — — — —

Yes Yes Yes Yes —

Tan et al. (2007) Cheong et al. (2007) Varghese et al. (2008c) Varghese et al. (2009) Song et al. (2005a)

3.4 ± 0.9 GPa — 15 ± 0.7 GPa 8–15 Gpa 10–20 GPa — — — —

— — — —

Feng et al. (2006) Ni and Li (2006) Tao and Li (2008) Tao et al. (2007) Tao et al. (2007) Chen et al. (2006) Bai et al. (2003) Liu et al. (2006) Yu et al. (2005a)

Yes Yes Yes —

Jing et al. take into account both the surface stress and the surface elastic constant to calculate the elastic constant of the nanostructure (Jing et al. 2006). The total surface stress of the nanostructure due to an applied strain of ε can be expressed as τ = τ 0 + Sε,

(17.13)

where τ is the surface stress ν is the Poisson’s ratio L is the length of the NW d is the diameter of the NW Thus, the elastic modulus of the NWs can be larger or lower than the corresponding bulk material depending on the positive or negative surface stresses.

(17.14)

where τ0 is the surface stress on the NW at ε = 0 and the elastic constant of NW can be expressed as Y − Yb 8 S 8L2 τ0 ≈ + . Yb d Yb 5d 3 Yb

By substituting Ib and Is, we get ⎡ ⎛ S ⎞⎛r r2 r3 r 4 ⎞⎤ Y = Yb ⎢1 + 8 ⎜ − 1⎟ ⎜ s − 3 s 2 + 4 s 3 − 2 s 4 ⎟⎥ , ⎝ Eb ⎠ ⎝ D D D D ⎠⎥⎦ ⎣⎢

Hardness

(17.15)

Although large amounts of contributions have been made to this topic during the last few years, a satisfactory theoretical framework to elucidate the size dependence on the mechanical properties of nanostructures is still lacking. This is fundamentally due to many contributing factors that determine the overall mechanics of nanostructured materials. Some of these factors are surface properties, internal microstructure (e.g., anisotropic growth direction), defects, and geometry of the nanostructure. Utilizing the mechanical properties of metal oxide nanostructures, many prototype devices are proposed and some of them have been demonstrated recently. Wang et al. demonstrated a direct-current nanogenerator making use of the piezoelectric properties of ZnO (Wang and Song 2006, Wang et al. 2007a). In their work, either an AFM tip or a micro fabricated zigzag electrode was used to deflect the free-end of vertically oriented ZnO NWs. The strain field induces charge separation on the piezoelectric ZnO and generates continuous direct current that would be adequate enough for powering various nanodevices.

17.3.3 Optical Properties The energy states near the band edges of low-dimensional materials are densely packed. This enhances the probability for optical transition to occur. Metal oxide nanostructures, in particular those having a direct band gap, are found to have attractive optical properties and could function as various nano-photonic components

17-17

Nanoribbon

Absorbance (a.u.)

2 3 4 5 6 Cycle number

0.6

1.6

0.5

1.2 0.8 0.4

Absorbance (a.u.)

Output

1

2.0

Absorbance (a.u.)

Input

PL intensity (a.u.)

Molecules

0.7

0.20 0.16 0.12 0.08 0.04 0.00 –0.004

Clean PL Sense PL Absorbance

Evanescent field

300

400

600

Absorbance Fluorescence

NR absorbance UV/Vis normalized

700

Thicker NR

450

Clean

0.1 25 μm

500

550

600

Wavelength (nm)

Sense

Light

0.2

1.0

(c)

25 μm

PL intensity (a.u.)

Thinner NR

0.2

–0.1 400

0.3

Absorbance (a.u.)

Absorbance (a.u.)

500

Wavelength (nm)

(b)

0.3

3 mM 1.8 mM 1 mM 0.75 mM Clean

0.6 0.5 0.4 0.3 0.2 0.1 0.5 1.0 1.5 2.0 2.5 3.0 [EITC] mM

0.0

0.0

(a)

0.4

Absorbance (a.u.)


(f )

(g)

0.0 400

(d)

450

500

550

Wavelength (nm)

600

400 450 500 550 600 650 700 750

(e)

Wavelength (nm)

800

(h)

FIGURE 17.13 (a) Schematic illustration of the SnO2 nanoribbon absorption sensor. (b–d) PL and optical absorption spectra of the nanoribbon waveguides immersed in solutions containing the dye molecules in different concentrations. (e) Absorption and fluorescence of the dye molecules containing solution using the nanoribbon waveguide and its photoluminescence images. (f–h) Photoluminescence images of λ-DNA flowing past the nanoribbon sensor. (From Sirbuly, D.J. et al., Adv. Mater., 19, 61, 2007. With permission.)

(Agarwal and Lieber 2006, Djuriŝić and Leung 2006, Pauzauskie and Yang 2006). Research activities on nano-photonics have been stimulated by the discovery of room temperature ultraviolet lasing in vertically oriented ZnO NWs (Huang et al. 2001a). Well-aligned ZnO NWs on sapphire substrates could function as natural laser cavities when excited with the fourth harmonic of the Nd:YAG laser. The excitation light was incident at an angle to the aligned NWs and the emission spectra were collected parallel to the long axis of the NWs. Stimulated emission from the NWs was observed when the excitation intensity was greater than a threshold power of ∼40 kW/cm2. The lasing is likely to be caused by an exciton recombination in ZnO. The threshold power for crystalline NW lasers are found to be significantly lower than that required for lasing for ZnO thin films (>300 kW/cm2). Later, the same group demonstrated the lasing action in an isolated ZnO nanobelt dispersed on a sapphire substrate (Yan et al. 2003). The low threshold lasing from ZnO nanorods synthesized via the solution growth technique was also reported recently (Hirano et al. 2005). Optical lasing action is observed in other metal oxide systems like SnO2 NWs (Liu et al. 2007a). One dimensional nanostructures of SnO2, ZnO, and Ga 2O3 are found to have sub-wavelength optical wave-guiding capability (Yan et al. 2003, Law et al. 2004, Sirbuly et al. 2005, Zhang et al. 2007c). Th is could be useful for the realization of integrated nano-optoelectronic devices, computing, and sensing. Recently, the wave-guiding ability of the SnO2 nanoribbons was

utilized for the demonstration of a prototype multifunctional optical sensor (Sirbuly et al. 2007). The sensor element consists of a SnO2 nanoribbon positioned across a microfluidic channel on a PDMS matrix. Th is geometry allows the measurement of the absorption/fluorescence spectra of the solution containing the analyte molecules excited by the evanescent wave of the guided light through the nanoribbon. A schematic illustration of the experimental technique and optical spectroscopic results obtained from the SnO2 nanoribbon evanescent sensor is displayed in Figure 17.13.

17.3.4 Field Emission Properties Field emission is the process of electron emission from a condensed phase to a vacuum by electron tunneling caused by the application of a high-intensity electric field (Gomer 1961, Fursey 2005). The applied field modifies and narrows the potential barrier of electrons at the solid–vacuum interface. This enables the electrons to tunnel through the barrier. Field emitters have applications in diverse fields including flat panel displays, x-ray sources, electron microscopes, etc. The field emission from metals is explained by the quantum mechanical tunneling theory developed by Fowler and Nordheim (Fowler and Nordheim 1928). Figure 17.14 shows a schematic of the energy diagram of electrons at the surface of a metal at 0 K. In the absence of the field, the potential energy of

17-18


The above integral can be simplified into the form of the Fowler– Nordheim (F–N) equation: e2 4z –eEz –

Energy (E)

φ

j=

Ef –

e

Metal

Vacuum

zc

0

Distance (z)

FIGURE 17.14 of a metal.

Potential energy diagram of electrons at the surface

the electron at a distance x from the surface is determined by the image potential (−e 2/4x). After the application of an electric field, the potential energy of the electron is represented by the solid curve and is given by

U (x ) = −

e2 − eFx , 4x

(17.16)

where e is electronic charge F is the local electric field at the metal surface The application of electric field reduced the work function of the electron by Δϕ =

e 3F . 4πε 0

(17.17)

Assuming a planar metal surface, the current density can be written as ∞

∫

j = e n(E ⊥ )D (E ⊥ , F )dE ⊥ ,

(17.18)

0

where E⊥ is part of the electron energy due to the momentum perpendicular to the metal surface n(E⊥) is the number of electrons with energies in between E and E⊥ + dE⊥; incident on the unit area of the potential barrier surface from the metal D(E⊥,F) is the tunneling probability of the incident electron with the energy component E⊥ due to momentum perpendicular to the metal surface

⎛ Bφ3/2 ⎞ A 2 . F exp ⎜ − φ F ⎟⎠ ⎝

(17.19)

According to Equation 17.19, the plot between ln(J/F2) versus 1/F (the Fowler–Nordheim (F–N) plot) gives a straight line. Although Equation 17.19 is derived for metal field emitters, it is often employed for most of the semiconducting field emitters as well. A useful field emitter should need to have attributes such as a low turn-on field (defi ned as the applied field at which the emission current density reaches 10 μA/cm2), high current emission capability, long-term emission stability, high mechanical and chemical stability, and the availability of cost effective fabrication techniques. Conventional field emitters are micro-fabricated sharp tips made of metals or semiconductors (Brodie 1994, Temple 1999, Xu and Huq 2005). The fabrication involves expensive multi-step lithography techniques including thin fi lm deposition, photolithography, etching, and liftoff procedures. Two types of design normally used in vacuum microelectronics are the gated and un-gated Spindt-type field emitters (Brodie 1994, Temple 1999, Xu and Huq 2005). In gated Spindt field emitters, the electric field is applied using a gate electrode that is fabricated ∼1 μm away from the emitter tip and an additional electrode is used for the collection of the emitted electrons. In such a design, a field of 100 V/μm is necessary to obtain appreciable current density. Greater efforts have been devoted for the realization of field emitting structures that can be fabricated by simple means and can be operated at low voltage conditions. The discovery of efficient field emission from CNT arrays (de Heer et al. 1995, Fan et al. 1999, Jonge and Bonard 2004) has stimulated intense efforts in the investigation of the field emission properties of other nanostructures. The CNT emitters exhibit a low turn-on field and long-term stability. However, the lack of structure-controlled growth strategies for CNTs is a challenging issue in device integration. Alternatively, FE emitters based on metal oxide nanostructures have attracted attention due to the feasibility of precise structural control during the growth process. In addition, the oxides emitters can be operated in the presence of oxygen. These factors guarantee a predictable voltage–current characteristic from nanostructured metal oxide field emitters. Typically, the field emission properties of nanostructures are characterized in vacuum conditions with a base pressure better than ∼10−6 Torr. Table 17.2 summarizes the field emission properties of various metal oxide nanostructure arrays. Nanostructures of zinc oxides, molybdenum oxides, tungsten oxides, and niobium oxides are found to exhibit efficient electron field emission capability. The field emission characteristics are sensitively dependent on the geometrical shape and areal density of nanostructures.

17-19

One-Dimensional Metal Oxide Nanostructures TABLE 17.2 Field Emission Properties of Various Metal Oxide Nanostructure Arrays Material ZnO

MoO3 MoO2 WO2.9 WO2 W18O49 CuO In2O3 RuO2 NiO Co3O4 Nb2O5 α-Fe2O3 TiO2 V2O5 · nH2O SnO2 AlZnO

Geometry

Turn-On field (V/μm)

Nanowires

6.0 (0.1 μA/cm2)

Nanoneedles Nanopencils Nanopins Nanobelts Nanowires Nanowires Nanorodes Nanorodes Nanowires/nanotubes Nanotips Nanowires Nanowires Nanopyramids Nanowires Nanorodes Nanorods Nanowalls Nanowires Nanowalls Nanowires Nanowires Nanowires Nanotubes Nanobelts Nanowhiskers Nanowires

2.4 (0.1 μA/cm2) 3.7 1.9 (0.1 μA/cm2) 8.7 3.5 2.4 1.2 >7 2.6 2.0 3.5–4.5 >7 2.7 10.3 11.5 7.4 6.4 ∼6.7 6.3 5.7 6.35 2.3–4.5(1 μA/cm2) 1.4 (0.1 μA/cm2) 2.9 (1 μA/cm2)

Threshold Field (V/μm)

Stability

11.0 (1 mA/cm2) 6.5 > Ts resistivity becomes thermally activated.

nanotechnology applications as polymers can be used in printable and wearable electronic devices.

20.3.1 Electrospinning Technique The synthesis of polymer nanofibers or nanowires is done mainly by template-free and template-based synthesis techniques. Among template-free techniques, the electrospinning technique is inexpensive and the most popular method for making nanofibers or nanowires from various types of polymers. The production techniques and applications of electrospun polymer nanofibers have been reviewed recently (Huang et al., 2003a,b; Ramakrishna et al., 2006). In this technique, the polymer melts/ solutions come out through a small diameter tube under the influence of an electric field kept between this tube and a metal collector. The electrically charged jet of polymer solution/melt gets deposited on the metal collector and takes the shape of a nanowire (or nanofiber) as the jet becomes long and thin due to instability and elongation processes. It is also possible to form core-shell nanowires by this electrospinning technique having two different types of co-axial polymers. One can also form polymer composite nanofibers using this technique, for example, 70 nm fibers of poly(vinylidene fluoride) (PVDF) and singlewalled carbon nanotubes (CNTs) have been produced using this technique for electrical applications (Seoul et al., 2003). It is difficult to control fiber orientation during electrospinning and also the doping, because the polymerization reaction takes place before the nanofiber formation.

20.3 Growth of Polymer Nanowires The fascinating world of nanoscience and nanotechnology has evolved due to the tunable properties of nanomaterials. The properties of nanomaterials change with shape and size, as the nanomaterial size becomes comparable to the molecular size. As the macromolecules are large in size, this effect becomes prominent in polymer materials even when the size in any of the three directions becomes 50 nm. For polymers, the size at which properties deviate from the bulk properties depends on the “radius of gyration,” rg, of the macromolecules. Long-chain macromolecules form randomly coiled structures in polymer materials and rg is defi ned as the root-mean-square distance of atoms to the center of gravity of the chain of length nl and can be given as n / 6l. Even for thin polymer fi lms when the thickness of the fi lms becomes comparable to rg, several anomalous properties are obtained (Sanyal et al., 1996) like layering of molecules and the reduction of the glass transition temperature (Bhattacharya et al., 2005). With the advancement of nanotechnology, it has now been possible to fabricate quasi-1D systems such as nanowires and nanotubes that are one of the most promising candidates for the future nano-electronic device. It is generally agreed now that nanowires will not only be used as inter-connectors of nanodevices, but they will also be used as active circuit elements. Among nanowires and nanotubes of various materials, polymer nanowires and nanotubes will have advantages in several

20.3.2 Template-Based Synthesis Template-based synthesis is one of the popular bottom-up techniques of nanostructure fabrication. Depending on the requirements and material properties, different types of templates have been used, such as the Stepped substrate (Barth et al., 2005), the Grooved substrate (Kapon et al., 1989), self-assembly using organic surfactant or a block copolymer (Thurn-Albrecht et al., 2000), biological macromolecules such as DNA or rod shaped viruses (Quake and Scherer, 2000; Yan et al., 2003; Ma et al., 2004; Mao et al., 2004; Nam et al., 2006), and porous membranes/materials (Martin, 1994; Wu and Bein, 1994; Hong et al., 2001; Wang et al., 2003). The main merits of the template-based synthesis are (a) fine control on the shape and size, (b) easy processing, (c) high yield cost-effectiveness, and (d) doping control. Conducting polymer nanowires have been synthesized both by template-based (Martin, 1994; Wu and Bein, 1994; Hong et al., 2001; Wang et al., 2003) and template-free (Huang et al., 2003a,b) methods. However, the template-based synthesis provides nearly monodisperse nanowires of a desired size and high-aspect ratio. After the synthesis, the nanowires can be easily separated from the template and the individual nanowires can be isolated and manipulated for further applications because they are not interconnected with each other, which is generally found in the template-free method. A synthesis of various kinds of nanowires of polymers (polypyrrole, polyaniline,

20-7

Polymer Nanowires

and poly[3,4-ethylenedioxythiophene]) using this technique by chemical and electrochemical routes has been done. A chemical synthesis of conducting polymer nanowires has been done inside the pores of several templates like polycarbonate membrane, alumina membrane, aluminosilicate MCM-41, etc. (Martin, 1994; Wu and Bein, 1994; Hong et al., 2001; Wang et al., 2003). Commercially available polycarbonate membranes are of various pore diameters ranging from 10 nm to more than a few μm. One can prepare conducting polymer nanowires with a diameter as low as ∼10 nm using these polycarbonate templates. The thickness of the templates are generally 6–20 μm and this determines the maximum length of the nanowires so the aspect ratio can be as large as ∼1000. The pore density in the polycarbonate membranes ranges from ∼107 to 109 pores cm−2 depending on the pore diameter. The pores inside the polycarbonate membrane were prepared using the track-etch method (Fleischer et al., 1975; Ferain and Legras, 1994, 1997) where a polycarbonate sheet of a few micron thickness is bombarded with energetic (∼MeV) heavy ions that lead to linear narrow paths of radiation damage called tracks. The tracks can be revealed by using a suitable chemical agent (like NaOH, HF, etc.) that selectively etches the latent track to create a hollow channel keeping the remaining part unaltered (Price and Walker, 1962). The pores created by this method are hydrophobic and this helps the attachment of the polymer chains on the pore walls. Alumina membrane is among other widely used templates that are commercially available. The pore diameter of the alumina membranes ranges from 20 nm to a few hundred nm and the pore density ranges from ∼1010 to 1012 pores cm−2 . The thickness of the membranes is ∼60 μm, hence, alumina templates are useful for synthesizing relatively long nanowires. However, the brittle nature of the alumina template deserves careful handling. The nanopores in the alumina are fabricated using the anodization process. Commercially, nanoporous alumina templates were manufactured by the sulfuric acid (H2SO4)-based hard anodization (HA) process of oxide fi lms. Using this method, pores having a diameter ranging from 5 to 200 nm and a density of 1010 –1012 pores cm−2 can be fabricated. The nanopores prepared in this method are disorderly organized. For the fabrication of well-ordered pores, one generally uses oxalic acid (H 2C2O4) and the potential for anodization is kept between 100 and 150 V (Masuda and Fukuda, 1995; Nielsch et al., 2002; Lee et al., 2006a,b).

The mechanical rigidity of the polycarbonate membrane is low compared to alumina membrane but alumina membranes are brittle. The pore density of alumina membranes is much higher than that of polycarbonate. It is also possible to fabricate well ordered pores in the alumina membrane. After the nanowire synthesis inside the porus membrane, it is necessary to remove the template in order to get the individual nanowires. A polycarbonate membrane can be dissolved using chloroform, N-mehyl-2 pyrrolidone, methylene chloride, etc. These solvents have no apparent effect on the doping level of conducting polymer nanowires so after removing the template, the properties of nanowires remain intact. On the other hand, the alumina membrane can be dissolved in concentrated (1–6 M) NaOH solution, HF, etc. The concentrated NaOH solution can alter the doping of the conducting polymer nanowires and may change the intrinsic properties of the nanowires. 20.3.2.1 Chemical Synthesis A chemical synthesis of conducting polymer nanowires of various doping concentrations can be done using the oxidative polymerization technique, where the monomers (pyrrole, aniline, etc.) get polymerized by an oxidizing agent (like ferric chloride, ammonium persulfate, potassium persulfate, etc.). For the synthesis, a porus membrane (polycarbonate or alumina) of a particular pore diameter is placed between a two-compartment glass cell with a rubber o-ring and clips (refer to Figure 20.7a and b). Before using the monomer, it is better to distill it under reduced pressure. An aqueous (a solvent other than water like acetonitrile, water/alcohol mixture can also be used) solution of monomer should be added in one compartment and the other compartment should have the solution of the oxidizing agent. Polymerization is done by the oxidizing agent (like FeCl3) that can also provide dopant counter anion (Cl−). Polymerization takes place within each pore as the oxidizing agent starts diffusing through the pores toward the compartment containing the monomer. This diff usion process of the oxidizing agent may create a profi le of doping concentration (the amount of the dopant can vary depending on the nanowire diameter or along the length of the nanowires) that helps one to fabricate conducting polymer nanowires with various average doping levels, and a doping profi le at one end of the nanowires (Rahman et al., 2006). As the pore walls of the polycarbonate are hydrophobic, the polymer chains prefer to be deposited on the walls and remain

Monomer

(a)

(b)

Oxidizing agent

Polymer

FIGURE 20.7 (a) Schematic of the porus membrane used for the nanowire synthesis. (b) Schematic of cell used for chemical synthesis; membrane is place in between the two cell and monomer is added in one compartment while oxidizing agent is added in another.

20-8


20.3.2.2 Electrochemical Synthesis For the electrochemical deposition of nanowires inside the porus membrane, a metal layer is deposited on the back side of the membrane that acts as a working electrode (anode) in a threeelectrode electrochemical cell. Ag/AgCl or saturated calomel (SCE) is used as a reference electrode and a platinum or gold foil is used as a counter electrode (see Figure 20.8). For the electropolymerization, lithium perchlorate, sodium perchlorate, sodium polystyrenesulfonate, dodecylbenzene sulfonic acid, tetrabutylammonium hexafluorophosphate, etc., are used as supporting electrolytes, and perchlorate, sulfonate, hexafluorophosphate, etc., act as dopant ions. Electrochemical deposition was done in constant potential (chronoamperometric) or in constant current (chronopotentiometric) mode using a potentiostat/galvanostat. In potentiostatic mode, the electrodeposition of various conducting polymers initiates at different voltages (with respect to the reference electrode); as an example, the electropolymerization of Counter electrode

Reference electrode

Monomer + electrolyte Porous membrane Clip O-Ring Connection to the back side metal electrode

FIGURE 20.8 Schematic of electrochemical cell used the electrodeposition of conducting polymer nanowires.

2.4

c

1.6 Current (mA)

extended, which is in contrast to the polymerization reaction in the hydrophilic substrate where they form spherical shapes due to larger covalent interactions among themselves (Wang et al., 1999). So polymer chains remain well aligned in nanowires synthesized inside the nanopores of membranes. The doping of the conducting polymer nanowires prepared by this technique can be very low. The doping concentration can be determined by the relative amount of the counterions, for example polypyrrole nanowires synthesized using an oxidising agent FeCl3, the atomic ratio of Cl to N determines the degree of doping. One can obtain two to three orders of magnitude lower doping (Rahman et al., 2006) as compared to a doping concentration (c) = 0.33 for fully doped polypyrrole (Armes, 1987; Wu and Chen, 1997). It is to be noted that lower doping concentrations are obtained for membranes with lower diameter nanopores.

b 0.8 a 0.0

0

200

400

600

Time (s)

FIGURE 20.9 Current vs. time (I–t) plot for electrochemical deposition of nanowires inside 15 nm pore diameter membrane showing the different regions of deposition.

pyrrole was done at +0.8 V with respect to the SCE reference. By monitoring the current vs. time (I–t) data (see Figure 20.9) during the potentiostatic deposition mode, one can get information about the growth process of the nanowires. As long as the nanowires are growing inside the pore, the current remains constant (region a) because the deposition area remains the same. When the pore fi lls up and the capping layer starts to form, the current increases due to an increase in the area (region b) and finally when they are all connected, the surface deposition area remains constant and the current also becomes constant (region c). The steady nature of the region (a) indicates a homogeneous deposition inside the pores. During the deposition of metal fi lm (gold or platinum) on the back side of the membrane, one should be very careful about the penetration of metal inside the pores of the membrane. This problem could be solved by keeping the membrane at a certain angle with respect to the deposition direction. In Figure 20.10, the secondary ion mass spectroscopic (SIMS) measurement on the gold-coated polycarbonate membrane has been presented to show the penetration of metal during the deposition process (done using magnetron sputtering). It has been observed that if the membranes were not rotated (static mode) and the membrane surface is held normal (i.e., the pores are parallel) to the deposition direction, then the gold goes through the pore to the other side and is deposited on the sample holder (see Figure 20.10a). In dynamic mode (the sample holder is rotated at maximum speed), if the membrane surface is not tilted up to a certain angle with respect to the target, then gold also enters up to a considerable depth inside the pores (see Figure 20.10b). This unwanted penetration of gold causes serious problems to the physical properties of the sample deposited inside the pore. It is easy to understand that for a fi xed pore diameter membrane the extent of region (a) in Figure 20.9 (i.e., time needed to fi ll the pores) indicates the length of the nanowires. It has been seen that region (a) is smaller, i.e., the length of the nanowires

20-9

Polymer Nanowires 1200 2

Intensity (cs–1)

Intensity (cs–1)

4000 800 2 400 1 0

2000

0

0 2 4 Length (μm)

(a)

1

6

0 (b)

2 4 Length (μm)

6

FIGURE 20.10 (a) SIMS measurements show that (1) gold deposited on 30 nm pore diameter membrane does not enter deep inside the pores when membrane was held at ∼30° angle with respect to the target and rotated at maximum speed. (2) Gold enters deep inside the pores and reaches on the other side when it was not rotated and not tilted. (b) SIMS profi le of gold, deposited at ∼30° angle with respect to the target on 10 nm diameter membrane. During deposition if the membrane was not rotated then gold enters up to a large extent inside the nanopores (2) compared to the deposition done at maximum speed (1). 3

2

Current (mA)

2

1

1

0

a

0

100 Time (s)

200

FIGURE 20.11 I–t plot for electrochemical deposition of nanowires inside two 10 nm pore diameter membrane. Length of nanowires are smaller (corresponds to extent of region (a)) in (2) where gold has entered deep inside the pore compared to (1).

is smaller, for those membranes where gold has entered up to a considerable length inside the pores (during sputtering deposition). An I–t curve shown in Figure 20.11 for nanowires synthesized inside a 10-nm pore diameter membrane supports this argument. Similar behavior is observed for higher pore diameter membranes.

20.4 Properties of Polymer Nanowires In the nanostructured form, conducting polymers show wide versatility and enhanced efficiency. Due to a large surface-tovolume ratio, a conducting polymer nanowire shows better sensitivity when used as an actuator or sensor (Smela et al., 1995; Swager, 1998; Jager and Edwin et al., 2000; Huang et al., 2003a,b; Hernndez et al., 2004; Ramanathan et al., 2004; Berdichevsky

and Lo, 2005; Ramanathan et al., 2005; Lee et al., 2008). Due to better alignment of the polymer chains and enhanced conjugation length, a conducting polymer nanowire shows higher electrical conductivity compared to bulk polymer (Cai and Martin, 1989; Cai et al., 1991; Wu and Bein, 1994; DemoustierChampagne and Stavaux, 1999; Choi and Park, 2000). Using electrochemical gating, a conducting polymer nanowire serves as a better filed effect transistor with large transconductance and on/off current ratio (Wanekaya et al., 2007). Functionalized polymer nanotubes show better chemical selectivity (Ramaseshan et al., 2006; Wang et al., 2006; Savariar et al., 2008). Also, an enhancement in the surface-to-volume ratio in this nanowire has made this material more effective in fi lter application as efficiency in this process is closely associated with the fineness of the fiber. The enhancement of the surface area has also made polymer nanowires ideal low-mass materials for several applications. The mechanical properties of polymer nanowires or nanotubes change dramatically in comparison to that of bulk (Bergshoef and Vancso, 1999; Cuenot et al., 2000; Qian et al., 2000; Ge et al., 2004; Sreekumar et al., 2004; Ye et al., 2004; Tan and Lim, 2005; Arinstein et al., 2007; Liu et al., 2007). It has been observed that the elastic modulus of the polypyrrole nanotube increases strongly when the thickness or outer diameter of the nanotubes decreases (Cuenot et al., 2000). The increase of mechanical strength is due to the better alignment of polymer chains and reduced voids in the nanowires.

20.4.1 Applications of Polymer Nanowires Several materials have been incorporated with polymer nanofiber to make composites that show better mechanical strength. CNTs have been widely used as reinforced materials in polymer nanofibers (Bergshoef and Vancso, 1999; Qian et al., 2000; Ge et al., 2004; Sreekumar et al., 2004; Ye et al., 2004; Liu et al., 2007). CNTs and poly(acrylonitrile) (PAN) composite nanofiber has been used for mechanical reinforcement (Ge et al., 2004; Sreekumar et al., 2004; Ye et al., 2004). The formation of CNTs and nanocomposite

20-10

wires through electrospinning is an ongoing research field and it is known that the enhancement of the physical properties like mechanical strength and electrical conductivity will depend on the even distribution and will control the alignment of CNTs in the nanowires of the polymer. Electrochemical supercapacitors are high-power density charge-storage devices. Several polymers have been used for making supercapacitors. Nanocomposites, nanofiber, or nanowires of polymers such as polypyrrole (Hughes et al., 2002; Li et al., 2002a), p-phenylenevinylene (Deng et al., 2002), polyaniline (Zhou et al., 2004) and polymethyl methacrylate (Sun et al., 2001), poly(3,4-ethylenedioxythiophene) (Cho and Lee, 2008; Liu et al., 2008), etc., show enhanced energy storage capacity and are suitable for fabrication of high-power density and long-life supercapacitors. Polymer nanofibers produced by this electrospun technique are finding wide applications in biomedical applications and tissue engineering (Deitzel et al., 2002; Li et al., 2002b; Khil et al., 2004; Smitha and Ma, 2004; Xu et al., 2004; Buttafoco et al., 2006). Biocompatible nanofibers of poly(d,l-lactide-co-glycolide) (PLGA) play an important role in modulating tissue growth. The nanofibers are capable of supporting cell attachment and guide cell growth (Li et al., 2002b). Poly(E-caprolactone) (PCL) nanofibers are used for the guiding and proliferation of fibroblasts and myoblasts cells (Williamson and Coombes, 2004). Collagen nanofibers and their composites have been found to be ideal for cell attachment and proliferation (Huang et al., 2001; Matthews et al., 2002).

20.4.2 Electronic Transport Properties of Conducting Polymer Nanowires Conducting polymer nanowires are quasi-1D systems composed of aligned polymer chains where charge carriers are created by doping. The electronic transport properties of such systems are strongly dependent on the detailed nature of interaction among charge carriers and with the environment and disorder present in the systems. Due to the presence of disorder, the charge carriers are localized and electron transport can take place by hopping from one localized site to another. In the absence of interaction, if the interchain electron hopping is negligible compared to intrachain, then one observes ln ρ ∝ T−1/2, which is a signature of 1D hopping. On the other hand, if interchain hopping plays a significant role, then 3D VRH behavior is observed (ln ρ ∝ T−1/4). In the presence of electron–electron interaction (EEI), the ES-type behavior (ln ρ ∝ T−1/2) is observed in all dimensions. However, recent studies on polymer nanowires at low temperatures have revealed exciting features of the 1D transport properties of interacting electrons. We have given a brief overview on the effect of interaction and disorder in 1D in Appendix 20.A. 20.4.2.1 Lüttinger Liquid Behavior in Polymer Nanowires The low temperature electronic transport study of helical polyacetylene (PA) fibers doped with iodine shows characteristics of LL behavior (Aleshin et al., 2004; Aleshin, 2006, 2007). The fibers


are nearly 10 μm long, have a thickness of a few tens of nanometers, and are composed of several polyacetylene chains. The low temperature I–V characteristics of these nanofibers are highly asymmetric and the asymmetry increases with decreasing temperature. The current–voltage characteristics show power-law behavior (I ∝ V1+β) in the temperature range 30 K < T < 300 K. The β value for R-helical-PA fibrils was found to be ∼1.0–4.7 depending on the diameter and the temperature. Power-law dependence was also observed in the conductance vs. temperature (G ∝ T α) and the α value increases from ∼2.2 to ∼7.2 with a decreasing cross-section of the nanofibers. Different temperature I–V characteristics show expected scaling to a master curve by plotting I/T 1+α versus eV/(kBT). The value of α ≠ β and the value of β is always less than α, however, up to a certain extent, the results indicate toward a LL-like behavior of helical polyacetylene nanofibers above ∼30 K (Aleshin et al., 2004; Aleshin, 2006, 2007). Below 30 K, the electronic transport characteristics of R-helical-PA nanofibers show Coulomb blockade behavior (Aleshin et al., 2005). 20.4.2.2 Wigner Crystal-Like Behavior of Polymer Nanowires Conducting polymer nanowires are disordered quasi-1D system where formation of Wigner crystal (WC) may be favored at a low carrier concentration. The disorder restricts the zeropoint motion of the electron, which in turn reduces the quantum fluctuation, and the quasi-1D nature restricts the increase of quantum fluctuation (due to the statistical averaging of fluctuations) even in the absence of disorder. Confinement (in 1D) also enhances the EEI; all these conditions favor the WC formation. It has been observed that conducting polymer nanowires synthesized by the template-based technique using the chemical method have very low charge carrier density compared to the electrochemically synthesized nanowires or bulk polymer (Rahman et al., 2006, 2007; Rahman and Sanyal, 2007b). Also, the polymer chains are well aligned in template-based synthesized nanowires, which helps to explore its quasi-1D nature. Due to the presence of the unavoidable defect, the charge carriers are localized and the carrier concentrations are further decreased with decreasing temperature; hence, the screening of interactions by charge carriers becomes less effective and the Coulomb interaction stars to play a significant role. For very low electron density (as in the case of chemically synthesized nanowires), the EEI becomes long ranged which favors the Wigner crystal formation. The low temperature I–V characteristics of these nanowires show characteristic features of a charge density wave (CDW) system that can arise due to the formation of 1D WC in these structurally disordered materials. At low temperatures, the system shows a gap in the dI/dV vs. V data, a powerlaw-dependent I–V was observed up to certain bias and above a threshold voltage the nanowires show switching transition and hysteresis. Current driven measurement shows the presence of negative differential resistance (NDR) and a huge enhancement of noise near the switching transition. The rapid decrease of the

20-11

Polymer Nanowires

10–5

dI/dV (s)

10–6 10–7 10–8 4K 6K 15 K

10–9 10–10 –2

–1

0 V (V)

1

2

FIGURE 20.12 Differential conductance (dI/dV) is plotted as a function of bias voltage (V) for 450 nm diameter nanowire at various temperatures.

7.2 β

10–3

6.4 5.6

10–5 Current (A)

gap with increasing temperature suggests the presence of strong EEI and a correlated nature of the system. The switching transition, NDR, and noise enhancement suggest the sliding motion of the WC. Below we will give a brief detail about the possible formation of WC in polymer nanowires. Low doped polymer nanowires with a quasi-1D nature and low electron density (rs >> 1) are a potential candidate to form Wigner crystals that can exhibit characteristics of a charge density wave state (Heeger et al., 1988; Schulz, 1993; Lee et al., 2000; Fogler et al., 2004; Aleshin et al., 2004; Aleshin, 2006, 2007). For weakly pinned Wigner crystals, the tunneling density of states shows a power law behavior with the applied bias (Maurey and Giamarchi, 1995) and the exponent ranges from ∼3 to 6 (Jeon et al., 1996; Lee, 2002). In the chemically synthesized nanowires, a gap (VG) was observed in the low temperature I–V characteristics. The gap shows strong temperature dependence and it vanishes at relatively high temperatures (see Figure 20.12), also VG decreases with the increasing diameter of nanowires (Rahman and Sanyal, 2007b). Strong temperature dependence of the gap suggests the presence of EEI and the collective nature of the charge carriers. I–V characteristics of all the nanowires show power law behavior (I ∝ V 1+β) above the gap. In Figure 20.13, one such bit of representative data has been shown for various diameter nanowires. The value of β increases with an increasing diameter (refer to the inset of Figure 20.13) and decreases with increasing temperature for all the nanowires (Rahman and Sanyal, 2007b; Rahman et al., 2007). It has been shown (Jeon et al., 1996) that for 1DWC with increasing pinning strength, β should decrease. So the decrease of VG with an increasing diameter of nanowires clearly indicates that pinning strength increases (Middleton and Wingreen, 1993) with a decreasing diameter. Hence, the reduction in β value with a decreasing diameter of nanowires is consistent with the 1DWC model.

40

10–7

80 d (nm)

120

T=2 K 30 nm 50 nm

10–9

70 nm 110 nm 10–11 1

Voltage (V)

10

FIGURE 20.13 I–V data for various diameter nanowires at T = 2 K is plotted in a double logarithmic scale. The data is fitted (solid line) by the power law I ∝ V 1+β . β value for various diameter nanowire has been shown in the inset.

Above a certain temperature (∼30 K), low bias resistance vs. temperature data shows a 3D VRH behavior (Rahman et al., 2006). However, power law gives a better fit for the data taken with a higher bias. A clean LL state predicts α = β and scaling of I–V curves of different temperatures to a master curve (Balents, 1999). Most of the nanowires do not show such collapse, also the exponents are of different values. It has been shown previously that above a certain temperature (>30 K), electronic transport properties of polymer nanowires can show LL-type behavior (Aleshin et al., 2004; Aleshin, 2006, 2007). The absence of a single master curve and unequal exponents in these wires show that the LL theory is not applicable to describe the electronic properties of these nanowires in the low temperature (VG) and the change in conductance is found to be more than three orders of magnitude (Rahman and Sanyal, 2007a). One such piece of representative data has been shown in Figure 20.14 for a nanowire with a diameter of 450 nm measured at 2.1 K. After switching, the current does not follow the same path with the reduction of voltage and the system returns to its low conducting state only below a certain threshold voltage VRe (|VTh| > |VRe| > |VG|) (see Figure 20.14). With increasing temperature, the hysteresis, defined as (ETh − ERe)/ERe (ETh,Re is the field corresponding to VTh,Re), decreases. The area under the hysteresis loop is independent of the scan speed of bias voltage (current) used in the experiment. The observed switching transition can be explained by considering the sliding motion of the WC that depinnes above the threshold voltage. This sliding motion of WC exhibits characteristics of depinning of the

20-12


been ruled out by taking different contact materials. The observed switching is obviously not due to dielectric breakdown because the length of higher diameter (>110 nm) nanowires are larger where low threshold voltages have been observed (Rahman and Sanyal, 2007a). In conclusion, all the observations like the existence of gap, the power-law behavior of I–V and R–T characteristics, switching transition, NDR, noise enhancement, etc., suggest the formation of Wigner crystal in these nanowires.

10

I (mA)

5

0

–5

VRe

VTh

–10 –4

–2

0 V (V)

2

4

FIGURE 20.14 I–V characteristics of 450 nm diameter nanowires measured at 2.1 K showing the threshold voltage and switching transition. Arrow indicates the direction of voltage scan.

pinned CDW state (Zettl and Grüner, 1982; Coleman et al., 1987; Grüner, 1988; Maeda et al., 1990; Levy et al., 1992). As the resistance of the nanowires in the highly conducting state is very less (a few Ohm), one cannot get information about the switched state from the voltage bias measurements (due to the presence of input impedance or the current limit of the source meters used). Current biased measurements show the existence of NDR in the switched state (refer to Figure 20.15). The switching and NDR were observed for all the nanowires of various diameters and the data were highly reproducible (Rahman and Sanyal, 2008). The sharp threshold and its time independence confirm that switching is not due to field heating. The reproducibility of observed switching and NDR rules out the burning of nanowires. The scan speed independence of the switching and zero crossing (zero current at zero voltage) rules out any capacitive effect. The possibility of any interface-dependent effect has

10 T = 3.5 K

V (V)

5

0

–5

–10 –3

–2

–1

0

1

2

3

I (mA)

FIGURE 20.15 Current driven I–V characteristics of 450 nm diameter nanowire measured at 3.5 K showing presence of NDR (arrows indicates the direction of current scan).

20.5 Conclusion Nanowires of polymers provide us unique systems to investigate various ideas of low dimensional physics. Moreover, the nanowires and nanofibers of the polymers have remarkable application potential in the emerging field of nanotechnology. It is known that the physical properties of materials change drastically as one of the three dimensions of a material becomes comparable to the molecular size. For polymers, which are macromolecules, the interesting changes in physical properties start happening even when the diameter of the nanowires is only 100 nm (0.1 μm). This provides us great technological potential as drastic changes in mechanical and electrical properties in these structurally disordered and easy-to-form nanomaterials can be exploited. We have presented a brief summary of growth, properties, and applications of the nanowires of polymers. We have also elaborated novel electronic transport properties of conducting polymer nanowires at the end of this chapter.

Appendix 20.A: Physics in One Dimension: Effect of Interactions and Disorder The electronic transport properties of 1D systems are very interesting as interactions and disorder play a very important role in determining them. In the absence of interactions, properties of the many-Fermion system can be described by the fermi gas model, which is a quantum mechanical version of an ideal gas model. To explain the properties of interacting fermions in higher (more than 1) dimensions, Landau gave a successful theory—the Fermi liquid theory, where elementary excitations can be considered as a collection of free quasi-particles obeying Fermi statistics (Abrikosov et al., 1963; Pines and Noziéres, 1966). Fermi liquid theory breaks down in 1D systems in the presence of interactions (Voit, 1994; Giamarchi, 2004). For the 1D electronic system, the ground state is strongly correlated in the presence of the EEI and the low energy excitations are bosonic sound-like density waves (plasmons). Depending on the range of the EEI, the properties of these systems can be described by Lüttinger liquid (for short-range interactions) or Wigner crystal (for long-range interactions). Lüttinger liquid theory: In the presence of short-range EEI, the electronic properties of a 1D system are described by the Lüttinger liquid theory. The most striking behavior, which

20-13

Polymer Nanowires

clearly distinguishes the Lüttinger liquid from the Fermi liquid, is the anomalous power-law dependence of various correlation functions in the low-energy region. For example, the density–density correlation function between two distant positions x and x exhibits the power-law dependence in the asymptotic region < ρe (x )ρe (x′) >∼ e2ikF ( x − x ′) (x − x′)−α

(20.9)

In contrast to the Fermi liquid, which has the finite density of states at the Fermi level, the tunneling density of states of these systems shows power-law energy dependence n(ε) ∝ εα . The differential conductance shows power-law dependence on either bias or temperature (Voit, 1994; Giamarchi, 2004) dI/dv ∝ T α

for low bias (V kBT/e)

(20.10)

dI/dV ∝ V β for low bias (V kBT/e)

(20.11)

Pure (in the absence of disorder) Lüttinger (LL) state predicts α = β and the bias (V) and temperature (T) dependence of the I–V curve can be expressed as (Balents, 1999; Giamarchi, 2004)

I = I 0T

1+α

⎛ eV ⎞ ⎛ α ieV ⎞ Γ 1+ + sinh ⎜ ⎝ 2kBT ⎟⎠ ⎜⎝ 2 2πkBT ⎟⎠

2

(20.12)

where I0 is a constant Γ(z) is the gamma function For LL, the I–V curves measured at different temperatures can be scaled to a master curve by plotting I/T 1+α versus eV/(k BT) (Balents, 1999). The value of the exponent α depends on the number of channels and on the strength parameter (g) of the LL. Depending on whether the electron tunnels into the bulk or the end of the LL, α can be expressed as (Matveev and Glazman, 1993) ( g −1 − 1) 4

(20.13)

( g −1 + g − 2) 8

(20.14)

α end =

α bulk =

Since g < 1, the exponent α is larger near the end, which is due to the fact that the spread away of an added charge at the end is small compared with the bulk. If g > 1 and this is the condition for WC formation. It has been found that (a) rs ≃ 36 for clean unbound system, (b) rs ≃ 7.5 for disordered unbound systems and (c) rs ≃ 7.4 for confined systems (Tanatar and Ceperley, 1989; Chui and Esfarjani, 1991; Chui and Tanatar, 1995; Reimann et al., 2000). The decrease of rs in disordered systems is due to the breaking of continuous translational invariance that stabilizes the crystalline state. It has been predicted theoretically that one dimensional WC exhibits the characteristic of a charge density wave (CDW) (Schulz, 1993). The charge–charge correlation function in 1D for long-range EEI can be written as

ρ(x )ρ(0) =

(

A1 cos(2kF x )exp −c2 ln(x )

)

x

(

)

+ A2 cos(4kF x )exp −4c2 ln(x ) +

(20.15)

where A1 and A2 are interaction-dependent constants. In the expression, other fast decaying Fourier components have been omitted. It is interesting to note that the 4kF component decays very slowly (much slower than any power law), showing an incipient charge density wave having a wave vector of 4k F (Schulz, 1993) (in a CDW system formed due to Peierls transition, one observes a 2kF periodicity). The decay of the 4k F component

20-14

observed in long-range interaction is much slower compared to short-range interactions where 2kF and 4kF components show power-law decay. As the 4k F oscillation period is the same as interparticle spacing, the structure is the same as 1D WC but a true long-range order does not exist in such systems because of its 1D nature. It should be mentioned that 4k F oscillation arises only due to the long-range nature of the EEI and can exist even for its extremely small value. It has been reported that the WC state may occur in quasi-1D systems (Wen, 1992; Boies et al., 1995; Arrigoni, 2000) or even in nanowires of structurally disordered materials (Fogler et al., 2004). This electron crystal is pinned by the impurities present in the system. Theoretically, it has been shown that if the impurities act as strong pinning centers then quasi-1D WC shows VRH conductivity with various exponents that depend on the impurity concentration (Fogler et al., 2004). At large impurity concentrations, conductivity follows Efros–Shklovskii law and at very low impurity concentrations, Mott’s 3D VRH-type behavior is predicted. For weakly pinned Wigner crystals, the tunneling density of states shows a power law behavior with the applied bias (Maurey and Giamarchi, 1995) and the exponent ranges from ∼3 to 6 (Glazman et al., 1992; Maurey and Giamarchi, 1995; Jeon et al., 1996; Lee, 2002). The impurities destroy the long-range order; however, for weak impurity strength, quasi-long-range orders may exist (Cha and Fertig, 1994) and a Wigner glass (Andrei et al., 1988; Li et al., 1995; Chakravarty et al., 1999; Fogler and Huse, 2000; Slutskin et al., 2003; Akhanjee and Rudnick, 2007) may form. Depending upon the pinning strength, a pinned WC becomes nonconducting below a certain threshold field. When an applied field is strong enough to overcome the pinning energy, the WC depins and the sliding motion starts giving rise to a switching transition similar to CDW systems (Zettl and Grüner, 1982, Grüner, 1988; Maeda et al., 1990; Levy et al., 1992).

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21 Organic Nanowires Frank Balzer Syddansk Universitet

Morten Madsen Syddansk Universitet

Jakob Kjelstrup-Hansen Syddansk Universitet

Manuela Schiek Syddansk Universitet

Horst-Günter Rubahn Syddansk Universitet

21.1 21.2 21.3 21.4 21.5

Introduction ........................................................................................................................... 21-1 Growth via Organic Molecular Beam Deposition ........................................................... 21-1 Growth on Microstructured Templates .............................................................................21-4 Embedding and Integration .................................................................................................21-6 Linear and Nonlinear Optical Properties .......................................................................... 21-7 Temperature-Dependent Spectroscopy • Determination of Molecular Orientations

21.6 Devices.....................................................................................................................................21-8 On the Way to Nanofiber Light Sources • Waveguides • Nanofiber Frequency Doublers

21.7 Conclusions........................................................................................................................... 21-10 Acknowledgments ........................................................................................................................... 21-11 References......................................................................................................................................... 21-11

21.1 Introduction Fiber-like, light-emitting nanoaggregates from small organic molecules, nanorods, nanowires, or nanofibers have evolved as a very active research field during the last years [1–3]. Depending on the material they are made of, the nanosizing in one or two dimensions leads to interesting new properties. To zeroth order, the surface-to-volume ratio is greatly enhanced and electrons and photons are much better confined as compared to micron-sized components, while the “long” axis allows easy connection to the macroscopic world. Among the properties that have found special interest are waveguiding [4], lasing [5], electrical transport [6,7], mechanical properties [8], and nonlinear optical properties [9–11]. The organic aggregates have either been grown directly on surfaces by organic molecular beam deposition (OMBD) [12], by hot-wall epitaxy [13], or by solvent vapor annealing [14], or they have been assembled in solution and then deposited onto a surface [15–18]. The formation of upright nanowires on a substrate has been facilitated by, e.g., fi lling of mesoporous substrates [19,20] or even by simple vapor phase deposition [21,22]. As for the organic nanofibers that are oriented parallel to the surface plane, the most detailed experimental data exists for the growth of nanofibers made of para-hexaphenylene (p-6P) molecules, Figure 21.1 [23,24]. Their growth is investigated in most detail on dielectric and metallic surfaces such as muscovite and phlogopite mica [23,25,26], KCl and NaCl [27–29], TiO2 [30], thin Au fi lms, Au foil and Au single crystals [31–33], and GaAs [34]. Micas as substrate surfaces are known to promote uniaxial growth. Examples include mesochannels in a mesoporous silica film [35], needles from organic molecules such as ph-thalocyanines [36,37] or anthraquinone [38], protein microfibrils [39], and cationic surfactants grown from solution [40]. On muscovite

mica, e.g., p-6P forms clusters as well as mutually parallel nanofibers from lying molecules, the fibers growing mainly by cluster aggregation. The first step is the formation of a wetting layer from lying molecules [41]. The wetting layer is crystalline, resulting in a clear low-energy electron diff raction (LEED) pattern [26]. On substrates such as KCl similar needles form, but without a wetting layer [42]. For another class of conjugated molecules, the α-thiophenes, α-quaterthiophene and α-sexithiophene, no wetting layer on muscovite has been detected by LEED yet [43]. Because of their fortunate optical and electrical properties phenylene/thiophene, cooligomers represent another class of interesting molecules [44–46]. Field-effect transistors [45], amplified spontaneous emission [47], and lasing [48,49] have, e.g., been demonstrated. Therefore, both from a fundamental point of view as well as from the application side, it is of interest to study the formation of nanofibers from thiophene/phenylene cooligomers on mica and on other dielectric surfaces such as KCl or NaCl. Choosing different sequences and numbers of thiophene and phenylene rings, more thiophene- or phenylene-like molecules can be tested, with different (e.g., zigzag or banana-like) shapes [50]. Reports have been published about the overall morphology of the cooligomers 5,5′-di-4-biphenyl-2,2′-bithiophene (PPTTPP) and 4,4′-di-2,2′bithienyl-biphenyl (TTPPTT) [51] as well as 2,5-di-4-biphenyl-thiophene (PPTPP) thin films on muscovite and on phlogopite mica.

21.2 Growth via Organic Molecular Beam Deposition One of the most prominent ways for the formation of organic nanofibers is their growth by OMBD on crystalline or rough substrates in high (1 × 10 −7 mbar) or even in ultrahigh vacuum 21-1

21-2


S S

S

S

S

S

S

S

S S

S

S

S S

S

S

S

FIGURE 21.1 Organic molecules used for producing nanofibers. From top to bottom, the molecules are para-hexaphenylene (p-6P); 4,4′-di-2,2′bithienyl-biphenyl (TTPPTT); 5,5′-di-4-biphenyl-2,2′-bithiophene (PPTTPP); 2,5-di-4-biphenyl-thiophene (PPTPP); α-quaterthiophene (α-4T); and α-sexithiophene (α-6T).

(1 × 10 −10 mbar). The proper cleaning of the substrates as well as of the organic molecules ensures well-defined chemical and morphological conditions during the deposition process for thermally and chemically stable molecules. Furthermore, many of the organic semiconducting molecules such as parahexaphenylene possess only a poor solubility, making growth from solution impractical, or even impossible. The resulting growth morphology and the question if the molecules form organic nanowires at all, however, depends on details of the molecule–substrate interaction together with surface

(a)

(b)

energies of adsorbate, substrate, and interface, and on details of the deposition process such as the level of supersaturation [52,53] or on the substrate temperature [29]. On the micas, e.g., a large deposition rate and/or a small adsorbate/substrate interaction does not lead to fibers, but to islands from upright molecules. In this section, we will exemplarily present growth by OMBD of needle-like aggregates from three different types of organic molecules: para-phenylenes, α-thiophenes, and thiophene/ phenylene cooligomers, Figure 21.1. As substrates, the KCl (0 0 1) surface together with the two micas phlogopite and muscovite [54] are used. Their surface symmetries are different, resulting in different simultaneous growth directions. Typical fluorescence microscope images of grown samples from para-hexaphenylene on these three substrates are shown in Figure 21.2. All fibers fluoresce after normal incidence UV irradiation due to their buildup from laying molecules on the substrate surface. Mean lengths of the fibers vary between a few hundred nanometers and a few hundred micrometers, depending on the deposition conditions. The widths and heights are typically in the range of a few hundred and a few ten nanometers, respectively. Differences in the mean length of the aggregates are due to different deposition conditions, but also due to details in the growth mechanism. Obviously, on KCl two simultaneous needle orientations evolve. On phlogopite mica, three needle orientations are present, whereas on muscovite only a single needle orientation evolves. Due to the quasi-singly crystalline nature of the fibers [24], the emitted light is polarized either along two (KCl), three (phlogopite), or one (muscovite) direction. Another example—the deposition of the thiophene/phenylene PPTPP on the same substrates as in Figure 21.2—is presented in Figure 21.3. Very similar to p-6P on KCl two needle directions evolve, whereas on muscovite only a single direction is present. On phlogopite, three growth directions are realized, but fibers start bending after they reach a certain length. In this manner, rings, loops, and ridgets with a few micrometers diameter evolve. On KCl, the growth directions are the 〈110〉 directions. This is a rather typical growth direction, in which fibers from many other molecules grow [44,55]. On phlogopite mica, the growth

(c)

FIGURE 21.2 100 × 100 μm2 fluorescence microscope images of p-6P on (a) KCl, (b) phlogopite mica, and (c) muscovite mica. White arrows mark the 〈110〉 directions for KCl, [100] and 〈110〉 for phlogopite, and a single 〈110〉 direction on muscovite.

21-3

Organic Nanowires

(a)

(b)

(c)

FIGURE 21.3 150 × 150 μm2 fluorescence microscope images of PPTPP (a) KCl, (b) phlogopite mica, and (c) muscovite mica. Arrows mark the same directions, as in Figure 21.2.

directions are the three high-symmetry directions [100] and 〈110〉, and on muscovite it is one of the 〈110〉 directions. Especially on muscovite the 〈110〉 is not necessarily a typical growth direction as demonstrated in Figure 21.4. For example, for the thiophenes α-4T and α-6T roughly three simultaneous growth directions are realized, whereas for PPTTPP two directions are present with one of the 〈110〉 substrate directions now serving as the bisecting line of the two needle growth directions. Note that for all three molecules, only two polarization directions of the fluorescence light are observed. The reason for the rather different morphologies on muscovite lies in the subtle interplay between epitaxial growth of the molecules, their bulk packing within the fibers (which might even be dictated by the substrate [56]), and energetically favorable molecule orientations on the substrate due to other-than-epitaxial interactions. The muscovite mica (001) plane possesses an uniaxial symmetry, with a single mirror plane along one of the 〈110〉 directions. Along this direction, grooves exist due to the stacking of the underlying crystal sheets [57]. Both the α-thiophenes as well as PPTTPP have their long molecular axes on the surface oriented along one out of two of the substrate high-symmetry directions, which is not the direction of the mirror axis. From this, the needle directions are determined by the packing of the molecules within the fibers. For PPTTPP, the angle between the

(a)

(b)

molecules long axis and the fiber long axis is close to 90°, while for the thiophenes, it is close to 70°. This way, either roughly three or two needle directions are realized. The para-phenylene p-6P as well as PPTPP grow with the short unit cell axis along the special 〈110〉 directions, which then results only in a single needle direction. For the micas, needle formation is not necessarily the initial growth stage. A number of AFM images (JPK NanoWizard in intermittent contact mode) for different growth stages of p-6P needles on phlogopite mica are shown in Figure 21.5. The first step is the formation of clusters, not of fibers. These clusters are also made of lying molecules, emitting light after normal incidence UV irradiation. Only after a critical coverage is reached, the clusters assemble into fibers, which then continue growing. Note that needle formation can also be induced from such a cluster fi lm by moving the fibers by an AFM tip. The fiber formation also shows a sensitive substrate temperature dependence. At room temperature, only a closed, packed fi lm of very short fibers forms. At elevated temperatures of about 450 K, the fibers are rather long and isolated, but again, only clusters grow at larger temperatures. The other two morphological parameters, that is, the mean width and height, only change marginally with deposition parameters. The width is increased from about 100 to 600 nm, whereas the mean height almost stays constant, because

(c)

FIGURE 21.4 100 × 100 μm2 fluorescence microscope images of (a) α-4T, (b) α-6T, and of (c) PPTTPP on muscovite mica. Whereas α-4T and α-6T grow roughly along the three muscovite high-symmetry directions, one of the muscovite 〈110〉 directions (white arrow) for PPTTPP serves as the bisecting line of the two needle orientations.

21-4


35 nm

0 nm

(a)

(b)

(c)

(d)

(e)

(f )

FIGURE 21.5 AFM images of p-6P growth on phlogopite mica as a function of the nominal thickness (upper panel) and of the deposition temperature (lower panel). Nominal thicknesses are 0.25, 0.5, and 1 nm at 440 K for the 20 × 20 μm2 images in (a), (b), and (c). For the 10 × 10 μm2 AFM images (d), (e), and (f), the deposition temperatures have been 313, 458, and 503 K, respectively. The arrows in (c) denote the three phlogopite high-symmetry directions.

of the growth by cluster agglomeration. This way, varying the deposition conditions allows an easy control over the fiber dimensions. The functionalization of the building blocks is a means to tune the fiber morphology, their linear and nonlinear optical properties, their chemical reactivity, electronic properties, etc. An overview over symmetrically functionalized, nonsymmetrically functionalized, and monofunctionalized para-quaterphenylenes in para positions are shown in Figure 21.6 [58,59]. Details of the growth vary between the molecules, but almost all of them emit blue light after UV excitation and form mutually parallel nanofibers on muscovite. Figure 21.7 demonstrates the variety in morphology, which can be achieved using different molecules and different growth conditions. The needle in Figure 21.7a from a di-chloro functionalized para-quaterphenylene (CLP4) is 700 nm long (rather short), only 25 nm tall, and 115 nm wide. The cross section is triangular. The fiber from α-6T in Figure 21.7b on the opposite is 16 μm long and 250 nm tall, with a width of 330 nm. In Figure 21.7c for a 3.5 μm long and 45 nm tall PPTPP fiber, single clusters as building blocks are still visible.

21.3 Growth on Microstructured Templates In most cases, fabrication of nanowires and their device integration are two separate processes, and that imposes strong constraints on the usefulness of the nanoaggregates. The most

obvious way of integrating fragile nanofibers into device platforms is to grow them on purpose directly at the place where they are supposed to act as new optoelectronic components. One way to achieve that goal is the selected deposition of growth catalysts, from which the nanowires would grow. This works sufficiently well in the case of carbon nanotubes. However, oriented growth of nanofibers from arbitrary organic molecules cannot be tailored in this manner. Since gold (Au) is an interesting surface for devices (bottom contact), let us begin with the direct growth of nanofibers on Au. Nanofibers from para-hexaphenylene molecules have been successfully grown on Au surfaces, but they did not show the strict alignment that has been observed on muscovite mica as the growth template. Instead, on Au (1 1 1), for example, the nanofibers have preferred growth directions along one of the three crystalline high-symmetry directions for deposition at elevated temperatures [33]. The reason for the oriented growth of organic nanofibers on muscovite mica is a combination of epitaxy and electric field-induced alignment of the initially deposited organic molecules. Such a combination is solely possible on a specific, single-crystalline growth substrate. Thus, until now, the mostly implemented way to generate oriented organic nanofibers on substrates that are interesting for device applications is via growth on a template and subsequent soft transfer [60,61]; see the below section. Such methodology has obvious drawbacks such as the fact that remnants from the transfer liquid perturb the functionality of the nanofibers on the surface; that the parallelism of the nanofibers over large areas is usually not retained; and that it is

21-5

Organic Nanowires

(a)

(b)

(c)

(d)

(e)

(f )

(g)

(h)

(i)

FIGURE 21.6 Fluorescence microscope images, 85 × 85 μm2, of functionalized para-quaterphenylenes on muscovite mica. In the upper panel fibers from symmetrically di-functionalized p-4P by O—CH3 (a), Cl (b), and CN (c) are shown, in the middle panel fibers from nonsymmetrically functionalized ones by O—CH3 on one side, by NH2 (d) and Cl (e), (f) on the other side. In the lower panel fibers from monofunctionalized p-4P are displayed, functionalized by O—CH3 (g), Cl (h), and CN (i). (From Schiek, M., Tomorrow’s Chemistry Today: Concepts in Nanoscience, Organic Materials and Environmental Chemistry, Pignatoro, B. (ed.), Wiley-VCH, Weinheim, Germany 2008.)

(a)

500 nm

(b)

2000 nm

(c)

2500 nm

FIGURE 21.7 AFM images of organic nanofibers from (a) a di-chloro functionalized quater-phenylene, (b) from α-sexithiophene, and (c) from PPTPP. Heights, widths, and lengths vary between 25 and 250 nm, 100 and 350 nm, and 700 nm and 16 μm, respectively.

21-6


extremely difficult to place them specifically on micrometersized regions defi ned by the device substrate. In order to meet the demands needed for devices, another method that relies on a combination between top-down fabrication of microstructured substrates and bottom-up growth of the organic nanofibers has been developed recently and is discussed here [62]. We note that the other possibilities for specific transfer such as scatter onto structured surfaces [16] or the alignment by electric fields between electrodes [63] might be useful for other classes of nanowires, but are in general not compatible with the specific material properties of organic nanowires as described in this chapter. As template, microstructured silicon is most conveniently applied. Silicon is structured by optical lithography and subsequent reactive ion etching. Afterward, a thin Au layer (55 nm) is deposited on the substrate. The resulting microstructured Au surface is then used as a template for growth of p-6P nanofibers. A nominal thickness of 5 nm p-6P is deposited at different substrate temperatures. Figure 21.8 shows a fluorescence microscopy image (Figure 21.8a) and a scanning electron microscopy image (Figure 21.8b) of p-6P nanofibers grown on a 5 μm wide Au-coated ridge at a substrate temperature of 388 K. As observed from the figure, the nanofibers are grown both on top of the Au-coated ridge and on the bottom of the substrate, since the whole template is coated with gold. Although this allows us to grow the nanofibers directly on prefabricated microstructures, there is at this substrate temperature no preferred growth direction of the nanofibers. However, when increasing the substrate temperature during growth, it is possible to grow the nanofibers almost perpendicular to the long axis of the Au-coated ridges. This is demonstrated in Figure 21.9, which shows a fluorescence microscopy image (Figure 21.9a) and a scanning electron microscopy image (Figure 21.9b) of p-6P nanofibers grown on a 5 μm wide Au-coated ridge at a substrate temperature of 435 K. Note also that the nanofiber length is increased with the increasing temperature as it is the case for nanofibers grown on mica [64]. One reason for this oriented growth on the microstructured Au ridges is temperature gradients near the edges of the microstructures that guide the nanofiber growth in preferred

10 μm (a)

4 μm (b)

FIGURE 21.8 (a) Fluorescence microscopy image and (b) tilted scanning electron microscopy image of p-6P nanofibers grown on a 5 μm wide Au-coated ridge at a substrate temperature of 388 K.

10 μm (a)

4 μm (b)

FIGURE 21.9 (a) Fluorescence microscopy image and (b) tilted scanning electron microscopy image of p-6P nanofibers grown on a 5 μm wide Au-coated ridge at a substrate temperature of 435 K.

directions. We have investigated this effect for several different substrate temperatures during growth, and it is seen that there is a strong correlation between substrate temperature and orientation of the nanofibers. At low substrate temperatures the nanofibers grow almost randomly on the Au-coated ridges (Figure 21.8), whereas nanofibers grown at high substrate temperatures result in growth nearly perpendicular to the long axis of the ridges (Figure 21.9). Since growth perpendicular to the ridge starts from the edges, the clusters that the nanofibers assemble from have to diff use to an edge. At low substrate temperatures the diff usion length of the clusters is smaller than at high temperatures, and growth in the center part of the ridges can take place. At high substrate temperatures the clusters can diff use to an edge and thus perpendicular growth takes place. Therefore, it is possible by controlling the substrate temperature during growth to achieve oriented organic nanofibers directly on prefabricated microstructures.

21.4 Embedding and Integration After growth on structured Au substrates (as detailed in Section 21.3), the fibers might be further contacted with a top electrode (see Section 21.6) or coated by a dielectric layer that protects them against oxygen and thus reduces problems arising from optical bleaching [65]. In general, growth on structured surfaces is still restricted by the fact that an ultrathin Au fi lm on the structured surface is needed. On silver or palladium, for example, oriented nanofibers cannot be grown. Therefore, there is still a need for transferring nanofibers individually or in arrays from the growth substrate to an arbitrary target substrate. Individual fibers can conveniently be transferred into a liquid and then drop casted or transferred into thin hollow fibers by applying capillary forces. Figure 21.10 shows an example. Alternatively, arrays of oriented nanofibers can be stamped using a delicate mixture of pressure and humidity. Figure 21.11 shows such an array of nanofibers, stamped from the single crystalline growth substrate to a glass plate. The transfer process does not change the photonic properties, here demonstrated via a conserved strong dichroism of the emitted blue light.

21-7

Organic Nanowires

21.5 Linear and Nonlinear Optical Properties

50 μm

An understanding of the optical properties of nanofibers is of importance for optical applications which include, e.g., superradiance [66], lasing, or electroluminescence [67,68].

21.5.1 Temperature-Dependent Spectroscopy (a)

(b)

FIGURE 21.10 (a) p-6P nanofibers inside a hollow fiber, excited by UV light. The nanofibers are aligned along the long axis of the hollow fiber. (b) p-6P nanofibers inside a glass micropipette. No alignment is visible.

100 μm

FIGURE 21.11 Array of blue-light-emitting nanofibers, stamped from the original growth substrate to a glass substrate. The image on the lefthand side has been taken with a polarization analyzer set parallel to the dipole emitter axis, the image on the right-hand side with the polarizer set perpendicular to that axis. The large dichroic ratio proofs that the transfer process does not damage the optical properties of the nanofibers.

In the case of the phenylenes, thiophenes, or phenylene/thiophene cooligomers, the UV excitation-induced fluorescence spectra are dominated by several excitonic transitions between the electronic ground state, S 0, and the first excited singlet state, S1, [69], that is, display vibronic progression series. For example, for p-6P, vibronic peaks from the S1 → S 0 transition with energetic differences of 160 ± 10 meV are clearly resolved, the energetic distance between the maximum in absorption and the first maximum in emission being approximately 0.7 eV. Samples of nanofibers emit at 3.09 eV (00), 2.94 eV (01), 2.77 eV (02), and 2.62 eV (03). These are transitions between the vibrational ground state of the S1 electronic state and the different vibrational states of the electronic ground state [69,71]. A detailed investigation as a function of surface temperature from 300 to 30 K reveals three classes of spectra for slightly different molecule–substrate systems: (a) well-resolved excitonic peaks, which shift to the blue up to 35 meV with decreasing temperature, (b) a similar spectrum with an additional intermediate broadening around 150 K, and (c) an excitonic spectrum similar to (b), but with an additional green defect emission band. Quantitative fitting of type (a) results in an exciton–phonon coupling factor of 80 ± 10 meV and an average phonon temperature of Θ = 670 ± 70 K. The Huang–Rhys factor decreases linearly from 1.2 to 1.0 with decreasing temperature. Fitting of type (b) spectra reveals that the apparent intermediate temperature broadening is due to the additional fluorescence peaks, the relative importance of which increases monotonically with decreasing temperature. In Figure 21.12, contour plots of the three types of spectra as a function of temperature are shown [70]. With decreasing temperature the overall intensity for all types of fluorescence spectra

300

Temperature (K)

250 200 150 100 50 2.2 (a)

2.4

2.6 2.8 3.0 Energy (eV)

3.2

2.2 (b)

2.4

2.6 2.8 3.0 Energy (eV)

3.2

2.2 (c)

2.4

2.6 2.8 3.0 Energy (eV)

3.2

FIGURE 21.12 Contour plots of the three types of temperature-dependent fluorescence spectra. The overall fluorescence intensity increases with decreasing temperature, whereas the green emission band at 2.6 eV from (c) shows only a very weak temperature dependence. (From Balzer, F. et al., J. Lumin., 129, 784, 2009.)

21-8


increases strongly, that is, an increase in the fluorescence yield is observed. However, the intensity of the green emission band from series (c) decreases slightly in intensity with decreasing temperature. Such a behavior has been observed before and hints, together with different decay constants from time-resolved measurements, to a different origin as compared to the regular excitonic peaks such as aggregate states [72–74]. Contamination of the material by, e.g., other organic molecules might be another possible reason. Besides investigations of arrays of nanofibers, also individual nanofibers have been spectroscopically investigated. Extended spectroscopic measurements along a single nanofiber have shown that the spectra depend on the morphology of the aggregates, and that they become significantly narrower if the nanofiber width decreases, e.g., at the tip of the nanofiber [75]. As a general result of the spectroscopic investigations, it is observed that the optical emission of nanofibers delicately depends on the growth conditions—even though the orientation and packing of the individual molecular emitters in the nanoscaled crystals is rather well defined. This—among with other findings—points to the importance of well-defined growth conditions especially for nanoaggregates that are supposed to serve in future devices.

21.5.2 Determination of Molecular Orientations As noted above, due to the quasi-singly crystalline nature of the fibers [24], the emitted light of arrays of p-6P nanofibers is polarized either along two (KCl), three (phlogopite), or one (muscovite) directions. For a single fiber the polarization vector of the emitted light is directed under a well-defined angle with respect to the characteristic axes of the nanofibers such as their long axes. Since in many cases the optical transition dipole moment of the molecules is along the long molecular axes, local measurements of the polarization of the emitted light will provide direct information about the local orientation of the molecules that build the nanofibers. The resolution of this method in a far-field optical approach is limited by the focus diameter of the excitation light, that is, of the order of λ/2 ≈ 200 nm. If one detects the light polarization in the near field by the use of, e.g., a SNOM (scanning near-field optical microscope), the resolution is eventually given by the effective diameter of the SNOM fiber tip. That could in principle be less than 50 nm. In experimental praxis, 400 nm is a more realistic value. An experimental implementation of the method uses two twodimensional scanning measurements of emitted light intensity, Ixy, behind a polarization analyzer set to a specific angle, and two cross-polarized excitation geometries. Here, the subscripts x and y stand for polarization orientations parallel to the transition dipole moment, p, and perpendicular to it, s. One finds, for example, that I sp = tan 4 θ, I pp

(21.1)

where θ is the molecular orientation angle with respect to the chosen coordinate system of the nanofiber on the surface. Besides the solution +θ, this equation allows also the solution −θ. One of the solutions can be removed by either rotating the analyzer (and obtaining a continuous polarization dependence) or by setting the sample at two different, fi xed angles with respect to the incoming and outgoing electric field vectors. Some recent polarized second-harmonic SNOM measurements [76,77] have shown that for para-hexaphenylene nanofibers on mica, the orientation angle, θ, is fairly constant along the fiber with small-scale deviations of less than 10°. These deviations are attributed to inhomogeneities of individual fibers as being confirmed by polarized linear fluorescence measurements. Since at present, crystallographic measurements with local molecular resolution are nearly impossible for single nanofibers, the discussed optical measurements represent still the only direct way of obtaining information about local molecular order in nanofibers.

21.6 Devices 21.6.1 On the Way to Nanofiber Light Sources Thin fi lms from para-hexaphenylene molecules have successfully been used as the light-emitting layers in blue-light-emitting devices (LEDs) [78]. It must therefore be expected that a nanoscale LED could be made using p-6P nanofibers as the light emitters. Electroluminescence in organic semiconductors originates from the radiative decay of an exciton, which is formed by the binding of an electron and a hole polaron. Th is process requires injection of both charge species from the cathode and anode, respectively, and their transport to the recombination zone by an applied electric field [79]. Optimizing the light generation process therefore involves balancing the electron and the hole current, which requires a detailed study of the charge injection and transport properties. In order to investigate this, a technique for electric contacting of the organic nanofibers was devised recently [80]: Nanofibers are transferred to a prefabricated silicon dioxide platform on a silicon chip either by the spreading of a small volume of water with dispersed nanofibers or by a stamping technique. A suitable nanofiber is located and a rigid silicon wire with a diameter of a few hundred nanometers is positioned on top of and perpendicular to the organic nanofiber. This silicon wire acts as a local shadow mask during the subsequent deposition of electrode material (typically gold) by electron beam evaporation. Upon removal of the silicon wire shadow mask, two electrodes are formed with a gap in between, the size of which is determined by the shadow mask wire diameter. This electrode gap is spanned only by the organic nanofiber, which can then be probed electrically. Figure 21.13a shows a scanning electron microscope image of a p-6P nanofiber with two gold contacts. Two-point measurements were performed by applying a DC voltage and recording

21-9

Organic Nanowires

Current density (A/cm2)

0.4

200 nm

0.3 0.2 0.1 0.0

(a)

0

1

2

(b)

3

4

Voltage (V)

FIGURE 21.13 (a) SEM image of a p-6P nanofiber with two gold electrodes. (b) Typical current density vs. voltage characteristic for a gold contacted p-6P nanofiber.

the resulting current. Figure 21.13b shows typical results. Here, the measured current has been scaled with the nanofiber crosssectional dimensions to provide the current density. In order to investigate this curve in more detail, the measured electrical characteristics were analyzed with the Mott–Guerney theory. Th is approach assumes that the current is bulk-limited and that the interface effects are not important. However, in a real sample such effects will be present. This theory can therefore only provide an intrinsic “theoretical” upper limit to the current flow. Contact effects, defects, traps, or any other influencing factors will always cause the current to be smaller than the prediction [81]. However, if the permittivity, εr ε 0, and device length, L, are known, an estimate of the minimum mobility, μmin, can be extracted from the current density, J, vs. voltage, V, characteristics by μ min =

8 JL3 . 9ε r ε 0V 2

(21.2)

Similar to the method used by de Boer et al. [81], the current measured at the maximum bias voltage was used in the calculation of μmin as indicated with an arrow in Figure 21.13b. Several nanofiber samples were investigated [82], which showed values with a significant spread over four orders of magnitude between 3 × 10−5 and 3 × 10−1 cm2/V s. This indicates that for the majority of the investigated samples interface effects play a significant role. The conclusion to be drawn from these results is therefore that the carrier mobility for para-hexaphenylene nanofibers is at least 3 × 10−1 cm2/V s. In addition, it was found that gold was a suitable electrode material for hole injection [82]. The next step toward realizing a nanofiber LED is then to locate a suitable electron injector material and to develop an efficient method capable of depositing two different electrode materials for the anode and the cathode. The shadow mask method described above can be extended to facilitate the deposition of different anode and cathode material through the use of two shadow masks and two metal deposition steps [80]. Figure 21.14 shows an example of a nanofiber contacted with two different contact materials: gold and titanium. Even though gold injects holes, titanium is apparently not an efficient

Ti

Au

200 nm

FIGURE 21.14 SEM image of a p-6P nanofiber contacted with Au and Ti for the anode and cathode, respectively.

electron-injecting material since no electroluminescence was observed from such devices. Thus, a more efficient electron injecting material is required. It is concluded that it is well possible to contact selected organic nanofibers with electrodes made from different materials. The current through the fibers is, however, limited by the injection through the contacts. If a too high voltage is applied, the breakdown threshold will be reached and the material will desorb. Thus, in contrast to thin fi lms made of the same material (where electroluminescence has easily been achieved) the very fact that the material is nanoscaled makes it necessary to develop other integration and contacting methods. In fact, as detailed above, a direct, sophisticated growth of nanoscaled material under appropriately clean conditions might result in a significantly lower threshold for electroluminescence.

21.6.2 Waveguides Nanofibers act as waveguides if the index of refraction of the underlying substrate or surrounding medium is smaller than their index of refraction ( ε iso = 1.7) and if a critical width is


overcome. Note that the cutoff wavelength for the guided modes in the nanofibers is [28]

60

λc =

2 ε⊥ a m

(21.3)

and the number of possible modes, m = 1, 2, 3…, is restricted by the condition

Intensity (arb. units)

21-10

MONHP4 MOClP4 p-6P

50 40 30 20 10 0

2a m< λ

ε⊥ ε

400

ε⊥ − ε s ,

(21.4)

with λ the wavelength, εs the dielectric constant of the substrate, ε|| and ε⊥ the components of the dielectric permittivity tensors, and a the fiber width. Hence, waveguiding occurs roughly for a ≈ λ/2. In the case of negligible scattering from surface irregularities, low absorption in the fiber, and if one assumes negligible damping of the evanescent field in the substrate, propagation losses in the fiber are very small. However, it is difficult to couple light to a nanofiber. Thus, the experiments performed so far have relied on the fact that the light to be guided can be generated inside the fiber via UV excitation. It then propagates along the fiber and is scattered at breaks, where it can be detected. Such a method, of course, suffers from reabsorption of the light that is generated in the nanofiber. As a result, the complex dielectric functions of nanofibers have been determined and a very good agreement with simple electromagnetic theory has been found [28,83]. Thus nanofibers can be treated and used as dielectric waveguides or connecting elements in opto-chips. Since the dielectric function of the nanofibers can be changed by changing the molecular basis units, future applications might even involve active waveguides, which alter the light that is propagating along the fibers. This might prove especially useful in the context of combined photonic and plasmonic circuits.

21.6.3 Nanofiber Frequency Doublers Recent developments of blue-light-based photonic devices such as fully integrated opto-chips have renewed strong interest in the development of small and efficient units for light conversion. A nanoscaled, integrated frequency doubler would allow one to use cheap, power-saving, and brilliant near-infrared laser sources as the main source for intense, coherent blue light sources. Such a frequency-doubling unit could rely on easily integrable nanofibers. Optimization of frequency-doubling efficiency asks for optimized hyperpolarizabilities of the involved molecular building blocks. para-Hexaphenylene molecules can be easily grown into nearly single crystalline nanoaggregates, and thus resemble the optimum configuration for the given molecular constitutes. They also show the optimum optoelectronic properties, just limited by the molecular building blocks themselves. However, their hyperpolarizabilities are very low due to a lack of donor

420

440

460

480

Wavelength (nm)

FIGURE 21.15 Optical two-photon spectra of nanofibers made from different molecules as a result of near-infrared (790 nm) femtosecond laser excitation. The nonsymmetrically functionalized nanofibers result in strong second-harmonic generation.

and acceptor groups. A way out of this dilemma bases on the nonsymmetrical chemical functionalization of a para-quaterphenylene (p-4P) block with electron push and pull groups, for example, methoxy, amino, or cyano groups. As shown above, these molecules also form oriented nanofibers. The generation of a strong second-harmonic signal intensity asks for an in-phase adding of the second-harmonic signals of the individual molecular emitters. So the phases of the fundamental and the second-harmonic light have to be matched within the nonlinear crystal (phase-matching condition). Fortunately, the use of doubling elements with dimensions significantly smaller than the wavelength of the doubled light makes such considerations of phase matching obsolete. However, aggregation of the molecules into nano-elements still imposes restrictions on doubling efficiency related to the aggregation state (single-crystalline, poly-crystalline, amorphous). In Figure 21.15, we demonstrate how the optical emission spectrum upon infrared excitation changes as one uses differently functionalized molecules. For p-6P, the spectrum consists solely of two-photon luminescence (TPL), while it consists solely of second-harmonic generated (SHG) light in the case of methoxy- and amino-functionalized p-4P (MONHP4). If one uses chlorine instead of the amino groups (MOCLP4), one obtains both SHG and TPL. Obviously, molecular tailoring provides one with high optoelectronic flexibility.

21.7 Conclusions Besides the above-mentioned device applications (lighting, waveguiding, and frequency doubling), there are many other areas where nanofibers might prove useful as nanoscaled integrated elements. For example, nanosensing is an obvious choice since the molecules that make up the nanofibers can easily be made sensitive to specific binding agents. Upon binding, the waveguiding and fluorescence properties of the nanofibers might be changed, which can readily be read out once the fibers are integrated into simple circuits. Obstacles here are low renewability of the sensing material and possibly insignificant surface-to-volume ratio; that is, the optical properties might be

Organic Nanowires

dominated by the volume rather than the surface of the nanofiber, which reduces the sensitivity and points to the use of very small nanofibers. In terms of photonic applications, nonlinear optical response and lasing [5,84] seem to be the most interesting. Both take advantage of the fact that nanofibers are made of active photonic material and do not simply resemble a passive dielectric slab. Optical gain and nonlinear response can be tailored by modifying the molecular units [10], which opens up a broad perspective for future application areas.

Acknowledgments We thank the Danish research foundations FNU and FTP as well as the Danish national advanced technology trust HTF for financial support.

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63. Talapin, D., Black, C., Kagan, C., Shevchenko, E., Afzali, A., and Murray, C. J. Phys. Chem. C 111, 13244–13249 (2007). 64. Balzer, F. and Rubahn, H.-G. Adv. Funct. Mater. 15, 17–24 (2005). 65. Maibohm, C., Brewer, J., Sturm, H., Balzer, F., and Rubahn, H.-G. J. Appl. Phys. 100, 054304 (2006). 66. Zhao, Z. and Spano, F. J. Phys. Chem. C 111, 6113–6123 (2007). 67. Stampfl, J., Tasch, S., Leising, G., and Scherf, U. Synth. Met. 71, 2125–2128 (1995). 68. Meghdadi, F., Tasch, S., Winkler, B., Fischer, W., Stelzer, F., and Leising, G. Synth. Met. 85, 1441–1442 (1997). 69. Guha, S. and Chandrasekhar, M. Phys. Stat. Solidi (B) 241, 3318–3327 (2004). 70. Balzer, F., Pogantsch, A., and Rubahn, H.-G. J. Lumin. 129, 784 (2009). 71. Graupner, W., Meghdadi, F., Leising, G., Lanzani, G., Nisoli, M., De Silvestri, S., Fischer, W., and Stelzer, F. Phys. Rev. B 56, 10128–10132 (1997). 72. Kadashchuk, A., Andreev, A., Sitter, H., and Sariciftci, N. Synth. Met. 139, 937–940 (2003). 73. Kadashchuk, A., Andreev, A., Sitter, H., Sariciftci, N., Skryshevski, Y., Piryatinski, Y., Blonsky, I., and Meissner, D. Adv. Funct. Mater. 14, 970–978 (2004). 74. Faulques, E., Wéry, J., Lefrant, S., Ivanov, V., and Jonussauskas, G. Phys. Rev. B 65, 212202 (2002). 75. Simonsen, A. and Rubahn, H.-G. Nano Lett. 2, 1379–1382 (2002). 76. Beermann, J., Bozhevolnyi, S., Bordo, V., and Rubahn, H.-G. Opt. Commun. 237, 423–429 (2004). 77. Beermann, J., Marquart, C., and Bozhevolnyi, S. Laser Phys. Lett. 1, 264–268 (2004). 78. Klemenc, M., Meghdadi, F., Voss, S., and Leising, G. Synth. Met. 85, 1243–1244 (1997). 79. Parker, I. J. Appl. Phys. 75, 1656–1666 (1994). 80. Kjelstrup-Hansen, J., Dohn, S., Madsen, D. N., Mølhave, K., and Bøggild, P. J. Nanosci. Nanotechnol. 6, 1995–1999 (2006). 81. de Boer, R., Jochemsen, M., Klapwijk, T., Morpurgo, A., Niemax, J., Tripathi, A., and Pflaum, J. J. Appl. Phys. 95, 1196 (2004). 82. Henrichsen, H., Kjelstrup-Hansen, J., Engstrøm, D., Clausen, C., Bøggild, P., and Rubahn, H.-G. Org. Electron. 8, 540–544 (2007). 83. Volkov, V., Bozhevolnyi, S., Bordo, V., and Rubahn, H.-G. J. Microscopy 215, 241–244 (2004). 84. Quochi, F., Cordella, F., Mura, A., Bongiovanni, G., Balzer, F., and Rubahn, H.-G. Appl. Phys. Lett. 88, 041106 (2006).

IV Nanowire Arrays 22 Magnetic Nanowire Arrays Adekunle O. Adeyeye and Sarjoosing Goolaup ......................................................... 22-1 Introduction • Synthesis of Magnetic Nanowires • Magnetic Properties • Summary • Acknowledgments • References

23 Networks of Nanorods Tanja Schilling, Swetlana Jungblut, and Mark A. Miller .................................................. 23-1 Introduction • Percolation • Nanorod Networks in Composite Materials • Networks of Biological Nanorods • Summary • References

IV-1

22 Magnetic Nanowire Arrays 22.1 Introduction ...........................................................................................................................22-1 22.2 Synthesis of Magnetic Nanowires .......................................................................................22-2 Electrodeposition • Electron Beam Lithography • Interference Lithography • Nanoimprint Lithography • Deep Ultraviolet Lithography

22.3 Magnetic Properties ..............................................................................................................22-5

Adekunle O. Adeyeye National University of Singapore

Sarjoosing Goolaup National University of Singapore

Demagnetizing Fields • Shape Anisotropy • Effects of Nanowires Th ickness • Coercivity Variations as a Function of Field Orientations • Pseudo-Spin Valve Nanowires • Alternating Width Nanowire Arrays

22.4 Summary ...............................................................................................................................22-17 Acknowledgments ...........................................................................................................................22-17 References.........................................................................................................................................22-17

22.1 Introduction Magnetic nanostructured materials are of scientific interest both from a fundamental point of view and because of their potential in a wide range of emerging applications. Nanomagnets, by virtue of their extremely small size, possess both static and dynamic properties that are quantitatively and qualitatively very different from their parent bulk material. The magnetization reversal mechanism can therefore be drastically modified in nanomagnets confined to sizes that preclude the formation of domain walls. Topical reviews on the magnetic properties of nanostructures can be found in articles by Bader (2006), Srajer et al. (2006), Bader et al. (2007), Adeyeye et al. (2008), and Adeyeye and Singh (2008). Magnetic nanostructures are the basic building blocks of various spintronic applications. In data storage for example, as the recording media rapidly approaches the superparamagnetic limit (whereby stored information is unstable due to thermal fluctuations), patterned magnetic media consisting of arrays of single domain nanomagnets have been proposed as a candidate for recording density up to 1 Tb/in.2 (Ross, 2001; Martin et al., 2003; Terris et al., 2007). Ferromagnetic (FM) nanowires are attracting considerable interest due to their unique and tunable magnetic properties and their potential in a wide range of applications. The shapeinduced magnetic anisotropy of planar magnetic nanowires creates a relatively simple magnetization structure that is being exploited for scientific and technological studies. The interplay between electronic transport and magnetic structure in nanowires has been the focus of many recent experiments. Arrays of giant magnetoresistive nanowires offer attractive potential to serve diverse applications such as high-density magnetic

recording devices and magnetic field sensors. Understanding the magnetic and transport properties of nanowires is important for the design and optimization of miniature magnetoresistance (MR) heads for ultra-high-density data storage (Yuan and Bertram, 1993). The movement of domain walls (DW) in planar magnetic nanowires forms the basis of several recently proposed technological applications from magnetic logic to magnetic memory devices (Allwood et al., 2002, 2004, 2005). In magnetic-field-driven nanowire devices, it is necessary to control the direction of DW motion. The time taken to change the direction of magnetization of a nanowire device is directly related to the writing and reading of such a device and therefore an understanding of both its static and dynamic properties is very important. In the field of biomagnetism, magnetic nanoparticles are used in a broad range of applications, including cell separation (Moore et al., 1998; Safarik and Safarikova, 1999), bio-sensing (Baselt et al., 1998), studies of cellular functions (MacKintosh and Schmidt, 1999; Alenghat et al., 2000), as well as a variety of potential medical and therapeutic uses. Most of the applications to date have used spherical magnetic nanoparticles consisting of a single magnetic species and a suitable coating to allow functionalization with bioactive ligands. It has been proposed that by using electrodeposited magnetic nanowires, a variety of functions would be possible (Fert and Piraux, 1999). Due to their large aspect ratios, FM nanowires have large remenant magnetizations and, hence, can be used in low-field environments where the superparamagnetic beads are unsuitable. Hultgren et al. (2003) have demonstrated the use of FM Ni nanowires in cell-sorting applications. For example, by precisely modulating the composition along the length of the nanowires and carefully selecting the ligands that bind selectively to different segments of 22-1

22-2

a multicomponent wire, it is possible to introduce spatially modulated multiple functionalization in these nanowires. Arrays of magnetic nanowires are being exploited in microrheology, an emerging technique for investigating the viscoelastic properties of complex fluids. In microrheology, nanometer or micrometer scale particles suspended in the fluid are used to probe the local mechanical environment. Recently, Anguelouch et al. (2006) have shown a microrheological approach that applies FM nanowires to the study of interfacial rheology and have described the application of this technique to the measurements of thin viscous oil fi lms on aqueous subphases. Magnetic nanowires are also being explored as artificial cilia to sense acoustic signals. Cilia are employed in nature for sensing sound, fluid flow, touch, and other stimuli (McGary et al., 2006). A better understanding of the fundamental properties of nanowires is important for the realization of these practical applications and novel devices. The ability to characterize and extract quantitative information about the magnetic properties and the reversal mechanisms of nanowire arrays is very crucial in the design of the various magnetoelectronic devices mentioned above. This chapter is organized as follows. Section 22.2 is devoted to a review of the various techniques for synthesizing ordered magnetic nanowire arrays. This includes the electrodeposition technique, the electron beam lithography, the nanoimprint lithography, the interference lithography, and the deep ultraviolet lithography. In Section 22.3, we focus on the magnetic properties of the nanowire arrays as a function of the various geometrical parameters. The chapter ends with a summary in Section 22.4.

22.2 Synthesis of Magnetic Nanowires The fabrication of uniformly distributed high-quality magnetic nanowires over a very large area is a major challenge. The key issues to be considered when developing a nanofabrication technique are critical dimension control, resolution, patterned area, size, and shape homogeneity. In the last decade, various techniques for synthesizing ordered magnetic nanowire arrays have been developed.

22.2.1 Electrodeposition Template synthesis using electrodeposition, first reported by Possin in 1970 (Possin, 1970), is a low-cost, high-yield technique for producing large arrays of nanowires in the fabrication of tin nanowires in tracked mica films. This technique has been used to synthesize a variety of metal (Liu et al., 1998; Sun et al., 1999) and semiconductor (Routkevitch et al., 1996; Ohgai et al., 2005) nanowires. It has been shown that it is possible to pattern wires of various materials with diameters as small as 5 nm (Zeng et al., 2002). A schematic illustration of the electro-chemical deposition technique is shown in Figure 22.1. A metallic fi lm serving as a cathode is normally evaporated on one side of the membrane


Potentiostat

Reference electrode Counter electrode Aqueous solution with material to be deposited Porous membrane Working electrode

FIGURE 22.1 A schematic illustration of the experimental setup for the electrodeposition of nanowires in membrane pores.

prior to the electrodeposition process. The growth of the material within the pores is monitored from the current response at a constant potential (Whitney et al., 1993). Nanowires form in the pores of the template as they are fi lled with the electrodeposited material. This method can also be used to make multilayer wires. Two materials can be deposited from a single bath by switching between the deposition potentials of the two constituents materials (Blondel et al., 1994; Piraux et al., 1994; Doudin et al., 1996; Heydon et al., 1997; Schwarzacher et al., 1997). The electrodeposition process is stopped when the wire emerges from the surface, leading to a sudden increase in the plating current. Several factors should be taken into account in choosing the template material: the uniformity in pore size and shape, pore density, pore orientation, surface roughness of the pores, and the template thickness. The two most commonly used templates are: nuclear track-etched polycarbonate membranes (Bean et al., 1970; Guillot and Rondelez, 1981; Fischer and Spohr, 1983) and self-ordered anodized aluminum oxide fi lms (Masuda et al., 1997; Ba and Li, 2000). This technique has been used to fabricate magnetic nanowire arrays from different materials (Ferre et al., 1997; Aranda and Garcia, 2002; Chien et al., 2002). Track-etched polymer membranes have been shown to possess good properties with respect to pore shape, size, and the parallel alignment of the pores (Ferain and Legras, 1997). The template is made using the nuclear track-etched technology, where a high-energy heavy ion accelerated in a cyclotron is used to bombard a fi lm of polymer polycarbonate. The irradiated template is then etched in an adequate solution. The etch rate of the damaged tracks is much higher as compared with that of bulk fi lm, leading to the formation of the pores. The diameters of the pores are dependent on the etching time and the etch selectivity between the bulk material and the tracks. The main limitation of this method is the random distribution of the pores on the membrane. The anodic anodized oxide templates are prepared by the anodization of high-purity aluminum in an acid electrolyte

22-3

Magnetic Nanowire Arrays

under a constant voltage. The templates have a packed array of columnar hexagonal cells with central, cylindrical, uniformly sized pores. Anodized aluminum oxide (AAO) fi lm templates are stable at high temperatures in organic solvent and the pore channels in AAO films are uniform, parallel, and perpendicular to the membrane surface. Long periods of anodization have been shown to improve the pore arrangement, with an almost ideal honeycomb lattice structure being possible (Masuda and Fukuda, 1995; Friedman and Menon, 2007). The pore sizes in the AAO templates are controlled by a further process that involves dipping the template into phosphoric acid. AAO fi lm is ideal for the electrodeposition of nanowire arrays. The pore sizes typically range from 4 to 200 nm. The method has been used by various research groups to fabricate nanowire arrays of various materials (Metzger et al., 2000; Nielsch et al., 2001; Sellmyer et al., 2001; Khan and Petrikowski, 2002; Chiriac et al., 2003; Chun-Guey et al., 2006; Vazquez et al., 2006; Wu et al., 2006; Napolskii et al., 2007). This template synthesis technique is however limited by the distribution in pore size and orientation making it difficult to control the period and uniformity of the nanostructures (Meier et al., 1996). Also, nanowires grown via electrodeposition processes exhibit the so-called skyscraper effect associated with a lack of length uniformity and control (Yin et al., 2001).

22.2.2 Electron Beam Lithography Electron beam lithography (EBL) is a versatile high-resolution technique for patterning magnetic nanostructures. The principle of the EBL technique is the direct writing of desired structures on a thin resist layer with a focused beam of electrons as shown in Figure 22.2. The e-beam can create extremely fi ne patterns due to the small spot size of the electron. The substrate to be patterned is coated with an electron beam sensitive polymeric resist fi lm, normally polymethylmethacrylate (PMMA). The PMMA resist is then exposed to high-beam energy with a small spot size. Following the exposure, the sample is usually developed in a methylisobutylketone (MIBK): isopropyl alcohol (IPA) (1:3) to form a resist template on the substrate. PMMA is a positive resist, so the regions that are exposed to the e-beam are removed after development. Following the development of the resist, the patterns in the resist may be transferred to the substrate by using an etching process (Shearwood et al., 1993; Adeyeye et al., 1996; Fraune et al., 2000; Katine et al., 2000; Remhof et al., 2008) with the PMMA acting as an etching mask. The developed patterns can also be converted into magnetic nanostructures by metal deposition and the lift-off process (Kirk et al., 1997; Cowburn, 2000; Castano et al., 2003; Tsoi et al., 2003; Vavassori et al., 2003; Adeyeye and White, 2004; Miyawaki et al., 2006) or electroplating (Chou et al., 1994; Obarr et al., 1997; Martin et al., 2002). For electroplating, a thin gold layer is deposited onto the substrate prior to the resist coating. After the development, the sample is immersed into a plating bath and the magnetic material is electrodeposited into the openings created in the resist. Adeyeye et al. (Adeyeye et al., 1997b) have shown that it is possible to combine

Substrate Resist coating

PMMA Substrate

Pre-cleaning Exposure (Direct writing using e-beam) and development Pattern transfer (Etching)

Material deposition

Lift-off

FIGURE 22.2 A schematic illustration of the fabrication processes of nanowires via electron beam lithographic combined with lift-off techniques.

the wet etching selectivity of different materials to enable the patterning of epitaxial magnetic fi lms. There are limitations, however, with the use of EBL in fabricating large area magnetic nanostructures. The writing process in EBL is serial and very slow, thus making large area fabrication extremely difficult, although it can be used in the preparation of masks for optical lithography. It is also very difficult to fabricate closely packed high aspect ratio nanostructure arrays due to proximity effects.

22.2.3 Interference Lithography Interference lithography is another method used by researchers to fabricate large area magnetic nanostructures. In this method, a resist layer is exposed by an interference pattern generated by two obliquely incident laser beams without the use of a mask, as shown in Figure 22.3. The interference pattern, due to the two laser beams, consists of standing waves whose intensity vary with a period of p = λ/(2 sin θ), where λ is the wavelength of the light and θ is the half angle at which the two beams intersect (Schattenburg et al., 1995; Spallas et al., 1996). This method of patterning allows for the fabrication of regular arrays of fi ne features, without the use of complex optical systems. This technique is particularly useful for patterning parallel arrays of lines. In order to pattern complex structures, the sample has to go through successive exposures. Arrays of rectangular dots may be patterned by rotating the sample by 90° followed by a second exposure. The patterned area is determined by the diameter of the

22-4


Mold Mold Polymer Substrate Interference region

Substrate

Resist

Substrate Substrate

FIGURE 22.4

FIGURE 22.3 A simplified illustration of the interference of two laser beams on a sample surface.

two laser beams. For processes where no alignment is required, the technique is relatively cheap and simple. Several groups have used this method to fabricate large arrays of magnetic nanostructures (Wassermann et al., 1998; Farhoud et al., 1999; Vogeli et al., 2001; Zheng et al., 2001; Heyderman et al., 2004). The minimum period that can be achieved in interference lithography has a lower limit of λ/2. For commonly used Argon-ion lasers, with a wavelength of 350 nm, the lower limit on the period is about 200 nm. Shorter wavelength lasers tend to generate a broad range of wavelengths to enable the formation of sufficiently sharp interference patterns (Savas et al., 1999). Achromatic interferometric lithography has been used to overcome this limitation, allowing for the patterning of structures with a period down to 100 nm (Savas et al., 1996). The main drawback of interference lithography is that it is limited to patterning arrays of highly symmetrical elements only. Hence, for fabricating patterns that are arbitrarily shaped, other lithography techniques have to be explored.

22.2.4 Nanoimprint Lithography Nanoimprint lithography (NIL) is one of the most promising low-cost, high-throughput technologies for patterning magnetic nanowires over a large area. NIL is based on a different fundamental principle compared with the conventional lithography process where the local chemical properties are changed using radiation. The patterned nanostructures are formed by physical deformation of a deformable material using a mold. This creates a thickness contrast in the materials. The best way to fabricate the mold is to select a mold substrate deposited with a suitable mold surface, and then use electron beam lithography to pattern a resist, followed by etching to transfer the patterns in the resist into the mold (Chou, 1997). In order to deform a resist on a substrate by a mold, the mold and the substrate must be pressed toward each other to have direct contact. To achieve good uniformity, the pressure should be uniform across the imprint area and should be sufficiently high to create the necessary deformation in either the mold or the substrate or both in order to make their surfaces conform. A good mold separation from the imprinted patterns is achieved by using good mold release

A schematic illustration of the NIL process.

agents, which reduce the bonding of the mold and the resist and also by reducing the stress between the mold and the resist. A typical NIL process is shown in Figure 22.4. Features as small as 10 nm have been replicated onto the resist layer. NIL can be implemented using various methods. In the thermal imprint process, a thermoplastic material starts as a solid fi lm, then becomes a viscous liquid when its temperature is raised higher than the glass transition temperature (Tg) and returns to a solid when its temperature is reduced to below Tg (Chou et al., 1995; Chou, 2001). Photo-NIL uses a photocurable material as a resist (Haisma et al., 1996; Chou, 2001). A photo-curable material is initially in liquid form but is cured photochemically using photons rather than heat. Like the thermal process, this is also an irreversible process. Step and flash imprint lithography is a photo-NIL in which drops of a resist liquid are dispensed and imprinted on one single die at a time. The process is repeated as the imprint mold is stepped from die to die across the wafer repeating the resist drop and imprint cycle (Bailey et al., 2001) This method of patterning allows for higher alignment accuracy, as only a small area is patterned at one time and more importantly, a small mold can be used to fabricate the nanostructures over a large area. Magnetic nanostructures can be fabricated using the resist as a deposition template or etch mask. This technique has been used to fabricate large area magnetic nanostructures from different materials.

22.2.5 Deep Ultraviolet Lithography An optical lithography system consists of a light source, a condenser lens, mask, an objective lens, and finally the resist-coated wafer. The mask image is projected onto the resist-coated wafer. In the process of imaging, the light source uniformly illuminates the mask through the condenser lens. The radiation passing through the transparent regions of the mask is partially diff racted before reaching the objective lens. The low spatial frequencies corresponding to the larger patterns appear closer to the lens center, whereas high frequencies corresponding to the smaller patterns and pattern corners fall toward the periphery of the lens pupil. The objective lens, being of finite size, cannot collect all of the light in the diffraction pattern. The diffracted radiation accepted by the pupil is collimated by the objective lens and interferes at the wafer plane to constitute the image. The loss of diff raction

22-5


information is the ultimate limiter of the image quality and resolution. The smallest features that can be printed, that is, the resolution is given by the relation R = K1

λ NA

where K1 is a process-dependent parameter usually ~0.6 for conventional lithography with a theoretical limit of 0.25 λ is the wavelength of the exposure tool NA is the numerical aperture In our fabrication technique, we have used a KrF exposing wavelength of 248 nm, and a DUV lithography scanner with a maximum NA of 0.68. The K1 factor is a measure of the degree of difficulty for printing a particular feature. There has been tremendous progress to reduce K1 through the use of resolution enhancement technology such as various phase shift mask approaches, off-axis illumination, optical proximity correction methods, and other approaches (Brunner, 2003). A process with K1 of 0.8 is considered easy; a process with K1 smaller than 0.5 is extremely difficult to achieve without any resolution enhancement techniques. For a general review of the application of deep ultraviolet lithography in the fabrication of magnetic nanostructures; the reader is referred to Adeyeye and Singh (Adeyeye and Singh, 2008). In order to pattern nanostructures below the conventional resolution limit of the optical exposure tool, the use of an aggressive resolution enhancement technique is necessary. Arrays of FM nanowires with lateral dimensions below the conventional resolution limit have been fabricated using resolution enhancement techniques such as alternating phase shift and chromeless phase shift masks (Singh et al., 2004). All the magnetic nanowire arrays presented in this chapter were fabricated using this technique.

22.3 Magnetic Properties Magnetic nanowires have attracted a lot of interest both from a fundamental viewpoint and because of their potential for use in various magnetoelectronic applications. A lot of research has focused on understanding the static and dynamic properties of homogeneous width FM nanowire arrays. Th is section will focus on some of the recent works. It will be shown that the reversal process evolved from a DW-dominated process to coherent spin rotation when the wire width is reduced to submicron.

22.3.1 Demagnetizing Fields Polycrystalline FM films consisting of small randomly distributed single crystalline grains will possess no crystal anisotropy. If such a film is patterned into spherical shaped magnets, the applied field will magnetize the sample to the same extent in any direction because there is no preferred direction of magnetization.

However, if the film is patterned into nanowires, magnetization prefers to lie along the length of the wire due to shape anisotropy. Shape anisotropy occurs because the magnetization vector prefers to lie along the long axis where the demagnetizing field is minimum and the magnetostatic energy is lowest. Demagnetizing fields hold the key to an understanding of the magnetic properties of nanowire arrays. When a magnetic wire of finite size is magnetized, the field experienced (Hexp) is different from the applied field (Happl). The difference that is usually known as the demagnetizing field (Hdem) is due to the presence of magnetic poles at the magnetized surface, which gives rise to a magnetic field that counteracts the applied field. H exp = H appl − H dem The magnitude of the demagnetizing field is a function of the magnetization in the material (i.e., pole strength) and the pole separation determined by the sample geometry. The demagnetizing field is both opposite and proportional to the magnetization. The constant of proportionality is known as the demagnetization factor, or more generally as the demagnetization tensor. H dem = − N d M The demagnetization factor for a normally magnetized disk is 1, while for an infinite cylinder magnetized along the long direction is 0. If the magnetization has components along more than one of the three axes of the sample, then a tensor relation is required. For a magnetic nanowire array, the strength of the demagnetizing field is determined by the ratio of the lateral dimension to the fi lm thickness. The computation of the demagnetizing field is rather complicated for nonellipsoidal shaped nanomagnets because the magnetization is nonuniform. There is a general theorem in micromagnetics that assumes that under rather relaxed conditions, a body of arbitrary shape is equivalent to an ellipsoid, both of them uniformly magnetized to saturation and of equal volume (Brown and Morrish, 1957; Brown, 1960). The concept of the equivalent ellipsoid is intuitively very appealing, but it is not straightforward to find the correct shape parameters of an ellipsoid in order to establish this equivalence (Beleggia et al., 2006b). It may be misleading, however, in some cases to assume that some symmetrical FM bodies would closely resemble an actual ellipsoid. The justification for the substitution of one shape (ellipsoid of revolution) for the other cylinder, for example, is due to the fact that until recently (Beleggia and De Graef, 2003; Millev et al., 2003), the demagnetizing factors for the cylinder have only been given in terms of elliptic integrals (Rhodes et al., 1962; Joseph, 1966) or numerically (Brown, 1962), while the counterpart expressions for the spheroid have been known for quite a while in terms of elementary functions (Osborn, 1945; Stoner, 1945). In addition, the difference in the demagnetization factor due to this swap of shapes has been considered to be negligible. However,

22-6


22.3.2 Shape Anisotropy

None

SEI

1.0 kV ×10,000

1 μm WD 6.9 mm

FIGURE 22.5 Scanning electron micrograph of 20 nm thick Ni80Fe20 nanowire array with width 185 nm and edge-to-edge spacing of 35 nm.

it has been shown in the case of very thin disks that the actual difference can be quite significant (Vedmedenko et al., 2003). Beleggia et al. (Beleggia et al., 2006a) have presented a self-contained method for the calculation of the demagnetization tensor for a uniformly magnetized ellipsoid based on a Fourier-space approach to the micromagnetics of magnetized bodies.

The concept of shape anisotropy in nanowire arrays can be illustrated by considering an array of 20 nm thick Ni80Fe20 nanowires with a width of 185 nm and edge-to-edge spacing of 35 nm, as shown in Figure 22.5. This nanowire array was fabricated using DUV lithography and the lift-off process. Experimentally, by measuring the magnetic properties of the nanowire arrays with the applied field parallel (θ = 0°) and perpendicular (θ = 90°) to the wire axis, it is possible to investigate the effects of shape anisotropy. Typical normalized hysteresis loops for fields applied along θ = 0° (long axis) and θ = 90° (short axis) are shown in Figure 22.6. For fields applied along θ = 0° (corresponding to the easy axis), the nanowire array displays a near rectangular hysteresis loop as expected with a large jump, as seen in Figure 22.6a. The nanowire array exhibits a high squareness ratio of 0.93, a coercive field of 220 Oe, and a saturation field of 350 Oe. For fields applied perpendicular to the wire axis, θ = 90° (the hard axis), however, a sheared M–H loop with a very small coercive field, 2 Oe and squareness ratio of 0.02 is obtained, as shown in Figure 22.6b. A significant increase in the saturation field of the M–H loop is obtained in comparison with the loops obtained for fields applied along the wire axis, with a field of 800 Oe. The observed changes in the magnetic properties for fields applied along long and short axes can be attributed to the shape anisotropy effects being the dominant anisotropy for

1 Magnetization (Norm.)

θ = 0°

θ = 90°

0.5

0 H

200 200

–0.5 H –1

–400

(a)

–200 0 200 Applied field (Oe)

400 (b)


2000


θ = 0° 0.5

0

–0.5

–1 (c)

θ = 90°

–40


(d)


40

FIGURE 22.6 M–H loops for 20 nm thick Ni80Fe20 nanowire arrays, w = 185 nm and s = 35 nm, with fields applied along (a) θ = 0° and (b) θ = 90°, with respect to the long axis. The corresponding M–H loops for the 20 nm Ni80Fe20 reference film with fields applied along (c) θ = 0° and (d) θ = 90°.

22-7

Magnetic Nanowire Arrays 1

t = 10 nm

0.5 0 –0.5

H

(a) t = 40 nm 0.5 0 Magnetization (Norm.)

the nanowire array, with the easy axis being along the long axis of the nanowire (Goolaup et al., 2005b). Th is is also in qualitative agreement with results of earlier published works (Adeyeye et al., 1997a). The corresponding normalized magnetization loops for 20 nm thick Ni80Fe20 reference (unpatterned) samples with fields applied along the θ = 0° and θ = 90° axes are shown in Figure 22.6c and d. This reference sample was deposited at the same time and under the same conditions as the nanowire arrays. The M–H loops are almost identical for both orientations of the applied field and display a very low coercivity of 2 Oe and a small saturation field of 20 Oe. This implies that, as expected, the reference sample has negligible intrinsic magnetic anisotropy as compared with the shape anisotropy of the nanowire. The difference between the M–H loops of the reference film and the nanowire array can be attributed to different mechanisms mediating the reversal process. For the continuous film, the reversal process is dominated by DW propagation.

–0.5 (b) t = 80 nm 0.5 0

22.3.3 Effects of Nanowires Thickness As shown earlier, the magnetic properties of nanowire arrays are strongly dependent on the spatially varying demagnetizing field, which is related to the geometric parameters. In order to understand the effects of the thickness of the Ni80Fe20 nanowire array on its magnetic properties, a series of experiments were performed. In order to investigate the effect of film thickness, the lateral dimensions of the wire arrays were fi xed while the Ni80Fe20 wire thickness (t) was varied from 10 to 150 nm.

–0.5

22.3.3.1 Easy Axis Behavior

–1 –800 (d)

The magnetization loops for fields applied along the easy axis (θ = 0°) for the nanowires of w = 185 nm and s = 35 nm, as a function of the Ni80Fe20 wire thickness, is shown in Figure 22.7. As expected, the M–H loops are markedly sensitive to the wire thickness. In general, for all the wire thicknesses, near rectangular M–H loops were obtained. For thin fi lms (t = 10 nm), as the field is increased from negative saturation, the maximum moment is retained until the applied field reaches zero, as shown in Figure 22.7a, followed by a gradual increase in the magnetization as the field is increased toward positive saturation. A further increase in the applied field, to 120 Oe, leads to an abrupt rise in the magnetization resulting in positive saturation. Th is field corresponds to the coercivity of the nanowire array. As the nanowire thickness is increased to 40 nm, there is a corresponding increase of the coercivity to 410 Oe, accompanied by a slight tilt, within the region where the magnetization changes direction, as shown in Figure 22.7b. The slight tilting in the M–H loops observed has been attributed to the switching field distribution and the dipolar coupling among the nanowires (Castano et al., 2001a,b; Gubbiotti et al., 2005; Vavassori et al., 2007). A further increase in the wire thickness to 80 nm results in a slight increase in the coercivity to 475 Oe, as shown in Figure 22.7c. The tilt in the M–H loop, first seen for t = 40 nm, becomes more pronounced. As the wire thickness is further increased to 150 nm, a drastic

(c) t = 150 nm 0.5 0 –0.5


800

FIGURE 22.7 Representative M–H loops for Ni80Fe20 nanowire arrays, of width = 185 nm and edge-to-edge spacing = 35 nm, with field applied along the easy axis for different wire thicknesses.

reduction in the coercivity to 140 Oe with a slight shearing in the M–H loop is observed, as shown in Figure 22.7d. The results clearly show that the easy axis coercive field of the Ni80Fe20 nanowire array of fi xed width is strongly dependent on the fi lm thickness. To gain an insight into the trend of the easy axis coercive field, as a function of Ni80Fe20 fi lm thickness, the coercivity was extracted from the M–H loops for all the thicknesses studied. A plot of the easy axis coercivity as a function of the thickness to width (t/w) ratio of the nanowires is shown in Figure 22.8. The coercive field follows a nonlinear increase with a t/w ratio, reaching a peak of 520 Oe when the t/w ratio = 0.54. This t/w ratio corresponds to a fi lm thickness of 100 nm. A further increase in the t/w ratio leads to a rapid reduction in the coercivity of the nanowires. This non-monotonic t/w ratio dependence suggests that there is a crossover in the magnetization reversal mechanism in the nanowires, as the thickness of the fi lm is increased. This has also been confirmed by others (Goolaup et al., 2005a; Vavassori et al., 2007).

22-8

Handbook of Nanophysics: Nanotubes and Nanowires 1

500

0.5 0

400

H –0.5

300

(a)

100

0

t = 40 nm

0.5


Coercive field (Oe)

t = 10 nm

0

0.2

0.4 0.6 Thickness/width

0.8

1

FIGURE 22.8 Easy axis coercive field of the Ni80Fe20 nanowire array of width = 185 nm and s = 35 nm as a function of thickness/width ratio. The dotted line is a visual guide.

–0.5 (b)

t = 80 nm

0.5 0 –0.5 (c)

22.3.3.2 Hard Axis Behavior

t = 150 nm

0.5 0 –0.5 –1 –3000 (d)

–1500 0 1500 Applied field (Oe)

3000

FIGURE 22.9 Representative M–H loops for Ni80Fe20 nanowire arrays of w = 185 nm and s = 35 nm, with field applied along the hard axis for different wire thicknesses.

5000

4000 Saturation field (Oe)

The magnetization loops for fields applied along the hard axis (θ= 90°) of the nanowire array as a function of the Ni80Fe20 fi lm thickness are shown in Figure 22.9. Again, an evolution in the hysteresis curve as a function of the Ni80Fe20 wire thickness is observed. The shape of the M–H loop changes from an almost sharp switching to a highly sheared curve as the wire thickness is increased. For t = 10 nm, a linear reversible M–H curve with a steep slope is obtained, as seen in Figure 22.9a. The M–H loop exhibits a small hard axis saturation field of 345 Oe. When t = 40 nm, there is a slight shearing of the M–H loop, resulting in an increase in the saturation field to 1160 Oe, as seen in Figure 22.9b. Interestingly, when t is increased to 80 nm, the M–H curve displays a small coercive field of 140 Oe, as shown in Figure 22.9c. Also, the hysteresis curve for t = 80 nm is less steep as compared with t = 40 nm, thus exhibiting a much larger saturation field of 1670 Oe. As t is increased to 150 nm, the M–H loop becomes highly sheared displaying an almost “S” shaped curve with a saturation field of 3500 Oe, as shown in Figure 22.9d. In order to investigate the effect of wire thickness on the hard axis behavior, we have extracted the saturation field of the nanowires from the M–H loops shown in Figure 22.9. A plot of the saturation field of the nanowires as a function of the Ni80Fe20 wire thickness is shown in Figure 22.10. It can be seen that the saturation field of the nanowire array increases monotonically as the Ni80Fe20 thickness is increased. This can be explained by the strong influence of the demagnetizing field across the wire width. The hard axis saturation field, Hs, is given by (Bajorek et al., 1974)

0

3000

2000

1000

0

0

40

80

120

160

Thickness (nm)

3 Hs ≈ Hk + Hd 2

FIGURE 22.10 Hard axis saturation field as a function of fi lm thickness for Ni80Fe20 nanowires with w = 185 nm and s = 35 nm.

22-9


where Hk is the magnetic anisotropy constant, which is negligible in our structures Hd represents the average demagnetizing field For wire arrays, the demagnetizing field is given by (Pant, 1996): μ o H d = M S wt α(r ) where α(r) is a function of the ratio of inter-wire spacing to the wire width (s/w). In the limit of (s/w) → 0, the factor α(r) → 0; in the opposite limit of (s/w) → ∞, α(r) → 1. The first limit corresponds to a continuous fi lm, where the wires are in physical contact and the second limit corresponds to the isolated wires. For the wire arrays investigated, s = 35 nm, w = 185 nm, and α(r) ≈ 0.34, a constant value. Thus, the demagnetizing field varies as t/w. As the thickness of the nanowire increases, the demagnetizing field across the wire increases leading to an increase in the saturation field, consistent with the experimental results obtained.

22.3.4 Coercivity Variations as a Function of Field Orientations The angular dependence of coercivity is known to provide information on the magnetization reversal mode in nanowires

(Lederman et al., 1995; Adeyeye et al., 1997a; Hao et al., 2001; Han et al., 2003; Perez-Junquera et al., 2003). To better understand the evolution of the magnetization reversal modes, the values of coercive fields were extracted from the hysteresis loops as a function of the orientation of the applied field for different Ni80Fe20 fi lm thicknesses. The variation of the coercivity (Hc) with the applied field orientation (θ) for various wire thicknesses is shown in Figure 22.11. Generally, for all the thickness ranges investigated, the coercivity is markedly sensitive to the orientation of the applied field and is symmetrical about field orientation of 180°. Several mechanisms may be responsible for magnetization reversal: coherent rotation, magnetization curling, magnetization buckling, or DW motion. For nanowires, two modes are considered as being important; coherent rotation (Stoner and Wohlfarth, 1991) and curling magnetization (Brown, 1957; Frei et al., 1957; Aharoni and Shtrikman, 1958; Shtrikman and Treves, 1959; Ishii, 1991). It is well-known that Ni80Fe20 is a soft magnetic material with low intrinsic magnetic anisotropy. Since all the wire dimensions are larger than the exchange length (Ha et al., 2003), we can expect deviations from uniform magnetization and as a result, the occurrence of DW nucleation at lower fields than the curling field or anisotropy field, for fields applied along the hard-axis. For t = 10 nm, the maximum coercive field occurs along the nanowire axis (θ = 0°), as shown in Figure 22.11a. As the field orientation (θ) increases, the coercive field decreases, reaching a minimum for fields applied along the hard-axis (θ = 90°). A bell-shape angular variation is observed as field 500 Coercive field (Oe)

Coercive field (Oe)

150

100

50

400 300 200 100

10 nm 0 –180

–120

(a)

40 nm –60 0 60 Field orientation (θ)

120

0 –180

180

–120

(b)

–60 0 60 Field orientation (θ)

120

180

–60

120

180

Coercive field (Oe)

Coercive field (Oe)

350 500 400 300

300 250 200 150 100

200 –180 (c)

150 nm

80 nm –120

–60

0

60

Field orientation (θ)

120

0 –180

180 (d)

–120

0

60


FIGURE 22.11 Coercive field as a function of the field orientation with respect to the nanowire (w = 185 nm and s = 35 nm) axis, for Ni80Fe20 wire thickness ranging from 10 to 150 nm.

22-10


orientation is varied from θ = −90° to θ = +90°. Th is angular variation of the coercive field is consistent with the coherent rotation reversal mode based on the Stoner–Wolhfarth model (Stoner and Wohlfarth, 1991). As the wire thickness is increased to 40 nm, a slight departure from coherent rotation is observed for the field orientation of 60° and 120°, as shown in Figure 22.11b. Th is departure from coherent rotation can be attributed to the onset of a curling mode of reversal. The coercive field vs. the θ curve for a nanowire array with t = 80 nm is shown in Figure 22.11c. A completely different field orientation dependence of the coercive field is observed when compared with the coherent mode for t = 10 nm. As the field orientation increases from θ = 0° to θ = 30°, a slight decrease in the coercive field is fi rst observed. With a further increase in field orientation, an increase in the coercive field with a peak at θ = 75° is seen. For θ ≥ 75°, the coercive field decreases reaching a minimum along the hard-axis (θ = 90°). Th is field orientation dependence of the coercive field suggests that the reversal process is dominated by a combination of coherent rotation and the curling mode of the magnetization reversal process. The region where there is an increase in coercivity with field orientation (i.e., 30° ≤ θ ≤ 75°) may be attributed to curling mode reversal (Brown, 1957; Frei et al., 1957; Aharoni and Shtrikman, 1958; Shtrikman and Treves, 1959). The coherent rotation reversal mode is present for fields applied close to the easy and hard axis. For t = 150 nm, the coercive field increases as the field orientation increases with respect to the wire axis, reaching a peak for field orientation θ = 75°, as shown in Figure 22.11d. Beyond this field orientation, θ > 75°, the coercive field decreases with increasing field orientation, reaching a minimum at θ = 90°. The curve displays a U-shape within the field orientation θ = −90° and θ = +90°. Th is angular dependence of the coercive field suggests that the magnetization reversal mechanism is dominated by the curling mode of the reversal process. To aid in the understanding of the reversal process of nanowire arrays with t = 150 nm, the experimental results were modeled using the theoretical prediction proposed in the article by Meier et al. (1996). For an infi nite cylinder with a curling mode of reversal, the coercive field is given as (Aharoni and Shtrikman, 1958)

450

μ oH c =

Ms 2

a(1 + a) a + (1 + 2a) cos 2 θ 2

where a = −1.08 (d0/d)2. The exchange length d0 = 2( A /M s ), where A is the exchange constant. This model is valid for cylindrical wires; however, the structures investigated in this work have rectangular geometry, which is not analytically solvable. Hence, the experimental results were fitted with this theoretical prediction. A plot of the experimental data and the theoretical prediction is shown in Figure 22.12; the dotted line is the predicted coercive field curve for the curling mode of magnetization reversal and the dots are the experimental points. There is a very good agreement between the measured coercivity and the theoretical prediction except for field orientation

Experiment Model

400

Coercive field (Oe)

350 300 250 200 150 100 50 –180

–135

–90

–45

0

45

90

135

180


FIGURE 22.12 Angular variation of coercive field together with theoretical prediction based on curling magnetization for 150 nm thick Ni80Fe20 nanowire array (w = 185 nm and s = 35 nm).

when θ = −90° and θ = +90°. For fields applied along the hard axis, a minimum is obtained as opposed to the theoretical prediction of the maximum coercive field. This is consistent with the results reported by Han et al. (2003), where curling magnetization is present for small θ angles and the coherent rotation occurs at larger θ values. Thus, for thick nanowires, t = 150 nm, the magnetization reversal is dominated by the curling mode except at fields applied along the hard axis, θ = 90°, where coherent rotation is responsible for the reversal.

22.3.5 Pseudo-Spin Valve Nanowires The effect of magnetostatic interactions in patterned heterostructures consisting of FM layers of different materials or two FM layers of the same material of different thicknesses, separated by a spacer layer is of fundamental interest. Patterned layered nanomagnets exhibit interesting magnetic properties because of interlayer magnetostatic coupling that can lead to a relative anti-parallel alignment of magnetization at remanence (Castano et al., 2001a). In addition, magnetotransport properties of patterned layered nanostructures have been shown to yield a higher giant magnetoresistance (GMR) when compared with the unpatterned fi lm (Hylton et al., 1995). Recent works on patterned heterostructures have typically been aimed at understanding the magnetization reversal process and transport properties of isolated pseudo-spin-valve (PSV) nanoelements (Castano et al., 2001b; Ross et al., 2005) and nanowires (Kume et al., 1996; Katine et al., 1999; Castano et al., 2002b; Morecroft et al., 2005). A systematic study of the magnetic properties of closely packed and isolated homogeneous width PSV nanowire arrays is presented in this section. PSV nanowires were fabricated by exploiting the differential thickness coercivity of single fi lm nanowires, shown in Figure 22.7. Nanowire arrays with a width of 185 nm and spacing (s) of 185 nm (isolated) and 35 nm

22-11


(closely packed) were patterned using DUV lithography and the lift-off process. The PSV structures consisting of (bottom to top) Ni80Fe20 (10 nm)/Cu(tCu nm)/Ni80Fe20 (80 nm) were fabricated using the electron beam deposition technique. As the inter-element spacing is reduced in arrays of magnetic nanostructures, the stray fields generated during the magnetization reversal process may influence the internal magnetic domain structure and reversal mechanisms of neighboring elements. The effect of the inter-element interaction is complicated by the fact that the dipolar fields depend on the magnetization state of each element, which in turn depends on the fields due to the adjacent elements. The thickness of the Cu spacer layer, tCu, was varied from 2 to 35 nm in order to investigate the effect of interlayer coupling on the overall magnetization reversal process. Scanning electron micrographs (SEM) of the closely packed and isolated PSV nanowires are shown in Figure 22.13a and b, respectively. A schematic of the PSV structure is shown in Figure 22.13c. The respective insets in Figure 22.13 show the tilted cross-sectional view of the PSV nanowire arrays with tCu = 35 nm. The nanowire arrays have uniform width and inter-wire spacing as can be clearly seen from the SEM micrographs.

ECE/ISML

SEI

3.0 kV ×20,000

1 μm

The normalized hysteresis loops for Ni80Fe20 (10 nm)/Cu(10 nm)/ Ni80Fe20(80 nm) nanowire arrays for fields applied along the easy axis, θ = 0°, of the nanowire arrays are shown in Figure 22.14. The respective arrows correspond to the possible magnetization alignment of the FM layers comprising the PSV nanowire arrays. Both the closely packed and isolated nanowire arrays display a doublestep reversal M–H loop. For the array with s = 35 nm, as the field is reduced from positive saturation, the first drop in magnetization occurs at an external field of −5 Oe, followed by a quasi-stable plateau-like region within the field range of −20 Oe to −190 Oe, as seen in Figure 22.14a. Further reduction in the external field leads to a gradual decrease in magnetization resulting in negative saturation at an external field of −720 Oe. For the PSV nanowire array with s = 185 nm, the first drop in magnetization occurs at the same external field as the wire array with s = 35 nm, as shown in Figure 14b. The quasi-stable plateau, however, occupies a larger field range of −30 Oe to −245 Oe. The magnetization then decreases to lead to negative saturation at an external field of −500 Oe. For both sets of wire arrays, the first drop in magnetization corresponds to the switching of the 10 nm thin layers, followed by the switching of the 80 nm thick layers leading to negative saturation. This is

WD 6.3 mm

θ

(a)

Ni80Fe20 (80 nm) tCu Ni80Fe20 (10 nm)

SEI

(b)

3.0 kV ×20,000

1 μm

S

WD 6.9 mm

(c)

FIGURE 22.13 Scanning electron micrograph of Ni80Fe20(10 nm)/Cu(35 nm)/Ni80Fe20(80 nm) spin valve nanowire arrays with width 185 nm, (a) edge-to-edge spacing = 35 nm and (b) edge-to-edge spacing = 185 nm; (c) schematic representation of the spin valve nanowires. (Reproduced from Goolaup, S. et al., J. Appl. Phys., 100, 114301, 2006. With permission.)

22-12

Handbook of Nanophysics: Nanotubes and Nanowires s = 185 nm

s = 35 nm


0.23 0.5

0

–0.5

–1 (a)

–800

–400

0

400

Applied field (Oe)

–400 (b)

0

400

800

Applied field (Oe)

FIGURE 22.14 M–H loops for Ni80Fe20(10 nm)/Cu(10 nm)/Ni80Fe20(80 nm) nanowire arrays with fields applied along the wire axis for (a) s = 35 nm and (b) s = 185 nm.

evidenced by the percentage drop in magnetization following the reversal of each layer comprising the PSV nanowire. For the reversal of the 10 nm thin layers, there is a drop of 0.23 in the magnetization. This is consistent with the percentage of magnetic moment of the 10 nm Ni80Fe20 thin layer as compared with the total magnetic moment of layers comprising the PSV nanowire. The quasi-stable plateau observed for both sets of PSV nanowire arrays correspond to the region of anti-parallel alignment of the magnetization of the two layers comprising the PSV nanowires. Interestingly, for both sets of PSV nanowires, the switching fields for the thin (10 nm) and thick (80 nm) layer do not coincide with the coercive field of the individual single-layer fi lms. This difference is due to the interlayer coupling between the two FM layers in the nanowires. Interlayer coupling between the FM layers results in a change of the effective field at which the magnetization of each layer reverses its orientation. As the exchange length of Ni80Fe20 is 5.4 nm (Ha et al., 2003), for tCu = 10 nm, the interlayer magnetostatic coupling is expected to dominate the reversal process in the PSV nanowires. The interlayer magnetostatic coupling tends to promote the anti-parallel alignment of the magnetization of the two layers to reduce the magnetostatic energy of the system during the reversal process. For s = 35 nm, following the plateau-like region, the reversal of the 80 nm Ni80Fe20 thick layer occurs over a field range extending from −190 Oe to −720 Oe. The 80 nm Ni80Fe20 thick layer for the PSV nanowire array with s = 185 nm switches over a smaller field range of −245 Oe to −500 Oe. Th is is attributed to the much larger switching field distribution (SFD) in the closely packed nanowire array, due to the dipolar coupling effect between neighboring nanowires. It has been shown that the dipolar contribution for a homogeneously magnetized cylindrical nanowire, with a density of magnetic poles +M and −M at the two edges of the wire, is given by (Velazquez et al., 2003)

H (r , z ) =

⎡ −MR 2L ⎢ s 2 − 2z 2 ⎢ 2 2 5 4 ⎢⎣ (s + z ) 2

⎤ ⎥ ⎥ ⎥⎦

where s corresponds to the edge-to-edge spacing between the wires z is the distance along the wire axis The radius and length of the cylindrical nanowire is denoted by R and L, respectively. The dipolar field varies both as a function of the edge-to-edge spacing, s, and along the wire length, z. The dipolar field has been shown by Velázquez et al. (2003) to decay rapidly as the distance between the wires is increased. As the equation above is not solvable for rectangular geometry, the dipolar field can be approximated by computing the ratio for different edge-to-edge wire spacing of cylindrical wires. By substituting s with 35 and 185 nm, the dipolar coupling is found to be reduced by a factor of 150 as the edge-to-edge spacing is increased from 35 to 185 nm. 22.3.5.1 Effects of Cu Spacer Layer Thickness To understand the effect of the interlayer coupling in the PSV nanowires, a systematic investigation of the effects of Cu spacer layer thickness has been conducted (Goolaup et al., 2006). The representative hysteresis loops for PSV nanowire arrays with s = 35 nm and s = 185 nm as a function of the spacer layer thickness, tCu, for fields applied along the easy axis of the wire are shown in Figure 22.15. The hysteresis loops are markedly sensitive to both the edge-to-edge spacing and the Cu spacer layer thickness. The evolution in the magnetization reversal process as a function of the spacer layer thickness can be attributed to the different coupling mechanisms between the two magnetic layers comprising the PSV nanowire arrays (Parkin et al., 1990; Barnas, 1992; Bruno, 1993; Bloemen et al., 1994; Castano et al., 2002a; Zhu et al., 2003). For tCu = 2 nm, the FM exchange coupling between the two FM layers dominates the reversal process in the PSV nanowire arrays. In FM coupling, the parallel alignment of the magnetization of the FM layers is favored and the field at which the magnetizations of the FM layers are anti-parallel aligned is reduced. When the FM coupling is very strong, the magnetization of the layers reverses their orientation simultaneously and a single rectangular loop is obtained. For exchange coupled FM layers, where each

22-13

Magnetic Nanowire Arrays 1 185 nm 35 nm

0.5 0 –0.5

tCu = 2 nm

(a)

0.5

Magnetization (Norm.)

0 –0.5 tCu = 5 nm

(b)

0.5 0 –0.5 tCu = 20 nm

(c) 0.5

HBottom

0

HLateral

–0.5 HTop –1 –800 (d)

tCu = 35 nm


800

FIGURE 22.15 Representative M–H loops for both the closely packed and isolated nanowire arrays with Ni80Fe20(10 nm)/Cu(tCu nm)/ Ni80Fe20(80 nm) fi lm as a function of the Cu spacer layer thickness, tCu. (Reproduced from Goolaup, S. et al., J. Appl. Phys., 100, 114301, 2006. With permission.)

layer is in the single domain state, the reversal field, Hr, is given by (Yelon, 1971)

Hr =

t i M Si H ir0 + t j M Sj H r0j t i M Si + t j MSj

where the i and j represent the two layers with small and large coercive fields, respectively. The thickness and saturation magnetization of the magnetic layers are denoted by tz z and M Sz , respectively, with z = i, j. H r0 denotes the field at which the magnetization of the respective single fi lm magnetic

layer switches. For the structures: ti = 10 nm, t j = 80 nm, and M Si = M Sj . The computed reversal fields are 292 Oe and 424 Oe for the PSV nanowire arrays with s = 35 nm and s = 185 nm, respectively. Due to the FM exchange coupling between the two FM layers, the switching of the 10 nm Ni80Fe20 layer exerts an additional field on the top 80 nm Ni80Fe20 layer. Th is field, coupled with the external applied field, causes the magnetic moment of the 80 nm Ni80Fe20 (thick) layer to rotate prior to switching, leading to a gradual decrease in magnetization as observed in Figure 22.15a. The slight deviation between the computed and experimental values may be due to the presence of additional FM coupling mechanisms such as pin holes, due to a break in the spacer layer and “orange peel” coupling (Neel, 1962). Th is will lead to a stronger FM coupling in the layers, resulting in a smaller reversal field. When tCu = 5 nm, a slight change in the M–H loop of the PSV nanowires is noted, due to the weakening of the FM exchange coupling between the two FM layers. For s = 35 nm, the switching of the 10 nm Ni80Fe20 layer is characterized by an almost abrupt drop in the magnetization. Th is is followed by the gradual decrease of magnetization until the 80 nm Ni80Fe20 layer reverses. For s = 185 nm, however, the switching of the 10 nm Ni80Fe20 (thin) layer is followed by a stable plateau-like region in the M–H loop, as shown in Figure 22.15b. As tCu is increased to 20 nm, both sets of nanowire arrays clearly exhibit two-step switching with the nanowire array with s = 185 nm displaying a slightly larger region of anti-parallel alignment. Interestingly, when the spacer layer thickness becomes equal to the wire edgeto-edge spacing of the closely packed PSV nanowire arrays, tCu = s = 35 nm, a totally different M–H loop is obtained, as shown in Figure 22.15d. Th is significant change in the magnetization behavior is attributed to the competition between the dipolar and inter-layer magnetostatic coupling in the nanowires. For the closely packed PSV nanowires, both the dipolar coupling between neighboring nanowires and the interlayer coupling between the thick and thin Ni80Fe20 layers are comparable in strength. The 80 nm thick Ni80Fe20 layer of the neighboring wire arrays may adopt an anti-parallel configuration due to dipolar coupling (Sampaio et al., 2000; Velazquez et al., 2003; Vavassori et al., 2007). Th is is evidenced by the appearance of an intermediate switching field HLateral, as indicated in Figure 22.15d. For PSV nanowire array with s = 185 nm however, no significant change in the M–H loop is observed as the spacer layer thickness is increased to 35 nm. The thin and thick layer reversal field, HTh in and HTh ick , are consistent with the reversal field obtained for smaller tCu thicknesses, suggesting that the neighboring wires are decoupled. These results clearly show that it is possible to laterally engineer the magnetic properties of nanowire arrays by carefully controlling the various geometrical parameters. To further elucidate our understanding of the different switching processes in the PSV system, the differential M–H loops, as a function of the applied field (dM/dH), were calculated for both sets of nanowire arrays from the measured M–H loops for Cu spacer layer thicknesses of 20 and 35 nm.

22-14


The curves for the 20 nm Cu spacer layer are used as a reference. The representative dM/dH curves for both sets of PSV nanowire arrays for Cu spacer layer thickness of 20 and 35 nm are shown in Figure 22.16. The peaks in the dM/dH curve correspond to the different switching processes occurring as the field is swept from positive saturation to negative saturation. For tCu = 20 nm, as shown in Figure 22.16a, both the PSV nanowire arrays display two distinct peaks corresponding to the switching of the two FM layers, 10 and 80 nm, respectively. For the isolated PSV nanowire array, the peaks occur at an external field of −20 Oe and −395 Oe, whereas for the closely packed nanowire array, the peaks occur at −25 Oe and −482 Oe, respectively. The switching of the thin 10 nm Ni80Fe20 T tCu = 20 nm

B

Isolated

B

dM/dH (arb. units)

T

Closely packed

(a) T

tCu = 35 nm B Isolated

T’ T B’

B

(b)

Closely packed

–600 –400 –200 0 200 400 Applied field (Oe)

600

FIGURE 22.16 Differentiated M–H loops for the isolated and closely packed PSV nanowire arrays with (a) tCu = 20 nm and (b) tCu = 35 nm. (Reproduced from Goolaup, S. et al., J. Appl. Phys., 100, 114301, 2006. With permission.)

layer and thick 80 nm Ni80Fe20 layer correspond to peak position B and T, respectively. The differences in the peak values for the two wire geometries can be attributed to the effect of dipolar coupling. When tCu is increased to 35 nm, the dM/dH curve of s = 185 nm displays the same two peaks response as observed for tCu = 20 nm, as shown in Figure 22.16b. The PSV nanowire array with s = 35 nm, however, displays four peaks. Two intermediate peaks, at position B’ and T’, are sandwiched between the peaks B and T, indicating the presence of two additional magnetic states in the PSV nanowire array. For the PSV nanowire array with s = 35 nm, as the spacer layer is comparable to the edge-to-edge spacing of the closely packed nanowires, tCu = s = 35 nm, the field acting on each FM layer is a result of the competition between the dipolar coupling from the FM layers in neighboring wires and the interlayer magnetostatic field. Due to the strong effect of the dipolar coupling, FM layers in neighboring wires may adopt an anti-parallel alignment.

22.3.6 Alternating Width Nanowire Arrays A lot of research has focused on understanding both the static (Goolaup et al., 2005b; Wegrowe et al., 1999; Wernsdorfer et al., 1997., Adeyeye et al., 1996, 1997a) and dynamic properties (Gubbiotti et al., 2004, 2005; Bayer et al., 2006) of homogeneous width FM nanowire arrays. It has been observed that the magnetic properties of nanowires are strongly dependent on the lateral size due to the spatially varying demagnetizing field (Adeyeye et al., 1996, 1997c). While most of the research has focused on vertically stacked multilayer nanowires with a view for application in miniaturized advanced read head sensor and nonvolatile magnetic random access memories, few works have exploited the lateral engineering of microwire arrays (Adeyeye et al., 2002; Husain and Adeyeye, 2003). The switching field of the wire arrays of a fi xed fi lm thickness is highly sensitive to the wire width and it increases as the wire width is reduced. By exploiting the width dependence of the coercivity, we have fabricated alternating nanowire arrays with unique magnetic properties. Alternating Ni80Fe20 nanowires with differential width, Δw = 200 nm consisting of nanowires with a width of w 1 = 330 nm; w2 = 530 m, and Δw = 570 nm constituting of nanowires with a width of w1 = 330 nm; w2 = 900 nm alternated in an array were fabricated on silicon substrate using deep ultraviolet lithography. A control experiment (reference nanowire) consisting of homogeneous nanowire arrays with a width of w = 330 nm was also patterned using the same technique. For all the geometry patterned, the length of all the nanowire arrays was maintained at 4 mm. In order to ensure that the nanowires are magnetostatically coupled, the edge-to-edge spacing for all the nanowire arrays patterned was maintained at 70 nm. To minimize the formation of end-domains during the reversal process, the nanowires were patterned with rounded edges. Polycrystalline Ni80Fe20 of thickness (t) in the range from 20 to 100 nm was deposited by DC magnetron sputtering at room temperature. The SEM images of the alternating and homogeneous width nanowire arrays are

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1 Δw = 0

Δw = 0 0.5

0

300 nm

H

–0.5 (a)

ECE/ISML

SEI

3.0 kV ×10,000 1 μm WD 6.7 mm

Δw = 200 nm θ

300 nm

dM/dH (arb. units)

(a)


Δw = 200 nm 0.5

0

H

–0.5 (b) Δw = 570 nm 0.5

(b)

ECE/ISML

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3.0 kV ×10,000 1 μm WD 6.7 mm

H

Δw = 570 nm –0.5

–1 300 nm

(c)

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FIGURE 22.17 Scanning electron micrographs of 40 nm Ni80Fe20 thick; alternating nanowire arrays with (a) Δw = 200 nm constituting of wires w1 = 330 nm; w 2 = 530 nm, (b) Δw = 570 nm consisting of wires w1 = 330 nm; w2 = 900 nm, (c) homogeneous width nanowire array with width = 330 nm. The edge-to-edge spacing for all the nanowire arrays is maintained at 70 nm. (Reproduced from Goolaup, S. et al., Phys. Rev. B, 75, 144430, 2007. With permission.)

shown in Figure 22.17. The large area view shows well-defined wires with uniform wire spacing and good edge definition. The representative M–H loops for 70 nm thick Ni80Fe20 nanowire arrays for fields applied along the long (easy) axis of the wires are shown in Figure 22.18a. Both alternating width nanowire arrays display a totally different M–H behavior as compared with the homogeneous width nanowire array. The homogeneous nanowire array with w1 = w2 = 330 nm, as expected, displays an

(c)

–300 –200 –100 0 100 200 300 Applied field (Oe)

FIGURE 22.18 Magnetic hysteresis and differentiated M–H loops of 70 nm thick Ni80Fe20 film for fields applied along the long axis (θ = 0°) for the alternating nanowire arrays, (a) Δw = 0 (w1 = 330 nm; w2 = 330 nm) (b) Δw = 200 nm (w1 = 330 nm; w2 = 530 nm), (c) Δw = 570 nm (w1 = 330 nm; w2 = 900 nm). (Reproduced from Goolaup, S. et al., Phys. Rev. B, 75, 144430, 2007. With permission.)

almost rectangular M–H loop with a coercivity of 170 Oe. For alternating width nanowire arrays with Δw = 200 nm, however, we observed a double-step hysteresis loop. As the applied field is reduced from positive saturation, a sharp drop in magnetization within the field range of −30 Oe to −100 Oe was observed. Beyond this field, a gradual decrease in magnetization is observed until an external field of −190 Oe. This is followed by an abrupt drop in magnetization leading to negative saturation. A similar trend was observed for alternating width nanowires with Δw = 570 nm, although, the switching fields were shifted to lower external fields due to the contribution from the larger width wire. The corresponding differentiated half M–H loop, for fields applied from positive to negative saturation, for the 70 nm Ni80Fe20 thick nanowire arrays is shown in Figure 22.18b. For the homogeneous wire, a broad base peak with the maximum at an

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external field of −175 Oe was observed. This value is consistent with the coercivity obtained from the M–H loop. As expected for Δw = 200 and 570 nm, two peaks, corresponding to the switching of the two sets of wires, w1 and w 2, comprising the array are seen. The fi rst peak corresponds to the low field switching of the larger width wire w 2, while the second peak corresponds to the high field switching of the smaller wire width, w1. For Δw = 200 nm, the switching of w 1 and w 2 occurs at an external field of −205 Oe and −70 Oe, respectively. For Δw = 570 nm, however, the switching of w 1 and w 2 are at −190 Oe and −25 Oe, respectively. As w1 is the same for both alternating width nanowire arrays, the difference in the switching fields may be attributed to the effects of the magnetostatic coupling, due to the small inter-wire spacing s = 70 nm, between the wires in the array, which greatly influences the reversal process. The clear and distinct differences between the two peaks in the M–H loops imply that a region of anti-parallel alignment in the magnetization of neighboring wires in the array exists. A schematic representation of the possible magnetic states in the 70 nm thick Ni80Fe20 alternating width nanowire arrays as the field is swept from positive to negative saturation are shown in Figure 22.19. At positive saturation, the magnetizations of all the wires are aligned along the field direction, as seen in Figure 22.20a. At the occurrence of the fi rst peak in the dM/ dH, in Figure 18b and c, the larger width (w 2) wire switches magnetization direction, as shown in Figure 22.19b. A further

(a)

(b)

(c)

FIGURE 22.19 Schematic representation of the different states of the 70 nm thick Ni80Fe20 alternating width nanowires.

decrease of the field leads to the switching of the smaller width wire, w1, as shown in the schematic in Figure 22.19c. 22.3.6.1 Effect of Thickness The effect of fi lm thickness on the magnetic properties of alternating width nanowire arrays has also been investigated. The Ni80Fe20 fi lm thickness was varied from 20 to 100 nm, while

1 Δw = 200 nm Δw = 570 nm w1 = w2 = 330 nm

0.5


0

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20 nm

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40 nm

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60 nm

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80 nm

0.5

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–100

0

100

200 –200 Applied field (Oe)

–100

0

100

200

300

FIGURE 22.20 Representative M–H loops for the alternating nanowire arrays, Δw = 200 nm, and Δw = 570 nm and reference nanowire arrays as a function of the Ni80Fe20 fi lm thickness. (Reproduced from Goolaup, S. et al., Phys. Rev. B, 75, 144430, 2007. With permission.)

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all other geometric parameters of the wire arrays were kept fi xed. The representative M–H loops for fields applied along the long (easy) axis of the nanowire arrays, as a function of the Ni80Fe20 fi lm thickness are shown in Figure 22.20. The evolution of the magnetization as a function of the Ni80Fe20 fi lm thickness can be attributed to the spatially varying demagnetizing field along the width of the alternating nanowire array. When the Ni80Fe20 fi lm thickness is increased, the thickness to width ratio of w1 and w 2, constituting the alternating nanowire array, will change at a different rate. Thus, as t is increased, the switching of the two sets of nanowires, w1 and w 2 , constituting the array becomes more distinct, resulting in a double-step reversal. The switching field distribution (SFD) was estimated by differentiating one branch of the hysteresis loop for all the thickness rang investigated. The peaks represent the switching of the respective wires, w1 and w2 , whereas the base reflects the SFD. It was observed that the homogeneous nanowire array and alternating width nanowire with Δw = 200 nm, exhibit a much larger SFD as compared with Δw = 570 nm. The SFD during the reversal of nanostructures is attributed to the process variation and the dipolar coupling between the elements. From the SEM image, the shape homogeneity of the nanowire array is confi rmed, thus, the broadening of the slope in the M–H curve can be attributed to the dipolar coupling between the nanowires in the array. For the larger width wires, we expect the formation of edge domains at the end of the wires as the field is relaxed along the wire axis. The edge domains inhibit the accumulation of magnetic charges at the wire end, thus reducing the effective coupling field between alternate wires. Th is is evidenced by the abrupt drop in the magnetic moment for Δw = 570 nm, for all thicknesses investigated. Collective spin-wave modes have also been observed in these structures using brillouin light scattering (Gubbiotti et al., 2007; Kostylev et al., 2008).

22.4 Summary In this chapter, the magnetization reversal mechanism in Ni80Fe20 nanowire arrays as a function of wire thickness has been presented. For fields applied along the nanowire array easy axis, a non-monotonic variation of the coercive field was observed due to the different mechanisms of magnetization reversal dominating the switching process in the nanowire arrays. The angular dependence of coercivity was used to map the reversal mechanism in the nanowires. A cross-over from coherent rotation to the curling mode of reversal was observed for a t/w ratio > 0.5. The question of how the magnetostatic interaction affects the reversal process in PSV nanowire arrays has been addressed. Closely packed and isolated homogeneous width Ni80Fe20(10 nm)/Cu(tCu)/Ni80Fe20(80 nm) PSV nanowire arrays with varied Cu spacer layer thicknesses were studied. The magnetization reversal process is strongly sensitive to the Cu spacer layer thickness. When the spacer layer thickness becomes comparable to the edge-to-edge spacing of the

nanowire array, a drastic change in the reversal mechanism was observed due to the competition between the dipolar coupling in the neighboring nanowires and the interlayer magnetostatic coupling between the two FM layers. By exploiting the width dependence of the coercive field in nanowires, we have shown that complex nanowires with unique magnetic properties can be engineered. Alternating width nanowires consisting of two sets of Ni80Fe20 nanowires differentiated by their width, which are alternated in an array, were fabricated and systematically studied. The magnetization reversal process in the alternating width nanowire arrays was found to be markedly sensitive to the Ni80Fe20 wire thickness and differential width between the two sets of nanowire arrays.

Acknowledgments The authors would like to thank Dr. N. Singh from the A*Star Institute of Microelectronics, Singapore, for his contributions to the fabrication of magnetic nanowires. This work was supported by the Ministry of Education, Singapore, under Grant No. R-263–000–437–112. The authors are grateful to Dr. D. Tripathy for proofreading the manuscript.

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23 Networks of Nanorods Tanja Schilling Université du Luxembourg

Swetlana Jungblut Universität Wien

Mark A. Miller University Chemical Laboratory

23.1 Introduction ...........................................................................................................................23-1 23.2 Percolation ..............................................................................................................................23-1 Connectivity Percolation • Rigidity Percolation

23.3 Nanorod Networks in Composite Materials .................................................................... 23-6 23.4 Networks of Biological Nanorods .......................................................................................23-8 23.5 Summary ...............................................................................................................................23-11 References.........................................................................................................................................23-11

23.1 Introduction The term nanorod may be applied to an unbranched macromolecule or supramolecular assembly whose diameter is a few nanometers or tens of nanometers and whose length is much longer than its diameter, giving a high aspect ratio. Additionally, nanorods are distinguished from long chain-like polymers by their stiff ness; a nanorod has a clear orientation in space, and although it may be somewhat flexible, it does not readily fold or form coils. Th is chapter provides an introduction to the physics of interconnected assemblies of nanorods. Although such networks are encountered in areas of science as apparently disparate as materials science and cell biology, their physics is based on some unifying underlying concepts that can be explained with a generic description of rod-like particles. Many of these concepts are geometrical in origin. One of the best known examples is Onsager’s explanation in 1949 of the spontaneous orientational ordering of long rods into a “nematic” phase, on the basis that the sacrificed orientational entropy is more than repaid by the gain in translational entropy (Onsager, 1949). Th is general theory explains results in a diverse range of specific experimental systems, including some that are apparently rather complex, such as suspensions of rod-like viruses (Bawden et al., 1936). In this chapter, the main underlying theme will be percolation theory, which describes the transition from isolated fi nite clusters of particles to a system-spanning network that produces connectivity on a macroscopic scale. Because of the generality of percolation theory, many inferences can be made about specific systems composed of nanorods based on knowledge gained from simple geometrical models. Indeed, the appealing simplicity of rod-like particles has led to simple models of rods being used as a forum for exploring fundamental physical properties of anisotropic objects such as phase transitions and scaling laws.

Following an outline of relevant aspects of percolation theory in Section 23.2, we turn in Section 23.3 to a survey of applications of carbon nanotube (CNT) networks in composite materials. In this field, the remarkable properties of individual CNTs are exploited to develop novel and useful materials. Although nanorods have many applications in materials science, such as liquid crystals, we have focused on cases where the involvement of a network is crucial. Most of our interest will lie in equilibrium properties, though it is important to bear in mind that nonequilibrium phenomena such as flow and gelation come into play, especially during the preparation of the composites. In Section 23.4, we address the importance of nanorod networks in cell biology, concentrating on the physics of cross-linked assemblies of actin fi laments, as found in the cytoskeleton. The cytoskeleton itself is a dynamic system, and its constant consumption of energy keeps it permanently out of equilibrium. However, in order to understand the rich physics of this complex network, simplified experimental systems have been extracted from it and explored under more controlled conditions in vitro. The analysis will bring us into contact with the highly developed field of semiflexible polymers, and we refer readers to the book by Doi and Edwards (1986) for further detail on this topic.

23.2 Percolation 23.2.1 Connectivity Percolation The physical networks that we are considering are collections of connected nanoparticles and can therefore be regarded as clusters that have grown to a very large size. In a system of finite clusters, the distribution of cluster sizes is governed by many factors, such as the strength and type of the interactions, the temperature, and the density. At certain combinations of these conditions, the particles form a cluster large enough to span 23-1

23-2


the system. The locus of conditions where spanning first occurs is known as the percolation threshold and it demarcates the divergence of the average cluster size, where the system is transformed from an ensemble of finite, disconnected clusters into a macroscopic network. The point at which a connected pathway first appears is important in a wide range of applications. An important example in the context of this chapter is a suspension of electrically conducting particles in an insulating medium, which is an insulator below the percolation density but suddenly becomes a conductor above the threshold. Similar considerations arise in quite different fields, such as porous materials, which only become permeable to fluids when the pores form a connected network on the scale of the sample. The connectivity percolation transition is one of the archetypal transitions in statistical physics. Although it is a purely geometrical problem, it shares certain properties with other transitions. It therefore serves as a useful model, in particular, for the mathematical treatment of critical phenomena, which is concerned with how properties change or diverge as a transition is approached. Introductions to percolation often start by considering bonds on a lattice for simplicity, but the theory and results (including the values of critical exponents) are universal for a given dimensionality of space and transfer directly from regular lattices to the continuum. Here, we will outline some continuum methods that are in current use and can be applied to networks of nanorods, referring the reader to the standard texts (Stauffer and Aharony, 1994, Grimmett, 1999, Bollobās and Riordan, 2006) for a more general treatment. For simplicity, we will start with a system of spherical particles to illustrate percolation theory, noting that the concepts transfer directly to the case of particles with anisotropic shape, such as rods. Consider N freely interpenetrable spheres of diameter σ. We place these spheres randomly in a cube of length L and volume V = L3 and apply periodic boundary conditions, as illustrated schematically in two dimensions in Figure 23.1. Periodic boundary conditions are useful for two reasons. First of all, they mitigate the effect of studying a finite system

by removing the exposed surfaces of an isolated box. Th is point is particularly important in computer simulations, since the number of particles that can be treated is usually much smaller than that encountered in an experiment. Secondly, as the system size L is increased, the remaining finite-size effects, associated with the fact that fluctuations larger than L are missing, scale in a well-understood way that allows the percolation threshold in the thermodynamic limit to be obtained relatively easily (Škvor et al., 2007). The most noticeable effect of studying a fi nite system is that percolation does not set in at a sharp threshold, but spanning clusters appear more gradually (i.e., with a smoothly increasing probability) over a narrow range of densities. For the purposes of defining clusters, we now consider two spheres to be connected if they overlap, i.e., if the distance between their centers is less than σ. We may then ask above what density threshold ρc = Nc/V a cluster of spheres is first observed to be connected to its own periodic image via the boundaries. At this point, the periodically repeated system appears to contain an infinite cluster. Below ρc, disconnected clusters with a distribution of sizes are found, and we may want to know the mean cluster size, defi ned as the mean number of particles in the cluster to which a randomly chosen particle belongs: S=

∑ sn ∑ sn 2

s

s

s

s

,

where ns is the number of clusters that contain s spheres. Later in this chapter, specific properties such as conductivity and mechanical response will be explored. Certain aspects of the internal structure of clusters are then of interest in addition to the cluster size distribution, and these will depend on the nature of the interactions between the particles even for clusters of a given size. These properties can be determined in various ways. For a mathematical treatment see, for example, the book by Meester and Roy (1996). Much progress can be made numerically by a direct implementation of the model in a computer simulation, generating many random configurations to improve statistics. It can be enlightening to follow a theoretical approach, and here we will describe one that makes use of a formal correspondence between the percolation theory and the statistical theory of fluids. A detailed understanding of the following analysis is not necessary for the subsequent discussion or for the other sections of this chapter. In order to show the connection between the percolation theory and the theory of fluids, we start with a brief summary of the key concepts in the latter (Hansen and McDonald, 2006). Consider the equation of state of an ideal gas βP = ρ,

(a)

(b)

FIGURE 23.1 Percolation in a system of freely interpenetrable spheres. The bold square marks the volume V, while the surrounding squares are the periodic images. (a) ρ < ρc, below the percolation threshold. (b) ρ > ρc, a cluster has formed that is connected to its own periodic images.

where P is the pressure ρ is the particle number density 1/β is the product of the Boltzmann constant k B with the temperature T

23-3

Networks of Nanorods

The equation of state of a real gas deviates from this simple relation because particles interact with each other. The higher the density is, the stronger the interactions and hence the greater the deviation from ideal gas behavior. This fact is expressed by writing the equation of state of an interacting gas as a power series in the density βP = 1+ ρ

∞

∑ B (T)ρ

i −1

i

,

i=2

which is known as the virial expansion with the virial coefficients, Bi(T). Computing virial coefficients is in general a difficult task. In order to do so systematically, one defi nes the Mayer f-function f (r ) = exp[−βυ(r )] − 1, where υ(r) is the potential energy due to the interaction of two particles separated by a distance r (assuming an isotropic potential here for simplicity). Using the f-function, the virial coefficients can be mapped to graphs that represent multidimensional integrals, with vertices standing for particle coordinates and edges for integration over the associated f-function. For short-ranged potentials, the f-function decays to zero rapidly with increasing distance, making a systematic evaluation of the graphs feasible. In this scheme the second virial coefficient, for example, is simply given by B2 (T ) = −

1 2

∫ f ( r ) dr .

While the virial expansion approach is rather intuitive, much more information about the liquid is contained in its structure. Hence, one usually approaches a liquid state theory problem using the pair-distribution function, g ( )(r ) = 2

1 N2

N

∑ δ( r − r i

j

direct interactions between the two particles; and the remaining indirect contributions that are mediated by other particles,

∫

h(2)(r ) = c(2)(r ) + ρ c ( r − r ′ )h(2)(r ′)dr ′.

The idea behind this strategy is to separate the part of the correlations that stays short-ranged even at criticality (Ornstein and Zernike, 1914). Equation 23.1 defines the direct correlation function, but it does not yet provide a route to the computation of, for example, an equation of state for a given system. In order to solve the Ornstein–Zernike Equation 23.1, one needs a closure relation, i.e., one further condition that relates h(2)(r) and c(2)(r). A variety of closures are available, and their suitability depends on the specific interactions between the particles and the relative difficulty of solving the resulting equations. One relation that is fairly robust for many types of interactions is the Percus–Yevick approximation, c (2 ) (r ) = g (2 ) (r ) ⎡⎣1 − exp (βυ(r ))⎤⎦ , which sets the direct correlation function to zero where the potential is zero. Finally, a density expansion similar to that described for the equation of state can also be made for the correlation functions. These concepts can now be transferred to the percolation problem. The idea goes back to work by Hill (1955) and more detailed subsequent work by Coniglio and coworkers (Coniglio et al., 1977), who suggested decomposing the partition function of the system into clusters of connected particles and unconnected particles and then applying a density expansion to a “pair-connectedness function” in analogy to the expansion of the pair-correlation function. First, we define an effective interaction that distinguishes between connected and unconnected particles. In the case of the freely interpenetrable spheres introduced above, this could be

− r) ,

⎧⎪0 if r < σ u + (r) = ⎨ ⎪⎩∞ otherwise

i , j =1

where ri is the position of particle i and the angle brackets denote a thermal average. g(2)(r) describes the probability of finding a particle at a given distance r from another particle, normalized with respect to the overall density. For short distances, g(2)(r) reflects the structure of the fluid that is due to interactions between the particles. In the limit of large distances g(2)(r) goes to 1, because there are no long-range correlations in a fluid. In order to focus on the structure, which is contained in the deviations from the overall density, one therefore often uses the total pair-correlation function instead h (2 ) (r ) = g (2) (r ) − 1. Ornstein and Zernike suggested splitting h(2)(r) into two parts: the direct correlation function c(2)(r), which accounts for the

(23.1)

⎧⎪∞ if r < σ u * (r ) = ⎨ ⎪⎩0 otherwise and f + (r ) = exp[−βu + (r )],

f * (r ) = exp[−βu* (r )] − 1,

known as the Mayer cluster functions (Bug et al., 1986, DeSimone et al., 1986). We then introduce the pair-connectedness function, H+(r1, r2), by analogy with the pair-correlation function, where ρ2 H + (r1, r2)dr1dr2

23-4


is the probability that sphere 1 is in dr1, sphere 2 is in dr2, and the spheres overlap. Analogously to the Ornstein–Zernike approach, we then defi ne the direct pair-connectedness function, C+(1, 2):

∫

C + (r1, r2) = H + (r1, r2) − ρ C + (r1, r3)H + (r2 , r 3)dr3. Given a closure condition we can now compute the percolation threshold as follows. We first note that C+ and H+ are translationally invariant, i.e., they can be written as functions of the difference r12 = r1 − r2, so that C+(r1, r2) ≡ C+(r12) and similarly for H+. We may then Fourier transform the Ornstein–Zernike equation, yielding the reciprocal space equation

+

+ (k ) = C (k ) , H 1 − ρC + (k ) where k is the wave number. At the percolation threshold, the mean cluster size S diverges. The mean cluster size is related to ~ C + by

∫

+ (0) = S = 1 + ρ H + (r12 )dr12 = 1 + pH

1 . 1 − ρC + (0)

Hence, S diverges at ρcC + (0 ) = 1, ~ where we must note carefully that C +(0) is dependent on the density. If we consider freely overlapping particles, the interpretation of C+(r1, r2) is particularly graphic. Within the second virial approximation, +

+

C (r1, r2 ) = f (r1, r2).

(23.2)

Hence, the percolation threshold, ρc , is inversely proportional to the so-called excluded volume, i.e., the volume around the center of mass of one particle into which a second particle’s center of mass has to enter in order to produce a connection (see Figure 23.2). The general approach described above for an ideal gas of spheres can also be applied to systems of interpenetrable nonspherical particles as well as particles that interact with each other. The particle shape and interaction potential enter in the cluster Mayer functions, f + and f *, leading to more complicated expressions, but the analysis proceeds analogously. The result of Coniglio theory in Equation 23.2 begins to explain a key observation in the percolation of rod-shaped objects, that the number density of particles at percolation is proportional to the inverse square of the rod aspect ratio, or equivalently, the fraction of the system volume occupied by the rods is inversely proportional to the length of the rods, since the volume of a single rod is proportional to its length (Balberg et al., 1984, Bug et al., 1985, 1986). The excluded volume of two

(a)

(b)

(c)

FIGURE 23.2 Illustration of the excluded volume (gray area) into which the center of the white particle may not enter if it is to avoid overlap with the black particle. (a) The excluded volume between two spheres is a sphere with twice the diameter. (b, c) The excluded volume between two rods depends on the relative orientation and is largest when the rods are perpendicular.

rods depends on their mutual orientations, as illustrated in Figure 23.2. For long rods, the average excluded volume over all orientations increases as the square of the rod length, producing a corresponding decrease in the percolation density. In order to account more thoroughly for the excluded volume of long rod-like polymers that do not freely interpenetrate due to a hard core, the approach by Coniglio has been combined with the reference interaction site model (RISM) (Leung and Chandler, 1991, Chatterjee, 2000, Wang and Chatterjee, 2003). Here, the pair-connectedness function takes into account the fact that the particles are joined to form molecules, using + ρ2 H αβ (r1, r2) dr1dr2

as the probability that site α of one molecule is in dr1, site β of another molecule is in dr2, and the sites are connected. In order + to compute H αβ the relevant Ornstein–Zernike-like equation is now a matrix equation containing both the intramolecular correlations and the site–site correlations (Chandler and Andersen, 1972). Using RISM, Leung and Chandler showed that, just as for ideal rods, the percolation threshold of rods with a hard core scales as the inverse aspect ratio. However, the larger the size of the hard core, the longer the rods have to be before the scaling regime is entered (Leung and Chandler, 1991). In a series of articles, Wang and Chatterjee extended the RISM approach to incorporate attractive interactions between the rods. The attraction can be a direct result of site–site interactions in the one-component fluid of rods (Wang and Chatterjee, 2001), or the indirect result of explicitly adding a second component to the suspension of rods (Wang and Chatterjee, 2002, 2003). In particular, these authors showed that the addition of coiled polymers (which can be regarded as small spheres) to the suspension lowers the percolation threshold of the rods. The polymers produce an effective attraction between the rods because closing the gaps between the rods increases the free volume (and thereby the entropy) of the polymers, as illustrated

23-5


(a)

(b)

(c)

FIGURE 23.3 (a) In a mixture of impenetrable rods and spheres, each rod is surrounded by an excluded volume (gray) into which the center of the spheres cannot enter. (b) When two rods approach, their excluded volumes overlap, increasing the free volume available to the spheres. (c) The greatest increase in free volume occurs when the rods are aligned as they approach.

in Figure 23.3. This depletion effect has also been studied in explicit Monte Carlo simulations using mixtures of hard rods and spheres (Schilling et al., 2007). The simulations are computationally challenging because they must equilibrate the configurations thoroughly, taking care to respect detailed balance (Frenkel and Smit, 2002), in order to capture the structural correlations introduced by the impenetrability of the particles. In a recent piece of work (Kyrylyuk and van der Schoot, 2008), a different theoretical approach was taken, by combining Coniglio theory with liquid state theory for rods on the level of the second virial approximation (which is exact for infinitely long rods interacting by a short-ranged, repulsive potential), fully including translation–rotation coupling. It was found that polydispersity in rod length decreases the percolation threshold—a counter-intuitive effect that has considerable advantages for the production of composite materials on a large scale. The same study also predicted that weak attractions decrease the percolation threshold and that flexibility of the rods increases it, in agreement with independent theoretical studies (Wang and Chatterjee, 2001). An increase in the percolation threshold has also been predicted for “wavy” nanorods, using a model where the lack of straightness is captured by considering rigid sinusoidal or helical rods (Berhan and Sastry, 2007). The inverse proportionality between the percolation threshold and the rod length transfers from straight to wavy rods, but the constant of proportionality becomes a function of the nanorod waviness.

23.2.2 Rigidity Percolation At the connectivity percolation threshold itself, the percolating cluster is only transient if the bonding is reversible, since the breaking of one bond may be enough to disconnect the cluster into two pieces. Further into the percolated regime, however, the greater number of connections leads to multiple independent paths through the cluster, so that the cluster continues to percolate even though its structure is changing dynamically. Even if the bonds are permanent, a network at the connectivity threshold will not be able to support stress if rotation at the

contacts is possible. The network is said to have floppy modes because it may deform without energetic penalty. If the number of connections is increased, some degrees of freedom will become constrained and floppy modes will be lost, leaving a rigid cluster that cannot be deformed without bending or stretching a rod. The point at which rigidity occurs on a macroscopic scale defines the rigidity percolation threshold, whose physics is independent of connectivity percolation (Jacobs and Thorpe, 1995). The concept and theory of rigidity percolation were introduced by Thorpe in the context of polymeric glasses (Thorpe, 1983). In such systems, isolated rigid regions may develop, but it is only when the average coordination number is increased that these regions join to make a true amorphous solid. The changes in mechanical properties at the rigidity threshold are also important in random fiber networks such as paper (Latva-Kokko et al., 2001) and the cytoskeleton in living cells (Head et al., 2003a). To illustrate the problem, imagine a network of rods in d dimensions and connected at M sites. The structure of the network is described by the positions of these sites, which collectively have Md degrees of freedom. Each rod represents a distance constraint between two sites and reduces the number of modes available for motion to the network by one, unless the sites were already constrained directly or indirectly by other rods, in which case the constraint in question is redundant. If all constraints were independent, simple counting would establish when no floppy modes remain. The difficulty comes in establishing the number of redundant constraints, since these do not reduce the number of floppy modes. Floppy modes can always, in principle, be identified by a standard normal mode analysis. A spring constant and equilibrium length are assigned to each rod, such that the network is at mechanical equilibrium. The Md × Md Hessian matrix of second derivatives of the potential energy with respect to site displacements is then assembled. Diagonalization of the Hessian leads to the normal mode frequencies, with each zero eigenvalue revealing a floppy mode. This method rapidly becomes impractical as M increases because of the time taken to diagonalize the Hessian (which typically increases as M3) and due to the numerical problem of obtaining unambiguously zero eigenvalues for the floppy modes. A less costly numerical method for detecting floppy modes that is also exact in principle involves relaxing the network after a perturbation (Chubynsky and Thorpe, 2007). Spring constants are assigned to the constraints as in the normal mode analysis, but then the connection sites are given small but finite random displacements. A local minimization of the energy is then performed. The network will be returned to equilibrium, but the position of equilibrium is not unique if floppy modes are present, since displacements along such modes do not affect the energy. Hence, the initial and relaxed configurations generally differ by a combination of displacements along the floppy modes, leaving the distances between mutually rigid sites unaffected. Hence, the mutual rigidity of sites can be tested systematically, and if there is any ambiguity due to numerical precision, an alternative set of random displacements can be tried.

23-6


A more rational approach to identifying rigid clusters and redundant constraints in two-dimensional networks, involving only integer arithmetic and avoiding artificial spring constants, has been developed by Thorpe and coworkers (Jacobs and Thorpe, 1995). This graph-theoretic method, called the pebble game, represents degrees of freedom by pebbles that may either be free or account for a constraint. The pebble game starts by assigning two free pebbles to each site (connection between rods). Each constraint (rod) is then accounted for by covering it at one of its ends using a free pebble from the site at that end. An example of such an arrangement is shown in Figure 23.4. To test whether an additional rod represents an independent or a redundant constraint, it is added to the network and the pebbles are shuffled around in an attempt to free two pebbles at each of the two sites connected by the test constraint while keeping all the existing constraints covered at one or other end. Each pebble may only be moved between being free at its original site on the one hand or covering an adjacent constraint on the other, but shuffling is possible by choosing to cover a constraint at the opposite end from the pebble currently covering it, thereby freeing the latter pebble. For example, if the test constraint labeled A in Figure 23.4a is added, the sequence of pebble movements indicated by the curved arrows produces two free pebbles at each end of A, while keeping all existing constraints covered. Rod A is therefore not redundant, and indeed rigidifies the rest of the network. In contrast, no sequence of pebble movements can free two pebbles at both ends of the test constraint B, and it is easy to see that B would be redundant due to the indirect rigidity of the two sites it connects. Redundant constraints are left uncovered, and testing of further constraints proceeds in the same way. The two-dimensional pebble game can be extended to “bondbending” networks in three dimensions (Jacobs, 1998). In such networks, not only do the rods generate distance constraints between the sites at which they are connected, but the angle

between rods meeting at any given site is also constrained. Constraining the bond angles is equivalent to inserting an implicit distance constraint between all next-nearest neighbor sites. The extended pebble game proceeds by freeing three pebbles at each end of a test constraint and then attempting to free a further pebble at each neighbor of both ends of the test constraint (Chubynsky and Thorpe, 2007). Three-dimensional bond-bending networks share an important property with the two-dimensional networks described above: rigid clusters are always contiguous, so that two mutually constrained sites always belong to the same rigid cluster. This is not true of non-bond-bending networks in three dimensions, where it is possible for the rigidity between nodes in the network to arise from connections to nodes that are not part of the cluster. An example is shown in Figure 23.4b. The three nodes denoted by the larger circles are mutually rigid, but are only connected via nodes that are not rigid with respect to all three of them (Jacobs, 1998). This sort of structure leads to the breakdown of some theorems on which the pebble game rests. Hence, the natural extension of the pebble game to three-dimensional non-bond-bending networks results in an algorithm that is, in principle, only approximate and can make errors in counting the number of floppy modes and in the rigid cluster analysis. Nevertheless, it turns out that the extent of these errors is very small for certain types of network (Chubynsky and Thorpe, 2007). For example, the approximate pebble game algorithm has been found to perform well for bond-dilution networks, which consist of a regular pattern of sites and connections with a specified fraction of connections (rods) removed at random, and some disorder in constraint length introduced to avoid the possibility of perfectly parallel bonds. Hence, the efficiency of the pebble game makes it attractive even in cases where it is not guaranteed to be exact.

23.3 Nanorod Networks in Composite Materials A

B (a)

(b)

FIGURE 23.4 (a) Illustration of the pebble game for rigidity analysis of a two-dimensional network of rods. The rigid cluster is shown by thick lines and pebbles, representing constraints, by open circles. It is possible to free two pebbles at each end of the test constraint labeled A by the sequence of pebble movements indicated by the arrows. This is not the case for test constraint labeled B, which is therefore redundant. (b) A noncontiguous cluster in three dimensions. The nodes indicated by the larger circles are mutually rigid, but the bipyramidal structures joining them are free to rotate.

The theory of connectivity percolation in rods and rod-containing mixtures finds a direct application in the development of lightweight electrically conducting materials and antistatic fi lms. Insulating materials such as plastics and ceramics can be made electrically conducting over macroscopic distances by embedding in them electrically conducting fibers that connect into a percolating network. As discussed in Section 23.2.1, highly elongated particles percolate at very low volume fractions (inversely proportional to their length), so that conducting composite materials can be obtained by adding only a small amount of conducting fi ller to an insulating matrix. The application of this idea using fibers with diameters on the micron scale goes back at least to the early 1980s, when stainless steel fibers or metalized glass fibers were employed as the fi ller (Crossman, 1985). The characterization of CNTs in 1991 (Iijima, 1991) paved the way for a new generation of conducting composites using networks of nanoscale conductors, since individual CNTs can be excellent electrical conductors (Saito et al., 1998). One of the first


applications to exploit this property was the doping of luminescent polymers to control their conductivity (Curran et al., 1998). By making the polymer more conductive, it was possible to achieve electroluminescence (the emission of light upon application of a potential difference) at lower current densities. The conductivity of the composites varies over several orders of magnitude as the density of CNTs is increased, but shows a sharp increase at the percolation threshold (Coleman et al., 1998), effectively transforming the material from an insulator to a conductor. In the vicinity of the percolation threshold, where macroscopically connected paths first appear, only a small fraction of CNTs contribute to the conductivity. Indeed, it has been argued that this regime provides an opportunity to access the intrinsic properties of individual nanotubes through macroscopic measurements (Benoit et al., 2002). Above the percolation threshold the “strength” of the network (the fraction of particles belonging to the infinite cluster) increases, and the conductivity continues to rise as (ρ − ρc)t, where t is a universal critical exponent (Stauffer and Aharony, 1994). Calculations and experiments on a variety of three-dimensional conducting networks (including model resistor networks) indicate that t ≈ 2 (Stauffer and Aharony, 1994). Although this approximate value has been confirmed in some experiments on CNT composites (Pötschke et al., 2003), considerably lower values (as low as t ≈ 1.4) and even a temperature dependence of the exponent have also been reported (Barrau et al., 2003). The experimental realization of theoretically predicted percolation behavior in CNT composites encounters a persistent obstacle. The nanotubes tend to bundle due to intermolecular attraction, including van der Waals forces, making it difficult to disperse them uniformly throughout the material. Incomplete dispersion leads to higher percolation thresholds than expected in the absence of bundling (Potschke et al., 2003, Sandler et al., 2005). There are both thermodynamic and kinetic aspects to achieving a proper dispersion; reducing the forces that lead to bundling makes the equilibrium state more dispersed, but it is still necessary to reach that state at a reasonable rate. Hence, the properties of the final composite can depend sensitively on the method of preparation and the conditions encountered in the process. Rapid stirring and sonication help to disperse the bundles, but a range of more invasive techniques have also been developed to assist the process. To improve the solubility of CNTs in organic solvents, one route is to modify the nanotubes chemically by attaching functional groups (Sun et al., 2002), while in mixtures of polymers and CNTs, the nature of the polymer plays an important role. CNT–polymer interactions can be strong (involving electrostatic attraction or even covalent attachment) or transient and weak, but in both cases, the interactions between polymer and CNT affect how the CNTs interact with each other (Szleifer and Yerushalmi-Rozen, 2005). For example, Du et al. have described a coagulation method, where precipitating polymers of poly(methyl methacrylate) entrap the nanotubes, preventing them from aggregating (Du et al., 2003). Naturally, these dispersion methods affect the individual and collective properties of CNTs, and the efficacy of the dispersion

23-7

must be balanced against retaining the desired properties of the resulting composite. In addition to determining the extent of nanotube dispersion, composite preparation methods influence the overall distribution of nanotube orientations. When exposed to a flow, elongated objects tend to align themselves with the direction of movement. The resulting anisotropy of the orientational distribution leads to electrical conductivity that is also anisotropic, being greater in the direction of the flow (Haggenmueller et al., 2000). Alignment of CNTs in a stream can be exploited to produce macroscopic (sub-millimeter scale) ribbons and fibers of CNTs that are so flexible that they can be tied into knots (Vigolo et al., 2000). Recent work provides evidence that the percolation threshold does not change monotonically with the extent of CNT alignment. It is certainly the case that strong alignment of rod-like objects leads to a relatively high percolation density (Du et al., 2005), since the excluded volume for parallel rods scales only as the length of the rods rather than its square (see Figure 23.2). However, it seems that the opposite extreme of a uniformly random distribution does not produce the lowest possible percolation density. A slight average alignment of the rods within 70°–80° of some space-fi xed axis is optimal (Li et al., 2008), a result that has been derived both from experimental investigation and from Monte Carlo simulations (Du et al., 2005). The anisotropic conductivity of composites with strongly aligned fibers may be advantageous in some applications. Indeed, anisotropic conducting adhesives are an important alternative to solder for connecting electronic components. However, the composites employed in such applications usually contain spherical conducting filler particles that become connected into chains that percolate in a particular direction by the application of pressure in that direction. Curing of the adhesive matrix then maintains the connected configuration (Lin and Zhong, 2008). Although an isotropic dispersion of long rods percolates at a low density, there are ways of decreasing the percolation threshold even further. One approach is to induce weak attractions between the rods so as to increase the number of contacts between them but without inducing strong bundling or introducing anisotropy in the overall distribution of rod orientations. This goal can be achieved by entropic means through the depletion mechanism that was described in Section 23.2.1 and Figure 23.3. The depletion effect in the context of CNT suspensions has been studied experimentally using micelles as the depletion agent (Wang et al., 2004, Vigolo et al., 2005). In these studies, micelles were self-assembled out of surfactant molecules, which, once formed can be regarded as impenetrable but otherwise noninteracting spherical particles. The resulting two-component fluid of hard rods and spheres has been modeled in computer simulations (Schilling et al., 2007). The simulations are able to probe the correlations that arise from the hard core excluded volume of both rods and depletants in detail, and show that depletion enhances the local alignment of the rods relative to their close neighbors (see Figure 23.3) without making the overall distribution of single particle orientations anisotropic (which would lead to a nematic liquid crystalline phase). Although this mutual

23-8

alignment is a weak form of bundling and alone would tend to raise the percolation density, the overall induced attraction between the rods leads to a greater number of contacts per rod, which overcomes the opposing effect of alignment and leads to a net reduction in the percolation threshold. A rather different approach to lowering the volume fraction of conducting fi ller that is necessary for percolation involves not requiring the conducting network to fi ll the entire volume of the sample. In segregated composites, CNTs or other conductive particles are coated onto the surface of the particles of a polymer powder. The coated powder is then compressed to mold it into a continuous material. The compression forces the powder particles into polyhedral shapes with the conductive fi ller confined to the interfaces between them (Gao et al., 2008). A conducting network consisting of a skeleton running between the polyhedra emerges, allowing percolation to be achieved at volume fractions of well under a tenth of one percent (Mamunya et al., 2008). Although the network ultimately spans three-dimensional space, its structure is effectively two-dimensional at a microscopic level. The critical exponent, t, for conductivity has been measured and is indeed compatible with the accepted range of values for two-dimensional connectivity, which is around 1.3, substantially lower than the value of 2 in three dimensions (Gao et al., 2008). A new range of applications of conducting CNT networks was opened up in 1999 by the discovery that the electrical and mechanical properties of CNTs are strongly coupled (Baughman et al., 1999). The injection of electrical charge into a CNT produces a mechanical deformation in the tube, primarily as a result of changes to its electronic structure rather than simple electrostatics. This electromechanical actuation can be used to extract mechanical work from CNT networks, as originally demonstrated in sheets of “nanotube paper”—entangled mats of nanotube bundles (Baughman et al., 1999). The reverse effect is also observed: the electrical properties of a CNT network are sensitive to mechanical deformation of the network by external forces. The resistivity of CNT fi lms has been shown to increase almost linearly with applied strain, making such fi lms attractive as strain sensors (Dharap et al., 2004). By embedding CNT films within a material, it is possible not only to measure strain, but also to detect and quantify damage due to mechanical loading and unloading. Microcracks in a laminated material lead to the onset of delamination and step-like increases in the resistivity of the material due to loss of conductivity in the CNT network. The resistance of a damaged sample therefore has a qualitatively different relationship to its stress from the smooth increase observed in an undamaged sample (Thostenson and Chou, 2006). The shift in the stress–resistivity relationship on repeated loading–unloading cycles is an indication of irreversible damage. The fact that a percolating network of CNTs can be embedded in a composite material with the need for only a small volume fraction of nanotubes means that the ability for strain and damage sensing can be built into a material in a minimally invasive way (Li et al., 2008).


Calculations and simulations based on the vibrational modes of CNTs predict that they have a high thermal conductivity (Che et al., 2000) in addition to their electrical conductivity. Th is property of individual particles should mean that the thermal conductivity of a composite material can be greatly enhanced by the addition of a CNT network, and that this enhancement should be achievable at the low volume fractions needed for percolation (Foygel et al., 2005). Percolation behavior of the thermal conductivity has indeed been observed experimentally, with a sharp jump over several orders of magnitude at the percolation threshold (Biercuk et al., 2002) and an increase thereafter that is well described by the same kind of power law, (ρ − ρc)t, as electrical conductivity (Foygel et al., 2005). Although vapor-grown carbon fibers with diameters in the micron range share and maybe even exceed the high thermal conductivity of CNTs, a much higher volume fraction of the fibers must be loaded into a matrix to enhance the conductivity, and the enhancement is weaker. The relatively poor performance of micron-diameter fibers has been attributed to their lower aspect ratio and the consequent difficulty of achieving a percolating network (Biercuk et al., 2002). One further remarkable property of CNTs is their high mechanical strength, leading to the expectation that they can be used to make exceptionally strong materials. Unfortunately, the strength of individual nanotubes does not transfer directly to bulk composites, since the mechanical load must be transferred between the host matrix and the nanotubes and, in the case of multiwalled nanotubes, between the layers of the tubes themselves. Both interfaces are prone to slippage, though load transfer is better in compression than in tension (Schadler et al., 1998). However, this disappointing result can be turned to advantage by exploiting the slippage in applications where strong mechanical damping is needed. Addition of CNTs to a polymer matrix in characteristically low volume fractions can dramatically increase their viscous loss factor with minimal penalty from the added weight (Suhr et al., 2005). This application of CNTs demonstrates once again the scope for designing novel materials based not only on the remarkable properties of individual CNTs, but also on the generic ability of nanorods to form percolating networks at very low densities.

23.4 Networks of Biological Nanorods A wide array of proteins can aggregate into elongated fibers, making nanoscale rod-like structures important in all biological matter. The formation of these structures sometimes takes place through a hierarchy of several levels. For example, collagen molecules, which are found abundantly in skin and bone, twist into helical ropes which aggregate into fibrils, which in turn bundle into fibers (Alberts et al., 2008). Other protein fibrils play a key role in disease when they form from incorrectly folded globular proteins that need to remain in solution in order to function properly (Chiti and Dobson, 2006). Living cells owe their shape, mechanical response, and some of their transport properties to their cytoskeleton, a complex

23-9


network of protein filaments. Three types of filament are usually found: actin filaments (two-stranded helices) and intermediate fi laments (rope-like fibers) play an important role in cell shape and structure, while microtubules (hollow cylinders) are involved in intracellular organization. A host of other proteins are involved in linking the network and in transporting loads along the fi laments. Furthermore, the network is dynamic, with individual fi laments growing or shrinking at both ends simultaneously through association and dissociation processes (Alberts et al., 2008). The cytoskeleton performs many remarkable tasks in cells, but to understand how such a complex structure works it can be fruitful to isolate a limited number of its components and probe their behavior in vitro. By distilling the essential physics of these simplified networks into suitable models, it may be possible not only to gain insight into the properties of the cytoskeleton in living matter, but also to exploit this knowledge in the design of artificial network-based materials. An important class of simplified system derived from the cytoskeleton consists of cross-linked networks of filamentous actin (F-actin). The mechanical behavior of actin networks depends both on the elastic properties of the individual filaments and on the architecture of the network. Actin filaments are semiflexible, which means that there are both enthalpic and entropic contributions to their elastic modulus. A very stiff rod would have purely mechanical (enthalpic) contributions to its stretching and bending moduli, as illustrated schematically Figure 23.5b. In the opposite extreme, the stretching modulus of a completely flexible polymer is entirely entropic and arises from the fact that there is only one fully stretched conformation but many coiled ones with a shorter end-to-end distance. Hence, stretching a flexible polymer leads to a loss of entropy without an increase in the contour length of the chain, as illustrated Figure 23.5a. Actin filaments have an intermediate behavior; they are not flexible enough to form loops or coils, but an unconstrained filament may still experience significant thermal fluctuations. The stiffness of a chain can be quantified by its persistence length, i.e., the distance over which its tangent vector becomes uncorrelated. For actin, the persistence length is somewhat longer than the filament itself. In turn, the filament length is longer than the scale of the mesh in the network. Actin networks are therefore rather different (MacKintosh et al., 1995) from some other elastic materials like rubbers, which consist of permanently cross-linked networks of flexible polymers (Doi, 1995). As might be expected, the density of cross-links in a suspension of actin fi laments strongly influences the elastic response of the resulting network (Janmey et al., 1990). In the absence of cross-linking proteins, an actin suspension subjected to a

constant shear stress responds by showing a large initial strain (deformation) which then continues to increase slowly. Th is creep is characteristic of viscoelastic fluids, as is the recovery of only part of the strain when the stress is released. In contrast, the addition of actin-binding proteins to the suspension results in a more elastic network, where the initial deformation upon shear strain is smaller, and is mostly recovered on release. Creep is practically eliminated by the mutual binding of the fi laments. In the cross-linked network the elastic shear modulus, which measures the strain required to produce a given deformation, rises with increasing strain, i.e., the network is strain stiffening, becoming progressively harder to deform further. Abrupt decreases in the modulus are observed at sufficiently large strains, and have been attributed to rupture of the fi laments rather than dissociation of the cross-links (Janmey et al., 1990). Both connectivity and rigidity percolation phenomena are relevant to the properties of actin networks. The onset of a significant elastic response is associated with the connectivity percolation threshold, where the combination of rod density and linker density first produces a macroscopically connected network (Åström et al., 2008). The rigidity threshold then depends on the type of protein involved in the cross-linking. One can imagine two extremes: free rotation at cross-links or a constrained angle at the links (Wilhelm and Frey, 2003). The latter case is the bond-bending model described in Section 23.2.2, and in a two-dimensional network of this type, connectivity and rigidity percolation coincide. At the rigidity threshold the generic behavior, as described by the critical exponents, seems to be insensitive to the mechanism of rigidification at the links. For example, suppressing the bending of adjacent segments of any given rod with a certain probability leads to a rigidity transition in the same universality class as the standard rigidity percolation model (Latva-Kokko et al., 2001). Simple models that capture the interplay between the bending and stretching of individual actin fi laments are able to explain and predict regimes of qualitatively different behavior in the networks. For example, Wilhelm and Frey considered a random network of rods in two dimensions, linked at the points where they cross. Spring constants for the bending and stretching of segments were assigned according to the thickness of the rods and the length of the segments (Wilhelm and Frey, 2003). For thick rods and high densities (leading to many contacts along the length of a rod), it is harder to bend rods than to stretch or compress them. In this regime, the network behaves like a homogeneous elastic medium with shear modulus proportional to the fi lament compressional stiff ness and the rod density. Local deformations of the network are affi ne, i.e., collinear sets of points remain collinear after the deformation and ratios of Stiff filament

Flexible coil

Bending deformation Stretched coil (a)

Stretching deformation (b)

FIGURE 23.5 (a) The elasticity of flexible polymers arises from the loss of thermal fluctuations, leading to a decrease in entropy when the coil is stretched. (b) For stiff polymers, the penalty for bending or stretching is predominantly enthalpic.

23-10


B A C

FIGURE 23.6 Example deformations of a square lattice. Deformation A is affine: it preserves collinearity and ratios of distances. Deformations B and C are non-affine due to explicit bending of the lines (B) or to loss of parallelism between the lines (C).

distances are preserved, as illustrated in Figure 23.6. In contrast, at lower densities or for thinner rods, bending deformations of individual fi laments dominate and the shear modulus becomes independent of the aspect ratio of the filaments at a value determined by the density. If thermal fluctuations are taken into account then it should be possible to distinguish two affi ne regimes (Head et al., 2003b). At the highest densities, the mechanical stiff ness is still expected to dominate. However, at somewhat lower rod densities, the separation between cross-links may be sufficiently large that thermal fluctuations cause a nonnegligible contraction of the network (a scenario intermediate between the two extremes depicted in Figure 23.5). These fluctuations can be drawn out by an applied strain or shear without increasing the contour length of the fi laments through stretching of the constituent subunits. Th is affine entropic regime is expected to be restricted to some maximum strain because of the limited size of the thermal fluctuations. The limit is reduced as the rod density increases, since the segments between cross-links become shorter and have less scope to fluctuate (Head et al., 2003b). The entropic contribution to the stiff ness of a segment of length l scales as l−4 and is thus much more sensitive to the density of cross-links than the mechanical stiffness, which goes only as l−1. This strong dependence also makes thermally fluctuating stiff networks highly sensitive to length polydispersity and to deviations from true randomness in the network architecture (Heussinger and Frey, 2007). Thermal fluctuations in actin fi laments are one of several characteristics that can explain the strain stiffening of networks, i.e., the increase in elastic modulus with the applied strain. The straightening of filaments accounts for the nonlinear elastic response at low strain until the point where this source of compliance has been exhausted (Storm et al., 2005). Beyond this point, enthalpic contributions must be considered to reproduce experimental modulus–strain curves. Nonlinearity in the enthalpic regime is influenced by the fact that, even at bulk mechanical equilibrium, individual segments are not at their equilibrium length due to the constraints imposed by the architecture of the

cross-linking network (Storm et al., 2005). A third contribution to strain stiffening comes from the cross-link proteins themselves, which may have both mechanical and thermal elasticity as well as a finite extensibility (Broedersz et al., 2008). Strain-stiffening can account for the apparent discrepancy between the elastic modulus of actin networks measured in vitro and in cells, where the latter is found to be much higher (Gardel et al., 2006). The higher modulus is a direct consequence of strain stiffening if cells prestress their cytoskeletal frameworks with a constant background force. The frequency-dependent elastic response of actin networks cross-linked by flexible filamin A proteins with a constant stress offset has been characterized experimentally in vitro and shown to be much closer to the mechanics of living cells than the behavior about zero initial stress (Gardel et al., 2006). In living cells, the architecture of actin networks is not static, since cross-linking proteins bind and unbind dynamically with characteristic rates. Transient cross-linking leads to a maximum and then a minimum in the viscous response of the network as a function of increasing frequency. The maximum coincides with a decrease in the elastic response at low frequencies. The effect of the unbinding rate on these properties has been studied experimentally in actin networks cross-linked by heavy meromyosin, whose unbinding rate can be increased by the addition of the nonhydrolyzable nucleotide analogue AMP-PNP (Lieleg et al., 2008). This work shows that the features in the viscous and elastic properties change in tandem, indicating a common origin of the effects. The unbinding of cross-links produces a local relaxation of the network that leads both to an increased viscous loss and to a decreased elastic response. The minimum in viscous response is a result of competition between stress release on unbinding and the friction induced by fi lament fluctuations. At sufficiently high frequencies of deformation, the unbinding rate is too slow to have an effect on the network properties. Åström et al. have developed a model that incorporates the fact that there is a limit to how far an individual cross-link can be stressed before it ruptures. The model consists of a random three-dimensional network of mechanically deformable rods cross-linked by springs at locations of close proximity (Åström et al., 2008). If the cross-links are not allowed to break then strain stiffening arises directly when the non-affi ne regime is reached. However, if the cross-links rupture beyond a specified extension, there must be a transfer of the supported strain to nearby cross-links. These in turn may rupture due to their increased stress, resulting in correlated avalanches of ruptures. These ruptures reduce the network’s elastic response, leading to a strain-softening regime. The model can be taken one step further by allowing new cross-links to form when segments are brought close together by the deformation. Th is dynamic crosslinking leads to bundling of the fi laments under strain. The new cross-links can be strong enough to retain the bundling even when the external strain is released, resulting in pronounced hysteresis in the strain–stress relationship. The survey of experiments and theoretical modeling on actin networks presented here has highlighted the rich properties of this particular example of biological nanorod network.


However, we have not even touched on the fact that in living cells, the cytoskeleton is constantly out of equilibrium. The constant consumption of energy makes the network into an “active gel.” Molecular motors travel along the network’s fi laments, performing mechanical work, transporting loads, and enabling cell motility. Active gels constitute an important new area of biophysics, though one which begins to depart from the theme of this chapter. We recommend two recent reviews (Liverpool, 2006, Jülicher et al., 2007) for a survey of the current state of the field.

23.5 Summary Explicit statistical modeling of composite materials or the cellular cytoskeleton with molecular detail would be a formidable task. Mastery of such systems at the molecular level can be important both for practical applications and for a deep understanding in specific cases. In this chapter, however, we have taken a highly coarse-grained approach to describing networks of rod-like particles. Retaining just the most essential detail has the advantage of highlighting unifying concepts that link some disparate systems. For example, we have emphasized the role of percolation theory, which lies at the heart of network formation and provides a general framework for analyzing physical properties of both finite and system-spanning clusters. The level of detail incorporated into this and other theories can be refi ned as needed, starting with ideal, fully penetrable rods, then proceeding in turn to particles that exclude volume through a hard core, that interact through longer-range forces, and that are flexible. Many challenges remain in the understanding, control, and development of nanorod networks. We have seen that network structure is sensitive to intermolecular forces and to the nature of cross-links, necessitating innovative preparation methods to produce composite materials with the desired properties. The high aspect ratio of some nanorods means that computer simulations must employ large periodic cells and a correspondingly large number of particles, making the simulations computationally costly. Here, new simulation methods and coarse-graining schemes will expand the range of problems that can be tackled. Purely theoretical approaches offer insight and semi-quantitative predictions, but must be extended to cover increasingly complex nanorod systems. All these approaches have made important advances in recent years, and there is every reason to expect that the field will continue to progress rapidly.

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V Nanowire Properties 24 Mechanical Properties of GaN Nanowires Zhiguo Wang, Fei Gao, Xiaotao Zu, Jingbo Li, and William J. Weber ................................................................................................................................................ 24-1 Introduction • Growth Methods and Shape Controlling • Mechanical Behavior Studied Experimentally and Limitation • Mechanical Behaviors Studied by Molecular Dynamics • Summary • Future Directions • Acknowledgments • References

25 Optical Properties of Anisotropic Metamaterial Nanowires

Wentao Trent Lu and Srinivas Sridhar ................ 25-1

Introduction • Negative Refraction and Metamaterials • Wave Propagation in an Anisotropic Planar Waveguide • Wave Propagation on Anisotropic Nanowire Waveguides • Energy Flow on Anisotropic Nanowires • Realization and Numerical Simulations • Light Coupling to Nanowire Waveguide • Applications of Nanowires Made of Indefinite Metamaterials • Conclusions • Acknowledgments • References

26 Thermal Transport in Semiconductor Nanowires Padraig Murphy and Joel E. Moore ...................................... 26-1 Introduction and Motivation • Phonons and Nanowires • Landauer Formula for the Thermal Conductance • Thermal Conductivity at Higher Temperatures • Localization • Conclusions • Acknowledgments • References

27 The Wigner Transition in Nanowires

David Hughes, Robinson Cortes-Huerto, and Pietro Ballone .....................27-1

Introduction • The Phase Diagram of the Q1D Electron Gas • Fabrication Techniques • Overview of Theoretical Studies • The Jellium Model of Metal Wires and the Density Functional Picture • Axially Symmetric Solutions • Iterative Minimization of the LSD Density Functional • Broken-Symmetry Solutions • Nano-Constriction: Preliminary DF Results • Discussion and Conclusive Remarks • References

28 Spin Relaxation in Quantum Wires

Paul Wenk and Stefan Kettemann .............................................................. 28-1

Introduction • Spin Dynamics • Spin Relaxation Mechanisms • Spin Dynamics in Quantum Wires • Experimental Results on Spin Relaxation Rate in Semiconductor Quantum Wires • Critical Discussion and Future Perspective • Summary • Symbols • Acknowledgments • References

29 Quantum Magnetic Oscillations in Nanowires A. Sasha Alexandrov, Victor V. Kabanov, and Iorwerth O. Thomas .............................................................................................................................................. 29-1 Introduction • Quantum Magnetic Oscillations in Bulk Metals • Combination Frequencies in the Quantum Magnetic Oscillations of Multiband Quasi-2D Materials • Quantum Magnetic Oscillations in Nanowires • Conclusion • Acknowledgment • References

30 Spin-Density Wave in a Quantum Wire

Oleg A. Starykh .................................................................................... 30-1

Introduction • Spin–Orbit-Mediated Interaction between Spins of Localized Electrons • Magnetized Quantum Wire with Spin–Orbit Interaction • Conclusions • Appendix 30.A: Bosonization Basics • Acknowledgments • References

31 Spin Waves in Ferromagnetic Nanowires and Nanotubes Hock Siah Lim and Meng Hau Kuok ......................... 31-1 Introduction • Isolated Cylindrical Nanowire • Collective Spin-Wave Modes in an Array of Nanowires • Hollow Cylindrical Nanowires • Summary • References

32 Optical Antenna Effects in Semiconductor Nanowires Jian Wu and Peter C. Eklund ....................................... 32-1 Introduction • Nanowire Synthesis and Characterization • Optical Backscattering Experiments • Classical Calculations of the Elastic Light Scattering from a Cylinder • Modeling the Backscattered Light from Semiconductor Nanowires • Optical Phonons, Raman Scattering Matrices, and Manipulations • Results and Discussion of Rayleigh Backscattering from Nanowires • Polarized Raman Backscattering from Semiconductor Nanowires • Summary and Conclusions • Acknowledgments • References

33 Theory of Quantum Ballistic Transport in Nanowire Cross-Junctions Kwok Sum Chan ................................. 33-1 Introduction • Theory and Model • Conductances • Summary and Future Perspective • Acknowledgment • References

V-1

24 Mechanical Properties of GaN Nanowires Zhiguo Wang University of Electronic Science and Technology of China and Chinese Academy of Sciences

Fei Gao Pacific Northwest National Laboratory

Xiaotao Zu University of Electronic Science and Technology of China

Jingbo Li Chinese Academy of Sciences

William J. Weber Pacific Northwest National Laboratory

24.1 24.2 24.3 24.4

Introduction ...........................................................................................................................24-1 Growth Methods and Shape Controlling ..........................................................................24-2 Mechanical Behavior Studied Experimentally and Limitation......................................24-3 Mechanical Behaviors Studied by Molecular Dynamics.................................................24-3 Molecular Dynamics • Stillinger–Weber Potential • Simulation Methodology • Simulation Results • Discussion

24.5 Summary ...............................................................................................................................24-11 24.6 Future Directions.................................................................................................................24-12 Acknowledgments ...........................................................................................................................24-12 References.........................................................................................................................................24-12

24.1 Introduction The further development of the modern semiconductor industry is based on the continued miniaturization of silicon-based devices; however, existing materials and technologies are approaching their physical limits, which will soon prevent the industry from maintaining similar rates of improvement (Peercy 2000). Finding new materials and technologies adequate for the fabrication of highly integrated devices within the nanometer regime could potentially overcome such limitations. One-dimensional materials like nanowires that can efficiently transport electrical carriers and optical excitations can meet these needs. A variety of freestanding semiconductor nanowires with controlled electrical and optical properties, including the group IV, III–V, and II–VI semiconductor nanowires, can now be synthesized as a result of recent advances in crystal growth technology (Dai et al. 1995; Han et al. 1997; Morales and Lieber 1998; Lexholm et al. 2006; Tragardh et al. 2007). These one-dimensional structures often show novel physical properties that are different from their bulk (Kanemitsu et al. 1993; Zach et al. 2000; Makhlin et al. 2001). As such, the nanowires are considered ideal building blocks for highly integrated nanoscale electronics and optoelectronics.

Wide band-gap semiconductors are of particular interest since their large energy gap allows for the possibility of tuning optoelectronic devices to work from infrared to ultraviolet at high temperatures and high frequency. Among these semiconductors, gallium nitride (GaN) is especially of interest for its wide direct band gap. After the successful synthesis of blue light–emitting GaN-based materials, research interest on GaN has intensified due to a great desire for high-efficiency blue light–emitting diodes or laser diodes. The fabrication of nanosized GaN materials has been a focused field of research, both due to the fundamental nanoscale mesoscopic physics and the developing nanoscale devices. Low-dimensional GaN nanostructures are expected to improve device characteristics and possess superior structural stability and mechanical properties at high temperatures; thus, they can be used as building blocks for nanodevices. In this chapter, a brief introduction to the growth methods and the shape controlling of GaN nanowires is provided. The limitations in determining the mechanical behavior of GaN nanowires experimentally are then presented. In Section 24.3, recent progress is described on the modeling and simulation of the nanomechanical properties and behavior of GaN nanowires. 24-1

24-2

24.2 Growth Methods and Shape Controlling Nanowires have small cross-sections, allowing them to accommodate much higher levels of strain without the formation of dislocations. The wires, furthermore, can rapidly exclude any dislocations that do form to the nearby sidewalls, where they terminate and cease to propagate through the crystal (Trampert et al. 2003). Transmission electron microscopy (TEM) performed on GaN nanowires detached from their substrate reveals that they are dislocation-free single crystals (Tchernycheva et al. 2007). This defect exclusion mechanism allows the growth of dislocation-free GaN nanowires on all types of substrates. In recent years, research interest in GaN nanowires has increased significantly because sufficiently small dimensions, such as those found in nanowires, promote quantum confi nement effects that are expected to lead to novel or enhanced physical properties for potential use in future nanotechnology. GaN nanowires are typically synthesized through vapor phase methods, in which the initial starting reactants for the wire formation are gas phase species. Numerous techniques have been developed to prepare precursors into the gas phase for nanowire growth, including laser ablation, chemical vapor deposition, chemical vapor transport methods, molecular beam epitaxy, and sputtering. It should be noted that the concentrations of gaseous reactants must be carefully regulated for nanowire synthesis in order to allow the nanowire growth mechanism to predominate and suppress secondary nucleation events. There are two approaches for vapor phase synthesis that have been used for GaN nanowire growth. The first one is based on the so-called vapor–liquid–solid (VLS) mechanism (Wagner and Ellis 1964), which uses a metallic catalyst (Ni, Fe, etc.) to promote onedimensional growth (Duan and Lieber 2000; Zhong et al. 2003). GaN nanowires grown by the VLS mechanism have been successfully used to fabricate nano-devices such as lasers or modulators from single nanowires (Gradecak et al. 2005; Greytak et al. 2005). The second approach to GaN nanowire synthesis relies on a catalyst-free growth mode, which consists of a spontaneous transition to one-dimensional growth when nitrogen-rich conditions are used (Calleja et al. 1999). One advantage is that catalyst-free synthesis excludes the possible incorporation of catalyst metallic impurities in the nanowire material. For heterostructure formation, this growth mode also provides better control of the elemental composition, since it does not depend on a complex interaction between the constituents in the vapor phase, the metallic catalyst, and the semiconductor solid-state. Many groups report the fabrication of dense GaN nanowires ensembles (Cerutti et al. 2006; Meijers et al. 2006), as well as GaN/AlGaN and InGaN/GaN heterostructures in nanowires (Ristic et al. 2003; Kikuchi et al. 2004) by molecular beam epitaxy (MBE) using this procedure, but the exact growth mechanism remains an open question. Due to its anisotropic and polar nature, GaN exhibits properties that depend on crystallographic orientation (Feng et al.


1999; Waltereit et al. 2000). Thus, controlling the growth direction of GaN nanowires is important for practical applications. Control of both the growth orientation, which has a strong effect on anisotropic properties, and the alignment are critical issues in nanowire growth. Control of the growth direction of GaN nanowires can be achieved through three methods demonstrated to date. The first method is to use heteroepitaxy on different single crystal templates, mediated by a catalyst cluster (Kuykendall et al. 2003). Epitaxial growth of wurtzite gallium nitride on Au- and Ni-coated sapphire substrates results in GaN nanowires with triangular cross-sections, with the wire axis oriented along the crystallographic [210] direction. However, the predominant nanowire growth direction is along the [110] direction instead of the [210] direction for nanowires grown on Fe-coated sapphire substrates. The second method to control the growth direction is choosing suitable substrates. For example, epitaxial growth of wurtzite gallium nitride on (100) γ-LiAlO2 and (111) MgO single crystal substrates with Au as the initiator results_in the selective growth of nanowires along the orthogonal [110] and [001] directions, exhibiting triangular and hexagonal cross-sections, respectively (Kuykendall et al. 2004). Crystallographic alignment occurs presumably due to the close symmetry and lattice match between the substrates and observed nanowire growth directions. The third method to control the growth direction of nanowires is based on controlling the Ga flux during direct nitridation in dissociated ammonia on an amorphous substrate (Li et al. 2006). The nitridation of Ga droplets at high flux leads to GaN nanowire growth along the [001] direction, while nitridation with a low Ga flux leads to growth in the [100] direction. The band gap of wurtzite GaN nanowires has been found to depend on the crystallographic orientation because of polarization in the [001] direction of GaN crystals (Waltereit et al. 2000). This is illustrated by the blue-shift in the bandgap of GaN nanowires grown in the [100] direction by about 100 meV, as compared with wires grown in the [001] direction at temperatures ranging from 0 to 300 K. Chin et al. (2007) also found the pho− toluminescence peak of [11 0]-oriented nanowires is blue-shifted ∼140 meV compared with that of the [001]-oriented nanowires. And they attributed the blue-shift to the surface states with act as traps of photoexcited carriers. The real mechanism of the blue shift of band gap needs further investigations. Based on the particular structure characteristics and size effects, much progress has been made in nanodevice applications using these GaN nanowires. For example, GaN nanowires have been reported in numerous works to show promise as elemental building blocks for photonic, electronic, and optoelectronic nanodevices including logic gates, field-effect transistors (FETs), light-emitting diodes (LEDs), subwavelength photonics components, and so-walled “nanolasers” (Huang et al. 2001; Johnson et al. 2002; Zhong et al. 2003; Stern, et al. 2005; Pauzauskie et al. 2006; Qian et al. 2008). For integration into true nanodevices, the controllable assembly and precise location of fabricated nanowires must be established in device architectures. A better understanding of mechanical behavior is just one of many aspects that require detailed study.

24-3

Mechanical Properties of GaN Nanowires

24.3 Mechanical Behavior Studied Experimentally and Limitation All of the promising future applications rely on the production of nanowires, in reasonable quantities, of controlled size, shape, and crystal structure. Ultimately, all applications will require that the nanowires be mechanically stable in the application environment. Measuring the mechanical properties of individual nanowires by conventional techniques is not trivial. Optical measurements used commonly in microelectromechanical systems are not readily applicable to nanowire resonators because the diameter is less than a visible wavelength. There are several approaches to probe the mechanical properties of nanoscale specimens experimentally. The most widely used approach is the bending test, in which a nanowire suspended over a trench is indented by an atomic force microscopy (AFM) tip until failure occurs. Although this measurement is relatively easy to set up, the resulting force-displacement curve is difficult to analyze and interpret. The main difficulty comes from the unknown contact behavior between the AFM tip and the nanowire and the resulting high-stress gradient and possible contact damage. Chen et al. (2007) have investigated the mechanical elasticity of hexagonal wurtzite GaN nanowires using this method with a digital-pulsed force mode AFM. For these GaN nanowires, the stiff ness and elastic modulus exhibit a dependence on diameter. With an increasing diameter, the elastic modulus decreases while the stiff ness increases. Elastic moduli for the tested nanowires are in the range between 218.1 and 316.9 GPa. The second method is electromechanical resonance analysis in a TEM, which is based on applying an actuating signal between the nanostructure and a counter-electrode. The elastic beam theory can then be employed to relate the observed resonance frequency to Young’s modulus (Nam et al. 2006). The modulus, E, obtained for an 84 nm nanowire was close to the theoretical bulk value; but the inferred E values decreased gradually for smaller diameters, which was difficult to resolve based on the present understanding of mechanics and materials at the nanoscale (Nam et al. 2006). Another approach is the tensile test, which measures the stress state throughout the entire specimen and thereby facilitates a direct comparison with theoretical models. The tensile test has been widely used to measure the mechanical properties of bulk materials. Unfortunately, tensile tests are difficult to perform at the nanoscale, mainly due to the challenge of manipulating the nanowire into the correct location and accurately applying and measuring the stress and strain. Recently, Hessman et al. (2007) have presented a stroboscopic detection method using an optical microscope that enables time-resolved imagining of the oscillating nanowire. Th is method may be a valuable tool for imaging and analyzing vibrating nanowires and needs further investigations. The mechanical properties of GaN nanowires are not well established due to the complexities of mechanical testing presented at the nanometer size regime. Currently, with recent

advances in computational power, atomistic simulations are the primary tools for investigating the mechanical deformation of nanowires and the associated mechanisms.

24.4 Mechanical Behaviors Studied by Molecular Dynamics 24.4.1 Molecular Dynamics For atomistic or molecular simulations, the most popular methods include quantum mechanic (QM), molecular mechanic (MM), molecular dynamic (MD), coarse-grained (CG), and Monte Carlo (MC) simulations. Among them, QM methods can account explicitly for bond forming and breaking but are limited to system sizes of up to several hundred atoms. The others, however, can deal with systems of thousands, and even millions, of atoms and predict the static and dynamic behaviors of materials at the molecular level. MD simulation is a technique for computing equilibrium and transport properties of a classical many-body system. It generates such information as atomic positions and velocities at the nanoscale level from which the macroscopic properties (e.g., pressure, energy, and heat capacities) can be derived by means of statistical mechanics. In MD, atoms move under the action of conservative forces that are additive, symmetric, and derived from intermolecular potential. The dynamic evolution of the system is governed by classical Newtonian mechanics, where for each atom i, the equation of motion is given by Fi = M i

d 2ri dt 2

(24.1)

The atomic force Fi is obtained as the negative gradient of the effective potential. A physical simulation involves the proper selection of a numerical integration scheme, employment of appropriate boundary conditions, and stress and temperature control to mimic physically meaningful thermodynamic ensembles.

24.4.2 Stillinger–Weber Potential The Stillinger–Weber (SW) potential (Stillinger and Weber 1985) is the most suitable potential for tetrahedral semiconductors like Si, and it is composed of both two- and three-atom contributions: −1 ⎧ ⎡⎛ r ⎞ ⎡ ⎛ rij ⎞ −4 ⎤ ⎤ ⎫⎪ ⎪ ij v2 (rij ) = εA ⎢ B ⎜ ⎟ − 1⎥ exp ⎨ ⎢⎜ ⎟ − α ⎥ ⎬ , ⎢⎣ ⎝ σ ⎠ ⎥⎦ ⎦ ⎪⎭ ⎪⎩ ⎣⎝ σ ⎠

rij Pcr, the column is in an unstable equilibrium in a straight position, which promotes buckling. Equation 24.5 shows that the critical load is inversely proportional to the square of the length. It can be clearly seen from Figure 24.10 that the longer the GaN nanowires, the smaller the critical stress (and corresponding critical strain) for buckling. The trend is in agreement with the Euler theory. There are two approaches to obtain the Young’s modulus, i.e., force approach and energy approach (Rafii-Tabar 2004). For the force approach, the Young’s modulus can be directly obtained from the ratio of stress to strain, whereas the energy approach calculates the Young’s modulus from the second derivative of strain energy with respect to the strain per unit volume. Using the force approach, Young’s modulus is determined from the results of the tension tests for strains Tc) and low temperatures (T < Tc), respectively. These features are qualitatively very similar to the experimental observations in InP (Suzuki et al. 1998), GaAs (Boivin et al. 1990), and SiC (Zhang et al. 2003). The BDT temperatures for the [001]-oriented nanowire deduced from Figure 24.12 are between 1500 and 1800 K, which is consistent with the atomic observations of rupture changes from a clean cut at low temperatures to necking at higher temperatures _ under tensile loading. Change in slopes for the [110]and [110]-oriented nanowires are not observed, which means that the nanowires fracture by a single mechanism. The mechanical behavior of GaN nanowires shows a significant dependence_on the crystallographic orientation. In particular, the [001], [110], and [110] directions represent three orthogonal crystallographic orientations within the wurtzite GaN crystal structure—the fi rst one is in a polar direction and the others are in nonpolar directions. It has been shown that the presence of spontaneous polarization in GaN has a drastic impact on electron-hole overlap, radiative lifetimes, and subsequent emission wavelength and quantum efficiencies for GaN (Waltereit et al. 2000). Spontaneous polarization can also have an effect on the mechanical behavior, as demonstrated in the present _studies. The unique isosceles triangular cross-section of the [110]- and [110]-oriented nanowires might also lead to the different tensile behavior compared with that of the [001]-oriented nanowires.

(c)

1/T (K–1)

0 (d)

1/T (K–1)

FIGURE 24.12 Evolution of ln(τc) as a function of reciprocal temperature for the nanowires with growth direction along (a) [001]-crystal direction with {100} side planes, (b) [001]-crystal direction with {110} – side planes, (c) [110]- and (d) [110] crystal direction, respectively. (From Wang, Z. et al., J. Mater. Sci.: Mater., Electron. 19, 863, 2008a. With permission.)

The effect of orientation and cross-section shape on the properties of the nanostructures needs further investigations.

24.5 Summary The fundamental deformation mechanisms in GaN nanowires with different orientations have been investigated by MD simulations. Due to its high crystal stability at low temperatures, the deformation behavior of [001]-oriented nanowires is characterized by brittle rupture. At higher temperatures, the crystal structure becomes less stable due to higher amplitudes of atomic vibrations around their equilibrium positions. The deformation of the nanowires changes from a brittle rupture to a ductile rupture with an increase in temperature. The mechanism of plastic deformation is through a phase transformation from a crystalline to an amorphous structure. The nanowires rupture in a brittle manner at a high strain rate and in a ductile manner at a low strain rate. Interestingly, the [110]-oriented nanowires slip in the {010} planes and there _ exists multiple yield stresses along the strain path, whereas the [110]-oriented nanowires fracture in a cleavage manner under tensile loading. It should be noticed that there are some drawbacks and advantages of this simulated method. The Stillinger–Weber

24-12

potential is “short-range potential,” which includes only the fi rst nearest neighbor interactions. Th is may cause the interactions between the atoms to abruptly vanish outside a certain radius, and this affects the results somehow. For example, the melting temperature of bulk GaN using this method is determined to be 3000 K (Wang et al. 2007a,b), where the experimental value is about 2773 K. However, this short-range potential should not affect the main conclusions, and simulation results can provide a qualitative level. Characterizing the mechanical properties of individual one-dimensional nanostructures is a challenge to many existing testing and measuring techniques. The method provides a powerful tool to quantify the mechanical properties of individual nanowires.

24.6 Future Directions Although classical MD simulations can predict some useful results, the interatomic potential employed is a “short-range potential” that only include the first nearest-neighbor interactions. This can cause the interactions between atoms to abruptly vanish outside a certain radius, which will affect the results to some degree. Ab initio or first principles molecular simulation should provide more accurate results, but these require extensive computational efforts. With the development of computer power, these will become useful methods to predict the properties of GaN nanowires. Until now, calculations on nanowires only involved very small systems with diameters of only several nanometers. At the nanoscale, the behavior can change dramatically as the diameters increase. It is essential to quantify the behavior of nanowires with large diameters and increased lengths. Quantification and evolution description requires a fundamental approach for transitioning from a molecular description to continuum descriptions.

Acknowledgments Z.G. Wang is grateful for the National Natural Science Foundation of China (10704014) and the Young Scientists Foundation of UESTC (JX0731). F. Gao and W. J. Weber were supported by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy under Contract DE-AC05-76RL01830. J. Li gratefully acknowledges financial support from the “One-Hundred Talents Plan” of the Chinese Academy of Sciences. The authors also wish to thank the Molecular Science Computing Facility in the Environmental Molecular Sciences Laboratory at the Pacific Northwest National Laboratory for a grant of computer time.

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25 Optical Properties of Anisotropic Metamaterial Nanowires 25.1 25.2 25.3 25.4

Introduction ...........................................................................................................................25-1 Negative Refraction and Metamaterials.............................................................................25-2 Wave Propagation in an Anisotropic Planar Waveguide ................................................25-2 Wave Propagation on Anisotropic Nanowire Waveguides .............................................25-3 TE Modes • TM Modes • Hybrid Modes

25.5 Energy Flow on Anisotropic Nanowires ............................................................................25-7 Energy Flow of TM Modes • Energy Flow of Hybrid Modes • Forward-Wave and Backward-Wave Modes

25.6 Realization and Numerical Simulations ............................................................................25-9 25.7 Light Coupling to Nanowire Waveguide .........................................................................25-11 25.8 Applications of Nanowires Made of Indefinite Metamaterials ....................................25-12

Wentao Trent Lu Northeastern University

Srinivas Sridhar Northeastern University

Phase Shifters • Longitudinal Electric Field Enhancement • Slow Light Waveguide • Light Wheel and Open Cavity Formation

25.9 Conclusions...........................................................................................................................25-14 Acknowledgments ...........................................................................................................................25-14 References.........................................................................................................................................25-14

25.1 Introduction The booming growth of nanoscience and nanotechnology is driven by our increasing capability of nanofabrication, new understanding of nanosized structures and materials, and new applications in material science and medicine. There are two methods for nanofabrication: top down and bottom up. The top-down method has been shrinking in size from the microscale to the nanoscale due to the use of shorter wavelengths in optical lithography. The bottom-up self-assembly method has been refined to fabricate more fancy and complicated structures. The electrons and photons, the two fundamental information carriers of optoelectronics, have typical wavelengths of a few nanometers and hundreds of nanometers or a few microns, respectively. Thus a nanostructure with an internal length scale of tens or hundreds of nanometers gives rise to strong scattering and induces an interaction between electrons and photons, which leads to new optical properties of the nanostructure. With Moore’s law for shrinking the size of the silicon-based electronics expected to come to an end, completely different approaches to design computer chips such as using all-optical circuitry [13,18] and carbon-based electronics and photonics [1] are being seriously pursued. Along with the advance of nanoscience and nanotechnology, new concepts in physics such

as negative refraction [57] and transformation optics [40] are being developed. Nanowires are long objects whose diameters are of nanometer size. Due to the confi nement of the motion of electrons, holes, photon, phonons, and other quasi-particles, many interesting features other than that of the bulk materials appear. For example for the quantum wires, the resistance will be quantized. Optical wave propagations in waveguides of nanometer size [49,50] have unique properties. Nanowires made of metal can be used to concentrate an electric field [48]. Some features can be advantageous and some can be adverse. Due to the shrinking size of metallic wires and the related overheating in computer chips, information exchange between different components of a chip is bottlenecked, preventing further increase in clock speeds. The unique properties of nanowires depend not only on their sizes, they also strongly depend on the materials used. Metallic nanowires are used in chips for electron transportation. For carbon nanotubes used as quantum wires, all the electronic transport properties are determined by the graphene layer [12], which is a single layer of a honeycomb lattice of carbon atoms. Carbon nanotubes can be metallic or semiconducting depending on the geometric arrangement, such as the chirality and the number of layers. Nanowires made of silicon in silicon-oninsulator platform are used to guide optical waves [36]. 25-1

25-2

Nanowires can be made of homogeneous and isotropic materials such as dielectrics or metals. It can also be made of inhomogeneous materials such as a grating. In this chapter, we focus on the optical properties of nanowires made of the so-called metamaterials.

25.2 Negative Refraction and Metamaterials Metamaterials are a broadly defined class of materials that are artificially designed and fabricated, inhomogeneous in the microscopic scale, and posses certain desirable properties that are not available in natural materials and can be treated homogeneously in the scale of interest. These metamaterials in common have periodic or quasi-periodic structures in certain dimensions. Due to this periodicity or quasi-periodicity and the artificially designed resonant properties, new features of physical properties appear. The concept of metamaterials is not new [5]. In radio frequency range, the frequency-selective surfaces can be viewed as metamaterials. Since the realization of negative refraction [56] in microwaves [41], there has been a renewed and intense interest in electromagnetic metamaterials. Negative refraction has added a new arena to physics, leading to new concepts such as perfect lens [22,33], superlens [11,21,33], and focusing by planoconcave lens [58,59]. Negative refraction has subsequently been achieved in microwaves [7,24,29,31,32], THz waves, and optical wavelengths [2,8,47], in metamaterials made of wire and splitring resonators [45] or photonic crystals [14,25,26]. Ordinary materials have both positive permittivity and permeability, thus they possess positive refractive indices. Metamaterials can be double-negative with both permittivity (ε) and permeability (μ) being negative [39]. Double-negative metamaterials (DNM) allow negative refraction according to Snell’s law, with a negative refractive index n = ε μ < 0 [43]. Metamaterials can be single-negative with either permittivity or permeability being negative. Single-negative materials such as ferrites and metals do not allow waves to propagate inside, and reflect waves back. Metamaterials can be periodic, such as photonic crystals [19]. They can also be non-periodic, such as the materials for cloaking [38]. The optical properties of metamaterials can be either isotropic or anisotropic. Among metamaterials, there is a subclass of materials called indefinite media [16,23,44,61] whose electromagnetic properties are extremely anisotropic. The permittivity and/or the permeability tensors are indefinite matrices, whose diagonal elements are not all positive if diagonalized. For ordinary anisotropic materials, the refractive index is given by an ellipsoid [4]. For an indefinite metamaterial (IM), the dispersion is hyperbolic for one polarization and elliptical for the other. Negative refraction, superlens imaging, and hyperlens focusing [20,46] can also be realized by using IMs. Th is broad range of properties opens infinite possibilities to use metamaterials in frequencies from microwave all the way up to the visible.


In this chapter, we further narrow our focus on nanowires made of IM [17]. To begin with, we consider a very general case in which the nanowire is made of a nonmagnetic material whose transverse permittivity component is different from the longitudinal one. In the case where the transverse permittivity is negative while the longitudinal one is positive (ε⊥ < 0, ε⏐⏐ > 0), these IM waveguides can support both forward- and backward-wave modes. A high effective phase index can be obtained for these modes. These waveguides can also support degenerate modes that can be used to slow down and trap light. Magnetic metamaterials and DNMs will not be considered, though our analysis can be easily extended to these cases. Due to the backward-wave nature of waves inside the waveguide, a nanowire waveguide made of DNM will have properties similar to that of a nanowire waveguide made of IM. For easy understanding, we first consider a free-standing planar waveguide. Then we consider a cylindrical nanowire waveguide. Other shapes of nanowires, such as rectangular shapes, can also be considered and may be more suitable for optical integrated circuits. Though numerical solutions must be sought, these waveguides will have similar interesting properties and their designs can be guided by the available analytical solutions for the slab and the cylindrical nanowire waveguides.

25.3 Wave Propagation in an Anisotropic Planar Waveguide We first consider a slab of planar waveguide made of anisotropic metamaterial in air. The wave propagation is along the z-direction with phase ei(βz−ωt) and the transverse direction is in the x-direction. We consider the case of an indefinite medium with ε z > 0, ε x < 0.

(25.1)

For the transverse magnetic (TM) modes, the magnetic field is in the y-direction. The transverse wave vector wave inside the metamaterial is kx = ε z k02 − β2/ ε x .

(25.2)

Here k0 is the wave number in vacuum. Due to the negativity of εx, the dispersion is hyperbolic instead of elliptic. Since the planar waveguide is symmetric, the magnetic field is −d κ x + d /2) H y (x ) = e 0 ( , x≤ , 2 = Ae ikx x + Be −ikx x , =e

−κ 0 (x − d /2)

, x≥

−d d ≤x≤ , 2 2 d . 2

(25.3)

where d is the slab thickness κ0 = β2 − k02 is the decay constant in the transverse direction in the air

25-3

Optical Properties of Anisotropic Metamaterial Nanowires

The tangential electric field is

Pz =

iκ 0 κ0 (x + d /2) −d e , x≤ , k0 2

Ez (x ) =

k = − x (Aeikx x − Be −ikx x ), ε z k0 =−

−d d ≤x≤ , 2 2

iκ 0 −κ0 (x − d /2) d e , x≥ . k0 2

(25.4)

The matching of the tangential electric and magnetic fields at boundary x = ±d/2 leads to the following eigen equation for the TMm modes

k0d =

⎛ ⎜ mπ + 2arctan 1 − np2 / ε x ⎜⎝

ε z np2 − 1 ⎞ ⎟. 1 − np2 / ε x ⎟⎠

1

εz

(25.5)

Here np ≡ β/k0 is the phase index of the guided mode. At a fi xed wavelength or frequency, εx and εz are constant. For easy plot of the phase index np for different waveguide thickness d, one can treat the thickness d as a function of np, which is chosen as a free parameter. The band structure for TM modes on a slab waveguide with εx = −3 and εz = 2 is plotted in Figure 25.1. For waves inside the IM, one has β = ε x k02 − kx2 / ε z .

∫

∞

−∞

(

Sz dx . A guided wave is forward (backward)

only if βPz is positive (negative). For the TMm modes on the planar waveguide, we have evaluated the total energy as

⎤

)⎥⎦ .

(25.7)

The first term is the energy flow in air and the second term is that inside the slab. Due to the negative sign of εx, the energy flow inside the waveguide is negative and contra-directional to that in the air. From the plotted example, a few salient features are evident. First unlike a conventional dielectric slab waveguide, most modes are backward-wave modes since dnp/dd < 0. Only for modes near the light line where np ~ 1, they are forward-wave modes since dnp/dd > 0. In Section 25.4, we will consider nanowire waveguide made of IMs. One will see that the modes on the cylindrical nanowire share most of the features of that on a slab waveguide. For the cylindrical nanowires, the determination of forward and backward wave will be discussed in details.

25.4 Wave Propagation on Anisotropic Nanowire Waveguides We now consider wave propagation in a cylindrical waveguide. The axis of the waveguide is along the z-direction as shown in Figure 25.2. The waveguide is nonmagnetic and has an anisotropic optical property ε x = ε y = εt ≠ ε z .

(25.6)

So for kx > ε z k0 , β will be real and negative if the imaginary part of the permittivity is ignored. So the waves inside the IM will be left-handed, βSz < 0 with Sz = ExHy, similar to that inside a double-negative metamaterial. However, for waves confined in the transverse direction, Sz is no longer uniform. The total energy, Pz, is the sum of energy carried inside and outside the waveguide, Pz =

np ⎡ 1 + (−1)m cos kx d d + 1 + (−1)m sinc kx d ⎢ 2 ⎣ κ0 εx

(25.8)

The waves propagate along the cylinder axis with E = E 0e (

i βz −ωt )

, H = H0 e (

i βz −ωt )

.

(25.9)

Here β is the propagation wave number along the nanowire waveguide. Though this waveguide still allows exact solutions for all the guided modes, the formulas are more involved and the features of the modes are not as apparent as that on a slab waveguide we have considered in Section 25.3. Nevertheless, we will arrive at the simplest expressions that can be easily generalized to more complex situations including magnetic waveguides. The optical properties

4

Phase index np

3.5

1

m=0

4

3

2

5

6

X εx,y < 0

3 2a

2.5

εz > 0

2 Y

1.5 1

Z

0

1

2

3

4

5

6

7

Reduced thickness k0d

FIGURE 25.1 The phase index np of the guided TM m modes on a freestanding planar waveguide of thickness d with εx = −3 and εz = 2.

FIGURE 25.2 A nanowire waveguide made of an indefi nite metamaterial. (From Huang, Y.J. et al., Phys. Rev. A, 77, 063836, 2008. With permission.)

25-4


of cylindrical nanowires will give guidelines for properties of other shapes of nanowires that may not allow exact solutions. Due to the symmetry of the waveguide, all the field components can be expressed in terms of the longitudinal components Ez and Hz . In the polar coordinate system, one has for the fields inside the waveguide r < a with a the radius, Er = i

1 1 ⎛ ⎞ β∂r Ez + k0 ∂ φ H z ⎟ , ⎠ r εt k02 − β2 ⎜⎝

Eφ = i

1 ⎛ 1 ⎞ β ∂ φ Ez − k0∂r H z ⎟ , ⎠ ε t k02 − β2 ⎜⎝ r

Hr = i

1 1 ⎛ ⎞ −ε t k0 ∂ φ Ez + β∂ r H z ⎟ , ⎠ εt k02 − β2 ⎜⎝ r

Hφ = i

1 1 ⎛ ⎞ εt k0∂r E z + β ∂ φ H z ⎟ . ⎠ εt k02 − β2 ⎜⎝ r

One can see that due to the anisotropic nature of the waveguide, Hz and Ez inside the waveguide will have completely different behaviors. The continuity of Ez and Hz at the interface r = a gives C J ( Ka ) D I ( κa ) , . = n = n A K n ( κ 0a ) B K n ( κ 0a ) The continuity of Eϕ at the interface gives inβ

(25.10)

2 x

⎛ β2 ⎞ + ∂2y Ez + ε z ⎜ k02 − ⎟ Ez = 0, εt ⎠ ⎝

(∂

2 x

g n (x ) = −

+∂

)

2 y

)H + (ε k

2 t 0

z

)

− β H z = 0. 2

Kn′ (x ) K n −1(x ) n = + , xK n (x ) xK n (x ) x 2

inβ

A J n (Ka) κ 2 − κ 20 ⎡ g n(κ 0a) − ε z fn (Ka)⎤⎦ = k0a2κ 20κ 2 B I n (κa) ⎣

with fn (x) =

Jn′ (x ) J n −1(x ) n = − . xJ n ( x) xJ n ( x) x 2

(25.12)

H z = BIn ( κr ) einφ , r < a, (25.13)

β2 , εt

25.4.1 TE Modes For the transverse electric (TE) modes, Ez = 0. The longitudinal magnetic field is given by Equation 25.13. One has Eφ = i

(25.14)

κ0 = β − k . 2 0

We only consider the extremely anisotropic case such that the longitudinal permittivity is positive while the transverse permittivity is negative ε z > 0, εt < 0.

(25.21)

In the following, we discuss different modes in detail.

The coefficients will be determined by matching the boundary conditions. Here

2

(25.20)

⎡⎣ g n (κ 0a) − ε z f n (Ka)⎤⎦ ⎡⎣ g n (κ 0a) + hn (κa)⎤⎦ = n2 ⎣⎡(κ 0a)−2 − (κa)−2 ⎦⎤ ⎡⎣(κ 0a)−2 − ε t (κa)−2 ⎤⎦ .

κ = β2 − εt k02 ,

(25.19)

Thus we obtain the equation for all the modes

and

K = ε z k02 −

(25.18)

I ′ (x ) In −1(x ) n hn (x ) = n = − . xIn (x ) xIn (x ) x 2

(25.11)

Ez = AJ n ( Kr )e inφ , r < a,

= DK n ( κ0r )einφ , r > a.

(25.17)

The continuity of Hϕ at the interface gives

The waveguide is free-standing in air, so the wave equations for r > a are given by the above equations with the permittivity replaced by unity. The solutions are expressed in terms of the Bessel functions of various kinds

= CK n ( κ0r )e inφ , r > a

B I n (κa) κ 20 − κ 2 ⎡ g n(κ 0a) + hn (κa)⎤⎦ = 2 2 2 A J n (Ka) ⎣ k0a κ 0κ

with the following defined functions

The wave equations for the longitudinal components inside the waveguide are

(∂

(25.16)

(25.15)

=i

k0 k ∂ r H z = i 0 BI1 ( κr ) , r < a, κ2 κ k0 k ∂ r H z = −i 0 DK1 ( κr ) , r > a. κ20 κ0

(25.22)

The continuity of Eϕ at the interface requires that h0 ( κa ) + g 0 ( κ0a ) = 0.

(25.23)

For materials without loss, each term on the left side is positive, thus there is no solution. The waveguide does not support TE

25-5


modes. This is exactly like that of a metallic wire, which does not support TE surface waves since current must flow along the waveguide. Only when εt > 1, the waveguide will support TE modes, like an ordinary dielectric fiber.

the property of the TM modes of the anisotropic waveguide is similar to that of an isotropic fiber with ε = 1 + ε z (1 − εt−1 ). In the limit of long wavelength or small waveguide radius, k 0 a a. κ0

(25.24)

The continuity of Hϕ leads to the equation ε z f 0 ( Ka ) = g 0 ( κ0a ).

(25.25)

The solutions to this equation give all the TM modes. We first consider the solutions for fi xed and real values of εz and εt. This is normally associated with a fi xed k0. It is convenient to consider a solution in the form of Ka or the reduced radius k0 a as a function of κ0a. The wave number along the wave1/ 2 guide can be obtained through β = (k02 + κ20 ) . Before we seek general solutions, it is better to consider the solutions in certain limits to reveal some important features of the TM modes on the anisotropic waveguide. For the TM modes close to the light line, κ0a → 0, one has

with η = −εt / ε z . This equation gives an infinite number of solutions κ0 a = ξ0,m, with m = 1, 2, 3, .… This indicates that the anisotropic waveguide supports infinite number of propagating modes, no matter how thin the waveguide is. For κ0a → ∞, since g0(κ0a) ≃ (κ0a)−1 →0, one has ξm ≃ ηx1,m.. Here xn,m is the mth zero of Jn(x) away from the origin. For the mth TM band, one has 0 ≤ κ0a ≤ ξ0,m. The mth band starts with k0a = x0,m ε z − ε z / εt when κ0a = 0 and ends at k0a = 0 when κ0a = ξ0,m. The modes with κ0 >> k0 have d(k0a)/d(κ0a) < 0, and are backward waves. It will be obvious if we include a small imaginary part in εt with ℑεt > 0. The equation will give β with the real and imaginary parts having opposite signs. The energy flow is opposite to the phase velocity, which will be discussed later. For arbitrary values of κ0 a, the solution must be sought numerically. Since the right-hand side of Equation 25.25 is always positive, the solution requires that J1(Ka) and J0(Ka) have different signs. For the mth band, since 0 ≤ κ0 a ≤ ξ0,m with ξ 0,m the solutions of Equation 25.26, one has x 0,m ≤ Ka ≤ ξ 0,m/η < x1,m. For each κ0 a value, the Ka value can be searched within [x 0,m,x1,m] to satisfy Equation 25.25. Once the corresponding Ka is found, the reduced radius can be obtained as k0a =

ε κa ⎞ 2⎛ Ka x0,m − z (κ 0a) ⎜ ln 0 + γ ⎟ . ⎝ ⎠ x 0,m 2 Here, we have used K0(x) = −ln(x/2) − γ for a small argument with γ the Euler constant. For complex εt with ℜεt < 0 and ℑεt > 0, the real and imaginary parts of β of the allowed modes will have the same signs. These modes are forward waves, similar to that of an ordinary optical fiber. We note that close to the light line,

(25.26)

(−εt / ε z )( Ka )2 − ( κ0a )2 1 − εt

.

For the mth band, the corresponding longitudinal electric field Ez will have m nodes. The band structure and the phase index np of the TM modes on a nanowire waveguide with εt = −3 and εz = 2 is shown in Figure 25.3. We next consider a waveguide of a fi xed radius a with the following permittivity

6

3

5 Phase index np

2.5 k0a

4 3 2

2

1.5

1 0 (a)

0

2

4

6 κ0a

8

10

1

12 (b)

1

2

3 4 Reduced radius k0a

5

6

FIGURE 25.3 Band structure (a) and phase index np (b) of the guided TM modes in a nanowire waveguide of radius a with εt = −3 and εz = 2. Open circles denote the degenerate points of forward-wave and backward-wave modes. The dashed lines are for a dielectric waveguide with ε = 1 + ε z (1 − εt −1 ) . (From Huang, Y.J. et al., Phys. Rev. A, 77, 063836, 2008. With permission.)

25-6


4

4

3.5 3.5 Phase index np

3 k0a

2.5 2 1.5 1

2.5 2 1.5

0.5 0

3

0

5

10 κ0a

(a)

15

1

20

1

1.5

(b)

2 2.5 3 Reduced wave number k0a

3.5

FIGURE 25.4 Band structure (a) and phase index np (b) of the guided TM modes on an anisotropic waveguide of radius a with εt and εz given by Equation 25.27 with a = 10/kp and εa = 2.25. (From Huang, Y.J. et al., Phys. Rev. A, 77, 063836, 2008. With permission.)

εt =

k 2p ⎞ 1⎛ 1 + εa − 2 ⎟ , ⎜ 2⎝ k0 ⎠

(25.27)

k02 − k 2p ε z = 2ε a 2 . k0 (1 + ε a )− k 2p

Here εa and kp are positive constants. The realization of this property will be discussed later in Section 25.6. If k0 < kp /(1 + εa)1/2, one has εt < 0 and εz > 0. The band structure of the TM modes on this waveguide is obtained by numeric means and plotted in Figure 25.4a with the corresponding phase index in Figure 25.4b. For this waveguide, there is no cutoff of κ0 a for each band. This is because that as k0 → 0, εz ≃ 2εa and εt → −∞, thus η = −ε t /ε z → ∞. The cutoff ξ0,m ≈ ηx1,m → ∞.

Here, we use the notation ξ = κ0 a and y = κa. Note that Ka = y/η with η = −εt / ε z . At a fi xed wavelength λ or wave number k0, εt, εz are constant. Since Ka = y/η, if we use ξ = κ0a as a free parameter, the eigen equation gives a single value of Ka or y for each ξ. Since y = ξ2 + (1 − εt )(k0a)2 , so the eigen equation actually gives the reduced radius k0 a for each ξ = κ0 a. Close to the light line, ξ → 0, the eigen equation can be simplified. Analytical solutions can be obtained [17]. These hybrid modes are all forward-wave modes. In the limit of long wavelength or small waveguide radius, k 0 a → 0, Equation 25.28 is reduced to ⎛ ξ⎞ ε z f n ⎜ ⎟ = g n (ξ). ⎝ η⎠

(25.29)

25.4.3 Hybrid Modes The modes with both Ez ≠ 0 and Hz ≠ 0 are called hybrid modes. Their dispersions are contained in the solutions of Equation 25.21 with n ≠ 0. We recast the equation in the following form: ε z fn ( Ka ) = g n (ξ ) −

n2 (ξ −2 − y −2 )(ξ −2 − εt y −2 ) . g n (ξ ) + hn ( y )

(25.28)

This will give a discrete set of solutions κ0 a = ξn,m for each n. The anisotropic waveguide supports infinite number of hybrid modes, no matter how thin the waveguide is. For the allowed eigenmodes of the mth hybrid band, one has the range 0 ≤ κ0 a < ξn,m with ξn,m the solutions of Equation 25.29. The solutions near both ends of the above range can be obtained analytically as we have done. For arbitrary ξ within 3

6 5 Phase index np

2.5

k0a

4 3 2

2

1.5

1 0 (a)

0

2

4

6 κ0a

8

10

1

12 (b)

0

1


4

5

FIGURE 25.5 Band structure (a) and phase index np (b) of the guided hybrid modes with n = 1 on an anisotropic waveguide with εt = −3 and εz = 2. Open circles denote the degeneracy of forward-wave and backward-wave modes. (From Huang, Y.J. et al., Phys. Rev. A, 77, 063836, 2008. With permission.)

25-7


this range, the solution must be obtained numerically. However, only when εt < −1, one can have eigenmodes with 0 < Ka < xn,1. For the mth band, one has x n,m < Ka ≤ ξn,m/η. Otherwise, the solution for the fi rst band requires xn,1 < Ka ≤ ξn,1/η with ξn,1 > ηx n,1. The band structure and the phase index, np, of the hybrid modes on a nanowire waveguide with εt = −3 and εz = 2 are shown in Figure 25.5.

For the field in the air r > a, one has Ez = CK0(κ0r), Er = i(β/κ0) CK1(κ0r), and Hϕ = i(k0/κ0)CK1(κ0r), thus 2

Sz =

For wave propagation on the cylindrical waveguide, the components of the Poynting vector are

Pzin =

1 (Er H φ* − Eφ H r* ), Sz = 4π

Pzout

Pzin =

β 4εt k0 (Ka)2

=− (25.31) out z

P

with

∫

∫

(25.32)

a

0

Following Ref. [53], the total energy flow is normalized as Pz =

P +P P +P in z in z

out z out z

.

(25.33)

Thus one has −1 < 〈Pz〉 < 1.

25.5.1 Energy Flow of TM Modes Within the waveguide, one has Ez = AJ0(Kr), Er = −i(εzβ/εtK) AJ1(Kr), and Hϕ = −i(εzk0/K)AJ1(Kr). So the Poynting vector component along the axis of the waveguide is 2

Sz =

β J1(Kr ) . εt k0a 2 J1(Ka)

∫ 0

∞

∫ a

2

J1(Kr ) r dr , J1(Ka) 2

(25.36)

K1(κ 0r ) r dr . K 1(κ 0a)

=−

⎡ 1 ⎤ 2 + + (Ka)2 ⎥ ⎢ 2 f Ka ( ) f Ka ( ) 0 ⎣ 0 ⎦

β ε 2z f 0′ (Ka) , 4k ε f (Ka) εt Ka 2 2 0 z 0

β = k 4 0 (κ 0a)2

∞

a

β = 2k0a 2

a

The above integrals can be carried out and more compact expressions for the energy flow can be obtained as

The physical Poynting vector is given by ℜS. The total energy flow along the waveguide is the sum of energy flow inside and outside the waveguide:

Pzin = 2π Sz r dr , Pzout = 2π Sz r dr .

β 2εt k0a 2

(25.30)

1 (Ez H r* − Er H z* ). Sφ = 4π

Pz = Pzin + Pzout

(25.35)

Here the coefficient C = −κ0/[k 0aK1(κ0 a)]. For the TM modes, one has

25.5 Energy Flow on Anisotropic Nanowires

1 Sr = − (Ez H φ* − Eφ H z* ), 4π

β K 1 ( κ 0r ) . k0a 2 K1(κ0a)

⎡ 1 ⎤ 2 + − (κ 0a)2 ⎥ ⎢ 2 g a κ ( ) g a κ 0 0 ⎣ 0( 0 ) ⎦

(25.37)

g 0′ (κ 0a) β . 4k0 g 02 (κ 0a) κ 0a

Here g 0′ (x ) and f 0′ (x ) are the derivatives of g0(x) and f0(x), respectively. For convenience, we set β > 0 throughout the paper. Since g 0′ (x ) < 0 and f 0′ (x ) < 0 , one has Pzin < 0 and Pzout > 0 . In this convention, if 〈Pz 〉 > 0, this indicates that the energy flow and the phase propagation are in the same directions and the mode is a forward-wave mode. Otherwise, if 〈Pz 〉 < 0, the group velocity and the phase velocity are in the opposite direction and the mode is a backward-wave mode. The normalized energy flow for TM modes on a waveguide with εt = −3 and εz = 2 is shown in Figure 25.6. We note that for some portion of the bands the value of 〈Pz 〉 is negative and thus these modes are backward waves.

(25.34)

Here, we set the coefficient A = K/[εzk0aJ1(Ka)]. Since εt < 0, the energy flow inside the nanowire is always opposite to the phase velocity.

25.5.2 Energy Flow of Hybrid Modes The energy flow can also be evaluated for hybrid modes. The expression for Sz is much more complex than that of the TM

25-8

Handbook of Nanophysics: Nanotubes and Nanowires 1 TM 0.5 0 −0.5 −1 1

n=1

Normalized energy flow (Pz)

0.5 0 −0.5 −1 1

n=2

0.5 0 −0.5 −1 1

n=3

0.5 0 –0.5 −1 0

1

2

3


6

7

8

FIGURE 25.6 Normalized energy flow 〈Pz 〉 for the first few bands of the TM modes and hybrid modes with n = 1, 2, 3 on an anisotropic waveguide with εt = −3 and εz = −2. Here we set β > 0. Open circles denote the degenerate points where 〈Pz 〉 = 0 and υg = 0. (From Huang, Y.J. et al., Phys. Rev. A, 77, 063836, 2008. With permission.)

modes. However, the final expression for Pz is much simpler than expected, once the integrals are all carried out. One has

Pzin =

β ⎧ ε 2z 1 ( g n + hn ) f n′ − ( g n − ε z fn )hn′ ⎨− 4k0 ⎩ εt Ka κa +

out z

P

2n2 ⎡ 1 + εt 2εt ⎤ ⎪⎫ − ⎬, 4 ⎢ 2 (κa) ⎣ (κ 0a) (κa)2 ⎥⎦ ⎭⎪

β = 4k0 −

of the Bessel functions are all real. We set A = g n + hn /(k0aJ n ). Explicitly, one has

f n′ (x ) = − xfn2 (x ) −

2 n2 1 f n (x ) + 3 − , x x x

h′n (x ) = − xhn2 (x ) −

2 n2 1 hn (x ) + 3 + , x x x

(25.38)

⎧ 1 (2 g n + hn − ε z f n ) g n′ ⎨− ⎩ κ 0a

2n2 ⎡ 2 1 + ε t ⎤ ⎪⎫ − ⎬. 4 ⎢ 2 (κ 0a) ⎣ (κ 0a) (κa)2 ⎥⎦ ⎭⎪

Here f n′, g′, n and h′n are the derivatives of f n, g n, and hn, respectively. In this derivation, we assume εt and εz are real, thus the arguments

g n′ (x ) = xg n2 (x ) −

(25.39)

2 n2 1 g n (x ) − 3 − . x x x

We point out that the above expressions for Pz can be readily modified for dielectric or metallic cylindrical waveguide with the exchange of fn(ix) = −hn(x) and ixf n′ (ix ) = − xhn′ (x ). The normalized energy flow on a waveguide with εt = −3 and εz = 2 for the hybrid modes with n = 1, 2, 3 is plotted in Figure 25.6.

25-9


25.5.3 Forward-Wave and Backward-Wave Modes There are three ways to determine whether a guided mode is a forward-wave mode or backward-wave mode. One is through the sign of the derivative d(k0a)/d(κ0 a). From the band structure, we notice that for the modes near the light line d(k0 a)/d(κ0 a) > 0. These modes are forward waves. For large κ0 a or small k0 a, one has d(k0 a)/d(κ0 a) < 0, these modes are backward waves. From the band structure shown in Figures 25.3 and 25.5, the majority of the modes are backward waves. The second way is through the sign of 〈Pz 〉. For the TM modes when κ0 a → ∞, one has f0(Ka) → 0 since g0(κ0a) → 0. One thus has Ka → x0,m. Th is solution leads to the divergence of Pzin which is negative, and the vanishing of Pzout which is positive, subsequently 〈Pz 〉 → −1, these modes are all backward waves. Correspondingly, one has d(k0 a)/d(κ0 a) → −∞ for κ0 a → ∞. This is evident from the band structure shown in Figures 25.3 and 25.5. One can prove that d(k0 a)/d(κ0 a) ≥ 0 leads to 〈Pz 〉 ≥ 0 and vice versa [17]. The degeneracy of forward- and backward-wave modes is located at d(k0 a)/d(κ0 a) = 0 or 〈Pz 〉 = 0. The third way to determine whether a mode is a forward or backward wave is through the relative sign of the real and 2

25.6 Realization and Numerical Simulations

Ez 1

imaginary parts of β if dissipation is included. For example, we consider εt = −3+0.05i and εz = 2. At k0 a = 1.6, the wave numbers of the first three eigenmodes are βa = ±(1.7112 + 0.0067i), ±(2.7250−0.0397i), ±(7.5756−0.0676i). Since the free space wave length is λ = 3.927a, this is a subwavelength waveguide. For the TM modes, except for the first mode, all the other modes are backward-wave modes since for those modes ℜβ and ℑβ have different signs. The normalized energy flow is 〈Pz 〉 = 0.5151 −0.0020i, −0.4002 − 0.0058i, − 0.8760 − 0.0078 for the above three modes, respectively. Here we set ℜβ > 0. The field and Poynting vector profi les are plotted in Figure 25.7. There is an interesting feature of the modes in the anisotropic waveguide. At a fi xed k0, for a < am ≡ −εt / ε z (1 − εt ) x0,m / k0 , the mth band TM modes are backward waves. If the radius a > am, the waveguide supports two TM modes for the mth band, one forward and one backward. At a = ac, these two modes become degenerate and the total energy flow is zero. This can be seen in Figures 25.3 and 25.6 where degenerate points are marked. Further increasing the radius, the waveguide will no longer support the mth band. The critical radius ac is located such that 〈Pz 〉 = 0, d(k0a)/d(κ0a) = 0 or dneff /da = ∞. These degenerate modes can be used to slow down and even trap light. Th is will be discussed in Section 25.8.3.

Sz

These extremely anisotropic media can be realized in a broad range of frequencies. For a multilayered structure of dielectric εa and metal εm, the effective permittivities can be obtained by using the effective medium theory [23,42],

0 −1 (a)

εt = f ε m + (1 − f )ε a ,

4 Ez 2

εz =

Sz

0

εa εm . f ε a + (1 − f )ε m

(25.40)

Here f is the filling ratio of the metal. For f > fmin ≡ εa/(εa − ℜεm), one has ℜεt < 0. A realization of the anisotropic nanowire is shown in Figure 25.8.

−2 (b) 10

X

0 −10 −20

Ez

−30

Sz 0

(c)

0.5

1

1.5 r/a

2

2a 2.5

Lattice spacing D

3

FIGURE 25.7 The longitudinal electric field and Poynting vector for the first three TM modes on an anisotropic waveguide of radius a with εt = −3 + 0.05i and εz = 2 at k 0 a = 1.6 with (a) βa = 1.7112 + 0.0067i, (b) βa = 2.7250−0.0397i, and (c) βa = 7.5756−0.0676i. The imaginary parts ℑE z and ℑS z are small and not plotted. (From Huang, Y.J. et al., Phys. Rev. A, 77, 063836, 2008. With permission.)

Y Z

FIGURE 25.8 A sketch of the realization of a nanowire made of alternative disks of metal and dielectric. (From Huang, Y.J. et al., Phys. Rev. A, 77, 063836, 2008. With permission.)

25-10

Handbook of Nanophysics: Nanotubes and Nanowires Abs (Ez)

X [nm]

50 0 –50 0

100

200

300 Z [nm]

400

500

600

400

500

600

Phase of Ez

X [nm]

50 0 –50 0

100

200

300 Z [nm]

FIGURE 25.9 (See color insert following page 20-16.) FDTD simulation of the amplitude and phase propagation of the longitudinal electric field Ez along the nanowire with radius a = 60 nm at λ = 488 nm. The metamaterial nanowire consists of alternative disks of silver and glass disks of thickness 10 nm. (From Huang, Y.J. et al., Phys. Rev. A, 77, 063836, 2008. With permission.)

10

8 Effective neff = β/k0

We first consider a metamaterial waveguide at a fi xed wavelength. For silver at λ = 488 nm, one has εm = −9.121 + 0.304i [28]. A nanowire made of alternative disks of silver and glass (εa = 2.25) of equal thickness will have εt = −3.436 + 0.152i and εz = 5.971 + 0.065i by using Equation 25.40. Here the disk thickness is 10 nm for both materials. For example, if one sets a = 60 nm, one has k0 a = 0.7725. The first three TM modes will have βa = 2.2525 − 0.0747i, 4.9318−0.1419i, 7.4031 − 0.2088i. Thus one has λβ ~ 167 nm and phase refractive index neff = 2.92 for the first TM mode. The decay length is 803, 423, and 287 nm, respectively. After traveling about 420 nm along the nanowire, only the fi rst one will survive. Finite-difference time-domain (FDTD) simulations [51] were performed to obtain the phase index np of modes on the metamaterial nanowire. The procedure is as follows. We illuminate the free-standing nanowire of finite length with a Gaussian beam, then get Ez after the termination of the simulation. The length of the waveguide is set to be larger than the decay length of the first TM mode. We get the phase from Ez, then determine β. Though the waveguide supports infinite number of modes including TM and hybrid modes, our method is legitimate due to the following two reasons: First, that the excitation of hybrid modes is small due to the profi le of the incident Gaussian beam. So mainly the TM modes are excited. Second, that due to the dissipation in the metamaterial, after certain distance, only the first TM mode will survive. Thus the phase propagation is mainly due to the fi rst TM mode. The amplitude and phase propagation of Ez along the above metamaterial nanowire is shown in Figure 25.9. The relation between the phase index np and the nanowire radius a is shown in Figure 25.10. Very good agreement between FDTD simulations and analytical results has been obtained.

6

4

2

0 20

30

40

50 60 Radius a [nm]

70

80

90

FIGURE 25.10 The phase index np of the fi rst TM band on a nanowire with a different radius at λ = 488 nm. The nanowire is made of alternative disks of silver and (see Figure 25.8). The disk thickness is 10 nm. Filled circle is obtained from FDTD simulations. The dashed line is the fitting of simulation data. The solid line is calculated from band equation with effective index εt = −3.436 + 0.152i and εz = 5.971 + 0.065i. (From Huang, Y.J. et al., Phys. Rev. A, 77, 063836, 2008. With permission.)

However, for small radius, there is noticeable discrepancy. This is expected since when the radius is comparable with the lattice spacing of the multilayered metamaterial, the effective medium theory will fail. We have also performed FDTD simulations for the nanowire with smaller lattice spacing. Better agreement is indeed obtained.

25-11


9

Effective index neff

8 7 6 5 4 3 2 1 0.2

0.3

0.4

0.5 λ [μm]

0.6

0.7

0.8

FIGURE 25.11 The phase index np for the TM modes on a nanowire with radius a = 40 nm. The nanowire is a stack of equally thick alternative disks made of a Drude metal εm = 1 − k 2p / k0 (k0 + iΓ) and glass. Here kpa = 1.64 and Γa = 0.0155. The analytical curves (solid) are calculated by using real εm. (From Huang, Y.J. et al., Phys. Rev. A, 77, 063836, 2008. With permission.)

We also consider the band structure for different frequencies. The permittivity given by Equation 25.27 can be realized through the multilayered heterostructure with Drude metal with εm = 1 − k 2p / k0 and dielectric εa . The nanowire is made of alternative disks of a Drude metal and a dielectric. The band structure and the phase index np of the TM modes are shown in Figure 25.4. One noticeable feature of these bands is the flatness of each band, which indicates small group velocity. We have performed the FDTD simulation for different frequencies for nanowire with a fi xed radius. The results are shown in Figure 25.11. Again a good agreement between FDTD simulations and analytical results is achieved. In the above example, we have shown only one realization of IM. Some other realizations may also be possible. For the metamaterials proposed and realized so far [40], most of them are anisotropic. Some of the metamaterials may have indefinite indices at some frequencies, such as the cloaking structures in Ref. [38]. A major barrier to the wide use of metamaterials is the absorptive loss. However, gain can be incorporated to reduce the loss. Active metamaterials have been pursued [3,6]. Since our focus is on the optical properties of nanowires, we will not discuss the details of active metamaterials here.

25.7 Light Coupling to Nanowire Waveguide Due to its subwavelength size, coupling light into the nanowire waveguide is a technical challenge. Currently there are four ways to couple light to waveguides [36]: butt coupling, end-fire coupling, prism coupling, and grating coupling. The coupling efficiency depends on the coupling method and the optical properties of the nanowire waveguide.

The end-fire coupling is a butt coupling with a focal lens. For the butt coupling we have used in our FDTD simulations in the previous section, multiple modes will be excited. However, for nanowire waveguide made of realistic metamaterial, loss is unavoidable, thus only one or two modes will survive over certain distance and eventually only one mode will survive after a certain distance. Though the nanowire waveguide supports infinite number of modes, selectively excitation of a single mode is possible. Thus, one can take the full advantage of the rich band structure provided by the nanowire waveguide. In order to excite a single mode, the prism coupling or grating coupling should be used. However, the phase-match condition must be satisfied for maximum energy transfer from the light source to the nanowire waveguide. For a simple illustration, we use the prism coupling to excite the TM modes in a slab waveguide made of IM at λ = 1.55/μm. The metamaterial is formed by using alternative layers of silver and MgF 2. At this wavelength, one has εm = −86.64 + 8.742i and εa = 1.9. Using Equation 25.40 with fi lling ratio f = 5.6%, we have εx = −3.0582 + 0.4896i and εz = 2.0153 + 0.0003i. For simplicity, we ignore the imaginary part of the permittivity. In the range 1.4 < n p < 2.2 and the slab thickness d between 225 and 240 nm, only the TM0 modes will be excited as shown in Figure 25.12 (see Figure 25.1 for band structure of similar parameters and Figure 25.3 for that on a cylindrical nanowire). The critical thickness, such that the forward TM0 and backward TM0 will be merged into a single mode of zero group velocity, is dc = 236.9 nm with np = 1.729. At the thickness d = 234 nm, two solutions are allowed, with np = 1.553, 1.950. The first one is a forward-wave mode while the second is a backwardwave mode. We shine a Gaussian beam into a silicon prism of refractive index n = 3.518. At an incident angle 26.19° inside the prism, the forward-wave mode will be excited while at an incident angle 33.66°, the backward-wave mode will be excited, which are shown in Figure 25.13.

2.1 2 Phase index np

10

1.9 1.8 1.7 1.6 1.5 0.225

0.23 0.235 Waveguide thickness d (μm)

0.24

FIGURE 25.12 The phase index np of the guided TM0 modes on a free-standing planar waveguide of thickness d with εx = −3.0582 and εz = 2.0153. The dashed line is for d = 234 nm. The circle marks the location of zero group velocity at the critical thickness dc = 236.9 nm.

25-12


26.19°

1.5 X (μm)

1 0.5 0 –0.5

Si Air

–1 Air

Forward wave

–1.5 (a) 2 33.66°

1.5

X (μm)

1 0.5

Si

0

Air

–0.5 –1 –1.5 –20

Air

Backward wave –15

–10

–5

(b)

0

5

10

15

20

Z (μm)

FIGURE 25.13 (See color insert following page 20-16.) Gaussian beam excitation through prism coupling of the forward-wave (a) and backwardwave mode (b) at incident angle 26.19° and 33.66°, respectively. The air gap between the prism and the waveguide (d = 234 nm) is 600 nm. Plotted is the absolute value of the magnetic field Hy.

Similarly, the prism coupling and grating coupling can be used to selectively excite the guided modes on the nanowire waveguide. We will not show examples here.

25.8 Applications of Nanowires Made of Indefinite Metamaterials 25.8.1 Phase Shifters One salient feature of the modes on nanowire made of IMs is the large phase index. The high phase index is due to the hyperbolic dispersion in the metamaterial. The nanowire can be used for phase shifters with small footprint in optical integrated circuits. The presence of loss in the metamaterials will restrict the use of long nanowire waveguides. However, for many applications other than the long-haul transportation, short waveguides have the advantage of small size and footprint. From our simulation, phase index np ~ 8 can be obtained. Thus to have a phase shift of π by a nanowire of radius 40 nm at λ = 488 nm, the length of the nanowire is about 41 nm. For excitation of such high phase index modes, grating coupling should be used.

25.8.2 Longitudinal Electric Field Enhancement Besides the large phase index, modes with large longitudinal electric field can be excited on the nanowire waveguide. Recently there is a strong interest in large longitudinal electric fields [10]. Large longitudinal electric field can have a lot of applications, such as superfocusing to beat the diffraction limit [9,55] and trapping metallic nanoparticles in optical tweezer [62].

Due to the confinement in the transverse direction, there is a π/2 phase difference between Ez and Er . Here, we consider the ratio s=

| Ez |max . | Er |max

(25.41)

As examples, we show the longitudinal and TE field of the first three TM mode on a nanowire of radius a with εt = −3 + 0.05i and εz = 2 at k0 a = 1.6 in Figure 25.14. One can see that s ~ 200%, which is far more stronger than that on a silicon nanowire waveguide [10,50]. If one shrinks the radius, an even larger ratio would be expected. As we pointed out in Section 25.4.2, the band structure of the TM modes near the light line is similar to that of an ordinary optical fiber with a core index ε = 1 + ε z (1 − εt−1 ). In the above example, one has ε = 1.9149. For such a low contrast waveguide, very small longitudinal electric field will be obtained for the guided modes [10]. The fact that the nanowire waveguide made of IM will have large longitudinal electric field is due to the hyperbolic dispersion Equation 25.2, which allows much stronger confinement of light in the transverse direction, thus stronger longitudinal electric field.

25.8.3 Slow Light Waveguide Recently Tsakmakidis et al. [54] proposed to trap light in a tapered waveguide with double-negative metamaterial as the core layer. The IM waveguide can also be used to slow down and trap light. These waveguides can thus be used as delay line in optical buffers [60]. The reason is that unlike the ordinary optical fiber, these waveguides support both forward and backward waves.

25-13

Optical Properties of Anisotropic Metamaterial Nanowires 1

X Re Ez

0.5

Im Er

0 −0.5 (a) 1 Re Ez

0.5

Z Y

Im Er

0

FIGURE 25.15 A sketch of the slow light waveguide made of an indefinite medium. (From Huang, Y.J. et al., Phys. Rev. A, 77, 063836, 2008. With permission.)

−0.5 (b) 1 Re Ez

0.5

waveguide radius, any branches of the guided modes can be accessed in principle.

Im Er

0

25.8.4 Light Wheel and Open Cavity Formation

−0.5 0 (c)

0.5

1 r/a

1.5

2

FIGURE 25.14 The longitudinal and transverse electric fields of the first three TM modes on a nanowire of radius a with εt = −3 + 0.05i and εz = 2 at k0 a = 1.6. One has βa = 1.7112 + 0.0067i, 2.7250−0.0397i, and 7.5756−0.0676i for (a), (b), and (c), respectively.

For the anisotropic waveguide we have considered, εt < 0, one has Pzin < 0 and Pzout > 0 if one sets β > 0. If Pz = Pzin + Pzout < 0, the mode is a backward mode since the total energy flow is opposite to the phase velocity. Otherwise, the mode is a forward mode. At the critical radius ac, the backward and forward modes become degenerate, and the energy flow inside the waveguide cancels out that in the air. One can prove that at the critical radius ac where Pz = 0, the group velocity is indeed zero. One does not need to know the material dispersion to locate the zero group velocity point. This is due to the fact that for these waveguides, the dispersion due to geometric confi nement dominates the material dispersion at and around the critical radius. The unique properties of the modes on anisotropic waveguide can be used to slow down and even trap light. Even though the waveguide supports infi nite number of both TM and hybrid modes at any fi xed radius and frequency, with appropriate laser coupling, the excitation of the hybrid modes in the waveguide can be suppressed or even eliminated. Among the TM modes, the fi rst TM mode will be more favorably excited. Furthermore, due to the material dissipation, the fi rst TM mode will propagate the longest distance. The rest of the TM modes will all decay out at about half the decay length of the fi rst TM mode. It is the fi rst TM band that can be used for slow light application. Unlike the double negative waveguide [54], the anisotropic waveguide will slow down and trap light if one increases the radius to the critical radius. A sketch of a slow light waveguide is shown in Figure 25.15. For the butt coupling we used in the FDTD simulations, multiple modes were excited. To excite a single mode, prism coupling or grating coupling can be used. By tapering the nanowire

Resonance is ubiquitous. They have many applications such as to store and confine energy in space, enhance the field concentration, and improve the defection accuracy. To have a resonance, a compact space such as cavities is required. Once they are made, cavities lack the translation symmetry in any direction. Examples are the quantum dots, microwave cavities, and photonic-crystal microcavities. Recently, negative-index metamaterials are also used to form open cavities [15,27,34,37], such as the checkerboard open resonators [35]. For optical integrated circuits, one of the most difficult tasks of nanofabrication is the alignment of different parts and devices, such as the alignment of active nanowires on photonic-crystal waveguide [30]. It would be more desirable to have an open cavity at any location. Recently, a new concept, a light wheel, has been developed [52]. This is formed in a composite waveguide, which is made of an ordinary slab waveguide coupled with a properly designed slab waveguide made of DNM. If these two waveguides are separated infinitely away, the ordinary waveguide support a single forwardwave mode. The DNM waveguide supports a single backward-wave mode with the same phase index. Once these two waveguides are placed in the vicinity of each other, the composite waveguide no longer support propagating modes at the same wavelength. Instead it will support complex-conjugate decay modes. These two decay modes will form the so-called light wheel [52]. In order to have complex-conjugate decay modes, the waveguide should first be able to support degenerate propagating modes. In the example we considered in Section 25.7, the slab waveguide is made of a metamaterial with εx = −3.0582 and εz = 2.0153 at λ = 1.55 μm. Below the critical thickness dc = 236.9 nm, the waveguide supports two modes of different phase indices (see Figure 25.12). One mode is a forward-wave TM0 mode and the other is a backward-wave TM0 mode. At the critical thickness (the effective thickness of the waveguide is zero due to the negative Goos– Hanchen lateral shift) a double light cone will be formed [54]. However, above the critical thickness, the waveguide supports no propagating modes. Instead, it supports complex-conjugate decay

25-14


2 1.5

X (μm)

1 0.5

Si

0 –0.5

Air

–1

Air –1.5 –20 –15

–10

–5

0 Z (μm)

5

10

15

20

FIGURE 25.16 Light wheel formation through a prism coupling to a slab waveguide of thickness d = 237 nm with εx = −3.0582 and εz = 2.0153 at λ = 1.55 μ. The incident angle of the Gaussian beam in the silicon prism is 29.4°. The air gap between the prism and the waveguide is 600 nm. Plotted is the magnetic field Hy.

modes. For example at d = 237 nm, one has np = 1.7286 ± 0.0373i. When an incident beam with β=1.7286k0 hits the waveguide, the two decay modes will be excited, one decays along and the other in the opposite direction of β, thus an open cavity will be formed. This cavity can be formed at any location along the waveguide, which is shown in Figure 25.16. When dissipation is present, the decay modes no longer form a complex-conjugate pair. However, the imaginary part of their complex phase indices will still have opposite signs, thus light wheel formation is still allowed. Light wheel and open cavity can also be formed on a nanowire waveguide made of IMs. If the radius of the nanowire is larger than the critical radius ac, the nanowire will support complexconjugate decay modes. Light confinement in all three dimensions can thus be realized on the nanowire at any location along the nanowire axis.

25.9 Conclusions Indefinite metamaterials can be used to achieve negative refraction [16] and hyperlensing [20,46]. They can also be used as superlens [23]. In this chapter, we consider the wave propagation along a nanowire waveguide with an anisotropic optical constant. We have derived the eigenmodes equation and obtained the solutions for all the propagation modes. The field profi les and the energy flow on the waveguide are also analyzed. Closed-form expressions for the energy flow for all the modes are derived. For an extremely anisotropic cylinder, where the transverse component of the permittivity is negative and the longitudinal is positive (ε⊥ < 0, ε⏐⏐ > 0), the waveguide supports TM and hybrid modes but not the TE modes. Among the supported TM modes, at most only one mode can be forward wave. The rest of them are backward waves. The possible realization of these extremely anisotropic nanowires is proposed by utilizing alternative layers of metal and dielectric. Extensive FDTD simulations have been performed and confirmed our analytical results.

Light couplings to the nanowire waveguide have also been discussed. To take full advantage of the rich band structure provided by the nanowire, prism coupling and grating coupling can be used to selectively excite the guided modes. Four unique properties have been revealed for the modes on nanowire waveguides made of IMs. The first is that the backwardwave modes can have very large phase index. These nanowires can be used as phase shifters and fi lters in optics and telecommunications. The second is the large longitudinal electric field of the modes due to the hyperbolic dispersion of the metamaterial. The third is that the waveguide supports modes of zero group velocity. This is due to the fact that the waveguide can support both forward- and backward-wave modes at a fi xed radius. If the nanowire waveguide is tapered, at certain critical radius, the two modes will be degenerate and carry zero net energy flow. At other radii, these waveguides support modes with small group velocity. These waveguides can thus be used as ultracompact delay lines in optical buffers [60]. The fourth is the formation of open cavity along the nanowire due to its support of complex-conjugate decay modes above the critical radius. The above features can lead to potential applications of these nanowires in optical integrated circuits.

Acknowledgments This work was supported by the Air Force Research Laboratories, Hanscom through FA8718-06-C-0045 and the National Science Foundation through PHY-0457002.

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26 Thermal Transport in Semiconductor Nanowires Padraig Murphy California College of the Arts and University of California, Berkeley

Joel E. Moore University of California, Berkeley and Lawrence Berkeley National Laboratory

26.1 Introduction and Motivation ...............................................................................................26-1 26.2 Phonons and Nanowires .......................................................................................................26-3 26.3 Landauer Formula for the Thermal Conductance........................................................... 26-4 26.4 Thermal Conductivity at Higher Temperatures .............................................................. 26-6 26.5 Localization ........................................................................................................................... 26-8 26.6 Conclusions............................................................................................................................ 26-8 Acknowledgments ............................................................................................................................ 26-8 References.......................................................................................................................................... 26-8

26.1 Introduction and Motivation Our motivation in this chapter is to understand the physics of thermal transport in semiconductor nanowires. These systems have the potential to revolutionize thermal management technology—a field which, if one looks over the entire course of human history, has been perhaps our most important manipulation of nature. As is so often the case, it is materials with extreme properties that are the most sought after. An extremely high thermal conductivity material could be used for cooling in microelectronic systems [26,28]. Materials with a low thermal conductivity could of course be used for thermal insulation [21], but also in the resurgent field of thermoelectrics [8,16]. A “thermoelectric” is a short-hand term for a material that can be used to make a solid-state energy conversion device. Consider the case of a heat engine operating between hot and cold heat reservoirs. Instead of the usual gaseous working substance, we imagine placing a piece of solid between the two reservoirs. The temperature difference can then be used to drive the electrons in the solid from the hot to the cold reservoir, and hence around a circuit. The current in the circuit can then be used to do work. Unfortunately, such a device is necessarily less efficient than a Carnot engine. The presence of the solid between the two reservoirs means that heat can leak from one reservoir to the other. Since this heat does no work, it reduces the overall efficiency of the engine. In the literature, the efficiency is parametrized by what is called the “figure of merit,” ZT, given by ZT =

T σS 2 . κ

(26.1)

Here T is the mean temperature of the engine σ is the electrical conductivity S is the thermopower κ is the thermal conductivity The thermopower is a measure of the electric field generated by a

E temperature gradient in the material. If is the electric field and

∇T the temperature gradient, then E = S∇T . The efficiency, η, is monotonic in ZT, for ZT = 0; the efficiency η = 0, as ZT → ∞; the efficiency approaches the Carnot efficiency, η → ηc. (The precise form of η as a function of ZT can be found in [25].) Creating materials with low thermal conductivity (while maintaining a high electrical conductivity and thermopower) is thus a crucial part of the thermoelectrics research program. Semiconductor nanowires are considered among the most promising materials currently under study [10]. For most semiconductors used in thermoelectric applications, the thermal conductivity is dominated by phonon transport, while the thermopower and electrical conductivity are dominated by electron transport. Hence, if the phonon mean free path (the distance between scattering events) can be reduced while electronic properties are unaffected, the figure of merit, ZT, will be increased. The versatility of semiconductors is an important part of their potential: doping can be used to manipulate thermopower and electrical conductivity, and various growth or nanostructuring processes can be used to modify electron and thermal transport via confinement and inhomogeneity effects. Nanowires can be grown of several semiconductor materials using a variety of techniques, and silicon in particular has been extensively explored, 26-1

26-2


although InSb and other compound semiconductors might have better thermoelectric properties and can also be grown as nanowires. Beyond conventional semiconductors, some correlated oxide materials show useful thermoelectric properties in bulk [30], but relatively few oxides have successfully been synthesized as nanowires. Other one-dimensional systems, such as carbon nanotubes, may be of interest for their high intrinsic thermal conductance [3,11,19]. As well as being of technological interest, the fundamental physics of nanowires has also attracted much attention. One of the great advances of nanoscience in recent times was the observation of the quantum of thermal conductance, g0 [23]. This is a thermal conductance constructed exclusively from the fundamental constants of nature: π2kB2T . 3h

(26.2)

Here kB is Boltzmann’s constant h is Planck’s constant T is the mean temperature of the system The thermal conductance of a system, G, can be related to the thermal conductivity, κ, through G = κA/L, where A is the crosssectional area, and L the length of the system. For most systems, κ is independent of A and L, and so G has the dependence shown, i.e., G ∝ A/L. However, since g0 is constructed exclusively from fundamental constants, it gives a thermal conductance independent of both the cross-sectional area and the length of the system. In addition, note that g0 is independent of the velocity of sound of the material, of the impurity density, etc. The thermal conductance quantum has been observed at very low temperatures T < 1K for a phonon waveguide system [27]. The four suspended phonon waveguides are made from silicon nitride, and each has a width of 200 nm at the narrowest point. As in the electronic case, where the fi rst measurement of the one-dimensional electrical conductance quantum e 2/h was made in quantum point contacts [29] rather than extended one-dimensional wires, observation of the thermal conductance quantum depends on quantum confi nement in the transverse directions rather than in having an extended wire. The device is shown in Figure 26.1. The catenoidal shape is chosen to maximize coupling to the reservoirs. Figure 26.2 shows the thermal conductance divided by 16g 0 as a function of temperature. Thus, per waveguide, the thermal conductance is 4g 0 for T < 1K . We will derive this result in detail in a later section.

At higher temperatures, the thermal conductivity of certain nanowires has an unexpected dependence on the temperature, T. One expects, for temperatures less than the Debye temperature (but not so small that one is in the quantum of thermal conductance limit), that the thermal conductivity, κ, should scale like κ ∼ T 3. Th is can be understood by recalling the result for the thermal conductivity of particles: κ = 1/3cυvl, where cυ is the heat capacity, v is the velocity, and l is the mean free path.

40 KV

1 μm × 6000

43 mm

FIGURE 26.1 The experimental set up for Schwab et al. [27]. During the experiment, the phonon cavity (seen here at the center of the figure) is heated. Th is heat is then carried off by a heat current that flows through the four catenoidal phonon waveguides, that can be seen in the figure as thin slivers between the dark regions. The temperature of the phonon cavity can be measured, and so the thermal conductance of the phonon waveguides can be determined from the ratio of the heat current to the temperature increase. 100

10 Gth (T)/16g0

g0 =

CIT

1

0.1 60

100

600 1000 Temperature (mK)

6000

FIGURE 26.2 The thermal conductance as measured by Schwab et al. [27]. On the y-axis, the thermal conductance G divided by 16g0 is plotted. Clearly for T < 1 K, G ≃ 16g0. Recall that the heat flux was through

four separate phonon waveguides. The data thus determines that, at sufficiently low temperatures, the thermal conductance of a single phonon waveguide is approximately 4g0. (From Schwab, K. et al., Nature, 404, 974, 2000. With permission.)

The heat capacity scales like T 3 for a semiconductor or insulator, and so one expects κ ∼ T 3. However, this has proven to not always be the case. For the thinnest silicon nanowires, with diameters of the order of 20 nm, the thermal conductivity has been shown experimentally to scale like κ ∼ T [6,14]; for wider wires, the T 3 dependence is obtained. In Figure 26.3, the points

26-3

Thermal Transport in Semiconductor Nanowires

Thermal conductivity (W/m K)

60 50

115 nm

Diameter

40 56 nm

30

37 nm

20 10 0

22 nm 3.00 nm 0

50

100

150 200 Temperature (K)

250

300

350

FIGURE 26.3 Here the thermal conductance is plotted as a function of temperature for wire diameters of 22, 37, 56, and 115 nm. The data is taken from Li et al. [14]. The solid lines are a fit to the data due to Mingo [18]. Note that the data and fit for the 22 nm wire do not agree. (From Murphy, P.G. and Moore, J.E., Phys. Rev. B, 76(15), 155313, 2007.)

are data, and the solid lines are from a model for the thermal conductivity that is adapted from the case of bulk silicon. (We will consider the model in more detail in a later section.) As one can see, the model gives a good fit to the data for larger diameters, but fails for the 22 nm diameter wire. Figure 26.3 gives much insight into the source of thermal resistance in nanowires. While the other scattering mechanisms are independent of the diameter of the nanowire, surface scattering gives a mean free path of order its width, at least in the simplest models. Thus the reduction in thermal conductivity, and so mean free path, with the decreasing wire diameter implies that surface scattering is dominant. This is confirmed by Figure 26.4, which shows the scattering rates for different processes in silicon nanowires. 0.0035

Boundary (width of 30 nm)

0.003 Umklapp (T = 300 K)

(ωDτ)–1

0.0025 0.002 0.0015

Boundary (width of 60 nm)

0.001

Impurity

0.0005

Umklapp (T = 100 K) 0.2

0.4

0.6

0.8

1

ω/ωD

FIGURE 26.4 The scattering rate divided by the Debye frequency for various processes in nanowires. The boundary scattering rates (for two cases, diameters 30 and 60 nm) are given by the labeled horizontal lines. The umklapp rates are shown for two temperatures, T = 300 K and T = 100 K, as light lines. The rate for scattering from isotopic impurities is shown (dark line).

FIGURE 26.5 A TEM image of a 22 nm silicon nanowire. The inset is a selected area electron diffraction pattern of the nanowire. (From Li, D.Y. et al., Appl. Phys. Lett., 83, 2934, 2003. With permission.)

It is also confirmed by transmission electron microscope (TEM) images of silicon nanowires, such as Figure 26.5 taken from Ref. [14]. These nanowires were grown using the vapor–liquid– solid (VLS) method, in which silicon is dissolved in a nanometer-scale gold cluster [15,31]. As the gold becomes saturated with silicon, the silicon is precipitated out. As more silicon is added, and so more is precipitated out, the nanowire grows. The length of the nanowires is typically of order a micron. The nanowires are clearly single crystals in the core, but with a rough boundary.

26.2 Phonons and Nanowires In this chapter we will consider only semiconducting nanowires. In these systems the transport of heat is primarily through vibrations of the crystalline lattice, rather than through the motion of electrons, as would be the case in a metal. To fully understand these systems, it is important to treat the lattice vibrations quantum mechanically [12]. The first step in this procedure is to find the normal modes of the nanowire. These fall into four classes: longitudinal modes, torsional modes, and two kinds of flexural modes. These are illustrated in Figure 26.6. The longitudinal modes consist of successive expansions and contractions of the nanowire; in the limit of infinite wavelength, the longitudinal mode becomes a translation of the entire nanowire along its “long” direction. A torsional mode consists of successive rotations of a cross section of the nanowire; in the infinite wavelength limit, the torsional mode becomes a rotation of the entire nanowire around its long axis. The flexural modes arise as motion of a cross section of the nanowire in one of the directions perpendicular to the long direction, rather like the motion of a snake. There are two kinds of flexural modes, just as there are two choices for the direction perpendicular to the long direction of the nanowire. Each mode has a particular wave vector, k, associated with it; the equation of motion for the atoms in the crystal then gives an angular frequency for each mode, ω(k). In a bulk crystal,

26-4


Note that for those modes with some transverse displacement, i.e., a nonzero k x and ky, then as kz → 0 one finds that ω tends to a constant greater than zero. We refer to those modes with a fi xed k x and ky as a subband. Each k x and ky is of order n/R, where n is some integer, and R is the length scale of the width of the nanowire. Thus the angular frequency gap between successive subbands will be of order v/R, as shown in Figure 26.7. Note also that due to the discrete nature of the lattice, there is a minimum allowed wavelength, and so a maximum allowed frequency ωD, known as the Debye frequency. Thus, each subband has a minimum and maximum frequency. For our purposes, the quantum mechanical nature of the system enters primarily through the Bose–Einstein formula for the number of phonons in a mode [12]: Longitudinal

Flexural

Torsional

FIGURE 26.6 The normal modes of a nanowire fall into four classes: longitudinal, which are analogous to the same modes found in bulk systems; flexural, which have a “snaking” motion and are analogous to transverse modes in a bulk solid; and torsional modes, which have no bulk analogue. There are two independent kinds of flexural modes, for the two directions perpendicular to the long axis of the rod into which the rod can move.

in the limit of small k , one typically finds relations of the form ω( k) = v k = v kx2 + k 2y + kz2 , where v is the velocity. In a nanowire, it is often convenient to isolate the long direction of the wire, which we will call kz. For different choices of fi xed kx and ky, we get different plots of ω(kz ). This is shown in Figure 26.7. kx = 2π/R, ky = 0 kx = 0, ky = 2π/R

φ

kx = 2π/R

kx = π/R, ky = π/R

x

k

x kx = 0

φ

≈ ν /R

kx = 0, ky = 0

φ

ω

kx = π/R kx = π/R, ky = 0 kx = 0, ky = π/R

x

FIGURE 26.7 On the left is shown a schematic representation of the dispersion, i.e., plot of ω versus k, for say the longitudinal modes of the nanowire. The variable k represents the wave vector in the long direction of the nanowire. The graphs on the right represent what the normal modes look like in a cross section: the ungapped mode (ω = 0 for k = 0) has zero transverse wave vector, whereas for the gapped modes k x and/or ky will be nonzero. Of course, if the shape of the cross section is not rectangular, the normal modes will not be labeled by k x and ky, but an analogous result will hold.

n(ω,T ) =

1 . eω /(kBT ) − 1

(26.3)

Here ω is the frequency of the mode ħ is defined as ħ = h/(2π), where h is Planck’s constant T is the temperature of the system In addition, we note that the energy of a single phonon is given by ħω. In practice, one can only find the normal modes of a “clean,” harmonic system, i.e., one without impurities or interactions between the phonons. In the presence of such perturbations, a normal mode of the clean system evolves over time into other modes. This can be thought of as a phonon being transferred from one mode to another. In the case of transport, the most important consequence of this is that a phonon may be backscattered: a phonon incident on the nanowire from the left may be turned around by impurities or anharmonicity. Thus any phonon incident on the nanowire has a probability, not necessarily equal to 1, of being transmitted across it. We will denote the transmission probability of a phonon in the ith subband with frequency ω as Ti (ω). If this probability of transmission is of order 1, and so independent of the length of the system, we refer to the phonon as ballistic; if the transmission probability is proportional to 1/L, where L is the length of the system, we say the phonon is ohmic or diff usive; and finally if the transmission probability is proportional to e−L/ξ, we say the phonon is localized with localization length ξ.

26.3 Landauer Formula for the Thermal Conductance Let us consider a nanowire that is contacted at the left with a heat reservoir at a temperature Th, and on the right with a heat reservoir at a temperature Tc, with Th > Tc. The mean temperature, T, is defi ned as T = (Th + Tc)/2, and the temperature difference, ΔT, is given by ΔT = Th − Tc. To find the total heat current, one takes the heat current that the left-hand side reservoir sends down the nanowire and subtracts from it the current being sent

26-5

Thermal Transport in Semiconductor Nanowires

by the right-hand side reservoir. The heat current due to a single mode in subband i with frequency ω and velocity v is given by the energy transmitted per second by that subband, and so by ħωn(ω,T) (v/L) Ti (ω), where Ti (ω) is the transmission probability. To find the total current one then sums over modes; this sum over modes can be expressed as [12]

∑⯝∫ dn = ∫ 2π dk. L

n

We can thus write the heat current from the left reservoir due to subband i as

∫

k 2T 2 IQ = B 2 π

L v (k ) dkωi (k)n(ωi (k),T ) i T i (ωi (k)) L 2π =

∫ 2 π v (k ) e dk

i

ω (k )

i ωi ( k ) /( kBTL )

−1

As in the electronic case [7], all of the details specific to a particular system relevant to transport (scattering rates, density of states, etc.) are now contained in the transmission function, T . Once this function is specified, the thermal conductance can be found by numerically integrating T against the function shown in Equation 26.6. In the ballistic regime, every phonon is transmitted across the nanowire with probability Ti = 1. Let us consider a one-dimensional chain with longitudinal phonons only, in which case T (ω) = 1 for 0 ≤ ω ≤ ωD. It is convenient to change the integration variable to x = ħω/(kBT); then the heat current is

T i (ωi (k)).

ω max /( kBT )

∫ 0

⎡ ⎤ 1 1 ⎥. − dx x ⎢ 1 1 ⎢ x ⎥ x 1 /(2 ) 1 /(2 ) + Δ T T − Δ T T ⎢⎣ e −1 e − 1 ⎦⎥

(26.7)

At low temperatures (i.e., temperatures well below the Debye temperature for the chain), the upper limit of integration can be taken as infinity. It is straightforward to show that

Then the total heat current from the left to the right is given by ∞

IQ =

∑ ∫ 2π ω (k)v (k) ⎢⎡⎣ e dk

i

i

1

ωi (k )/(kBTh )

i

−1

−

1 ωi (k )/(kBTc )

e

× Ti (ωi (k)) =

2π

e

1 1 ⎡ ⎤ − ω /(kBTc ) dω ωTi (ω) ⎢ ω /(kBTh ) , −1 e s − 1 ⎥⎦ ⎣e min

ωimax

ω ω

∑ ∫ dω 2π k T ωimin

2

B

eω /(kBT ) T i (ω). (e − 1)2 ω /( kBT )

(26.5)

If we define each of the functions Ti (ω) to be zero for ω < ω min i and max T = T , which we will refer to as ω > ω i , and we also define i i the transmission function, then

∑

Gth (T ) =

1 1+ ΔT /(2T )

−1

π2 ⎛ ΔT ⎞ 1+ . 6 ⎜⎝ 2T ⎟⎠

IQ =

kB2T 2 π2 ΔT π2kB2T 2 = ΔT . 3h h 6 T

(26.8)

ωi

where we have used that dω/dk = v; as before, i labels the subband of the system, and Ti (ω) is the transmission probability for a phonon of frequency ω in the ith subband. The index i runs over all possible subbands, which are labeled by the allowed wave vectors in the two directions transverse to the long direction of the wire and by the nature of the branch (i.e., longitudinal, etc.). We now Taylor expand the heat current to first order in ΔT; the thermal conductance is given by G th = IQ /ΔT, and so can be expressed as

i

x

=

Therefore

(26.4)

Gth (T ) =

0

ωimax

∑ ∫ i

∫

⎤ ⎥ − 1⎦

2

x

dx

2

⎛ ω ⎞ kB eω /(kBT ) dω ⎜ T (ω). ⎟ ω /(kBT ) ⎝ kBT ⎠ (e 2π − 1)2

∫

(26.6)

We have shown that the thermal conductance of the exactly onedimensional chain—for temperatures well below the Debye temperature, and for ballistic transport—is given by Gth =

π2kB2T = g 0. 3h

(26.9)

As mentioned before, it is independent of all material properties of the system. The Debye frequency was eliminated by choosing the low temperature limit, and the sound velocity vanished due to a cancellation between a velocity term in the current and an inverse velocity term in the density of modes. Of course, because of the assumption of ballistic transport, there was no dimensionful quantity associated with the scattering of the phonons. Note also that we have a finite thermal resistance even though the phonons are not scattered inside the chain. This is the analogue of the “contact resistance” familiar from one-dimensional electronic systems. It is interesting that the identical result for the thermal conductance can be derived for fermions, and for particles that satisfy fractional exclusion statistics [24]. Th is universality appears to be related to theorems limiting the information flow in a channel [4,22].

26-6


We are now in a position to understand the result seen by Schwab et al. [27]. For sufficiently low temperatures, kBT 1 for attractive interactions. Also, T is the transmission probability, often assumed to have value of unity. As we saw above, recent experiments (Thomas et al. 1996, Reilly et al. 2002, Bird and Ochiai 2004, Yoon et al. 2007, Danneau et al. 2008) have observed intermediate plateaus in the conductance, in the range 0.5–0.7 × 2e2/h. Such deviations from the traditional Landauer picture have been explained by the formation of spin moments, arising due to confi nement. A recent review paper by Kawabata (2007) summarizes many of the standard features of modern transport simulations. We also highlight the work of Todorov and his group, who have performed a number of time-dependent tightbinding simulations, focusing on the effects of heating and power dissipation due to current flow in nanometric wires (Montgomery et al. 2002). Indeed, transport computations constitute a large portion of all modern literature concerning nanowires. As we have indicated previously, our primary area of interest is the low charge density regime. In this respect, we highlight the work of Lenac (2005), who has modeled the interaction of the phonons of a 2D WC with the electromagnetic field. This work has been used to model dispersion relations for polaritions and riplons, that might be observed experimentally, for example, on the surface of liquid He (Grimes and Adams 1979). Gold et al. have used generalized, frequency-dependent linear response computations to model the excitation spectra of CDWs and SDWs in Q1D systems (Gold and Calmels 1998). The effects of varying the background density as well as the polarization of the system have been incorporated.

27.5 The Jellium Model of Metal Wires and the Density Functional Picture A broad view of the different phases and properties relevant for nanometric conducting wires across a wide density range can be obtained by computational investigations based on the DF theory (Hughes and Ballone 2008). The jellium model and the DF methods used in this and in the following section are closely related to general models and computational techniques widely used in condensed matter physics, (Ashcroft and Mermin 1976, Brack 1993, de Heer 1993, Martin 2004) and are briefly described here mainly for completeness and for didactical purposes. We consider a highly idealized model of nanowires consisting of a rod-like distribution of positive charge, and of N electrons moving in the electrostatic field due to all the charges in the system. The wire segment has length L, it is globally neutral, and is oriented along the z direction (see Figure 27.7). Cylindrical coordinates (r, ϕ, z) are used throughout the chapter, together with Hartree atomic units. The simplest model is obtained by considering the following distribution of positive charge:

φ

L

Rc

Z r

FIGURE 27.7

Schematic drawing of a cylindrical jellium nanowire.

⎧⎪ρb ρ+ (r) = ρ+ (r , z ) = ⎨ ⎪⎩0

for r ≤ Rc , 0 ≤ z < L for r > Rc , 0 ≤ z < L

(27.1)

where Rc is the radius of the cylindrical charge density, and, in what follows, Rc 30) local spin polarization is found also in nominally paramagnetic samples at ζ = 0.

27.8 Broken-Symmetry Solutions Computations have been performed for a series of wires of length L = 32rs periodically repeated along z, consisting of N = 240 electrons moving in the electrostatic potential of a cylindrical distribution of positive charge whose plasma parameter rs spans the range 1 ≤ rs ≤ 100. At rs = 1, therefore, the radius of the positive charge is Rc = 3.162 a.u., or 1.673 Å, and the wire segment explicitly included in the computation is 16.93 Å long. The radius reaches 167.3 Å at the lower density range (rs = 100), and in such a case the periodicity along z is 1693 Å or 0.1693 μm. All wires have the same aspect ratio, L/Rc = 10.12.

Plane wave computations have been performed using a cubic simulation box of side L, and, therefore, the background density occupies ∼3% of the simulation box. Because of the fairly large size of our samples, and considering also their low average density, the sampling of the system BZ in the plane wave computations has been restricted to the Γ-point only. The role of kz points is decreased by the fact that the systems we are primarily interested in (i.e., the low-electrondensity wires) are in fact insulators. The full range of spin polarizations, 0 ≤ ζ ≤ 1, has been explored by varying the relative number of spin-up (N+) and spin-down (N−) electrons, starting from the paramagnetic case (N+ = N− = 120), and progressively increasing (decreasing) N+(N−) in steps of 10 electrons up to N+ = 240 (N− = 0). We use here N+ and N− instead of n+ and n− to indicate that the net spin imbalance, Nspin = N+ − N−, is restricted to integer values.

27.8.1 Plane Wave Computations Electron localization is fully accounted for by the 3D-plane wave computations, whose results display the same magnetic transition at nearly the same density of the radial-cylindrical computations. More precisely, up to rs = 30 the solutions found by the plane wave code also display translational invariance along z at all ζ, and, apart from occasional interchanges of nearly degenerate states, the corresponding sequence of KS orbitals agrees with that of the radial computations for full cylindrical symmetry. As anticipated in Section 27.5, the convergence of the plane wave expansion is confirmed by the good agreement of the total energy obtained by the two computational approaches. Sizable differences between the solutions of the radial scheme and those of the plane wave computation first appear at rs = 30. Inspection of the electron density found by the unconstrained plane wave minimization (see Figure 27.10) reveals that an apparent localization transition involving all coordinates (i.e., now including z) takes place in the samples of highest spin polarization (ζ ≥ 0.5). Localization can be described as partial because the overlap of different electron-density peaks is significant, and the density at local minima is still a sizable fraction of ρb (see the inset in Figure 27.10). We verified that the density modulation along z remains nearly unchanged when the sampling of the BZ is extended to more kz points. Further analysis described below suggests that the electron configuration consists of a majority of delocalized states similar to those found at high density, coexisting with localized states whose energy is at the bottom and at the top of the occupied (valence) band. We remark that up to rs = 40 localization in the radial direction, giving rise to a sequence of well-defined electron-density shells, is more marked than localization within each of the radial shells. Moreover, localization is stronger in the inner region of the wire, and somewhat attenuated in the outer electron shell, as apparent in Figure 27.10. Comparison of the different energy contributions for the z-invariant and for the localized states show that, as expected, localization is driven by a gain in correlation energy, only partly

27-13

The Wigner Transition in Nanowires 3

ρ/ρb

2 1 0 –5 (b)

0 τ/Rb

5

3 5

0 –2 0

–1

—x Rh

—z Rh

ρ 2 — ρb 1

0 1

–5

(a)

2

FIGURE 27.10 Electron density of a fully spin-polarized jellium wire obtained by the plane wave method. N = 240, rs = 30. Panel (a): 2D plot of the density ρ(r, z) on the axial plane ϕ = 0. Panel (b): 1D plot of the electron density, ρ(z), along the line parallel to the cylindrical axis of coordinates: x − a = y = 0, where a = 2.25 × rs = 0.71Rc. (From Hughes, D. and Ballone, P., Phys. Rev. B, 77, 245314, 2008. With permission.) 1.0 30 20 rS — × E(ζ) e2

0.5 rs — × Etot e2

compensated by the kinetic energy term, which increases upon localization. This energy balance, in turn, explains why localization takes place at first in spin-polarized samples, since the kinetic energy (Ashcroft and Mermin 1976) of ferromagnetic states is higher than that of the paramagnetic state, while their correlation energy is lower. On the other hand, the kinetic energy of the WC is nearly independent of spin, and thus, the kinetic energy cost of localized states is less relevant for the ferromagnetic configuration, while the potential gain in correlation energy is comparatively larger. Both energy terms, therefore, point to high spin configurations as the first candidates for localization. Despite the energy gain provided by localization, at rs = 30 the ferromagnetic configuration is still slightly higher in energy than the paramagnetic, z-invariant state. The energy difference between the two, however, is very small, and, in fact, the ground state energy is almost constant over the entire 0 ≤ ζ ≤ 1 range, suggesting that in the vicinity of the localization transition spin glass features might arise from the near degeneracy of several different spin states. This same near degeneracy with respect to changes of ζ makes it difficult to provide an accurate determination of the net ground state spin polarization at densities close to the transition point (see the inset in Figure 27.11). Nevertheless, plane wave computations confirm the stability of partially polarized stated at rs slightly higher than 30, as already anticipated by the radial-cylindrical computations. Of course, these observations imply that the transition to the spin-polarized state is at most weakly first order. The fully ferromagnetic state becomes the state of lowest energy for rs ≥ 35, the shift in the transition point from rs = 27 estimated in the radial-cylindrical method being due to the discretization

10 0 –10

0.0

–20 –30 0.0

0.2

0.4

–1.0

0.6

0.8

1.0

ζ

–0.5

0

20

40

60

80

100

rs

FIGURE 27.11 Total energy per electron of jellium wires computed by the radial-cylindrical method (full line) and by the 3D-plane wave method (full dots). At each rs, the energy of the lowest energy spin configuration is reported. The arrow marks the transition between paramagnetic and partially spin-polarized configurations. The polarization energy per electron, ΔEs(ζ), at six background densities is shown in the inset. Full dots: rs = 1; circles: rs = 20; squares: rs = 30; fi lled squares: rs = 40; diamonds: rs = 70; triangles: rs = 100. (From Hughes, D. and Ballone, P., Phys. Rev. B, 77, 245315, 2008. With permission.)

of the DOS, and to the fi nite plane wave expansion of orbitals and electron density, which affects spin-polarized systems slightly more than the paramagnetic ones. In the low-density regime at rs ≥ 40, localization is apparent in all systems, irrespective of spin polarization, and becomes progressively more marked with increasing rs. Plots of constant electron-density surfaces (isosurfaces, in what follows) provide a direct and intuitive view of

27-14


ρ/ρb

3 2 1

(b)

0 –1.5 –1.0 –0.5 0.0 0.5 (c) x/Rb

1.0

1.5

FIGURE 27.12 Electron-density contour plot for the ferromagnetic ground state of the rs = 70, N = 240 wire. Panel (a): side view. Panel (b): transversal view. Panel (c): radial density profi le obtained upon averaging the 3D density over ϕ and z. (From Hughes, D. and Ballone, P., Phys. Rev. B, 77, 245316, 2008. With permission.)

the localization extent in low-density systems, as apparent from Figure 27.12) showing the ρ = 1.6ρb isosurface for the ferromagnetic ground state at rs = 70. Shell effects are apparent from the transversal view of the density distribution (Figure 27.12b and c), and might be seen as the oversized and frozen-in version of the charge (Friedel) oscillations already present in the high density liquid phase (see Figure 27.8). The perspective view of the same isosurface (Figure 27.12a) clearly shows that the system consists of an assembly of well-defined charge droplets. In what follows, these droplets will sometimes be referred to as charge blobs, to account for their somewhat irregular shape. The gradual organization of charge into shells with increasing rs followed by the breakdown of shells into one-electron droplets is qualitatively similar to the two-stage freezing (or, equivalently, melting) observed in 2D circular quantum dots (Filinov et al. 2001), whose radial and orientational order are set in at different densities and/or temperatures. Only for sufficiently large systems, the two localization processes merge into a unique freezing transition. The fairly regular pattern displayed by the droplets distribution in low-density samples suggests that a geometrical lattice, possibly closely related to the bcc structure of the extended WC, might underlay the ground state charge configuration. To identify this ideal geometry, the continuous density distribution provided by LSD is mapped onto a particles configuration by (1) first identifying connected regions whose density is higher than ρcut = 2ρb, and then (2) associating one particle to each of these domains, and locating it at the center of mass of the corresponding charge distribution. Th is procedure provides a fully unambiguous result only for systems of fairly high rs (rs > 50). For these low-density systems, the number of connected regions (and thus the number of associated particles) is always very close to the number N of electrons in the system, the difference

r

∫

nc (r ) = 4πρb r ′ 2 g ( r ′ )dr ′

(27.13)

0

and displayed in the inset of Figure 27.13. Despite the unambiguous mapping of charge blobs into particles, the identification of the ideal structure underlying the ground state charge distribution of low-density wires is made 4 30 25 20

3 n(r)

4

15

14 12

10 g (r)

(a)

being at most a few units in all samples at rs ≥ 70, thus lending a reality flavor to the representation of the electron density by particles. The configuration obtained in this way closely resembles the low-temperature structure of classical particle models such as the one-component plasma (OCP), as obtained by slowly annealing liquid samples. In this respect, it is interesting to note that the radial distribution function, g(r), of the representing particles belonging to the inner radial shells of the computed structures displays the same characteristic features found in the glassy state of the classical OCP (Tanaka and Ichimaru 1987), consisting in an asymmetric first peak and a split-second peak (see Figure 27.13). The radial distribution function of particles representing charge blobs depends only weakly on spin polarization (see below) and on density for rs ≥ 70, apart from a trivial scaling of all distances. The dependence on sample size is also very weak up to the second peak of g(r), while it becomes important at larger distances. These results are reflected in the weak density and size dependence of the running coordination number, nc(r), defined as

8

5

2

0 1.0

1.5

2.0

2.5

3.0

3.5

r/rs 1

0

1

2

3

4

5

r/rs

FIGURE 27.13 Radial distribution function of particles representing charge blobs (see text) for a wire of N = 240 electrons at rs = 70 and ζ = 1. Inset: running coordination number, nc(r), of particles representing charge blobs. Full line: N = 240 electrons, rs = 70, ζ = 1; dash line: N = 480 electrons, rs = 70, ζ = 1; dotted line: N = 240 electrons, rs = 100, ζ = 1. The horizontal lines correspond to full shells of neighbors in the bcc (nc = 8 and nc = 14) and in the fcc (nc = 12) lattice. The vertical line corresponds to the minimum of g(r), and defines the cutoff radius for the computation of the average coordination number. (From Hughes, D. and Ballone, P., Phys. Rev. B, 77, 245317, 2008. With permission.)

27-15

–

+

Jmol

The Wigner Transition in Nanowires

(a)

(b)

FIGURE 27.14 Defective configurations in the ground state distribution of charge droplets of fully spin-polarized wires at rs = 70: (a) pair (+, −) of miscoordinated droplets in the middle shell of the N = 480 sample; (b) grain boundary on the inner shell of the N = 240 sample. (From Hughes, D. and Ballone, P., Phys. Rev. B, 77, 245318, 2008. With permission.)

difficult by the unavoidable distortions imposed by the finite sample size, and by the likely mismatch of the optimal lattice parameter with the other length scales entering the definition of our model, such as the background radius and the wire length. However, the major difficulty in characterizing the particles’ geometry arises from a variety of point and extended defects such as dislocations and grain boundaries that are distributed in the structures produced by our energy optimization (see Figure 27.14) both for the N = 480 (Figure 27.14a) and for the N = 240 systems (Figure 27.14b). These defects are likely to result, at least to some extent, from limitations of our computational scheme, unable to reach the absolute minimum of the DF energy within an acceptable number of iterations. More importantly, the disorder in the electron droplets distribution certainly reflect a real and relevant property of these low energy systems, having a vast number of similar but different configurations of nearly equal energy. In turn, the positional disorder frozen into low energy configurations is likely to affect the properties of electrondensity systems measured in experiments, and to give rise to glass-like features in the thermodynamics and real-time dynamics of low-density wires. Despite the unavoidable uncertainties due to the intrinsic disorder of the structures resulting from our computations, information on the underlying ground state structure can be obtained from average quantities such as the radial distribution function, g(r), and the coordination number, nc, defined as the value of the running coordination number up to a distance corresponding to the first minimum of the radial distribution function (rmin = 2.3rs for wires of rs ≥ 50). The similarity of the particles’ g(r) with those of the OCP already pointed out above clearly suggests a close relation with a bcc lattice. The coordination number, nc, of the charge blobs residing in the inner shell of the computed structures is close to but nevertheless systematically lower than the nc = 14 value that corresponds to the number of first and second neighbors in the bcc structure (see Figure 27.13). However, the absence of the shell closing at nc = 8 also expected for bcc, once again prevents a fully unambiguous identification. A detailed analysis of the structures found for the N = 240 and N = 480 samples however suggest that it might be more appropriate to characterize the computed geometries as being intermediate between an fcc and a bcc lattice.

At the highest rs’s considered in our study (rs ≥ 70), the separation of the density peaks is so marked that it is possible and even easy to identify the spatial domain occupied by each blob. Th is allows us to verify that not only the number of droplets corresponds to the number N of electrons, but, in addition, the integral of the charge density for each blob is very close to one, the standard deviation amounting to only 5%. At low density, therefore, blobs can be identified with electrons, even though they should not be identified with KS orbitals, as it will be discussed later. The remarkable correspondence of density blobs and electrons arises from well-known anomalies in the response functions, that in reciprocal space identify the BZ of the WC, and in real space delimit the lattice unit cell, thus determining the size, charge, and spin of the basic building block. Individual charge blobs always display partial (at rs < 70) or full (rs ≥ 70) spin polarization for all systems in which localization is apparent, irrespective of the average polarization ζ and including nominally paramagnetic samples. Needless to say, this implies that the electronic structure of low-ζ systems includes a spin-compensating mechanism, such as antiferromagnetic ordering or a more general spin wave, bringing the net spin to the value imposed by the N+ and N− values. This is apparent in Figure 27.15, displaying the spin polarization of charge blobs for the wire rs = 70, ζ = 0 wire. The spin configuration of the outermost shell is fairly disordered, while the inner shell displays a regular helical pattern (not very clear in Figure 27.15, as in any 2D representation, but apparent in computer visualizations

(a)

(b)

FIGURE 27.15 Spin polarization of charge droplets for the rs = 70, ζ = 0 wire. Light droplets: spin up electrons. Dark droplets: spin down electrons. The inner (a) and outer (b) shells are shown separately. (From Hughes, D. and Ballone, P., Phys. Rev. B, 77, 245319, 2008. With permission.)

27-16


that allow one to rotate the isosurface) probably related to the enhancement of the spin-spin response function at kz = 2K F and kz = 4KF (Pouget and Ravy 1997). The disordered spin distribution of the outer shell suggests that the spin–spin coupling constant is fairly small, as confi rmed by the computation of the spin-resolved radial distribution functions g++(r) and g+−(r) that show only a slight predominance of antiferromagnetic coupling in the first coordination shell. Th is could be seen as the expected consequence of a nearly disjoint charge and spin blobs, reducing also the exchange interaction. It is important to realize, however, that localization concerns the density, not necessarily the KS orbitals. We verified, in fact, that even in samples displaying the most apparent charge localization, each KS orbital contributes to the density of several, widely spaced blobs, as can be seen in Figure 27.16. The qualitative information contained in this figure is confi rmed by a quantitative measure of localization provided by the computation of the inverse participation ratio, that, apart from a few cases, points to a remarkably low localization for KS orbitals, as discussed below. Charge and spin localization are nevertheless clearly reflected into basic properties of the KS orbitals, affecting, for instance, the system DOS. As can be seen in Figure 27.17, the density of states for the rs = 30, ζ = 1 sample shows a deep minimum (pseudogap) at the Fermi energy, pointing to an incipient metal–insulator

FIGURE 27.16 Density isosurface, ρ1 = 0.05 ρb, for a KS orbital whose eigenvalue is close to the top of the occupied band. rs = 70, N = 240, fully spin-polarized ground state. Solid particles mark the center-of-mass position of individual charge blobs. (From Hughes, D. and Ballone, P., Phys. Rev. B, 77, 245320, 2008. With permission.)

transition driven by localization. The rs = 40, ζ = 1 sample is clearly an insulator, and at rs = 70, ζ = 1 the energy gap separating occupied and unoccupied states is as wide as the total width of the occupied bands. The localization of individual orbitals is measured by computing the inverse participation ratio, defined as follows. First of all, we compute the z-dependent planar average of each orbital, defi ned as

ψ iplane (z ) =

∫ ψ (x, y, z)dx dy i

πRc2

(27.14)

then, the inverse participation ratio, pi−1, is computed according to

pi−1 = L

∫ ⎡ ψ ⎢⎣∫

4

ψ iplane (z ) dz plane i

2 (z ) dz ⎤⎥ ⎦

2

(27.15)

The definition implies that localized states correspond to p−1 ∼ L, and delocalized states to p−1 ∼ 1. The results, reported in Figure 27.17, show that the central and major portion of the occupied band is made of fairly delocalized states, while the most localized orbitals are found at the low- and high-energy band edges. While the localization of these states could have been expected, the relative delocalization of all the other states is more surprising, given the apparent strong localization of the charge. The different behavior of the density and of KS orbitals with respect to localization might be related to the invariance of LSD with respect to unitary transformation of the occupied states, that, by definition, leave the electron density and the kinetic energy unchanged. All the effects described in this section are likely to affect the transport properties of nearly 1D, low-carrier density conductors. As a last observation, we would like to mention that at the lowest densities explored by our computations (80 ≤ rs ≤ 100), the plane wave energy optimization gives rise to surprising new structures, especially for the low-spin samples, as shown in Figure 27.18 for the N = 240 wire of rs = 100, ζ = 1.

rs = 70 DOS, Pi–1

12 rs = 30

rs = 40

8 4 0 –3

–2

–1

0

1

2

–3.0 –2.0 –1.0 0.0 1.0 rs — × (ε – εF) e2

2.0 –3.0 –2.0 –1.0 0.0

1.0

2.0

3.0

FIGURE 27.17 Density of states and inverse participation ratio, pi−1, for the fully spin-polarized wires at rs = 30, 40, and 70, N = 240. (From Hughes, D. and Ballone, P., Phys. Rev. B, 77, 245321, 2008. With permission.)

27-17


FIGURE 27.18 Density isosurface, ρ = 2ρb, for the sample of N = 240 electrons, rs = 100, ζ = 0. (From Hughes, D. and Ballone, P., Phys. Rev. B, 77, 245322, 2008. With permission.)

FIGURE 27.20 Magnetization density close to a nanoconstriction, m(r) = 0.20 ρb (darker surface), m(r) = −0.20 ρb (lighter surface), background density rs = 25, nominally paramagnetic sample of N = 230 electrons.

27.9 Nano-Constriction: Preliminary DF Results

constriction. Work is in progress to assess the effect of these modulations of the spin density on properties such as DC and AC conductivity.

The experiments of Bird et al. (Bird and Ochiai 2004, Yoon et al. 2007) point toward exotic magnetic behavior in QPC. By introducing a neck, or narrowing into the jellium background, we have been able to model such systems (Cortes-Huerto et al. 2009). Once again, the background density is a constant within the wire and zero outside. The form of the neck is as follows: ⎧ρb if γ (z ) ≤ r ≤ Rb and z − z 0 ≤ D ⎪ ρN (r) = ⎨ ⎪⎩ 0 otherwise

(27.16)

where D = 0.336 Rb. Moreover, γ(z) = r0 + α(z − z0)2, with r0 being the minimum radius at the constriction, and α = (Rb – r0)/D2. A plot of the corresponding background density is shown in Figure 27.19. The corresponding Fourier transform is 2πρb ρN (G) = V

z0 + D

∫

dz cos(Gz z ) I (Gr , z )

(27.17)

z0 − D

where ⎧1⎡ 2 2 if Gr = 0 ⎪ 2 ⎣Rb − γ (z ) ⎤⎦ ⎪ I (Gr , z ) = ⎨ ⎪ 1 [R J (G R ) − γ (z ) J (G γ (z ))] otherwise r 1 ⎪⎩ Gr b 1 r b (27.18) and Gr2 = Gx2 + G 2y . We shall not describe in detail the results of these computations that are still partially preliminary. However, we include a plot of the magnetization isosurfaces, where the positiondependent magnetization is defined by m(r) = n↑(r) − n↓(r), in an rs = 20 system (Figure 27.20). Our findings reveal the formation of complex magnetic structures, such as rings close to the

FIGURE 27.19 constriction.

Electron density for a wire with a narrow geometric

27.10 Discussion and Conclusive Remarks Experimental and technological advances have raised great expectations on the potential applications of systems of nanometric size and on nanostructured materials. Among the systems of interest in this domain, nanowires represent a particularly important case, since every functional device will require the electrical connectivity of a vast amount of more elementary and equally nanometric components. Moreover, at the level of complexity achievable at this length scale, a network of conducting wires may acquire new functionalities (Yanushkevich and Steinbach 2008), going far beyond the passive role of traditional circuits. Besides being important from the technological point of view, the conductivity of nanowires contains many features of interests from a more fundamental perspective. First of all, experimental measures for thin wires show that conductivity is quantized at values 2e2/ħ (van Wees et al. 1988). This behavior is explained by standard scattering theory (Landauer 1970), but violations to this rule have been observed (Bird and Ochiai 2004, Yoon et al. 2007), pointing to spontaneously spin-polarized electronic configurations in otherwise fully paramagnetic systems (Thomas et al. 1996). More generally, the Tomonaga–Luttinger (Tomonaga 1950, Luttinger 1963) theory predicts that the normal FL theory familiar from 3D metals (Giuliani and Vignale 2005) does not apply to conductors all the way down to atomically thin wires, and has to be replaced by a different picture in which the dynamics of charge and spin degrees of freedom becomes progressively more decoupled (Fiete 2007), opening the way to a wealth of new phenomena. Finally, and perhaps more importantly for this chapter, a metal–nonmetal transition is predicted to take place with decreasing density of the charge carriers, due to the localization of electrons and/or holes under the effect of their mutual Coulomb interactions (Schulz 1993). This transition, conceptually similar to the well-known Wigner transition predicted long ago for the 3D homogeneous electron gas (Wigner 1934), is comparably more relevant in the case of Q1D conductors, since confinement and low dimensionality enhance the role of electron correlation, and amplify the effect of localization, which can completely block conduction even when it takes place in a finite portion of the entire system.

27-18

In this chapter, we have provided a brief description of the phases believed to occur in 1D and Q1D systems. These include the “normal” regime, believed to be LL, which is contrasted to the regular FL; a magnetically ordered phase; a low-density, crystalline regime; and finally, other even more exotic phases identified by dynamical or excited-state properties. Moreover, we have summarized a number of theoretical studies, ranging from the use of simple idealized models to state-of-the-art correlated methods. A short summary of the experimental methods that have been used to fabricate wires of nanometric diameter over a wide carrier density range has been provided in Section 27.3, The summary aims at emphasizing that the low-density nanometric wires that are the subject of this chapter are not just theoretical or computational abstractions. In the second half of our review we introduce an idealized (jellium) model of metal wire, and we investigate the localization transition by a conceptually simple and computational tractable approach based on density functional (DF) theory (Hohenberg and Kohn 1964, Kohn and Sham 1965). The LSD exchange-correlation approximation we use has been specifically devised for electron systems in a low correlation regime. Therefore, the quantitative predictions of the method for the highly correlated states found in low-density nanowires have to be taken with some caution. However, the method and the results it provides have a great didactical value, since they describe the system and its localization transition in simple and intuitive terms, especially taking advantage of the independentelectron picture underlying DF. The computations described in Sections 27.5 through 27.9 offer a wealth of information on the ground state structure of the broken-symmetry phase. At sufficiently low density (rs ≥ 40), electron localization is complete and the ground state is a collection of charge droplets, each corresponding to one electron and 1/2 Bohr magneton. The broken-symmetry state is an insulator, and therefore the transition could be detected by measurements of the low-frequency electric conductivity. Other general spectroscopic properties such as the Raman spectrum might also be significantly affected by the transition that is also likely to change the frequency, strength, and dispersion relation of plasmon excitations. The distribution of the charge droplets is fairly regular, and defines a lattice whose structure is intermediate between bcc and fcc. Nevertheless, several defects are distributed in the lowest energy structures found by our numerical minimization. Disorder is apparent also in the spin distribution, which shows only a weak preference for antiferromagnetic coupling. Either positional and spin disorder might give rise to nonlinear features in the conductivity and in the Raman excitation spectrum of low-density wires. Both spontaneous spin polarization and electron localization found in our computations have a counterpart in the results of recent experiments (Rahman and Sanyal 2007, Deshpande and Bockrath 2008, Danneau et al. 2008). Noncollinear spin states, excluded by our simple LSD scheme, might instead appear as


broken-symmetry solutions both in real systems and in more sophisticated determinations of the nanowire electronic structure. In this respect, the complex density distribution found by our energy minimization at rs = 100, ζ = 0 provides clear evidence that intriguing surprises may still be expected even from the simplest jellium model of nanowires.

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28 Spin Relaxation in Quantum Wires 28.1 Introduction ...........................................................................................................................28-1 28.2 Spin Dynamics .......................................................................................................................28-1 Dynamics of a Localized Spin • Spin Dynamics of Itinerant Electrons

28.3 Spin Relaxation Mechanisms .............................................................................................. 28-6 D’yakonov–Perel Spin Relaxation • DP Spin Relaxation with Electron–Electron and Electron–Phonon Scattering • Elliott–Yafet Spin Relaxation • Spin Relaxation due to Spin–Orbit Interaction with Impurities • Bir-Aronov–Pikus Spin Relaxation • Magnetic Impurities • Nuclear Spins • Magnetic Field Dependence of Spin Relaxation • Dimensional Reduction of Spin Relaxation

28.4 Spin Dynamics in Quantum Wires ..................................................................................28-10 One-Dimensional Wires • Spin Diff usion in Quantum Wires • Weak Localization Corrections

Paul Wenk Jacobs University Bremen

Stefan Kettemann Jacobs University Bremen and Pohang University of Science and Technology

28.5 Experimental Results on Spin Relaxation Rate in Semiconductor Quantum Wires ..................................................................................28-13 Optical Measurements • Transport Measurements

28.6 Critical Discussion and Future Perspective ....................................................................28-15 28.7 Summary ...............................................................................................................................28-15 Symbols .............................................................................................................................................28-16 Acknowledgments ...........................................................................................................................28-16 References.........................................................................................................................................28-16

28.1 Introduction The emerging technology of spintronics intends to use the manipulation of the spin degree of freedom of individual electrons for energy-efficient storage and transport of information.72 In contrast to classical electronics, which relies on the steering of charge carriers through semiconductors, spintronics uses the spin carried by electrons, resembling tiny spinning tops. The difference to a classical top is that its angular momentum is quantized, it can only take two discrete values, up or down. To control the spin of electrons, a detailed understanding of the interaction between the spin and orbital degrees of freedom of electrons and other mechanisms that do not conserve its spin is necessary. These are typically weak perturbations, compared to the kinetic energy of conduction electrons, so that their spin relaxes slowly to the advantage of spintronic applications. The relaxation or depolarization of the electron spin can occur due to the randomization of the electron momentum by scattering from impurities, and dislocations in the material, and due to scattering with elementary excitations of the solid, such as phonons and other electrons, when it is transferred to the randomization of the electron spin due to the spin–orbit interaction. In addition, scattering from localized spins, such as nuclear spins and magnetic impurities, are sources of electron spin relaxation. The electron spin relaxation can be reduced by constraining the electrons in low-dimensional structures, quantum

wells (confined in one direction, free in two dimensions), quantum wires (confined in two directions, free in one direction), or quantum dots (confined in all three directions). Although spin relaxation is typically smallest in quantum dots due to their discrete energy level spectrum, the necessity to transfer the spin in spintronic devices recently led to intense research efforts to reduce the spin relaxation in quantum wires, where the energy spectrum is continuous. In the following, we review the theory of spin dynamics and relaxation in quantum wires, and compare it with recent experimental results. Section 28.2 provides a general introduction to spin dynamics. In Section 28.3, we discuss all relevant spin relaxation mechanisms and how they depend on dimension, temperature, mobility, charge carrier density, and magnetic field. In Section 28.4, we review recent results on spin relaxation in semiconducting quantum wires, and its influence on the quantum corrections to their conductance. These weak localization corrections are thereby a very sensitive measure of spin relaxation in quantum wires, in addition to optical methods, as we see in Section 28.5. We set ħ = 1 in the following.

28.2 Spin Dynamics Before we review the spin dynamics of conduction electrons and holes in semiconductors and metals, let us first reconsider the spin dynamics of a localized spin, as governed by the Bloch equations. 28-1

28-2


28.2.1 Dynamics of a Localized Spin A localized spin sˆ, like a nuclear spin, or the spin of a magnetic impurity in a solid, precesses in an external magnetic field B due to the Zeeman interaction with Hamiltonian Hz = −γgsˆB, where γg is the corresponding gyromagnetic ratio of the nuclear spin or magnetic impurity spin, respectively, which we will set equal to one, unless needed explicitly. This spin dynamics is governed by the Bloch equation of a localized spin, ∂t sˆ = γ g sˆ × B.

(28.1)

This equation is identical to the Heisenberg equation ∂tsˆ = −i[sˆ, Hz] for the quantum mechanical spin operator sˆ of an S = 1/2-spin, interacting with the external magnetic field B due to the Zeeman interaction with Hamiltonian Hz. The solution of the Bloch equation for a magnetic field pointing in the z-direction is sˆz(t) = sˆz(0), while the x- and y-components of the spin are precessing with frequency 𝛚0 = γgB around the z-axis, sˆx(t) = sˆx(0) cos ω0t + sˆy(0) sin ω0t, sˆy(t) = − sˆx(0) sin ω0t + sˆy(0) cos ω0t. Since a localized spin interacts with its environment by exchange interaction and magnetic dipole interaction, the precession will dephase after a time τ2, and the z-component of the spin relaxes to its equilibrium value sz0 within a relaxation time τ1. This modifies the Bloch equations to the phenomenological equations: 1 ∂t sˆx = γ g (sˆy B z − sˆz B y ) − sˆx τ2 1 ∂t sˆy = γ g (sˆz B x − sˆx B z ) − sˆy τ2

(28.2)

1 ∂t sˆz = γ g (sˆx B y − sˆy B x ) − (sˆz − sz 0 ). τ1

28.2.2 Spin Dynamics of Itinerant Electrons 28.2.2.1 Ballistic Spin Dynamics The intrinsic degree of freedom spin is a direct consequence of the Lorentz invariant formulation of quantum mechanics. Expanding the relativistic Dirac equation in the ratio of the electron velocity and the constant velocity of light, c, one obtains in addition to the Zeeman term, a term that couples the spin s with the momentum p of the electrons, the spin–orbit coupling H SO =

1 μB ˆ − sp × E = −ˆsBSO (p), 2 mc 2

(28.3)

where we set the gyromagnetic ratio γg = 1. E = −∇V, is an electrical field, and BSO(p) = (μB/2mc2)p × E. Substitution into the Heisenberg equation yields the Bloch equation in the presence of spin–orbit interaction: ∂t sˆ = sˆ × BSO (p),

(28.4)

so that the spin performs a precession around the momentumdependent spin–orbit field BSO(p). It is important to note that the spin–orbit field does not break the invariance under time reversal (sˆ → −sˆ, p → −p), in contrast to an external magnetic field B. Therefore, averaging over all directions of momentum, there is no spin polarization of the conduction electrons. However, by injecting a spin-polarized electron with given momentum p into a translationally invariant wire, its spin precesses in the spin–orbit field as the electron moves through the wire. The spin will be oriented again in the initial direction after it has moved a length LSO, which is the length of the spin precession. The precise magnitude of LSO does not only depend on the strength of the spin–orbit interaction but may also depend on the direction of its movement in the crystal, as we will discuss below. 28.2.2.2 Spin Diffusion Equation Translational invariance is broken by the presence of disorder due to impurities and lattice imperfections in the conductor. As the electrons scatter from the disorder potential elastically, their momentum changes in a stochastic way, resulting in diff usive motion. This results in a change of the local electron density ρ(r, t ) =

∑

α=±

| ψ α (r, t )|2 , where α = ± denotes the orientation

of the electron spin, and ψα(r, t) is the position- and timedependent electron wave function amplitude. On length scales exceeding the elastic mean free path, l e, this density is governed by the diff usion equation ∂ρ = D∇ 2 ρ ∂t

(28.5)

where the diff usion constant, D, is related to the elastic scattering time, τ, by D = v F2 τ/dD , where v F is the Fermi velocity, and dD the diff usion dimension of the electron system. This diff usion constant is related to the mobility of the electrons, μe = eτ/m by the Einstein relation μeρ = e2νD, where ν is the density of states per spin at the Fermi energy EF. Injecting an electron at position r0 into a conductor with previously constant electron density ρ0, the solution of the diff usion equation yields that the electron density spreads in space according to ρ (r, t ) = ρ0 + exp(−(r − r0 )2/4Dt )/(4π Dt )dD /2, where dD is the dimension of diff usion. This dimension is equal to the kinetic dimension d, dD = d, if the elastic mean free path, l e, is smaller than the size of the sample in all directions. If the elastic mean free path is larger than the sample size in one direction, the diff usion dimension accordingly reduces by one. Thus, on average, the variance of the distance the electron moves after time t is 〈(r − r0)2〉 = 2dDDt. This introduces a new length scale, the diff usion length LD (t ) = Dt . We can rewrite the density as ρ = 〈ψ†(r, t)ψ(r, t)〉, where ψ † = (ψ †+ , ψ †− ) is the two-component vector of the up (+) and down (−) spin fermionic creation operators, and ψ the two-component vector of annihilation operators, respectively, 〈…〉 denoting the expectation value. Accordingly, the spin density, s(r, t), is expected to satisfy a diff usion equation, as well. The spin density is defined by

28-3

Spin Relaxation in Quantum Wires

s(r, t ) =

1 † ψ (r, t )σψ(r, t ) , 2

(28.6)

where σ is the vector of Pauli matrices ⎛0 σx = ⎜ ⎝1

1⎞ ⎛0 , σy = ⎜ ⎟ 0⎠ ⎝i

H D[001] = α1 (−σx kx + σ y k y ) + γ D (σ x kx k 2y − σy ky kx2 )

−i⎞ ⎛1 , σz = ⎜ ⎟ 0⎠ ⎝0

0⎞ . −1⎟⎠

Thus, the z-component of the spin density is half the difference between the density of the spin-up and -down electrons, sz = (ρ+ − ρ−)/2, which is the local spin polarization of the electron system. Thus, we can directly infer the diff usion equation for sz and, similarly, for the other components of the spin density, yielding, without magnetic field and spin–orbit interaction,63 ∂s s = D∇ 2 s − . ∂t τˆ s

(28.7)

Here, in the spin relaxation term we introduced the tensor τˆs, which can have nondiagonal matrix elements. In case of a diagonal matrix, τsxx = τsyy = τ2 is the spin dephasing time and τszz = τ1 is the spin relaxation time. The spin diff usion equation can be written as a continuity equation for the spin density vector by defining the spin diff usion current of the spin components si, J si = − D∇si .

(28.8)

Thus, we get the continuity equation for the spin density components si, ∂si + ∇J si = − ∂t

well. Grown in [001]-direction, one gets, taking the expectation value of Equation 28.10 in the direction normal to the plane, noting that kz = kz3 = 0,17

∑τ

sj

j

.

(28.9)

sij

28.2.2.3 Spin–Orbit Interaction in Semiconductors While silicon and germanium have in their diamond structure an inversion symmetry around every midpoint on each line connecting nearest-neighbor atoms, this is not the case for III–V semiconductors like GaAs, InAs, InSb, or ZnS. These have a zincblende structure that can be obtained from a diamond structure with neighbored sites occupied by the two different elements. Therefore, the inversion symmetry is broken, which results in spin–orbit coupling. Similarly, this symmetry is broken in II–VI semiconductors. This bulk inversion asymmetry (BIA) coupling, often called the Dresselhaus coupling, is anisotropic, as given by17 H D = γ D ⎡⎣σ x kx (k 2y − kz2 ) + σ y k y (kz2 − kx2 ) + σ z kz (kx2 − k 2y )⎤⎦ , (28.10) where γD is the Dresselhaus spin–orbit coefficient. Confinement in quantum wells with width a on the order of the Fermi wavelength λF accordingly yields a spin–orbit interaction where the momentum in growth direction is of the order of 1/a. Because of the anisotropy of the Dresselhaus term, the spin–orbit interaction depends strongly on the growth direction of the quantum

(28.11)

where α1 = γ D kz2 is the linear Dresselhaus parameter. Thus, by inserting an electron with momentum along the x-direction, with its spin initially polarized in the z-direction, it will precess around the x-axis as it moves along. For narrow quantum wells, where kz2 ~1/a2 ≥ kF2, the linear term exceeds the cubic Dresselhaus terms. A special situation arises for quantum wells grown in the [110]-direction, where the spin–orbit field points normal to the quantum well, as shown in Figure 28.1, so that an electron whose spin is initially polarized along the normal of the plane remains polarized as it moves in the quantum well. In quantum wells with asymmetric electrical confinement, the inversion symmetry is broken as well. Th is structural inversion asymmetry (SIA) can be deliberately modified by changing the confinement potential by the application of a gate voltage. The resulting spin–orbit coupling, the SIA coupling, also called Rashba spin–orbit interaction58 is given by H R =α2 (σ x k y −σ y kx ),

(28.12)

where α2 depends on the asymmetry of the confinement potential, V(z), in the direction z, the growth direction of the quantum well, and can thus be deliberately changed by the application of a gate potential. At first sight, it looks as if the expectation value of the electrical field εc = −∂zV(z) in the conduction band state vanishes, since the ground state of the quantum well must be symmetric in z. Taking into account the coupling to the valence band,43,66 the discontinuities in the effective mass,46 and corrections due to the coupling to odd excited states,6 yields a sizable coupling parameter depending on the asymmetry of the confinement potential.23,66 This dependence allows one, in principle, to control the electron spin with a gate potential, which can therefore be used as the basis of a spin transistor.14 We can combine all spin–orbit couplings by introducing the spin–orbit field such that the Hamiltonian has the form of a Zeeman term H SO = − sBSO (k),

(28.13)

where the spin vector is s = σ/2. But we stress again that since BSO(k) → BSO(−k) = −BSO(k) under the time reversal operation, spin–orbit coupling does not break time reversal symmetry, since the time reversal operation also changes the sign of the spin, s → − s. Only an external magnetic field B breaks the time reversal symmetry. Thus, the electron spin operator sˆ is for fi xed electron momentum k governed by the Bloch equations with the spin–orbit field,

28-4

[010]


0

BIA[001]

0

BIA[111]

0

BIA[110]

0

0 [00

0

SIA

1]

[110]

0 [100]

[1–1 0 0]

0

FIGURE 28.1 The spin–orbit vector fields for linear BIA spin–orbit coupling for quantum wells grown in [001]-, [110]-, and [111]-direction, and for linear-structure inversion asymmetry (Rashba) coupling, respectively.

∂sˆ ˆ 1 = s × (B + BSO (k )) − ˆs . ∂t τˆ s

(28.14)

The spin relaxation tensor is no longer necessarily diagonal in the presence of spin–orbit interaction. In narrow quantum wells where the cubic Dresselhaus coupling is weak compared to the linear Dresselhaus and Rashba couplings, the spin–orbit field is given by ⎛ −α1 kx + α 2k y ⎞ BSO (k ) = − 2 ⎜ α1 k y − α 2k x ⎟ , ⎜ ⎟ ⎜⎝ ⎟⎠ 0 which

changes

both

its

direction

and

(28.15)

its

amplitude,

| BSO (k )| = 2 (α12 + α22 ) k 2 − 4α1α2 kx k y , as the direction of the momentum k is changed. Accordingly, the electron energy dispersion close to the Fermi energy is in general anisotropic, as given by E± =

1 2 αα k ± αk 1 − 4 1 2 2 cos θ sin θ , 2m* α

(28.16)

where k = |k| α = α12 + α22 k x = k cos θ Thus, when an electron is injected with energy E, with momentum k along the [100]-direction, k x = k, ky = 0, its wave

function is a superposition of plain waves with the positive −αm* + m*(α2 + 2E/m*)1/2. The momentum differmomenta k± = + ence, k− − k+ = 2m*α, causes a rotation of the electron eigenstate in the spin subspace. When the electron spin was polarized up spin at x = 0 with the eigenvector ⎛ 1⎞ ψ (x = 0) = ⎜ ⎟ ⎝ 0⎠ and its momentum points in x-direction, at a distance x, it will have rotated the spin as described by the eigenvector ψ(x ) =

1 ⎞ ⎛ 1 ⎛ ⎞ 1⎜ 1 ik x ik x α1 + iα 2 ⎟ e + + ⎜ α1 + iα 2 ⎟ e − . ⎟ 2⎜ 2⎜− ⎟ ⎝ ⎝ α ⎠ α ⎠

(28.17)

In Figure 28.2, we plot the corresponding spin density as defined in Equation 28.6 for pure Rashba coupling, α1 = 0. The spin will point again in the initial direction, when the phase difference between the two plain waves is 2π, which gives the condition for spin precession length as 2π = (k− − k+)LSO, yielding for linear Rashba and Dresselhaus coupling, and the electron moving in [100]- direction, LSO = π /m* α.

(28.18)

We note that the period of spin precession changes with the direction of the electron momentum since the spin–orbit field, Equation 28.15, is anisotropic.

28-5

Sz


1

Also, we get (s(x, t + Δt) − s(x, t) )/Δt → ∂ts(x, t) for Δt → 0, and

0

Δxi2 = 2DΔt where D is the diff usion constant. While the disorder average yields 〈Δx〉 = 0, and 〈BSO(k(t))〉 = 0, separately, for isotropic impurity scattering, averaging their product yields a finite value, since Δx depends on the momentum at time t, k(t), yielding 〈ΔxBSOi(k(t))〉 = 2Δt〈vFBSOi(k(t))〉, where 〈…〉 denotes the average over the Fermi surface. In this way, we can also evaluate the average of the spin–orbit term in Equation 28.19, expanded to first order in Δx, and get, substituting Δt → τ the spin diff usion equation,

–1 LSO /2 x

0

LSO

FIGURE 28.2 Precession of a spin injected at x = 0, polarized in z-direction, as it moves by one spin precession length L SO = π/m*α through the wire with linear Rashba spin–orbit coupling, α2.

28.2.2.4 Spin Diffusion in the Presence of Spin–Orbit Interaction As the electrons are scattered by imperfections like impurities and dislocations, their momentum is changed randomly. Accordingly, the direction of the spin–orbit field, BSO(k), changes randomly as the electron moves through the sample. This has two consequences: The electron spin direction becomes randomized thus dephasing the spin precession and relaxing the spin polarization. In addition, the spin precession term is modified as the momentum k changes randomly, and has no longer the form given in the ballistic Bloch-like equation, Equation 28.14. One can derive the diff usion equation for the expectation value of the spin, the spin density Equation 28.6, semiclassically47,60 or by diagrammatic expansion.65 In order to better understand this equation, we provide a simplified classical derivation in the following. The spin density at time t + Δt can be related to the one at the earlier time t. Note that for ballistic times Δt ≤ τ, the distance the electron has moved with a probability pΔx, Δx, is related to that time by the ballistic equation Δx = k(t)Δt/m when the electron moves with the momentum k(t). On this timescale, the spin evolution is still governed by the ballistic Bloch equation (Equation 28.14). Thus, we can relate the spin density at the position x at the time t + Δt, to the one at the earlier time t at position x − Δx: s(x , t + Δt ) =

∑p

Δx

Δx

⎛⎛ ⎞ 1 ⎞ × ⎜ ⎜ 1 − Δt ⎟ s(x − Δx , t ) − Δt[B + BSO (k(t ))] × s(x − Δx , t )⎟ . ˆ τs ⎠ ⎝⎝ ⎠ (28.19) Now, we can expand in Δt to first order and in Δx to second order. Next, we average over the disorder potential, assuming that the electrons are scattered isotropically, and substitute

∑ Δx p = ∫ (dΩ / Ω)… where Ω is the total angle, and ∫ dΩ denotes the integral over all angles with ∫ (dΩ / Ω) = 1. Δx

∂s 1 = −B × s + D∇2s + 2τ (∇v F )BSO (p) × s − s, ˆ ∂t τs

(28.20)

where 〈…〉 denotes the average over the Fermi surface. Spinpolarized electrons injected into the sample spread diff usively, and their spin polarization, while spreading diff usively as well, decays in amplitude exponentially in time. As the spins precess around the spin–orbit fields between scattering events, one also expects an oscillation of the polarization amplitude in space. One can find the spatial distribution of the spin density, which is the solution of Equation 28.20 with the smallest decay rate Γs. As an example, the solution for linear Rashba coupling is60 s(x , t ) = (eˆq cos qx + Aeˆz sin qx ) e −t / τs,

(28.21)

with 1/τs = 7/16τs0, where 1/τ s0 = 2τkF2α 22 and where the amplitude of the momentum q is determined by Dq2 = 15/16τs0, and A = 3/ 15 , and êq = q/q. This solution is plotted in Figure 28.3 for eˆq = (1, 1, 0)/ 2 . In Figure 28.4, we plot the linearly independent solution obtained by interchanging cos and sin in Equation 28.21, with the spin pointing in z-direction, initially. We choose êq = êx. Figure 28.4 shows that the period of precession is enhanced by the factor 4/ 15 in the diff usive wire, and that the amplitude of the spin density is modulated, changing from 1 to A = 3/ 15 when compared with the ballistic precession of the spin. Injecting a spin-polarized electron at one point, say x = 0, its density spreads the same way it does without spin–orbit interaction, ρ(r, t ) = exp(−r 2/4 Dt )/(4 πDt )dD /2, where r is the distance to the injection point. However, the decay of the spin density is periodically modulated as a function of 2π 15/16 r /LSO .25 The spin–orbit interaction together with the scattering from impurities is also a source of spin relaxation, as we discuss in the Section 28.3 together with other mechanisms of spin relaxation. We can find the classical spin diffusion current in the presence of spin–orbit interaction in a similar way as one can derive the classical diffusion current: The current at the position r is a sum over all currents in its vicinity, which are directed toward that position. Thus, j(r, t) = 〈vρ(r − Δx)〉 where an angular average over all possible directions of the velocity v is taken. Expanding in Δx = lev/v, and noting that 〈vρ(r)〉 = 0, one gets j(r, t) = 〈v(−Δx) ∇ρ(r)〉 = − (vFle/2)∇ρ(r) = − D∇ρ(r). For the classical spin diff usion

28-6


Sy

–0.5

0

0.5

0

0

Sz

0.5

–0.5

( 15/4)LSO/2 x

( 15/4)LSO

FIGURE 28.3 The spin density for linear Rashba coupling, which is a solution of the spin diff usion equation with the relaxation rate 7/16τs. The spin points initially in the x–y plane in the direction (1,1,0). |S|

1

<S>z

1 — 2 0.89 0

1 –— 2

0.77

0

LSO/2 x

LSO

FIGURE 28.4 The spin density for linear Rashba coupling, which is a solution of the spin diff usion equation with the relaxation rate 1/τs = 7/16τs0. Note that, compared to the ballistic spin density (Figure 28.2), the period is slightly enhanced by a factor 4/ 15 . Also, the amplitude of the spin density changes with the position x, in contrast to the ballistic case. The color is changing in proportion to the spin density amplitude.

current of spin component Si, as defined by jSi (r , t ) = vSi (r , t ), there is the complication that the spin keeps precessing as it moves from r − Δx to r, and that the spin–orbit field changes its direction with the direction of the electron velocity v. Therefore, the 0th order term in the expansion in Δx does not vanish, rather, we get jSi (r, t ) = vSik (r, t ) − De∇Si (r, t ) where Sik is the part of the spin density that evolved from the spin density at r − Δx moving with velocity v and momentum k. Noting that the spin precession on ballistic scales t ≤ τ is governed by the Bloch equation, Equation 28.14, we find by the integration of Equation 28.14 that Sik = −τ(BSO (k ) × S)i , so that we can rewrite the fi rst term yielding the total spin diff usion current as jSi = − τ v F (BSO (k ) × S)i − D∇Si .

(28.22)

Thus, we can rewrite the spin diffusion equation in terms of this spin diffusion current and get the continuity equation

∂s i 1 = − D∇jSi + τ ∇v F (BSO (k ) × S)i − sj. ˆ ∂t τ sij

(28.23)

It is important to note that in contrast to the continuity equation for the density, there are two additional terms due to the spin–orbit interaction. The last one is the spin relaxation tensor, which will be considered in detail in the next section. The other term arises due to the fact that Equation 28.20 contains a factor 2 in front of the spin–orbit precession term, while the spin diff usion current, Equation 28.22, does not contain that factor. Th is has important physical consequences, resulting in the suppression of the spin relaxation rate in quantum wires and quantum dots as soon as their lateral extension is smaller than the spin precession length, LSO, as we will see in the subsequent chapters.

28.3 Spin Relaxation Mechanisms The intrinsic spin–orbit interaction itself causes the spin of the electrons to precess coherently, as the electrons move through a conductor, defining the spin precession length LSO, Equation 28.18. Since impurities and dislocations in the conductor randomize the electron momentum, the impurity scattering is transferred into a randomization of the electron spin by the spin–orbit interaction, which thereby results in spin dephasing and spin relaxation. This results in a new length scale, the spin relaxation length, L s, which is related to the spin relaxation rate 1/τs by Ls = Dτ s .

(28.24)

28.3.1 D’yakonov–Perel Spin Relaxation D’yakonov–Perel spin relaxation (DPS) can be understood qualitatively in the following way: The spin–orbit field BSO(k) changes its direction randomly after each elastic scattering event from an

28-7


[001]-direction, Equation 28.11, and linear Rashba coupling, Equation 28.12, are the dominant spin–orbit couplings. The energy dispersion is anisotropic, as given by Equation 28.16, and the spin–orbit field, BSO(k), changes its direction and its amplitude with the direction of the momentum k: ⎛ −α1kx + α 2k y ⎞ BSO (k ) = −2 ⎜ α1k y − α 2kx ⎟ , ⎜ ⎟ ⎜⎝ ⎟⎠ 0 FIGURE 28.5 Elastic scattering from impurities changes the direction of the spin–orbit field around which the electron spin is precessing.

impurity, that is, after a time of the order of the elastic scattering time τ, when the momentum is changed randomly as sketched in Figure 28.5. Thus, the spin has the time τ to perform a precession around the present direction of the spin–orbit field, and can thus change its direction only by an angle of the order of BSOτ by precession. After a time t with Nt = t/τ scattering events, the direction of the spin will therefore have changed by an angle of the order of BSO τ N t = BSO τt . Defining the spin relaxation time, τs, as the time by which the spin direction has changed by an angle of order one, we thus fi nd that 1/τs ~ τ 〈BSO(k)2〉, where the angular brackets denote integration over all angles. Remarkably, this spin relaxation rate becomes smaller the more scattering events take place, because the smaller the elastic scattering time τ is, the less time the spin has to change its direction by precession. Such a behavior is also well known as motional or dynamic narrowing of magnetic resonance lines.9 A more rigorous derivation for the kinetic equation of the spin density matrix yields additional interference terms, not taken into account in the above argument. It can be obtained by iterating the expansion of the spin density Equation 28.19 once in the spin precession term, which yields the term Δt

Δt

∫

∫

s(x , t ) × dt ′BSO (k (t ′))× dt ′′BSO (k (t ′′)) , 0

(28.25)

0

where 〈…〉 denotes the average over all angles due to the scattering from impurities. Since the electrons move ballistically at times smaller than the elastic scattering time, the momenta are correlated only on timescales smaller than τ, yielding 〈ki(t′) kj(t″)〉 = (1/2)k2δijτδ(t′ − t″). Noting that (A × B × C)m = εijkεklmAiBjCl and ∑ εijk ε klm = δ il δ jm − δ imδ jl , we find that Equation 28.25 simplifies to − ∑i (1 / τ sij )S j , where the matrix elements of the spin relaxation terms are given by18

(

)

1 = τ BSO (k )2 δij − BSO (k )i BSO (k ) j , τ sij

(28.26)

where 〈…〉 denotes the average over the direction of the momentum k. These nondiagonal terms can diminish the spin relaxation and even result in vanishing spin relaxation. As an example, we consider a quantum well where the linear Dresselhaus coupling for quantum wells grown in the

(28.27)

with BSO (k ) = 2 (α12 + α 22 )k 2 − 4α1α 2kx k y . Thus we fi nd the spin relaxation tensor as ⎛ 1 2 α ⎜ 2 ⎜ 1 (k) = 4τk 2 ⎜ −α1α 2 τˆ s ⎜ ⎜ 0 ⎜⎝

−α1α 2 1 2 α 2 0

⎞ ⎟ ⎟ 0⎟. ⎟ α2 ⎟ ⎟⎠ 0

(28.28)

Diagonalizing this matrix, one fi nds the three eigenvalues (1/τs)(α1 ± α2)2/α2 and 2/τs where α2 = α12 + α22, and 1/τs = 2k2 τα2. Note that one of these eigenvalues of the spin relaxation tensor vanishes when α1 = α2 = α0. In fact, this is a special case, when the spin–orbit field does not change its direction with the momentum: ⎛ 1⎞ BSO (k ) α1 =α2 =α0 = 2α 0 (kx − k y ) ⎜ 1⎟ . ⎜ ⎟ ⎜⎝ 0⎟⎠

(28.29)

In this case, the constant spin density given by ⎛1⎞ ⎜ ⎟ S = S0 ⎜ 1 ⎟ ⎜0⎟ ⎝ ⎠

(28.30)

does not decay in time, since the spin density vector is parallel to the spin–orbit field BSO(k), Equation 28.29, and cannot precess, as has been noted in Ref. [4]. It turns out, however, that there are two more modes that do not decay in time, whose spin relaxation rate vanishes for α1 = α2. These modes are not homogeneous in space, and correspond to precessing spin densities. They were found previously in a numerical Monte Carlo simulation and found not to decay in time, therefore being called persistent spin helix.7,53 Recently, a long-living inhomogeneous spin density distribution has been detected experimentally in Ref. [64]. We can now get these persistent spin helix modes analytically by solving the full spin diff usion equation, Equation 28.20, with the spin relaxation tensor given by Equation 28.28. We can diagonalize that equation, noting that its eigenfunctions are plain waves S(x) ∼ exp(iQx–Et). Thereby one finds, first of all, the mode with eigenvalue E1 = DQ2, with the spin density

28-8

Handbook of Nanophysics: Nanotubes and Nanowires 0 x

LSO/2

LSO

0

<S> z

√2 S0

–√ 2 S0

0

<S> y

S0

–S0

FIGURE 28.6 Persistent spin helix solution of the spin diff usion equation for equal magnitude of linear Rashba and linear Dresselhaus coupling (Equation 28.33).

⎛ 1⎞ S = S0 ⎜ 1⎟ exp(iQx − DQ2t ). ⎜ ⎟ ⎜⎝ 0⎟⎠

(28.31)

Indeed for Q = 0, the homogeneous solution, it does not decay in time, in agreement with the solution we found above, Equation 28.31. There are, however, two more modes with the eigenvalues E± =

(

)

1 2

x − Q

y , Q +2±2Q τs

(28.32)

~ ~ ~ where Q = LSOQ/2π. At Q x = −Q y = ±1, these modes do not decay in time. These two stationary solutions are ⎛ 1⎞ ⎛ 0⎞ ⎛ 2π ⎞ ⎛ 2π ⎞ ⎜ ⎟ S = S0 −1 sin ⎜ (x − y )⎟ + S0 2 ⎜ 0⎟ cos ⎜ (x − y )⎟ , ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ LSO LSO ⎜⎝ 0 ⎟⎠ ⎜⎝ 1⎟⎠ (28.33) and the linearly independent solution obtained by interchanging cos and sin in Equation 28.33. The spin precesses as the electrons diffuse along the quantum well with the period LSO, the spin precession length, forming a persistent spin helix, as shown in Figure 28.6.

28.3.2 DP Spin Relaxation with Electron–Electron and Electron– Phonon Scattering It has been noted, that the momentum scattering which limits the D’yakonov–Perel mechanism of spin relaxation is not restricted to impurity scattering, but can also be due to

electron–phonon or electron–electron interactions.19,26,27,57 Thus, the scattering time τ is the total scattering time as defined, 1/τ = 1/τ0 + 1/τee + 1/τep, where 1/τ0 is the elastic scattering rate due to scattering from impurities with potential V, given by 2 1/τ 0 = 2πνni ∫(dθ/2π)(1 − cos θ) V (k , k ′) , where v is the density of states per spin at the Fermi energy, ni is the concentration of impurities with potential V, and kk′ = kk′ cos(θ). In degenerate semiconductors and in metals, the electron–electron scattering rate is given by the Fermi liquid inelastic electron scattering rate 1/τee ∼ T 2/εF. The electron–phonon scattering time, 1/τep ∼ T 5, decays faster with temperature. Thus, at low temperatures, the DP spin relaxation is dominated by elastic impurity scattering τ0.

28.3.3 Elliott–Yafet Spin Relaxation Because of the spin–orbit interaction, the conduction electron wave functions are not eigenstates of the electron spin, but have an admixture of both spin-up and spin-down wave functions. Thus, a nonmagnetic impurity potential V can change the electron spin, by changing their momentum due to the spin–orbit coupling. Th is results in another source of spin relaxation which is stronger, the more often the electrons are scattered, and is thus proportional to the momentum scattering rate 1/τ.21,68 For degenerate III–V semiconductors one finds13,56 1 Δ 2SO Ek2 1 , ∼ 2 τ s (EG + Δ SO ) EG2 τ(k )

(28.34)

where EG is the gap between the valence and the conduction band of the semiconductor, and ΔSO is the spin–orbit splitting of the valence band. Thus, the Elliott–Yafet spin relaxation (EYS) can

28-9


be distinguished, being proportional to 1/τ, and thereby to the resistivity, in contrast to the DP spin scattering rate, Equation 28.26, which is proportional to the conductivity. Since the EYS decays in proportion to the inverse of the band gap, it is negligible in large band gap semiconductors like Si and GaAs. The scattering rate 1/τ is again the sum of the impurity scattering rate,21 the electron–phonon scattering rate,29,68 and the electron– electron interaction,10 so that all these scattering processes result in EY spin relaxation. In nondegenerate semiconductors, where the Fermi energy is below the conduction band edge, 1/τs ∼ τT 3/EG attains a stronger temperature dependence.

28.3.4 Spin Relaxation due to Spin–Orbit Interaction with Impurities The spin–orbit interaction, as defi ned in Equation 28.3, arises whenever there is a gradient in an electrostatic potential. Thus, the impurity potential gives rise to the spin–orbit interaction VSO =

1 ∇V × k s 2m2c 2

(28.35)

Perturbation theory yields, then, directly the corresponding spin relaxation rate 1 = πνni τs

dθ

∑ ∫ 2π (1 − cos θ) V

2

SO

(k , k ′)αβ ,

(28.36)

α ,β

proportional to the concentration of impurities, ni . Here α, β = ± denotes the spin indices. Since the spin–orbit interaction increases with the atomic number Z of the impurity element, this spin relaxation increases as Z 2, being stronger for heavier element impurities.

28.3.5 Bir-Aronov–Pikus Spin Relaxation The exchange interaction, J, between electrons and holes in p-doped semiconductors results in spin relaxation, as well.8 Its strength is proportional to the density of holes p and depends on their itinerancy. If the holes are localized, they act like magnetic impurities. If they are itinerant, the spin of the conduction electrons is transferred by the exchange interaction to the holes, where the spin–orbit splitting of the valence bands results in fast spin relaxation of the hole spin due to the Elliott–Yafet, or the Dyakonov–Perel mechanism.

28.3.6 Magnetic Impurities Magnetic impurities have a spin S that interacts with the spin of the conduction electrons by the exchange interaction J, resulting in a spatially and temporarily fluctuating local magnetic field B MI (r) = −

∑ J δ(r − R )S, i

(28.37)

i

where the sum is over the position of the magnetic impurities Ri. This gives rise to spin relaxation of the conduction electrons, with a rate given by

1 τMs

= 2πnM νJ 2S(S + 1),

(28.38)

where nM is the density of magnetic impurities, ν is the density of states at the Fermi energy. Here, S is the spin quantum number of the magnetic impurity, which can take the values S = 1/2, 1, 3/2, 2…. Antiferromagnetic exchange interaction between the magnetic impurity spin and the conduction electrons results in a competition between the conduction electrons to form a singlet with the impurity spin, which results in enhanced nonmagnetic and magnetic scattering. At low temperatures, the magnetic impurity spin is screened by the conduction electrons, resulting in a vanishing of the magnetic scattering rate. Thus, the spin scattering from magnetic impurities has a maximum at a temperature of the order of the Kondo temperature, TK ∼ EF exp(−1/νJ), where ν is the density of states at the Fermi energy.50,52,71 In semiconductors, TK is exponentially small due to the small effective mass and the resulting small density of states, v. Therefore, the magnetic moments remain free at the experimentally achievable temperatures. At large concentration of magnetic impurities, the RKKYexchange interaction between the magnetic impurities quenches however the spin quantum dynamics, so that S(S + 1) is replaced by its classical value S2. In Mn-p-doped GaAs, the exchange interaction between the Mn dopants and the holes can result in compensation of the hole spins and therefore a suppression of the Bir-Aronov–Pikus (BAP) spin relaxation.3

28.3.7 Nuclear Spins Nuclear spins interact by the hyperfine interaction with conduction electrons. The hyperfine interaction between nuclear spins, Î, and the conduction electron spin, sˆ, results in a local Zeeman field given by54 ˆ (r) = − 8π g 0μ B B N 3 γg

∑ γ Î, δ(r − R ), n

n

(28.39)

n

where γn is the gyromagnetic ratio of the nuclear spin. The spatial and temporal fluctuations of this hyperfine interaction field result in spin relaxation proportional to its variance, similar to the spin relaxation by magnetic impurities.

28.3.8 Magnetic Field Dependence of Spin Relaxation The magnetic field changes the electron momentum due to the Lorentz force, resulting in a continuous change of the spin– orbit field, which similar to the momentum scattering results in motional narrowing and thereby a reduction of DPS:12,34,56 1 τ . ∼ τ s 1 + ω 2c τ 2

(28.40)

28-10

Another source of a magnetic field dependence is the precession around the external magnetic field. In bulk semiconductors and for magnetic fields perpendicular to a quantum well, the orbital mechanism is dominating, however. Th is magnetic field dependence can be used to identify the spin relaxation mechanism, since the EYS does have only a weak magnetic field dependence due to the weak Pauli-paramagnetism.

28.3.9 Dimensional Reduction of Spin Relaxation Electrostatic confinement of conduction electrons can reduce the effective dimension of their motion. In quantum dots, the electrons are confined in all three directions, and the energy spectrum consists of discrete levels like in atoms. Therefore, the energy conservation restricts relaxation processes severely, resulting in strongly enhanced spin relaxation times in quantum dots.1,39 Then, spin relaxation can only occur due to the absorption or the emission of phonons, yielding spin relaxation rates proportional to the inelastic electron–phonon scattering rate.39 Quantitative comparison of the various spin relaxation mechanisms in GaAs quantum dots resulted in the conclusion that the spin relaxation is dominated by the hyperfine interaction.22,37,38 A similar conclusion can be drawn from experiments on low temperature spin relaxation in low density n-type GaAs, where the localization of the electrons in the impurity band results in spin relaxation dominated by hyperfi ne interaction as well.20,62 For linear Rashba and linear Dresselhaus spin–orbit coupling, we can see from the spin diff usion equation (Equation 28.20) with the DP spin relaxation tensor (Equation 28.28) that the spin relaxation vanishes, when the spin current (Equation 28.22) vanishes, in which case the last two terms of Equation 28.20 cancel exactly. The vanishing of the spin current is imposed by hard wall boundary condition for which the spin diff usion current vanishes at the boundaries of the sample, jSi n |Boundary = 0, where n is the normal to the boundary. When the quantum dot is smaller than the spin precession length LSO, the lowest energy mode thus corresponds to a homogeneous solution with vanishing spin relaxation rate. Cubic spin–orbit coupling does not yield such a vanishing of the DP spin relaxation rate. Only in quantum dots whose size does not exceed the elastic mean free path, le, the DP spin relaxation from cubic spin relaxation becomes diminished. In quantum wires, the electrons have a continuous spectrum of delocalized states. Still, transverse confinement can reduce the DP spin relaxation as we review in the next section.


spin precession, so that the D’yakonov–Perel-spin relaxation is absent in one-dimensional wires.49 In an external magnetic field, the precession around the magnetic field axis, due to the Zeeman interaction, is competing with the spin–orbit field, however. Then, as the electrons are scattered from impurities, both the precession axis and the amplitude of the total precession field is changing, since |B + BSO(−p)| = |B − BSO(p)| ≠ |B + BSO(p)|, resulting in spin dephasing and relaxation, as the sign of the momentum changes randomly.

28.4.2 Spin Diffusion in Quantum Wires How does the spin relaxation rate depend on the wire width W when the quantum wire has more than one channel occupied, W > λF? Clearly, for large wire widths, the spin relaxation rate should converge to a finite value, while it vanishes for W → λF. It is both of practical importance for spintronic applications and of fundamental interest to know on which length scales this crossover occurs. Basically, there are three intrinsic length scales characterizing the quantum wire relative to its width W. The Fermi wavelength λF, the elastic mean free path le, and the spin precession length LSO (Equation 28.18). Suppression of spin relaxation for wire widths not exceeding the elastic mean free path, le, has been predicted and obtained numerically in Refs. [11,16,35,40,47,55]. Is the spin relaxation rate also suppressed in diff usive wires in which the elastic mean free path is smaller than the wire width as in the wire shown schematically in Figure 28.7? We will answer this question by means of an analytical derivation in the following. The transversal confinement imposes that the spin current vanishes normal to the boundary, jSi n Boundary = 0 . For a wire grown along the [010]-direction, n = êx is the unit vector in the x-direction. For wire widths W smaller than the spin precession length L SO, the solutions with the lowest energy have thus a vanishing transverse spin current, and the spin diff usion equation (Equation 28.20) becomes ∂si = − D∂ y jS i y + τ ∇ VF (BSO (k ) × S )i − ∂t

∑ τˆ

1

j

sj

(28.41)

sij

28.4 Spin Dynamics in Quantum Wires 28.4.1 One-Dimensional Wires In one-dimensional wires, whose width W is of the order of the Fermi wavelength λF, impurities can only reverse the momentum p → −p. Therefore, the spin–orbit field can only change its sign, when a scattering from impurities occurs. BSO(p) → BSO(−p) = −BSO(p). Therefore, the precession axis and the amplitude of the spin–orbit field does not change, reversing only the

FIGURE 28.7 Elastic scatterings from impurities and from the boundary of the wire change the direction of the spin–orbit field around which the electron spin is precessing.

28-11


with x = ±W /2

(

= −τ υ x (BSO (k ) × S)i − D∂ x Si

)

x = ±W /2

= 0,

⎛ α12 1 ⎜ ∂t S = − −α1α 2 τsα 2 ⎜ ⎜ 0 ⎝

−α1α 2 α 22 0

0⎞ ⎟ 0 ⎟ S. α 2 ⎟⎠

(28.43)

Thus, we can conclude that the boundary conditions impose an effective alignment of all spin–orbit fields, in a direction identical to the one it would attain in a one-dimensional wire, along the [010]-direction, setting k x = 0 in Equation 28.27,

(28.45)

which therefore does not change its direction when the electrons are scattered. This is remarkable, since this alignment already occurs in wires with many channels, where the impurity scattering is two dimensional, and the transverse momentum, k x, actually can be finite. Rather, the alignment of the spin–orbit field, accompanied by a suppression of the DP spin relaxation rate occurs due to the constraint on the spin dynamics imposed by the boundary conditions as soon as the wire width, W, is smaller than the length scale which governs the spin dynamics, namely, the spin precession length, LSO. It turns out that the spin diff usion equation (Equation 28.41) has also two long persisting spin helix solutions in narrow wires36,65 that oscillate periodically with the period, LSO = π/m*α. In contrast to the situation in 2D systems we reviewed in Section 28.3, in quantum wires of width W < LSO these solutions are long persisting even for α1 ≠ α2. These two stationary solutions, ⎛ α1 ⎞ ⎜ α⎟ ⎜ α ⎟ ⎛ 2π S = S0 ⎜ − 2 ⎟ sin ⎜ ⎝ LSO ⎜ α⎟ ⎜ 0 ⎟ ⎜⎝ ⎟⎠

⎞ y⎟ ⎠

⎛ 0⎞ ⎛ 2π + S0 ⎜ 0⎟ cos ⎜ ⎜ ⎟ ⎝ LSO ⎜⎝ 1⎟⎠

π — 4

–1

1

(28.44)

⎛ α2 ⎞ BSO (k ) = −2k y ⎜ α1 ⎟ , ⎜ ⎟ ⎜⎝ 0 ⎟⎠

0

0

0

Indeed this has one persistent solution given by ⎛ α2 ⎞ S = S 0 ⎜ α1 ⎟ . ⎜ ⎟ ⎜⎝ 0 ⎟⎠

S z/S 0

(28.42) where W is the width of the wire. One sees that this equation has a persistent solution, which does not decay in time and is homogeneous along the wire, ∂yS = 0. In this special case, the spin diff usion equation simplifies to60

π — 2

1

x

LSO 2

0 LSO –1

FIGURE 28.8 (See color insert following page 20-16.) Persistent spin helix solution of the spin diff usion equation in a quantum wire whose width, W, is smaller than the spin precession length, LSO, for varying ratio of linear Rashba, α2 = α sin φ, and linear Dresselhaus coupling, α1 = α cos φ (Equation 28.46) for fi xed α and L SO = π/m*α.

and the linearly independent solution, are obtained by interchanging cos and sin in Equation 28.46. The spin precesses as the electrons diff use along the quantum wire with the period LSO, the spin precession length, forming a persistent spin helix, whose x-component is proportional to the linear Dresselhaus coupling, αx, while its y-component is proportional to the Rashba coupling, α2, as seen in Figure 28.8. A similar reduction of the spin relaxation rate is not effective for cubic spin–orbit coupling for wire widths exceeding the elastic mean free path, le. One can derive the spin relaxation rate as a function of the wire width for diff usive wires, le < W < LSO. The total spin relaxation rate, in the presence of both the linear Rashba spin–orbit coupling, α2, and the linear and cubic Dresselhaus coupling α1 and γD, respectively, is as function of wire width W given by36 2

1 1 ⎛W ⎞ 2 1 δ SO + D(m*2 ε F γ D) 2 , (W ) = ⎜ τs τs 12 ⎝ LSO ⎟⎠

(

(28.47)

)

where 1/τ s = 2 pF2 α 22 + (α1 − m * γ D εF /2)2 τ. We introduced the 2 2 dimensionless factor δ SO = (QR2 − QD2 )/QSO with QSO = QD2 + QR2 , where QD depends on Dresselhaus spin–orbit coupling, QD = m* (2α1 − m*εFγ). QR depends on Rashba coupling: QR = 2m*α2. Thus, for negligible cubic Dresselhaus spin–orbit coupling, the spin relaxation length increases when decreasing the wire width W as Ls (W ) = Dτ s (W ) ∼

⎞ y⎟ , ⎠

Sy / S0

jSi x

L2SO . W

(28.48)

(28.46) This can be understood as follows:24,36,59 In a wire whose width exceeds the spin precession length LSO, the area an electron covers by diff usion in time τs is WLs. To achieve spin relaxation,

28-12


this area should be equal to the corresponding 2D spin relaxation area Ls(2D)2, where L s(2D) = LSO/(2π). Thus, the smaller the wire width, the larger the spin relaxation length becomes, L s ∼ (LSO)2/W in agreement with Equation 28.48. For larger wire widths, the spin diff usion equation can be solved as well, and one finds that the spin relaxation rate does not increase monotonously to the 2D limiting value but shows oscillations on the scale LSO, which can be understood in analogy to Fabry–Pérot resonances.36 For pure linear Rashba coupling, that behavior can be derived analytically, in the approximation of a homogeneous spin density in transverse direction, yielding a relaxation rate given by 1 D 2 ⎛ sin(QSOW ) ⎞ (W ) = QSO ⎜⎝ 1 − Q W ⎟⎠ , τs 2 SO

(28.49)

where Q SO = 2π/LSO. Furthermore, taking into account the transverse modulation of the spin density by performing an exact diagonalization of the spin diff usion equation with the transverse boundary conditions (Equation 28.42), one fi nds for W > LSO modes which are localized at the boundaries and have a lower relaxation rate than the bulk modes.60,65 For pure Rashba spin relaxation, we find that there is a spin helix solution located at the edge whose relaxation rate 1/τs = 0.31/τs0 is smaller than the spin relaxation rate of bulk modes, 1/τs = 7/16τs0.

28.4.3 Weak Localization Corrections Quantum interference of electrons in low-dimensional, disordered conductors results in corrections to the electrical conductivity, Δσ. Th is quantum correction, the weak localization effect, is known to be a very sensitive tool to study dephasing and symmetry breaking mechanisms in conductors.2 The entanglement of spin and charge by spin–orbit interaction reverses the effect of weak localization and thereby enhances the conductivity, the weak antilocalization effect. The quantum correction to the conductivity, Δσ, arises from the fact that the quantum return probability to a given point x0 after a time t, P(t), differs from the classical return probability, due to quantum interference. As the electrons scatter from impurities, there is a finite probability that they diffuse on closed paths, which

(a)

(b)

does increase the lower the dimension of the conductor. Since an electron can move on such a closed orbit clockwise or anticlockwise, as shown in gray and black in Figure 28.9, with equal probability, the probability amplitudes of both paths add coherently, if their length is smaller than the dephasing length, Lφ. In a magnetic field as indicated by the big arrow in the middle of Figure 28.9, the electrons acquire a magnetic flux phase. This phase depends on the direction in which the electron moves on the closed path. Thus, the quantum interference is diminished in an external magnetic field since the area of closed paths and thereby the flux phases are randomly distributed in a disordered wire, even though the magnetic field can be constant. Similarly, the scattering from magnetic impurities breaks the time reversal invariance between the two directions in which the closed path can be transversed. Therefore, magnetic impurities diminish the quantum corrections in proportion to the rate at which the electron spins scatter from them due to the exchange interaction, 1/τMs (Equation 28.38). Thus, the quantum correction to the conductivity, Δσ, is proportional to the integral over all times smaller than the dephasing time, τφ , of the quantum mechanical return probability, P(t ) = λ dFρ(x , t ), where d is dimension of diff usion and ρ is the electron density. In the presence of spin–orbit scattering, the sign of the quantum correction changes to weak antilocalization as was predicted by Hikami et al.30 for conductors with impurities of heavy elements. As conduction electrons scatter from such impurities, the spin–orbit interaction randomizes their spin, Figure 28.10. The resulting spin relaxation suppresses interference of time-reversed paths in spin triplet configurations, while interference in singlet configuration remains unaffected, as indicated in Figure 28.10. Since singlet interference reduces the electron’s return probability it enhances the conductivity, the weak antilocalization effect. Weak magnetic fields suppress also these singlet contributions, reducing the conductivity and resulting in negative magnetoconductivity. If the host lattice of the electrons provides spin–orbit interaction, the spin relaxation of DP or EY type does have the same effect of diminishing the quantum corrections in the triplet configuration. When the dephasing length, L φ , is smaller than the wire width W, the quantum corrections are determined by the interference of 2-dimensional closed diff usion paths, and as a result, the

(c)

FIGURE 28.9 (a) Electrons can diff use on closed paths, orbit clockwise or anticlockwise as indicated by the gray and black arrows, respectively. (b) Closed electron paths enclose a magnetic flux from an external magnetic field, indicated as the big arrow, breaking time reversal symmetry. (c) The scattering from a magnetic impurity spin breaks the time reversal symmetry between the clock- and anticlockwise electron paths.

28-13


the change in sign in the weak localization correction. The other three terms are suppressed by the spin relaxation rate, since they originate from interference in triplet states S = 1; m = 0 = ↑↓ + ↓↑ 2, S = 1; m = 1 , S = 1; m = −1 , which do not conserve the spin symmetry. Thus, at strong spin–orbit-induced spin relaxation, the last three terms are suppressed and the sign of the quantum correction switches to weak antilocalization. In quasi-1-dimensional quantum wires that are coherent in transverse direction, W < L φ , the weak localization correction is further enhanced, and increases linearly with the dephasing length, L φ . Thus, for WQ SO W, the spin relaxation rate is suppressed in analogy to the flux-cancellation effect, which yields the weaker rate, 1/τ s = (W /Cle )(DW 2 /12L4SO) where C = 10.8.5 The dimensional crossover from weak antilocalization to weak localization, seen in Figure 28.11, as the wire width W is reduced, has recently been observed experimentally in quantum wires as we will review in Chapter 29.

28.5 Experimental Results on Spin Relaxation Rate in Semiconductor Quantum Wires 28.5.1 Optical Measurements Optical time-resolved Faraday rotation (TRFR) spectroscopy61 has been used to probe the spin dynamics in an array of n-doped InGaAs wires by Holleitner et al. in Refs. [31,32].

28-14


0 5

ΔG 2e2 /h

–0.5 –1

4

–1.5

B/H

2

0

SO

1

WQ

SO

3 –1

1

FIGURE 28.11 The quantum conductivity correction in units of 2e2/h as function of magnetic field B (scaled with bulk relaxation field HS) and the wire width W (scaled with LSO/2π) for pure Rashba coupling, δSO = 1.

The wires were dry etched from a quantum well grown in the [001]-direction with a distance of 1 μm between the wires. Spinaligned charge carriers were created by the absorption of circularly polarized light. For normal incidence, the spins point then perpendicular to the quantum well plane, in the growth direction [001]. The time evolution of the spin polarization was then measured with a linearly polarized pulse (see inset of Fig. 1c of Ref. [31]). The time dependence fits well with an exponential decay, ∼ exp(−Δt/τs). As seen in Fig. 2a of Ref. [31], the thus measured lifetime, τs, at fi xed temperature, T = 5 K, of the spin polarization is enhanced when the wire width, W, is reduced31: While for W > 15 μm it is τs = (12 ± 1) ps, it increases for channels grown along the [100]-direction to almost τs = 30 ps, and in the [110]-direction to about τs = 20 ps. Thus, the experimental results show that the spin relaxation depends on the patterning direction of the wires: wires aligned along [100] and [010] show equivalent spin relaxation times, which are generally longer than the spin relaxation times of wires patterned along [110] and [1−10]. The dimensional reduction could be seen already for wire widths as wide as 10 μm, which is much wider than both the Fermi wavelength and the elastic mean free path, le, in the wires. This agrees well with the predicted reduction of the DP scattering rate (Equation 28.47) for wire widths smaller than the spin precession length, LSO. From the measured 2D spin diff usion length, L s(2D) = (0.9 − 1.1) μm, and its relation to the spin precession length (Equation 28.18), LSO = 2πL s(2D), we expect the crossover to occur on a scale of LSO = (5.7 – 6.9) μm as observed in Fig. 2a of Ref. [31]. From LSO = π/m*α, we get, with m* = .064me, a spin–orbit coupling α = (5 – 6) meVÅ. According to Ls = Dτ s , the spin relaxation length increases by a factor of 30 /12 = 1.6 in the [100]-direction, and by 20 /12 = 1.3 in the [110]-direction. The spin relaxation time has been found to attain a maximum, however, at about W = 1 μm ≈ L s(2D), decaying

appreciably for smaller widths. While a saturation of τs could be expected according to Equation 28.47 for diffusive wires, due to cubic Dresselhaus coupling, a decrease is unexpected. Schwab et al., Ref. [60], noted that with wire boundary conditions that do not conserve the spin of the conduction electrons, one can obtain such a reduction. A mechanism for such spin-flip processes at the edges of the wire has not yet been identified, however. The magnetic field dependence of the spin relaxation rate yields further confi rmation that the dominant spin relaxation mechanism in these wires is DPS: It follows the predicted behavior Equation 28.40, as seen in Fig. 3a of Ref. [31], and the spin relaxation rate is enhanced to τs (B = 1 T) = 100 ps for all wire growth directions, at T = 5 K and wire widths of W = 1.25 μm.

28.5.2 Transport Measurements A dimensional crossover from weak antilocalization to weak localization and a reduction of spin relaxation has recently been observed experimentally in n-doped InGaAs quantum wires,48,67 in GaAs wires,15 as well as in AlGaN/GaN wires.44 The crossover indeed occurred in all experiments on the length scale of the spin precession length, LSO. We summarize in the following the main results of these experiments. Wirthmann et al. [67] measured the magnetoconductivity of inversion-doped InAs quantum wells with a density of n = 9.7 × 1011/cm 2, and a measured effective mass of m* = 0.04me. In the wide wires, the magnetoconductivity showed a pronounced weak antilocalization peak, which agreed well with the 2D theory, 33,41 with a spin–orbit-coupling parameter of α = 9.3 meVÅ. They observed a diminishment of the antilocalization peak that occurred for wire widths W < 0.6 μm, at T = 2 K, indicating a dimensional reduction of the DP spin relaxation rate. Schäpers et al. observed in Ga x In1−x As/InP quantum wires a complete crossover from weak antilocalization to weak localization for wire widths below W = 500 nm. Such a crossover has also been observed in GaAs quantum wires by Dinter et al.15 Very recently, Kunihashi et al.42 observed the crossover from weak antilocalization to weak localization in gate-controlled InGaAs quantum wires. The asymmetric potential normal to the quantum well could be enhanced by application of a negative gate voltage, yielding an increase of the SIA-coupling parameter, α, with decreasing carrier density, as was obtained by fitting the magnetoconductivity of the quantum wells to the theory of 2D weak localization corrections of Iordanskii et al.33 Thereby, the spin relaxation length, L s = LSO/2π, was found to decrease from 0.5 to 0.15 μm, which according to LSO = π/m*α corresponds to an increase of α from (20 ± 1) meVÅ at electron concentrations of n = 1.4 × 1012/cm 2 to α = (60 ± 1) meVÅ at electron concentrations of n = 0.3 × 1012/cm 2. The magnetoconductivity of a sample with 95 quantum wires in parallel showed

28-15


a clear crossover from weak antilocalization to localization. Fitting the data to Equation 28.51, a corresponding decrease of the spin relaxation rate was obtained, which was observable already at large widths of the order of the spin precession length, LSO, in agreement with the theory Equation 28.47. However, a saturation as obtained theoretically in diff usive wires, due to cubic BIA coupling was not observed. Th is might be due to the limitation of Equation 28.47, to diff usive wire widths, le < W, while in ballistic wires a suppression also of the spin relaxation due to cubic BIA coupling can be expected, since it vanishes identically in 1D wires (see Section 28.4.1). Also, an increase of the spin scattering rate in narrower wires, W < L s(2D), was not observed in contrast to the results of the optical experiments (Ref. [31]) reviewed above. The dimensional crossover has also been observed in the heterostructures of the wide-gap semiconductor GaN.44 The magnetoconductivity of 160 AlGaN/GaN-quantum wires were measured. The effective mass is m* = .22me, all wires were diffusive with le < W. For electron densities of n ≈ 5 × 1012/cm2, an increase from L s(2D) ≈ 550 nm to Ls(W ≈ 130 nm) > 1.8 μm, and for densities n ≈ 2 × 1012/cm2 an increase from Ls(2D) ≈ 500 nm to Ls (W ≈ 120 nm) > 1 μm was observed. Using Ls(2D) = 1/2m*α, one obtains for both densities n, the spin–orbit coupling α ≈ 5.8 meVÅ. A saturation of the spin relaxation rate could not be observed, suggesting that the cubic BIA coupling is negligible in these structures. We note, that an enhancement of the spin relaxation rate as in the optical experiments of narrow InGaAs quantum wires (Ref. [31]) was not observed in these AlGaN/GaN wires.

28.6 Critical Discussion and Future Perspective The fact that optical and transport measurements seem to fi nd opposite behavior, enhancement and suppression of the spin relaxation rate, respectively, in narrow wires, calls for an extension of the theory to describe the crossover to ballistic quantum wires. Th is can be done using the kinetic equation approach to the spin diff usion equation, 60 a semiclassical approach,69,70 or an extension of the diagrammatic approach. 65 In particular, the dimensional crossover of DPS due to cubic Dresselhaus coupling, which we found not to be suppressed in diff usive wires, needs to be studied for ballistic wires, l e > W, as many of the experimentally studied quantum wires are in this regime. Furthermore, using the spin diff usion equation, one can study the dependence on the growth direction of quantum wires, and fi nd more information on the magnitude of the various spin–orbit coupling parameters, α1, α2 , γD, by comparison with the directional dependence found in both the optical measurements31 of the spin relaxation rate, as well as in recent gate-controlled transport experiments.42 In narrow wires, corrections due to electron–electron interaction can become more important and influence especially the temperature dependence. Ref. [32] reports a strong temperature dependence of the spin relaxation rate in narrow quantum

wires. As shown in Ref. [57], the spin relaxation rates obtained from the spin diff usion equation and the quantum corrections to the magnetoconductivity can be different, when corrections due to electron–electron interaction become important. As the DPS becomes suppressed in quantum wires, other spin relaxation mechanisms like the EYS may become dominant, since it is expected that the dimensional dependence of EYS is less strong. In more narrow wires, disorder can also result in Anderson localization. Similar as in quantum dots, 38,39 this can yield enhanced spin relaxation due to hyperfi ne coupling (Equation 28.39). The spin relaxation in metal wires is believed to be dominated by the EYS mechanism, which is not expected to show such strong wire width dependence, although this needs to be explored in more detail. Even dilute concentrations of magnetic impurities of less than 1 ppm do yield measurable spin relaxation rates in metals and allow the study of the Kondo effect with unprecedented accuracy. 50,71

28.7 Summary The spin dynamics and spin relaxation of itinerant electrons in disordered quantum wires with spin–orbit coupling is governed by the spin diff usion equation (Equation 28.20). We have shown that it can be derived by using classical random walk arguments, in agreement with more elaborate derivations.60,65 In semiconductor quantum wires, all available experiments show that the motional narrowing mechanism of spin relaxation, the DPS, is the dominant mechanism in quantum wires whose width exceeds the spin precession length, LSO. The solution of the spin diff usion equation reveals the existence of persistent spin helix modes when the linear BIA- and the SIA-spin–orbit coupling are of equal magnitude. In quantum wires that are more narrow than the spin precession length LSO, there is an effective alignment of the spin–orbit fields giving rise to long-living spin density modes for arbitrary ratio of the linear BIA- and the SIA-spin–orbit coupling. The resulting reduction in the spin relaxation rate results in a change in the sign of the quantum corrections to the conductivity. Recent experimental results confi rm the increase of the spin relaxation rate in wires whose width is smaller than LSO, both the direct optical measurement of the spin relaxation rate as well as transport measurements. These show a dimensional crossover from weak antilocalization to weak localization as the wire width is reduced. Open problems remain, in particular in narrower, ballistic wires, where optical and transport measurements seem to find opposite behavior of the spin relaxation rate: enhancement, suppression, respectively. The experimentally observed reduction of spin relaxation in quantum wires opens new perspectives for spintronic applications, since the spin–orbit coupling and therefore the spin precession length remains unaffected, allowing a better control of the itinerant electron spin. The observed directional dependence moreover can yield more detailed information about the spin–orbit coupling, enhancing the spin control for future spintronic devices further.

28-16

Symbols τ0 τee τep τ τˆs D le LSO QSO Ls α1 α2 γD γg

elastic scattering time scattering time due to electron–electron interaction scattering time due to electron–phonon interaction total scattering time 1/τ = 1/τ0 + 1/τee + 1/τep spin relaxation tensor diff usion constant, D = vF2 τ /d D , where dD is the dimension of diff usion elastic mean free path spin precession length in 2D. The spin will be oriented again in the initial direction after it moved ballistically LSO = 2π/LSO spin relaxation length Ls(W) = Dτ s (W ) with Ls(W) |w →∞ = L s (2D) = LSO/2π linear (bulk inversion asymmetry (BIA) = Dresselhaus)parameter linear (structural inversion asymmetry (SIA) = BychkovRashba)-parameter cubic (bulk inversion asymmetry (BIA) = Dresselhaus)parameter gyromagnetic ratio

Acknowledgments We thank V. L. Fal’ko, F. E. Meijer, E. Mucciolo, I. Aleiner, C. Marcus, T. Ohtsuki, K. Slevin, J. Ohe, and A. Wirthmann for helpful discussions. This work was supported by SFB508 B9.

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29 Quantum Magnetic Oscillations in Nanowires 29.1 Introduction ...........................................................................................................................29-1 29.2 Quantum Magnetic Oscillations in Bulk Metals..............................................................29-2 Fock–Landau Quantization • Quantum Magnetic Oscillations

29.3 Combination Frequencies in the Quantum Magnetic Oscillations of Multiband Quasi-2D Materials .......................................................................................29-7

A. Sasha Alexandrov Loughborough University

Victor V. Kabanov Josef Stefan Institute

Iorwerth O. Thomas Loughborough University

Introduction • dHvA Oscillations • SdH Oscillations

29.4 Quantum Magnetic Oscillations in Nanowires..............................................................29-13 Introduction • Statement of the Problem • Weak Field Limit • Beyond the Weak Field Limit

29.5 Conclusion ............................................................................................................................29-19 Acknowledgment.............................................................................................................................29-20 References.........................................................................................................................................29-20

29.1 Introduction Periodic oscillations in the magnetization and magnetoresistance of metals in a magnetic field as the strength of the field is varied—known as the De Haas–van Alphen (dHvA) and the Shubnikov–De Haas (SdH) effects, respectively, and arising from the Fock–Landau quantization of the electron spectrum—have, since their discovery in the 1930s, become one of the standard techniques in the analysis of the structures of the Fermi surfaces (Shoenberg 1984a). A summary of the fi rst 38 years of the field, focusing on the dHvA effect, may be found in Gold (1968), while more recent reviews relevant to our topic are Singleton (2000) and Kartsovnik (2004), which focus on magneto-oscillations in the two-dimensional (2D) and the quasi-two-dimensional (q2D) metals, such as the organic charge-transfer salts. Both the dimensionality and the physical confinement of electrons may affect the quantum magneto-oscillations of a system, and it is therefore important to understand how these effects may manifest so that the structure of the Fermi surface may be properly understood. The attempt to define the effects of physical confinement (which is of paramount importance in the description of the quantum magnetic oscillations of nanowires) has a long history in this field, since a correct treatment of the boundary conditions is important even in bulk metals as the density of states of the system is finite (e.g., see the following classic papers: Fock 1928, Landau 1930, Darwin 1930, Dingle 1952b,c). Some attention, however, has also been

paid to the case of electrons confined within a cylindrical wire exposed to a longitudinal magnetic field. Much of the theoretical work has focused on the weak field case—as in Dingle (1952c), Bogacheck and Gogadze (1973), Aronov and Sharvin (1987), and Gogadze (1984)—where there is only a small perturbation to the energy spectrum. In these cases, oscillations arise as a result of size quantization or the Aharonov–Bohm effect. In order to treat a system confined within a narrow wire at stronger fields, Alexandrov and Kabanov (2005a) have modeled the confinement using the Fock–Darwin model (Fock 1928, Darwin 1930), and in Alexandrov et al. (2007) this was extended to a stronger variety of confinements. In both these cases, a complex pattern of oscillations due to the interaction of size and Fock–Landau quantization was predicted. In this chapter, we first introduce, in Section 29.2, the theory describing quantum magnetic oscillations in three-dimensional (3D) bulk metals. This is largely well understood, and functions as a foundation for the following discussion. Next, in Section 29.3, we describe some interesting properties of q2D metals. While this might seem irrelevant to the topic at hand, they serve as an important reminder that certain properties of 3D metals do not hold in systems of reduced dimensionality and are illustrative of some important new phenomena that might arise in such cases (the difference between the quantum magneto-oscillations of closed and open multiband systems in 2D and q2D metals does not exist in 3D bulk metals, for example). Finally, in Section 29.4, we review in detail the theory of quantum magnetic oscillations in nanowires. 29-1

29-2


29.2 Quantum Magnetic Oscillations in Bulk Metals 29.2.1 Fock–Landau Quantization It has been known since the early twentieth century (Fock 1928, Landau 1930, Darwin 1930) that the spiral motion of electrons in a magnetic field is quantized, and that this has profound consequences on the behavior of metallic compounds that are exposed to such fields. We begin with an overview of this Fock–Landau quantization, as discussed by Landau (Landau 1930). Accessible treatments of this topic may be found in textbooks such as Landau and Lifschitz (1977), Abrikosov (1988), and Hamguchi (2001), though in this section we mainly follow the account of Abrikosov (1988). The presence of a magnetic field modifies the momentum operator pˆ to pˆ−eA, where A is the vector potential of the field. We may therefore write the Schrödinger equation for a free electron in a magnetic field parallel to the z-axis in the Landau gauge (Ax = Az = 0, Ay = Bx):

We now turn to the density of states. This requires some care, as we must take into account the boundary conditions of our system in order to avoid an infi nite degeneracy of states for each Fock–Landau level n. Let us assume that we have a box whose sides are of different lengths Lx, Ly, Lz . We also assume that these lengths are very large, so that any effects due to the size quantization are negligible (see the discussion in Landau and Lifschitz (1977)); situations where this simplification cannot be applied, such as nanowires, are treated later on. Now, the position of the center of the oscillator is dependent on the value of py. In order to be contained within the box, it must therefore be within the interval 0 < p y < eBLx .

(29.5)

The number of states in the infinitesimal intervals dpz and dpy are dny =

dp y L y , 2π

dnz =

dpz Lz . 2π

(29.6)

2

−

⎞ 2 ∂2 ∂ 2 ∂2 1 ⎛ ψ+ −i − eBx ⎟ ψ − ψ = E ψ , (29.1) 2 2 ⎜ ∂y ⎠ 2m* ∂x 2m* ⎝ 2m* ∂z

eB

(29.8)

where V = LxLyLz is the sample volume. The density of states of the system is hence given by g (E ) =

2 ∂2 1 ψ − Kx 2 ψ = ⑀ψ , 2 2m* ∂x 2

(29.3)

where K is a quantity related to the oscillation frequency ω, so that ω = (K/m*)1/2. This equation has the energy spectrum ϵn = ћω(n + 1/2), where n is an integer greater than or equal to 0. If we shift the center of the oscillation in Equation 29.2 by py/eB and set ω = eB/m* ≡ ωc (we call ωc the cyclotron frequency) and ⑀ = E − pz2 2m* , we find that they are identical. Our spectrum is then ⎛ pz2 + ω c ⎜ n + 2m* ⎝

1⎞ . 2 ⎟⎠

(29.4)

∑ δ (E − E

pz , n

)

n, pz , p y

This is similar to the equation of motion of a one-dimensional (1D) harmonic oscillator:

E pz ,n =

Vdpz , (2π )2

z

∂2 1 pz2 2 ψ ( x ) + ( p − eBx ) ψ ( x ) + ψ (x ) = E ψ (x ). y 2m* ∂x 2 2m* 2m* (29.2)

−

(29.7)

which when multiplied by dnz gives us the number of states between pz and pz + dpz:

We solve this through the separation of variables: our soluip y ip z tion has the form ψ (x , y , z ) = e e ψ (x ) (py and pz are the respective y and z components of p). Substituting this into the above wave equation we acquire −

Lx L y eB , 2π

ny =

where ћ = h/2π (h is Planck’s constant) m* is the mass of the electron B is the magnetic field e is the charge of the electron E is the energy

y

Overall, then, the total number of states in the interval py will be

= eB

V (2π)2

∑ ∫ dp δ(E − E z

pz , n

),

(29.9)

n

where δ(E − E pz , n ) is a delta function. Changing the variables so that we are integrating over E pz,n gives us g (E ) = eB

V 2m* (2π )2

∑ ∫ dE

δ ( E − E pz , n ) pz , n

n

E pz ,n − ω c (n + 1/2)

(29.10)

and performing this integration gives us the density of states g (E) = eB

V 2m* (2π )2

∑ n

1 . E − ω c (n + 1/2)

(29.11)

29-3

Quantum Magnetic Oscillations in Nanowires

29.2.2 Quantum Magnetic Oscillations This quantization has many intriguing consequences. The focus of this chapter is the oscillations it induces in the magnetization (de Haas–van Alphen or dHvA oscillations) and the conductivity (Shubnikov–de Haas or SdH oscillations) (Shoenberg 1984a), and how these manifest in low-dimensional systems. These oscillations, periodic in 1/B, were first predicted by Landau (1930) (who did not think that they would be observable), and are now commonly used as a technique for examining the properties of the Fermi surface of metallic compounds. Before we embark on an overview of the typical methods used to derive quantitative expressions for these effects, we shall fi rst examine how they arise. At zero temperature, a 3D electron gas will fi ll all the energy levels up to the Fermi energy, which describes a spherical surface (the Fermi surface) in momentum space. Upon application of the magnetic field, the orbits of the electrons in the plane perpendicular to the field become quantized, and we have an allowed distribution of electrons in momentum space that resembles a set of concentric cylinders, whose axes are ordered in parallel with the direction of the field. The separation of these cylinders is governed by the strength of the magnetic field, and the highest occupied energy level is that closest to the Fermi energy. As the magnetic field is varied, the separation of the cylinders and the associated energy levels change. Eventually, the topmost occupied energy level will pass above the Fermi energy—and therefore all the electrons occupying it will fall to the next lowest energy level, resulting in a discontinuous change in the properties of the system that rely on the electron occupation. Th is repeats as the successive highest occupied energy levels pass above the Fermi energy, leading to oscillations in those quantities (Gold 1968, Shoenberg 1984a, Abrikosov 1988, Hamguchi 2001).

Here, we outline the general procedure for calculating the dHvA oscillations of a free electron gas in a magnetic field. The same steps can be used in order to calculate the oscillations for a variety of systems, provided that one has an appropriate model dispersion relation for the system. (Here we follow the treatment in Abrikosov (1988), based on that of Lifshitz and Kosevich (1968); similar treatments may be found in Gold (1968) and Shoenberg (1984a).) We begin with the formula for the energy of a free electron gas in a magnetic field B oriented along the z-axis, pz2 1⎞ ⎛ + ω c ⎜ n + ⎟ , ⎝ 2⎠ 2m*

(29.12)

and the formula for the thermodynamic potential Ω = −kBT

∑ ln(1 + e β

(μ − ⑀ β )/T

),

Summing over all identical states gives us Ω=−

kBTVeB (2π )2

∑ ∫ dp ln(1 + e z

( μ − E (n, pz )) / kBT

).

(29.14)

n, σ

For reasons of clarity, we neglect spin splitting, and so the summation over the spin index σ merely results in the right-hand side of the above equation being multiplied by 2. In order to proceed, we shall need to make use of Poisson’s summation formula ∞

∑

∞

∞

n0

∞

∑ ∫ dn f (n)e

∫

f (n) = dn f (n) +2ℜ

i 2π r n

,

(29.15)

r =1 a

a

where f(n) is an arbitrary, real function, n0 − 1 < a < n0 r is the summation index ℜ indicates that we take the real portion of the subsequent quantity We are interested in the oscillatory portion of Ω, which after the application of Poisson’s formula is given by ∞

∑I ,

= −4ℜ Ω

r

(29.16)

r =1

with Ir being given by

29.2.2.1 De Haas–Van Alphen Oscillations

E pz ,n =

where the sum over β indicates summing over all states T is the temperature kB is Boltzmann’s constant μ is the chemical potential ϵβ is the energy of the rth state

(29.13)

Ir =

∞

∞

a

−∞

∫ ∫

VeBkBT dn dpz ln(1 + e (μ − E (n, pz ))/ kBT )e 2 π ir n. (2π )2

(29.17)

We change the integration so that it is over E: ∞

pzmax

⎛ ∂n( pz , E) ⎞ ( p ,E) ln(1 + e( μ − E )/T )e i 2 π r n z . ∂E ⎟⎠

∫ ∫ dp ⎜⎝

VeBkBT Ir = dE (2π )2 0

z

pzmin

(29.18) The lower limit of the integral is set to 0 for convenience; since the dominant contribution to the phenomenon of interest is from states near the Fermi energy, its precise value is of little import. The values of pzmax and pzmin are the maximum and minimum possible values of pz for a given E. We now observe that e i 2 πrn( pz , E ) is a rapidly oscillating function. If n(pz, E) obtains an extremal value nα at pzα, we can expand it as

29-4


1 ⎛ ∂2n ⎞ n( pz , E) = nα (E) + ⎜ 2 ⎟ ( pz − pzm )2 . 2 ⎝ ∂pz ⎠ p α

(29.19)

z

(There is no first-order term in the expansion since ∂n/∂pz = 0 at an extremum.) We can use this to integrate over pz near the extrema, and the integrals over pz can be well approximated by ⎛ ∂n ⎞ i 2 πrnm ⎜⎝ ∂E ⎟⎠ e m

∞

⎛ ⎞ ⎛ ∂2n ⎞ exp ⎜ − iπr ⎜ 2 ⎟ z 2 ⎟ dz , ⎝ ∂pz ⎠ m ⎠ ⎝ −∞

∫

(29.20)

where x = E − μ. We may integrate over this using the standard ∞ integral dy exp(iαy )/(exp y + 1) = −iπ /sinh(απ) , where α sig0 nifies a quantity of the correct form, and so acquire

∫

Ir = −

VeBkBT ω cm * ei 2 πrn(μ ) 2(2π )2 r 3/2 sinh(2π 2rkBT/ω c )

Making use of the Onsager relation between n(μ) and the crosssectional area of the Fermi surface SF (Onsager 1952) nm ≈

which is a standard integral, ∞

⎛ ⎞ ⎛ ∂2n ⎞ ⎛ ∂2n ⎞ exp ⎜ − iπr ⎜ 2 ⎟ z 2 ⎟ dz = e ± iπ /4 ⎜ r 2 ⎟ ⎝ ∂pz ⎠ m ⎠ ⎝ ⎝ ∂pz m ⎠ −∞

∫

−1/2

.

(29.21)

If there is more than one extremal surface, we must sum over them. For simplicity, however, we shall assume only the existence of one such surface and that it has a simple parabolic band structure given by Equation 29.12. (For more complicated structures, one may use the more general approach given in Abrikosov (1988).) Hence Ir following our integration is ∞

ln(1 + e(μ − E )/ kBT ) i2 πn( E ) ± iπ /4 ⎛ ω cm* ⎞ Vk TeB I r = B 2 dE e ⎜ ⎟ (2π ) ω c ⎝ r ⎠

∫

,

(29.22) since (∂2n / ∂pz2 ) = 1 / ω cm* and (∂n/∂E) = 1/ћωc. We now integrate this by parts and neglect the non-oscillatory term, obtaining

∫

(29.27)

, we and inserting our results into our overall expression for Ω obtain the following expression: 5/2 3/2

= V ω c (m* ) Ω 4π 41/2

∞

∑ r =1

RT (r λ) π⎞ ⎛ S cos ⎜ r F ± ⎟ , (29.28) ⎝ eB 4 ⎠ r 5/2

where RT(x) = x/sinh(x) is the Lifshitz–Kosevich (LK) temperature damping factor λ = 2π 2kBT/ω c We may further derive expressions for the oscillatory magneti through the use of zation M˜ and the magnetic susceptibility χ

the relations M = −∂Ω / ∂B and χ = ∂M / ∂B , where we keep only the most rapidly oscillating parts. It is also useful to consider the relationship when given in terms of the Fermi energy. In this case, n(μ)=(μ/ћωc) − 1/2, and we find

= V ω c (m* ) Ω 4π 41/2 5/2

i 2 πrn( E )

VeB ω cm ±iπ /4 e Ir = e dE ( E −μ )/ kBT . e +1 (2π )2 cr 3/2 2πi

Sm , 2πeB

1/2

0

∞

(29.26)

(29.23)

3/2

∞

∑ r =1

⎛ 2πμ π ⎞ (−1)r RT (r λ) ± cos ⎜ r . (29.29) ⎝ ω c 4 ⎟⎠ r 5/2

0

The dHvA effect is a phenomenon of the Fermi surface, so the dominant contribution should be from values of E close to μ. We need to account for the rapid oscillation of the exponential, however, so we expand n(E) for E ≈ μ: ⎛ ∂n ⎞ n(E) = n(μ) + ⎜ ⎟ (E − μ). ⎝ ∂E ⎠ μ

(29.24)

We then find that the rapidly varying part of the integral becomes

∞

∫ dE e 0

e

∞

i 2 πrn

( E −μ )/ kBT

+1

∫

= ei 2πrnm (μ) dx −∞

⎛ i2πrx ⎞ exp ⎜ ⎝ ω c ⎟⎠ e x / kBT + 1

,

(29.25)

Thus far, we have ignored the effects of the splitting of energy levels by the spin quantum number. This is easy enough to repair. The energy spectrum incorporating the spin splitting is Eσ = E(n, pz ) + βBσ,

(29.30)

where σ = ±1 β = eћ/2me c is the Bohr magneton (me is the mass of the free electron, c the speed of light in a vacuum) This functions as a straightforward shift of the chemical potential in the exponential term to μσ = μ + βBσ. Since βBσ > Δ(E), the Green’s functions are strongly peaked at A = 0. This implies that replacing (E − A)3/2 with E3/2 and extending the lower limit of integration to −∞ is a good approximation (Altland and Simons 2006). We then close the contour of integration in the upper complex plane of A and through use of the residue theorem compute ∞

− E 3/2

∫

dA

Δ(E)2 πE 3/2 =− . 2 2 2 Δ(E) [ A + Δ (E) ] 2

(29.44)

Gathering these results together, we have

(29.39)

∑ f ⎛⎜⎝ n + 2 ⎞⎟⎠ = ∑ (−1) ∫ dnf (n)e n

2 m* ⑀

Now we perform the integration over ϵ. We have

−∞

In order to proceed, we use the following generalization of the Poisson summation formula (Dingle 1952a, Hamguchi 2001):

1

1 2m*ω c

2

v z (⑀, n) ⎣⎡ ℑG R (⑀, E)⎦⎤ .

n

∞

For the r = 0 term, we find

∞

∑ ∫ dn(−1)

r

f (n) ei 2 πrn .

r =1 0

(29.41) Since the Green’s function in our approximation is no longer dependent on n, we may integrate the velocity alone, and in the 3D case we need only keep the contribution from the r = 0 term, since it is dominant over all the others. In general (such as in q2D systems), this will not be so. In addition, in the analytic structure of the Green’s functions at r > 0 may give rise to an additional contribution from the second-order pole, which also cannot be ignored in such cases (Grigoriev 2002, 2003).

σ zz =

e2 ⎡ ∂f (E ) ⎤ (2m* E )3/2 dE ⎢ − . ⎥ 2 6π m* ⎣ ∂E ⎦ Δ ( E ) 2

∫

(29.45)

In this case, the dominant effect giving rise to the SdH oscillations is due to the effects of scattering on Δ(E). Our result is equivalent to that derived from the semiclassical generalization of the Boltzmann equation that is mentioned earlier. As noted before, this is not always true in materials of lower dimensionality or if the scattering effects are otherwise suppressed, such as in the presence of a large external reservoir of states. How we next proceed depends on how we handle the behavior of the self-energy. As noted earlier, in 3D electron gases, the dominant form of scattering is normally due to impurities. We choose the self-consistent Born approximation (see Bruus and Flensberg (2004), for example) as described previously for what follows. In this approximation and in three dimensions, as before, we may ignore the real portion of the self-energy in our calculations. The imaginary portion of the self-energy is proportional to the density of states multiplied by the square of the scattering amplitude W (whose value depends on the concentration of impurities and the strength of their interaction with the electrons, and into which we enfold the factor of 2 due to summation over spins) and may thus be written

29-7


Δ (E ) =

W π

∑

ℑG R (⑀β , E),

(29.46)

β

where β corresponds to the set of free electron quantum numbers defined earlier. Using Poisson’s formula as above, we may calculate this to be 1/2 ⎡ ⎛ ω ⎞ Δ (E) = Wg (E ) ⎢1 − ⎜ c ⎟ ⎢ ⎝ 2E ⎠ ⎣

∞

∑ r =1

⎤ ⎛ 2πrE π ⎞ ⎥ (−1)r R ( r , E )cos − , D ⎜⎝ ω r 1/2 4 ⎟⎠ ⎥ c ⎦ (29.47)

where g (E) = E (2m*/ 2 )3/2/2π 2 is the 3D density of states for zero magnetic field (Kubo et al. 1965, Hamguchi 2001) R D(r, E) = exp(−2π|r|Δ(E)/ћωc) is similar to the Dingle damping factor Since the average value of Δ(E) is directly proportional to the average Dingle temperature TDπ, we may then write 1/2 ⎡ ⎛ ω ⎞ Δ (E) = π kB TD ⎢1 − ⎜ c ⎟ ⎢ ⎝ 2E ⎠ ⎣

In Section 29.3.3 we turn to specific applications of these methods to low-dimensional systems.

⎤ ⎛ 2πrE π ⎞ ⎥ (−1)r RD (r , E )cos ⎜ − ⎟ . ⎝ ω c 4 ⎠ ⎥ r =1 ⎦ (29.48)

29.3 Combination Frequencies in the Quantum Magnetic Oscillations of Multiband Quasi-2D Materials 29.3.1 Introduction Quasi-two-dimensional (q2D) materials are those in which the ease of electron transport in the x–y plane is greater than in the z direction. Examples of compounds where this is the case include the organic charge-transfer salts* [extensively reviewed in Singleton (2000) and Kartsovnik (2004)] and cuprate superconductors—however, the phenomena typical to q2D systems are more easily observed in the former case than in the latter. Figure 29.1 shows the structure of the Fermi surface: a cylinder with periodic bulges. A typical approach to such systems is to use a tight-binding model spectrum, which in the presence of a magnetic field is 1⎞ ⎛ ⎛ p a⎞ E = ω c ⎜ n + ⎟ − 2t cos ⎜ z ⎟ , ⎝ ⎝ ⎠ 2⎠

∞

∑

We may then substitute this into the expression for the conductivity, obtaining (when μ/ћωc >> 1) 1/2 ⎡ ⎡ ∂f ( E ) ⎤ ⎢1 + ⎛ ω c ⎞ σ zz = dE ⎢ − σ ( E ) ⎥ ⎢ ⎜⎝ 2E ⎟⎠ ⎣ ∂E ⎦ ⎣

∫

∞

(−1)r

∑r r =1

1/2

(29.51)

where ωc = eB cos(Θ)/m is the cyclotron frequency t = t0 J0 (kFa tan (Θ)) is the interlayer transfer integral J0(x) is the zeroth-order Bessel function a is the interlayer separation Θ is the angle between the direction of the field and the normal to the x–y planes

⎛ 2πrE π ⎞ ⎤⎥ RDr cos ⎜ − , ⎝ ω c 4 ⎟⎠ ⎥ ⎦

(29.49) where σ(E) = N(E)ћe2/m*πkBTD = N(E)τe2/m* is the zero-field conductivity N(E) = 2(2m*E)3/2/(6π2ћ3) and we have made the approximation RD(r, E) ≈ R Dr, R Dr being given by Equation 29.32 Integrating over dE, we use the delta-function-like behavior of −∂f(E)/∂E near the Fermi energy for kBT B, F = fn + f b = 2mμ/eћ cos(Θ), and fn,b = m(μ ∓ 2t)/eћ cos(Θ) are the frequencies corresponding to the neck and the belly of the Fermi surface, respectively (see Figure 29.1). In principle, therefore, one should be able to observe in the Fourier Transforms of systems of this type a small amplitude between the two peaks corresponding to the second harmonics of the neck and belly frequencies. However, this amplitude is very small, and depending on the size of the magnetic field window used experimentally, it may prove difficult to detect.

29.3.3 SdH Oscillations 29.3.3.1 Phenomenological Differences from dHvA Oscillations Shubnikov–de Haas oscillations in q2D systems exhibit a number of features that are quite unusual. These include both the presence of an unusual phase shift in the beating of the oscillations due to the warping of the Fermi surface and the appearance of additional slow oscillations, both of which are absent in the dHvA oscillations. Phase shifts: From the asymptotic form of the Bessel function in the intermediate case, J 0 (x ) ≈ 2 / πx cos(x − π /4), it follows that one would typically expect that in that limit, the frequencies corresponding to the neck and belly oscillations would have a phase shift of ±π/4, in accordance with the conventional 3D theory. As it happens, this is true for the dHvA effect but not for the SdH effect. In q2D organic salts, it has been observed that the phase shift in the SdH effect has been as high as ±π/2 (Weiss et al. 1999, Schiller et al. 2000), even in measurements strongly dominated by the first harmonic, where one would expect the conventional theory to apply very well. Slow oscillations: The appearance of additional slow oscillations (Singleton 2000, Grigoriev 2002, Kartsovnik 2004, and the references therein) in measurements of the SdH effect has been attributed to the presence of small electron pockets in these materials. However, these pockets are not predicted by band structure calculations and, moreover, do not appear in the dHvA oscillations, whereas they are very pronounced in those of the SdH effect. This seems to argue against this interpretation.

Explanations: Grigoriev and collaborators (Grigoriev et al. 2002, Grigoriev 2002, 2003) have provided theoretical descriptions of the SdH effect in single-band q2D materials, using both the semiclassical Boltzmann formula and the linear response theory using the Born approximation. While there are some slight quantitative differences between the two accounts, they are qualitatively similar and provide the same explanations for the phenomena in question. Phase shifts arise due to the importance of the oscillations in the mean free velocity, which becomes as important as the oscillations in the density of states or self-energy (depending on the approach taken) when ћωc is of the order of t (Harrison et al. 1996, Grigoriev 2002, Grigoriev et al. 2002). Whereas the effect of warping in the latter enters in a zeroth-order Bessel function J0(4πrt/ћωc), it enters into the former as a first-order Bessel function J1(4πrt/ћωc). In the limit where these can be replaced with the equivalent cosine or sine functions, we obtain (Grigoriev et al. 2002, Grigoriev 2002, 2003) (after integration over the energy) ⎛ ⎞ ⎛ ⎞

zz ∝ cos ⎜ 2μπ ⎟ cos ⎜ 4 πt − π + φ⎟ , σ ⎝ ω c ⎠ ⎝ ω c 4 ⎠

(29.60)

with ϕ = arctan(a) where a = ћωc/2πt if the Boltzmann approach is used (Grigoriev et al. 2002), or a = ћωc/2πt(1 + (2π2kBTD/ћωc)) (Grigoriev 2003) in the case of the linear response or Kubo formalism. While the latter gives closer agreement to experiment at magnetic fields of around 10 T, for higher fields it has been found that the measured maximum of the phase shift exceeds the theoretical maximum of π/2 (Kartsovnik et al. 2003). Slow oscillations also arise due to the interference of fast quantum oscillations of the various quantities on which the conductivity depends, such as the mean free velocity and the various mechanisms implicitly included in τ. Taking the product of two such oscillations has the following result: (1 + A cos x)(1 + B cos x) = 1 + (A + B) cos x + (AB/2) cos 2x + AB/2. The final AB/2 term corresponds in this case to the slow oscillations, which contains the slowly oscillating terms proportional to the Bessel functions. Taking 4πt ≥ ћωc, so that we may use the sinusoidal asymptotics of the Bessel functions, we obtain (Grigoriev et al. 2002, Grigoriev 2002, 2003): ⎡ ⎛ 2πμ ⎞ ⎛ 4 πt π ⎞ ⎛ 2π2 kBT ⎞ ω c (1 + a2 ) cos ⎜ σ zz = σ0 ⎢1 + 2 2 ⎟ cos ⎝⎜ ω − 4 + φb ⎠⎟ RT ⎜ ω ⎟ RD ω 2 t π ⎝ ⎠ ⎝ c c c ⎠ ⎢⎣ +

⎛ ⎡ 4 πt π φ s ⎤ ⎞ 2 ⎤ ω c 1 + as2 cos ⎜ 2 ⎢ − + ⎥⎟ RD∗ ⎥ , 2 π 2t ⎝ ⎣ ω c 4 2 ⎦ ⎠ ⎥⎦

(29.61)

where we have omitted the second harmonics and the constant terms, and σ0 = e2m*td2/π2ћ2kBTD. ϕs = arctan(as) and ϕb = arctan(a), and R D* is a Dingle factor that differs slightly from the expected factor RD—see below. For derivations using both the Kubo formula (Grigoriev 2003) and Boltzmann semiclassical approximation (Grigoriev et al. 2002), as = ћωc/2πt. For the Boltzmann derivation, a = a s, whereas for the Kubo derivation

29-11


a = ћωc/2πt(1 + (2π2kBTD/ћωc)), the difference between the two being due to the additional term corresponding to the secondorder poles in the integrations over the Green’s functions in the linear response calculation that cannot be neglected in a q2D system (Champel and Mineev 2002, Grigoriev 2003). The second term gives rise to the fast SdH oscillations, the third to the slow oscillations. It should be apparent that the slow oscillations have no temperature damping, which follows from their lack of dependency on μ—they depend on the energy spectrum, not on the distribution function. This entails that they can be larger than the fast oscillations at T > TD. However, one would intuitively expect

(and experiment shows (Kartsovnik et al. 2002) ) that there would be some damping of these oscillations at reasonably high temperatures—this would likely be due to electron–electron and electron–phonon scattering, which has not been included in this calculation but can be expected to play a role. Grigoriev (2002) includes a brief account of how this might transpire, but to our knowledge a detailed calculation has yet to be carried out. It is likely that the temperature dependence of the oscillations would carry important information about the strength of the electron– electron and electron–phonon interaction in these materials. In addition, the slow oscillations also have a different Dingle damping factor (RD*) from the fast oscillations (which have R D). This is because the traditional Dingle factor includes all temperature-independent mechanisms of Fock–Landau level smearing, from microscopic causes such as scattering to more macroscopic inhomogeneities in the sample that can cause spatial variations in the electron energy equivalent to local shifts of the chemical potential μ. Since the measurements of the SdH oscillations are in effect an averaging over the entire sample, these variations normally result in increased damping of the oscillations. However, since the slow oscillations are not dependent on μ, they are sensitive only to the smearing introduced by microscopic mechanisms, and are thus damped less than they otherwise would be. One can estimate the degree to which the macroscopic inhomogeneities contribute to the damping of the SdH effect by taking the ratio of the two Dingle temperatures (Kartsovnik et al. 2002). The above discussion is relevant to low and intermediate values of the magnetic field. At high values of the field where ћωc/t >> 1, we enter the 2D limit discussed in Section 3.5.2 of Kartsovnik et al. (2003). This is largely beyond the scope of this chapter, however. We should observe, though, that in this limit, it is likely that the model of point-like impurities assumed in the above calculations is no longer valid (Raikh and Shahbazyan 1993, Champel and Mineev 2006), and that various theoretical difficulties conspire so as to make the description of real materials in this limit nontrivial. 29.3.3.2 Scattering and Combination Frequencies As in the dHvA effect, we may also observe combination SdH frequencies in multiband metals. In addition to the possible causes listed for the dHvA, all of which apply in this case, there are two additional causes: the Shiba–Fukuyama–Stark effect and interband scattering.

Area A

Path a

Path b

FIGURE 29.3 Schematic diagram showing the Shiba–Fukuyama– Stark effect. An electron can move along two possible paths, one of which encloses an orbit of area A. (Simplified from Singleton, J., Rep. Prog. Phys., 63, 1111, 2000, Figure 10.)

Shiba–Fukuyama–Stark effect: This effect (Shiba and Fukuyama 1969, Stark and Friedberg 1974, Stark and Reifenberger 1977, Morrison and Stark 1981) produces subtractive combination frequencies in the conductivity that are not permitted by the traditional form of magnetic breakdown. In this case, breakdown occurs between two Fermi surfaces along which electrons are traveling in the same direction—they can go either by route a or b on the pathways described in Figure 29.3. As a result of this, we obtain oscillations in the resistivity that are periodic in B −1 with a frequency proportional to that of the area A (Singleton 2000). Since this effect requires no circulation of the electrons around the closed Fermi surface, the free energy of the system is not affected, and so this effect gives rise to no further oscillations in Ω, and hence no oscillations in the dHvA effect. In addition, since these effects do not arise from the Fock–Landau quantization of the system, they are not so strongly damped by the temperature as compared to the SdH oscillations. Interband scattering: Thomas et al. (2008) have generalized Grigoriev’s linear response theory (Grigoriev 2003) results to the case of multiple bands, examining the effects of chemical potential oscillations and scattering on these systems. In addition to the expected mixing effect that gives rise to combination frequencies, additional slow oscillations may be observed, as predicted in the single-band case by Grigoriev. The most dramatic effect arises if impurities are permitted to scatter electrons between bands. In that case, the self-energy of the system must include oscillatory contributions from other bands, and so interference effects that give rise to combination frequencies arise. Restricting ourselves to the case of two bands, and taking R D l R >> 2πћ/(m*v F); here and previously, v F is the Fermi velocity and m* is the band mass in the bulk metal. We also define a new frequency ωs = πv F/2R, which will account for some aspects of the confi ning effects later on. The Schrödinger equation in polar coordinates for an electron moving in a longitudinal magnetic field B parallel to the

29-14


z-axis, where the vector potential has the form Aϕ = Bρ/2 and Az = Aρ = 0, is

−

2 ⎡ 1 ∂ ⎛ ∂ψ ⎞ ∂2 ψ 1 ∂2 ψ ⎤ ρ + + ⎢ ⎥ 2m * ⎣ ρ ∂ρ ⎜⎝ ∂ρ ⎟⎠ ∂z 2 ρ2 ∂φ2 ⎦ +

E=

iω c ∂ψ m * ω 2cρ2 + ψ = Eψ. 2 ∂φ 8

(29.72)

This is easy to solve (see Landau and Lifschitz 1977), for example, though one should be careful of conventions regarding e. The solution can be written in the form ψ=

1 R(ρ)e imφe ipz z / , 2π

(29.73)

where R(ρ), following a redefinition in terms of ξ = (m*ωc/2ћ)ρ2 is the radial function: ξ 2

(29.74)

as −a (here we follow the notation of Abramowitz and Stegun

If the wave function is fi nite everywhere, then −a is an integer, and the eigenfunctions of the equation are the Laguerre polynomials with the Fock–Landau eigenvalues given by Equation 29.4. However, if it becomes zero at some finite radius R, then −a is a positive, real number. Th is can lead to significant differences in the behavior of the system relative to the unconfi ned case (see Figure 29.6). In order to proceed analytically in these cases, particularly where R is small, we must therefore obtain some form of approximate energy spectrum.

29.4.3 Weak Field Limit In the fourth part of his seminal series of papers on magnetization in metals, Dingle (1952c) treated the case of an electron gas confi ned within a thin cylindrical wire in a weak field. In this case, the influence of the field can be treated as a perturbation of the zero-field case, provided that for a given energy level of interest E0 , πω 2c /(8ω 2s ) v F and dimensionless interaction parameter Kρ = 1 − (2U(0) − U(2kF))/(2πv F) < 1. This important result is the starting point for many interesting features of strongly correlated one-dimensional electrons, as it allows for an essentially exact calculation of the low-energy properties of these electrons in terms of free bosons φρ and θρ. The spin Hamiltonian does not have simple harmonic appearance because the product of spin currents in (30.A.15) includes very nonlinear cos [ 8πϕ σ ] term. Nevertheless, the progress is possible by attacking (30.A.15) using perturbative (in small U(2kF)/v F ratio) RG, as described in the main text. The key idea is to exploit the charge-spin separation to the fullest: by its very derivation, the spin backscattering correction (30.A.15) involves only spin modes φσ and θσ. It is then allowed to disregard the charge part, Hint,ρ, altogether and treat Hint,σ as the only perturbation to the free Hamiltonian (30.27). The charge part of kinetic energy Hkin,ρ, which is contained in (30.27), is guaranteed (by the independence of charge and spin bosons) not to affect the result of such calculation in any way. The end result is that one can formulate perturbation theory in question in terms of weakly interacting fermions again! The bosonization is used here only to make the phenomenon of the charge-spin separation explicit on the level of operators. This observation provides for a convenient short-cut in deriving the operator-product relations between spin currents that are required for the perturbative RG (see Starykh et al. (2005) for more details). In general, one is interested in RG equation for the anisotropic current–current interaction term, a a a R L

Renormalization is possible because the last term in the brackets contains terms that reduce to the second term, i.e., to H σ′ . This follows from the nontrivial property of the product of spin currents

∑∫

∑ ∫

(30.A.18)

(30.A.19)

This result follows from applying Wick’s theorem (justified here because the unperturbed Hamiltonian is just Hkin) and making all possible fermion pairings. These are constrained by the fact that the only nonzero correlations are between like fermions (right with right, and left with left) of the same spin, as dictated by the structure of (30.27). Thus, the singular denominator in the equation above is just the Green’s function of right fermions. Another ingredient of (30.A.19) is the well-known Pauli matrix property: σa σb = δ ab + i

∑⑀

abc

σc

(30.A.20)

c

The product of left currents has similar expansion with the obvious replacement v Fτ − ix → v Fτ + ix, as appropriate for the left-moving particles. With the help of these operator-product expansions, the last term (denoted V here) simplifies to V = const −

1 ⑀abc ⑀abd J Rc (x , τ)J Ld (x , τ) g a g b dxdτdx ′dτ ′ 2 2 a ,b , c , d 4 π (vF τ − ix)(vF τ + ix)

∑

∫

(30.A.21) where the constant is the contribution from the first term in (30.A.19). Switching to the center-of-mass and relative coordinates, we finally have V=

1 2

∑ δg ∫ dxdτ J (x, τ)J (x, τ) c R

c

c L

(30.A.22)

c

where a′

1 g a g b dr 1 g a g b ⎛ a′ ⎞ ln δg c = − =− 2 a ,b ≠ c 2πv F a r 2 a ,b ≠ c 2πv F ⎜⎝ a ⎟⎠

∑

∑

∫

(30.A.23)

is given by the integral over relative coordinate. In terms of RG scale ℓ = ln(a′/a), the differential change in the coupling gc follows

∑

dg c ga gb =− πv F d 4 a ,b ≠ c

(30.A.24)

This, of course, contains three equations g ygz dg x =− d 2πvF dg y g g =− x z d 2 πv F gxg y dg z =− d 2 πv F

(30.A.25)

30-13

Spin-Density Wave in a Quantum Wire

RG equations in the main text follow from the ones above by simple rearrangements. To obtain bosonic representation of spin in (30.55), one needs to start with the definition of 2kF component of spin density, 1 S2kF (x ) = 2

∑R σ + s

L e −i 2kF x + L+s σ s , s ′ Rs ′e i 2kF x

s,s ′ s ′

−iθR σ y /2

σ xe

iθL σ y /2

=e

−i ( θR +θL ) σ y /2

σx = σx

e

− iθR σ y /2

σ ye

iθL σ y /2

=e

− i (θR −θL )σ y /2

σy = e

− i χσ y /2

σy

⎡χ⎤ + sin ⎢ ⎥ ⎣2⎦

ρ2k F = −

(30.A.27)

s

+

y − i 2 kF x s,s ′ s ′

L′ e

s,s ′

⎡χ⎤ 1 + sin ⎢ ⎥ ⎣2⎦2

∑ {iR′ L′e s

+

s,s ′

s

−i 2 kF x

}

+ h.c.

This

We now bosonize primed fermions using (30.A.2, 30.A.3, 30.A.6, and 30.A.9), as well as the Baker–Hausdorff formulae e Ae B = e Be Ae[ A, B] , e Ae B = e A + Be[ A,B]/2

(30.A.30)

In this way, we obtain S2ykF =

1 2

1 M = 2

(30.A.29)

1 cos ⎡ 2πϕρ + 2kF x ⎤ ⎣ ⎦ πa0 ⎛ ⎞ ⎡χ⎤ ⎡χ⎤ ⎡ ⎤ ⎡ ⎤ ⎜⎝ cos ⎢ 2 ⎥ cos ⎣ 2πθσ ⎦ + sin ⎢ 2 ⎥ cos ⎣ 2πϕ σ + t ϕ x ⎦ ⎟⎠ ⎣ ⎦ ⎣ ⎦ (30.A.31)

The appearance of the position-dependent phase tφ x in the last term in the brackets is due to the shift (30.47), where we need to use t ϕ = hR /vF = 4 Δ 2s − o + Δ 2z /v F in order to account for the mutually orthogonal orientation of the spin–orbital and magnetic field axes. The other, unrotated, components of the spin density vector are obtained by similar calculations, resulting in (30.55) of the main text. Calculation of the backscattering (2kF) component of the density ρ2k F proceeds very similarly:

+

s

−i2k F x

}

+ h.c.

s

∑ {−iR′ σ s

+

−i2k F x y s, s ′ s

L′e

s, s ′

}

+ h.c.

(30.A.32)

2 sin ⎡ 2πϕρ + 2kF x ⎤ ⎣ ⎦ πa0

Bosonized form of the nonlinear Cooper term M R+ M L+ follows from

+ L

}

s

(30.A.33)

(30.A.28)

+ h.c.

∑ {R′ L′e

⎛ ⎞ ⎡χ⎤ ⎡χ⎤ ⎡ ⎤ ⎡ ⎤ ⎜⎝ cos ⎢ 2 ⎥ cos ⎣ 2πϕ σ + t ϕ x ⎦ + sin ⎢ 2 ⎥ cos ⎣ 2πθσ ⎦ ⎟⎠ ⎣ ⎦ ⎣ ⎦

Hence,

∑ {R′ σ

+ L+s Rse i 2 kF x

Subsequent bosonization leads to

M R+ =

⎡χ⎤ 1 S2ykF = cos ⎢ ⎥ ⎣2⎦2

+ − i 2 kF x s s

⎡χ⎤ → cos ⎢ ⎥ ⎣2⎦

(30.A.26)

where we made use of (30.A.20) and the fact that in the current geometry, θR = −θL. Hence we find that Sx and Sz components of the spin density are not affected by the rotation, while Sy component does change. Indeed,

∑R L e s

s,s ′

Next, we need to account for the rotation (30.41). Th is leads us to consider the following objects e

ρ2k F =

∑ R′ (σ s

+

x

+ iσ y )s ,s′ Rs′′ =

s ,s′

∑ s ,s′

i e −i 4 πa0

2 π ( ϕσ −θσ )

(30.A.34) i Ls′ (σ x + iσ y )s ,s′ Ls′′ = ei 4 πa0 +

2 π (ϕσ +θσ )

together with (30.A.4), implies that M M + h.c. ~ cos ⎡ 8πθσ ⎤ . Note also that Equations 30.A.34 ⎣ ⎦ explain the effect of the shift (30.47) on transverse components ± M R/L of the magnetization, Equation 30.48. + R

result,

+ L

Acknowledgments I would like to thank my collaborators on this project, Jianmin Sun and Suhas Gangadharaiah, for their invaluable contributions to this topic. I also thank Suhas Gangadharaiah for careful reading of the manuscript. I am deeply grateful to Andreas Schnyder and Leon Balents for the collaboration on quantum kagome antiferromagnet where the trick of chiral rotations of spin currents has originated. I would like to thank I. Affleck, T. Giamarchi, K. Matveev, E. Mishchenko, M. Oshikawa, M. Raikh, and Y.-S. Wu for discussions and suggestions at various stages of this work. I thank Petroleum Research Fund of the American Chemical Society for the financial support of this research under the grant PRF 43219-AC10.

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Levitov, L.S. and E.I. Rashba, 2003. Dynamical spin-electric coupling in a quantum dot. Phys. Rev. B 67: 115324. Lin, H.-H., L. Balents, and M.P.A. Fisher, 1997. N-chain Hubbard model in weak coupling. Phys. Rev. B 56: 6569. Moriya, T. 1960a. New mechanism of anisotropic superexchange interaction. Phys. Rev. Lett. 4: 228. Moriya, T. 1960b. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120: 91. Moroz, A.V. and C.H.W. Barnes, 1999. Effect of the spin-orbit interaction on the band structure and conductance of quasione-dimensional systems. Phys. Rev. B 60: 14272. Moroz, A.V. and C.H.W. Barnes, 2000. Spin-orbit interaction as a source of spectral and transport properties in quasi-onedimensional systems. Phys. Rev. B 61: R2464. Moroz, A.V., K.V. Samokhin and C.H.W. Barnes, 2000. Theory of quasi-one-dimensional electron liquids with spin-orbit coupling. Phys. Rev. B 62: 16900. Mugarza, A. and J.E. Ortega, 2003. Electronic states at vicinal surfaces. J. Phys.: Condens. Matter 15: S3281. Mugarza, A., A. Mascaraque, V. Repain et al., 2002. Lateral quantum wells at vicinal Au(111) studied with angle-resolved photoemission. Phys. Rev. B 66: 245419. Orignac, E. and T. Giamarchi, 1997. Effects of disorder on two strongly correlated coupled chains. Phys. Rev. B 56: 7167. Ortega, J.E., M. Ruiz-Osés, J. Gordón et al., 2005. One-dimensional versus two-dimensional electronic states in vicinal surfaces. New J. Phys. 7: 101. Pereira, R.G. and E. Miranda, 2005. Magnetically controlled impurities in quantum wires with strong Rashba coupling. Phys. Rev. B 71: 085318. Schnyder, A.P., O.A. Starykh, and L. Balents, 2008. Spatially anisotropic Heisenberg kagome antiferromagnet. Phys. Rev. B 78: 174420. Shahbazyan, T.V. and M.E. Raikh, 1994. Low-field anomaly in 2D hopping magne-toresistance caused by spin-orbit term in the energy spectrum. Phys. Rev. Lett. 73: 1408. Starykh, O.A., D.L. Maslov, W. Häusler, and L.I. Glazman, 2000. Gapped phases of quantum wires. In Low-Dimensional Systems: Interactions and Transport Properties, ed. T. Brandes, pp. 37–78. Lecture Notes in Physics, Vol. 544, Springer, Berlin, Germany. Starykh, O.A., A. Furusaki, and L. Balents, 2005. Anisotropic pyrochlores and the global phase diagram of the checkerboard antiferromagnet. Phys. Rev. B 72: 094416. Sun, J., S. Gangadharaiah, and O.A. Starykh, 2007. Spin-orbitinduced spin-density wave in a quantum wire. Phys. Rev. Lett. 98: 126408. Trif, M., V.N. Golovach, and D. Loss, 2007. Spin-spin coupling in electrostatically coupled quantum dots. Phys. Rev. B 75: 085307. Wu, C., B.A. Bernevig, and S.-C. Zhang, 2006. Helical liquid and the edge of quantum spin hall systems. Phys. Rev. Lett. 96: 106401. Xu, C. and J.E. Moore, 2006. Stability of the quantum spin hall effect: Effects of interactions, disorder, and Z2 topology. Phys. Rev. B 73: 045322.

31 Spin Waves in Ferromagnetic Nanowires and Nanotubes

Hock Siah Lim National University of Singapore

Meng Hau Kuok National University of Singapore

31.1 31.2 31.3 31.4

Introduction ........................................................................................................................... 31-1 Isolated Cylindrical Nanowire ............................................................................................ 31-3 Collective Spin-Wave Modes in an Array of Nanowires ................................................. 31-7 Hollow Cylindrical Nanowires ............................................................................................ 31-9 Isolated Hollow Cylindrical Nanowires • Array of Hollow Cylindrical Nanowires

31.5 Summary ............................................................................................................................... 31-12 References......................................................................................................................................... 31-12

This chapter begins with a brief overview of the historical development of the theory of spin waves in magnetic nanostructures. State-of-the-art calculations for dipole–exchange spin waves in a ferromagnetic nanowire and a hollow ferromagnetic nanowire, both of cylindrical cross-sections, are presented. Additionally, the treatment of collective spin-wave modes in ordered or disordered nanowire arrays, within the multiple scattering framework, is discussed. Here we adopt a tutorial style approach, interspersed with examples, to illustrate how the calculations are performed. In particular, we show how the frequencies of spin waves and the corresponding eigenmode profi les are determined. The aim is to provide a solid foundation in concepts and techniques that will enable the reader to make his own independent calculations.

31.1 Introduction In 1930, Bloch introduced the concept of a magnon in order to account for the reduction of spontaneous magnetization in a ferromagnet as a function of temperature T. At T = 0 K, all atomic spins point along the same direction so that the total energy is at a minimum. As the temperature increases, the spins begin to deviate randomly at an increasing rate from the common direction, thereby reducing the spontaneous magnetization. At sufficiently low temperatures, the low-lying energy states of spin systems have only a few misaligned spins and can be treated as a gas of quasiparticles called magnons or quantized spin waves. Because of exchange interactions, the disturbance arising from spin misalignments propagates through the spin system, as a collective motion of the atomic spins, as waves with discrete energies. Thus, spin waves can be viewed as the analog for magnetically ordered systems of lattice waves in solid systems. Based on this spin wave theory, Bloch derived the T 3/2 law for the

decrease of spontaneous magnetization. The quantitative theory of quantized spin waves was developed further by Holstein and Primakoff (1940) and Dyson (1956). For a comprehensive treatment of spin waves, the reader is referred to a book by Kittel (2004). The theory of magnetostatic modes of an isotropic ferromagnetic slab, in the absence of exchange interactions, was fi rst developed by Damon and Eshbach (1961). Within the long-wavelength approximation, a semi-classical continuum model (see, e.g., Cottam and Lockwood 1986) is applicable and the theory yields two types of spin waves: backward volume waves and nonreciprocal surface waves (commonly called “Damon–Eshbach” waves). As the mode frequency is much less than the corresponding electromagnetic frequency for a given wave number (and hence retardation effects can be ignored), these propagating modes are known as magnetostatic waves. When both dipolar and exchange interactions are important, the problem becomes more complicated. Kalinikos and Slavin (1986) have proposed a perturbation approach to treat this problem and have obtained analytical expressions for the dipole–exchange spin wave spectrum. Kalinikos et al. (1990) extended their theory to treat anisotropic ferromagnetic fi lms, while Rado and Hicken (1988) and Hicken et al. (1995) considered surface as well as bulk anisotropies. Hillebrands (1990) investigated the magnetic field dependence of the spin mode frequencies for magnetic multilayers. The reader is referred to Demokritov and Tsymbal (1994) and Hillebrands (1999) for a review of spin waves in structured fi lms. In the 1990s, the quantization of spin waves was observed in Permalloy micron-sized stripes (Mathieu et al. 1998). The modes observed are identified as surface Damon–Eshbach waves, which are quantized due to lateral confi nement. For a finite, nonellipsoidal, micron-sized magnetic thin fi lm element, 31-1

31-2

it was experimentally and theoretically demonstrated that the spin waves can exhibit strong spatial localization near the edge of the element due to the formation of a potential well for spin waves (Jorzick et al. 2002, Demokritov 2003). For a review of the theory and experiments on the propagation of linear and nonlinear spin waves in magnetic films and arrays of micron-sized magnetic dots and wires, the reader is referred to a review article by Demokritov et al. (2001). If two ferromagnetic stripes are brought sufficiently close together, a dynamic magnetic dipole field generated in the spatial region outside each stripe by the precession of the magnetization will couple with that generated by the other stripe, resulting in the formation of collective magnetostatic modes. This “crosstalk” between magnetic elements in densely packed arrays (e.g., in magnetic data storage) can be especially important as it can limit device performance based on these arrays. Kostylev et al. (2004) and Bayer et al. (2006) developed a method based on the Green’s function approach to treat collective magnetostatic modes in a one-dimensional (1D) array of ferromagnetic stripes. Gubbiotti et al. (2007) extended the theory in order to interpret the observed frequency dispersion of collective spin modes in dense arrays of magnetic stripes. In 2001, Arias and Mills (2001, 2002) formulated an exact theory within the long-wavelength approximation to treat dipole– exchange spin waves in a ferromagnetic cylindrical nanowire, where the magnetization is parallel to the axis of the wire. An approximate analytical expression based on this theory was used to interpret the quantization of spin waves in isolated nickel nanowires observed by Brillouin light scattering (BLS) spectroscopy (Wang et al. 2002). Arias and Mills (2003) extended the theory for a single isolated cylindrical nanowire to disordered and ordered arrays of these nanowires based on a real-space multiple scattering approach. Nguyen and Cottam (2006) used a microscopic theory to study spin waves in a single ferromagnetic nanotube, and Das and Cottam (2007) investigated the magnetostatic modes in an antiferromagnetic nanotube. Arias et al. (2005) have developed an analytical theory for dipole–exchange spin waves in a ferromagnetic sphere and the response of a uniformly magnetized sphere to a spatially inhomogeneous microwave field. The theory was subsequently extended to treat spin-wave collective modes in a two-dimensional (2D) square lattice of ferromagnetic nanospheres (Arias and Mills 2004, Chu et al. 2006) and to calculate the BLS crosssection for the various spin-wave modes of a single metallic nanosphere (Chu and Mills 2007). Wang et al. (2006) performed Brillouin studies of the magnetic field dependence of collective spin waves in hexagonally ordered 2D arrays of vertically oriented Fe48Co52 nanowires. The arrays comprised wires with fi xed diameters of 20 nm and wire spacing-radius ratios ranging from 3 to 5.5. The calculated frequencies of the collective spin modes as a function of interwire separation based on the multiple scattering theory (Arias and Mills 2003) showed excellent agreement with the experimental BLS data. The influence of neighboring wires in the arrays is manifested as a depression of the frequency of the lowest-energy


collective spin-wave mode relative to that of the isolated wire. This frequency depression becomes progressively more pronounced with decreasing interwire spacing, which shows that interwire dipolar coupling plays an important role in highdensity 2D arrays of nanomagnets. The calculation of the spin-wave modes of small magnetic particles, in general, is a complicated problem when both dipolar and exchange contributions are taken into consideration. A numerical approach may be necessary and it often requires the solving of the Landau–Lifshitz–Gilbert equation (see for example, Aharoni 1996) dM (r , t ) α dM (r , t ) = −γM (r , t ) × H eff (r , t ) + M (r , t ) × , (31.1) dt Ms dt where Heff is the effective magnetic field acting on the magnetization M(r, t) Ms is the saturation magnetization γ is the gyromagnetic ratio α is the Gilbert damping parameter The damping corresponds to the energy dissipated during the precessional motion of the magnetization and allows M(r, t) to turn towards the effective field until both vectors are parallel in the static solution. The energy minimum of a magnetic system can be obtained using the freeware OOMMF, which is available from the National Institute of Standards and Technology. To find the dynamic spin-wave modes, the damping is next reduced substantially and the system is given a small perturbation. The system is allowed to evolve in time and the time evolution of the average magnetization of the particle is then tracked. The Fourier transform of the time-dependent magnetization subsequently gives the spinwave frequencies (Grimsditch et al. 2004a). Using this numerical procedure, Wang et al. (2005) studied the spin dynamics of a high-aspect-ratio nickel nanoring (Wang et al. 2005) in an external magnetic field. Grimsditch et al. (2004b) developed an alternative and powerful method for the calculation of magnetic normal modes of nanometer-sized particles based on the “dynamical matrix” approach. This has the advantage that a single calculation can yield the frequencies and eigenvectors of all the modes. As the size of magnetic particles decreases, it is important to have access to experimental tools that can probe magnetic properties on the submicron scale. In this regard, BLS spectroscopy (Patton 1984, Demokritov and Tsymbal 1994, Carlotti and Gubbiotti 1999) provides an ideal tool to probe the spin-wave dynamics of magnetic nanostructures (Himpsel et al. 1998, Martín et al. 2003), such as dots, dot arrays, wires, and arrays of wires (Demokritov and Hillebrands 1999, Skomski 2003, Liu et al. 2005). In this chapter, we shall discuss the low-frequency magnetic excitations that are accessible by BLS. A quantum mechanical

31-3

Spin Waves in Ferromagnetic Nanowires and Nanotubes

Assuming that the fluctuations in M associated with the spin waves are small compared with the static values, we may write M as

TABLE 31.1 Unit Conversion Table Quantity

Symbol

CGS

SI

Magnetic field

H

Oe

103 A/m 4π

Flux density Magnetization Surface anisotropy Exchange stiffness

B M Ks A

G emu/cc erg/cm2 erg/cm

10−4 T 103 A/m 10−3 J/m2 10−5 J/m

M(r , t ) = M se z + m(r , t ),

formalism is necessary at low temperatures or for fi lms that are only a few atomic-layer thick. However, as the wavelengths of these spin waves are generally much larger than interatomic distances, a continuum description of dipole–exchange spin waves will therefore be adequate. In addition, we shall limit our discussion to infinitely long cylindrical nanowires and hollow cylindrical nanowires as well as arrays of these structures. Also presented are several original works that have not been previously published. All equations in this chapter are expressed in the centimeter, gram, second (cgs) system of units in which the fundamental magnetic equation is B = H + 4πM, where the flux density B is in gauss (G), the magnetic field strength H is in oersted (Oe), and the magnetization M is in emu/cc. Table 31.1 lists conversion factors for commonly used magnetic quantities. For further information on the various systems of units used in magnetism, the reader is referred to an article by Scholten (1995).

31.2 Isolated Cylindrical Nanowire

(31.2)

By neglecting the magnetic volume anisotropy field, we can write Heff as H eff (r , t ) = H 0e z + h(r , t ) +

2A 2 ∇ M(r , t ), Ms2

where m(r, t) is the dynamic component of the magnetization, and |m| = Ms. We shall consider the time-harmonic solution, so that the time-dependence part of the spin wave is of the form e−iΩt. This allows us to write m(r, t) and h(r, t) as a product of a spatial and a time-dependent part, namely, m(r , t ) = m(r )e −iωt ,

(31.5)

h(r , t ) = h(r )e −iωt .

(31.6)

Now, the magnetic dipolar field h(r) must satisfy the magnetostatic equations ∇ × h(r) = 0 and ∇ · b = ∇ · [h(r) + 4πm(r)] = 0, where b is the magnetic field induction. Thus, h(r ) = −∇Φ(r ),

(31.7)

where Φ(r) is a magnetostatic potential. Using Equations 31.3 through 31.7, we linearize Equation 31.2 by ignoring all the terms that are of quadratic and higher order in the components of h and m. This leads to the following relations: iΩmx = (H 0 − D∇2 )my + M s

∂Φ , ∂y

(31.8)

−iΩmy = (H 0 − D∇2 )mx + Ms

∂Φ , ∂x

(31.9)

and

Let us consider an infi nitely long ferromagnetic cylindrical wire of radius R, with saturation magnetization Ms, and an exchange stiff ness constant A. The easy axis of magnetization and the externally applied magnetic field H0 are aligned parallel to the axis of the wire. We shall assume that Ms is uniform throughout the wire and also that spin waves propagate without attenuation. Thus, ignoring the damping term, the Landau–Lifshift z equation of motion (see Equation 31.1) can then be written as dM (r , t ) = −γM (r , t ) × H eff (r , t ). dt

(31.4)

(31.3)

where ez is a unit vector along the z direction (or symmetry axis of the wire) h(r, t) is the dynamic dipolar field, and the last term describes the exchange field

where Ω = Ω/γ and D = 2A/Ms. In this linear approximation, we have set m(r) · ez = 0. In addition, within the material, the condition ∇ · b = 0 becomes ⎛ ∂m x ∂m y ⎞ ∇2Φ − 4π ⎜ + ⎟ = 0. ∂y ⎠ ⎝ ∂x

(31.10)

To derive a differential equation for Φ, we proceed as follows. We differentiate Equations 31.8 and 31.9 with respect to y and x to obtain ⎛ ∂m ⎛ ∂m ∂m y ⎞ ∂m y ⎞ iΩ ⎜ x − = ( H 0 − D∇ 2 ) ⎜ x + + M s∇2⊥ Φ, ⎟ ∂x ⎠ ∂y ⎠⎟ ⎝ ∂y ⎝ ∂x (31.11) where ∇ ⊥ = (∂2/∂x 2 ) + (∂2/∂y 2 ). Applying the operator (H0 − D∇ 2) to both sides of Equation 31.10, and using Equation 31.11, we arrive at ⎛ ∂mx ∂my ⎞ − 4πiΩ ⎜ = (H 0 − D∇2 )∇2Φ + 4πMs ∇2⊥ Φ. ∂x ⎟⎠ ⎝ ∂y

(31.12)

31-4


By differentiating Equations 31.8 and 31.9 with respect to x and y, and using Equation 31.12, we finally obtain the following homogeneous equation that the magnetic potential Φ must satisfy, namely, ∂2 ⎡(D∇2 − H 0 )(D∇2 − B0 ) − Ω2 ⎤ ∇2Φ + 4πMs (D∇2 − H 0 ) 2 Φ = 0, ⎣ ⎦ ∂z

−

∂Φ wire ∂Φ out =− . + 4 πmρ ∂ρ ∂ρ ρ= R

(31.19)

Additionally, if we assume that the surface anisotropy contributes energy −Ks (m ρ/Ms)2, where Ks is in units of erg/cm2, then the exchange boundary conditions require that m(r) at the surface of the wire satisfies (Arias and Mills 2001)

(31.13) where B 0 = H0 + 4πMs. Arias and Mills (2001) derived Equation 31.13 using a different approach. Using cylindrical polar coordinates, we seek solutions of Equation 31.13 for which the magnetic potential has the form Φ(ρ, φ, z ) = J n (κρ)exp(inφ + ikz ),

Substitution of Equation 31.14 into Equation 31.13 yields D 2 (κ 2 + k 2 )3 + D(H 0 + B0 )(κ 2 + k 2 )2

(31.20)

⎛ ∂mρ ⎞ − bpinmρ |ρ= R = 0, ⎜⎝ ∂ρ ⎟⎠ ρ= R

(31.21)

and

(31.14)

where Jn(κρ) is a Bessel function of order n k is the wave vector of the mode in the direction parallel to the z-axis

⎛ ∂mφ ⎞ = 0, ⎜⎝ ∂ρ ⎟⎠ ρ= R

where bpin = 2Ks/AMs is the pinning parameter. When bpin = 0, there is no surface spin pinning. When bpin is large, the spins are fully pinned. To evaluate these boundary conditions, we need explicit expressions for the radial and azimuthal components of the dynamical magnetization. They are obtained as follows. From Equations 31.8 and 31.9, we have (Ω + H 0 − D∇2 ) m+ = Msh+ ,

+ (H 0 B0 − Ω2 − 4πMs Dk 2 )(κ 2 + k 2 ) − 4πMs H 0k 2 = 0. (31.15)

(31.22)

where m+ = mx + imy and h+ = h x + ihy. Let us assume that This equation is cubic in κ , so that for each choice of n and k, we have three linearly independent solutions of Equation 31.14. Thus, the magnetic potential within the cylinder can be written as 2

3

Φ wire (ρ, φ, z ) =

∑A J (κ ρ)e i n

i

i ( nφ+ kz )

.

(31.16)

3

3

Outside the cylindrical wire, the potential satisfies the Laplace equation, so that

∑Q (κ

To determine the four coefficients A1, A2, A3, and C, we note that m(r) and h(r) have to fulfi ll the boundary conditions at the surface of the wire. Two of these boundary conditions are the continuity of Φ, namely,

2 i

i

+ k 2 )J n +1(κ iρ)ei ((n +1)φ+ kz ) ,

i =1

so that

(31.17)

where Kn(kρ) are the modified Bessel functions of the second kind C is an arbitrary constant

∑

(κ iρ)ei(nφ+ kz ) ,

where Qi are constants to be determined. Then ∇2m+ = −

Ai J n (κ i R) = CK n (kR),

i n +1

i =1

i =1

Φ out (ρ, φ, z ) = CK n (kρ)ei(nφ+ kz ) ,

∑Q J

m+ =

3

(Ω + H 0 − D∇2 )m+ =

∑Q ⎡⎣Ω + H i

0

+ D(κ i2 + k 2 )⎤⎦ J n +1

i =1

(κ iρ)e i((n +1)φ+ kz ) .

(31.23)

Since ⎛ ∂ i ∂ ⎞ wire h+ = −e iφ ⎜ + Φ ⎝ ∂ρ ρ ∂φ ⎟⎠ 3

(31.18)

i =1

and the continuity of the normal component of the magnetic field induction

=

∑A κ e i

i

i((n +1)φ+ kz )

J n +1(κ iρ),

(31.24)

i =1

substituting Equations 31.23 and 31.24 into Equation 31.22 allows us to determine Qi, so that we can write

31-5


3

m+ = M s

∑ i =1

Ai κ i J n +1 (κi ρ). Ω + H 0 + D (κ2i + k 2 )

(31.25)

A similar procedure yields 3

m − = −M s

∑ −Ω + H A+ κD(κ + k ) J i

2 i

0

i =1

i

2

n −1

(κi ρ),

(31.26)

where m− = mx − imy. Finally, the radial and azimuthal components of the dynamical magnetization are, respectively, given by 1 mρ (ρ, φ, z ) = ⎡⎣e −iφm+ + e iφm− ⎤⎦ 2 1 = Ms 2

3

⎡

∑A κ ⎢⎣ D(κ i

i

i =1

−

2 i

J n +1(κ i ρ) + k2 ) + H0 + Ω

⎤ i(nφ+ kz ) J n −1(κ iρ) e D(κ i2 + k 2 ) + H 0 − Ω ⎥⎦ (31.27)

and i mφ (ρ, φ, z ) = ⎡⎣ −e −iφm+ + eiφm− ⎤⎦ 2 i = − Ms 2

3

⎡

∑A κ ⎢⎣ D(κ i

i

i =1

+

2 i

J n +1(κ iρ) + k2 ) + H0 + Ω

⎤ i (nφ+ kz ) J n −1(κ iρ) e . D(κ i2 + k 2 ) + H 0 − Ω ⎥⎦

determinant to determine if its magnitude “vanishes” within a preset accuracy. If it vanishes, a solution has been found; if not, a new trial frequency is attempted and the process repeated. Once the correct value of the frequency has been determined, the corresponding eigenvector (i.e., the four coefficients) can also be found, hence the corresponding mode profi les can be obtained. Thus, the spatial distribution of the dynamic potential can be calculated from Equations 31.16 and 31.17. The spatial profi les of the dynamical magnetization can be similarly obtained from Equations 31.27 and 31.28. For illustration purposes, we shall assume that the magnetic material is Permalloy (Ni80Fe20) and use the following parameters in our calculations: saturation magnetization Ms = 800 × 103 A/m, exchange constant A = 10−11 J/m, and gyromagnetic ratio γ = 190 GHz/T. As Permalloy also exhibits negligible anisotropy, we may set the surface anisotropy energy K s = 0, and hence bpin = 0. Additionally, we shall also assume that this Permalloy material is amorphous or polycrystalline so that the uniaxial anisotropy constant K1 = 0. However, for accurate studies, it is important to assign appropriate values to these parameters. They can be normally obtained from a fit to the experimental data, for instance, the dispersion relations of both Damon–Esbach spin waves and the standing bulk spin-wave modes of a reference thin Permalloy fi lm. We shall consider the case where the azimuthal quantum number is n = 1. In Figure 31.1, we show the magnitude of the determinant as a function of trial frequencies for a nanowire with a radius of 20 nm at an applied field of 0.3 T and k = 107 m−1. Within the considered trial frequency range, four frequencies at 23.74, 49.57, 116.33, and 219.42 GHz satisfy the boundary conditions. The corresponding mode profi le for the lowest eigenfrequency,

(31.28)

105 104

These expressions allow us to write the boundary conditions as a set of four linear homogeneous equations, and the condition for the existence of nontrivial solutions can be written as (31.29)

where M(Ω , κi) is a 4 × 4-matrix. It should be noted that the determinant, called the boundary-value determinant, is a complex function (with real and imaginary parts) of two unknowns. In general, the algebraic complexity makes it impractical to derive an analytical expression for the spin wave frequency by solving Equations 31.15 and 31.29 simultaneously. A better approach is to solve the problem numerically by developing a computer code based on the following procedure. As the determinant is complex, we search for simultaneous zero crossings of the real and imaginary parts of the determinant. This can be accomplished if we consider the magnitude of the determinant. Thus, for a given trial frequency Ω, the three linearly independent κ are calculated from Equation 31.15. Both Ω and κ are then substituted into the boundary-condition

102 |Determinant| (arb. unit)

det M (Ω, κi ) = 0,

103

101 100 10–1 10–2 10–3 10–4 10–5 10–6

0

50

100

150

200

250

Trial frequency (GHz)

FIGURE 31.1 Computed plot of the magnitude of the boundary-value determinant versus trial frequency.

31-6

4


×10–6

4

×10–6

4

3

3

3

2

2

2

1

1

1

0

0

0

–1

–1

–1

–2

–2

–2

–3

–3

–3

–4 –4 (a)

–3

–2

–1

0

1

2

3 4 ×10–6

–4 –4 (b)

–3

–2

–1

0

1

2

3 4 ×10–6

×10–6

–4 –4 (c)

–3

–2

–1

0

1

2

3

4 ×10–6

FIGURE 31.2 Mode profi les of an isolated Permalloy nanowire of radius 20 nm for an azimuthal quantum number (a) n = 1, (b) n = 2, and (c) n = 3. The arrows indicate the orientation of the dynamic magnetization m and the gray scale depicts the variation of the magnetic potential. The unit of the spatial dimension is centimeters.

23.74 GHz, is presented in Figure 31.2a. Similar calculated mode profi les for n = 2 and 3 of the lowest eigenfrequencies are also shown in Figure 31.2. These modes, as expected, exhibit the dipolar, quadrupolar, and sextupolar characteristics for n = 1, 2, and 3, respectively. The arrows indicate the orientation of the dynamical magnetization over the cross-section of the nanowire while the gray-scale depicts the variation of the magnetic scalar potential within and outside the nanowire. The magnetic field dependence of the spin-wave frequencies for n = 1, 2, and 3 is shown in Figure 31.3, which reveals that the frequencies increase linearly with the magnetic field, with a slope given by γ/2π. This linear dependence can be understood as follows. While the full theory requires solving Equation 31.15 together with the required boundary conditions, an approximate analytical expression can be obtained that is applicable to

⎡ (κ 2 + k 2 ) ⎢ D 2 (κ 2 + k 2 )2 + D(H 0 + B0 )(κ 2 + k 2 ) ⎣ + (H 0 B0 − Ω2 − 4πMs Dk 2 ) −

4 πMs H 0k 2 ⎤ ⎥ = 0. κ2 + k2 ⎦

(31.30)

The bulk standing modes of the nanowires correspond to real values of κ. If we make the assumption that κ >> k, the spin-wave frequencies are then given by

f =

140

Frequency (GHz)

either small or large pinning. We note that Equation 31.15 can be factorized to give

1/2 γ ⎡ H 0 (H 0 + 4 πMs ) + Dκ 2 (Dκ 2 + 2H 0 + 4 πMs )⎤ ⎣ ⎦ 2π

120

=

1/2 γ ⎡(H 0 + 2πMs + Dκ 2 )2 − (2πMs )2 ⎤ ⎣ ⎦ 2π

100

=

1/2 γ ⎡(H 0 + Dκ 2 )(H 0 + Dκ 2 + 4πMs )⎤ , ⎣ ⎦ 2π

(31.31)

80

where f is the frequency, in Hz, of the mode. This equation is similar to the following Herring–Kittel formula (Herring and Kittel 1951) for the dipole–exchange spin-wave spectrum in an infinite ferromagnetic medium

60 40 20

f = 0 0.0

0.2 0.4 0.6 0.8 Applied magnetic field (T)

1.0

FIGURE 31.3 Magnetic field dependence of the spin wave frequencies for an isolated Permalloy nanowire of radius 20 nm and k = 107 nm−1. Solid, long-dashed and short-dashed lines denote n = 1, 2 and 3, respectively.

1/2 γ ⎡(H 0 + Dq 2 )(H 0 + Dq 2 + 4πMs sin2 θq )⎤ , ⎣ ⎦ 2π

where θq is the angle between the direction of the wave vector q and that of the magnetization. Thus, Equation 31.31 represents the propagation of a spin wave, on the xy plane of the nanowire, with a wave vector κ whose values are quantized by the finite radius of the nanowire.

31-7


For the applied field H0 = 0, Equation 31.31 reduces to f=

γ Dκ 2[Dκ 2 + 4πMs ] 2π

{

}

1/2

.

(31.32)

If the pinning is small, the radial function Jn(κρ) will have antinodes (Cottam and Lockwood 1986, Wang et al. 2002) at ρ = R, such that the first three values of κ are 1.84/R, 3.05/R, and 4.20/R for n = 1, 2, and 3, respectively. It is obvious from Equation 31.31 that for H0 to be sufficiently large, the frequency varies linearly with the field, with a slope of γ/2π.

where we have defined Bn(0) = An(0)Sn and Sn, in an analogy with the T-matrix of the scattering theory, describes the response of the nanowire to a driving field. Sn relates the incident amplitude of the driving field to the amplitude of the scattered field of a single nanowire. Sn can be determined as follows. The potential outside the nanowire, consisting of the incident and scattered fields, can be written as ∞

Φ total =

∑ ⎡⎣ A (0)I (kρ) + B (0)K (kρ)⎤⎦ e n

n

n

Φ wire =

n

(31.36)

3

∑∑A J (κ ρ)e n i n

i

inφ

.

(31.37)

n = −∞ i =1

Continuity of the magnetic potential across the surface of the nanowire therefore gives An (0)I n (kR ) + B n (0)K n (kR ) = −Γn ,

(31.38)

where 3

Γn = −

∑A J (κ R). n i n

i

(31.39)

i =1

In addition, continuity of the normal component of the magnetic field induction yields −

∂Φ total ∂Φ wire =− + 4 πmρ . ∂ρ ∂ρ

(31.40)

By substituting Equations 31.27, 31.36, and 31.37 into Equation 31.40, we obtain kAn (0)I n′ (kR) + kBn (0)K n′ (kR) = − Gn ,

∞

n

,

n= −∞

∞

If two infinitely long parallel ferromagnetic nanowires are brought close to each other, the dipolar fields generated, due to the spin precession within each nanowire, will couple so that collective spin-wave modes are formed. These modes are different from those of an isolated nanowire and cannot be described using a single azimuthal quantum number. In the following, we shall calculate the collective spin-wave modes of the array following the method described by Arias and Mills (2003). We employ the multiple scattering theory to account for the interwire interactions. We now consider an array of infinitely long cylindrical wires, each of radius R and of saturation magnetization Ms, and with their symmetry axes parallel to the z axis. Adjacent wires are sufficiently far apart so that interwire exchange interactions can be neglected and the wires only interact through dipolar coupling. As the magnetization precesses in one wire, a dynamic magnetic dipole field h is generated outside this wire. Firstly, we need to derive an expression for the response of a single nanowire to an external driving field. Consider the wire to be at the origin of a coordinate system, with the wire axis parallel to the z axis. Then, this field in the xy plane, which must satisfy the Laplace equation in the spatial region outside the wire, can be described by a potential Φ of the form

∑A (0)I (kρ)exp(inφ),

inφ

while the potential within the nanowire (summing over the azimuthal quantum number n) is

31.3 Collective Spin-Wave Modes in an Array of Nanowires

Φ (ρ, φ) =

n

(31.41)

(31.33) where the prime on the various Bessel functions indicates derivatives with respect to their respective arguments and

n=−∞

where In(kρ) is the modified Bessel function of the first kind. This external potential will excite the magnetization of the nanowire at the origin, resulting in the precession of the magnetization, which in turn will generate a dynamic dipole field. From Equation 31.17, the field generated outside the nanowire can be written as

3

Gn =

⎡

⎧

∑κ A ⎢⎣2πM ⎨⎩ D(κ i

n i

s

i =1

−

2 i

J n +1(κ i R) + k2 ) + H0 + Ω

⎤ ⎫ J n −1(κ i R) ⎬ − J n′ (κ i R)⎥ . 2 D(κ + k ) + H 0 − Ω ⎭ ⎦ 2 i

(31.42)

∞

Φ (0) (ρ, φ) =

∑B (0)K (kρ)exp(inφ) n

n

(31.34)

n = −∞

From Equations 31.38 and 31.41, we finally obtain the expression for Sn:

∞

Φ (0) (ρ, φ) =

∑A (0)S K (kρ)exp(inφ), n

n = −∞

n

n

(31.35)

Sn =

B n (0) G I (kR ) − k Γ n I n′ (kR ) =− n n . An (0) G nK n (kR ) − k Γ nK n′ (kR )

(31.43)

31-8


By replacing ν by −ν in the above equation and noting that K−ν(x) = Kν(x), we obtain

y fj Q

e

x

Cylinder j

∞

∑e

i ψ j (ν− n )

K ν − n (kd j )I n (k ρ)e inφ .

(31.45)

yj

When Equation 31.45 is combined with Equation 31.44, we have

ρj P

K ν (k ρ j ) = (−1)ν

n = −∞

y

dj

i νφ j

f

ρ

x

O

M = Φ

∞

∑A (0)I (k ρ)e n

n

inφ

,

(31.46)

n = −∞

where FIGURE 31.4 Coordinate system and quantities used in the multiple scattering theory.

The coefficients Ain appearing in the expressions for Gn and Γn are determined as follows. For a given n, we first set Ain = 1, and determine the two remaining coefficients by requiring that Equations 31.20 and 31.21 be satisfied. These two inhomogeneous equations will then yield the values of A2n and A3n . We are now in a position to consider the collective spin waves in an array of nanowires. In Figure 31.4, we have a nanowire at the origin of a coordinate system and one of the nanowires in the array, wire j. The magnetization of the wire at the origin is driven by the field generated by all the other wires in the array, and can be calculated from the following magnetic potential

n (0) = A

∞

∑∑ (−1) e

m i ( m − n )ψ j

B m ( j )K m − n (kd j ).

j ≠ 0 m = −∞

M excites the magnetization in the nanoThis driving potential Φ wire at the origin, and using Sn, we can relate Bn(0) to Ãn(0), i.e.,

n (0). B n (0) = S n A Thus, we have the result S n−1B n (0) =

∞

∑∑ (−1) e

m i ( m − n )ψ j

K m − n (kd j )B m ( j ).

(31.47)

j ≠ 0 m = −∞

M = Φ

∞

∑∑B ( j)K (k ρ )exp(inφ ), n

n

j

j

(31.44)

j ≠ 0 n = −∞

where the quantities ρj and ϕj are defined in Figure 31.4. Now, we want to express the incident wave, arising from the scattered waves of all the other nanowires, in the local cylindrical eigenfunction expansion. This can be accomplished by using the Graf addition theorem for Bessel functions given in the article by Abramowitz and Stegun (1964, Eq. 9.1.79). That is, this theorem allows us to relate the dynamic dipolar fields, of all but the nanowire at the origin, to the field expanded around this nanowire. By applying Graf’s theorem to the triangle OPQ of Figure 31.4, we obtain e i νςK ν (k ρ j ) =

∞

∑K

ν+ n

(kd j )I n (k ρ)e inϕ ,

n = −∞

which is valid provided that ρ < dj. φ is the angle between vectors r and dj and ς is the angle opposite to r. Now ς = ψj − (Φj − π), so that

e

− i νφ j

K ν (k ρ j ) = e

− i νψ j − i νπ

∞

∑e

n = −∞

− inψ j

K ν+ n (kd j )I n (k ρ)e inφ .

For clarity, and the fact that the method can easily incorporate variations in radii and magnetic properties from nanowire to nanowire, we rewrite Equation 31.47 as S n−1( p )B n ( p ) =

∞

∑∑ (−1) e

p m i ( m − n )ψ j

K m − n (kd jp )B m ( j ), (31.48)

j ≠ p m = −∞

where p = 1, 2,…, L, L is the total number of nanowires in the array, S n−1( p ) describes the scattering property of the nanowire p, and ψ pj and d jp are determined with respect to nanowire p. Thus, variations in the physical properties of the nanowires can be taken into account simply by using the appropriate expressions for Sn for each nanowire. For numerical calculations, the infinite sums in Equation 31.48 have to be truncated. In practice, N is chosen to be sufficiently large so that a finite sum, taken from −N to N, gives converged results. The frequencies of the various modes are determined by finding the zeros of the determinant of the coefficients formed from Equation 31.48. For a periodic array, the Bloch theorem requires that the coefficients Bn( j) of nanowire j are related to those of the nanowire at the origin by Bn( j) = Bn(0) exp(ik⊥ . dj), where k⊥ is the Bloch wave vector in the xy plane, i.e., perpendicular to wires’ axes. When an eigenfrequency has been calculated from Equation 31.48, the corresponding Bn coefficients of all the nanowires in the array can also

31-9


sufficiently large interwire separations, the eigenfrequencies of the lower branch correspond to the n = 1 mode of the dipolar character of an isolated nanowire. The higher branch, on the other hand, corresponds to the n = 2 mode of the quadrupolar character. As the interwire separation decreases, the branches begin to split, with the lower branch exhibiting the largest splitting. Additionally, the lower branch shows significant splitting over a wider range of separations, in comparison with the higher branch. This can be understood as follows. In the multiple scattering theory, modified Bessel functions Kn(kdj) appear in Equation 31.48. Because Kn(x) diverges as x−n as x → 0, it is clear that the interaction associated with the n = 1 dipolar mode is comparatively longer in range, and the splitting, thus, decreases slowly with increasing interwire separation, compared to that of the n = 2 branch. Figure 31.6 displays the mode profiles of the lowest-frequency eigenmodes of the respective branches.

Frequency (GHz)

27

26

24

23 100

200 300 w (nm)

400

FIGURE 31.5 Frequencies of collective spin wave modes of an array of four parallel identical nanowires of radius 30 nm, placed at the corners of a square of side length w at H0 = 0.3 T.

be evaluated. Hence, the magnetic potential in the spatial region outside the nanowires can be computed using Equation 31.36 with An(0) replaced by Ãn(0), while inside the nanowire located at the origin, the potential is given by Equation 31.37. In addition, the dynamical magnetization m of this nanowire at the origin can be determined from Equations 31.27 and 31.28, and summing over n. The potential and dynamical magnetization inside the other nanowires in the array can also be similarly calculated. As an illustration, we consider an array of four identical nanowires with a radius of 30 nm in an external field of 0.3 T and k = 107 m−1. The wires are located at the corners of a square of side w, and the frequencies of the collective spin-wave modes of the array are calculated as a function of w. In Figure 31.5, we show the first few lowest eigenfrequencies. We note that, at

1

×10–5

31.4 Hollow Cylindrical Nanowires 31.4.1 Isolated Hollow Cylindrical Nanowires We can easily generalize the formalism outlined in Section 31.2 to the case of a hollow cylindrical nanowire. Let us consider an infinitely long hollow cylinder with an outer radius of Ro and an inner radius of Ri. The linearized equations, Equations 31.8 through 31.10, remain the same but we need to replace Equation 31.14 with Φ(ρ, φ, z ) = [AJ n (κρ) + BYn (κρ)]exp(inφ + ikz ), where Yn(κρ) is a Bessel function of the second kind A and B are arbitrary constants For a solid cylinder, it is obvious that B is zero as Yn(κρ) diverges at the origin. Thus, the magnetic potential within the annulus of the hollow cylinder, i.e., for R i ≤ ρ ≤ Ro, is

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

–0.2

–0.2

–0.4

–0.4

–0.6

–0.6

–0.8

–0.8

–1 –1 (a)

–0.6

–0.2

0.2

0.6

1 ×10–5

(31.49)

×10–5

–1 –1 (b)

–0.6

–0.2

0.2

0.6

1 ×10–5

FIGURE 31.6 Mode profiles of an array of four identical nanowires of Figure 31.5. The arrows indicate the orientation of m and the gray scale depicts the variation of the magnetic potential for interwire separation w of 80 nm. Profi les of the lowest-frequency mode of the (a) n = 1 and (b) n = 2 branches. The unit of the spatial dimension is centimeters.

31-10


3

Φ

hollow wire

(ρ, φ, z ) =

∑ ⎣⎡A J (κ ρ) + B Y (κ ρ)⎦⎤ e i n

i

i n

i

i ( nφ+ kz )

. (31.50)

20

i =1

⎧⎪ C 1I n (k ρ)e i (nφ+ kz ) Φ(ρ, φ, z ) = ⎨ i ( nφ+ kz ) ⎪⎩C 2K n (k ρ)e

for ρ < R i , for ρ > Ro .

Frequency (GHz)

Outside the wire and within its bore, the potential satisfies the Laplace equation, so that Equation 31.17 is replaced by

(31.51)

0 0.0

3

J n +1(κ iρ) ⎪⎧ ⎡ κ i ⎨ Ai ⎢ 2 D ( κ + k2 ) + H0 + Ω i ⎩⎪ ⎣ i =1

∑

−

⎤ J n −1(κ iρ) D(κ i2 + k 2 ) + H 0 − Ω ⎥⎦

⎤ ⎫⎪ i (nφ+ kz ) Yn −1(κ iρ) , ⎬e 2 D(κ + k ) + H 0 − Ω ⎦⎥ ⎪⎭ 2 i

(31.52) and i mφ (ρ, φ, z ) = − Ms 2

3

⎧

⎡

∑κ ⎪⎨⎪⎩A ⎢⎣ D(κ i

i

i =1

+

2 i

J n +1(κ i ρ) + k2 ) + H0 + Ω

⎤ J n −1(κ iρ) D(κ i2 + k 2 ) + H 0 − Ω ⎥⎦

⎡ Yn +1(κ iρ) + Bi ⎢ 2 2 ⎣ D(κ i + k ) + H 0 + Ω +

0.1 0.2 0.3 Applied magnetic field (T)

0.4

FIGURE 31.7 Magnetic field dependence of eigenfrequencies of spin wave modes in a ferromagnetic tube with inner radius 500 nm and thickness 10 nm.

⎡ Yn +1(κ i ρ) + Bi ⎢ 2 2 D ( κ i + k ) + H0 + Ω ⎣ −

10

5

Similarly, it can be shown that the respective expressions for the radial and azimuthal components of the dynamical magnetization are given by 1 mρ (ρ, φ, z ) = Ms 2

15

⎤ ⎫⎪ i(nφ+ kz ) Yn −1(κ iρ) . ⎬e D(κ i2 + k 2 ) + H 0 − Ω ⎥⎦ ⎭⎪

the spin-wave modes is similar to that detailed in Section 31.2, and the corresponding mode profi les can be obtained when the eight coefficients are known. As an illustration, we consider a tube with an inner radius of 500 nm with a thickness of 10 nm and set k = 0. In Figure 31.7, we show the first few calculated eigenfrequencies as a function of the external field, which is applied along the axis of the tube. The lowest-frequency eigenmode is nondegenerate, while higherfrequency modes appear to be two-fold degenerate in the plot as the difference in frequencies is less than 0.03 GHz. The two closely spaced eigenfrequencies correspond to spin waves propagating on the outer and inner surfaces of the nanotube. With a fixed thickness, and as Ri → ∞, the two frequencies become identical. For an understanding of the field dependence behavior of the eigenfrequencies, we may regard the cylindrical tube as locally equivalent to a planar fi lm. This is reasonable as the thickness of the fi lm is constant and small compared to the radius of the nanotube. According to Kalinikos and Slavin (1986), the frequencies of dipole–exchange Damon–Eshbach modes propagating on a ferromagnetic thin film are given by

(31.53) The eight coefficients, A1, A 2, A3, B1, B2, B3, C1, and C2, are determined by requiring that m(r) and h(r) satisfy the boundary conditions at the inner and outer surfaces of the hollow nanowire. Four of the boundary conditions are the continuity of Φ and the continuity of the normal component of the magnetic field induction at both these surfaces. The remaining four conditions require m(r) at the inner and outer surfaces to satisfy Equations 31.20 and 31.21. The pinning parameters at these surfaces may differ, however. These boundary conditions lead to a set of eight linear homogeneous equations, and the condition for the existence of nontrivial solutions can be written as det M(Ω,κ) = 0, where M(Ω,κ) is now a 8 × 8 matrix. The numerical procedure to solve for the eigenfrequencies of

f=

{

γ ⎡ 2 Akp2 + Ms (H 0 + 2πMskp L) 2πMs ⎢⎣

{

}

}

1/2

× 2 Akp2 + Ms (H 0 − 2πMs (−2 + kp L)) ⎤⎥ , (31.54) ⎦ where kp is the magnitude of the wavevector perpendicular to the applied field L is the film thickness However, in the tube, the wavelength of the spin wave propagating around the circumference of the wire cannot assume any arbitrary value. Only an integral number of standing waves can

31-11


fit into the circumference of the tube, and the following condition must therefore be satisfied

22.5

2π n = , λ Ro

22.0

where n = 0, 1, 2, …, is the azimuthal quantum number and Ro = 510 nm. If the discrete wavevector values are inserted into Equation 31.54, we reproduce exactly the same results shown in Figure 31.7. For n = 0, Equation 31.54 reduces to the well-known Kittel formula for the uniform precession mode in which all the moments precess together in phase with the same amplitude for an external field applied parallel to the surface of the thin film. Hence, we have identified the first few lowest eigenfrequencies as those of the uniform Kittel mode and the Damon–Eshbach surface spin-wave modes of the tube.

31.4.2 Array of Hollow Cylindrical Nanowires The collective spin-wave modes in an array of hollow nanowires can also be calculated following the approach detailed in Section 31.3. The multiple scattering approach also yields Equation 31.48 with Sn (Equation 31.43) retaining the same form, except that Γn and Gn are now given respectively by 3

Γn = −

∑[A J (κ R ) + B Y (κ R )], n i n

i

n i n

o

i

o

(31.55)

i =1

and 3

Gn =

⎡

⎧

∑κ A ⎢⎣2πM ⎨⎩ D(κ i

n i

s

i =1

−

J n +1(κ i Ro ) + k2 ) + H0 + Ω

i

i =1

n i

⎡

s

⎧

Yn +1(κ i Ro ) + k2 ) + H0 + Ω

2 i

⎤ ⎫ Yn −1(κ i Ro ) − ⎬ − Yn′(κ i Ro )⎥ . 2 2 D(κ i + k ) + H 0 − Ω ⎭ ⎦ The seven coefficients, A1n , A2n , A3n , B1n , B2n , B3n , and C1, are obtained as follows. We set A1n = 1 and solve for the other six coefficients by requiring that six boundary conditions are satisfied: four from the exchange boundary conditions (at the inner and outer surfaces), and two from the continuity of both the magnetic potential and the normal component of the magnetic field induction at the inner surface of the nanowire. The frequencies of the collective spin-wave modes are then given by the solutions of Equation 31.48, and the corresponding mode profi les can be generated once the required coefficients are known. As an illustration, we consider a 2D simple cubic array of parallel Permalloy nanotubes with an inner radius of 40 nm and a thickness of 5 nm. We confine our treatment to wave propagation

21.0

20.0 19.5 M

Γ

X

M

FIGURE 31.8 Dispersion curves of a 2D square array of Permalloy nanotubes, of inner radius 40 nm and thickness 5 nm, with a lattice constant of 94 nm for an applied field H0 = 0.3 T. The band structure is plotted in the three high-symmetry directions ΓXM of the first Brillouin zone.

perpendicular to the axis of the wires, and set k = 0, H0 = 0.3 T and the interwire separation a = 94 nm. Figure 31.8 shows the band structure of this square array of nanotubes, calculated along the path M = π/a(1,1) → G = π/a(0,0) → X = π/a(1,0) → M = π/a(1,1) in the irreducible part of the first Brillouin zone. A direct band gap of width 0.75 GHz exists between the first and second bands at the M point. The mode profi le for the lowestfrequency collective spin excitation at the X point is presented in Figure 31.9.

⎤ ⎫ J n −1(κ i Ro ) ⎬ − J n′ (κ i Ro )⎥ 2 D(κ + k ) + H 0 − Ω ⎭ ⎦

∑κ B ⎢⎣2πM ⎨⎩ D(κ

21.5

20.5

2 i

3

+

2 i

Frequency (GHz)

kp =

23.0

×10–6 5 4 3 2 1 0 –1 –2 –3 –4 –5 –5 –4 –3 –2 –1

0

1

2

3

4 5 ×10–6

FIGURE 31.9 Profi le of the lowest-frequency collective spin-wave mode at the high-symmetry X point. A unit cell of a 2D square lattice containing a nanotube is shown. The unit of the spatial dimension is centimeters.

31-12

Interestingly, the square array of ferromagnetic nanotubes may be regarded as a magnonic crystal, which is the magnetic analog of photonic crystals, with spin waves acting as information carriers. A magnonic crystal can have a complete band gap within which the propagation of spin waves is prohibited in any direction inside the crystal (Krawczyk and Puszkarski 2008). Recently, Wang et al. (2009) have observed frequency band gaps in a nanostructured magnonic crystal in the form of a 1D periodic array of alternating Permalloy and cobalt nanostripes. Dispersion relations of spin waves in the magnonic crystal have been mapped by Brillouin spectroscopy, and the results obtained accord with theoretical calculations.

31.5 Summary In summary, we have discussed the dynamics of dipole–exchange spin waves in solid as well as hollow ferromagnetic nanowires of cylindrical cross-sections. By solving the linearized Landau– Lifshitz equation and the Maxwell equation, ∇ . b = 0 in a quasistatic approximation simultaneously, we obtain an important equation: Equation 31.15. In addition, by imposing the magnetostatic and the exchange boundary conditions at the surfaces of nanowires, we arrive at a system of homogeneous equations, whose determinant must vanish in order to yield the resonant frequencies of the spin-wave modes of an isolated nanowire. In a densely packed array, provided the nanowires are sufficiently far apart that interwire exchange coupling is negligible, the dynamic dipolar coupling between nanowires is treated within the multiple scattering approach. The collective spin wave excitations can be calculated for any 2D array of infi nitely long nanowires with different diameters and magnetic properties by using the expressions for the scattering matrix for each nanowire. Several examples have been provided to illustrate the various methods of calculations. We have also demonstrated, in a tutorial style approach, how frequencies of spin wave excitations and their profi les are computed.

References Abramowitz, M. and Stegun, I., eds., 1964. Handbook of Mathematical Functions. Dover Inc., New York. Aharoni, A. 1996. Introduction to the Theory of Ferromagnetism. Oxford University Press, New York. Arias, R. and D. L. Mills. 2001. Theory of spin excitations and the microwave response of cylindrical ferromagnetic nanowires. Physical Review B 63: 134439-1–134439-11. Arias, R. and D. L. Mills. 2002. Erratum: Theory of spin excitations and the microwave response of cylindrical ferromagnetic nanowires. Physical Review B 66: 149903-1. Arias, R. and D. L. Mills. 2003. Theory of collective spin waves and microwave response of ferromagnetic nanowire arrays. Physical Review B 67: 094423-1–094423-15. Arias, R. and D. L. Mills. 2004. Theory of collective spin-wave modes of interacting ferromagnetic spheres. Physical Review B 70: 104425-1–104425-7.


Arias, R., P. Chu, and D. L. Mills. 2005. Dipole exchange spin waves and microwave response of ferromagnetic spheres. Physical Review B 71: 224410-1–224410-12. Bayer, C., M. P. Kostylev, and B. Hillebrands. 2006. Spin-wave eigenmodes of an infinite thin film with peroidically modulated exchange bias field. Applied Physics Letters 88: 1125041–112504-3. Carlotti, G. and G. Gubbiotti. 1999. Brillouin scattering and magnetic excitations in layered structures. Rivista Italiana del Nuovo Cimento 22: 1–60. Chu, P. and D. L. Mills. 2007. Theory of Brillouin light scattering from ferromagnetic nanospheres. Physical Review B 73: 054405-1–054405-8. Chu, P., D. L. Mills, and R. Arias. 2006. Exchange/dipole collective spin-wave modes of ferromagnetic nanosphere arrays. Physical Review B 73: 094405-1–094405-8. Cottam, M. G. and D. J. Lockwood. 1986. Light Scattering in Magnetic Solids. Wiley, New York. Damon, R. W. and J. R. Eshbach. 1961. Magnetostatic modes of a ferromagnet slab. Journal of Physics and Chemistry of Solids 19: 308–320. Das, T. K. and M. G. Cottam. 2007. Magnetostatic modes in nanometer-sized ferromagnetic and antiferromagnetic tubes. Surface Review and Letters 14: 471–480. Demokritov, S. O. 2003. Dynamic eigen-modes in magnetic stripes and dots. Journal of Physics: Condensed Matter 15: S2575–S2598. Demokritov, S. O. and E. Tsymbal. 1994. Light scattering from spin waves in thin films and layered systems. Journal of Physics: Condensed Matter 6: 7145–7188. Demokritov, S. O. and B. Hillebrands. 1999. Inelastic light scattering in magnetic dots and wires. Journal of Magnetism and Magnetic Materials 200: 706–719. Demokritov, S. O., B. Hillebrands, and A. N. Slavin. 2001. Brillouin light scattering studies of confined spin waves: Linear and nonlinear confinement. Physics Reports 348: 441–489. Dyson, F. J. 1956. General theory of spin-wave interactions. Physical Review 102: 1217–1230. Grimsditch, M., G. K. Leaf, H. G. Kaper, D. A. Karpeev, and R. E. Camley. 2004a. Normal modes of spin excitations in magnetic nanoparticles. Physical Review B 69: 174428-1–174428-12. Grimsditch, M., L. Giovannini, F. Montoncello, F. Nizzoli, G. K. Leaf, and H. G. Kaper. 2004b. Magnetic normal modes in ferromagnetic nanoparticles: A dynamical matrix approach. Physical Review B 70: 054409-1–054409-7. Gubbiotti, G., S. Tacchi, G. Carlotti et al. 2007. Collective spin modes in monodimensional magnonic crystals consisting of dipolarly coupled nanowires. Applied Physics Letters 90: 092503-1–092503-3. Herring, C. and C. Kittel. 1951. On the theory of spin waves in ferromagnetic media. Physical Review 81: 869–880. Hicken, R. J., D. E. P. Eley, M. Gester, S. J. Gray, C. Daboo, A. J. R. Ives, and J. A. C. Bland. 1995. Brillouin light scattering studies of magnetic anisotropy in epitaxial Fe/GaAs films. Journal of Magnetism and Magnetic Materials 145: 278–292.


Hillebrands, B. 1990. Spin-wave calculations for multilayered structures. Physical Review B 41: 530–540. Hillebrands, B. 1999. In Light Scattering in Solids VII, M. Cardona and G. Güntherodt, eds., Springer, Berlin, Germany, pp. 174–269. Himpsel, F. J., J. E. Ortega, G. J. Mankey, and R. F. Willis. 1998. Magnetic nanostructures. Advances in Physics 47: 511–597. Holstein, T. and H. Primakoff. 1940. Field dependence of the intrinsic domain magnetization of a ferromagnet. Physical Review 58: 1098–1113. Jorzick, J., S. O. Demokritov, B. Hillebrands, et al. 2002. Spin wave wells in nonellipsoidal micrometer size magnetic elements. Physical Review Letters 88: 047204-1–047204-4. Kalinikos, B. A. and A. N. Slavin. 1986. Theory of dipole-exchange spin wave spectrum for ferromagnetic films with mixed exchange boundary conditions. Journal of Physics C: Solid State Physics 19: 7013–7033. Kalinikos, B. A., M. P. Kostylevi, N. V. Kozhus, and A. N. Slavin. 1990. The dipole-exchange spin wave spectrum for anisotropic ferromagnetic films with mixed exchange boundary conditions. Journal of Physics C: Condensed Matter 2: 9861–9877. Kittel, C. 2004. Introduction to Solid State Physics, 8th edn. Wiley, Hoboken, NJ. Kostylev, M. P., A. A. Stashkevich, and N. A. Sergeeva. 2004. Collective magnetostatic modes on a one-dimensional array of ferromagnetic stripes. Physical Review B 69: 0644081–064408-7. Krawczyk, M. and H. Puszkarski. 2008. Plane-wave theory of three-dimensional magnonic crystals. Physical Review B 77: 054437-1–054437-13. Liu, H. Y., Z. K. Wang, H. S. Lim et al. 2005. Magnetic-field dependence of spin waves in ordered Permalloy nanowire arrays in two dimensions. Journal of Applied Physics 98: 046103-1–046103-3.

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Martín, J. I., J. Nogués, K. Liu, J. L. Vicente, and I. K. Schuller. 2003. Ordered magnetic nanostructures: Fabrication and properties. Journal of Magnetism and Magnetic Materials 256: 449–501. Mathieu, C., J. Jorzick, A. Frank et al. 1998. Lateral quantization of spin waves in micron size magnetic wires. Physical Review Letters 81: 3968–3971. Nguyen, T. M. and M. G. Cottam. 2006. Spin-wave excitations in ferromagnetic nanotubes. Surface Science 600: 4151–4154. OOMMF package is available at http://math.nist.gov/oommf. Patton, C. E. 1984. Magnetic excitations in solids. Physics Reports 110: 251–315. Rado, G. T. and R. J. Hicken. 1988. Theory of magnetic surface anisotropy and exchange effects in the Brillouin scattering of light by magnetostatic spin waves. Journal of Applied Physics 63: 3885–3889. Scholten, P. C. 1995. Which SI? Journal of Magnetism and Magnetic Materials 149: 57–59. Skomski, R. 2003. Nanomagnetics. Journal of Physics: Condensed Matter 15: R841–R896. Wang, Z. K., M. H. Kuok, S. C. Ng et al. 2002. Spin-wave quantization in ferromagnetic nickel nanowires. Physical Review Letters 89: 027201-1–027201-3. Wang, Z. K., H. S. Lim, H. Y. Liu et al. 2005. Spinwaves in nickel nanorings of large aspect ratio. Physical Review Letters 94: 137208-1–137208-4. Wang, Z. K., H. S. Lim, L. Zhang et al. 2006. Collective spin waves in high-density two-dimensional arrays of FeCo nanowires. Nano Letters 6: 1083–1086. Wang, Z. K., V. L. Zhang, H. S. Lim et al. 2009. Observation of frequency band gaps in a one-dimensional nanostructured magnonic crystal. Applied Physics Letters 94: 0831121–083112-3.

32 Optical Antenna Effects in Semiconductor Nanowires

Jian Wu The Pennsylvania State University

Peter C. Eklund The Pennsylvania State University

32.1 Introduction ...........................................................................................................................32-1 32.2 Nanowire Synthesis and Characterization ........................................................................32-2 32.3 Optical Backscattering Experiments ..................................................................................32-3 32.4 Classical Calculations of the Elastic Light Scattering from a Cylinder ........................32-5 32.5 Modeling the Backscattered Light from Semiconductor Nanowires ............................32-6 32.6 Optical Phonons, Raman Scattering Matrices, and Manipulations ..............................32-7 32.7 Results and Discussion of Rayleigh Backscattering from Nanowires ......................... 32-10 32.8 Polarized Raman Backscattering from Semiconductor Nanowires ............................ 32-11 32.9 Summary and Conclusions ................................................................................................32-13 Acknowledgments ........................................................................................................................... 32-14 References......................................................................................................................................... 32-14

32.1 Introduction The past two decades have witnessed many rapid advances in the science and technology of submicron-scale solid state objects (Kittel 2005; Rahman 2008). It is now possible to synthesize many forms of electronic and electro-optic grade materials at the nanoscale, such as two-dimensional (2D) quantum wells (Yu and Cardona 2001) and graphene (Hashimoto et al. 2004; Novoselov et al. 2005; Zhang et al. 2005), one-dimensional (1D) single-walled carbon nanotubes (SWNTs) (Dresselhaus et al. 2001; O’Connell 2006; Popov et al. 2006) and semiconductor nanowires (SNWs) (Morales and Lieber 1998; Duan and Lieber 2000; Lu and Lieber 2007), and zero-dimensional (0D) quantum dots (Yoffe 1993, 2001, 2002). These nanostructures, to mention a few outstanding examples, provide ideal prototypes to explore the electrical, optical, thermal, and mechanical properties that originate from their dimensionality and scale. A variety of elemental, binary, and ternary compositions have been synthesized in the nanowire (NW) form and, in some cases, with diameters, d, down to a few nanometers. NWs can be prepared with lengths of ∼100’s of μm to short segments of < 300 nm, the latter by cutting a long master NW using a focused ion beam (FIB) (Wu et al. 2009). SNWs have been demonstrated to be building blocks for future nanoscale electronics (Lieber and Wang 2007; Lu and Lieber 2007), thermoelectronics (Boukai et al. 2008; Hochbaum et al. 2008), photonics (Pauzauskie and Yang 2006; Lieber and Wang 2007), and biochemical sensors (Patolsky et al. 2006a–c). Phonon confinement (Piscanec et al. 2003; Adu et al. 2005, 2006; Fukata et al. 2006) effects inside the waist of small diam− eter Si NWs (d < 10 nm) have been demonstrated by several

groups via Raman scattering. Surface optical (SO) phonons that lie between the usual transverse optic (TO) and the higher frequency longitudinal optic (LO) phonons of binary semiconductors have been observed by Raman scattering in SNWs and proposed to be activated by quasiperiodic modulation of the surface potential via diameter modulations from a growth instability and from faceting (Gupta et al. 2003; Xiong et al. 2004). Indeed, the observation of a sharp SO peak in the Raman spectrum requires a periodic spatial perturbation of the surface potential of the semiconductor. Photon confinement effects in NWs are the focus of this chapter. Considerable effort has been expended to understand photon confinement in spherical particles, e.g., their cavity resonances, the enhanced internal electromagnetic (EM) fields, the enhanced Raman scattering, and the photoluminescence emission have been the subject of extensive research in the 1990s (Chew and Wang 1982; Yamamoto and Slusher 1993; Slusher 1994; Chang and Campillo 1996). Photon confinement effects in spherical particles have also produced interesting nonlinear optical phenomena associated with the high quality factor (high Q) “whispering gallery modes” (Vahala 2003), where the photon circulates around the interior of the particle near to the surface. Application of the classical EM theory to the long line antenna gives rise to the well-known dipolar intensity pattern, I(θ) ∼ cos2θ, for both the absorbed and emitted intensity, where θ is the angle between the incoming or outgoing electric field and the antenna axis. The antenna problem was first published (for the infinitely long antenna) by Lord Rayleigh in 1918 using Bessel functions to solve the EM boundary value problem (Rayleigh 1918). The “nano” 32-1

32-2

antenna effect was first predicted by Ajiki and Ando (1996) for SWNT. By calculating the electric dipole matrix elements for this molecular wire, they predicted that the optical absorption would be strongly suppressed if the incident electric field is perpendicular to the nanotube axis. This is a manifestation of the symmetry of the electronic wavefunctions of the small-diameter 1D nanotube. Four years later, the “Raman antenna effect” for SWNTs was reported by Duesberg et al. (2000). They observed that the Raman scattering intensity from both radial and tangential displacement phonons exhibits a classic cos2θ dependence, where θ is the angle between the incident laser electric field and the nanotube axis, i.e., a dipole pattern (caution: the functional form of the experimental polar intensity plot depends on whether an analyzer is used when observing the scattered radiation). Their antenna results were found to be in apparent contradiction with the expectations, which were based on group theory (Duesberg et al. 2000; Saito et al. 2001). However, since the SWNT Raman scattering is resonant, this contradiction is to be expected, and we will discuss this in more detail below. We have reported strong antennae resonances at critical NW diameters associated with both Rayleigh (elastic) (Zhang et al. 2009) and Raman scattering (Chen et al. 2008). Strongly polarized resonant Raman backscattering has also been reported for 15–25 nm diameter WS2 nanotubes (Rafailov et al. 2005); a detailed theory connecting the backscattering with the details of the nanotube antenna effect was not presented in this work. Finally, experimental studies of polarized Raman scattering from optical phonons in tapered Si NWs has been reported and analyzed in terms of the calculated fields internal to the wire (Cao et al. 2006). An array of multiwalled carbon nanotubes (MWNT) on a substrate has demonstrated the ability to absorb light as an antenna when the length of the MWNTs in the array matches the wavelength of the light (Wang et al. 2004). Relatively little systematic work on the antenna effect from individual SNWs has been reported to date. To pin down the physical mechanisms behind the polarized scattering from these systems, studies should be made over an interesting range of materials as a function of their length, diameter, and variable photon wavelength. Indeed, the antenna effect is strongly connected with the details of the geometrical cavity resonances that can be excited by incident light with the proper wavelength. The geometric enhancement of the incident and the re-radiated fields from small particles has been known for some time (Chew and Wang 1982). It can be thought of in terms of classical EM fields (Bohren and Huffman 1983) or in terms of excitation and emission from quasi-stationary cavity modes of the nanoparticle or NW (Chang and Campillo 1996). In this chapter, we review a systematic investigation of polarized elastic (Rayleigh) and inelastic (Raman) backscattering from individual GaP NWs (Chen et al. 2008; Zhang et al. 2009). The experiments and calculations are carried out for visible laser radiation on crystalline GaP NWs with diameters in the range 40 < d < 600 nm. In the zinc-blende phase, GaP possesses a cubic lattice symmetry that greatly simplifies the analysis of the data. We first give a brief description of NW synthesis followed by a discussion of the experimental setup that we have used for polarized light scattering studies. Subsequently, we describe our theoretical model


for the polarization dependence of the elastic (Rayleigh) backscattered radiation from individual NWs via the polarizability tensor and examine the experimental results in light of this theory. Next, we proceed to the case of inelastic (Raman) backscattered light from individual wires via the Raman scattering tensor. Finally, we present concluding remarks and give some suggestions for directions for future research into NW antenna effects.

32.2 Nanowire Synthesis and Characterization Extensive literature in the past 10 years have reported conditions for the synthesis of SNWs with diameters in the range ∼4 < d < 500 nm and that grow via the vapor–liquid–solid (VLS) mechanism. The VLS mechanism was discovered in the context of micron-diameter single crystal “whisker” growth (Wagner and Ellis 1964; Givargizov 1975). In 1998, the VLS process was first used to create nano-sized whiskers or NWs (Morales and Lieber 1998). The VLS process generally involves exposing the vapors of a semiconductor (e.g., Si) to small liquid metal droplets (e.g., Au), whereby a solid NW is observed to grow from the surface of the droplet. The process, therefore, requires a sufficiently high temperature to create liquid metal or liquid metal-semiconductor alloy droplets. This liquid state is usually achieved by growing the NWs inside a quartz tube placed in a tube furnace capable of ∼1200°C. In slightly more detail, the VLS process proceeds via a steady flux of semiconductor vapors striking the droplet surface. The atoms stick and rapidly diffuse, eventually saturating the droplet. Saturation then leads to the surface precipitation of a solid that seeds the growth of the SNW. The NW grows in the steady state as long as the supply of the semiconductor remains; the droplet surface is not poisoned and the temperature remains sufficiently high to maintain the liquid droplet state consistent with a high rate of atomic diffusion. Usually, one NW grows from each particle and the diameter of the NW is approximately the same as that of the droplet. The SNWs can then be harvested from the quartz tube into lab air when the furnace is cooled. Scanning electron microscopy (SEM) and/or transmission electron microscopy (TEM) will then show that most NWs still have the “seed particle” attached to the NW. Two main approaches have been devised for the production of the semiconductor vapor (Xia et al. 2003): (1) pulsed laser vaporization (PLV) (Morales and Lieber 1998; Duan and Lieber 2000) of a semiconductor material via the ejected plume from a solid target and (2) the introduction of reactant vapors, i.e., via gases (e.g., SiH4, GeH4), or sublimation as practiced in sources used for the chemical vapor deposition (CVD) of thin films. For the same reason that pulsed laser deposition (PLD) has been found successful in growing a large variety of thin films, PLV has been similarly found successful in growing a large variety of NWs. This success exploits the general finding that the plume generated by the laser focused on a fresh target surface apparently has nearly the same stoichiometry as that of the target. PLV then provides a simple means to control the stoichiometry of the NW. The GaP NWs studied here were grown by laser vaporization of a (GaP)0.95Au0.05 target in a flowing argon gas, as shown in the

32-3

Optical Antenna Effects in Semiconductor Nanowires

Argon

Furnace

Nd:YAG

Target To vacuum (a)

Furnace

Lens

(b)

FIGURE 32.1 (a) Real image of our PLV apparatus showing the tube furnace, vacuum/gas handling components, Nd:YAG laser and optics for separating various harmonics housed inside the box on the upper right; (b) schematic of the PLV oven. Two quartz tubes are centered inside a furnace. A carrier gas (e.g., Ar) is introduced as shown, passing the preheated target down the central tube. For GaP NWs, the target containing GaP and Au was positioned just outside the furnace. The laser ablates the target to produce hot vapor that will condense into small clusters/ nanoparticles. The furnace temperature is controlled to maintain the clusters in a liquid state.

schematic illustration and image of our PLV apparatus (Figure 32.1). In the PLV method, one can also mix metal powder (e.g., Au) with the semiconductor powder in the target. The success of an Au:GaP target suggests that there is an early formation of Au droplets in the vapor plume. In this case, the NWs grow from the droplets as they drift down the quartz tube entrained in Ar gas and the semiconductor vapor. Alternatively, metal nanoparticles can be deposited on a substrate that is positioned downstream from the target inside the growth furnace. At elevated temperatures, these particles melt and form droplets that then capture the semiconductor vapor and initiate the VLS growth. In our apparatus, the target is vaporized using focused pulses from an Nd:YAG laser (Spectron Laser Systems Inc., Model SL803) outputting pulsed radiation at either a wavelength of 1.064 μm (fundamental, ∼13 ns pulse width, 10 Hz repetition rate at 850 mJ/ pulse) or at 0.532 μm (second harmonic, 10 Hz repetition rate at 320 mJ/pulse). The focal spot is scanned over the surface of the target to continually present a fresh target surface to the beam. GaP NWs grown in our PLV apparatus were harvested from the walls of the quartz tube and transferred to Si substrates or TEM grids (Ted Pella, Inc.) for study. For deposition on a Si substrate, the NWs were first dispersed in ethanol or isopropyl alcohol using mild ultrasound and then a small drop of the suspension was immediately placed on the substrate. The substrates often had lithographic markers to locate the NW position. These markers and the small diameter NWs could both be seen in a SEM or an atomic force microscopy (AFM) image. However, if the NWs were less than ∼40 nm in diameter, they could not be seen in the spectrometer microscope image. However, the optical experiments could be carried out by superimposing the SEM and optical images of the large metal markers. NWs were usually deposited on TEM grids by gently rubbing the TEM grid over a substrate containing many NWs. AFM was used to measure the NW diameter when they were supported on substrates. The diameter of NWs on TEM grids could be obtained from calibrated TEM magnification and the crystallographic growth axis was determined using selected area electron diffraction (SAD) patterns. The same NW could then be studied optically as the TEM grid pattern was indexed.

As we shall see, the experimental results are expected to be somewhat sensitive to the local NW diameter, where the excitation beam falls. One, therefore, has to be careful to notice possible fluctuations in the NW diameter along its length. Real NWs can also have rough surfaces and/or be a tapered cylinder instead of a constant diameter cylinder. For most of our NWs, we estimate that the diameter we report in connection with the optical data is accurate to ±2 nm. Figure 32.2a shows an AFM topographic image of a GaP NW on an Si substrate with lithographic markers; Figure 32.2b shows a low magnification TEM image of an individual 80 nm GaP NW protruding over a hole in a TEM grid with a SAD pattern indicating that this − particular NW grew along direction > (the NW growth axis will be indicated by double brackets in the chapter). High resolution TEM (HRTEM) images can also be used to indicate the uniformity of the NW. In this case, lattice images (Figure 32.2c) indicate where the wire is crystalline and if twinning (Xiong et al. 2006) has occurred. All of our GaP NWs examined by TEM were found to be crystalline. SAD patterns indicate that they have the cubic zinc-blende structure. Most have been observed by TEM to be single crystalline over lengths greater than several microns. We have observed several growth directions {>, >, >} for PLV-grown GaP NWs. From TEM observations, the > growth direction seem to be slightly preferred, followed closely in popularity by >, with > being significantly less likely.

32.3 Optical Backscattering Experiments Polarized Rayleigh (elastic) and Raman (inelastic) backscattering experiments were carried out using either a single-grating Renishaw in VIA micro-Raman spectrometer or a triple-grating Jobin-Yvon T64000 micro-Raman spectrometer. These spectrometers are referred to as “micro” because the incident and backscattered light pass through a microscope before entering the spectrometer. The microscope stage and objective lens carousel are shown in Figure 32.3a for the in VIA spectrometer. The focal

32-4


(a)

– 022

202 – B = [111]

– – 202 500 nm

[111]

(b)

5 μm

– 022 220 – [022] (c)

10 nm

FIGURE 32.2 (a) AFM topographic image of a GaP NW on an Si substrate with Au/Cr markers to defi ne NW positions. (b) TEM image of a −>> direction. (c) HRTEM lattice image GaP NW protruding out over a hole in a TEM grid. SAD pattern shows that this wire grew along > Q⊥. ˜ will also be a function of the position inside the In general, Q NW or nanoparticle. It can be shown, however, that the volume average of the tensor components (Q || ,Q⊥) is all that is required to compute the scattering. Values of Q || and Q⊥ could be obtained from the calculated polar plots of |E(θ)|, as shown in Figure 32.5, or they can be obtained by experiments as shown below. The scattered vector field, Es′, inside the NW is defined by the application of a scattering matrix, S˜, to the internal field, E0′ . That is, we have (32.4) Es ′ = S ⋅ E0 ′ . The NW then emits an external field given by the product of an ˜ e, and the internal scattered field, emission tensor, Q

e ⋅ Es ′ . (32.5) Es = Q ˜ (Chew and ˜e ≃ Q We also make the usual simplification, i.e., Q Wang 1982; Cao et al. 2006; Cardona and Merlin 2007). The polarization of the detected scattered electric field, ês, is parallel to the incident field, êi (this follows from the experimental conditions; Section 32.3). Then, the external scattered intensity associated with this polarization can be written as

(32.6)

NW

⋅ α I ⋅ Q

⋅ eî I 0 = ω 4α 2 (Q 2 cos 2 θ + Q⊥2 sin2 θ)2 . I Rayleigh ∝ ω 4 eˆs ⋅ Q (32.7) 2

32.6 Optical Phonons, Raman Scattering Matrices, and Manipulations In the case where there are two atoms per primitive cell (Ga and P), we require six degrees of freedom to describe the atomic displacements of these atoms. Therefore, there will be three acoustic and three optic phonon branches (Yu and Cardona 2001). At zero phonon wave vector (q = 0), two acoustic phonon modes are transverse (TA) and one is longitudinal (LA). These phonon modes describe a rigid translation of the crystal lattice. The frequency of these modes is zero because there is no restoring force. Of the three q = 0 optic branches, two are transverse (TO) and one is longitudinal (LO). The eigenvectors of the q = 0 LO and TO modes for GaP are shown in Figure 32.8. These optic modes would be degenerate without the presence of a long range EM force that induces a small splitting between them (Yu and Cardona 2001). First-order Raman scattering can occur from optical phonons when they have the proper symmetry (Hayes and Loudon 1978; Loudon 2001; Yu and Cardona 2001). For GaP (zinc blende), both the LO and TO modes are Raman-active and can be observed for specific experimental conditions that prescribe the wave vectors, the polarization of the phonon and incident and scattered photons. The frequencies of the TO and LO phonons are obtained from the Raman spectrum of an individual GaP NW, as shown in Figure 32.8. The SO phonons appear as a low frequency shoulder on the higher frequency LO peak. The so-called selection rules that determine the scattering geometry where the TO and LO q = 0 phonons can be observed are summarized by the Raman tensor (Loudon 2001; Yu and Cardona 2001) TO, LO

⋅ eˆ 2 I , I Bulk Raman ∝ ω 4 eˆs ⋅ ℜ i 0

(32.8)

where ê i and ê s are unit vectors for the incident (i) and scattered (s) fields inside the material ˜ is the Raman tensor ℜ Equation 32.8 ignores the transmission coefficients (Bohren and Huff man 1983) for the incident and scattered waves into and out

32-8


qz ~ 0 TO Mode

ˆk and ˆk , respectively, are both parallel to the xˆ axis. To comply i s 1 with the photon–phonon wave vector conservation rule (Yu and Cardona 2001), the wave vector of the optical phonon, qˆ, excited in the Raman process must also be parallel to the xˆ1 axis (the ˆj phonon wave vector is q = 2ki). We consider the case where xˆ j x′ (j = 1,2,3). Therefore, the displacement of the LO phonons must be along the xˆ1 axis and the displacement of the TO phonons must be perpendicular to the xˆ1 axis. Then from Equations 32.8 and 32.10, we have for scattering from the bulk that:

GaP TO

Intensity (a.u.)

qz ~ 0 LO Mode (z) P Ga P

qz

LO

Ga qz

320

340

360 380 400 Raman shift (cm–1)

420

440

˜ , is related of the bulk sample. The form of the Raman tensor, ℜ to the symmetry of Raman-active phonons. For semiconductors with the zinc-blende structure, the Raman tensors for optic phonons with atomic displacements along the [100], [010], and [001] crystallographic axes are 0 0 a

0⎞ a⎟⎟ 0⎟⎠

⎛0

ℜ(eˆ2 ) = ⎜⎜ 0 ⎜⎝ a

0 0 0

a⎞ 0⎟⎟ 0⎟⎠

⎛0

ℜ(eˆ3 ) = ⎜⎜ a ⎜⎝ 0

a 0 0

0⎞ 0⎟⎟ , 0⎟⎠

(32.9) where the unit vectors, êm (m = 1,2,3), represent the [100], [010], and [001] axes, respectively. We use the subscript, m, to distinguish them from the polarizations of incident and scattered light, êi and ês. The constant tensor elements a take on different values for TO and LO phonons. The forms of the Raman tensors above apply only to the case of nonresonant scattering, i.e., the incident photon must not participate in strong electronic absorption. It is interesting to compare Equation 32.8 for bulk Raman scattering and Equation 32.6 for SNWs. Scattering from SNWs introduces ˜. the matrix, Q We are primarily interested in the change in the backscattered intensity as the incident polarization is rotated by θ about the incident photon wave vector, k⃗i (c.f., Figure 32.4). Depending on the orientation of the HWP, the polarization direction for the incident, êi, and backscattered radiation, ês, is ⎛ 0 ⎞ eî = eˆs = ⎜ cos θ⎟ = eˆ. ⎜ ⎟ ⎜⎝ sin θ ⎟⎠

(32.11a)

and

FIGURE 32.8 Typical Raman spectrum from an individual GaP NW showing TO and LO frequencies at 358 and 391 cm−1, respectively. The insets present the vibrational eigenvectors for both TO and LO phonons. Ga and P atoms are indicated in the insets, respectively.

⎛0

ℜ(eˆ1 ) = ⎜⎜ 0 ⎜⎝ 0

(xˆ ) ⋅ eˆ 2 , I LO (θ) ∝ eˆ ⋅ ℜ 1

(32.10)

The experimentally collected backscattered polarization component, ês = êi, is consistent with the discussion in Section 32.3 and involves the HWP polarization rotator and analyzer. The wave vectors (k = 2π/λ) for the incident and backscattered radiation,

2

2

1 ˆ ˆ 1 ˆ ˆ I TO (θ) ∝ eˆ ⋅ ℜ (x2 ) ⋅ e + eˆ ⋅ ℜ (x3 ) ⋅ e , 2 2

(32.11b)

where the two terms in Equation 32.11b can be identified with two orthogonal TO phonons. These results predict the polarization dependence of the bulk TO and LO scattering. Next, we demonstrate how to calculate the Raman tensors of cubic GaP NWs with different growth directions and with a specific orientation, φ, about the NW axis (Figure 32.4). The scattering geometry in Figure 32.4 is defi ned by introducing two coordinate systems: the lab coordinate system (xˆ1,xˆ2,xˆ3) and the crystal coordinate system (xˆ1′ , xˆ2′ , xˆ 3′ ) attached to the NW. xˆ 2′ (xˆ2) is parallel to the NW growth axis; xˆ1′ and xˆ3′ are the other two crystallographic axes of the NW. First, we suppose that xˆ k′ =

∑α

eˆ ,

km m

xˆ k =

m

∑β

xˆ ′ ,

(32.12)

km m

m

where k,m = 1,2,3 the êm are defined as before Then, we can define two transformation matrices, T′ and T based, respectively, on the coefficients, αkm and βkm: ⎛ α11 T′ = ⎜ α 21 ⎜ ⎜⎝ α 31 ⎛ β11 T = ⎜ β21 ⎜ ⎜⎝ β31

β12 β22 β32

α12 α 22 α 32

β13 ⎞ ⎛ cos ϕ β23 ⎟ = ⎜ sin ϕ ⎟ ⎜ β33 ⎟⎠ ⎝⎜ 0

α13 ⎞ α 23 ⎟ , ⎟ α 33 ⎟⎠

− sin ϕ cos ϕ 0

(32.12a)

0⎞ 0⎟ . ⎟ 1⎠⎟

(32.12b)

The Raman tensors, ℜ(xˆ′k ), for optical phonons displaced along ˆ k can then be written as the directions, x′,

32-9


(xˆ ′ ) = ℜ k

∑α

km

ℜ(eˆm ),

(32.13)

m

˜(ê ), are given by Equation 32.9. where the Raman tensors, ℜ m Then the Raman tensors for phonon displacements in the direcˆ k } are ˆ k in the coordinate system {x′ tions, x′,

′(xˆ ′ ) = ℜ k

∑α

km

(eˆ ) ⋅ T′ −1. T′ ⋅ ℜ m

θ

(32.14)

m

TO

Similarly, we have the Raman tensors for optical phonons disˆ k and placed along the direction, xˆk, in the coordinate system, x′ xˆk, given by

′(xˆ ) = ℜ k

∑β

′(xˆ ′ ), ℜ k

(32.15a)

′(xˆ ) ⋅ T −1. T⋅ℜ k

(32.15b)

km

m

(xˆ ) = ℜ k

∑β

km

LO

FIGURE 32.9 Calculated θ-dependence of the contribution to the backscattered Raman intensity as seen at the detector for a GaP NW oriented along the ˆx2-axis and taking into account only the contribution from the Raman tensor. Results are calculated according to Equation 32.8 for angles φ between the direction and the xˆ1-axis. It is shown from the calculation that the shape of both TO and LO polar plot remains the same, independent of φ. Since the patterns do not include a contribution from Q and Q⊥, the results do not depend on the diameter of the NW.

m

Following the strategy above, we can calculate the Raman ten˜ (xˆ ), for NWs with growth axis along , , sors, ℜ k and and for specific choices of the NW rotation angle φ. The bulk scattering TO and LO intensities vs. θ (polar plots) can be calculated by applying Equations 32.11 and 32.15b. Th is ˜ , as discussed calculation leaves out the effects of the tensor, Q above in Equation 32.6. For pedagogical reasons, we nevertheless present the calculated tensor-based polar backscattering patterns as the effects assignable to the Raman tensor alone. The Raman tensor-based polar patterns are shown in Figures 32.9 through 32.11, respectively. These patterns correspond to bulk scattering for a specific orientation of the crystal axes

relative to the lab axes or they can be thought of as the result for Q⊥ = Q = 1 (no antenna effect). They are calculated for ês êi. The NW is represented schematically in Figure 32.9; it is permanently oriented along θ = 0°, where θ increases with counterclockwise rotation of the radial coordinate in the polar plot. The tensor-based polar plots for GaP NW are the simplest, as shown in Figure 32.9. The shapes of both TO and LO patterns stay the same, except for changes in intensity. For GaP NWs shown in Figure 32.10, the polar patterns associated with LO phonon scattering do not change, while the TO patterns display some variety for 0° ≤ φ ≤ 30°; the polar patterns just repeat themselves for 30° ≤ φ ≤ 60° and again in 60° ≤ φ ≤90°. Figure 32.11 shows the TO and LO polar patterns of GaP

TO

LO

= 0°

10°

20°

30°

FIGURE 32.10 Calculated θ-dependence of the contribution to the backscattered Raman intensity as seen at the detector for a GaP NW oriented along the ˆx2-axis and taking into account only the contribution from the Raman tensor. Results are calculated according to Equation 32.8 – for angles φ between the direction and the xˆ1-axis. It can be seen that the shape of the LO polar plot remains the same, independent of φ. The TO patterns change only from 0°10 μm) GaP NWs illuminated by a ∼1 μm laser spot behaves similar to the prediction for infinite wires. This agreement also

suggests that we can also use the calculated results for L = ∞ (I int /I ⊥int ) to predict values for (Q 2 / Q⊥2 ).

32.8 Polarized Raman Backscattering from Semiconductor Nanowires In Figure 32.14a and b, we display the measured integrated Raman band intensity ratio (I X /I ⊥X ) for X = TO, LO phonons vs. d (Chen et al. 2008). Data in Figure 32.14a and b were obtained, respectively, using 514.5 or 488 nm excitation. For clarity, the TO data in each case have been displaced vertically. The data shown in these panels were collected from NWs lying either on an Si (100) substrate (with 200 nm oxide) or on a TEM grid. Notice that the smallest diameter wires studied, i.e., d ∼ 45 nm, exhibit a very large anisotropy in the scattering, i.e., I||X / I ⊥X ∼ 30 for 488 nm excitation and even larger, i.e., I||X / I ⊥X ∼ 100, for 514.5 nm excitation. These peaks are examples of strong Raman antenna effects coming from photon confinement in the waist of the NW. This antenna behavior is always observed for both LO and TO phonon Raman scattering in very small diameter NWs. As we shall see, this behavior is theoretically expected for TO scattering but not anticipated for LO scattering. As we have shown in Section 32.6, the scattering matrix, S˜, ˜. Then appropriate for Raman scattering is the Raman tensor, ℜ ˜ can be written as the Raman intensity of NW including Q

160

λ = 514.5 nm TO

120

96

80

140

40

400

Intensity ratio

GaP:Rayleigh 30 20

I||/I Ratio

300

10 200

40 LO

100 140

90 150

TO

100

120

140

160

180

170

260

260

(a) 0

0 80

170

λ = 488 nm

40 LO

100

96

150

0

0

0

50

100

150 200 Diameter (nm)

250

300

FIGURE 32.13 Ratio of the external backscattered Rayleigh radiation vs. GaP NW diameter d. Light incident at 514.5 nm perpendicular to the NW axis. Data are represented by (+) and the solid curve is the calculated result for an infi nite GaP NW using the results of internal EM intensity (Figure 32.6 and Equation 32.7).

(b)

100 200 Diameter (nm)

300

FIGURE 32.14 Ratio of the observed backscattered Raman intensity (LO and TO phonon backscattering) for incident polarization parallel and perpendicular to the NW axis: (a) with 514.5 nm excitation and (b) with 488 nm excitation. Peaks in the Raman intensity ratio are associated with the antenna effect (see text). (From Chen, G. et al., Nano Lett., 8, 1341, 2008. With permission.)

32-12


NW

⋅Q

⋅ℜ

⋅ eî I 0 . I Raman ∝ ω 4 eˆs ⋅ Q

GaP (cubic): Raman backscattering

2

(32.16)

After transformations of T and T′ in Equation 32.15, the Raman tensor has the general form TO

⎛ R11

= ⎜R ℜ ⎜ 21 ⎜⎝ R31

R12 R22 R32

R13 ⎞ R23 ⎟ . ⎟ R33 ⎟⎠

Using Equation 32.16, the scattered Raman intensity for SNWs then takes the form NW I Raman ∝ (R22Q 2 cos2 θ + R23Q Q⊥ cos θ sin θ

+ R32Q Q⊥ cos θ sin θ + R33Q⊥2 sin2 θ)2

(32.17)

The values of Rij depend on the “tipping” angle φ. Equation 32.17, together with the appropriate values for the transformed Rij can be used to predict the backscattered Raman intensity pattern. In principle, values of Q and Q⊥ can be obtained from the Rayleigh backscattering data and fitting to Equation 32.7, or the values can be obtained by calculation (L = ∞). If the Raman scattering is nonresonant, the Rij are derived from the ˜ (ê ) (Equation 32.9), via a suitable coordinate transtensors, ℜ m formation by Equation 32.9 that depends on the growth direction and the particular axial rotation angle, φ. If the scattering is resonant, i.e., via the production of an exciton or via higher direct interband transitions, the form of the Raman matrix may be altogether different than the nonresonant forms found in Section 32.6. For example, a strong antenna effect was reported for WS2 nanotube where strong resonant Raman scattering was expected. In this case, the Raman polar pattern was explained in terms of a Raman tensor that has the same form as the anisotropic polarizability tensor of the WS2 nanotube (Rafailov et al. 2005). As we have discussed above, SNWs at antenna resonance exhibit the property Q >> Q⊥. In this case, Equation 32.17 can be simplified by neglecting terms involving Q⊥. The Raman intensity for SNWs at strong antenna resonance is then TO, LO

2 I NW Raman ∝ R22 Q 4 cos 4 θ, (Q Q⊥ ; any hkl ).

(32.18)

This is a very important result. It tells us that for transformed LO or TO Raman tensors that exhibit a nonzero R 22 element, the Raman intensity polar patterns for these phonons should both follow the ∼cos4θ form. In Figure 32.15, we show Raman results for two GaP NWs that exhibit a strong antenna effect for both TO and LO phonon scattering. The 50 nm wire (Figure 32.15a) was supported on an Si substrate. If this wire grows along the direction, the R 22 element for both TO and LO are nonzero and the cos4θ fit agrees with Equation 32.18. In Figure 32.15b, we show results for a slightly larger diameter d = 56 nm GaP NW on TEM grid whose SAD pattern indicates a

LO

50 nm (a)

(b)

56 nm with [111] growth direction

FIGURE 32.15 (a) Experimental data for the TO and LO polar plots for a 50 nm wire on Si substrate (unknown growth axis). Solid curves represent fits of a cos4(θ–θ0) to the data, where θ0 is added as a parameter to account for a small rotation of the NW relative to the ˆx2 axis. (b) TO and LO antenna polar plots for a 56 nm wire over a hole on a TEM grid; SAD indicates a growth axis. Data are indicated by the points, and solid curves for TO and LO are predictions according to Equation 32.16. The dashed curve in the LO polar plot is a cos4θ fitting to the data points and is not predicted by theory. For the NW in (b), the theory does not predict the observed LO pattern. This suggests that the form of the Raman tensor used to analyze the data for the 56 nm wire may be incorrect (see text).

growth direction. From the results calculated in Section 32.6, it can be shown that the transformed Raman tensor of a TO GaP NW has nonzero R22 element for all tipping angles, φ. The cos4θ fit in Figure 32.15a agrees with Equation 32.18. However, the transformed LO Raman tensor elements of a GaP LO LO LO LO ≠ 0. Therefore, NW must satisfy R22 = R23 = R32 = 0, and R33 from Equation 32.17, without any restrictions on Q and Q⊥, we have for LO phonons LO 2 4 4 I NW Raman ∝ R33Q⊥ sin θ, ( 111 ).

(32.19)

In Figure 32.15b, the dashed curve is the cos4θ fitting to LO polar pattern, which is not the theoretical prediction. The rotated solid curve is the predicted LO polar pattern as indicated in Equation 32.19 for the GaP NW with growth axis. In Figure 32.16, we display typical experimental Raman polar patterns, I(θ) (dots), for LO and TO backscattering collected from various individual GaP NWs using 514.5 nm excitation. Data in the figure (dots) are taken on wires either

32-13

Optical Antenna Effects in Semiconductor Nanowires GaP (cubic): Raman backscattering

TO

LO

80 nm with direction (a) at possible ~ 92°

133 nm with direction (b) at possible ~ 105°

265 nm on Si substrate with possible (c) at possible ~ 162°

562 nm with direction at possible (d) ~ 0°, 60°, 120°

FIGURE 32.16 Experimental polar (θ) plots (dots) of the TO (upper) and LO (lower) polarized Raman backscattering from individual GaP NWs collected with 514.5 nm excitation (dots). Solid curves calculated according to Equation 32.16 for angles φ indicated. Column (a) d = 80 nm wire on TEM grid, column (b) d = 133 nm wire on a TEM grid, column (c) d = 265 nm on a Si substrate, column (d) d = 562 nm on TEM grid. The labels found below the diameter values indicate the growth axis and tipping angle φ measured between xˆ1 and xˆ1' . See text for method to obtain Q 2 /Q⊥2 .

suspended over a hole in a TEM grid (c.f., d = 80, 133, and 562 nm) or supported on an Si substrate (d = 265 nm). As already discussed, for wires on a TEM grid, the TEM image and SAD pattern were used to determine the wire diameter and the growth axis; for NWs on Si substrates, an AFM z-scan was used to measure the diameter, and the growth direction, , is not known. The theoretical curves are calculated on the basis of Equation 32.17 using either experimental values of (Q 2/Q⊥2 ) (from Rayleigh scattering) or results calculated from the infi nite long wire (I int / I ⊥int ) = (Q 2/Q⊥2 ) . For two NWs on TEM grids, we know the from SAD. These NWs have d = 80 and 133 nm and they both grew along . Both the experimental TO and LO patterns of a d = 80 nm wire (Figure 32.16a) can be described as an “open dipole” (i.e., the patterns are dipole-like and open at the origin); the solid curve is calculated from Equation 32.17 for a wire with a Q /Q⊥ ∼ 1.77 (based on the infi nite wire calculation) and φ = 92°. For the d = 133 nm NW (Figure 32.16b), the experimental TO pattern is also an “open dipole,” while the experimental LO pattern is nearly elliptical and oriented at 90° with respect to the TO pattern. Our best model calculation (solid curve) for a wire uses Q /Q⊥ ∼ 1.24 (infi nite long wire calculation) and φ = 105°. For the d = 265 nm NWs supported on Si, we observe nearly elliptical LO and TO patterns, oriented along θ = 90°; the solid curve is for a possible wire with Q /Q⊥ ∼ 0.89 (value from Rayleigh scattering on a wire with same diameter) and φ = 162°. Finally, for the d = 562 nm GaP NW, the solid curve is for Q /Q⊥ ∼ 0.96 (from Rayleigh scattering) and φ ∼ 0°, 60°, or 120°.

32.9 Summary and Conclusions Using GaP as a model NW system, we have presented results that indicate how polarized photons “squeeze” into individual NWs and how their antenna behavior can be observed. GaP (zinc blende) exhibits a cubic space lattice. This symmetry simplifies considerably the analysis of the data. We have endeavored to show that when the wavelength of the incident light can drive a strong antenna resonance, the NW responds like a classical line antenna. For fi xed wavelength and photons incident at right angles to the NW axis, a series of antenna resonances are predicted and observed. Rayleigh fi rst predicted these resonances in 1918, but he probably was thinking about wires and wavelengths at a scale much larger than the tens to hundreds of nanometers applicable to the antennae we have considered here. During an antenna resonance, only the component of the incident and scattered E-field that is parallel to the NW determines the scattering from the wire. In our theoretical picture, this behavior is introduced via a diagonal ten˜ ) that describes the transmission of the EM fields into and sor (Q out of the NW that are polarized parallel () and perpendicular (⊥) to the NW axis. We have shown that Rayleigh scattering via the polarizability tensor can be used to provide an experimental value for the ratio (Q /Q⊥). This ratio can be obtained theoretically and provides a means of validating the model calculations. For conditions driving a strong antenna resonance, we have shown that Q >> Q⊥; for conditions far away from resonance, we observe that Q ∼ Q⊥. Large values of this ratio were then shown to distort the Raman backscattering polar pattern from

32-14

what one observes in the bulk solid. However, this distortion is ˜ and the Raman selection rules that apply for the just due to Q bulk still apply to the NW. As matters now stand, the theoretical apparatus that we have used to explain the experimental results is doing a reasonably good job, except in the case of the smallest diameter NWs. There, we find trouble with the LO phonon backscattering pattern. Our best explanation (at this time) is that these NWs may have intragap optical adsorption that may invalidate the use of the nonresonant form of the Raman tensors. We are currently doing experiments to investigate this explanation.

Acknowledgments This work was supported, in part, by the National Science Foundation grapheme NIRT program and the Materials Research Institute at Pennsylvania State University (PSU). The authors would like to thank Professor D. H. Werner, Dr. Gugang Chen, Dr. Qihua Xiong, Dr. M. E. Pellen, and J.S. Petko from PSU for their early, important contributions to this work. We are also indebted to Qiujie Lu and Duming Zhang for the helpful discussions as well as for having shared preliminary results of their work on Rayleigh scattering from NWs. We also gratefully acknowledge Dr. H. Gutierrez for his help with the NW growth and the TEM characterization of the GaP NWs.

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33 Theory of Quantum Ballistic Transport in Nanowire Cross-Junctions 33.1 Introduction ...........................................................................................................................33-1 33.2 Theory and Model..................................................................................................................33-3 Transmission Function of a Cross-Junction • Green’s Function • Tight-Binding Models • Numerical Calculation of Green’s Function • Transverse Modes • Projected Green’s Function

33.3 Conductances .......................................................................................................................33-12 Conductances of Square Wire Junctions • Conductances of a Circular Wire Junction • Energies and Probability Densities of Quasi-Bound States

Kwok Sum Chan City University of Hong Kong

33.4 Summary and Future Perspective .....................................................................................33-16 Acknowledgment.............................................................................................................................33-17 References.........................................................................................................................................33-17

33.1 Introduction There is a widespread and intense research interest in the fabrication, characterization, and manipulation of semiconductor nanowires (Duan and Lieber 2000, Duan et al. 2000, Lee et al. 2001, Zhang et al. 2001, Bjork et al. 2002), with the aim to develop the novel “bottom up” approach to microelectronic circuit fabrication. In this approach, nanodevices and nanocircuits are assembled from the bottom up, using nano-size building blocks, such as carbon nanotubes (Rao et al. 1997, 2003, Tans et al. 1998, Liang et al. 2001) or semiconductor nanowires. This is a paradigm shift from the traditional “top down” approach, in which nano-size devices are etched into a bulk material using lithographic techniques (Kalliakos et al. 2007, Perez-Martinez et al. 2007). As it is now easy to produce nano-building blocks in large numbers with relatively inexpensive equipment, current research focus is also on developing techniques for manipulating the nano-building blocks (Whang et al. 2003a,b, Pauzauskie et al. 2006, Jamshidi et al. 2008), novel device structures (Cui and Lieber 2001, Duan et al. 2001, Huang et al. 2001b), and circuit architectures (Dehon 2003, Dehon et al. 2003, Zhong et al. 2003) that suit the “bottom up” approach. Recently, fabrication of nanowire devices using the “bottom up” approach has been demonstrated experimentally (Cui and Lieber 2001, Duan et al. 2001, Huang et al. 2001b). In these experiments, nanowires were put on top of nanowires so that electrical contacts could be formed between some of the wires.

These contacts were then used to define or form the devices and circuits. The device structures of these novel nanodevices are different from the traditional ones fabricated using the “top down” approach as nano-size junctions play an important role in determining the device characteristics. In combination with some nanowire manipulation techniques, such as the solutionbased Langmuir–Boldgett (LB) fi lm technique (Whang et al. 2003b), the new device structure enables the fabrication of arrays or circuits of nanowire devices. For example, nanowires can be aligned to form a nanowire array, which is then rotated along a desirable direction and put on top of another nanowire array to form a nanowire grid structure (Huang et al. 2001a). It is possible to stack several layers of nanowires with this technique and construct 3D electronic circuits in the stack of nanowires. In Figure 33.1, we show schematically how a p–n junction is constructed from two nanowires by putting an n-type wire on top of a p-type wire. Figure 33.2 shows the scanning electron micrograph of a silicon nanowire cross-junction and the I–V characteristics of p–n, p–p, and n–n nanowire junctions obtained by Lieber’s group in Harvard University (Cui and Lieber 2001). The nanowire cross-junction was formed by a sequential deposition of solutions of n- and p-type nanowires, and contacts to the nanowires were defined by electron beam lithography. The rectifying behavior is clearly seen in the I–V characteristics, while approximately linear I–V behavior is found in p–p and n–n junctions. Similarly, nanowire p–n–p bipolar junction transistors can be made from three nanowires (2 p-type wires and 1 n-type wire) 33-1

33-2


Scattering region

FIGURE 33.1 The schematic diagram of a nanowire p–n junction (cross-junction) formed by putting an n-type nanowire (light grey) on top of a p-type nanowire (dark grey). As an illustration, a dashed rectangle is also shown surrounding the scattering region of a quantum transport model. The wires outside the dashed rectangle are leads connected to the scattering region in the model.

400 Current (nA)

2

1

3

4

200 0 –200 –400

(a)

(b)

–4

–2 0 2 Voltage (V)

4

Current (nA)

Current (nA)

40 50 0 –50

20 0 –20 –40

(c)

–4 –2 0 2 Voltage (V)

4 (d)

–1.0 –0.5 0.0 0.5 1.0 Voltage (V)

FIGURE 33.2 (a) shows the electron micrograph of a silicon nanowire cross-junction. (b), (c), and (d) show the I–V characteristics of silicon nanowire cross-junctions. One of the I–V characteristics shown in (b) shows the rectifying behavior of a p–n junction; others resemble Ohmic junctions. (Reprinted from Cui, Y. and Lieber, C.M., Science, 291, 851, 2001. With permission.)

as shown schematically in Figure 33.3. The experimental demonstration of nanowire bipolar junction transistors as well as of an inverter circuit using crossed silicon nanowires has also been reported (Cui and Lieber 2001). Nanowire logic circuits and nanowire computation were also demonstrated by Lieber’s research group (Huang et al. 2001b). Figure 33.4 shows the scanning electron micrographs of the three

FIGURE 33.3 The schematic diagram of a nanowire p–n–p bipolar junction transistor formed from 2 p-type nanowires (dark grey) and one n-type nanowire (light grey).

logic gates, an OR gate, an AND gate, and a NOR gate, constructed from two nanowire p–n junctions. In the OR gate, three nanowires, two p-doped and one n-doped, are used to form two p–n junctions. In both the AND gate and the NOR gate, only two p–n junctions are formed from three of the four wires used in the circuit. The fourth wire is used as a gate to modify the resistivity of one of the nanowires to form a resistor in the circuit. Figure 33.5 shows a scanning electron micrograph of a nanowire field effect transistor decoder consisting of eight nanowires studied by Zhong et al. (2003). Four nanowires are used as inputs to control the other four output wires. The diagonal cross-junctions are specially modified so that the input wires I1, I2, I3 and I4 can be used to control the output wires O1, O2, O3 and O4. From these experimental results, it is quite clear that nanowire cross-junctions play a key role in nanowire devices and circuits; an understanding of the transport properties of nanowire cross-junctions is crucial to the development of nanowire technology. There are a number of theoretical studies (Büttiker 1986, Ravenhall et al. 1989, Gaididei et al. 1992) of quantum wire (or quantum waveguide) cross-junctions in the literature, but most of these studies considered quantum wires lying on the same plane and cannot be applied to nanowire junctions with wires lying on different planes. The exceptions are the studies by Takagaki and Ploog (1993) and the author’s group (Wei and Chan 2005, Chan and Wei 2007). Takagaki and Ploog (1993) used a continuum model and the wave-matching method to fi nd the transmission functions between two square nanowires lying on different planes. It is difficult to apply this approach to circular wire junctions with different interwire coupling strengths. We have adopted a tight-binding model approach that can overcome all these shortcomings. It is, therefore, useful to present in this chapter a tutorial review of the theory and physics of quantum ballistic transport in nanowire cross-junctions for researchers with different backgrounds. We have included in some subsections background material that is not suitable for a journal article but useful for beginners in the field, such as the Green’s function technique and the quantum transport theory;

33-3

Theory of Quantum Ballistic Transport in Nanowire Cross-Junctions Vc1

Vi1 Vi2

Vo

R

Vc1 Vi1

Vi2

Vi1 V0

p

n p Vi1 V0 Vi2 Vc2 Silicon oxide

Vo

Vo

n Vi1 Vi2 Silicon oxide

Vi2

OR

(a)

Vo

Vc2 Vi1 Vi2

Vc1

AND

(b) Vc1

Vo Vc1

Vi1

G Vi1

Vi2

Vc2

n

Vo

Vi1 Vi2

Vc2

Vc1

Vi2 Vo R

p NOR

Silicon oxide (c)

FIGURE 33.4 Electron micrographs and schematic diagrams of nanowire logic gate circuits. (a) OR gate. (b) AND gate. (c) NOR gate. (Reprinted from Huang, Y. et al., Science, 291, 630, 2001a. With permission.)

33.2.1 Transmission Function of a Cross-Junction

O1 I2 O2

I1

I3 O3

O4

I4

FIGURE 33.5 Electron micrograph of a nanowire decoder formed from eight nanowires. I1, I2, I3, and I4 are input wires. O1, O2, O3, and O4 are output wires. (Reprinted from Zhong, Z. et al., Science, 302, 1377, 2003. With permission.)

so, beginning researchers can readily learn how to model crossjunctions from this chapter. Experienced researchers, however, may want to skip these subsections and jump to those that deal directly with issues pertaining to nanowire cross-junctions.

33.2 Theory and Model In this section, the background theory and the tight-binding models for determining the quantum ballistic transmission coefficients of an electron traveling through a nanowire crossjunction are discussed.

In nanowire devices or circuits built from cross-junctions, the transmission of an electron between wires is determined by the quantum scattering of the electron by the cross-junction as phase coherence is maintained when the electron travels through the crossjunction. The cross-junction is regarded as the scattering region (surrounded by the dotted rectangle in Figure 33.1) in a quantum transport model, which scatters electrons incident from one of the nanowires connected to the junction into the other nanowires. The nanowires connected to the cross-junction are usually referred to as the “terminals” or “leads” in the language of quantum transport theory. In the cross-junction shown in Figure 33.1, there are four terminals or leads connected to the scattering region in the center. The electron transmission probability from one terminal into the other terminals, or back to the incoming terminal, is given by the scattering wave function. The scattering wave function in terminal, β, for an incident electron in mode, n, of terminal, α, is given by ψ(xβ , yβ , z β ) =

∑S

φ (xβ , yβ )exp(ikmz β )

β, m; α , n m

m

+ δ αβ φn (x α , y α )exp(−ikn z α ),

(33.1)

where (xβ , y β , z β) are local coordinates defined for terminal, β, as in Figure 33.6 ϕm(xβ, yβ) and ϕn(xα, yα) are wave functions of the transverse modes (subbands), m and n, of terminals, β and α, respectively Here, different x–y–z coordinates are defi ned for the terminals and they are distinguished by the Greek letter subscripts.

33-4


where vm(n) is the electron velocity in mode, m(n) → ρβ(α) = x β(α)î + y β(α) ˆj is a position vector in the transverse plane of lead, β(α) Aβ(α) is the cross-sectional area of the lead

xβ

R Gβα (E , ρβ , zβ , ρα , z α ) is the retarded Green’s function of the juncˆ →), by tion at energy, E, related to the junction’s Hamiltonian, H(r

zβ

yβ

⎡ E − Hˆ (r )⎤ G R (E, r , r ′) = δ(r − r ′). ⎣ ⎦

(33.5)

→→

FIGURE 33.6 Schematic diagram of a nanowire cross-junction showing the coordinate axis used in the present chapter.

S β,m;α,n(E) is the scattering matrix (in short, the S-matrix) relating the incident and outgoing wavefunctions. It depends on the incident energy, E, and is related to the transmission function, Tβα (E), by Tβα (E) =

∑

2

m, n

Sβ,m; α ,n (E) . The transmission function

can be used to calculate the current at lead, β, in a biased junction with different Fermi energies in the leads according to Iβ =

∫hT 2e

βα

(E )[ f β (E ) − f α (E )]dE ,

(33.2)

where fβ(α ) (E) = 1 / (exp((E − EFβ(α ) ) / kT ) + 1) is the Fermi–Dirac distribution for lead, β(α), with the Fermi energy, EFβ(α ) . Here, we assume that an external bias voltage produces different Fermi energies in the leads without any modification of the potential profi le of the scattering region. In the linear response regime, the bias voltage in the junction, which equals −(E Fβ − EFα )/e, is small. The current can be approximated by Iβ =

2e ⎛ ∂f ⎞ Tβα (E ) ⎜ − α ⎟ dE × (EFβ − EFα ). ⎝ ∂E ⎠ h

∫

(33.3)

At low temperatures, the Fermi–Dirac distribution resembles a step function, f α (E) ≈ θ(EFα − E) ; only electrons at the Fermi energy contribute to the current because the derivative of the Fermi–Dirac distribution sharply peaks at the Fermi energy, ∂f − α ≈ δ(EFα − E). If the transmission function has a weak ∂E dependence on energy around the Fermi energy, the conductance of the junction between leads, α and β, is approximately given 2e 2 by Gβα (EFα ) = Tβα (EFα ) . h

33.2.2 Green’s Function The S-matrix is related to the retarded Green’s function of the junction by Sβ,m; α ,n (E ) = −δ βa + i vmvn R φ*m (ρβ )Gβα (E , ρβ , z β , ρα , z α )φn (ρα )d 2ρβd 2ρα , ×

∫∫

Aβ Aα

(33.4)

The retarded Green’s function, GR(E,r ,r ′), is a Green’s function of the time-independent Schrodinger equation, which can be used to find the stationary scattered wave solution of the timeindependent Schrodinger equation (Schiff 1968). As a consequence, the S-matrix can be expressed in terms of the retarded Green’s function as in Equation 33.4. It is difficult to fi nd a simple analytical expression for the Green’s function of a structure with complex geometry. Nevertheless, the following approaches can be used to fi nd the Green’s function numerically. If the eigen→ functions, ψE′ ( r ), of the Hamiltonian are available, the Green’s function can be obtained by using the expression, G R (E , r , r ′) = (ψ E*′ (r )ψ E ′ (r ′))/(E − E ′ + iγ) , where iγ (γ > 0) E′ is a small imaginary part. To obtain the advanced Green’s function, γ < 0 is used. Both the retarded and advanced Green’s functions are Green’s functions of the time-independent Schrodinger equation, which can be used to find scattered wave solutions of the Schrodinger equation, but they correspond to imposing different boundary conditions on the solutions. The retarded Green’s function corresponds to an outgoing wave solution, while the advanced Green’s function corresponds to an incoming wave solution. For the present purpose, it is sufficient to consider the retarded Green’s function only. For a system which has a perturbation, Vˆ, the Green’s function can be found by solving the Dyson equation

∑

G R (E , r , r ′) = G 0R (E , r , r ′) +

∫∫ G (E, r , r ″)Σ(E, r ″, r ″′)G (E, r ″′, r ′)d r ″d r ″′. R 0

R

3

3

(33.6)

where G0R is the Green’s function of the unperturbed Hamiltonian, Hˆ0 = Hˆ −Vˆ → → ∑(E, r ″, r ′″) is the self energy due to the perturbation, Vˆ The wave functions or the Green’s function, which can be used to determine the observable properties of a physical system, are continuous functions of the spatial coordinates in continuum models. However, to determine numerically the physical observables, knowing the wave function or the Green’s function at discrete points in space is already sufficient, provided that the separations between the discrete points are small enough to have the desired accuracy. Moreover, the task of finding the Green’s

33-5

Theory of Quantum Ballistic Transport in Nanowire Cross-Junctions

function can be simplified by discretizing the coordinates, which converts the Hamiltonian operator and the Green’s function into matrices and operator equations into matrix equations. The matrix equation for the Green’s function after discretization, which can be solved by matrix inversion, is

∑ ⎡⎣E − H (l, k) + iγ ⎤⎦ G (E, k, j) = δ R

l, j

(33.7)

k

or, in the matrix notation,

R

⎤G

⎡(E + iγ) I − H ⎣ ⎦ (E) = I .

(33.8)

Here, l, j, k are indexes used to denote the discrete points in space and I˜ represents the identity matrix. The small imaginary part, iγ (γ > 0), is added to the matrix equation to obtain the retarded Green’s function. Dyson’s equation is also a matrix equation instead of an integral equation in discrete coordinates,

R (E ) = G

0R (E) + G

0R (E)Σ

(E)G

R (E). G For finite-size systems, the Green’s functions are finite matrices; ˜ so they can be found from the inversion of the matrix, (E + iγ)I˜ − H R −1 ˜ ˜ ˜ as G (E) = [(E + iγ)I − H] , or from the following equation derived

R (E ) = ( I − G

0R ( E )Σ

(E ))−1G

0R ( E ). from Dyson’s equation: G However, a nanowire cross-junction is an open system, which is infinite in size, and hence is not suitable for a straightforward application of these two equations. For an infinitely large structure, the problem of finding the Green’s function is simplified by using a tight-binding Hamiltonian and dividing the infinite structure into a scattering region and leads connected to the scattering region. With this approach, a finite-size matrix equation for the Green’s function, which is similar in structure to Equation 33.8 given above, can be derived from Dyson’s equation ˜ s]G ˜ s is the modified Hamiltonian ˜ R(E) = I˜; here, H as [(E + iγ)I˜ − H of the scattering region, which includes the effect of the leads as self-energies. The Green’s function can then be easily found by ˜ s. In Section 33.2.4, we inverting the finite matrix (E + iγ)I˜ − H describe in detail how the modified Hamiltonian is derived using Dyson’s equation for tight-binding models. When the dimensions of the wires are large, the number of tight-binding sites in the scattering region is large and the inversion of the matrix, ˜ s, is computationally slow because the size of the (E + iγ)I˜ − H matrix is large. For this situation, we have developed a modular approach based on Dyson’s equation, which reduces the sizes of the matrices inverted in computation. In Section 33.2.4, this modular approach will be described in detail.

as self-energy functions in the tight-binding formulation. Before we discuss in detail how the Green’s function is calculated numerically in Section 33.2.4, it is necessary to describe the tight-binding models of nanowire junctions. Two tight-binding models are discussed; one is suitable for wires with polygonal cross section, and the other is suitable for wires with circular cross section. The simple tight-binding (TB) models, which are widely used to approximate continuum Hamiltonians, consist of a number of tight-binding sites (or artificial atoms), each of which has a single electron s-orbital that can accommodate one electron if spin is ignored. Electrons in the s-orbitals can hop to the s-orbitals in neighboring TB sites. The spatial arrangement of the TB sites depends on the geometry or boundary of the system considered. For example, the TB sites of a circular wire are arranged in concentric circles, which is different from square or rectangular wires to approximate the circular boundary. The simplest TB model has the TB sites occupying the lattice points of a simple cubic lattice (Figure 33.7), and all the hopping couplings between neighboring sites are identical. The TB sites are denoted by three integral indices, l, m, and n. The TB Hamiltonian is written in terms of the creation, al+,m,n, and annihilation, al,m,n, operators of the s-orbital as Hˆ =

∑ea

+ l , m, n l , m, n

l , m, n

a

+

∑ ta

+ < l , m, n > l , m, n

a

(33.9)

l , m, n < l , m, n >

where denotes the neighboring TB sites of site (l,m,n) e and t are respectively the orbital energy and the hopping matrix element between neighboring orbitals This TB Hamiltonian can be obtained by replacing differential operators in the continuum kinetic energy opera− 2 2 2 tor, ((∂ /∂x ) + (∂2/∂y 2 ) + (∂2/∂z 2 )), by finite difference 2m

33.2.3 Tight-Binding Models As pointed out above, the calculation of the Green’s function of a finite region in a nanowire cross-junction is simplified by using a tight-binding Hamiltonian for the junction, because the Green’s function for the fi nite region can be obtained by inverting a finite-size matrix as in a finite-size system. The effect of the infinite region outside the finite region can be included

FIGURE 33.7 Tight-binding sites in a simple cubic tight-binding model.

33-6

Handbook of Nanophysics: Nanotubes and Nanowires x

z y

4

3

2

1

FIGURE 33.8 Tight-binding sites in a square nanowire crossjunction. Each square wire has a 2 × 2 cross section. 1, 2, 3, and 4 are labels of the leads.

∂f ≈ ( f l +1 − f l ) / Δ; here, Δ is the dis∂x tance between two neighboring TB sites, t = −ħ 2/(2mΔ2) and e = −6ħ 2/(2mΔ2). In Figure 33.8, a schematic diagram is shown of a cross-junction formed by 2 square wires (each has a 2 × 2 square cross section). The dotted lines joining the two nanowires represent the hopping coupling between the two wires. In reality, there is a thin layer of oxide separating the two wires in a junction and the probability of electron tunneling between the two nanowires is determined by the thickness of the oxide layer. Here, we use the simple model of hopping coupling between the two wires to represent the effect of the oxide layer. The strength of the hopping coupling can be varied to mimic the effect of oxide thickness. When the hopping coupling between two wires equals the hopping coupling between sites within the same wire, it is the strong coupling regime, in which there is no oxide layer hindering electron transfer between wires, and the hopping of electrons between the two wires is as easy as hopping within the same wire. The total Hamiltonian of the nanowire junction is Ηâ + Ηˆc + Ηˆb. ˆ Ηa is the Hamiltonian of the upper wire with a+(a) representing the creation (annihilation) operator, and Ηˆb is the Hamiltonian of the lower wire with b+(b) representing the creation (annihilation) operator. Ηˆc is the coupling operator between two wires approximations such as

given by Hˆ c = t c

∑

q = Nc q =1

aq+bq + bq+ aq , where tc is the interwire

coupling constant, and q(Nc) is the index (number) of the peripheral TB sites in the wires that participate in the interwire coupling. The simple cubic TB model is not suitable for modeling a circular wire, although one can use a potential to confi ne the electron in a circular region in the center as in Figure 33.9, where sites inside the grey circle have lower potential energy than those

FIGURE 33.9 Schematic diagram showing how a simple cubic tightbinding model can be used to approximate a nanowire with circular cross section. Tight-binding sites within the grey circle have lower site energy, leading to electron confinement within the grey circle. As the number of the tight-binding sites is small, it is an approximation of a polygonal cross section.

outside the circle. These central lattice points resemble a circular region only when the number of sites is very large. When the number of sites is small, the cross section defined by the grey circle resembles a polygon, which still has the fourfold symmetry of a square. In actual calculation, it is impractical to use a large number of TB sites to approximate a circular cross section, as computation time required is long. To develop an accurate TB model with a small number of sites for circular wires, we need to use cylindrical coordinates and arrange the TB sites in concentric circular rings (a nonCartesian lattice) as in Figure 33.10. To ensure the Hermitian property of the TB model defined on a non-Cartesian lattice, the TB Hamiltonian is derived by a variant approach, in which the energy of an electron in a circular wire defi ned on a circular grid is minimized. The total energy of an electron expressed in cylindrical coordinates is Etotal =

∫∫∫

ψ * (ρ, φ, z )

− 2 ⎡ 1 ∂ ∂ 1 ∂2 ∂ ⎤ ρ + 2 2 + 2⎥ ⎢ 2m ⎣ ρ ∂ρ ∂ρ ρ ∂φ ∂z ⎦

ψ(ρ, φ, z )ρ dρ dφ dz.

(33.10)

When the electron wave function, ψ, is confined to a wire of diameter, D, it satisfies the boundary condition, ψ(ρ = D,ϕ,z) = 0. Before minimizing the total energy, the energy expression is rewritten in a more symmetrical form as

Etotal =

2 2m

∫∫∫

⎧⎪⎛ ∂ψ * ⎞ ⎛ ∂ψ ⎞ 1 ⎛ ∂ψ * ⎞ ⎛ ∂ψ ⎞ ⎛ ∂ψ * ⎞ ⎛ ∂ψ ⎞ ⎫⎪ ⎨⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎬ ⎟+ 2⎜ ⎟ +⎜ ⎪⎩⎝ ∂ρ ⎠ ⎝ ∂ρ ⎠ ρ ⎝ ∂φ ⎠ ⎝ ∂φ ⎠ ⎝ ∂z ⎠ ⎝ ∂z ⎠ ⎪⎭

ρ dρ dφ d z .

(33.11)

33-7

Theory of Quantum Ballistic Transport in Nanowire Cross-Junctions x

solving (∂Etotal / ∂ψ * (i, j, k)) − ζρn ΔρΔφΔz ψ(i, j, k) = 0, where ζ is the Lagrange multiplier. The matrix eigenequation thus obtained is

y

2 ⎧⎪ 1 ⎨ 2m ⎪⎩ Δρ2

⎡ ρi +1/2 ⎤ ρi −1/2 ⎢ − ρ ψ(i + 1, j, k) + 2ψ(i, j, k) − ρ ψ(i − 1, j, k)⎥ i i ⎣ ⎦

+

1 ⎡ −ψ(i, j + 1, k) + 2ψ(i, j, k) − ψ(i, j − 1, k)⎤⎦ ρi2 Δφ2 ⎣

+

1 ⎫ ⎡ −ψ(i, j, k + 1) + 2ψ(i, j, k) − ψ(i, j, k − 1)⎤⎦ ⎬ = ζψ(i, j, k). Δz 2 ⎣ ⎭ (33.13)

Converting the matrix eigenequation into the TB format, the following TB model is obtained Nρ

Hˆ circular = FIGURE 33.10 Tight-binding sites in the cross section for modeling a circular nanowire.

Nφ

∑ ∑ ∑ (e k

+t

i =1

z k

+ eiρ + e φj )ai+, j , k ai , j , k + t iρ,i −1ai+, j , k ai −1, j , k

j =1

ρ + i ,i +1 i , j , k i +1, j , k

a

a

+ t φj , j−1 ai+, j , k ai , j −1, k + t φj , j +1ai+, j , k ai , j +1, k

+ t z (ai+, j, k ai , j, k −1 + ai+, j, k ai , j, k +1 ).

(33.14)

To avoid the singular term, (∝ (1/ρ )) in the Laplacian opera2

tor, the discrete coordinates are chosen to be (ρi , φ j , z k ) = ⎛⎡ 1⎤ ⎞ ⎜⎝ ⎢i + ⎥ Δρ, jΔφ, k Δz ⎟⎠ . i, j = 0, 1, 2, 3…, and k = 0, ±1, ±2, ±3… ⎣ 2⎦ are the integral indices, and Δρ, Δϕ, and Δz are the grid sizes. By replacing the differential operators by their finite difference expressions and integration by discrete summation, the approximate expression of the total energy is

where ai+, j , k (ai , j , k ) is the creation (annihilation) operator of the orbital at the TB site, i, j, k, of the circular grid. The TB parameters are given by tz =

tiρ,i ±1 = ΔρΔφΔz 2m 2

Etotal =

⎡∑ ρ ⎣

i +1/2

i, j,k

(ψ * (i + 1, j, k) − ψ * (i, j, k))(ψ(i + 1, j, k) − ψ(i, j, k)) Δρ2

+

1 (ψ * (i, j + 1, k) − ψ * (i, j, k))(ψ(i, j + 1, k) − ψ(i, j, k)) ρi Δφ2

+

(ψ * (i, j, k + 1) − ψ * (i, j, k))(ψ(i, j, k + 1) − ψ(i, j, k)) . Δz 2

⎤⎦

Note that the approximation of ∂ψ / ∂ρ is taken at ρi+1/2 = (i + 1) Δρ in the expression to make the TB model Hermetian. The eigenvalues and the eigenfunctions of the Hamiltonian are given by minimizing the energy of the electron subjected to the nor-

∫∫∫ ρ dρ dφ dz ψ

2

(33.15)

− 2 ρi ±1/ 2 , 2mΔρ2 ρi ±1ρi

(33.16)

2 , 2mρi2 Δφ2

e φj =

eiρ =

2 , mΔρ2

2 , mρi2 Δφ2

(33.17)

As in square wire junctions, the total Hamiltonian of the crossjunction is the sum of three parts, ΗˆL + ΗÛ + ΗˆT, where ΗˆL , ΗÛ, and ΗˆT are, respectively, the lower wire Hamiltonian, the upper wire Hamiltonian, and a hopping coupling between the two wires. The coupling term is similar to that in the square wire (33.12)

malization condition,

t φj , j ±1 = −

− 2 2 , ekz = −2t z = , 2 2mΔz mΔz 2

= 1, or, equivalently,

junction, H T = t C

∑

p=Q p =1

(a +p bp + bp+ a p ), where a and b denote

annihilation operators of the two wires of the junction, p denotes the indices of sites coupled by HT, and Q is the number of sites in each wire participating in the interwire coupling. Figure 33.11 shows schematically how the TB sites are coupled through the interwire coupling. In the figure, four out of the sixteen TB sites in the circumference are coupled to another wire through ΗˆT.

33-8


Lead: nanowire

Scattering region of junction

FIGURE 33.12 Schematic diagram showing how a lead is connected to the scattering region.

FIGURE 33.11 Tight-binding sites participate in interwire coupling in the cross section of a circular wire cross-junction.

33.2.4 Numerical Calculation of Green’s Function The nanowire junction is an open system with no boundary, which means the TB Hamiltonian is an infinite matrix. It is not possible to find the Green’s function by simply inverting the matrix, ˜ , as no numerical packages can handle infinite matrices. EI˜ − H The trick to circumvent this problem is to partition the system into a scattering region and the leads (the region outside the scattering region is the leads, which is also referred to as contacts), so that a finite-size, effective Hamiltonian matrix can be constructed for the scattering region, which can then be used to calculate the Green’s function. The derivation of the effective Hamiltonian (Datta 1995) is as follows. First, the Hamiltonian is partitioned into two parts: the Hamiltonian for the scattering region, ΗˆS, and the Hamiltonian for the leads, ΗˆL . The coupling between the two regions isVˆC. The equation for the Green function (Equation 33.8) is rewritten in terms of these sub-Hamiltonians or sub-matrices as

S ⎡ E + iγ − H ⎢ + V C ⎣⎢

S ⎤ ⎡ I ⎤ ⎡G = ⎥ ⎢

⎥ ⎢

L ⎥ ⎢G E + iγ − H ⎦ ⎣ C ⎦⎥ ⎣⎢0 V C

0 ⎤ .

I ⎥⎥ ⎦

(33.18)

˜ C from the equations, one can get Eliminating G

S − V C

S ]G [E + i γ − H

I

+

L ) VC GS = I (E + i γ − H

(33.19)

or

S = I

S − Σ

]G [E + i γ − H

(33.20)

LV C+ being the self-energy

L ))V C+ = V CG with Σ = V C ( I /( E + iγ − H

L = ( I /(E + i γ − H

L )) is the due to the coupling to the leads. G ˜ S is given by the inverse of Green’s function of the leads. Now, G

S = [ E + i γ − H

S − Σ

]−1 . In a tight-binding a finite-size matrix, G ∼ model, the coupling VC couples the neighboring atomic sites on the surfaces of the lead and scattering region as shown schematically in Figure 33.12. So, we only need to know the Green’s functions on the surfaces of the leads, i.e., the surface Green’s functions of the leads. If the transverse and the longitudinal motions are separable in the leads, the surface Green’s function of a lead can be written in terms of the transverse mode wave functions as GL (l, j) =

∑

m

e ikma φ*m (l )φm ( j), where l and j denote the atomic

sites on the surface of the lead, m denotes the transverse mode of the lead, a is the lattice spacing of the TB model, and km is the wave vector of mode, m. km is related to energy, E, by E = Em + 2t(1 − cos kma), where Em is the mth subband edge energy and t is the hopping strength along the longitudinal direction of the lead. To approximate the continuum, the number of TB sites in the model cannot be too small. Usually, a transverse dimension of 8–10 TB sites is sufficient for a good approximation of the continuum in an energy range covering the fi rst six subbands. If the size of the scattering region is very large, for example, in wires with large diameters or scattering regions with more than one junction, the calculation of the Green’s function as the inverse of the effective Hamiltonian matrix is not computationally efficient. This problem can be circumvented using a modular approach that considers the junction as the coupling of two free-standing wires, and determines the junction’s Green’s function in terms of the free-standing wires’ Green’s function using Dyson’s equation. For example, to find the scattering matrix between terminals, 1 and 3, shown in Figure 33.13, we need to find the Green’s function between the two cross sections, 1 and 3, denoted as G(1,3). This Green’s function can be written in terms of the free-standing wire’s Green’s function by solving the following Dyson’s equations: G(1,3) = G 0 (1, P )VPQG(Q,3),

(33.21)

G(Q,3) = G 0 (Q,3) + G 0 (Q, Q)VQPG(P ,3),

(33.22)

33-9


sites, because the sizes of the matrices used are the number of TB sites in either the cross sections or the interfaces between wires. Moreover, the Green’s function of a free-standing wire is computationally easy to find (no inversion of large matrices is needed) because it can be expressed in terms of the transverse mode eigenfunctions and the Green’s function of a one-dimensional TB chain as follows:

2

Q

1

G 0 (E , l1 , x1 , l2 , x 2 ) =

P

∑ φ* (l )G α 1

1D

(E − Eα , x1 , x 2 )φα (l2 ).

(33.29)

α

G1D can be obtained from the Green’s function of a semi-infinite TB chain GS1D (E ) = ⎡⎢ −(1 − E / 2t ) ± (1 − E / 2t )2 − 1 ⎤⎥ by the fol⎣ ⎦ lowing relation, which can be derived by using Dyson’s equation: 3

G1D (E , x1 , x2 ) =

FIGURE 33.13 Schematic diagram showing the labeling of the cross sections and surfaces in a cross-junction used in Equations 33.21 through 33.28.

G(P ,3) = G 0 (P , P )VPQG(Q,3),

(33.23)

where G 0 is the Green’s function of a free-standing wire VPQ is the tunneling coupling between the two wires through the surfaces, P and Q Equation 33.23 can be substituted into Equation 33.22 to fi nd G(Q,3), which can then be substituted into Equation 33.21 to obtain G(1,3) as G(1,3) = G 0 (1, P )VPQ (1 − G 0 (Q, Q)VQPG 0 (P , P )VPQ )−1G 0 (Q,3). (33.24) To find the Green’s function between cross sections 1 and 2, G(1,2), we can use Dyson’s equations shown below: G(1,2) = G 0 (1,2) + G 0 (1, P )VPQG(Q,2),

(33.25)

G(Q,2) = G 0 (Q, Q)VQPG(P ,2),

(33.26)

G(P ,2) = G 0 (P ,2) + G 0 (P , P )VPQG(Q,2).

(33.27)

Substituting Equation 33.27 into Equation 33.26, we can obtain G(Q,2), which can be substituted into Equation 33.25 to obtain G(1,2) as G(1,2) = G 0 (1,2) + G 0 (1, P )VPQ[I − G 0 (Q, Q)VQPG 0 (P , P )VPQ ]−1 G 0 (Q, Q)VQPG 0 (P ,2).

(33.28)

This approach is very efficient for calculating the Green’s functions of scattering regions consisting of a large number of TB

GS1D (−tGS1D ) x1 − x2 . 1 − t (GS1D )2

(33.30)

33.2.5 Transverse Modes The transport properties of a nanowire junction depend on the coupling of the electron wave functions of the constituting wires. The transverse mode probability densities or mode profi les have a strong effect on this coupling. As bound and quasi-bound states, which lead to dips and peaks in the junction conductances, are formed at the junction by this coupling, knowledge of the transverse modes of a nanowire is crucial to the analysis of the features in the conductances. In this section, a discussion is given on the transverse modes, which forms the basis for the discussion of junction conductances in the following sections. For a square or rectangular wire, the transverse x and y-direction wave functions are separable. A transverse mode is denoted by (α,β), where α is the quantum number for the y-direction wave function and β is that for the x-direction wave function, and the wave function is φαβ (l , j) =

2sin[lβπ /(N + 1)]sin[ jαπ /(M + 1)] , N +1 M +1

(33.31)

where l and j are, respectively, the site indices along the x and y directions N and M are the total number of sites along these two directions Here, the x-direction is perpendicular to both wires in the junction, and the y-direction is parallel to one of the wires in the junction. The energies and the symmetry properties of the fi rst six transverse modes are given in Table 33.1 for a rectangular wire. Note that the first and the fourth transverse modes are nondegenerate while the other transverse modes have degenerate modes in a square wire with N = M. In Figure 33.14, the transverse mode profiles of the first six modes of a square wire are shown schematically.

33-10


and an angular part, Θk, as Φ i , j, k = Π i Λ jΘ k / ρ, which satisfies respectively the following equations:

TABLE 33.1 The First Six Transverse Modes of an N × M Rectangular Wire and Their Subband Energies and Symmetry Properties Mode (1,1) (1,2) (2,1) (2,2) (3,1) (1,3)

Subband Energy E1 = 4t − 2t cos[π/(N + 1)] − 2t cos[π/(M + 1)] E2 = 4t − 2t cos[2π/(N + 1)] − 2t cos[π/(M + 1)] E3 = 4t − 2t cos[π/(N + 1)] − 2t cos[2π/(M + 1)] E4 = 4t − 2t cos[2π/(N + 1)] − 2t cos[2π/(M + 1)] E5 = 4t − 2t cos[π/(N + 1)] − 2t cos[3π/(M + 1)] E6 = 4t − 2t cos[3π/(N + 1)] − 2t cos[π/(M + 1)]

Symmetry about X-Axis +

Symmetry about Y-Axis +

2 [Θ j −1 − 2Θ j + Θ j +1] = ωΘ j , 2m Δφ2

+

−

−

+

⎞ ω 2 ⎛ ρi −1/2 ρ − Λ[ω ],i −1 + 2Λ[ω ],i − i +1/2 Λ[ω],i +1 ⎟ − 2 Λ[ω],i 2 ⎜ 2mΔρ ⎝ ρi −1ρi ρiρi +1 ⎠ ρi

−

−

+

+

+

+

= E(ω)Λ[ω ],i ,

(33.33)

− 2 (Π k −1 − 2Π k + Π k +1 ) = EΠ k , 2mΔz 2

(33.34)

ω is a constant introduced in the separation of the radial and angular parts of the discretized Schrodinger equation and is related to the angular momentum through Lz = 2mω . For a full circle with Nϕ TB sites, the angular eigenfunction and eigenvalue for angular quantum number, n, are Θ[n],i = ⎣⎡1/ N φ Δφ ⎦⎤

Mode (1,1)

(33.32)

x

Mode (1,2)

y

Mode (2,1)

Mode (2,2)

Mode (1,3)

Mode (3,1)

FIGURE 33.14 The transverse mode profi les of the lowest six subbands of a square nanowire.

The Schrodinger equation of a circular wire is separable into the longitudinal, angular, and radial parts, when expressed in cylindrical coordinates. Similarly, the TB eigenfunctions are also separable into a longitudinal part, ∏i, a radial part, Λj,

ωn =

1/ 2

(

exp[in2πj / N φ ],

)

2 ⎡⎣cos 2nπ /N φ − 1⎤⎦ . (mΔφ2 )

(33.35)

(33.36)

The radial function, Λi, has nρ nodes, where nρ = 0, 1, 2,… (the position, ρ = 0, is not included in the counting of nodes), and the transverse modes are denoted by [nρ,n]. The two transverse modes, [nρ,+n] and [nρ,−n], are degenerate; so, they can both be denoted by [nρ,|n|], where |n| is the absolute value of n. To approximate the continuum, it is sufficient to use a model with Nρ = 4 and Nϕ = 20, which has 8 TB sites along the cross-section diameter. The eigenenergies of the lowest six transverse modes of the wire are given in Table 33.2. The first and sixth subbands have rotational symmetry for nρ = 0 and, as a consequence, are symmetric with respect to reflection about the x–z plane shown. Any linear combination of the second and the third subbands, which are degenerate, is also an eigenfunction. So, the linear combination can be either symmetric or antisymmetric with respect to reflection about the x–z plane. This way of symmetrizing or antisymmetrizing the eigenfunctions can also be applied to the degenerate third and fourth subbands. In general, symmetrized wave functions have high electron density at sites involved in interwire coupling and therefore have stronger coupling to the other wire. Antisymmetrized modes have a node and thus a lower electron density near the coupling sites; so, they have weaker interwire coupling. In Figure 33.15, the transverse mode profiles of the first six modes of a circular wire are shown schematically.

33.2.6 Projected Green’s Function The coupling between two nanowires in a cross-junction lowers the electron energy and causes the formation of bound and quasibound states in the junction. The bound and quasi-bound state

33-11


TABLE 33.2 Subband Energies, Mode Indexes and Mode Labels of the Transverse Mode Wave Functions of the Lowest Six Subbands of a Circular Wire. Energy Is Expressed in Unit of tz Subband 1st 2nd 3rd 4th 5th 6th

Subband Energy (in Unit of tz)

Number of Node along Radial Direction nρ

0.2777 0.6985 0.6985 1.2296 1.2296 1.3200

0 0 0 0 0 1

Angular Quantum Number n

Mode Label [nρ,|n|]

0 ±1 ±1 ±2 ±2 0

[0,0] [0,1] [0,1] [0,2] [0,2] [1,0]

Source: From Chan, K.S. and Wei, J.H., Phys. Rev. B, 75, 125310, 2007. With permission.

x Mode [0, 0]

Mode [0,1] antisymmetrized

y

lower subbands and consequently a quasi-bound state is formed. Prominent features, such as peaks and dips, in the conductances of a nanojunction can be explained in terms of the quasi-bound states; so, it is useful for analyzing the nanojunction conductance to have a means of finding out how subbands are coupled in a quasi-bound state. Chan and Wei (2007) have devised a mathematical tool called the projected Green’s function for identifying the main subband components of a quasi-bound state. In this section, we discuss the definition of the projected Green’s function and how it can be used to analyze quasi-bound states. The projected Green’s function for modes, m and n, of leads, β and α, at longitudinal positions, zβ and zα, of a junction are defined as GP (m, n, zβ , z α ) =

∫∫ φ* (ρ )G m

Aβ Aα

Mode [0,1] symmetrized

Mode [0,2] symmetrized

Mode [0,2] antisymmetrized

Mode [1,0]

FIGURE 33.15 The transverse mode profi les of the lowest six subbands of a circular nanowire.

wave functions are formed by mixing the electron wave functions of the two wires. When the two lowest subbands of the two wires are coupled together in a junction, they form a bound state. However, if two higher subbands are coupled in the junction, the bound state formed can be coupled to the unbound states of some

β

R βα

(E, ρβ , zβ , ρα , z α )φn (ρα )d 2ρβd 2ρα . (33.37)

This defi nition applies to both cylindrical and square wires. Actually, the defi nition of the projected Green’s function is related to the scattering matrix. The projected Green’s function is obtained from the scattering matrix by removing the first term, −δαβ , and the factor, i vmvn , in the second term from Equation 33.4. If the subbands, ϕm and ϕn, are components of the quasi-bound state, rapidly changing features can be found near the quasi-bound state energy. Chan and Wei (2007) have given a detailed mathematical analysis of the projected Green’s c + id function. They proved that GP (m, n, zβ , z α ) = , where E − E B + iΓ m and n are subbands with subband-edge energy higher than the quasi-bound state energy and c + id = φm χ χ φn , where χ is the bound state wave function formed from the m and n subbands by ignoring the coupling to the lower subbands. As χ is a bound state, it is localized around the junction. A quasi-bound state is formed from χ when χ is coupled to some lower subbands. The quasi-bound state formed is an unbound state, which gives rise to a resonance feature in the transmission or reflection coefficients. When the quasi-bound state has a long enough lifetime (or a small enough Γ) and ϕm and ϕn are important components of the bound state, χ, the complex number, c + id, is not small c + id and the function, , has rapidly changing features in E − EB + iΓ the real and imaginary parts at energy near EB. These features

33-12


can be readily identified in the graph and used to confirm that ϕm and ϕn, are important components of the quasi-bound state. Chan and Wei (2007) have shown several examples of how to use the projected Green’s function to analyze the quasi-bound states in a cross-junction. The imaginary part of the projected Green’s function, (cΓ /((E − EB )2 + Γ 2 )) + (d(E − EB ) / ((E − EB )2 + Γ 2 )), is not a Lorentzian peak as it is not a diagonal matrix element of the Green’s function. Nevertheless, the projected Green’s function can be symmetrized so that the imaginary part is a Lorentzian peak according to GP (m, n, zβ , z α ) + GP (n, m, z α , zβ ) c = . 2 E − E B + iΓ

5

8 × 8 Square t = tc = 0.67to

4 3 2

(1,1)

Peak

(3,1) (2,2) (1,3) (2,1) (1,2) Peak

1 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

5 4

(33.38)

8 × 8 Square t = 0.67to, tc = 0.7t

3 Peak 2

Peak

33.3 Conductances In this section, the conductances of nano-junctions formed from square and circular wires are presented and discussed. In particular, the relationship between the features (dips and peaks) in the conductances and the quasi-bound states formed at the junction are analyzed in detail.

Conductance (2e2 / h)

1 0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

5 4 3

8 × 8 Square t = 0.67to, tc = 0.5t G12 G13

2 Peak

33.3.1 Conductances of Square Wire Junctions In Figure 33.16, which is reprinted from Chan and Wei (2007), the interwire (G13) and intrawire (G12) conductances of an 8 × 8 square wire cross-junctions for several interwire coupling strengths are shown. In this figure, the energy is expressed in unit of t o = ( 2 /2ma2 ), the hopping constant of a cubic lattice TB model. In this energy unit, the hopping constant of the model has the numerical value of 1 in the calculation. However, the results shown in the figure are obtained by using a numerical value of 0.67 for t. So, in SI unit, t equals 0.67to or t = 0.67 ( 2/2ma 2 ) = ( 2/2m(1.22a)2 ) , which is equivalent to enlarging the lattice spacing to a′ = 1.22a. For the 8 × 8 square wire considered, the transverse dimension is 9a′ = 9 × 1.22a ≈ 11a. Although the TB model has 8 TB sites along the transverse direction, the dimension is 11a, which is equivalent to 10 TB sites along the transverse direction with a lattice spacing of a. In the adoption of TB models to approximate the continuum, there is always the question of how small the lattice spacing, a, should be for a desirable degree of accuracy. Th is question has been studied by Chan and Wei (2007) by comparing the conductances of models with different numbers of TB sites (including 4 × 4, 6 × 6, 8 × 8, and 10 × 10). It was found that an 8 × 8 model can already give good quantitative agreement with a 10 × 10 model for the lowest 6 subbands. So, although the number of sites in the transverse dimension is 8, the use of t = 0.67 allows the model to mimic a 10 × 10 square wire. The use of a smaller number of TB sites can reduce the computational time required for calculating the Green’s function by matrix inversion. When the interwire coupling is strong (here, strong coupling means the interwire hopping strength equals the intrawire hopping strength, tc = t = 0.67t0), the intrawire conductance (G12)

0.9

1 0 0.1

Peak 0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0.7

0.8

0.9

5 4 3

8 × 8 Square t = 0.67to tc = 0.3t

2 1 0 0.1

Peak Peak 0.2

0.3

0.4

0.5

0.6

Energy/to

FIGURE 33.16 Intrawire (G12) and interwire (G13) conductances of an 8 × 8 square wire cross-junction. (Reprinted from Chan, K.S. and Wei, J.H., Phys. Rev. B, 75, 125310, 2007. With permission.)

near the subband edges (at E ≈ 0.16to, 0.39to, 0.63to, 0.74to) is suppressed. The intrawire conductance increases with electron incident energy measured from the subband edges. This means, near the subband edges, there is a strong scattering of the incident electron by the junction. There are dips in the intrawire conductance just below the subband edges, which are due to the quasibound states formed below the subband edges. Bound states and quasi-bound states are formed at a junction between two wires, because the junction allows an electron to move from one wire to another wire and lowers its energy. Firstly, bound states, which are localized states, are formed from the states of the two wires by the interwire coupling. If not forbidden by symmetry, these bound states may have strong enough coupling to the unbound states of lower subbands (those

33-13


with subband edges at lower energies) and form delocalized quasi-bound states. For the lowest subband (1,1), the bound states formed cannot form quasi-bound states because there is no lower subband providing any unbound states for coupling. The dip and peak features in the intrawire and interwire conductances depend on the nature of the quasi-bound states. To explain the relationship, we fi rst classify the intrawire conductance dips into two groups: Group A are dips in the intrawire conductance associated with corresponding peaks in the interwire conductance at the same energy, and Group B are dips in the intrawire conductance not associated with peaks in the interwire conductance. Quasi-bound states giving rise to Group A dips are delocalized in both wires; so, they can enhance interwire transmission and give rise to peaks in the interwire conductances at the same energy. The dip in the intrawire transmission is caused by this increase in interwire transmission due to the quasibound state. Quasi-bound states giving rise to Group B dips are localized in one of the two wires; they do not form transmission channels between wires and do not lead to features in the interwire conductances. Nevertheless, these quasi-bound states provide alternative channels for electron transmission between the two terminals of the same wire (here terminals 1 and 2). The electron waves going through the two transmission channels (one through the quasi-bound state and one not through the quasi-bound state) can interfere destructively, leading to transmission dips in intrawire transmission. Similar phenomena have been investigated by Shao et al. (1994) in quantum waveguides. The characteristics of the quasi-bound states depend on the symmetry of the transverse modes involved. For example, below the second and third subbands (they are subbands with transverse modes, (1,2) and (2,1), hereafter we call them subbands (1,2) and (2,1)), the bound state formed from subbands (1,2) of both wires, has the same symmetry as the (1,1) transverse modes of both wires. So it can couple to the subbands (1,1) of the two wires and form quasi-bound states delocalized in both wires (Group A). On the other hand, the bound state formed from the coupling of subbands (1,2) to (2,1) can only couple to the subband (1,1) of one wire and the quasi-bound state formed is delocalized in one wire and localized in another wire (Group B). The Group A quasi-bound state from (1,2) × (1,2) forms a conduction channel and enhances electron transmission between two wires. So, there is a peak in G13 and a dip in G12 at energy around 0.32. Th is symmetry argument can be strictly applied to the second and third subbands because states of subband (1,1) are the only continuum states needed to be considered. For higher quasi-bound states, symmetry is not the only factor considered as continuum states of different symmetries are present; in these cases, coupling strength also plays an important role in determining the quasi-bound state characteristics. In Table 33.3, we give a summary of the conductance features and the related quasi-bound states for tc = to, which can be explained in terms of the two groups of quasi-bound states discussed above. As to the features at E ≈ 0.6 and 0.74 in Table 33.3, there are some subtleties, which require further discussion. The dip around E ≈ 0.6 actually consists of two dips from two quasi-bound states.

TABLE 33.3 Features in the Conductances of a Square Wire Cross-Junction (tc = to) Energy

G12

G13

0.32 0.38 0.6

Dip Dip Dip (two close dips)

Peak No feature No feature

0.65 0.74

Dip Dip

Peak No feature

Coupling of Subbands in the Quasi-Bound State (1,2) × (1,2) (1,2) × (2,1) (2,2) × (2,2) (2,2) × {(1,1)+(1,2)} (1,3) × (1,3) (1,3) × (3,1)

The subband (2,2) of one wire can form a quasi-bound state with the subband (2,2) of another wire at energy around 0.627. This (2,2) subband could form another quasi-bound state with the (1,1) and (1,2) subbands of another wire at energy around 0.61. The (2,2) × (2,2) quasi-bound state is very close to the subband edge and the interwire transmission peak is probably very close to the subband edge and cannot be resolved from the subband edge. The second quasi-bound state could be delocalized in both wires as they can interact with continuum states with the appropriate symmetry. However, the interwire transmission peak is not very clear and this quasi-bound state has a Group B behavior. The dip in G12 around energy 0.741 comes from the quasibound state formed from the mixing of (1,3) and (3,1) subbands. Symmetry does not preclude the formation of a quasi-bound state delocalized in both wires and a transmission peak between the two wires. However, no clear interwire transmission peak can be identified, indicating a strong Group B behavior. The interwire conductance, G13, is high when the electron incident energy is near a subband edge and G13 decreases with increasing energy. When tc decreases, G13 at high incident energy decreases and the peaks shift toward the subband edges so that significant interwire transmission can only be found very close to the subband edges. For example, when tc = 0.3t, the interwire conductance looks like an symmetric peak located at the subband edge.

33.3.2 Conductances of a Circular Wire Junction In this section, we discuss the conductances of a circular wire cross-junction. As the conductance features are determined by the symmetry of the transverse mode profi les, there are some similar features between the square and circular wire junctions. So, a comparison with square wire junctions will be made to highlight the similarities. The conductances of a circular wire cross-junction are shown in Figure 33.17 with tc = 0.6, 0.8, 1.0 (reprinted from Chan and Wei 2007). In general, the interwire conductance between two circular wires is weaker than that between two square wires because circular wires have smaller contact area in between. So, there is a weaker suppression of intrawire transmission near the subband edges. The transverse mode profi les of the degenerate second and third subbands (modes [0,1]) resemble the mode profi les of the (1,2) and (2,1) modes of the square wire. So, the nearby peak and dip structure

33-14


tc (0.8 and 1.0), the peak and dip are not clear because, probably, the interwire coupling is so strong that the dip and peak features are significantly distorted.

6 5 4

Circular cross-section tc = 0.6

3

G12 G13

2

33.3.3 Energies and Probability Densities of Quasi-Bound States

1 0 0.2

0.4

0.6

0.8

1.0

1.2

1.4

6 Conductance (2e2/h)

5 4

tc = 0.8

To understand the characteristics of the conductance features of the junction, it is important to understand various factors that affect the quasi-bound state energy, which can be found from the local density of states obtained from the imaginary part of the Green’s function. The local density of states at site J at energy E is defined as

3

ρ( J , E ) = Im[G( J , J , E )] =

2 1

0.4

0.6

0.8

1.0

1.2

1.4

5 4 3

tc = 1

k

k

1 0 0.4

ρ(J,E) is the probability density of states with energy, E, at a particular site, J. Since the quasi-bound states of the junction have higher probability densities in the scattering region of the junction, we can use the total probability density in the scattering region (hereafter referred to as the scattering region probability density) to find the quasi-bound state energies. The scattering region probability density is ρS (E) =

2

0.2

k

k

0 0.2

∑ ψ *(J )ψ (J )δ(E − E ).

0.6

0.8 1.0 Energy/tz

1.2

1.4

FIGURE 33.17 Intrawire (G12) and interwire (G13) conductances of a circular wire cross-junction. (Reprinted from Chan, K.S. and Wei, J.H., Phys. Rev. B, 75, 125310, 2007. With permission.)

should resemble that below the second and third subbands in the square wire junction. However, we can only find one dip and one peak, with the second dip missing. This difference is due to the fact that the second quasi-bound state has a smaller binding energy in the circular wire junction and thus the dip cannot be resolved from the subband edge. The mode profi les of the symmetrizied mode of the fourth and fi ft h degenerate subbands (modes [0,2]) resemble the mode profi le of (1,2) or (1,3) of a square wire because the TB sites participating in the interwire coupling concentrate in a small peripheral region in the wire. The electron probability distribution in this small region resembles that of mode (1,2) or (1,3) of a square wire. The antisymmetrized [0,2] mode resembles the (2,2) or (2,1) modes. As a result, the peak and dip structure just below the fourth and fi ft h subbands also resembles the peak and dip structure below the second and third subbands of the square wire junction. For the subband with the edge at around E = 1.3to, the mode profi le is symmetrical with respect to the x–z plane. So, the quasi-bound state formed resembles that of the symmetrized mode of [0,1] or the square wire (1,2) mode. A dip in G12 and a peak in G13 are expected. In Figure 33.17, a dip in G12 and a small peak in G13 are identified when tc = 0.6 is used. For larger

∑ ρ(J , E), where J is a site J

in the scattering region. Here, the scattering region in a junction is defined as sections of the two wires that have some peripheral sites coupling to the other wires. For example, in an 8 × 8 square wire junction, the scattering region is an 8 × 8 × 16 rectangular prism, consisting of 8 layers of TB sites of each wire. For a circular wire junction, when the number of peripheral sites coupling to the other wire is 4, the scattering region consists of 4 layers of TB sites from each of the two wires. In Figure 33.18, the layers of TB sites in the scattering region of a square wire junction and a circular wire junction are shown schematically. The square wire junction has 4 TB layers, so the transverse dimension is 4 × 4. For the circular wire junction, it is usually assumed that 1/5 of the peripheral sites participate in the interwire coupling; so, 4 sites couple to another wire in a ring of 20 sites. At the quasi-bound state energies we expect to find peaks of the scattering region probability density in a plot as a function of the incident energy. The quasi-bound state energies are usually defined as the energy positions of the peaks. In Figure 33.19, we show an example of the plot of total probability density versus incident energy. The peaks in the scattering region probability density are indicated by arrows. In the work of Chan and Wei (2007), factors that determine the quasi-bound state energies were investigated. It was found that the quasi-bound state energy decreases with increase in the transverse dimension of the wire. Th is means that stronger confi nement in the wire and larger separation between subbands lead to stronger electron binding in the junction region. Th is trend is similar to that found in the exciton-binding energy of quantum wells and wires: decrease in the confi nement dimensions leads to increase in binding energy. The only exception

33-15


FIGURE 33.18 Layers of tight-binding sites in the scattering regions of a square wire junction and a circular wire junction. The square wire has a 4 × 4 cross section, so there are 4 layers in each wire. In the circular wire, 4 tight-binding sites participate in interwire coupling, so there are 4 layers of TB sites in each wire.

Probability density (a.u.)

250

200

150

100 Total probability density in the scattering region 8 × 8 square wire junction tc = t = 0.67t0

50

0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Energy (t0)

FIGURE 33.19 Total probability density in the scattering region plotted as a function of the electron incident energy.

is the quasi-bound state formed from subband [1,0] in a circular wire junction. In this case, the quasi-bound state energy is very small and is strongly affected by the coupling between the bound state of the [1,0] subbands and the unbound states of the subbands lying below. When the diameter of the circular wire is increased, the subband separation is reduced, resulting in an increase in the coupling between the bound state and unbound states. As a result, the quasi-bound state energy is increased, because the decrease in the quasi-bound state energy due to lower quantum confi nement in larger wires is not large enough to cancel the effect. The quasi-bound state energy also depends on the transverse mode profi le along the x- and y-directions in square wire junctions (along the x- and y-directions in the upper wire, along the x- and z-directions in the lower wire). When the node number along the x-direction increases, the quasi-bound state energy increases. For example, the quasi-bound state, (1,3) × (1,3), has an energy about three times the energy of (1,1) × (1,1) and 20%

higher than that of (1,2) × (1,2). The reason is that the electron probability density at the coupling sites is higher in mode, (1,3), than in other two modes. On the other hand, the quasi-bound state energy decreases with increase in the node number along the y-direction. For example, the quasi-bound state, (2,2) × (2,2), has an energy that is about a thousand times smaller than that of (1,2) × (1,2). The reason is that in subband, (2,2), states near the subband edge with k ≈ 0 have weak interwire coupling. Owing to the transverse mode profi le, more effective coupling is to higher states with k ≈ 2π/L. As a result, the quasi-bound state (2,2) × (2,2) energy is much smaller than that of (1,2) × (1,2). For the same reason, the quasi-bound state, (2,1) × (1,2), also has a smaller energy, which is about one-fi ft h of the energy of (1,2) × (1,2). In circular wire junctions, angular momentum is an important factor that determines the quasi-bound state energy as higher angular momentum pushes the electron toward the boundary of the wire and enhances interwire coupling. For example, the quasibound state, [0,0] × [0,0], has an energy of about one-seventh of the quasi-bound state, [0,1]s × [0,1]s. Nevertheless, the quasi-bound state, [0,2]s × [0,2]s, has energy close to [0,1]s × [0,1]s. The difference is only a few percent. This is the result of the transverse mode distribution in the [0,2]s mode, in which the electron distribution is divided into four sections, while there are two sections in the [0,1] s mode, which have higher probability density at sites coupled to the other wire in the junction. The probability densities of some bound and quasi-bound states have also been obtained and studied by Chan and Wei (2007) using the Green’s function. For quasi-bound states, the probability density is given by the local density of states (LDOS), ρ( J , E ) = Im[G( J , J , E )] =

∑ ψ *(J )ψ (J )δ(E − E ). However, for k

k

k

k

a bound state below the lowest subband edge, the imaginary part of the retarded Green’s function is zero when the small imaginary number, δi, is set to be zero. To find the probability density of a bound state, the small imaginary part, δi, is not set to be zero so that a nonzero imaginary part is obtained. When the energy is close to the bound state energy, EB(|E − EB | < δ), and δ is small,

33-16

the contribution to the Green’s function from the unbound states can be ignored and the Green’s function is approximately given by G( J , J ′, E ) = ( f B* ( J ) f B ( J ′))兾(E − EB + iδ). The probability density of the bound state, |f B(J)|2, can be obtained from the Green’s function as |f B(J)|2 = iδG(J,J,EB). Chan and Wei (2007) have investigated the probability densities of a bound state and two quasi-bound states in a squarewire junction. They have plotted the electron probability densities on a cross section of a square wire junction (the cross section considered is shown schematically in Figure 33.20). The bound state studied is one formed from the two (1,1) subbands of the wires. When the coupling is strong (tc = 1), it was found that a peak of electron density is found at the interface between the two wires. Significant density of electron is found around the interface region. When the interwire coupling is reduced, the peak in the probability density is split into two, with each peak located in one wire. In the quasi-bound state formed by two (1,2) subbands, there are three peaks of electron density in the cross section and they are of the same magnitude when the interwire coupling is strong. The middle density peak centers around the interface. Without the interwire coupling, there should be four peaks in the electron density in the cross section according to the mode density profi le shown in Figure 33.14. The LDOS cannot be used to determine the probability of all quasi-bound states because some quasi-bound states have degenerate states which cannot be distinguished by this approach. When a quasi-bound state is formed from two subbands with different transverse modes of the two wires, there is a degenerate quasi-bound state with the same energy. For example, a degenerate state, formed from (1,2) × (2,1) exists for the quasi-bound state, (2,1) × (1,2). In the LDOS approach, these two states are not distinguished and the probability density found is the sum of these two states. One can make use of the symmetry properties of the quasi-bound states to distinguish them. For example, one of the two states is symmetrical with respect to the x–z plane and antisymmetrical with respect

FIGURE 33.20 Cross section of a junction considered by Chan and Wei (2007) for plotting the electron probability density.


to the y–z plane, while the degenerate quasi-bound state has a different symmetry, antisymmetrical with respect to the x–z plane and symmetrical with respect to the y–z plane. It is possible to use a projection operator to distinguish these two states. The projection operator is an operator which can be used to project out wave functions with the desired symmetry as follows: Pˆ ψ = ψ , if ψ has the symmetry of Pˆ Pˆ ψ = 0, if ψ does not have the symmetry of Pˆ For example, the projection operator for wave functions symmetric with respect to reflection about the x–z plane is defi ned as Pˆsψ = (ψ + ψ′)/2, where ψ′ is obtained from ψ by reflection about the x–z plane. The projection operator for antisymmetric wave functions is defi ned as Pâψ = (ψ − ψ′)/2. We can then use the appropriate projection operator to project the desired wave functions in the Green’s function as ˆ ˆ= G ′ = PGP (Pˆ ψ *k Pˆ ψ k )/(E − Ek + iδ). G′ consists of only k wave functions with the desired symmetry and so can be used to fi nd the probability density of the quasi-bound state with the desired symmetry. Chan and Wei (2007) have used this approach to fi nd the probability density of the quasi-bound state, (2,1) × (1,2), the probability density profi le resembles the mode, (2,1), in the upper wire, but with the density peaks shifted significantly toward the interface.

∑

33.4 Summary and Future Perspective Th is chapter gives a tutorial review of the theory of quantum ballistic transport in nanowire cross-junctions. The basic theory required for the determination of the charge conductance of cross nanowire junctions are presented and discussed. The features of the inter and intrawire conductances of cross-junctions are analyzed and explained in terms of the symmetry properties of the quasi-bound states formed at the junctions. Th is is also related to the symmetry of the transverse mode profi les involved. Theoretical tools, such as projected Green’s function and local density of states, for the analysis of the characteristics of the quasi-bound states are also presented and discussed. Th is chapter provides the background for further study of the physics and device characteristics of nanowire cross-junction devices, which is a topic of current widespread research interest. Despite the successful demonstration of nanowire crossjunction devices, our understanding of the physics and device characteristics of nanojunction devices are not complete, both theoretically and experimentally. There are still many important questions to be answered for the full development of these nanojunction devices. Topics to be explored in this area include the effects of electron–electron interaction, charge transport dynamics, contact effects, etc.


Acknowledgment The work described in this chapter is fully supported by a grant of the Research Grants Council of Hong Kong SAR, China (Project No. CityU 100303/03P).

References Björk, M. T., B. J. Ohlsson, T. Sass et al. 2002. One-dimensional heterostructures in semiconductor nanowhiskers. Applied Physics Letters 80: 1058–1060. Büttiker, M. 1986. Four-terminal phase-coherent conductance. Physical Review Letters 57: 1761–1764. Chan, K. S. and J. H. Wei. 2007. Quantum ballistic transport in nanowire junctions. Physical Review B 75: 125310. Cui, Y. and C. M. Lieber. 2001. Functional nanoscale electronic devices assembled using silicon nanowire building blocks. Science 291: 851–853. Datta, S. 1995. Electronic Transport in Mesoscopic Systems. Cambridge, U.K.: Cambridge University Press. Dehon, A. 2003. Array-based architecture for FET-based nanoscale electronics. IEEE Transactions on Nanotechnology 2: 23–32. Dehon, A., P. Lincoln, and J. E. Savage. 2003. Stochastic assembly of sublithographic nanoscale interfaces. IEEE Transactions on Nanotechnology 2: 165–174. Duan, X. and C. M. Lieber. 2000. General synthesis of compound semiconductor nanowires. Advance Materials 12: 298–302. Duan, X., J. Wang, and C. M. Lieber. 2000. Synthesis and optical properties of gallium arsenide nanowires. Applied Physics Letters 76: 1116–1118. Duan, X., Y. Huang, Y. Cui et al. 2001. Indium phosphide nanowires as building blocks for nanoscale electronic and optoelectronic devices. Nature 409: 66–69. Gaididei, Yu. B., L. I. Malysheva, and A. I. Onipko. 1992. Electron scattering and bound-state energies in crossed N-chain wires. A comparative study of discrete and continuous models. Journal of Physics: Condensed Matter 4: 7103–7114. Huang, Y., X. Duan, Q. Wei et al. 2001a. Directed assembly of one-dimensional nanostructures into functional networks. Science 291: 630–633. Huang, Y., X. Duan, Y. Cui et al. 2001b. Logic gates and computation from assembled nanowire building blocks. Science 294: 1313–1317. Jamshidi, A., P. J. Pauzauskie, P. J. Schuck et al. 2008. Nature Photonics 2: 86–89. Kalliakos, S., C. P. Garcia, V. Pellegrini et al. 2007. Photoluminescence of individual doped GaAs/AlGaAs nanofabricated quantum dots. Applied Physics Letters 90: 181902.

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Lee, M. W., H. Z. Twu, C. C. Chen, and C. H. Chen. 2001. Optical characterization of wurtzite gallium nitride nanowires. Applied Physics Letters 79: 3693–3695. Liang, W., M. Bockrath, D. Bozovic et al. 2001. Fabry-Perot interference in a nanotube electron waveguide. Nature 411: 665–669. Pauzauskie, P. J., A. Radenovic, E. Trepagnier et al. 2006. Optical trapping and integration of semiconductor nanowire assemblies in water. Nature Materials 5: 97–101. Perez-Martinez, F., I. Farrer, D. Anderson et al. 2007. Demonstration of a quantum cellular automata cell in a GaAs/AlGaAs heterostructure. Applied Physics Letters 91: 032102. Rao, A. M., P. C. Eklund, S. Bandow et al. 1997. Evidence for charge transfer in doped carbon nanotube bundles from Raman scattering. Nature 388: 257–259. Rao, S. G., L. Huang, W. Setyawan et al. 2003. Large-scale assembly of carbon nanotubes. Nature 425: 36–37. Ravenhall, D. G., H. W. Wyld, and R. L. Schult. 1989. Quantum Hall effect at a four-terminal junction. Physical Review Letters 62: 1780–1783. Schiff, L. I. 1968. Quantum Mechanics. New York: McGrawHill. Shao, Z., W. Porod, and C. S. Lent. 1994. Transmission resonances and zeros in quantum waveguide systems with attached resonators. Physical Review B 49: 7453–7465. Takagaki, Y. and K. Ploog. 1993. Quantum transmission in an out-of-plane crossed-wire junction. Physical Review B 48: 11508–11511. Tans, S. J., A. R. M. Verschueren, and C. Dekker. 1998. Roomtemperature transistor based on a single carbon nanotube. Nature 393: 49–52. Wei, J. H. and K. S. Chan. 2005. Electron transport in crossed nano-wire junctions. Physics Letters A 341: 224–250. Whang, D., S. Jin, and C. M. Lieber. 2003a. Nanolithography using hierarchically assembled nanowire masks. Nanoletters 3: 951–954. Whang, D., S. Jin, Y. Wu et al. 2003b. Large-scale hierarchical organization of nanowire arrays for integrated nanosystems. Nanoletters 3: 1255–1259. Zhang, R. Q., T. S. Chu, H. F. Cheung et al. 2001. Mechanism of oxide-assisted nucleation and growth of silicon nanostructures. Materials Science and Engineering C 16: 31–35. Zhong, Z., D. Wang, Y. Cui et al. 2003. Nanowire crossbar array as address decoders for integrated nanosystems. Science 302: 1377–1379.

VI Atomic Wires and Point Contact 34 Atomic Wires Nicolás Agraït................................................................................................................................. 34-1 Introduction • Experiments on Atomic Wires and Atomic Contacts of Gold • Conductance of Atomic Contacts and Wires • Elastic Scattering: Conductance Oscillations • Mechanical Properties of Atomic Wires • Inelastic Scattering and Dissipation • Numerical Calculations of the Properties of Atomic Wires • Summary • Acknowledgments • References

35 Monatomic Chains

Roel H. M. Smit and Jan M. van Ruitenbeek .......................................................................... 35-1

Introduction • The Initial Experiments • Distances beyond Expectation • Why Do Chains Form? And for Which Metals? • The Conductance Revisited • Conclusions and Outlook • References

36 Ultrathin Gold Nanowires Takeo Hoshi, Yusuke Iguchi, and Takeo Fujiwara ..................................................... 36-1 Introduction • Basic Properties of Solid Gold • Metal Nanowire and Quantized Conductance • Helical Multishell Structure • Summary and Future Aspect • Appendix 36.A: Note on Quantum-Mechanical Molecular Dynamics Simulation • References

37 Electronic Transport through Atomic-Size Point Contacts

Elke Scheer .............................................................37-1

Introduction • The Landauer Approach to Conductance • Fabrication of Atomic-Size Contacts • Conductance of Atomic-Size Contacts • Conclusions and Outlook • References

38 Quantum Point Contact in Two-Dimensional Electron Gas Igor V. Zozoulenko and Siarhei Ihnatsenka ...................................................................................................................... 38-1 Introduction • Conductance Quantization in the Quantum Point Contact • Quantum Point Contact in Magnetic Field • 0.7 Anomaly and Many-Body Effects in the Quantum Point Contact • Conclusion and Outlook • References

VI -1

34 Atomic Wires 34.1 Introduction ...........................................................................................................................34-1 34.2 Experiments on Atomic Wires and Atomic Contacts of Gold .......................................34-1 34.3 Conductance of Atomic Contacts and Wires ................................................................... 34-4 A Simple Free-Electron Model of an Atomic Wire • Landauer Theory of Conductance • Tight-Binding Models

Nicolás Agraït Universidad Autónoma de Madrid and Instituto Madrileño de Estudios Avanzados en Nanociencia

34.4 Elastic Scattering: Conductance Oscillations ...................................................................34-7 34.5 Mechanical Properties of Atomic Wires ............................................................................34-9 34.6 Inelastic Scattering and Dissipation .................................................................................34-11 34.7 Numerical Calculations of the Properties of Atomic Wires .........................................34-15 34.8 Summary ...............................................................................................................................34-17 Acknowledgments ...........................................................................................................................34-18 References.........................................................................................................................................34-18

34.1 Introduction An atomic wire consisting of metal atoms forming a single fi le is certainly the thinnest wire that you could imagine. Atomic wires are classical textbook examples often used to illustrate many fundamental topics in condensed matter physics. The experimental realization of atomic wires has been possible recently (Ohnishi et al. 1998, Yanson et al. 1998). The electronic and mechanical properties of these exceptional systems are quite different to those of their macroscopic counterparts, directly reflecting the quantum nature of the atoms. For example, they can carry enormous current densities of up to 8 × 1014 A/m 2, and they are much stronger than bulk. These systems are very attractive because they are ideally suited for investigating in atomic detail the theories of electronic transport in the nanoscale and the mechanical behavior of matter with atomic detail. These topics are of fundamental interest in nanotechnology and have implications for the miniaturization of electronic components. Here, we will only discuss atomic wires freely suspended between metallic electrodes, which make it possible to perform detailed transport experiments. Atomic wires supported on a substrate (Segovia et al. 1999, Nilius et al. 2002) can be considered to be a completely different system due to the strong coupling to the substrate. In addition, the lack of electrodes does not make it possible to perform transport experiments. We will start by considering the basic experimental results in atomic contacts and atomic wires of different metals in Section 34.2. The conductance of these systems will be discussed theoretically in Section 34.3 starting with a simple free-electron model (Section 34.3.1), followed by the general

approach of describing the conductance of a nanoscopic system (Section 34.3.2) and a review of the tight-binding method (Section 34.3.3), which will be used to shed light on the elastic scattering processes in atomic wires in Section 34.4. In Section 34.5, we will investigate the mechanical properties of atomic wires. The inelastic scattering of electrons in an atomic wire, which causes dissipation and heating but also makes it possible to obtain structural information will be considered in Section 34.6. Finally, we will discuss the theoretical calculations for the structure of the wires in Section 34.7. For an in-depth review of atomic contacts and wires, the interested reader can consult the article by Agrait et al. (2003).

34.2 Experiments on Atomic Wires and Atomic Contacts of Gold Atomic wires form in the last stages of the breaking of a metallic contact of certain metals, particularly gold. The scanning tunneling microscope (STM) is a convenient tool to study a metallic contact because it allows for fine positional control between a metal tip and a metal substrate through the use of piezoceramics, and it has high mechanical stability. Using a gold tip and a gold substrate, it is possible to make a metallic contact on a given spot of the substrate by carefully touching the surface. If the contacting surfaces of the tip and the substrate are clean, the two pieces of metal will strongly adhere or cold weld. Now if the tip is retracted, the contact will first deform plastically forming a neck, and becoming thinner as its cross-sectional area decreases. Just before separation occurs, the contact will consist, in most cases, of a single atom and sometimes an atomic wire could form as shown schematically in Figure 34.1. 34-1

34-2


FIGURE 34.1 Metallic contact formation. As the metallic tip of an STM touches the metallic substrate, the atoms at the clean contacting surfaces become intimately joined forming a contact. Retraction of the tip results in a controlled thinning of the contact. Just before separation a single atom contact forms, which can eventually lead to the formation of an atomic wire.

The formation and evolution of the contact is monitored by measuring the current at a low fi xed bias voltage, typically expressed as the conductance G = I/V, which is the inverse of the resistance. Figure 34.2a and b shows the evolution of the conductance for a gold contact as the tip retracts (black curve) and then advances to reform the contact (gray curve). Notice that in these and all subsequent figures, the conductance will be expressed in units of G 0 = 2e2/h, where e is the charge of the electron and h is Planck’s constant. Th is is the quantum unit of conductance and is the natural units conductance in the atomic scale as we will see in Section 34.3. The inverse of G 0 has units of resistance; its value is 12,906 Ω. In these conductance curves, two clearly different regimes can be observed: the contact regime and the tunneling regime. In the contact regime, the conductance is larger than G 0 and changes in steps. As we 4

Contact regime

will see when we discuss the mechanical properties, the contact deforms elastically, which results in a continuous variation in the conductance, until the accumulated stress is relaxed by sudden atomic rearrangement that is manifested as an abrupt jump in the conductance. A similar behavior is observed during contact formation. In the tunneling regime, the conductance is one or two orders of magnitude smaller and depends exponentially on the tip displacement, as evidenced by plotting the logarithm of the conductance. The conductance of the smallest contact of gold is given by the last conductance plateau before the transition to the tunneling regime. Th is smallest contact consists of just one atom and its conductance is one in units of G 0. The conductance of a one-atom contact of gold is very well defi ned and results in a sharp peak in the conductance histogram as shown in Figure 34.2c. × 104

Tunneling regime

16

3

G/G0

14 2 12 One-atom contact

1

10

Counts

0 (a)

log G/G0

0

6

–1

4

–2

–3 (b)

8

2

0

0.5

1 1.5 Displacement (nm)

0

2 (c)

0

1

2 G/G0

3

4

FIGURE 34.2 Conductance evolution during contact breaking (black curve) and contact formation (gray curve) for two different Au contacts, in linear (a) and logarithmic scales (b) at low temperature (4.2 K) using an STM. Note that both contacts show plateaus of conductance at G0 = 2e2 / h, the quantum unit of conductance. (c) Histogram obtained from all the measured conductance data points (conductance histogram) for 6000 contacts.

34-3

Atomic Wires 4 Counts (arb. units)

1

3.5 3

G/G0

2.5 2

0.5

0

0.25 0.5 0.75 1 Plateau length (nm)

8 Atom wire length = 2 nm + Elastic deformation

1.5 1 Plateau length

0.5 0 –1 (a)

Return distance –0.5

0

0.5

Displacement (nm)

Return length 7 × 0.25 = 1.75 nm

1 (b)

FIGURE 34.3 (a) A long plateau at a conductance of one G 0 indicates the formation of an atomic wire. (Inset) A histogram of plateau lengths shows peaks separated by an interatomic distance. (Adapted from Untiedt, C. et al., Phys. Rev. B, 66, 085418, 2002.) (b) The scheme shows how an 8-atom wire could give the observed return length taking into account that the atoms of the collapsed wire stay at the contact and the elastic deformation of the electrodes is relaxed after breaking.

Typically, the length of the last conductance plateau is less than 0.5 nm, however, sometimes it is possible to observe much longer plateaus, as the one shown in Figure 34.3a. These plateaus, which can be as long as 2.5 nm, correspond to the formation of a wire of single gold atoms. The probability of forming these atomic wires decreases rapidly with length, being less than 10−4 for plateaus longer than 2 nm (Yanson et al. 1998). A histogram of the length of the conductance plateaus, see the inset in Figure 34.3a, shows a preference for lengths that are multiples of 0.25 ± 0.2 nm, suggesting that that distance corresponds to the interatomic separation of the atoms in the atomic wire. This value is smaller than the bulk nearest neighbor interatomic distance of 0.288 nm, and in good agreement with the theoretical calculations, as we will see in Section 34.7. But how does the plateau length relate to the real length of the wire? We may argue that the return length, that is the tip displacement required to reform the contact, would give a better indication of the length of the wire because the plateau length would depend on the way the atomic wire forms. After breaking, the atoms of the collapsed wire will remain on the surface possibly forming two atomic layers (see the scheme in Figure 34.3) and consequently the return length would underestimate the real length of the 2 atomic diameters. However, we must also take into account that due to the elastic deformation of the electrodes at the point of maximum wire elongation, the return length will tend to give an overestimate of the real length. As we will see in Section 34.5, this elastic deformation will be typically 0.25 nm or 1 atomic diameter. Thus, we can conclude that the length of the atomic wire will be approximately equal to the return length plus an atomic diameter. The experimental curves in Figures 34.2 and 34.3 were measured at cryogenic temperatures (4.2 K). At these temperatures, there is no atomic diff usion on the surface of the metal, the thermal drift of the STM is negligible, and the high energy resolution

is in spectroscopic measurements. In addition, as we mentioned above, the contact formation requires the contacting surfaces to be clean. In this respect, the low temperature environment is also convenient because it provides a cryogenic ultrahigh vacuum environment and residual adsorbates are effectively frozen. For many metals it suffices to prepare the contact in situ by making a number of large contacts, this effectively brings fresh metal to the surface that then remains clean. The cleanliness of the surface can be checked in the tunneling regime. From the theory of quantum tunneling, we should have G ∝ e−αz where z is the displacement of the tip, and α = 1.025 φ where ϕ is the apparent tunneling barrier. From the slope of the log G versus z, we obtain ϕ ≈ 5 eV. This high value of the apparent tunneling barrier indicates that the contacting surfaces are clean, since in this case the apparent tunneling barrier should equal the work function, which for Au is 5.1 eV. In Figure 34.4a and b, we show a typical conductance cycle at room temperature and in ambient conditions. The steps in the conductance are not as well defined as they are at low temperatures due to the much larger mobility of atoms. As a consequence, the peak in the conductance histogram at G0 (Figure 34.4c) is not so sharp. In the tunneling regime, the tunneling barrier is much lower (~1 eV) and it is not uncommon to see some small steps due to the presence of adsorbates on the surface. Another tool that has also served to gain a wealth of information on these systems is the mechanically controlled break-junction (MCBJ) (Agrait et al. 2003) in which the sample is a notched wire glued on a flexible substrate. Once the sample is cold and in a cryogenic vacuum, the notch is broken by bending the substrate with a piezo element resulting in the separation of the wire into two electrodes with fresh surfaces. This can be very advantageous for studying metals that oxidize in ambient conditions. As in STM, the separation of the electrodes can be controlled with

34-4

Handbook of Nanophysics: Nanotubes and Nanowires 4 6000 G/G0

3 5000

2 1 0

1 2 Displacement (nm)

(a)

3

4

Counts

4000 0

log G/G0

0

3000 2000

–1 1000

–2 –3

0

1 2 Displacement (nm)

(b)

3

0

4 (c)

0

1

2 G/G0

3

4

FIGURE 34.4 Conductance evolution during contact breaking (black curve) and contact formation (gray curve) for two different Au contacts, in linear (a) and logarithmic scales (b) at room temperature in ambient conditions using an STM. The right panel (c) shows a histogram of the measured conductance values or conductance histogram for 6000 contacts.

a resolution of the order of picometers, but the mechanical stability can be somewhat higher. Many other metals have been investigated experimentally, in most cases using the MCBJ, as shown in Figure 34.5. The conductance traces and histograms are characteristic of each material. In most cases, the conductance of the last contact, the one-atom contact, is around G0 but not so sharply defined as it is in gold. Atomic 6

Normalized counts

4 Ag 3 Cu 2 Al 1 Pt 0

1

2

3

34.3 Conductance of Atomic Contacts and Wires As we saw in Section 34.2, in the case of gold, the conductance of a one-atom contact has a particular value of 2e2/h, which is independent of temperature, and this is also the value of the conductance of an atomic wire independently of length. This is quite surprising and shows that Ohm’s law is not valid for atomic wires. Indeed Ohm’s law states that the conductance G would be proportional to the transverse area S and the conductivity σ, a strongly temperature-dependent material property, and inversely proportional to the length L. The reason is that the dimensions of atomic wires are too small. Ohm’s law results from a semiclassical description of the electron motion and is applicable to macroscopic conductors in which electrons scatter many times and lose all memory of their phase. In contrast, the dimensions of an atomic wire are just a few nanometers, and of the order of the electron wavelength λF. A full quantum description of electron transport is required.

5

0

wire formation, signaled by the appearance of long conductance plateaus, have been observed only in gold, platinum, and iridium. For other metals, the length of the one-atom conductance plateau is just of a few tenths of nanometer. In Section 34.7, we will discuss why atomic wires form only in certain metals.

4

5

G(G0)

FIGURE 34.5 Normalized histograms for Cu, Ag, Pt, and K taken at low temperature using a MCBJ. The histograms for each metal are different and characteristic. (Adapted from Yanson, A., Atomic chains and electronic shells: Quantum mechanisms for the formation of nanowires, PhD thesis, Universiteit Leiden, Leiden, the Netherlands, 2001; Agrait, N. et al., Phys. Rep., 377, 81, 2003.)

34.3.1 A Simple Free-Electron Model of an Atomic Wire In quantum mechanics, the current through a conductor can be written as the probability of an electron to transmit through it. Consider a uniform cylindrical wire with a radius R and a length L with free and independent electrons connected to two bulk reservoirs (electrodes) (Agrait et al. 2003). If we neglect the effect

34-5

Atomic Wires

of the reservoirs, it is easy to obtain the electronic states in the conductor solving Schrödinger’s equation:

−

2 2 ∇ ψ(r ) = E ψ(r ), 2m *

Each eigenstate can carry in the axial direction an amount of current Imnk given by the integral over the transverse section of the conductor of the probability current density times the electron charge

I mnk = with the boundary condition ψ[r = R] = 0. We find that the eigenstates are given by

ψ mnk (r , φ, z ) = ψ mn (r , φ)eikz =

J m (γ mnr / R)e imϕ πLRJ m +1(γ mn )

e ikz ,

where the z coordinate is taken along the cylinder axis m = 0, ±1, ±2, ±3,… and n = 1, 2, 3,… are the quantum numbers γmn is the nth zero of the Bessel function of the order m, Jm The energies of the eigenstates are

Emn (k) =

States that carry a net current μL

ψ01

E (eV)

L ψ11

L

R

εF

μR

eV

3 E c11

2 1

E c01

(a)

V

0 –10 (b)

–5

e L

∑v ( f (ε ) − f (ε ))= π ∫ dkv ( f (ε ) − f (ε )), e

k

L

k

R

k

k

L

k

R

k

kσ

where L is the length of the conductor σ is the electron spin

since J−m(r) = (−1)m Jm(r), the states m and −m are degenerate. The electron states are divided into a set of parabolic one-dimensional subbands with the bottom of each subband located at an c energy E mn as shown in Figure 34.6.

4

∫

Since there is a degenerate left-moving mode that carries the same current in the opposite direction for each right-moving mode, to have a net current flowing in the conductor there must be an imbalance between the population of the left-moving mode (fi xed by the Fermi distribution on the left electrode, f L) and of the right-moving mode (fi xed by the Fermi distribution on the right electrode, f R). Such an imbalance is provided by the voltage bias V applied between the electrodes. Each subband contributes to the current with I mn =

2 2k 2 2 γ mn 2k 2 c + = + Emn , 2 2m * 2m * R 2m*

5

e i ⎛ ∂ψ *mnk ∂ψ mnk ⎞ e hk e ψ mnk − ψ *mnk dS = = vk . L 2m* ⎜⎝ ∂z ∂z ⎟⎠ L m* L

R = 0.45 nm 0 5 10 k (nm–1)

FIGURE 34.6 (a) Cylindrical wire of radius R = 0.45 nm and length L with an electronic charge density corresponding to Au, connected to two electrodes with an applied potential difference V. (b) For this small diameter two modes carry the current because only two of the onedimensional electronic subbands are below the Fermi level εF. Subbands E 01 is nondegenerate and E11 is doubly degenerate, and the Fermi level is determined by the condition that the wire remains charge neutral. The applied potential V unbalances the chemical potentials for rightgoing electrons μR and left-going electrons μL , resulting in a net current through the wire.

The total current will be given by the sum of the contributions c < εF , where εF of all subbands that are populated, that is E mn is the Fermi level of the wire. For a long conductor, one can replace the sum over the allowed k values by an integral over k. As we are dealing with a one-dimensional system, the density of the states is ρ(ε) = 1/vk ħ and then I sub =

∫

2e d ε( f L (ε) − f R (ε)). h

At zero temperature, f L(ε) and f R(ε) are step functions, equal to 1 below their respective chemical potentials μL = εF + eV/2 and μR = εF − eV/2, and 0 above. Thus, the contribution to the conductance of each subband is identical Gsub = Isub/V = 2e2/h. These results show that a perfect single mode conductor between two electrodes has a finite conductance, given by the universal quantity 2e2/h (Datta 1997).

34.3.2 Landauer Theory of Conductance A general approach for describing the conductance of a nanoscopic system like an atomic contact or an atomic wire is the scattering approach depicted schematically in Figure 34.7 (Datta 1997). The system is assumed to be connected to ideal electron reservoirs, the electrodes, by perfect leads. The basic idea is to relate the transport properties with the transmission and reflection probabilities for carriers incident on the system. The electrodes are assumed to

34-6


μL

NL

NR

System

μR

Perfect lead Electrode

FIGURE 34.7 Scattering approach. The electrodes are in thermal equilibrium and have well-defi ned chemical potentials μL and μR. The perfect leads that connect the system to the left and right electrodes have NL and NR conducting modes, respectively.

be in a thermal equilibrium with a well-defined temperature and chemical potential. Phase coherence is assumed and inelastic scattering may occur only in the reservoirs. In these leads, the electrons propagate as plane waves along the longitudinal direction, while its transverse momentum is quantized due to the lateral confinement. This defines a number of conducting modes in these leads, say NL and NR for the left and right lead, respectively, in the two-terminal configuration. The use of perfect leads, an auxiliary construction that greatly simplifies the formalism, does not affect the results as long as the number of modes considered is sufficiently large. In the scattering formalism, at a temperature of zero, the conductance of the system can be expressed as G=

2e 2 Tr(tˆ†tˆ), h

(34.1)

where ˆt is an NR × NL matrix whose elements tmn give the probability amplitude for an electron in mode n on the left to be transmitted into mode m on the right. Equation 34.1 is known as the Landauer formula. The matrix ˆt†ˆt is a hermitic NL × NL matrix. Consequently, it has real eigenvalues of τi, i = 1,…, NL which can be shown to satisfy 0 ≤ τi ≤ 1. The eigenvectors of ˆt†ˆt are called eigenchannels and correspond to particular linear combinations of the incoming modes that remain invariant upon transmission through the system. On the basis of the eigenchannels, the transport problem becomes a simple superposition of independent mode problems without any coupling, and the conductance can be written as G=

2e 2 h

∑τ . i

(34.2)

i

The set of eigenvalues {τi} fully determines the transport properties of the junctions and is known as the PIN code of the junction. An experimental determination of these eigenvalues can be done in the superconducting state exploiting the nonlinearities of the current-voltage characteristics for contacts in the superconducting state (Scheer et al. 1997, 1998). In the case of long wires, all nonzero eigenvalues can be close to unity and the value of the conductance will be a multiple of the quantum of the conductance, G = NcG0, where Nc is the number of

modes in the system. In a metallic wire of radius R, as discussed in Section 34.3.1, the number of modes is given by the number of subbands whose band bottom is lower than the Fermi level, c < εF , where εF. Taking into account that the number of that is, E mn zeros of the Bessel functions Jm below a certain value x is approximately equal to x2/4, we find that Nc ≈ (πR/λF)2, where R is the wire radius. This quantization effect was first observed in two-dimensional electron gases (2DEG) in the form of clear conductance plateaus at integer values of G0. In metal contacts, the fact that λF is of the order of the size of the atom obscures the effect of conductance quantization. The conductance traces show plateaus at stable structural configurations that might not coincide with an integer value of the conductance in units of G 0 (see the histograms and conductance traces in Figure 34.5). As we have seen above, one-atom contacts of gold have a well-defined conductance of 1G 0, which reflects the fact that one atom of gold provides a single completely open quantum channel and results in a sharp peak in the conductance histogram. Th is is also the case for silver and copper; in contrast, the one-atom contacts of other metals like aluminum and platinum are close to 1G0 but not so well defined, as evidenced in the broad peaks in the histogram. The explanation to this observation is that the atomic contacts of monovalent metals, like the noble metals and alkali metals, provide a single quantum channel for transmission, which is almost completely open (the corresponding eigenvalue is one), while in sp-metals and transition metals, which have more complicated electronic structures, several partially open eigenchannels contribute to the conductance. The transmissions of these partially open eigenchannels are quite sensitive to the detailed atomic arrangement.

34.3.3 Tight-Binding Models As we saw in Section 34.3.2, calculating the conductance of an arbitrarily shaped conductance connected to electrodes requires computing its transmission matrix ˆt . In order to do this, it is necessary to solve Schrödinger’s equation for the system composed of the conductor and the electrodes. A convenient way to do this is by using the tight-binding method in combination with Green’s functions (Datta 1997). In its simplest version, the tight-binding model uses an orthogonal basis {|i>} corresponding to a spherically symmetric local orbital at each atomic site in the system. Within this basis, the model Hamiltonian adopts the form Hˆ =

∑ε | i〉〈i | +∑t i

i

ij

| i 〉〈 j |,

i≠j

where εi corresponds to the site energies tij denotes the hopping elements between sites i and j, which are usually assumed to be nonzero only between the nearest neighbors

34-7

Atomic Wires

The retarded and advanced Green operators are defi ned as −1

−1

Gˆ r (E ) = lim ⎡ E + i η − Hˆ ⎤ , Gˆ a (E ) = lim ⎡E − i η − Hˆ ⎤ . ⎦ ⎦ η→ 0 ⎣ η→ 0 ⎣ The matrix elements of Gˆ (E) are directly related to the local densities of states (LDOS) by

where we implicitly take the limit η → 0. Gˆ Cr is the Green operator for the central region modified due to the coupling with the leads. It can be written explicitly as Gˆ Cr = (E − Hˆ C − Σˆ rL − Σˆ rR )−1 ,

(34.4)

r

ρi (E ) =

1 1 Im〈i | Gˆ r (E ) | i 〉 = − Im〈i | Gˆ a (E ) | i 〉. π π

(34.3)

In order to study the conductance of a fi nite system, like an atomic wire connected to semi-infi nite electrodes, the total Hamiltonian can be decomposed as (see Figure 34.8) ⎛ Hˆ L ⎜ Hˆ = ⎜ VˆCL ⎜ ⎝ 0

VˆLC Hˆ C

VˆRC

0 ⎞ ⎟ ˆ VCR ⎟ , ⎟ Hˆ R ⎠

where Σˆ rL = VˆCL gˆ Lr VˆL C and Σˆ rR = VˆCR gˆ Rr VˆR C are self-energy operators introducing the effect of the coupling with the r −1 leads. We have introduced gˆ L = lim η→0(E + i η − Hˆ L ) and gˆ Rr = lim η→0(E + i η − Hˆ R )−1 as the Green operators of the uncoupled electrodes. The expressions for the advanced Green operators are obtained through the substitution η → −η. The zero-temperature linear conductance is given in terms of the Green operators as functions by the following expression (see Datta 1997, Todorov et al. 1993 for more details). G=

where Hˆ L and Hˆ R describe the electronic states in the uncoupled left and right leads, respectively Hˆ C corresponds to the central region Vˆ LC, Vˆ CL , Vˆ RC , and Vˆ CR describe the coupling between the central region and the left lead and the central region and the right lead, respectively, and the corresponding retarded Green operator

8e 2 ⎡ ˆ a Tr ImΣ L (E F )Gˆ Ca (E F )ImΣˆ rR (E F )Gˆ Cr (E F )⎤ , ⎣ ⎦ h

(34.5)

are self-energy operators introducing the effects on the dynamics of the electrons in the central region due to the coupling with the leads, and Gˆ Cr and Gˆ Ca are the Green operators projected onto the central region. The expression (34.5) can be written in the usual form G = (2e 2/h)Tr[tˆ(EF )tˆ† (EF )],

(34.6)

where 1/2

1/2

tˆ(E ) = 2 ⎡ImΣˆ aL (E )⎤ G Cr (E ) ⎡ImΣˆ aR (E )⎤ . ⎣ ⎦ ⎣ ⎦ ⎛ Gˆ Lr ⎜ r Gˆ r (E ) = ⎜ Gˆ CL ⎜ ˆr ⎝ GRL

r Gˆ LC Gˆ r C

r Gˆ RC

⎛ (E + iη) − Hˆ L ⎜ =⎜ −VˆCL ⎜ 0 ⎝

ˆ H

r ⎞ Gˆ LR ⎟ r Gˆ CR ⎟ r ⎟ ˆ GR ⎠

(34.7)

The knowledge of the ˆtˆt † matrix in terms of Green functions allows the determination of the conduction channels for a given contact geometry. −VˆLC

(E + i η) − Hˆ C −Vˆ RC

−1

⎞ ⎟ CR ⎟ , ⎟ ˆ (E + i η) − H R ⎠ 0 −Vˆ

HˆL

HˆR HˆC

LC V

CR V

FIGURE 34.8 Decomposition of the Hamiltonian for an atomic wire ˆ is the Hamiltonian of the whole system; H ˆ L, connected to electrodes. H ˆ R, and H ˆ C are the Hamiltonian for the uncoupled left and right electrodes, H and atomic wire, respectively; and Vˆ LC and Vˆ CR describe the coupling of the atomic wire to the electrodes.

34.4 Elastic Scattering: Conductance Oscillations As we have seen, a perfect atomic wire with a single quantum channel would have a conductance of 2e 2/h. However, an imperfect coupling of the wire to the electrodes would result in reflections at the end of the wire leading to interference effects and a reduction of the conductance. The Au atomic wires in Figures 34.3, 34.11, and 34.13 indeed show a maximum conductance of 2e 2/h but lower values are also possible. In order to investigate the effect of the coupling to the electrodes, we can apply the tight-binding scheme in conjunction with Green operators as detailed in Section 34.3.3. Consider the atoms with a single electronic state. For an infinite chain, we have a single subband with E = ε0 + 2tcoska where a is the separation between the atoms. The wave vectors extend

34-8


As a further simplification that still preserves the main features of the problem, we can describe the electrodes by semiinfinite one-dimensional wires with fi xed hopping between the atoms t. In this case, the wire is connected by its first atom to only one atom of the electrode and by its last atom to only one atom of the right electrode. As a consequence, the coupling matrices have only one element different from zero, and we only need to know the Green function for the end element of the electrodes, which can be found analytically (see Todorov et al. 1993):

from (−π/a) to (π/a). If we assume that the each atom has a single electron, then the Fermi level is located at ε0, the Fermi wave number is k F = π/2a, and the Fermi velocity v F = 2ta/ħ. However, we are interested in the effect of coupling a finite wire to the electrodes. First, we take the Hamiltonian of the isolated wire as ⎛ ε0 ⎜ −t Hˆ C = ⎜ ⎜ ⎜0 ⎝

−t ε0 0

0

0

−t 0

0 −t

0⎞ 0⎟ ⎟. ⎟ ε 0 ⎟⎠

g R (E ) = g L (E ) =

To determine the effect of the coupling on the conductance of the wire, we need to obtain the expression for the Green operators for the central region Gˆ Cr and Gˆ Ca , given by Equation 34.4, and for the self-energy operators, Σˆ rR , Σˆ rL , Σˆ aR , and Σˆ aL, which in turn requires knowledge of the Green operator for the uncoua r r a pled electrodes gˆ R , gˆ L , gˆ R , and gˆ L , and the coupling matrices VˆRC , VˆCR , VˆLC , and VˆCL.

E + s E 2 − 4t 2 , 2t 2

where s = 1 for E > 0 and s = −1 for E < 0 and the advanced is its complex conjugate, and we have taken ε0 = 0. In Figure 34.9a through c, we show the resulting LDOS for a 5-atom wire calculated using Equation 34.3. The discrete levels of the isolated wire broaden progressively as the coupling increases. Note that for t1 = t2 = t, there will be no distinction between the wire and the electrodes and the LDOS is that corresponding to an infinite wire as shown in Figure 34.9c.

2

LDOS

1.5

t1 = 0.5t

t1 = 0.8t

t1 = t

t2 = 0.5t

t2 = 0.8t

t2 = t

1

0.5

0

–2

0 E/t

(a)

2

–2

0 E/t

(b)

2

–2

0 E/t

(c)

2

1 t2 = t1 G/G0

Transmission

1

0.8 0.6

0.5

0.4

t2 = t

0.2 t2 = t1

0 –3 (d)

–2

–1

0 E/t

1

2

0

3 (e)

0

0.2

0.4 0.6 Coupling t1/t

0.8

1

FIGURE 34.9 5-Atom wire coupled to semi-infi nite 1d electrodes. The hopping within the wire and within the electrodes is t. (a and b) Local density of states (LDOS) for at the fi rst and second (gray lines) and third or central (black line) atoms for a coupling to the electrodes of 0.5t (a) and 0.8t (b). Note that the peaks which correspond to the discrete states of the isolated wire broaden as a consequence of the coupling. (c) When the coupling to the electrodes is t the LDOS of all atoms is that of the infinite wire. (d) Transmission as a function of energy for a 5- and 6-atom wires, in both cases t1 = t2 = 0.8t. For a coupling t1 = t2 = t the transmission equals one in all the band and is independent of the number of atoms (black curve). (e) Conductance as a function of coupling to the electrodes. For 5-atom wire the conductance is always G 0 if the coupling to the electrode is identical on both sides (upper curve). In contrast for a 6-atom wire the conductance decreases with decreasing coupling (lower curve). Note that for a 5-atom wire the conductance also decreases with decreasing coupling if the coupling is not symmetric (middle curve).

34-9

Atomic Wires

In Figure 34.9d, we show the transmission of the wire, that is Tr[tˆ(E)tˆ†(E)], where ˆt (E) is given by Equation 34.7 for a 5-atom wire and the 6-atom wire for t1 = t2 = 0.8t and for perfect coupling. For this system, the Fermi level is at E = 0, and consequently the low voltage conductance will be given by the transmission at E = 0 multiplied by G 0. It is important to realize that the conductance depends on the number of atoms; for an odd number of atoms, the conductance will be G0 even for imperfect coupling as long as the coupling to the electrodes is symmetrical, while for an even number of atoms the conductance will decrease with decreasing coupling even in the case of symmetrical coupling, as illustrated in Figure 34.9e. In other words, the conductance is much more sensitive to the coupling to the electrodes if there is an even number of atoms in the wire as opposed to an odd number of atoms. Th is even–odd effect is not directly observable in the conductance plateaus of atomic wires, however, after averaging many conductance traces, the oscillations become evident (Smit et al. 2003) as shown in Figure 34.10. In Au wires, the half-period of the oscillation is in agreement with the interatomic distance of the atoms in the wire as it is to be expected from the discussion above. For Pt and Ir, a similar periodicity is found but it is accompanied by a continuous decrease in conductance from ~2.5G 0 to ~1G 0 in Pt and from ~2.2G 0 to ~1.8G 0 in Ir. Th is behavior can be explained (de la Vega et al. 2004) by the existence in Pt and Ir of a partially fi lled 5d band. Indeed the presence of several channels in the wire gives rise to a more complex oscillatory pattern with a different wavelength for each channel. 1.05 1

Au

0.95

G/G0

0.9 2 Pt 1.5 1 2.2 Ir

2 1.8 1.6

0

0.2

0.4

0.6 0.8 Length (nm)

1

1.2

FIGURE 34.10 Averaged plateaus of conductance for atomic wires of Au, Pt, and Ir. Each of the curves is made by the average of many individual traces of conductance starting from the moment that an atomic contact is formed. The measurements were performed at low temperature using an STM in the case of Au and an MCBJ for Pt and Ir. (Adapted from Smit, R.H.M. et al., Phys. Rev. Lett., 91, 076805, 2003.)

34.5 Mechanical Properties of Atomic Wires How do the mechanical properties of atomic wires affect their transport properties? What is the force needed to break an atomic bond? Can we understand the results of the experimental conductance measurements without taking into consideration the mechanical processes taking place during the elongation of the contact? In this section, we will see how the experiments in which conductance and forces are measured simultaneously can provide answers to these questions and also to a more fundamental question: how do the mechanical properties of matter change as the size is reduced down to the atomic scale? We would like to remark that this question is not only academic, but has implications in nanotechnology and also in many technologically important problems like adhesion, friction, wear, and lubrication as all are related to small size contacts. In Figure 34.11b, we can see the forces during the evolution of a gold contact. The forces and conductance were measured simultaneously using two STMs mounted in a series as sketched in Figure 34.11a. The top STM is used to make a metal contact as in the experiments described in the previous sections. The sample is a thin gold wire supported on one side. The forces exerted on the contact by the STM tip cause a deflection on this cantilever beam. An auxiliary STM is used to detect this deflection with picometer resolution. The experiments are performed at low temperatures, which ensure clean conditions and a negligible drift. Let us follow the evolution of the contact shown in Figure 34.11. Consider the zero of the horizontal axis as the starting point. At this point, the contact consists of several atoms (G ≈ 6G 0). In order to elongate the metallic contact, it is necessary to exert a tensile force, which in our plot is negative. In response to this force, the contact deforms elastically accumulating stress. Th is elastic deformation is naturally accompanied by a reduction in the cross-section, which results in a conductance plateau with a negative slope. Stress accumulation leads to the instability of the contact, which suddenly relaxes as the atoms in the contact rearrange to a new thinner and longer stable configuration. Th is force relaxation reflects as an abrupt decrease in the conductance. Th is process repeats several times leading to the formation of an atomic wire. During the elongation of this wire, the conductance variations are just a fraction of G 0 while the force shows the same sawtooth behavior observed for larger contacts. The yield force, that is the magnitude of the force right before relaxation, decreases as the size of the contact diminishes. During atomic wire elongation, the yield force remains approximately constant, being somewhat larger at the breaking point. After breaking, the atomic wire collapses and the tip has to cover the return distance to reform the contact. Surprisingly, forces at contact formation are also negative (that is tensile) while one would expect compressive forces. Only much larger contact forces would be compressive. In general, atomic wires are always in tensile stress. This can be understood as a surface tension effect.

34-10


0 1 0.8 0.6

4

–1 0.8

1

1.2

F (nN)

G (G0)

6

–2

0

Breaking force

–3

2

0

0.2

(a)

0.4 0.6 0.8 1 Displacement (nm)

–4

1.2

Yield force 0

(b)

0.2

0.4 0.6 0.8 1 Displacement (nm)

1.2

Atomic contact is formed here

STM tip

Sample (thin wire)

Sample support

Auxiliary STM tip (c)

FIGURE 34.11 Simultaneous measurement of forces and conductance in gold. Notice the correlation between the sawtooth behavior of the force and the stepwise evolution of the conductance: the sudden force relaxations lead to the abrupt changes in conductance. During the elongation of the atomic wire the yield force are of the order of the breaking force, although smaller in magnitude. Force relaxations are always accompanied by an abrupt change in the conductance, of the order of G 0 for the contact and of only a fraction of G 0 for the atomic wire. The return length of this wire is 0.75 nm, about 3 atomic diameters, form the previous discussion this could correspond to 4 or 5 atoms. The measurements were performed at low temperatures using the setup sketched on the right. The force exerted on the contact by the (a) STM tip is obtained from the deflection of the sample, which is a thin gold wire, using the (b) auxiliary STM. We have used a similar procedure to that described in Rubio-Bollinger et al. (2001).

the atoms in the electrodes is smaller than the breaking force, the chain can grow in length by incorporating atoms from the electrodes. The breaking force of gold atomic wires is 1.5 nN with a rather small dispersion (see Figure 34.12). This is in fact the force required to break an atomic bond. This force is found to be independent of

50

10

40

8 Keff (N/m)

Counts

In a metallic contact of atomic dimensions, force relaxations reflect the atomic rearrangement processes that take place at the thinnest part of the contact, where the stress is largest. However, as the atomic wire starts forming, the atomic rearrangements do not occur in the atomic wire itself but in the electrode region close to the wire. As long as the force required to rearrange

30 20 10 0

(a)

6 4 2

0

0.5

1 1.5 2 Breaking force (nN)

Km

0

2.5 (b)

K

0

0.5 1 Chain length (nm)

1.5

1 1 N = + Keff Km K

FIGURE 34.12 (a) Breaking force of gold atomic wires. (b) Effective stiff ness, Keff, measured in the experiments. The experimental system can be represented by a series of springs with stiff ness K, representing the interatomic bonds, in series with a spring Km, representing the electrodes.

34-11

Atomic Wires

the length of the wire and is considerably larger than the force required to break individual bonds in bulk gold, which is estimated at only 0.8–0.9 nN using density functional theory (DFT) calculations, as we will discuss below. This is direct experimental evidence that bonds of low-coordinated metal atoms are considerably stronger than bonds in the bulk (Rubio-Bollinger et al. 2001). The slope of the force as a function of displacement defines the stiff ness of the contact. However, to obtain the stiff ness of an atomic wire, it is necessary to take into account the stiff ness of the electrodes. In the bottom panel of Figure 34.12, the atomic wire and electrodes are shown schematically. The atomic wire is represented as a number of springs, each one having a stiff ness K, connected in series, and the electrodes as another spring with a stiff ness Km. The effective compliance 1/Keff of the system is the sum of the compliances of the different elements, 1 1 N = + . K eff K m K

(34.8)

In Figure 34.12, the stiffness of atomic wires as a function of their length is plotted. Notice that the stiffness of the wires decreases with their length and that the electrodes (chain length zero) is typically of the order of 6 N/m. This plot shows that the atomic wire is unusually stiff: it takes a wire of three atoms in length to equal the compliance of the electrodes, giving a value of about 18 N/m for the stiffness of a single atomic bond. This low compliance of the electrodes implies that a large fraction of the elastic deformation takes place at the electrodes next to the wire. The breaking point will be of the order of 0.25 nm, which must be taken into account in the estimate of the length from the return distance.

34.6 Inelastic Scattering and Dissipation As we have discussed in Section 34.3, atomic wires of gold are perfect one-dimensional conductors. The current is carried by a single quantum mode whose transmission probability deviates from unity only due to elastic scattering at points where the wire joins the electrodes. Th is description of transport through atomic wires is correct only if both temperature and bias voltage are low enough. Indeed, the thermal vibrational motion of the atoms in the wire alters the potential seen by the electrons and causes them to scatter inelastically. At room temperature, this is the mechanism that is responsible for most of the resistance in metals. In contrast, at low temperatures, the motion of the atoms in the wire is frozen and at low bias voltages, there is no inelastic scattering and consequently no dissipation. However, at higher voltages, the electrons will interact with the quantized vibrations or phonons of the atomic wire, which will result in a decrease in the conductance and heating of the atomic wire. The sensitivity to voltage of this electron– phonon interaction can be used for spectroscopy and to deduce structural information. This inelastic electron spectroscopy

(IES) usually receives different denominations depending on the transmission regime, being termed inelastic electron tunneling spectroscopy (IETS) in the low transmission regime and point contact spectroscopy (PCS) in the high transmission regime. The differential conductance dI/dV as a function of the voltage measured in a gold atomic wire at low temperatures is shown in Figure 34.13b, and its derivative or IES spectrum is shown in Figure 34.13c and the inset of Figure 34.13a. We can see the conductance is close to unity at low voltages and drops by about 1% for voltages above a certain threshold in the range of 12–20 mV. This drop in the conductance, which is symmetric with respect to voltage, is due to the onset of the excitation of a particular vibrational mode in the wire with a well-defined energy ħω = eVp, where Vp is the position of the peak in the IES spectrum. Stretching the atomic wire results in a shift of the peak to lower voltages, which reflects the softening of the vibrational modes due to the weakening of the interatomic bonds. In Figure 34.14a and b, we show the peak position Vp and amplitude Ap, respectively as the wire is stretched for different wires of different lengths. Figure 34.14c and d shows that the dependence of the peak amplitude with peak position and with length is proportional to L/ω2. Note that the symmetry of the conductance drops in the conductance curve or antisymmetry of the peaks in the IES spectra is a signature of the phonon signal and serves in practice to differentiate them from the variations in the conductance due to elastic scattering in the electrodes or at the ends of the wire, which are typically asymmetric. The experiments in Figure 34.13 show that for voltages below a certain threshold Vp the electrons cannot lose their energy through the interaction with the ions in the wire, that is, there is no dissipation in the wire. The dissipation of the excess energy of the electrons will take place inside the electrodes. Only for voltages above Vp will the electrons be able to give part of their energy to the ions and the atomic wire will start heating. Interestingly, as we will see in this section, the electrons traversing the wire both heat up and cool down the wire through phonon emission and absorption, respectively, resulting in a steady-state temperature as a function of voltage. To gain insight into the fundamental mechanisms taking part in inelastic scattering processes in atomic wires, we will consider in detail the electron–phonon interaction in an ideal onedimensional wire of length L at zero temperature. In an infinite atomic wire, the electron–phonon interaction is given by Hˆ ep =

⎛ ⎞ qVq ⎜ ⎟ ⎝ 2 MN ω q ⎠

∑ qk

1/2

(aˆqcˆk++ qcˆk + aˆq+cˆk+− qcˆk ) ,

where M is the mass of the ions N is the number of ions ωq is the frequency of the phonon mode q + ĉk and cˆk are the annihilation and creation operators, respectively for the electrons in state k + âq and aˆ q are the annihilation and creation operators, respectively for the phonons in state q

34-12


3

e

c

1 d2I (1/V) G0 dV 2

3

1

d b a

2

2.5

1 dI G0 dV

3.5

1

0.99 d

0.98

e

c

–

b a

2 G/G0

0

0.97 0

(b)

10 20 Bias voltage (mV)

2

a b c d e 1

1 d 2I (1/V) G0 dV 2

1.5 1 0 –1

0.5 Plateau length 0 –0.5 (a)

–2

Return distance 0

0.5 1 Displacement (nm)

1.5

–50

0 Bias voltage (mV)

(c)

50

FIGURE 34.13 Inelastic scattering in Au atomic wires. (a) Shows the low bias conductance of a gold atomic contact. The conductance plateau indicates that an atomic wire of 5–6 atoms has formed. Note that the plateau presents quite a lot of structure, which is a result of elastic scattering possibly due to disorder at the point where the atoms are being extracted. The tip displacement was stopped before the wire broke and the bias voltage was ramped with a small superimposed ac modulation to obtain the differential conductance dI/dV curves shown in (b). These conductance curves present a marked symmetric drop in conductance for voltages of 10–20 mV. Between the different conductance curves the tip was displaced by a fraction of a nanometer stretching the atomic wire. In (c) the derivative of the differential conductance d2I/dV 2 or IES spectrum is shown. In these IES spectra the conductance drop appears as an antisymmetric peak. Stretching results in a displacement of the peak in the spectra to lower frequencies as can be clearly observed in (c) and the inset in (a). These measurements were performed at 0.3 K using an STM and the procedure in Agrait et al. (2002a,b). +

+ The term aˆq cˆk − q cˆk represents the scattering of an electron from state k to state k − q with the emission of a phonon q, and the + term aˆq cˆk +q cˆk corresponds to the scattering of an electron from state k to state k + q with the absorption of a phonon q. We can write the probabilities per unit of time for an electron k to emit or absorb a phonon q, using Fermi’s Golden Rule as

implies a net flow of electrons from the left electrode to the right electrode. Then, taking the sum of all the allowed electron and phonon states we have W em =

∑∑M (n q

k

em wkq =

2π | 〈k − q, nq + 1| Hˆ ep | k, nq 〉 |2 δ(ε k − ω q − ε k − q )

= Mq (nq + 1)δ(ε k − ω q − ε k − q ), 2π w = k + q, nq − 1 Hˆ ep k, nq ab kq

2

δ(ε k + ω q − ε k + q )

= Mqnq δ(ε k + ω q − ε k + q ), where nq is the occupation of the vibrational mode q and Mq = πVq2q 2/MN ω q . To find the number of electrons that are inelastically scattered in the wire, we need to consider the scattering probabilities of all the states in the wire taking into account that the connection to the electrodes imposes a chemical potential μL for right-going electrons and μR for left-going electrons. In the following discussion, we will take μL > μR, which

W ab =

q

+ 1)δ(ε k − ωq − ε k −q )f L (ε k )[1 − f R (ε k −q )],

q

∑∑M n δ(ε q q

k

k

+ ωq − ε k + q )f L (ε k )[1 − f R (ε k +q )].

q

These summations can be easily performed by taking into account the following considerations. The kinetic energy of phonons is typically more than two orders of magnitude smaller than the energy of electrons at the Fermi level, and since energy is conserved in the scattering process, as expressed by Dirac’s delta function, this implies that εk ≈ εk−q and consequently |k| ≈|k − q|, that is, either q ≈ 0 or q ≈ −2k (see Figure 34.15a). Both for phonon emission and for the absorption the final electronic states must be unoccupied. Emission is possible for electrons whose energy is εk > μR + ħω because this provides unoccupied left-going states (see Figure 34.15b). As all right-going states at lower energies are occupied, the electron must always backscatter in this process and consequently q ≈ −2k. For low

34-13

Atomic Wires 0.4 22 0.2 log (Ap)

Vp (mV)

20 18 16 14 (a)

0 –0.2 –0.4 1.1

12

1.2 log (Vp)

(c)

2.5

1.3

2

Ap (1/V)

Ap (1/V)

1.5 1.5 1

0.5

0.5 0 0.5

1

Vp = 18 mV 1

(b)

1.5 L (nm)

2

0

2.5 (d)

0

0.5

1

1.5 L (nm)

2

2.5

FIGURE 34.14 Frequency and amplitude of phonon scattering in Au wires. (a and b) Peak position V p and amplitude A p in the IES spectra as a function of length L for different atomic wires. Each color represents a different wire. As the wire is elastically stretched the peak position changes to lower values. In some of the wires, atomic rearrangements leading to a sudden increase of V p and a sudden decrease in A p are clearly observable. (c) Log-log plot of the peak amplitude versus peak position. The slope of the curve shows that Ap ∼ Vp−2 . (d) Peak amplitude A p for V p = 18 mV versus atomic wire length L, showing A p ~ L. These results are similar to those reported in Agrait et al. (2002a,b).

bias voltages, all electrons involved in transport will have k = ± k F, which for an atomic wire with one conduction electron per atom is kF = π/2a and consequently the wave number of the vibrational mode is q = π/a, which lies at the zone boundary. This vibrational mode qB = π/a has a wavelength equal to 2a—it is the alternating bond length mode (see Figure 34.16). Then, for the phonon emission rate, we have

W

em

0, ⎧ ⎪ = ⎨ ⎛ eV − ω B ⎞ ⎟⎠ (nB + 1) ⎪LC ⎜⎝ ω B ⎩

for eV < ω B , for eV ≥ ω B ,

scattered by phonons with q ≈ 0, but it is negligible. Note that for eV < ħωB it is possible to have phonon absorption by both right-going and left-going electrons, this process is possible even at V = 0 and provides a mechanism for phonon damping in the absence of a bias voltage (damping by electron-hole pair generation). The rate of phonon absorption in the atomic wire is then ⎧ ⎛ ω B + eV ω B − eV ⎞ + nB , ⎪ LC ⎜ ω B ⎟⎠ ⎪ ⎝ ω B ab W =⎨ ⎛ eV + ω B ⎞ ⎪ LC ⎜ nB , ⎪⎩ ⎝ ω B ⎟⎠

for eV < ω B , for eV ≥ ω B .

where C = Vq2B qBame/πM 2v F me is the mass of the electron v F is the Fermi velocity of the electrons in the wire Phonon absorption will be possible for electron εk > μR − ħω (see Figure 34.15b). Again in this case this implies backscattering and q = π/a = q B (note that qB and −q B represent the same mode). In principle, there could be a contribution from electrons with energies very close to μL that could be forward

It is remarkable that due to the one-dimensional character of the one-dimensional wire, electrons can interact only with one phonon mode, the qB mode of energy ħωB. Thus, the passage of current through the wire creates and annihilates phonons in this mode, changing the occupation number nB according to the emission and absorption rates Wem and Wab −2LCnB , ⎧ dnB ⎪ = ⎨ ⎛ eV − ω B ⎞ eV + ω B nB , dt ⎪LC ⎜ ⎟⎠ (nB + 1) − LC ω ⎝ ω B B ⎩

for eV < ω B , for eV ≥ ω B ,

34-14

Handbook of Nanophysics: Nanotubes and Nanowires States available for ε phonon emission

ε

Absorp ti

on

ћωB

μL

μL

Emission

eV

eV

ћωB

μR

Right-going states

Left-going states

k

(a)

ћωB

μR Left-going states

μR

μL

Right-going states

ћωB

States available for ε phonon absorption

fR

fL

fR

fL

(b)

1.5 1

nB

G/G0

1 0.5 0

0

10

(c)

20 30 V (mV)

40

50

0

10

(d)

20 30 V (mV)

40

50

FIGURE 34.15 Inelastic effects in one-dimensional ideal wires. (a) Phonon emission and absorption processes in a one-dimensional conductor. The left electrode is at a chemical potential μL higher the chemical potential of the right electrode μR, such that μL − μR = eV. In the emission process the electron is backscattered and gives and energy ħωB to the emitted phonon. In the absorption process the electron is backscattered and receives and energy ħωB from the absorbed phonon. (b) The rightgoing-states available for phonon emission have energies between and μR + ħωB and μL , while those available for absorption are more numerous and have energies between μR − ħωB and μL . (c) Dependence of the occupation nq of mode qB (number of phonons) on voltage, in a wire in two different stretch states, with ħωB values of 12 and 20 meV. (d) Total conductance of the atomic wire taking into account inelastic effects for the same two stretch states considered in (c). The full line is for the externally undamped limit and the dashed line is for the externally damped limit.

M

K

a

qB-mode: q = ± π a

4K M

ω

–π a

q

π a

FIGURE 34.16 Simple mechanical model of an atomic wire. The interatomic interaction is modeled by a first-neighbor interaction with represented by spring with stiff ness K, with a constant value for a given value of the interatomic separation a. M is the mass of the atoms. The dispersion relation is given by Equation 34.12. For a wire with a single half-fi lled band, the zone-boundary mode qB = ±π/a will be the only mode capable of interacting with the electrons, and as a consequence the frequency detected in the IES spectra will be given by ω B = 4K / M . Note that the wavelength of the qB-mode is precisely 2a, which results in an alternating bond character.

whence nB attains a steady-state value at any voltage (see Figure 34.15c) given by 0, ⎧ ⎪ nB = ⎨ eV − ω B ⎪ 2ω , B ⎩

Both phonon emission and absorption cause backscattering of the electrons, these backscattered electrons return to the electrode from which they originated and as a consequence they do not contribute to the current. The total backscattering current is given by

I b = eW

em

0, ⎧ ⎪ 2 2 + eW = ⎨ ⎛ e V − 2ω 2B e 2V 2 ⎞ + 2 2 ⎟, ⎪eCL ⎜⎝ 2 2ω 2 2 ω B ⎠ B ⎩

for eV < ω B ,

ab

for eV ≥ ω B .

where we have used the steady-state value for nB. The differential backscattering conductance is then 0, ⎧ dI b ⎪ Gb = =⎨ ⎛ 2eV ⎞ dV ⎪e 2CL ⎜ 2 2 ⎟ ⎝ ωB ⎠ ⎩



(34.9)

The drop in conductance occurs at eV = ħωB (see Figure 34.15) and has a magnitude of

34-15

Atomic Wires

ΔG = −[Gb ]eV =ωB =

2e 2CL . ω B

(34.10)

Half of the drop is due to the electrons backscattered in the emission process and the other half is in the absorption. So far we have considered that electrons once emitted could leak out of the wire, that is, the energy transferred from the electrons to the vibrations remains in the wire, this is the externally undamped limit. In general, the wire vibrations would couple to the electrodes to some degree. We can consider the limiting case in which the energy is instantaneously absorbed into an external heat bath, then the occupation would remain nB = 0 for all voltages, this is the externally damped limit. The backscattered current will be due only to the emission process and the conductance will be 0, ⎧ dI b ⎪ Gb = =⎨ ⎛ eV ⎞ dV ⎪e 2CL ⎜ 2 2 ⎟ ⎝ ωB ⎠ ⎩

ω=2

K 1 sin qa M 2

(34.12)


and the drop in conductance at the threshold ΔG = −[Gb ]eV =ω B =

more phonon emission but this is compensated by more phonon absorption. Electrons both heat and cool the wire preventing a catastrophic increase in the number of phonons in the wire. The coupling of the electronic current with the vibrational modes of the wire makes it possible to obtain structural information from the IES spectra. In order to illustrate how the mechanical and structural properties reflect on the electronic properties, we will consider the simplest model: an infinite monoatomic wire. If the atoms are separated a distance a, have a mass M, and their interaction is represented by a spring with spring constant K, as represented in Figure 34.16, the dispersion relation would be given by

e 2CL . ω B

(34.11)

In the case of arbitrary coupling to the electrodes, the results would be between the externally damped and undamped limits. According to Equations 34.10 and 34.11 for a given vibrational frequency, the magnitude of the conductance drop depends linearly on the length of the wire L, which agrees with the experimental observation (see Figure 34.14d). However, the observed experimental dependence (Figure 34.14c) is on 1/ω2B rather than on 1/ωB. Now we can consider the effect of this exchange of energy between the electron and phonon systems on the temperature of the wire. According to Equation 34.9, for voltages above the threshold for the excitation of mode qB, the onset of dissipation gives rise to a steady-state occupation of this mode. This occupation is completely out of equilibrium because modes of lower energies are unoccupied, and the temperature is not defined. However, we can get an equivalent temperature Teq by equating the voltage-dependent occupation in Equation 34.9 to the equilibrium occupation eV − ω B 1 = , for eV ≥ ω B , 2ω B exp(ω B / k BTeq ) + 1 where k B is the Boltzmann constant. We find that for ħωB = 20 meV at V = 60 mV, the occupation is one and Teq = 335 K; and for ħωB = 12 meV at V = 36 mV, the occupation is also one and Teq = 200 K. Note that these equivalent temperatures are independent of the wire length because in a longer wire there is

where q is the eigenmode wave number. The highest frequency ω B = 2 K /M corresponds to the zone boundary mode qB = π/a = 2kF. In the case of a half-fi lled single band wire, this is the only mode that interacts with the electrons. From the position of the peak in the IES spectra that are in the range 10–22 meV, and taking into account the atomic mass of gold, we find values for K in the range of 18–90 N/m. These values should be considered just as a rough approximation, since this mechanical model is far too simple because we cannot approximate the metallic interaction by a simple first-neighbor interaction. A first-principles calculation as discussed in Section 34.7 will be necessary for a realistic quantitative comparison. The preceding basic theoretical considerations for the inelastic scattering and dissipation in a one-dimensional wire and the coupling between the electrons and the atomic vibrations give a semiquantitative account of the experimental observations. For a more detailed treatment using a first-principles approach and with application to nanoscale devices, see the articles by Frederiksen et al. (2004, 2007).

34.7 Numerical Calculations of the Properties of Atomic Wires A fully realistic computation of the properties of atomic wires taking into account an adequate number of atoms, dynamic structural arrangements, applied bias, temperature, and the effect of the electrodes is still beyond the capabilities of any theoretical model. Nevertheless, theoretical methods can be used to investigate the influence of different factors on the system (Agrait et al. 2003). First principles or ab initio methods attempt a full quantummechanical treatment of both nuclear and electronic degrees of freedom. In order to obtain realistic configurations, the atoms are allowed to move in response to the forces they experience. These forces are obtained from the evaluation of the potential energy in a quantum-mechanical description of the system in which certain approximations must unavoidably be included.

34-16


Within this approach, the total energy of a system of ions and valence electrons can be written as Etotal ({rI },{r I }) =

∑ 2 m | r | +∑ | r − r | + E 1

ZI ZJ

2

I

I

I

I

I>J

elect

({rI }), (34.13)

J

where rI, mI, and ZI are the position, mass, and charge of the Ith ion, respectively E elect ({rI}) is the ground-state energy of the valence electrons evaluated for the ionic configuration {rI} The first two terms in this equation correspond to the ionic kinetic and interionic interaction energies, respectively. In this equation, we have considered that the electrons follow the instantaneous configuration of the ions (Born-Oppenheimer approximation). The major task in first principles or ab initio methods is to calculate the ground-state electronic energy, which is typically done via the Kohn–Sham (KS) formulation of the DFT of many-electron systems within the local density (LDA) or localspin-density (LSD) approximation. The ground-state energy of the valence electrons, according to the Hohenberg–Kohn theorem, depends only on the electronic density. In the KS method, the many-body problem for the ground-state electronic density n(r) of an inhomogeneous system of N electrons in a static external potential (due to the positive ions) is reduced to solving selfconsistently the independent-particle Schödinger equation ⎡ −∇2 + v eff (n)⎤ ψ j = ε j ψ j , ⎣ ⎦ with the electronic density given by n=

∑ f |ψ | , 2

j

j

j

where f j is the occupation of the jth (orthonormal) orbital. The KS effective potential veff is given by the functional derivative of the electronic energy and since it depends on the electronic density it must be obtained self-consistently. In ab initio methods, the computational requirements are so high that systems that can be studied this way are limited to a small number of atoms although this number is increasing due to the increasing power of computers and new schemes. Ab initio methods have been used to investigate the strength of the atomic bonds in gold (Rubio-Bollinger et al. 2001). Interestingly, the bonds in the atomic wire were found to be much stronger than those in the bulk, which in fact makes the formation of the wires possible. The agreement with the experimentally measured breaking forces is quite good as mentioned in Section 34.5. Although, as we have seen above, atomic wires have been found experimentally only for few metals (Au, Pt, and Ir); atomic wires of many different metals have been studied. Calculations have been performed for infinite wires using cells with one or several atoms and periodic boundary conditions, and for finite wires supported between pyramidal tips, which are generally considered to have a negligible effect on the structure and stability of the wires. These calculations show that the atomic wires are linear only if they are overstretched. As the available state per atom decreases for all metals, a zigzag structure appears (see Figure 34.17). Taking the z-axis along the axis of the chain and defining dz as the projection of the interatomic distance over this axis, the energy per atom as a function of dz, for calculations with two atoms per cell, has one or two minima depending on the metal. The absolute minimum corresponds to atomic wires where the interatomic bonds with the z-axis make an angle of about 60°. The 5d elements, in particular in Au, Pt, and Ir, show a second minimum corresponding to a stable zigzag configuration. This is rather surprising since it implies that a force will be needed to either stretch or contract the wire. The angle subtended with the z-axis for Au, Pt, and Ir are 20°, 25°, and 45°, respectively. 3.5

Linear wire Zigzag wire

2-Atom

Zigzag

E (eV)

2.5

Linear

Double

3

Double wire

2

1.5 1 0.5 1

1.5

2

2.5

3

dz (Å)

FIGURE 34.17 DFT calculated energy per atom of atomic wires of Au using a 2-atom cell (right) and corresponding atomic configurations. The absolute minimum corresponds to a double wire configuration, and the second minimum to a stable zigzag configuration. The atomic wire is linear only for bond lengths above 0.275 nm. (Adapted from Fernández-Suivane, L. et al., Phys. Rev. B, 75, 075415, 2007, and consistent with the results of Sánchez-Portal et al., 1997, 2001.)

34-17

Atomic Wires

The origin can be explained as a reduction in the transverse kinetic energy for the electrons due to the increased effective wire width. This mechanism is of the same nature as the shell structure observed for alkali metal nanowire. For Au, the bond lengths are consistent with those observed experimentally. The wire is stretched to a linear configuration only for bond lengths above about 0.275 nm, shortly before breaking. There is no experimental evidence of this zigzag structure, because as we saw in Section 34.5, the experimentally realized atomic wires are always under tensile stress. The magnetic properties of atomic wires have also been investigated because the partially fi lled d orbitals might lead to magnetic states in a one-dimensional geometry. The calculations indicate that Au atomic wires would not show magnetism but Pt and Ir seem to have magnetic anisotropies (Fernández-Suivane et al. 2007). Th is has not been confi rmed experimentally. Calculating the conductance of an atomic wire using ab initio methods poses additional problems because it requires the inclusion of the electrodes in the calculations. In general, it is not enough to count the number of bands crossing the Fermi level in an infinite wire because the electrodes, which are strongly coupled to the wire, will modify the band structure and may also cause an interference effect. A possible approach is to use a combination of ab initio and Green’s functions methods (de la Vega et al. 2004, García-Suárez et al. 2005, Jelínek et al. 2006). In order to investigate the transport properties or the formation dynamics, it is necessary to include many more atoms to describe the electrodes. One possibility is to use the classical MD for the problems that require a large number of atoms and simulation time. Another possibility, in between first principles and empirical methods, is the tight-binding molecular dynamics (TBMD) method, which is more accurate than empirical potential methods because it explicitly includes the electronic structure and is much faster than the first principles methods. Conventional MD simulations use phenomenological inter-atomic potentials to model the energetics and dynamics of the system (Agrait et al. 2003). Pair potentials are not adequate for an accurate description of metallic systems that require potentials that include manybody interactions. These potentials contain the physics of the model systems and their functional form is selected on the basis of theoretical considerations and is typically fitted to a number of experimental or theoretically calculated data. The embedded atom method (EAM) and effective medium theory (EMT) potentials derived from the DFT are often used to model metallic systems. In these models, the potential energy of the system is written as a sum of a short-range pair-interaction repulsion and an embedding energy for placing an atom in the electron density of all the other atoms:

E pot =

∑

Fi [ρh , i ] +

i

1 2

∑∑

Vij (rij ),

i

j ≠i

(34.14)

where Vij (rij) is a two-body potential that depends on the distance rij between atoms i and j Fi[ρh,i] is the embedding energy for placing an atom at position i, where the host electron density due to the rest of the atoms in the system is ρh,i The latter is given by ρ h ,i =

∑ ρ(r ),

(34.15)

ij

j ≠i

where ρ(rij) is the “atomic density” function. The first term in Equation 34.14 represents, in an approximate manner, the many-body interactions in the system. These potentials provide a computationally efficient approximate description of bonding in metallic systems, and have been used with significant success in different studies. Closely related are the Finnis–Sinclair (FS) potentials, which have a particularly simple form ⎡1 FS E pot = ε⎢ ⎢2 ⎣

∑∑V (r ) − c ∑ ij

i

j ≠i

i

⎤ ρi ⎥ , ⎥ ⎦

(34.16)

∑

with V(rij) = (a/rij)n and ρi = (a /rijm ), where a is normally taken j ≠i to be the equilibrium lattice constant, m and n are positive integers with n > m, and ε is a parameter with the dimensions of energy. For a particular metal, the potential is completely specified by the values of m and n, since the equilibrium lattice condition fixes the value of c. It is important to remark that the predictions of molecular dynamics (MD) simulations using empirical potentials may be inaccurate in systems of atomic dimensions like atomic wires, since the usual potentials fitted to bulk properties do not describe adequately lowcoordinated atoms. This situation may be improved by expanding the database used for the fitting to include a set of atomic configurations calculated by ab initio methods. Nevertheless, MD simulations present a picture of the events that lead to wire formation in gold contacts. Deformation involves a series of elastic stretching stages, each terminated by a change in the atomic arrangement, which involves the breaking of atomic bonds. Whether or not a chain is formed depends on the relative strength of the bonds for different atomic configurations. The way a surface atom gets incorporated into a chain is by keeping the bond with a low coordinated chain atom while breaking the bonds to more highly coordinated atoms in the shoulders of the contact. This shows that chain formation can occur if the bonds in the chain are much stronger than the bonds in the bulk so that it is harder to break the chain than to pull out an atom of the electrodes.

34.8 Summary We have given a brief account of the basic transport and mechanical properties of atomic wires and atomic contacts. These systems due to their simplicity are ideally suited to investigate

34-18

the fundamental properties of matter at the atomic scale, and serve as benchmark systems for theories that describe nanoscale systems. In addition, many of the techniques and concepts described in this chapter in relation to atomic wires can be applied to other nanosized systems.

Acknowledgments This work was supported by MEC, Spain (MAT2004-03069); MICINN, Spain (MAT2008-01735 and CONSOLIDERINGENIO-2010 CSD-2007-00010); and Comunidad de Madrid (Spain) through program Citecnomik (P-ESP-0337-0505).

References Agrait, N., Untiedt, C., Rubio-Bollinger, G., and Vieira, S. 2002a, Electron transport and phonons in atomic wires, Chem. Phys. 281, 231–234. Agrait, N., Untiedt, C., Rubio-Bollinger, G., and Vieira, S. 2002b, Onset of dissipation in ballistic atomic wires, Phys. Rev. Lett. 88, 216803. Agrait, N., Yeyati, A., and van Ruitenbeek, J. 2003, Quantum properties of atomic-sized conductors, Phys. Rep. 377, 81–279. Datta, S. 1997, Electronic Transport in Mesoscopic Systems, Cambridge University Press, Cambridge, U.K. de la Vega, L., Martín-Rodero, A., Yeyati, A. L., and Saúl, A. 2004, Different wavelength oscillations in the conductance of 5d metal atomic chains, Phys. Rev. B 70(11), 113107. Fernández-Suivane, L., García-Suárez, V. M., and Ferrer, J. 2007, Predictions for the formation of atomic chains in mechanically controllable break-junction experiments, Phys. Rev. B 75, 075415. Frederiksen, T., Brandbyge, M., Lorente, N., and Jauho, A. 2004, Inelastic scattering and local heating in atomic gold wires, Phys. Rev. Lett. 93(25), 256601. Frederiksen, T., Paulsson, M., Brandbyge, M., and Jauho, A.-P. 2007, Inelastic transport theory from first principles: Methodology and application to nanoscale devices, Phys. Rev. B 75, 205413. García-Suárez, V. M., Rocha, A. R., Bailey, S. W., Lambert, C. J., Sanvito, S., and Ferrer, J. 2005, Conductance oscillations in zigzag platinum chains, Phys. Rev. Lett. 95, 256804. Jelínek, P., Pérez, R., Ortega, J., and Flores, F. 2006, Hydrogen dissociation over au nanowires and the fractional conductance quantum, Phys. Rev. Lett. 96, 046803.


Nilius, N., Wallis, T., and Ho, W. 2002, Development of onedimensional band structure in artificial gold chains, Science 297(5588), 1853–1856. Ohnishi, H., Kondo, Y., and Takayanagi, K. 1998, Quantized conductance through individual rows of suspended gold atoms, Nature 395, 780–785. Rubio-Bollinger, G., Bahn, S., Agraït, N., Jacobsen, K., and Vieira, S. 2001, Mechanical properties and formation mechanisms of a wire of single gold atoms, Phys. Rev. Lett. 87, 026101. Sánchez-Portal, D., Artacho, E., Junquera, J., Garcia, A., and Soler, J. 2001, Zigzag equilibrium structure in monatomic wires, Surf. Sci. 482, 1261–1265. Sánchez-Portal, D., Untiedt, C., Soler, J., Sáenz, J., and Agraït, N. 1997, Nanocontacts: Probing electronic structure under extreme uniaxial strains, Phys. Rev. Lett. 79, 4198–4201. Scheer, E., Agraït, N., Cuevas, J., Levy Yeyati, A., Ludoph, B., Martn-Rodero, A., Rubio Bollinger, G., van Ruitenbeek, J., and Urbina, C. 1998, The signature of chemical valence in the electrical conduction through a single-atom contact, Nature 394, 154–157. Scheer, E., Joyez, P., Esteve, D., Urbina, C., and Devoret, M. 1997, Conduction channel transmissions of atomic-size aluminum contacts, Phys. Rev. Lett. 78, 3535–3538. Segovia, P., Purdie, D., Hengsberger, M., and Baer, Y. 1999, Observation of spin and charge collective modes in one-dimensional metallic chains, Nature 402, 504–507. Smit, R. H. M., Untiedt, C., Rubio-Bollinger, G., Segers, R. C., and van Ruitenbeek, J. M. 2003, Observation of a parity oscillation in the conductance of atomic wires, Phys. Rev. Lett. 91, 076805. Todorov, T., Briggs, G., and Sutton, A. 1993, Elastic quantum transport through small structures, J. Phys.: Condens. Matter 5, 2389–2406. Untiedt, C., Yanson, A. I., Grande, R., Rubio-Bollinger, G., Agraït, N., Vieira, S., and van Ruitenbeek, J. 2002, Calibration of the length of a chain of single gold atoms, Phys. Rev. B 66, 085418. Yanson, A. 2001, Atomic chains and electronic shells: Quantum mechanisms for the formation of nanowires, PhD thesis, Universiteit Leiden, Leiden, the Netherlands. Yanson, A., Rubio Bollinger, G., van den Brom, H., Agraït, N., and van Ruitenbeek, J. 1998, Formation and manipulation of a metallic wire of single gold atoms, Nature 395, 783–785.

35 Monatomic Chains 35.1 Introduction ...........................................................................................................................35-1 Motivation • Outline • One-Dimensional Conductance

35.2 The Initial Experiments ........................................................................................................35-3 TEM Imaging • Low Temperature Break Junctions

35.3 Distances beyond Expectation ........................................................................................... 35-6 The First Experimental Results • Stretching the Bond • More Recent Experiments

35.4 Why Do Chains Form? And for Which Metals? ..............................................................35-7 The Problem • The Physics behind the Bond Strength • Relativistic Effects of Atomic Chains

Roel H. M. Smit Universiteit Leiden

Jan M. van Ruitenbeek Universiteit Leiden

35.5 The Conductance Revisited ................................................................................................35-10 Plane Wave Model for an Atomic Chain • Observation of Parity Effects in Real Chains • Conductance Oscillations for Other Materials

35.6 Conclusions and Outlook ...................................................................................................35-12 References.........................................................................................................................................35-12

35.1 Introduction

frustrations of the students and allow the readers to enjoy the beauty of quantum mechanics.

35.1.1 Motivation In the study of quantum mechanics, there are only a very limited number of realistic examples that can be used as illustrations, much to the frustration of undergraduate students. The complexity of solving Schrödinger’s equation often leads to a necessary simplification to one dimension, and one-dimensional structures appear to be minimally realistic. When one starts studying the behavior of metallic point contacts on the atomic scale (for a general introduction to metallic point contacts, see (Agraït et al., 2003)), one begins to explore computer simulations with realistic interatomic potentials. When these calculations indicate that gold (Sørensen et al., 1998) and platinum (Finbow et al., 1997) contacts form one-dimensional chains of single atoms upon stretching, the authors were implying that this could be an artifact of the model. It did not take long, however, before experiments confirmed the formation of chains. Experiments at both room temperature (Ohnishi et al., 1998) and liquid helium temperature (Yanson et al., 1998) showed the existence of strands of single atoms with lengths up to seven atoms. From this point on, many theory groups have used the metallic chains to fine-tune their theories. Having a reduced number of atoms and interactions, they form ideal test candidates, even for present-day models (Rocha et al., 2006). A discussion of all models is beyond the scope of this chapter, but as we will see, the basic quantum mechanics of the planewave model is often enough to understand the physics of these metallic chains. These examples may reduce at least some of the

35.1.2 Outline Monatomic chains have not only been studied upon the breaking of metallic contacts, they have also been studied on surfaces. Two different techniques, namely, single atom manipulation with the use of the tip of a scanning tunneling microscope (Wallis et al., 2002) and the self-assembly of surface atoms (Segovia et al., 1999) have been used for this purpose. For a recent review on these studies, see Oncel (2008). This chapter only discusses freestanding monatomic chains suspended between two electrodes of the same material. In this case, the absence of interaction with a substrate simplifies the electron structure. This field of study has already led to many interesting physical discoveries and we will have to limit ourselves to only a few in this chapter. Following the chronology of the references, we begin in Section 35.2 with a description of the experimental setups in which they were produced. All experimental details necessary for the understanding of the remaining experiments of this chapter are provided in this section. Once the basics are treated, Section 35.3 continues with a discussion of the atom-to-atom distance in the chain. Although this appears to be a minor detail at first, it has led to many imaginative proposals for alternative structures. Since it proved difficult to bridge the gap between experiment and theory, a significant number of papers has appeared on this problem. A small matter as the atom-to-atom distance provided the fuel for many other interesting studies on chains. 35-1

35-2


In Section 35.4, we discuss the reasons for the formation of chains and, especially, the reason for their forming only on certain metals. Although many models are able to reproduce the formation of chains by stretching metallic contacts, a fundamental comprehension was lacking for several years. As the key arguments are presented, we see that the relativistic corrections to the electron orbitals are vital in the formation of chains. In Section 35.5, we address once again the conductance properties of chains. Within the framework of the plane-wave model, we see that it is possible to include interferences of the wave function. These interferences can be observed directly in the conductance of the chain, forming a powerful illustration of the wave character of electrons. Many other interesting phenomena have been investigated for monatomic chains. Among others, it was found that the electron– phonon interaction inside chains reflects the one-dimensional character of the lattice (Agraït et al., 2002). There are studies of the influence of current and voltage on the stability of chains, where both electro migration forces (Todorov et al., 2001) and local heating (Frederiksen et al., 2007), due to the passing of electrons, lead to breakdown. In order to introduce the reader to monatomic chains, it was decided not to systematically discuss all the subjects, but to limit the discussion to a few key items, where we focus on the unique physics of these one-dimensional conductors.

Ψ(x , z ) = A sin

−

2 ∇ Ψ( x , z ) + U ( x , z )Ψ( x , z ) = EΨ( x , z ) 2m

(35.1)

where ħ is Planck’s constant divided by 2n. For smoothly shaped boundaries of the potential, the wave functions for the electrons can be separated into two components. For hardwall boundaries, the wave function should be zero at the boundary, resulting in

W

FIGURE 35.1 Sketch of a two-dimensional wire of width W with parallel hard wall boundaries, which give rise to a set of transverse modes for the electron wave function.

(35.2)

where W represents the width of the structure n is the transverse quantum number For the component parallel to the boundaries of the structure, the solution is a running wave eikz, such that the corresponding energy of every mode n is given by En (k) =

2k 2 2 ⎛ nπ ⎞ + 2m 2m ⎜⎝ W ⎟⎠

2

(35.3)

Let us take the constriction sufficiently narrow, such that only n = 1 for the transverse component of the wave function is of importance at the scale of the Fermi energy of the metal. When connecting the wire to macroscopic leads, as in Figure 35.2, matching of the wave functions gives rise to a partial reflection of the waves, which can be represented by a transmission probability amplitude t1. The transmitted partial wave becomes ψ t (z ) =

35.1.3 One-Dimensional Conductance Many of the conclusions about the properties of metallic chains have been drawn on the basis of their electrical conduction. Before starting our journey through this manifold of experimental and theoretical studies, we therefore need to have a basic understanding of conductance at the relevant scale. Atomic chains of gold have a diameter of the size of the Fermi wavelength. Macroscopic classical theories will therefore not suffice; we need to take the wave character of the electrons into account. Let us therefore consider a two-dimensional wire, as shown in Figure 35.1. The wave function, f, of any electron with mass, m, inside the confining potential, (U), should obey Schrödinger’s equation

nπx ⋅ ψ (z ) W

1 t1e −ik1z L

(35.4)

In general, t1 depends on energy or wave vector, but for small energies we can ignore this dependence. The probability current for one electron wave to be transmitted is then given by P (k1 ) =

⎛ ∂ψ(z , k1 ) ∂ψ * (z , k1 ) ⎞ k1 ψ * (z , k1 ) − ψ(z , k1 ) ⎟⎠ = mL t1* t1 2mi ⎜⎝ ∂z ∂z (35.5) Electrode L

Electrode R

Ballistic contact

z

x μL

μR

E

k

FIGURE 35.2 Graph showing the dispersion relation for the parallel wave vectors in the electrodes and the contact. The finite width of the contact only permits a finite number of bands. Both forward and backward moving states are occupied up to the chemical potential. At zero bias this leads to zero current.

35-3

Monatomic Chains

The total electrical current through the constriction can now be given by a sum over all vectors k1 multiplied by the charge carried per electron wave. Note that this value is 2e instead of e, assuming spin degeneracy of the wave function. I1 = 2e

∑ k1

2e P (k1 ) = T1 mL

∑k

∫

2e L I1 = T1 kdk h 2π

(35.7)

Substituting the dispersion relation E = ħ2k2/2m leads to an integration over energies: μ R = EF + eV /2

∫

dE =

μ L = EF − eV /2

2e 2 T1V h

(35.8)

For a realistic three-dimensional contact, the confining potential U will be more complex, resulting in a more complex transverse solution of the wave function. Let us, therefore, briefly look into the situation where more than one transverse wave function starts to play a role in the conduction. There will be a finite number of incoming waves from theleft of (right), amplitudes which can be represented by a vector lL (lR ). Equivalently, oR (oL ) represents the outgoing amplitudes to the right (left). All possible scattering processes in the system are now described by the equation

⎛ oL ⎞ ⎛ s ⎜

⎟ = ⎜ LL ⎜ o ⎟ ⎜⎝ s RL ⎝ R⎠

⎞ s LR ⎟ ⎟ s RR ⎠

⎛ lL ⎞ ⋅ ⎜ ⎟ ⎜l ⎟ ⎝ R⎠

(35.9)

where the ˆsij matrices describe the amplitude for scattering from the incoming waves from lead j to the outgoing waves in lead i. While the general form of the matrix may be quite complicated, it can be simplified. As all wave functions are linearly independent and the scattering is linear, we can diagonalize the matrix. In other words, we can find a transformation to a new basis. In the new basis, the eigenstates are a property of the junction, such that ⎛ r1 ⎜ ⎜ ˆ =⎜0 S' ⎜t ⎜ 1 ⎜ ⎜ ⎝0

0 rN 0 tN

t1 0 r1 0

0⎞ ⎟ ⎟ t N ⎟ ⎛ rˆ ≡⎜ 0 ⎟⎟ ⎝ tˆ ⎟ ⎟ rN ⎠

G=

1 2e 2 = V h

(35.6)

1

Here we define T1 ≡ t1* t1 as the transmittance of the contact for the transverse mode n = 1. Figure 35.2 shows the dispersion relation of these k1 values. As the contact is connected by macroscopic leads, the wave vectors form a continuum. The summation over k-vectors in Equation 35.6 can thus be replaced by an integral. This allows us to rewrite the current as

2e I1 = T1 h

As electron waves can only be transmitted or reflected into the same eigenmodes, Equation 35.8 can be reduced to

t'ˆ ⎞ ⎟ ˆ⎠ r'

(35.10)

N

∑T

n

(35.11)

n =1

where we define Tn ≡ (tˆ†tˆ)nn ′ , similarly as in Equation 35.8. Equation 35.11 is known as the Landauer expression. The number of relevant channels (Tn ≠ 0) is determined by the atoms forming the narrowest part of the junction. In the simplest case of a monovalent single atom chain there is only one channel with T1 close to 1, as will be discussed in the following text.

35.2 The Initial Experiments The one-dimensional structure of the monatomic chain was shown to be real by the experiments performed by Ohnishi et al. (1998) and Yanson et al. (1998). In the following paragraphs, we discuss these two experiments separately.

35.2.1 TEM Imaging Of course, the structure is too small for most imaging techniques to be detected, but Ohnishi et al. (1998) showed that a transmission electron microscope (TEM) does meet the demands. They started with a very thin gold fi lm, mounted in an ultra-high vacuum chamber (p ≈ 10−8 Pa) of a TEM. The fi lm was sufficiently thin such that a high-intensity electron beam (∼100 A/cm2) was able to create holes in it. This gave them the opportunity to shape a freely hanging gold bridge by forming two holes in close proximity to each other. Further illumination of this bridge resulted in additional thinning such that a desired width of four atomic strands (≈1 nm) was easily obtained. When the beam intensity was reduced, it appeared possible to image the final structure without influencing it any further. As such, the evolution of the structure over time could be studied. The structure continues thinning due to the diff usion of atoms at room temperature. In the final stages of this breaking process, the remaining structure frequently consists of a single strand of atoms, as shown in Figure 35.4d. This provides very appealing visual information on the existence of monatomic chains. Within the same TEM, the authors also performed an experiment using an STM-like configuration, comparable to the one shown in Figure 35.3. Instead of a single gold film, they now started with a mechanically sharpened gold tip positioned close to an evaporated gold counter electrode. The gold tip was mounted on a shear-type piezoelectric transducer, allowing for movement in all three directions. The tip could therefore be dipped into and pulled from the counter electrode under computer control. Although this configuration does not provide as much mechanical stability, hampering atomic resolution on the chains, it provides an opportunity to measure conductance simultaneously with the imaging. In Figure 35.4, we see how the

35-4


x

200 kV Electron beam

z

20 mm

Tube-type piezo

y

Guide-pin

o-ring

Motor

Screw z

x

Specimen A (needle)

200 kV Electron-beam

y Specimen B

Specimen mount (mobile side)

Specimen mount (fixed side)

11

)

)

11

(1

(200)

1.5

1

1

0.5

0.5

0

(a)



1.5

1

1

0.5

0.5

0

1

0.5

0.5

0

1.5

1

1

0.5

0.5

0

(d) Single line Double lines

3 2 1 0

0

1

2 3 4 Conductance (G/G0)

Conductance (G/G0)

Number of counts (103)

1.5

1

(c)

(b)

(e)


(1


FIGURE 35.3 Schematic overview of a sample holder inside a TEM, including the piezoelectric element necessary for the contact breaking. (Reprinted from Kizuka, T. et al., Phys. Rev. B, 55(12), R7398, 1997. With permission.)

5 B

3

C

2

D

1 0

(f )

A

4

0

1

2 Time (s)

3

FIGURE 35.4 Figure summarizing the results of the TEM experiments with simultaneous conductance measurements. (a)–(d) Images taken by the TEM during the thinning, with an accompanying cross section of the image intensity. At the left bottom (e) the evolution of the conductance observed simultaneously with these images. At the right bottom (f) a histogram resulting from 50 of those evolutions. (Reprinted from Kurui, T. et al., Phys. Rev. B, 77(16), 161403R, 2008. With permission.)

35-5

Monatomic Chains

The second article in that same issue of Nature showed independent evidence of linear gold chains (Yanson et al., 1998). It showed that atomic chains are not limited to the parameters in the TEM experiment, but that they also appear under cryogenic conditions. Since surface diffusion at these temperatures is absent, the contacts need to be thinned down by stretching. The fact that chains also form under such circumstances is even more surprising. At room temperature, the contact starts as a bridge of several atomic strands in parallel, which break one by one. In this case, the final structure is already present at the start of the experiment. At low temperatures, however, the breaking of the contact is an irregular process during which surface diffusion plays no role. At low temperatures, most of the experiments were performed either by the previously discussed STM-like technique or by mechanically controllable break junctions (MCBJ) presented in Figure 35.5. This second technique has the advantage of producing clean contacts by very simple means. One starts with a macroscopic wire (about 100 μm in diameter and 15 mm long) and cuts a notch halfway its length. Locally, the diameter is thus reduced to less than 50 μm, providing a weak spot. Th is wire is glued on a phosphor bronze bending beam (20 × 4 × 1 mm), having a Kapton foil as insulating layer. This gluing is done with two droplets on either side of the notch, in order to keep the two wire ends fi xed with respect to each other. After the insertion of the substrate in a vacuum chamber, it is cooled down to 4.2 K, providing a cryogenic vacuum. A piezoelectric element, situated at the middle of the bending beam, opposite to the metallic wire, can now be used to bend the substrate, resulting in the stretching of the wire. With an appropriate coarse movement (often provided by a mechanical drive) the incision can be stretched and broken down to a microscopic d+δ

d w

Δ L

FIGURE 35.5 Schematic side view of the MCBJ. The left-hand image depicts the situation were the substrate is relaxed. At the right-hand side the piezo is pushed against the substrate resulting in bending of the substrate and stretching of the contact.

8 25 7

Return distance (Å)

35.2.2 Low Temperature Break Junctions

size. Note that the surfaces are now freshly exposed. In combination with the cryogenic vacuum, this provides an extremely clean environment for the contact. Afterward, the piezoelectric element can be used for the fi ne control, resulting in an atom-byatom thinning of the contact. During such a stretching of a gold wire, one obtains evolutions of the conductance similar to the one presented in Figure 35.6. The information in this graph appears insufficient to prove the existence of monatomic chains, but one already notices the relatively long plateau situated around a conductance of 2e2/h. In Equation 35.11, we saw that this information agrees with the maximum transmission for a single channel, and for a monovalent atom such as gold, this represents a contact of one atom in diameter. To prove that this plateau is indeed caused by the formation of an atomic chain, the authors performed several tests. In one of these tests, they measured the distribution of the lengths of this last plateau, defined as the number of data points between 1.2 and 0.5G 0, over many conductance-breaking traces. In Figure 35.7, this distribution shows a clear periodic structure. The chain thus periodically experiences an increased chance to break. As the chain consists of an integer number of atoms, the periodicity in this distribution directly reflects the atom-to-atom distance in the chain, confirming its existence. By now, many other experiments at low temperatures have been performed and the observed results are in accordance with the properties of a

6

Conductance (2e2/h)

conductance changes in steps of the order of 2e 2/h as the contact is broken. The size of these steps is in agreement with Equation 35.11 provided that the different modes close one by one. The images taken for the structures and their density profi les show how the number of strands is proportional to the conductance. Although the individual atoms cannot be identified in these images, this is convincing evidence that the conductance of a chain of gold atoms is close to 2e2/h.

5

20 15 10 5

4

0 3

4

8

12

16

20

Plateau length (Å)

2 Plateau length 1 Return distance 0 0

4

8

12 16 20 24 Electrode displacement (Å)

28

32

FIGURE 35.6 The conductance of a gold contact recorded while it is stretched and pushed back into contact. The conductance decreases stepwise, one step for every reconfiguration. Finally, the conductance reaches a value close to 2e2/h before breaking. The length of this plateau reflects the one-dimensional character of the chain. After breaking a similar stepwise increase is found upon moving the electrodes back into contact. (Reprinted from Yanson, A.I. et al., Nature, 395(6704), 783, 1998. With permission.)

Number of times each length is observed

35-6


this distance was 0.36 ± 0.11 nm, which was later corrected to 0.26 ± 0.02 nm (Untiedt et al., 2002).

200

35.3.2 Stretching the Bond

×10

150

100

14

16

18

20

22

6 8 10 12 14 16 Length of the last plateau (Å)

18

20

22

50

0

0

2

4

FIGURE 35.7 Distribution of lengths for the last conductance plateau before breaking. This result is obtained by collecting 10,000 breaking curves for Au contacts at 4.2 K. It shows a number of maxima at multiples of a fi xed distance. Note that distance between peaks (and thus between atoms in the chain) was later corrected to 0.26 ± 0.02 nm. (Reprinted from Yanson, A.I. et al., Nature, 395(6704), 783, 1998. With permission.)

monatomic chain. It goes beyond the scope of this chapter to include all the evidence, but interested readers are encouraged to read reference (Yanson et al., 1998) and others mentioned in this chapter.

Because of the small number of atoms in the chain and the reduced number of interactions, the chain forms a perfect litmus test for developing numerical models describing the physics of atomic contacts and their conductance. Almost all calculations predicted the atom-to-atom distance in the chains to be smaller than the nearest neighbor distance of 0.288 nm in the bulk (Sørensen et al., 1998). This value agrees with the corrected atom-to-atom distance at low temperatures, but is in strong disagreement with the TEM values at room temperature. To bridge the gap to the 0.4 nm reported in experiment, several creative propositions were given. One was an imaginative idea by Sánchez-Portal et al. (1999). Their calculations indicated that the chain only becomes linear at higher tensile stresses. At lower tensile stress it has a zigzag structure, as this allows the electrons to delocalize more, reducing their energy. At a certain finite temperature (∼40 K) thermal excitation induces a rotation of the chain around its axis, fi xed by the anchoring points to the electrodes. For a chain with an odd number of atoms, all the odd-numbered atoms lie on the axis of rotation, such that only the even-numbered atoms will move. This movement results in a difference in contrast, which depends on the zigzag angle, but can be as large as a factor of 5, see Figure 35.8. The contrast for

35.3 Distances beyond Expectation

4.63

4.73

4.65

It came as a big surprise that breaking gold contacts spontaneously formed freely suspended, linear chains of a single atom wide. In hindsight, simulations proved to be more correct than they were expected to be. Interestingly, there were also many differences between theory and experiment. Even when looking at the atom-to-atom distance in the chain, there was already a disagreement. The images made with TEM, such as Figure 35.4, leave little doubt about the Au-to-Au distance in the chain. The value can be directly extracted from the images and the first values represented in literature reported distances of 0.35–0.40 nm. The actual value can be slightly bigger when the chain is not perfectly perpendicularly oriented to the electron beam, but it can never be smaller. It is interesting to note that this interatomic distance varies between pairs of atoms. Even within one image, the atomto-atom distance varies along the chain. For the experiments performed at low temperatures, the atom-to-atom distance is not directly visible. Nevertheless, one can still extract its average value. The distribution of the plateau lengths depicted in Figure 35.7 shows a periodicity, which must coincide with the average atom-to-atom distance in the chain. Note that this distribution forms a statistical description over a large population of chains. Individual distances can deviate from the most frequently occurring value. The fi rst value reported for

3.6

5.0

35.3.1 The First Experimental Results

FIGURE 35.8 Two different proposed configurations to explain the interatomic distances observed in the room temperature TEM experiment. The top configuration of Legoas et al. has interstitial carbon atoms and carbon dimers inserted within the chemically reactive gold chains. Since the contrast of the electron beam is low on these light elements, atom-to-atom distances measured up to 0.5 nm represent the Au–Au distances, which can be explained. (Reprinted from Legoas, S.B. et al., Phys. Rev. Lett., 88(7), 076105, 2002. With permission.) The bottom panel gives a configuration by Sánchez-Portal et al., which proposes the gold atoms to spin in a zigzag configuration such that the contrast for half of the atoms is smeared. Atom-to-atom distances up to 0.47 nm can then be explained. (Reprinted from Sánchez-Portal, D. et al., Phys. Rev. Lett., 83(19), 3884, 1999. With permission.)

35-7

Monatomic Chains

35.3.3 More Recent Experiments A second step toward the verification of these models appeared quite a challenge. For the spinning zigzag geometry, one would have to freeze the structure below the activation energy of the spinning motion (∼40 K). Obtaining such low temperatures inside the experimental chamber of the TEM is certainly not straightforward. But without performing the experiment, calculations on the contrast were already sufficient to indicate that the actual signal to noise ratio of commercial TEMs should be good enough to observe also the spinning atoms (Koizumi et al., 2001). As of yet, no experimental evidence for the zigzag configuration in Au has been presented. The hypothesis of this structure, however, shows how rich the properties of metals can be, even at these smallest scales. Although the structure might prove unrealistic in the end, the investigations on its plausibility have been very fruitful. The interstitial atoms of light elements inside metallic chains have been the subject of intense debate as well. For the experimentalists the complication was that even the best cleanliness of an ultra-high vacuum could prove insufficient. The best opportunity was therefore to start from the cryogenic vacuum and deliberately contaminate the space with the desired molecular species. For both hydrogen (Csonka et al., 2006) and oxygen (Thijssen et al., 2006) it has been shown convincingly that atoms and even molecules can act as interstitial structures. At room temperature, however, the hydrogen is probably bound to loosely to stay bound during the atomic reconfigurations (Anglada et al., 2007). More recent experiments in TEM (Takai et al., 2001) show several examples of chain structures having interatomic distances of 0.29 nm at room temperature in agreement with calculations. It is thus very well possible that the first reports discussed contaminated structures. Interestingly, Thijssen et al. were also able to induce the formation of chains in silver, by the admission of oxygen molecules.

1.0 Normalized number of counts

half of the atoms could thus be too low to be imaged in the TEM, resulting in almost double atom-to-atom distances. Other theoretical studies showed that atoms of light elements such as carbon, oxygen, and sulfur could put themselves in interstitial positions between the atoms of a gold chain (Häkkinen et al., 1999; Legoas et al., 2002; Daniel and Astruc, 2004). As the contrast of an electron beam of a TEM goes as Z2/3, with Z the atomic number, its sensitivity for these interstitial lightweight elements could be too low to detect them. The positioning of interstitial carbon atoms and dimers can explain distances up to the incidentally reported 0.5 nm, as shown in Figure 35.8. Although the TEM experiments were carefully performed in an ultra-high vacuum, these conditions could still not prove clean enough. While bulk gold is relatively inert, nano-sized gold particles have such strong interactions that they are widely explored as a catalyst (Daniel and Astruc, 2004). As also monatomic chains are shown to react easily with light elements (Häkkinen et al., 1999), a single atom diff using over the electrodes toward the contact can be enough to contaminate the chain.

Ag + O2 Ag

0.8

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8 1.0 1.2 Plateau length (nm)

1.4

1.6

1.8

FIGURE 35.9 Length distribution for the last plateau in silver after the admission of oxygen gas into the vacuum. The gray line serves as a reference and indicates the length of the last plateau before oxygen admittance. (Reprinted from Thijssen, W.H.A. et al., Phys. Rev. Lett., 96(2), 026806, 2006. With permission.)

First, they started from a clean Ag contact and obtained the length distribution for the last plateau depicted in Figure 35.11. In contrast to gold, these short plateaus indicate the absence of chains in silver. After adding 10 μmol of O2 in the cryogenic vacuum, however, they repeated the experiment and found a spectacular change in behavior (Thijssen et al., 2006). The silver after exposure to oxygen, reproduced in Figure 35.9, shows plateaus up to 1.5 nm long, with many peaks in the distribution. As the diameters of O and Ag are different, it is understandable that these peaks are not equidistant anymore.

35.4 Why Do Chains Form? And for Which Metals? 35.4.1 The Problem The fact that the numerical models predicted atomic chains (Finbow et al., 1997) does not imply that it was immediately obvious why some metals have the tendency to form these one-dimensional structures. The ingredients deposited into the models were approximate effective interaction potentials between atoms, which were optimized to reproduce bulk behavior, but not expected to be reliable at this scale. More elaborate density functional theory (DFT) calculations confirmed the stability of atomic chains for Au (Häkkinen et al., 1999; Portal et al., 1999; Sánchez-Portal et al., 1999; Bahn and Jacobsen, 2001; Palacios et al., 2002; da Silva et al., 2004). To be able to determine whether these chains also form during the breaking of a point contact, one needs to perform a dynamic simulation. This performance increases the computational demands greatly and only few studies have been performed within the DFT framework to reproduce the spontaneous chain formation (Krüger et al., 2002; Anglada et al., 2007).

35-8


With their one-dimensional structures, atomic chains account for a large surface area, making the whole structure energetically expensive. Each atom in a chain has only two nearest neighbors, while every atom on the surface has many more. In order to understand the stability of the chains we should actually look into the dynamics of stretching (Gall et al., 2004). Especially at low temperatures, the system does not have the internal energy to explore all possible configurations. It will simply be frozen to the first energetic minimum it encounters, irrespective of whether this minimum is more general or local. Let us start from a contact that has been pulled until it reaches a single atom at its narrowest constriction. For this atom to be able to pull additional atoms into a chain, we require that the bonds of the atoms with lower coordination are stronger than those with higher coordination. Although the structure would lower its energy more effectively by breaking of the chain, the barrier for this is too high to overcome. Rubio-Bollinger et al. have succeeded in measuring the forces in gold contacts directly (Rubio-Bollinger et al., 2001). They replaced one of the two electrodes of the contact by a small gold cantilever, of which they determined the deflection with sub-nm resolution. Hooke’s law states that this deflection is directly proportional with the force exerted on the contact. This initiative opens the possibility to follow the force in parallel with the conductance of the contact as shown in Figure 35.10. The authors estimate that the force required to break an individual bond in the bulk amounts to 0.8–0.9 nN. For the breaking force of the final contact, however, they found an average value of 1.5 nN. On average, the barrier against breaking the chain is thus indeed much higher.

Conductance (2e2/h) Force (nN)

Auxiliary STM 2

(b)

Cantilever

STM tip

1

1.0 5×

0.9

0 0 –1 –2 –3 0.0

In order to understand the spontaneous formation of atomic chains in gold we need to understand why bonds between lowly coordinated atoms are stronger than bonds between highly coordinated atoms. To some extent, all metals share this property. For the transition metals with an almost full d-shell, however, it is especially large due to the interplay of d-electrons and spelectrons at the Fermi energy. The d-electrons have a localized character and their interaction is thus more bond-like. As the d-band is gradually fi lled from 0 to 10 electrons per atom, this leads to a subsequent occupation of the bonding, nonbonding and antibonding states. The force of these bonds therefore has a parabolic dependence on the fi lling. The sp-electrons, on the contrary, have a spatially delocalized character and can be modeled as a Fermi gas. As the Fermi energy of this gas is dependent on the density of these sp-electrons, the electrons provide a large (Fermi-) pressure. For a free electron gas, this pressure can be deduced from the total energy of the electrons (Kittel, 2004): 3 2 ⎛ 3 π2 N ⎞ U0 = N ⎜ ⎟ 5 2m ⎝ V ⎠

0.5

1.0

1.5

2.0

STM tip displacement (nm)

FIGURE 35.10 A combined plot of the evolution of the conductance of a gold contact and the force exerted on it during stretching. The discontinuities in force coincide with the breaking of metallic bonds. The fi nal minimum in the force curve is on average 1.5 nN. (Reprinted from Rubio-Bollinger, G. et al., Phys. Rev. Lett., 87(2), 026101, 2001. With permission.)

2/3

(35.12)

Here N represents the number of electrons V is the volume The pressure of this gas can thus be obtained following p=−

3

(a)

35.4.2 The Physics behind the Bond Strength

∂U 2 U 0 2 2 = = n (3π2n)2/3 ∂V 3 V 5 2m

(35.13)

where n is the density of electrons. This amounts to a pressure of 2.1 × 1010 Pa for an s-electron density equal to that of the 6s-electrons in gold. At the surface of the metal, the equilibrium of these opposite forces is influenced by the boundary to vacuum. The s-electrons can spill out into the vacuum, releasing the pressure resulting in an increased bonding due to the d-electrons. In atomic chains, where a lot of vacuum is available for the spill-out of the s-electrons the bonding due to the d-electrons increases considerably. However, as was mentioned already in Section 35.4.1, the spontaneous formation of chains is almost absent in pure silver (Rodrigues et al., 2002; Thijssen et al., 2006). Chains are seldom seen for this material and if they appear, they have short lifetimes. Where most chemical properties are shared between materials in the same column of the periodic table, this effect proved to be an exception. When asked for the difference between gold and silver, most will answer as the color. Interestingly, that is exactly where we need to look for the answer. Gold has a higher atomic number and therefore has a larger relativistic correction to the mass of its electrons (Pyykkö, 1988). Due to this enhancement of the mass, m, the Bohr radius for the inner s-electrons, is given by

35-9

Monatomic Chains

a0 =

4πε0 2 me 2

(35.14)

with ε0 the permittivity of vacuum, will be smaller for gold than it is for silver. This contraction will also appear for the outer s-electrons. Their orbitals must be orthogonal to the lower ones, and they feel the same relativistic correction directly as well. The consequence of this smaller radius is that the s-electrons come closer to the nucleus, lowering their energy. The behavior of the d-electrons sharply contrasts to this. They are already orthogonal to the inner s-states by their angular momentum and their orbitals never come close to the nucleus. The s-band therefore shifts to lower energy with respect to the d-band bringing the d-band closer to the Fermi energy. This electronic effect causes the difference in colors for gold and silver. And if the d-band is more than half fi lled, as it is the case for gold, the charge transfer from d-states to s-states leads to a depopulation of antibonding states. The attractive force of the d-bonds in gold is thus further increased by the same relativistic

effects. In bulk gold, the stronger d-bond is compensated by a larger Fermi-pressure. At the surface, this s-pressure can be relieved and the stronger d-bond gives rise to enhanced binding, most prominently in atomic chains.

35.4.3 Relativistic Effects of Atomic Chains In order to verify that chains are the result of relativistic corrections to the electronic density, one would like to switch off relativity and set the speed of light to infi nity. But there are other tests, which are experimentally accessible, because when we look at the argumentation for chain formation, the arguments should equally apply for other heavy transition metals. As long as their d-band is more than half fi lled, depopulation will lead to a larger bonding, strengthening the bond. Smit et al. produced length distributions for the 4d-transition metals Rh, Pd, and Ag, as well as the 5d-transition metals Ir, Pt, and Au (Smit et al., 2001). The results, obtained with a similar technique as explained in Section 35.2.2, are shown in Figure 35.11.

60 Rhodium

40

Counts (a.u.)

Counts (a.u.)

50

30 20 10 0

0

5 10 Plateau length (Å)

15

40 35 30 25 20 15 10 5 0

Iridium

0


15

25 20 Platinum

Palladium

Counts (a.u.)

Counts (a.u.)

20 15 10

10 5

5 0

0 0


15

0

80 70 Silver

60

Counts (a.u.)

Counts (a.u.)

15

50 40 30 20 10 0 0


15

90 80 70 60 50 40 30 20 10 0


15

Gold

0


15

FIGURE 35.11 Length distributions measured for the 4d transition elements Rh, Pd, and Ag in comparison to their 5d counterparts (Ir, Pt, and Au). Where the 4d-transition metals have the length of the final plateau limited to a single peak, attributed to a dimer forming the bridge, the 5d metals have multiple, equidistant peaks, indicating the formation of monatomic chains.

35-10


The clear difference between the length distributions for the 5d-elements and the 4d-elements shows that only the heavier transition metals show spontaneous chain formation. This chemical effect is not shared between elements of the same column in the periodic table, but by elements in the same row. In parallel with this experimental paper, Bahn and Jacobsen (2001) published a DFT-based calculation, where they compare the bonding between bulk atoms and chain atoms. They agree that the stronger relativistic corrections in the 5d-transition metals cause relatively stronger chain bonds, resulting in an increased possibility for chain formation.

35.5 The Conductance Revisited

D=

(k1 + k2 )(k2 + k3 )e

4k1k2 + (k1 − k2 )(k2 − k3 )eiL( k3 + k2 )

iL ( k3 − k2 )

(35.16)

One then obtains for the transmission of the system T (k1 , k2 , k3 ) =

16k12k22 F + G cos2k2 L

(35.17)

Where we used the substitution F = (k1 + k2 )2 (k2 + k3 )2 + (k1 − k2 )2 (k2 − k3 )2

(35.18)

G = (k1 + k2 )(k2 + k3 )(k1 − k2 )(k2 − k3 )

(35.19)

35.5.1 Plane Wave Model for an Atomic Chain In Section 35.1.3, we used elementary quantum mechanics to show that the conductance of a metallic contact at the scale of the Fermi wavelength is independent of its length. In this section, we show that with almost equally elementary quantum mechanics one can also fi nd the first correction to this answer. Let us consider the model depicted in Figure 35.12. Here the different dimensionality of the chain and the electrodes is reflected by a different bandwidth for the conduction electrons. This inference leads to different values for the k-vectors. Although the Fermi energy is situated above the energy barrier, there still occurs scattering of the electronic plane waves at both ends of the chain, situated a distance L apart. The wave functions in the three different regions of this potential are given by ⎧ ΨI ( x , t ) ⎪ ⎨ ΨII (x , t ) ⎪ ⎩ΨIII (x , t )

= ei (k1x −ωt )

+ A e −i(k1x +ωt )

= B ei (k2 x −ωt )

+C e −i (k2 x +ωt )

=De

k1 x

i ( k3 x −ωt )

k2 < k1 x=0

4(k2 / k1 )(k2 / k3 ) [(k2 / k1 ) + (k2 / k3 )]2

Tmax =

(35.15)

In order to obtain a wave function describing the total system one needs to match the wave functions for the different regions, which results in restrictions for A, B, C, and D. The result for D is given by

E

Although the full result of this transmission looks rather cumbersome, the most important physics is dominated by the cosine term. Th is model is the electronic analogue of the optical Fabry–Perot interferometer. In that case, a coherent beam of light is partially reflected by two parallel mirrors. The light is scattered backward and forward between these mirrors, leading to an interference pattern. Th is pattern can be destructive or constructive depending on the matching of the wavelength of the light with the length of the cavity. In case of the metallic chains, the interference in partial electron wave functions will result in a variation of the conductance as a function of length (Sim et al., 2001). The extreme values of the transmission in Equation 35.17 are given by

k3 x=L

FIGURE 35.12 Elementary one-dimensional plane-wave model for the potential representing the contact of a chain of length L, to leads at both ends.

(35.20)

in case of constructive interference (k2 = n · π/L) and

Tmin =

4(k2 / k1 )(k2 / k3 ) [1 + (k2 / k1 )(k2 / k3 )]2

(35.21)

in case of destructive interference. If k1 = k3 in Equation 35.17, which is easily defensible when both electrodes are from the same wire, the maximum transmission in Equation 35.20 becomes Tmax = 1. The cosine in Equation 35.17 depends only on the length of the chain and its k-vector at the Fermi energy. To calculate this Fermi k-vector we need to divide the volume inside the Fermi-sphere (or Fermi line in one dimension) by the volume per k-point, Δk = 2π/L, which will give us the number of occupied states, N =2

2kF (2π/L)

or kF, L =∞ =

πN π = 2L 2d

(35.22)

35-11

Monatomic Chains

35.5.2 Observation of Parity Effects in Real Chains The evolution of conductance during the formation of a gold chain as the one presented in Figure 35.6 does not show a parity effect as predicted. The fact that individual contacts hardly ever present this behavior is explained by the irregular behavior of the contacts. In order to be able to extract the result of the parity effect on the conductance of chains one needs to average over atomic-scale configurations by obtaining enough statistics. In order to have a statistical description on the length dependence of the conductance of a monatomic chain, Smit et al. (2003) repetitively broke a gold contact for over 105 times. The conductance of the data points, Gi of the last plateau of each curve were labeled by an index i, starting at 1 and increasing with successive digitized points for further stretching. Because the probability for the chain to survive decreases with stretching, the number of data points at each index, ni, decreases monotonically with i. The average conductance 〈Gi〉 = (ΣiGi)/ni then takes the shape of the top panel in Figure 35.13. After an initial decrease, due to tunneling through additional states at short distances, this curve indeed shows that the average conductance is oscillating as a function of length. Fitting to a sine function gives a periodicity of 0.47 ± 0.03 nm, which agrees within the error bar with twice the interatomic distance (0.50 ± 0.05 nm). Note that the background of this oscillation at larger lengths is indeed flat, agreeing with Equation 35.11. It is important to note that the curve obtained for the average conductance in Figure 35.13 cannot be directly compared to Equation 35.17. In contrary to the model, the maxima of Figure 35.13 are not situated at 1 G 0. A realistic contact, namely, will be different in several aspects from our model. First of all, the contact will not be perfectly symmetric, leading to a decrease in the maximum conductance. Furthermore, there will be additional

Conductance (2e2/h)

0.95

0.90

0.85

Population

105 104 103

80 Counts (a.u.)

Here, d represents the atom-to-atom distance in the chain. To the subscript of the Fermi k-vector we added L = ∞ to indicate that this expression is deduced for the limit of an infinite chain. The result of Equation 35.22 has an elegant effect on the total transmission of the chain. For chains with an even number of atoms, the transmission in Equation 35.17 will be maximal, while for an odd number of atoms, it will be minimal. The conduction of a one-dimensional chain therefore shows a so-called parity effect. The effect is independent of the interatomic distance so that stretching will not be able to change it. As L increases due to the elasticity of the chain, the atom-to-atom distance d changes likewise. The absolute length is not of importance, only the length expressed in the number of atoms. The interference pattern in the conductance was shown to remain valid also in more realistic numerical simulations (Sim et al., 2001; de la Vega et al., 2004; García-Suárez et al., 2005). The precise length, at which the maximum conductance occurs, however, depends on the precise geometry of the electrodes. Th is can be modeled effectively by including an additional phase shift at the scatter points at both end of the chain (Major et al., 2006).

60 40 20 0 0.0

0.5 1.0 Length (nm)

1.5

FIGURE 35.13 Th ree panels constructed from a series of conductance evolutions of a gold contact. The lower panel shows a length distribution similar as the one in Figure 35.7. The middle panel plots the integral of this length histogram, indicating how frequently a conductance value for a certain length was measured. The upper panel shows the average value of the conductance as a function of length, giving evidence for even–odd oscillations. Note that data points for short lengths have thus been calculated over a population of more than 105 data points, while this number drops toward the right. (From Smit, R.H.M. et al., Phys. Rev. Lett., 91(7), 076805, 2003.)

scatterers close to the contact, which lead to a general decrease of the average conductance. And finally, a plateau of a given length will not always be representing a chain consisting of the same number of atoms, resulting from the irregular process of the chain formation. Although these effects are strong enough to suppress the parity oscillations in the conductance evolution of a single contact, the statistics over several thousands of contacts verify the existence of these oscillations.

35.5.3 Conductance Oscillations for Other Materials Besides gold, parity oscillations were also seen in the conductance of Pt chains (Smit et al., 2003). The experimental result for this material shown in Figure 35.14 does not fit as well to a sine wave but shows a well-defined oscillation. Its periodicity of 0.50 ± 0.04 nm falls within the error bar of twice the atom-toatom distance in the chain (0.46 ± 0.04 nm).

35-12


35.6 Conclusions and Outlook

G (2e2/h)

2.5 2.0

Pt

1.5 1.0

Population

105 104 103 102

Counts (a.u.)

60 40 20 0 0.0

0.5 Length (nm)

1.0

1.5

FIGURE 35.14 The average conductance, number of occurrences and length distribution as a function of length, presented as in Figure 35.13, but now for platinum contacts. (From Smit, R.H.M. From quantum point contacts to monatomic chains: Fabrication and characterization of the ultimate nanowire. PhD Thesis, Leiden University, Leiden, the Netherlands, 2003.)

This is surprising as the theory of Section 35.5.1 is only valid for monovalent metals. The pinning of the k-vector inside the chain leading to Equation 35.22 is not valid for a multivalent metal. For a multivalent metal, the electron energy is not simply given by a 1D parabolic dispersion relation. One needs to consider the k-vectors for all the electronic modes responsible for the conductance (Palacios et al., 2002) to be able to predict the interference pattern. The conductance as a function of length for platinum chains is therefore much more involved. This can also be judged from the slope in the average conductance in Figure 35.14. A strongly decreasing conductance as a function of length is not expected based on Landauer’s expression, Equation 35.11. Two interesting interpretations for this slope have been proposed. De la Vega et al. (2004) state that the slope is part of a very long wavelength period. In their calculations, one of the conduction bands passes the Fermi energy at a small k-value leading to such long wavelengths. GarcíaSuárez et al. (2005), however, are of the opinion that this decrease is a fingerprint of the zigzag configuration of platinum chains. For longer chains, their calculations indicate a stronger zigzag configuration accompanied by a reduction in the conduction. Both these propositions remain to be verified experimentally, but show that the monatomic chains still provide us with interesting problems.

The 5d-transition metals Ir, Pt, and Au show the spontaneous formation of chains of single atoms upon the thinning and stretching of contacts. This chapter presented the TEM and break junction experiments, which lead to their discovery. Considering the experimental data carefully and taking interstitial atoms into account, the atom-to-atom distance for pure metallic chains was proven to be slightly smaller than the bulk distance. A combination of theoretical and experimental investigations leads to the conclusion that these chains are formed due to a transfer of charge from the sp to the d-bands. This transfer is larger for the heavier 5d metals than for their 4d counterparts, due to the stronger relativistic corrections. The result is a stronger interatomic bond at lower coordination, increasing the possibility of chain formation in comparison to an immediate rupture of the contact. The change in bonding strength for atoms with a lower coordination also explains why the bond distance is shorter inside the chain than it is for the bulk. While the discovery of the chains stimulated the development of advanced computational methods, many of the observed phenomena in chains of Au can be explained by simple and elegant plane-wave models. It has been shown that the quantization of conductance in units of 2e2/h follows from the transverse confinement of the wave function of the conduction electrons. If one also takes into account the partial reflection of wave functions at both ends of the chain, the resulting interferences lead to a small oscillation on top of this value. For monovalent metals, this leads to a parity effect, illustrating the wave character of the conduction electrons. For the more complicated electronic structure of Pt and Ir, a plane-wave model is insufficient. Describing the same phenomena appropriately in these systems is a challenge even for the most advanced numerical models. There are, for instance, indications that chains of Pt should spontaneously show magnetic anisotropy (Delin et al., 2004; Smogunov et al., 2008). Experimental studies have not yet been able to verify the presence of magnetism in chains, as it is not directly reflected in the conductance of the contact (Untiedt et al., 2004). A further challenge for the experimentalists is to provide longer chains. For longer chains, the electron–phonon and electron– electron interactions should play a more important role. This might lead to more exotic effects such as a Peierls transition or even the formation of a Luttinger liquid phenomena outside of the independent electron models presented here. Even though monatomic chains appear to be an elementary structure, they provide both experiment and theory with advanced and interesting problems.

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Monatomic Chains

Anglada, E., Torres, J.A., Yndurain, F., and Soler, J.M., 2007. Formation of gold nanowires with impurities: A firstprinciples molecular dynamics simulation. Physical Review Letters, 98(9), 096102. Bahn, S.R. and Jacobsen, K.W., 2001. Chain formation of metal atoms. Physical Review Letters, 87(26), 266101. Csonka, S., Halbritter, A., and Mihály, G., 2006. Pulling gold nanowires with a hydrogen clamp: Strong interactions of hydrogen molecules with gold nanojunctions. Physical Review B, 73(7), 075405. da Silva, E.Z., Novaes, F.D., da Silva, A.J.R., and Fazzio, A., 2004. Theoretical study of the formation, evolution, and breaking of gold nanowires. Physical Review B, 69(11), 115411. Daniel, M.C. and Astruc, D., 2004. Gold nanoparticles: Assembly, supramolecular chemistry, quantum-size-related properties, and applications toward biology, catalysis, and nanotechnology. Chemical Reviews, 104(1), 293–346. de la Vega, L., Martín-Rodero, A., Yeyati, A.L., and Saúl, A., 2004. Different wavelength oscillations in the conductance of 5d metal atomic chains. Physical Review B, 70(11), 113107. Delin, A., Tosatti, E., and Weht, R., 2004. Magnetism in atomicsize palladium contacts and nanowires. Physical Review Letters, 92(5), 057201. Finbow, G.M., Lynden-Bell, R.M., and McDonald, I.R., 1997. Atomistic simulation of the stretching of nanoscale metal wires. Molecular physics, 92(4), 705–714. Frederiksen, T., Paulsson, M., Brandbyge, M., and Jauho, A.P., 2007. Inelastic transport theory from first principles: Methodology and application to nanoscale devices. Physical Review B, 75(20), 205413. Gall, K., Diao, J.K., and Dunn, M.L., 2004. The strength of gold nanowires. Nano Letters, 4(12), 2431–2436. García-Suárez, V.M. et al., 2005. Conductance oscillations in zigzag platinum chains. Physical Review Letters, 95(25), 256804. Häkkinen, H., Barnett, R.N., and Landman, U., 1999. Gold nanowires and their chemical modifications. Journal of Physical Chemistry B, 103(42), 8814–8816. Kittel, C., 2004. Introduction to Solid State Physics. John Wiley & Sons Ltd., New York. Kizuka, T. et al., 1997. Cross-sectional time-resolved high-resolution transmission electron microscopy of atomic-scale contact and noncontact-type scannings on gold surfaces. Physical Review B, 55(12), R7398–R7401. Koizumi, H., Oshima, Y., Kondo, Y., and Takayanagi, K., 2001. Quantitative high-resolution microscopy on a suspended chain of gold atoms. Ultramicroscopy, 88(1), 17–24. Krüger, D. et al., 2002. Pulling monatomic gold wires with single molecules: An ab initio simulation. Physical Review Letters, 89(18), 186402. Kurui, Y., Oshima, Y., Okamoto, M., and Takayanagi, K., 2008. Integer conductance quantization of gold atomic sheets. Physical Review B, 77(16), 161403R.

35-13

Legoas, S.B., Galvão, D.S., Rodrigues, V., and Ugarte, D., 2002. Origin of anomalously long interatomic distances in suspended gold chains. Physical Review Letters, 88(7), 076105. Major, P. et al., 2006. Nonuniversal behavior of the parity effect in monovalent atomic wires. Physical Review B, 73(4), 045421. Ohnishi, H., Kondo, Y., and Takayanagi, K., 1998. Quantized conductance through individual rows of suspended gold atoms. Nature, 395(6704), 780–783. Oncel, N., 2008. Atomic chains on surfaces. Journal of Physics: Condensed Matter, 20, 393001. Palacios, J.J. et al., 2002. First-principles approach to electrical transport in atomic-scale nanostructures. Physical Review B, 66(3), 035322. Pyykkö, P., 1988. Relativistic effects in structural chemistry. Chemical Reviews, 88(3), 563–594. Rocha, A.R. et al., 2006. Spin and molecular electronics in atomically generated orbital landscapes. Physical Review B, 73(8), 085414. Rodrigues, V. et al., 2002. Quantum conductance in silver nanowires: Correlation between atomic structure and transport properties. Physical Review B, 65(15), 153402. Rubio-Bollinger, G. et al., 2001. Mechanical properties and formation mechanisms of a wire of single gold atoms. Physical Review Letters, 87(2), 026101. Sánchez-Portal, D. et al., 1999. Stiff monatomic gold wires with a spinning zigzag geometry. Physical Review Letters, 83(19), 3884–3887. Segovia, P., Purdie, D., Hengsberger, M., and Baer, Y., 1999. Observation of spin and charge collective modes in onedimensional metallic chains. Nature, 402(6761), 504–507. Sim, H.S., Lee, H.W., and Chang, K.J., 2001. Even-odd behavior of conductance in monatomic sodium wires. Physical Review Letters, 87(9), 096803. Smit, R.H.M., 2003. From quantum point contacts to monatomic chains: Fabrication and characterization of the ultimate nanowire. PhD Thesis, Leiden University, Leiden, the Netherlands. Smit, R.H.M. et al., 2003. Observation of a parity oscillation in the conductance of atomic wires. Physical Review Letters, 91(7), 076805. Smit, R.H.M., Untiedt, C., Yanson, A.I., and van Ruitenbeek, J.M., 2001. Common origin for surface reconstruction and the formation of chains of metal atoms. Physical Review Letters, 87(26), 266102. Smogunov, A. et al., 2008. Colossal magnetic anisotropy of monatomic free and deposited platinum nanowires. Nature Nanotechnology, 3(1), 22–25. Sørensen, M.R., Brandbyge, M., and Jacobsen, K.W., 1998. Mechanical deformation of atomic-scale metallic contacts: Structure and mechanisms. Physical Review B, 57(12), 3283–3294. Takai, Y. et al., 2001. Dynamic observation of an atom-sized gold wire by phase electron microscopy. Physical Review Letters, 87(10), 106105.

35-14

Thijssen, W.H.A., Marjenburgh, D., Bremmer, R.H., and van Ruitenbeek, J.M., 2006. Oxygen-enhanced atomic chain formation. Physical Review Letters, 96(2), 026806. Todorov, T.N., Hoekstra, J., and Sutton, A.P., 2001. Currentinduced embrittlement of atomic wires. Physical Review Letters, 86(16), 3606–3609. Untiedt, C. et al., 2002. Calibration of the length of a chain of single gold atoms. Physical Review B, 66(8), 085418. Untiedt, C., Dekker, D.M.T., Djukic, D., and van Ruitenbeek, J.M., 2004. Absence of magnetically induced fractional quantization in atomic contacts. Physical Review B, 69(8), 081401.


Wallis, T.M., Nilius, N., and Ho, W., 2002. Electronic density oscillations in gold atomic chains assembled atom by atom. Physical Review Letters, 89(23), 236802. Yanson, A.I. et al., 1998. Formation and manipulation of a metallic wire of single gold atoms. Nature, 395(6704), 783–785.

36 Ultrathin Gold Nanowires Takeo Hoshi Tottori University and Japan Science and Technology Agency

Yusuke Iguchi The University of Tokyo*

Takeo Fujiwara The University of Tokyo and Japan Science and Technology Agency

36.1 36.2 36.3 36.4

Introduction ...........................................................................................................................36-1 Basic Properties of Solid Gold .............................................................................................36-1 Metal Nanowire and Quantized Conductance ................................................................ 36-4 Helical Multishell Structure................................................................................................ 36-6 Overview • Two-Stage Formation Model • Simulation of Formation Process • Analysis of Electronic Structure • Discussions

36.5 Summary and Future Aspect .............................................................................................36-12 Appendix 36.A: Note on Quantum-Mechanical Molecular Dynamics Simulation ...............36-14 References.........................................................................................................................................36-15

36.1 Introduction Ultrathin nanowires of gold (Au) and other metals have been studied intensively, particularly from the 1990s, as a possible foundation of nano electronics (see Agraït et al. (2003) for a review). They are fabricated as nano-meter-scale contacts within two electrode parts and are composed of a couple of atoms in their wire length and/or diameter. Monoatomic chain, the thinnest wire, was also fabricated. The metal nanowires can show quantized conductance, even at room temperature, which is completely different from Ohm’s law in macroscale samples. Structural and transport properties of nanowires were investigated by (1) fi ne experiments, such as high-resolution transmission electron microscopy (HRTEM), scanning tunneling microscope (STM), and atomic force microscopy (AFM) and (2) atomistic simulations with quantum-mechanical theory of electrons. Many experiments and simulations lead us to several common understandings among metal nanowires of various elements (Agraït et al. 2003). This chapter focuses on the unique properties of Au nanowires. Common properties between Au and other nanowires are also discussed so as to figure out the uniqueness of Au nanowire from a general scientific viewpoint. As a unique and fascinating property of Au nanowire, helical multishell structure was reported in 2000 (Kondo and Takayanagi 2000), as in carbon nanotube (Iijima 1991, Dresselhaus et al.

2001). A shell of helical Au nanowires consists of a folded hexagonal sheet, while carbon nanotube (Iijima 1991, Dresselhaus et al. 2001) consists of a folded graphene sheet. This chapter focuses particularly on the helical multishell nanowire of Au both for phenomena and for proposed formation mechanisms. The discussion is based on electronic structure and covers a wide range of nanostructures of Au and other metals. This chapter is organized as follows: Section 36.2 is devoted to a tutorial for atomic structure and electronic wave functions of solid Au with other materials. Section 36.3 summarizes the structure and transport properties of Au and other metal nanowires. Section 36.4 focuses on the helical multishell Au nanowire, as the main topic of this chapter. Finally, in Section 36.5, a summary of this chapter is given and a future aspect is addressed for establishing the foundation of nano electronics. The appendix is devoted to quantum-mechanical molecular dynamics (QM-MD) simulation, which is important for understanding points of this chapter.

36.2 Basic Properties of Solid Gold The basic properties of Au are summarized with those of other elements. Figure 36.1a shows periodic table up to period 6 elements and Figure 36.1b a part of periodic table that includes Au. Several elements in Figure 36.1b are discussed in this chapter. Among the elements, copper (Cu), silver (Ag), and Au are called

* He is currently affl iated with NEC Corporation, Kanagawa, Japan.

36-1

36-2


H

He

Li Be

B C N O F Ne

Na Mg

Al Si P

S Cl Ar (a)

K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe Fr Ba La Hf Ta W Rc Os Ir Pt Au Hg Tl Pb Bi Po At Rn (a) 27

Co 3d7 4s2

Rh 4d8 5s1

5d9

Ni

3d8 4s2

45

Ir

28

46

Pd 4d10

77

Pt

78

5d9 6s1

(b)

29

(d)

Cu

(c)

3d10 4s1

47

Ag

4d10 5s1

79

(e)

Au

(f )

5d10 6s1

(b)

FIGURE 36.1 (a) Periodic Table up to Period 6 elements. (b) A part of Periodic Table that contains gold (Au). The atomic number is plotted at the upper right corner of each box and the valence electron configuration is plotted at the lower area of each box. All the elements shown in (b) form FCC structure in solid state, except cobalt (Co).

noble metals. They have 10 n d electrons and 1 (n + 1) s electron (nd10ns1) as valence electrons at each atom, where n = 3, 4, 5 for Cu, Ag, and Au, respectively. The labels of “s” and “d” indicate the character of atomic orbitals. In general, an atomic orbital is described as R (r )Q (θ, φ )

(36.1)

in polar coordinate. The nonspherical distribution comes from the part of Q(θ, ϕ). The wave function of s electron is spherical (Q(θ, ϕ) = const) and that of d electron is not. Figure 36.2 illustrates the (non-)spherical distribution Q(θ, ϕ) of s and d wave functions by plotting the function of r = r(θ, ϕ) = |Q(θ, ϕ)|. For example, the xy orbital spreads mainly on the xy plane. Another type of nonspherical atomic orbital, “p” orbital, also appears later in this chapter. For details of the atomic orbitals, the interested reader can refer to elementary textbooks on quantum mechanics. All the elements in Figure 36.1b form solid in facecentered-cubic (FCC) structure, shown in Figure 36.3a, except cobalt (Co) that forms close-packed hexagonal (HCP) structure. The lattice constant, the distance between the A and B atom in Figure 36.3a, is 3.61, 4.09, and 4.08 Å for Cu, Ag, and Au, respectively. In this chapter, as in many textbooks, x, y, z axes are defined and normalized so that the cubic box shown in

FIGURE 36.2 Schematic figure of the nonspherical forms |Q(θ, ϕ)| of (a) s orbital and (b)–(f) five d orbitals; (b) xy orbital, (c) yz orbital, (d) zx orbital, (e) x2–y 2 orbital, (f) 3z2–r2 orbital. The xy orbital spreads mainly on the xy plane.

Figure 36.3a gives the cubic region of 0 ≤ x, y, z ≤ 1. In Figure 36.3a, the eight corner atoms labeled by A, B, C, D, E, F, G, and H are described, for example, by (1,0,0) and (0,1,0) and the six face-center atoms, such as the atom I, are described, for example, (1/2,1/2,0), (0,1/2,1/2) in the normalized coordinate. FCC and HCP structures are ones in close packing (highest average density) as three-dimensional lattice and each atom has 12 nearest neighbor atoms. For example, the atom placed at (0,0,0) has the 12 nearest neighbor atoms that are placed at (±1/2, ±1/2, 0), (±1/2, 0, ±1/2), (0, ±1/2, ±1/2). Several planes in the FCC structure are drawn in Figure 36.3b through d. Here the notations of planes are explained. A plane of (x/l) + (y/m) + (z/n) = 1 is indicated by an index of (l, m, n). The negative surface index is written with a bar. For example, the index of (1–11) means the plane of (x/(−1) ) + (y/1) + (z/1) = 1 or −x + y + z = 1. A direction vector is denoted as [abc] and a “[abc] nanowire” is a wire of which axis is in the [abc] direction. Moreover, equivalent planes are called by “type” in this chapter. For example, the lattice structures on (111) and (1–11) planes of FCC structure are equivalent to those shown in Figure 36.3d and these planes are called “(111)-type” planes. The (001)-type, (110)type, and (111)-type planes are drawn in Figure 36.3b through d, respectively. A (111)-type plane has a hexagonal structure with triangular tiles, as shown in Figure 36.3d, and is one in close packing as two-dimensional lattice. Electronic states among noble metals in FCC solid are described in Figures 36.4 and 36.5. The data are given by

36-3

Ultrathin Gold Nanowires D

[1

10

]

C

A B

D

C

A

B

[010]

I

H G

E

[001]

F

(a)

A

[100]

(b)

C I

A

G

E

C

[001]

I

F – [111]

[110]

– [110] (c)

(d)

FIGURE 36.3 (a) FCC structure. “Bonds” are drawn between nearest neighbor atoms. (b) (001)-type, (c) (110)-type, and (d) (111)-type planes of FCC structure. The atoms marked as A, B, C, D, E, F, G, H, and I are common among (a)–(d). Shapes drawn by lines in (b), (c), and (d) will appear later in this chapter. In (c), two successive atomic layers are drawn and the atoms in the two atomic layers are depicted by open and filled circles, respectively.

Wave function (e/Å–3/2)

5

Cu

Ag

Au

4 3 2 1 0 –1 –2 0

(a)

0.5

1

0 (b)

0.5

1

Distance (Å)

1.5 0

0.5

1

1.5

(c)

FIGURE 36.4 Radial wave functions R(r) of noble metals, (a) copper, (b) silver, and (c) gold, in the bulk FCC structure. See text for details.

a modern quantum-mechanical calculation for electronic structure. The calculations are carried out by first-principles theory, the density-functional theory with the linear-muffin-tin-orbital method (Andersen and Jepsen 1984). See the textbook by Martin (2004) for an overview of modern electronic structure theories. Since Au is a heavy element, relativistic effect is included in these calculations, as scaler-relativistic formulation. See textbooks of quantum mechanics, such as Schiff (1968), for relativistic effect

and scaler-relativistic formulation. Figure 36.4 shows the radial wave functions R(r) of the d orbitals. An (nd) orbital has (n–2) nodes in the radial wave function R(r). The Cu wave function, a 3d wave function, does not have a node and the Ag and Au wave functions, 4d and 5d wave functions, have one and two node(s), respectively. Figure 36.5 shows the electronic density of states (DOS), or the energy spectrum of electronic states. The dashed line in Figure 36.5 indicates the Fermi level EF, which means that

36-4

Handbook of Nanophysics: Nanotubes and Nanowires EF

DOS (states/eV cell)

5

EF

EF

Cu

Ag

Au

4 3 2 1 0

–10

–5

(a)

0

–10 (b)

–5 Energy (eV)

0

–10

–5

0

(c)

FIGURE 36.5 Density of states, or the energy spectrum of electronic states, in bulk FCC solids among noble metals, (a) copper, (b) silver, and (c) gold. See text for details.

the electrons occupy the energy region of E < EF. As a common feature of the three elements, the d band is narrow and fully occupied, while the s band is broad and partially occupied. In the case of Au, for example, the d band lies in the narrow energy region of −10 eV ≤ E ≤ −4 eV, which is within the occupied energy region. The s band, on the other hand, lies in the region from −13 eV up to the right end of the graph. The above feature indicates that the 10 d electrons form a closed electronic shell and are nearly localized at atomic regions, while the 1 s electron is extended and can contribute to electrical current. Therefore, these solids are metallic and electrical current is observed with Ohm’s law.

36.3 Metal Nanowire and Quantized Conductance Nanowires or nanoscale contacts of metals are fabricated by deformation processes with, for example, STM tip or mechanically controllable break junction (Agraït et al. 2003). Figure 36.6 illustrates the fabrication process by a STM tip, in which a nanowire is formed between two electrode parts. Real-space image, like the one in Figure 36.6, is obtained by HRTEM. A formation process of a Au nanocontact was observed as successive HRTEM images (Kizuka et al. 1997, Kizuka 1998). Quantized conductance of metal nanowires was reported, even at room temperature, and relatively clear quantized values were observed among noble metals and alkali metals (Agraït et al. 1993, Krans et al. 1993, Pascual et al. 1993, Olesen et al.

(a)

(b)

1994, Brandbyge et al. 1995, Krans et al. 1995, Muller et al. 1996, Rubio et al. 1996, Costa-Krämer et al. 1997, Hansen et al. 1997, Yanson et al. 1999, Yanson et al. 2000, Yanson et al. 2001, Agraït et al. 2003, Smit et al. 2003, Mares et al. 2004, Bettini et al. 2005, Mares and van Ruitenbeek 2005). The quantized conductance G is defi ned as G = nG0 ,

(36.2)

2e 2 h

(36.3)

where n is an integer and G0 ≡

is the conductance unit defined by the charge of electron e (>0) and Planck’s constant h. Figure 36.7 is a schematic figure of conductance trace in thinning process, such as the process of Figure 36.6c and d. In Figure 36.7, conductance plateaus of G ≈ 2G 0, 1G 0 appear and the conductance reaches G ≈ 0, when the wire is broken. In experimental research, the histogram of conductance values is constructed from many independent samples and a sharp peak at an integer value (G = nG 0) in the histogram is assigned to be a quantized conductance. A simultaneous observation of structure and transport was realized by a combined experiment of HRTEM and STM (TEMSTM) (Ohnishi et al. 1998, Erts et al. 2000, Kizuka et al. 2001a), which is crucial for understanding nanowires, because the conductance value, in general, does not determine atomic structure

(c)

(d)

FIGURE 36.6 Schematic figures for the snapshots, (a)–(d), during the fabrication process of nanowire using a STM tip. Atoms are depicted as balls.

36-5

Ultrathin Gold Nanowires 3

Conductance (G/G0)

Plateau with G = 2G0

2

Plateau with G = 1G0

1

Wire is broken (G = 0)

0 Time

FIGURE 36.7 process.

Schematic figure of conductance trace in thinning

uniquely. A TEM-STM experiment in 1998 (Ohnishi et al. 1998) obtained a direct image of a monoatomic Au chain with the length of several atoms and observed a quantized conductance in G = G 0. Force measurement was realized by a combined experiment of AFM and STM (Rubio et al. 1996) and shows that a jump in conductance trace occurs with a jump in the force. The result implies that a jump in conductance trace is caused by the plastic deformation of nanowire. Stress–strain curve was obtained by the combined experiment of TEM and AFM (Kizuka et al. 2001b, Erts et al. 2002, Kizuka 2008) and the above statement was confirmed. In addition, the simultaneous measurement of stiff ness and conductance was realized by mechanically controllable break junction with a force sensor and applied to Au and platinum nanowires (Rubio-Bollinger et al. 2004, Valkering et al. 2005, Shiota et al. 2008). The quantum mechanical theory of electrical current has a rigorous foundation of nonequilibrium Green’s function, which can be seen in the textbook by Datta (1995). Here only a resultant formulation is briefly explained. In quantum mechanics, electrons are described as “waves” with discretized energy levels and electrical current is given by the “wave modes” that are extended through nanowires. Such extended “wave modes” are usually called channels. The integer n in Equation 36.2 is the number of the channels. A general expression of the conductance is G = G0

∑τ , i

(36.4)

i

where τi is the transmission rate of the ith “wave mode” (0 ≤ τi ≤ 1) (Datta 1995). Equation 36.4 will be reduced to Equation 36.2, when all the channels are perfectly ballistic (τi = 1). The simulation of atomic structure is crucial for metal nanowires, since the transport properties are sensitive to atomic structure. A pioneering theoretical work in 1993 (Todorov and Sutton 1993) investigated the relation between the atomic structure and the conductance, in which the dynamics of atomic structure is determined by a classical potential. Nowadays, quantum

mechanical molecular dynamics (QM-MD) simulation is a standard simulation tool for nanowires and other nanostructures. In QM-MD simulations, an effective Schrödinger equation is solved at each time step and atomic structures are updated under the change of electronic states. See Appendix 36.A for a tutorial of QM-MD simulation. QM-MD simulation enables us to investigate (1) simultaneous discussion of structure and electronic properties in a single theoretical framework with quantum mechanics of electrons and (2) systematic research among various elements. Many simulations were carried out for monoatomic Au chain and related materials (Brandbyge et al. 1995, 1999, Sorensen et al. 1998, Okamoto and Takayanagi 1999, Sánchez-Portal et al. 1999, da Silva et al. 2001, Agraït et al. 2003, Fujimoto and Hirose 2003, da Silva et al. 2004). After intensive works of simulation and experiment, it is established that a monoatomic Au chain is formed and shows a quantized conductance of G = 1G 0, since only one s orbital contributes to the current. “Thicker” nanowires, nanowires thicker than monoatomic chains, can show complex structural and/or transport properties. An interesting viewpoint is shell effect that enhances the stability of specific integer values in the conductance. The conductance histograms in alkali and noble metals were analyzed from the viewpoint of the shell effect (Yanson et al. 1999, 2000, 2001, Mares et al. 2004, Mares and van Ruitenbeek 2005). In the context of the shell effect of alkali and noble metals, the stability of elliptical metal nanowires was investigated quantum mechanically within a free-electron model in continuum media (Urban et al. 2004). Other theoretical investigations with free-electron models are found in the reference lists of the above papers or a review (Agraït et al. 2003). It is noteworthy that a difference of alkali and noble metals is the fact that the valence band of alkali metals consists of one s electron, while that of noble metals consists of 1s electron and 10d electrons, as shown in Figure 36.5. A TEM-STM experiment was carried out with thicker [110] Au nanowires for the direct relation between the structure (the shape of the cross section) and the conductance (Kurui et al. 2007). The appearance of helical multishell Au nanowire (Kondo and Takayanagi 2000) is a fascinating property of “thicker” nanowires and will be discussed in the rest of this chapter. As another systematic research of “thicker” nanowires, a TEM-STM experiment was carried out for Au nanocontacts parallel to the [001], [111], and [110] directions (Oshima et al. 2003a). The paper suggests that the variety in the crystalline orientation of the contacts can be the origin of the observed variety in the conductance trace among samples. The sliding of a twin boundary in a Au nanocontact was reported by direct HRTEM observation (Kizuka 2007). A recent TEM-STM experiment (Kurui et al. 2008) found (111)-type and (001)-type atomic sheets between electrode parts and they show quantized conductance. To conclude this section, brief comments are made on metal nanowires in which p and d orbitals contribute to current. Such nanowires exhibit a complex transport behavior, even in monoatomic chains. For example, nanowires of aluminum has s and p valence orbitals and show that plateau structure in conductance trace is less regular but the plateaus-like structures are still

36-6

observed nearly at integer conductance values (Krans et al. 1993, Cuevas et al. 1998, Yanson et al. 2008). In nanowires of nickel, a transition metal, the quantized conductance changes systematically under varying external parameters, the temperature (above and below the Curie temperature), and the applied magnetic field (Oshima and Miyano 1998).

36.4 Helical Multishell Structure 36.4.1 Overview In 2000, helical multishell Au nanowires were synthesized and observed by HRTEM image (Kondo and Takayanagi 2000). They were fabricated by focusing an electron beam on a thin fi lm (Kondo and Takayanagi 1997). The wire axis of the helical nanowires is the [110]-type direction of the original FCC geometry (see Figure 36.3b and c). The outermost shell is a folded (111)-type (hexagonal) atomic sheet (see Figure 36.3d) and helical around the nanowire axis. A single shell helical structure was synthesized later (Oshima et al. 2003b). Experiments also observed the thinning process (Oshima et al. 2003c) and the conductance (Oshima et al. 2006) of the multishell helical structures. Hereafter multishell structures are denoted by the numbers of atoms in each shell. For example, a “14-7-1 nanowire” is a rod with three shells in which the outer, middle, and inner shells have fourteen, seven, and one atom(s) in the section view, respectively. The shape of the “6-1 nanowire” is depicted by lines in Figure 36.3c. The observed multishell configurations are quite specific and are characterized by “magic numbers.” The multishell structures in the 7-1, 11-4, 13-6, 14-7-1, 15-8-1 structures were experimentally observed. These numbers are called “magic numbers,” since the difference of numbers between the outermost and the next outermost shells is seven, except the cases of the 7-1 structure. Such a rule of the observed multishell configurations implies that there is an intrinsic mechanism to form the specific multishell configurations. The transport property of helical multishell Au nanowire was investigated both in theory (Ono and Hirose 2005) and in experiment (Oshima et al. 2006), which will be discussed in Section 36.5. The appearance of helical metal wires was investigated by simulations with different methodologies. Before the experimental report of helical Au nanowire (in 1998), various “exotic” structures of nanowires were predicted with potentials of Al and Pb and one of them is helical structure (Gülseren et al. 1998). After the experimental report, several investigations, such as Bilalbegović (2003) and Lin et al. (2005), were carried out with classical potentials. Quantum mechanical simulations were also carried out (Tosatti et al. 2001, Senger et al. 2004, Yang 2004). A first principles calculation reported that the tension of nanowires gives the minimum values when the number of atoms of the lateral atomic row on the outermost shell is seven and the nanowire is helical (Tosatti et al. 2001). It is also showed that the tension does not have the minimum in model Ag nanowires. Another first principles calculation


was carried out for nanowires with atoms from three to five on the lateral atomic row on the outermost shell, and showed that helical nanowires are not the configuration of the minimum energy but of the minimum tension (Senger et al. 2004). Although these studies gave important progress on theory, they did not explain why the helical structures are formed in the specific multishell configurations with “magic numbers.” Pt nanowires were also synthesized with the same type of helicity (Oshima et al. 2002). Pt is placed at a neighbor of Au in Periodic Table (see Figure 36.1b) and its electronic configuration (5d96s1) is similar to that of Au (5d106s1). The above experiment implies that the mechanism of the formation of helical nanowires is generic between Au and Pt and may be inherent among some other elements. In 2007, a theory (Iguchi et al. 2007) was proposed for the formation model of helical multishell nanowires. In this model, the transformation consists of two stages. At the first stage, the outermost shell is dissociated from the inner shell to move freely. At the second stage, an atom row slips on the wire surface and a (001)-type (square) face (Figure 36.3b) transforms into a folded (111)-type (hexagonal) one (Figure 36.3d) with helicity. The driving force for the helicity comes from the nature of nonspherical 5d electrons and a (111)-type surface structure is energetically favorable for 5d electrons. The theory contains the following points: (1) The theory explains the observed multishell configurations with “magic numbers” systematically. (2) The theory was validated by QM-MD simulations for Au and Cu with tight-binding form Hamiltonian (see Appendix 36.A). (3) The theory gives a general understanding among helical nanowire structures of Au and Pt and several reconstructed structures of equilibrium surfaces. After the proposal of the two-stage model (Iguchi et al. 2007), several related simulations were carried out for the formation of helical Au nanowire within tight-binding form Hamiltonian (Amorim and da Silva 2008, Fujiwara et al. 2008).

36.4.2 Two-Stage Formation Model Hereafter, the multishell helical structures are systematically constructed, according to the proposed theory of two-stage formation model (Iguchi 2007; Iguchi et al. 2007). First, a set of ideal [110] nanowires are prepared under two conditions: (a) there is no acute angle on the surface because of diminishing surface tension, and (b) there is no (001) side longer than any (111) side since the surface energy of a (001) surface is higher than that of (111). These structures are called “reference” structures in this chapter. Figure 36.8 shows the section views of the “reference” structures in the 6-1, 10-4, and 12-6 multishell configurations. Among them, the outermost shell has six more “bonds” on the lateral row than the inner shell, shown by bold lines in Figure 36.8. The formation process of the helical 7-1 structure is depicted schematically in Figure 36.9. When an atom row appears at a (001)type surface on the outermost shell of the reference 6-1 structure (Figure 36.9a), the structure turns into a 7-1 structure (Figure 36.9b)

36-7

Ultrathin Gold Nanowires

A and C are (111)-type (hexagonal) ones. Dashed lines connect the same atoms at the right ends and the left ends. When the number of atoms on the lateral row in the outermost shell is odd, the surface reconstruction brings the helicity to the nanowire inevitably. A slip of atom row transforms the surface region B from (100)-type surface into a folded (111)-type one with helicity. As a result, the helical 7-1 structure is formed, as in Figure 36.9d.

6-1

(b)

36.4.3 Simulation of Formation Process

10-4

[001]

(a)

– [ 110] [110] (c)

12-6-1

FIGURE 36.8 Section view of a set of “reference” [110] nanowires. The (a) 6-1, (b) 10-4, and (c) 12-6-1 structures are shown. See text for details. The shape of the 6-1 structure can be found in Figure 36.3c.

and is called “initial” structure in this chapter. The outermost shell has seven more atoms on a lateral atom row than the inner shell and the outermost shell can have room for atom row slip. This is the origin of the “magic numbers.” Figure 36.9c shows an expanded lateral surface of the initial 7-1 structure. In Figure 36.9c, the surface region of B is (001)-type (square) lattice, while the surface regions of

A QM-MD simulation was carried out, so as to confirm the proposed process model for forming helical structures. The simulation was carried out with a tight-binding form Hamiltonian that was used for several simulations of Au nanowire (da Silva et al. 2001, 2004). Details and theoretical foundations of the QM-MD simulations are described in Appendix 36.A. A Relaxation process with thermal fluctuation was simulated. The samples in the simulations were finite [110] nanowires with the initial structures of the ideal 7-1, 10-4, 11-4, 12-6-1, 13-6-1, 15-7-1, and 15-8-1 structures. The simulation results are summarized in Figure 36.10 as section views, except those of the 10-4 and 12-6-1 structures. The initial structures appear in the left panels of Figure 36.10a through e. They are “reference” structures with one additional atom row on the outermost shell, as explained in Section 36.4.2. The additional atom rows are marked by arrows. The rod length is nine layers in the 7-1, 10-4, and 11-4 nanowires, seven layers in the 12-6-1 and 13-6-1 nanowires, or six layers in the 14-7-1, and 15-8-1 nanowires, respectively. Here the layer unit is defined as the periodic unit of the ideal FCC structure and is composed of two successive atomic layers shown in Figure 36.3c. The numbers of atoms in the simulations are from 76 to 156.

“Reference” (001)-type surface

Initial

A

(a) A Initial

(c)

B

(001)-type surface B

C

(b) C

Helical

(d)

A

B

C

FIGURE 36.9 Schematic figure of the formation of the helical 7-1 structure: (a) “Reference” 6-1 structure. Two (001)-type surfaces appear and the other surfaces are (111)-type ones. (b) Initial 7-1 structure. The areas A and C are the (111)-type surfaces and the area B is the (001)-type surface. (c) Expanded lateral surface of the initial 7-1 structure of which the section view is given in (b). Dashed lines connect the same atoms at the right and the left ends. (d) Expanded lateral surface of the helical 7-1 structure transformed from the ideal non-helical structure of (c).

36-8

(a)


4Å

(b) 4 Å

(a)

(b)

(c)

(d)

–445

(c) 4 Å

Energy (eV)

–450

(a)

–455 –460

First stage (dissociation of the outermost shell)

–470 100 (e)

FIGURE 36.10 Section views in the relaxation process of the Au nanowires into helical structures; (a) 7-1 structure at the initial (left), 400-th (middle), and 5000-th iteration steps (right); (b) 11-4 structure at the initial (left), 400-th (middle), and 6000-th iteration steps (right); (c) 13-6-1 structure at the initial (left), 750-th (middle), and 9000-th iteration steps (right); (d) 14-7-1 structure at the initial (left), 400-th (middle), and 7000-th iteration steps (right); and (e) 15-8-1 structure at the initial (left), 400-th (middle), and 5000-th iteration steps (right). An atom row is marked by arrow for each system. See text for details.

The boundary condition is imposed by fixing the center of gravity of the top and bottom layers of the nanowires. The finite-temperature dynamics is realized by the thermostat technique and the simulations were carried out at T = 600 and 900 K, which are lower than the melting temperature (1337 K). One iteration step corresponds to δt = 1 fs. Helical structures appear in all cases except the 12-6-1 nanowire at 600 K. The results shown in Figure 36.10 are those at T = 600 K for (a), (b), (c), and (e) and at T = 900 K for (d). As a typical case, Figure 36.11a through d shows a set of side views in the case of the 11-4 structure at T = 600 K. Figure 36.11a through d indicates that the square tiles on the (001)-type surface at the initial structure are transformed into the hexagonal (111)-type surface, as in Figure 36.9c and d. The transformation propagates from the top and bottom of the nanowire. In the snapshot of Figure 36.11c, for example, the transformation has been completed except the region near the third lowest layer and the transformation. The energy of the nanowire is plotted in Figure 36.11e as the function of iteration step and decreases almost monotonically after the 1000-th iteration step.

(c)

–465

(d) 4 Å

(e) 4 Å

Second stage (atom slip on nanowire surface)

(d)

(b) 200

500

1,000 2,000 Iteration steps

5,000

10,000

FIGURE 36.11 Formation of helical 11-4 Au nanowire in relaxation process. Side views at the (a) 500-th, (b) 2000-th, (c) 4000-th, and (d) 6000-th iteration step. Dark atoms are those that are placed initially on the (001)-type (square) surface and are transformed into a (111)-type (hexagonal) surface. (e) Change of the energy during the formation process. The iteration steps of (a), (b), (c), and (d) are indicated by arrows.

It should be noted that the present simulation is different from experiment in several points. For example, the time scale of the process is quite short, on the order of 10 ps, owing to the practical limit of computational resource. Therefore, the simulation result should be understood so that it captures an intrinsic energetical mechanism of the real process.

36.4.4 Analysis of Electronic Structure The mechanism of the two-stage formation process is investigated by analyzing local electronic structure. A typical case of the 11-4 structure with T = 600 K is picked out for explanation. Local density of states (LDOS) and “local energy” are used throughout the analysis. In general, LDOS is defined for each atom and can be decomposed into the contributions of each orbital. The profile of LDOS means the energy spectrum of electronic states at a local region near the atom. “Local energy” is defi ned for each atom or orbital by the energy integration of LDOS within the occupied energy region. A decrease of the local energy during the process means that the atom gains the energy by a binding mechanism. Figure 36.12 shows the LDOS of specific atoms at specific iteration steps. The section views of Figure 36.12a are identical to those of Figure 36.10b. The LDOS for the atoms A, B, and C in

36-9

Ultrathin Gold Nanowires Initial

First stage

B

B

B

A (a)

A

C Atom A

1.5

Second stage

A

C Atom C

C

Atom B

s

0.14 eV

s

–0.59 eV

s

–0.38 eV

d

0.02 eV

d

–0.42 eV

d

–0.15 eV

Density of states (states/eV)

1 0.5 0 4 3 2 1 0 4

Total

0.21 eV

Total

–1.16 eV

Total

–0.68 eV

3 2 1 0 (b)

–6 –4 –2

0

2

Energy (eV)

4

6

–6 –4 –2

0

2

Energy (eV)

4

6

–6 –4 –2

0

2

4

6

Energy (eV)

FIGURE 36.12 Analysis of the two-stage formation process in the 11-4 Au helical nanowire. (a) The section views with specifying the atoms A, B, and C. The snapshots are the same as those in Figure 36.10b. (b) Local density of states of the atoms A, C, and B at different iteration steps. The solid and broken lines in (b) are at the initial state and at the 500-th iteration step respectively for the atoms A and C, and at the 500-th and 5000-th steps for the atom B. The difference of the local energy between the two iteration steps is written at the right corner of each panel. The vertical broken line indicates the Fermi level. The upper, middle, and lower panels of (b) are the partial densities of states of s orbital, d orbital, and the total density of states, respectively. In each panel, the difference of the local energy between the two iteration steps is written at the upper right corner of each panel.

Figure 36.12a are plotted in Figure 36.12b. The solid and broken lines are the LDOS profile at the initial and 500-th steps for the atoms A and C, and are at the 500-th and 5000-th steps for the atom B, respectively. The vertical broken line indicates the Fermi level and the positions of the Fermi level are almost the same in the two iteration steps and indistinguishable in the graphs. The top panel of Figure 36.12b is the LDOS for the s orbital and the middle panel is the LDOS for the d orbitals. The p orbitals are also included in the simulation but their contribution in the energy range of Figure 36.12b is small. The bottom panel is the total LDOS value that is contributed by the s, p, and d orbitals. In each panel of Figure 36.12b, the difference of the local energy between the two iteration steps is written at the upper right corner. A negative or positive value of the difference means an energy gain or loss during the process, respectively. The LDOS for each d orbital is also plotted Figure 36.13, which will be key for understanding the mechanism. The shape of the d orbitals is shown in Figure 36.2 Here a local coordinate system is defined for each atom as follows. The local x-axis is the nanowire axis, [110], the local y-axis is along the lateral direction and the

local z-axis is perpendicular to the surface. Therefore, the wire surface corresponds to the xy plane at each atom. The first stage of the process can be explained by the change of LDOS of the atoms A and C. Figure 36.12 indicates that the dissociation of the atom A occurs with an energy loss by 0.21 eV, because of the reduction of its coordination number. It is remarkable, however, that the energy loss of the d electrons is quite small (0.02 eV), when it is compared with that of the s electron (0.14 eV). Here one should remember that the number of d electrons is larger, by about 10 times, than that of s electrons (d10s1). The left column of Figure 36.13, the data of the d orbitals at the atom A, indicates that the energy gain and loss among the d orbitals are on the order of 0.1 eV but they cancel with each other. The energy gain or loss mechanism of each orbital can be explained by the spatial spread of each orbital; The local energy of the yz orbital of the atom A increases, since the orbital extents perpendicularly to the wire surface and the nearest neighbor distance increases. The local energy of the xy orbital decreases, since the orbital can expand more to another (111)-type sheet through the

36-10

Handbook of Nanophysics: Nanotubes and Nanowires Atom A 1.5

Atom C

Atom B

xy

–0.25 eV

xy

–0.37 eV

xy

–0.21 eV

yz

0.14 eV

yz

0.18 eV

yz

0.03 eV

zx

0.13 eV

zx

0.09 eV

zx

–0.01 eV

x2–y2

–0.03 eV

x2–y2

–0.06 eV

x2–y2

–0.12 eV

3z2–r2

0.02 eV

3z2–r2

–0.26 eV

3z2–r2

–0.08 eV

1 0.5 0 1.5


1 0.5 0 1.5 1 0.5 0 1.5 1 0.5 0 1.5 1 0.5 0

–6 –4 –2

0

2

4

6

–6 –4 –2

0

2

4

6

–6 –4 –2

0

2

4

6

Energy (eV)

FIGURE 36.13 LDOS for each d orbital of the atoms A, C, and B at the two iteration steps in the relaxation process of the 11-4 Au nanowire. The definitions and notations are the same as in Figure 36.12. Local coordinate system for each atom is written in text.

atom C because of flattening two (111)-type sheets and the bandwidth becomes wider. It is also remarkable in Figure 36.12 that the atom C has a large energy gain, by 1.16 eV, at the first stage. The middle column of Figure 36.13 indicates that the largest energy gain comes from that of the xy orbital (0.37 eV). Therefore, the local energy of the d orbitals decreases on the atom C. This energy gain of d orbitals can be attributed to the flattened surface structure around the atom C after the dissociation between the atom A and the inner shell. The energy of s orbital of the atom C also decreases appreciably but it may be not associated primarily with the dissociation, since the s orbital does not always favor the flatter atomic configurations. The second stage is explained by the change of LDOS of the atom B. The local energy of the s orbital decreases by 0.38 eV (Figure 36.12b), since the coordination number of the atom B increases like an atom depicted by filled square in Figure 36.9c and d. The local energy of the d orbital of the atom B also decreases by 0.15 eV (Figure 36.12b). In particular, the xy orbital gives the largest energy gain among the d orbitals, by 0.21 eV (Figure 36.13), since the surface transforms to (111)-type and is flattened.

Here, the slip deformation is essential since it widens the area of the (111)-like hexagonal surface and the LDOS of the xy orbital transfers its weight from the antibonding region (the high energy region) to the bonding region (the low energy region) (Figure 36.12). Here one should recall that the atom A is connected with the atom B and can move relatively freely from the inner shell, after the dissociation at the first stage. Therefore, the atom B slips easily and introduces the helicity, since the atom B can trail other atoms without dissociating their bonds. The above analysis is concluded that the first and second stages are governed by the energy gain mechanism of d orbitals among atoms on or near the (001)-type surface area. If a wire has a larger diameter than those in this section, the ratio of the energy gain to the total energy will be smaller and the transformation will not occur.

36.4.5 Discussions Three points are discussed for the two-stage formation model. First, the 10-4 and 12-6-1 nanowires are discussed. They are ones of the “reference” structures (see Figure 36.8) and

36-11


(a)

(b)

FIGURE 36.14 Relaxation process of the (a) 10-4 and (b) 12-6-1 Au nanowires into helical structures. The panels of (a), from right to left , are the snapshots at every 500 step from the initial one and the panels of (b) are those at every 2500 step from the initial one. An atom marked by arrow indicates the atom row that moves from an inner shell into the outermost shell.

do not have an additional atom row on the outermost shell unlike the nanowires in Figure 36.10. Figure 36.14 shows the relaxation process of the 10-4 nanowire at T = 600 K and the 12-6-1 nanowires at T = 900 K. These nanowires are transformed into helical ones. In the transformation process, an atom row indicated by arrows in Figure 36.14a or b moves from the inner shell into the outermost one, unlike the cases in Figure 36.10. The fi nal snapshots in Figure 36.14a and b are the helical 11-3 and 13-5-1 (or 13-6) structures, respectively. Figure 36.15 shows the transformation process of the 12-6-1 case from a different viewpoint, so as to clarify the atom row movement. The two-stage formation model holds also in these nanowires. The present results show that the inserted atom rows are supplied possibly from outer and inner regions. For example, in the 13-6-1 nanowire (see Figure 36.10c), an atom row is supplied from the outer region into the shell with 12 atoms, while, in the 12-6-1 nanowire (see Figure 36.14b), an atom row is supplied from the inner region into the shell with 12 atoms. Both nanowires show surface reconstruction into helical structure with the outermost shell of 13 atoms. Second, QM-MD simulation for Cu nanowire was carried out for comparison with Au nanowires. The calculation of the Cu 11-4 nanowire was simulated and temperature was set to be 600 and 900 K, which is lower than the melting temperature (1358 K). As result, helical wire appeared at 900 K but did not at 600 K, since the surface atom did not dissociate from the inner shell at

(a)

(b)

600 K. These results lead us to the conclusion that Cu nanowire is more difficult than Au one to be transformed into helical structure, which is consistent to the fact that no helical Cu nanowire was observed in experiment. Local electronic structure is analyzed in the 11-4 Cu nanowire at T = 600 K. Figure 36.16 shows the LDOS calculations for the atoms A, C at the initial structure and the 500-th iteration step. The definitions and notations are the same as in the Au case (Figure 36.12b). The two vertical broken lines are the values of the Fermi level at the two iterations and the higher value is that at the 500-th iteration step. Figure 36.17 shows LDOS for each d orbital of the atoms A, C at the two iteration steps. The definitions and notations are the same as in Figure 36.13. As common features of the LDOS of Cu nanowire (Figures 36.16 and 36.17) and Au nanowire (Figures 36.12b and 36.13), the energy gain of d orbitals at the atom C is seen and the maximum energy gain is given by the xy orbital among d orbitals. The theory in the previous sections explains the above results; Cu, Ag, and Au have 10 d electrons (d10s1) but the d bandwidth of Au is wider than that of Cu and Ag (see Figure 36.5). Therefore, the energy gain mechanism for helical transformation is inherent in Cu but its effect is weaker than that in Au. Hence, helical structure appears in Au nanowire but not in Cu nanowire. The above theory also explains why Pt helical nanowire can be formed, since Pt and Au has the 5d band (see Figure 36.1b), which is wider than the 3d and 4d band in the lighter elements of Figure 36.1b. The energy gain mechanism in the two-stage

(c)

(d)

FIGURE 36.15 Relaxation process of 12-6-1 Au nanowire into helical structures: (a) 750-th, (b) 5000-th, (c) 9500-th, and (d)14500-th iteration steps. The atoms in the outer and inner shells are shown as dark and light balls, respectively.

36-12

Handbook of Nanophysics: Nanotubes and Nanowires Atom A 1.5

Atom C –0.01 eV

s

–0.29 eV

s

1


0.5 0 4

–0.16 eV

d

0.03 eV

d

Total

0.05 eV

Total

3 2 1 0 4

–0.57 eV

3 2 1 0

–6

–4

–2 0 2 Energy (eV)

4

6

–6

–4

–2 0 2 Energy (eV)

4

6

FIGURE 36.16 Local density of states in the 11-4 Cu nanowire for the atoms A and C. The solid and broken lines are at the initial state and at the 500-th iteration step, respectively. The difference of the local energy between the two iteration steps is written at the right corner of each panel. The definitions and notations are the same as in Figure 36.12b.

formation model is governed by the d bandwidth and the helical nanowires appear only among metals with a wider d band. Third, the similarity in mechanism is discussed between helical nanowires and equilibrium surfaces. Since the present mechanism is based on the electronic structure of 5d band, the mechanism is inherent not only with nanowires but also with bulk surfaces of FCC 5d metals, Au, Pt, and iridium (Ir) (see Figure 36.1b). For example, the equilibrium (110) surfaces of these elements reconstruct into a 2 × 1 “missing row” structure (Fedak and Gjostein 1967, Binnig et al. 1983, Ho and Bohnen 1987), in which (111)-type hexagonal surfaces appear as successive nanofacets. Figure 36.18a and b shows the top and side views of the missing row structure. The cubic cell of the ideal FCC structure, identical to Figure 36.3a, is also shown in Figure 36.18c for tutorial. The (110)-2 × 1 missing row structure appears, when one removes every other row in the [001] direction from the ideal (110) surface. The atom position of missing rows is drawn by the crosses in Figure 36.18b. In the ideal FCC structure of Figure 36.18c, the ideal (110) surface corresponds to the plane with the atoms R, P, S, and U. The atom positions of the “missing row” are those on the atom row that contains the atoms S and U. In Figure 36.18a and b, (111)-type facets appears, such as one that contains the atoms P, V, T, W, and R. One can confi rm in Figure 18c that the atoms P, V, T, W, and R are on a (111)-type plane. Moreover, the equilibrium (001) surfaces of these elements also show reconstruction in which a surface layer with (111)type hexagonal regions is placed on the (001)-type square layers (Van Hove et al. 1981, Binnig et al. 1984, Abernathy et al. 1992, Jahns et al. 1999).

It should be noted that, an experimental paper of [110] Au nanowire (Kondo and Takayanagi 1997), earlier than the report of helical structure (Kondo and Takayanagi 2000), suggests that (001)-type regions on the nanowire surface reconstruct into (111)-type ones, as on equilibrium surfaces. The suggested reconstruction mechanism is consistent with the present one.

36.5 Summary and Future Aspect This chapter gives a review of ultrathin Au nanowire and focuses two properties: (1) monoatomic chain with quantized conductance and (2) helical multishell structures with “magic numbers.” The first property is general among different metal nanowires. The second property is unique for Au and Pt. The proposed theory with the two-stage formation process and the analysis of electronic structure explains how and why these helical structures appear. The appearance of helical multishell Au nanowire should be understood by the following two points: (1) This is a typical nanoscale effect, in which the numbers of atoms in surface (outermost) region and bulk (inner) region are comparable and the structure is determined by the energy gain mechanism of the surface region. (2) This is a typical quantum mechanical effect, in which the electronic structure, the nonspherical property of 5d band, governs the phenomena. As a future aspect, structural and transport properties should be investigated further for “thicker” nanowires, so as to establish the foundation of nano electronics. The investigation of helical multishell nanowires is a typical one and other

36-13

Ultrathin Gold Nanowires Atom A 1.5 xy

investigations were explained in Section 36.3. The above two points should be of general importance among the investigations of “thicker” nanowires. Moreover, “thicker” nanowires can show complex transport property, since the interference of electron “wave” is influenced by the structures of nanowires and electrodes and the character of valence orbitals. Among such cases, the conductance value is, in general, not quantized and a more fundamental expression of Equation 36.4 should be considered. An example is found in theoretical (Ono and Hirose 2005) and experimental (Oshima et al. 2006) papers of helical multishell Au nanowires. These papers point out that the structure of electrode parts varies conductance value. For another example, transport behavior was investigated for model “thicker” nanowires (Shinaoka et al. 2008). The paper focuses on electrode effect and d orbital effect on conductance and local current. The theoretical investigation of “thicker” nanowires requires quantum-mechanical simulation with a larger number of atoms. For an example, Figure 36.19 shows a simulation of a 15-8-1 nanowire with the wire length of 12 nm (Fujiwara et al. 2008). The number of atoms is 1020. The resultant nanowire contains multiple helical domains on wire surface with well-defi ned domain boundary, which cannot be obtained in smaller samples. More systematic investigations will be given in near future. When one would like to compare simulations with experiment, the above system size is still insufficient and electrode parts are missing in the simulation. A promising theoretical approach for larger quantum-mechanical simulations is “order-N” method, in which the computational time is “order-N” or proportional to the system size N. See articles cited in Hoshi and Fujiwara (2006) or a recent journal volume that includes Fujiwara et al. (2008) and focuses on the order-N methods.

Atom C

–0.12 eV

xy

–0.12 eV

yz

0.06 eV

yz

0.01 eV

0 1.5 zx

0.08 eV

zx

0.02 eV

–0.03 eV

x2–y2

–0.02 eV

0.03 eV

3z2–r2

–0.02 eV

1 0.5 0 1.5


1 0.5

1 0.5 0 1.5 x2–y2 1 0.5 0 1.5 3z2–r2 1 0.5 0

–6 –4 –2 0 2 Energy (eV)

4

6

–6 –4 –2 0 2 Energy (eV)

4

6

FIGURE 36.17 LDOS for each d orbital of the atoms A, C at the two iteration steps in the relaxation process of the 11-4 Cu nanowire. The solid and broken lines are at the initial state and at the 500-th iteration step, respectively. The definitions and notations are the same as in Figure 36.13.

– [110]

R W T R

V P

(a) [110]

X

[001]

Q P

– [110] (b)

1] [11

[110]

P

×

(S)

× W

V X

[001]

V U

T (c)

S

FIGURE 36.18 (a) Top and (b) side views of the “missing row” structure in FCC(110) surface. (c) Ideal FCC structure. In (a), three successive atomic layers are drawn. Atoms are distinguished by larger fi lled circles, open circles, or smaller fi lled circles, for the first, second, third surface layers, respectively. In (b), two successive atomic layers are drawn. Atoms are distinguished by fi lled and open circles. The crosses in (b) indicate the site of the “missing row.” The lines in (b) indicate (111)-type surface regions as successive nanofacets. The atoms marked as P, R, R, S, T, U, V, and W appear among (a)–(c). The atom marked X appears only in (a) and (b).

36-14


The energy E[{RI}] and the force {FI} depend on the solutions {ϕi(r)} of Equation (36.A.1). A well-established method in QM-MD simulation is the first principles molecular dynamics that is realized by plane-wave bases and density functional theory (Car and Parrinello 1985, Payne et al. 1992, Martin 2004). The procedure of QM-MD simulation within one time step is summarized as {R I } ⇒ {εi , φi(r )} ⇒ E ,{FI } ⇒ update{R I };

(a)

(b)

FIGURE 36.19 Transformation of 15-8-1 helical structure of a Au nanowire with the length of 12 nm. Initial (left) and fi nal (right) snapshots are shown. A dashed line is drawn for an eye guide commonly on the two snapshots.

Appendix 36.A: Note on Quantum-Mechanical Molecular Dynamics Simulation This note is devoted to a tutorial of QM-MD simulation. See a textbook (Martin 2004) for general introduction of quantum mechanical (electronic structure) calculations. QM-MD simulation method is based on the quantum mechanical description of electrons and the classical description of atomic nuclei. Atomic structure is determined with the electronic wave functions. Since the mass of atomic nucleus is heavier by more than 1000 times than that of the electron, the motion of atomic nucleus is treated adiabatically. In many QM-MD simulations, only valence electrons are treated explicitly and nucleus and core electrons are treated as “ions.” Electronic wave functions are determined with the given position of the atomic nuclei or ions {R I} and the effective Schrödinger equation is obtained as H φi = ε i φi,

(36.A.1)

where H is an effective Hamiltonian ϕi ≡ ϕi(r) and εi are electronic wave function and its energy level, respectively The motion of atomic nuclei or ions is described by their position {R I} and velocity {VI} and the Newton equation is derived MI

dVI = FI , dt

(36.A.2)

where the force FI is given by the derivative of the energy E[{R I}]; FI = −

∂E . ∂R I

(36.A.3)

(36.A.4)

(1) With given positions {R I}, the effective Schrodinger equation of Equation (36.A.1) is solved and the energy levels and wave functions for electrons ({εi,ϕi(r)}) are obtained. (2) The total energy E and the forces {FI} are obtained. (3) The positions are updated (R I (t) ⇒ R I (t + δt)) by a numerical time evolution of Equation (36.A.2) with a tiny time interval δt. The time interval δt is usually on the order of femtosecond. The QM-MD simulation shown in Section 36.4 was realized by the simulation code “ELSES” (=Extra-Large-Scale Electronic Structure calculations). See the web page (http://www.elses.jp/) or the papers (Hoshi and Fujiwara 2000, 2003, Geshi et al. 2003, Takayama et al. 2004, Hoshi et al. 2005, Hoshi and Fujiwara 2006, Takayama et al. 2006, Iguchi et al. 2007, Fujiwara et al. 2008, Hoshi and Fujiwara 2009). In the simulation, Slater–Koster or tight-binding form Hamiltonian are used. In general, a QM-MD simulation with tight-binding form Hamiltonians enable a much faster simulation than the first principles molecular dynamics, although it has not yet established to construct tight-binding form Hamiltonians among general materials from the first principles. In the simulations of Au and Cu in Section 36.4, the energy E[{R I}] and the Hamiltonian H are written within the tight-binding form developed in Naval Research Laboratory (Mehl and Papaconstantopoulos 1996, Kirchhoff et al. 2001, Papaconstantopoulos and Mehl 2003, Haftel et al. 2004). The form contains several parameters and they are determined to represent electronic structures of bulk solids, surfaces, stacking faults, and point defects. Au nanowires were calculated by the present Hamiltonian (da Silva et al. 2001, 2004, Haftel and Gall 2006, Iguchi et al. 2007). Figure 36.20 shows the energy band diagrams of FCC Au calculated by the present tight-binding form Hamiltonian and the first-principles Hamiltonian with linear muffin-tin orbital theory (Andersen and Jepsen 1984). In general, the energy band diagram is a standard method for visualizing electronic state in solids. The diagram describes the relation between energy e and wave vector k of electronic states (e = e(k)). Each point in the horizontal axis indicates a wave vector k. Several specific wave vectors are labeled, such as “Γ.” See a textbook (Bradley and Cracknell 1972) for their definitions. In Figure 36.20, the energy band diagram by the tight-binding form Hamiltonian reproduces well that by the first-principles Hamiltonian, particularly in the occupied energy region (E < EF). The electronic structure in the occupied energy region contributes to the cohesive mechanism and determines atomic structures.

36-15

Ultrathin Gold Nanowires 30 TB FP 25

20

Energy (eV)

15

10

EF

5

0

–5

–10

L

Γ

X

W

L

K

Γ

FIGURE 36.20 Band diagrams of bulk gold that are calculated by the tight-binding form Hamiltonian (TB) and the first-principles one (FP). See text for details.

Tight-binding form Hamiltonians for QM-MD simulation were developed also by many other groups, which can be found, for example, in a review paper (Goringe et al. 1997). Some of them are developed for specific elements, such as ones for C (Xu et al. 1992) and Si (Kwon et al. 1994) and some of them, Calzaferri et al. (1989), Nath and Anderson (1990), Calzaferri and Rytz (1996) for example, were developed for more general materials. The analysis method of electronic wave function is also important for understanding phenomena. A bond between two atoms can be determined by electronic wave function through the theory of crystal orbital Hamiltonian population (COHP) (Dronskowski and Blöchl 1993), a well-defined energy spectrum of bond. In Section 36.4, the dissociation of two atoms was ascertained in two ways: (1) the interatomic distance increases by more than 20% and (2) the peak height of the COHP decreases down to 1/5 of that of the initial state.

References Abernathy, D. L., S. G. J. Mochrie, D. M. Zehner, G. Grübel, and D. Gibbs. 1992. Orientational epitaxy and lateral structure of the hexagonally reconstructed Pt(001) and Au(001) surfaces. Phys. Rev. B 45: 9272–9291. Agraït, N., J. G. Rodrigo, and S. Vieira. 1993. Conductance steps and quantization in atomic-size contacts. Phys. Rev. B 47: 12345–12348.

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37 Electronic Transport through Atomic-Size Point Contacts 37.1 Introduction ........................................................................................................................... 37-1 37.2 The Landauer Approach to Conductance .......................................................................... 37-2 37.3 Fabrication of Atomic-Size Contacts .................................................................................. 37-3 Scanning Tunneling Microscope Technique • TEM and Dangling Wires • Mechanically Controllable Break-Junctions • Electro-Migration Technique • Recent Developments

37.4 Conductance of Atomic-Size Contacts ..............................................................................37-6 Opening and Closing Traces • Conductance Histograms • Determination of Conduction Channels • Atomic Contacts of Conventional Metals • Atomic Contacts of Semimetals • Atomic Contacts of Magnetic Metals

Elke Scheer University of Konstanz

37.5 Conclusions and Outlook ................................................................................................... 37-16 References......................................................................................................................................... 37-17

37.1 Introduction In this chapter, we will describe the electronic transport properties of a particular type of mesoscopic structure, namely, point contacts with lateral dimensions of a single or a few atoms. These structures, first realized in the early 1990s, are still attracting much interest because they represent test beds for important concepts of quantum mechanics, quantum chemistry, atomic physics, and solid-state physics. While the first years of research were devoted to the understanding of the intrinsic transport properties of atomic-size contacts (APCs) of normal metals, new fields of interest have now been entered, for example, the application of APCs to provide highly-spin polarized resistors or atomically sharp electrodes for contacting individual nanoobjects in the context of molecular electronics. Although manifold transport properties like noise, thermopower, and thermal conductance were studied and they revealed rich behavior, we will concentrate here on the understanding of the most basic transport property, namely, the linear electrical conductance, i.e., the conductance measured with a small voltage bias and the current-voltage characteristic as linear. A more comprehensive review, including the mechanical properties and the more complex transport quantities such as noise or thermopower has been given by Agraït, Levy Yeyati, and van Ruitenbeek (Agraït et al. 2003). Furthermore, we exclude the discussion of the so-called atomic nanowires because they are the point of focus in several other chapters of this handbook. Atomic nanowires are constrictions

with atomic size lateral dimensions and have been produced from manifold materials. Many properties of nanowires share the same concepts as those used to describe the physics of APCs. The term nanowire usually implies that the constriction has a length of nanometer scale and gives rise to a well pronounced one- or two-dimensional transport behavior. The term APC suggests that the constriction is short in the sense that any voltage applied at both ends drops locally at the position of the APC. Nevertheless, there is no sharp criterion distinguishing APCs from atomic nanowires. Finally, the term “quantum point contact” bears the risk of being mistaken with “quantum points,” which is equivalent to “quantum dots.” The latter terms denote structures that have finite size in all three dimensions. For this concept, manifold realizations exist as, among others, semiconductor quantum dots, clusters, or fullerenes. For example, a quantum point can be formed by two QPCs in a series. The physics of quantum points are addressed in several chapters of this handbook as well. This chapter is organized as follows: In Section 37.2, we summarize the concept of conductance as a quantum mechanical wave scattering problem, we discuss its application to quantum point contacts (QPCs), and we point out the main differences between APCs and QPCs made from two-dimensional electron gases (2DEGs), which are described in detail in Chapter 38 (Zozoulenko and Ihnatsenka 2010). In Section 37.3, we first briefly describe the main fabrication schemes of APCs and then in Section 37.4 we comment on their electrical conductance. Section 37.5 gives a summary and an outlook of the field.

37-1

37-2


37.2 The Landauer Approach to Conductance

T=

The theory of electron transport pioneered by Rolf Landauer (1957, 1970) treats an electronic transport experiment on a mesoscopic sample as an ordinary wave scattering experiment. Electrons are sent from a reservoir into the sample, which scatters them (Figure 37.1). A fraction of the electrons is reflected back and the rest is collected by another reservoir. Each reservoir plays the role of a source and a detector (for a review, see Zozoulenko and Ihnatsenka 2010, Beenakker and van Houten 1991, or Beenakker 1997). To establish a well defined set of electronic quantum states with which to describe the scattering experiment, the theory introduces “leads” connecting the mesoscopic sample to the reservoirs. In the leads, a finite number N of independent electronic ingoing and outgoing modes propagate. These modes constitute a finite and complete basis of states for the scattering problem. The scatterer connects in principle any incoming mode with all outgoing modes. In theory, this is represented by a transmission matrix t and a reflection matrix r. For example, the matrix element tnm gives the probability amplitude for the left incoming mode m to be transmitted into the right outgoing mode n. These matrices carry all the information on the scatterer and, thus, determine all the transport properties of the mesoscopic sample under investigation. For example, the total conductance of a mesoscopic structure is given by the Landauer formula: G = G0T where T = Tr{t†t} is the total transmission of the structure G 0 = 2e2/h is the conductance quantum The factor 2 arises from the two possible spin orientations, assuming spin degeneracy. The quantity e 2/h also occurs in the context of the quantum Hall effect and describes the contribution of one (spin-split) electronic state to the Hall conductance. However, as the trace of a matrix remains invariant through any unitary basis transformation, there is no preferred basis to calculate the conductance. In other words, it is possible to chose a suitable basis in which the matrix tnm has a simple form in order to calculate the eigenvalues τi of the matrix t†t from which one can then deduce the total transmission

∑

(t †t )nn =

N

∑τ

i

i =1

with the transmission coefficients τi of the N individual transport channels. Conversely, a measurement of the conductance does not provide enough information to determine the ensemble {τi}, unless the scatterer has only one mode. In fact, not even the total number N of transmission eigenmodes—nicknamed “channels”—of the scatterer can be extracted from the knowledge of the conductance. Methods for the determination of the channel ensemble will be described in Section 37.4.3. Before going into this, we will briefly comment on the most important differences between the quantum point contacts of two-dimensional electron gases (2DEGQPC) and the atomic point contacts (APCs) (Figure 37.2). The transport through 2DGQPCs is easily interpreted in terms of the scattering theory of transport and is described in detail in Chapter 38 (Zozoulenko and Ihnatsenka 2009). A QPC, regardless of its realization, is a constriction between two metallic reservoirs of lateral size comparable to the Fermi wavelength λF of the electrons. A 2DEGQPC is fabricated on two-dimensional electron gases in epitaxially grown heterostructures of semiconductors. These gases are low electronic density (1012 cm−2) metals in which the Fermi wavelength of the electrons can be as large as 30 nm. The constriction is defined and its width D is adjusted by means of nanofabricated electrostatic gates that deplete locally the electron layer. Experimentally, as the width is increased continuously, the conductance increases in steps of nearly G0: this is the phenomenon of conductance quantization (van Wees et al. 1988, Wharam et al. 1988). The experiments are performed on very good realizations of ideal ballistic leads and QPCs. Moreover, the connection between the leads and the sample is almost perfectly adiabatic: the eigenmodes of both the leads and the QPC are of the same nature and they are perfectly matched. For these structures, the choice of a basis of laterally confined plane waves that are funneled through the constriction is quite natural. In this language, the conductance staircase has a very simple interpretation: if the width of the constriction is much smaller than λF, the conductance of the QPC is zero, i.e., all τi = 0. Then, as the width is increased continuously, each time D reaches an integer multiple of λF/2, a new mode starts being transmitted (τi = 1). Although, as explained before, a conductance measurement does not provide information on the individual eigenvalues, λF ~ D ~ a

λF ~ D >> a

tnm m Reservoir 1

Lead 1

Elastic scattering potential rnm

Lead 2

Reservoir 2

n

FIGURE 37.1 Illustration of the Landauer approach to electrical conductance.

(a)

τ1 = 1, τ2 = ... = τN = 0

(b)

τi = ?

FIGURE 37.2 Comparison between a 2DEGQPC (a) and an APC (b). The dark gray lines in the left visualize the confi ning potential induced by the action of the gate.

37-3

Electronic Transport through Atomic-Size Point Contacts

the fact that in the experiments at least 15 conductance plateaus are observed in a reproducible and reversible fashion undoubtedly supports the view of channels opening one by one and reaching perfect transmission. Moreover, measurements of the shot-noise (de Jong and Beenakker 1997, Blanter and Büttiker 2000) of the current going through a 2DEGQPC (Reznikov et al. 1995, Kumar et al. 1996) have shown that the noise power at the conductance plateaus is drastically reduced with respect to its full value, thus confirming the fact that the channel transmissions are either 0 or 1. The main reason for achieving perfect transmission is the long Fermi wavelength, which makes the motion of conduction electrons quite insensitive to atomic defects. In contrast APCs, which nowadays can be fabricated using several techniques, provide contacts between two reservoirs of conventional three-dimensional metals with high electron density, like Au, Na, etc. In this case, λF is comparable to the interatomic distance and the conduction electrons are sensitive to all defects at this scale. The exact geometry is usually not known in the experiments, but obviously there is a region around an atomic contact in which the electrons experience boundary conditions that are rough at the atomic scale. There is a “central cluster” where many atoms are in fact surface atoms that cannot establish all their possible metallic bonds. This fact limits the number of the modes that can be transmitted through the constriction. In other words, the modes of the cluster do not connect adiabatically to the modes of the reservoirs (Brandbyge et al. 1997). An adequate language for the transport eigenmodes of the cluster is the one of linear combinations of the valence atomic orbitals. The set {τi} of the transmission eigenvalues can be constructed as superpositions from these orbitals as basis functions. It is thus determined both by the chemical properties of the atoms forming the contact and by their geometrical arrangement (Cuevas et al. 1998a,b, Scheer et al. 1998, Agraït et al. 2003).

37.3 Fabrication of Atomic-Size Contacts 37.3.1 Scanning Tunneling Microscope Technique As introduced in the chapters by N. Agraït (2010) and J. Kröger (2010), the scanning tunneling microscope (STM) (for a review, see Wiesendanger 1994) is a suitable tool for the fabrication of atomic-size contacts and atomic chains and has been used for that purpose from the very beginning of its invention (Gimzewski and Möller 1987). While in the standard application of an STM a fine metallic tip is held at distance from a counter electrode (in general a metallic surface) by making use of the exponential distance dependence of the tunneling current, the tip can also be indented into the surface and carefully withdrawn until an atomic-size contact or short atomic wire forms (see Figure 37.3). For many metals it has been shown that the tip will be covered by several atomic layers of the metal of the counter electrode upon repeated indentation such that clean contacts may be formed

Motion

200 μm

FIGURE 37.3 Working principle of the fabrication of APCs with an STM. The electron micrograph shows a W STM tip the width at half length is in the order of 100–200 μm. The lower inset gives an artist’s view of the atomic arrangement of an APC. (Courtesy of C. Bacca.)

consisting of the same metal for both electrodes. The main advantages of the STM in this application are its speed and versatility to also form hetero-contacts, i.e., contacts between two different metals. When the electrodes forming the contacts are prepared in ultra high vacuum conditions, the STM allows us to gather information about the topography of the two electrodes on a somewhat larger scale than the atomic scale before or after the formation of the contact. Spectroscopic measurements on the scale of electron volts allow one to deduce information about the cleanliness and the electronic structure of the metal. The main drawbacks are its limited stability with respect to the change of external parameters such as the temperature or magnetic fields and the short lifetime of the contacts in general because of the sensibility of STM to vibrations.

37.3.2 TEM and Dangling Wires Another interesting method for preparing and imaging atomic contacts are transient structures forming in a transmission electron microscope (TEM) when irradiating thin metal fi lms on dewetting substrates (Kondo and Takayanagi 1997, Rodrigues and Ugarte 2001). The high energy impact caused by the intensive electron beam locally melts the metal fi lm causing the formation of constrictions that eventually shrink down to the atomic size and finally pinch off building a vacuum tunnel gap. A typical system for these studies is Au on glassy carbon substrates. Several variations of this principle have been developed that allow one to contact both electrodes forming the contact. Because of the locally high temperatures, the typical lifetime of these contacts is very short, but they offer the unique possibility to simultaneously perform conductance measurements and imaging with atomic precision. Similar results have been obtained with variations of the STM inside a TEM (Kizuka 1998). This method enabled us to directly prove the existence of single-atom contacts and single-atom wide nanowires as well

37-4


Tip

Vacuum Sample (a)

(b)

(c)

(d)

1 nm (e)

(f )

FIGURE 37.4 High resolution TEM images of short atomic wires fabricated with an STM inside the vacuum chamber of the TEM. The arrows indicate the number of atomic rows. In panel (f) the contact is broken to a tunnel contact. (Reprinted from Ohnishi, H. et al., Nature, 395, 780, 1998. With permission.)

as to establish a correlation between contact size and conductance. For Au and Ag contacts, it has been shown that preferably well ordered contacts in high symmetry growing directions are formed. The method is particularly fruitful for studying atomic nanowires (see Figure 37.4). Transient nanowires and APCs with lifetimes in the millisecond range can also be fabricated in a table-top experiment first demonstrated by N. Garcia and coworkers (Costa-Krämer et al. 1995), which we call “dangling-wire contacts.” Two metal wires in loose contact to each other are excited to mechanical vibrations so that the contact opens and closes repeatedly. One end of each wire is connected to the poles of a voltage source and the current is recorded with a fast oscilloscope. This method is in principle particularly versatile because it enables the formation of heterocontacts between various metals. However, in order to provide clean metallic contacts, a thorough cleaning of the wires would be required, similar to the tip and surface preparation in a STM. Another drawback is the lack of control of the distance of the electrodes. It is thus mostly used as a demonstration experiment in schools with Au-Au contacts.

37.3.3 Mechanically Controllable Break-Junctions Already before the development of the first STM another technique enabling the fabrication of atomic-size contacts and tunable tunnel-contacts has been put forward. The fi rst realizations

include the needle-anvil or wedge-wedge point contact technique pioneered by Yanson and coworkers (for a review, see Naidyuk and Yanson 2005) and the squeezable tunnel junction method described by Moreland and Hansma (1984) and Moreland and Ekin (1985) who used metal electrodes on two different substrates that have then carefully adjusted with respect to each other. The needle-anvil technique was mainly used to form contacts with diameters of typically several nanometers and thus having hundreds to thousands of atoms in the narrowest cross-section. These two techniques formed the starting point for the development of the mechanically controllable break-junctions (MCBJ) by Muller and coworkers (Muller et al. 1992), which nowadays is applied for the fabrication of APCs in various subforms, the most common of which are the so-called notched-wire (Krans et al. 1993) and thin-fi lm MCBJs (van Ruitenbeek et al. 1996). The working principle is the same for both variations (Figure 37.5): A suspended metallic bridge is fi xed on a flexible substrate, which itself is mounted in a threepoint bending mechanism consisting of a pushing rod and two counter-supports. The position of the pushing rod relative to the counter supports is controlled by a motor or piezo drive or combinations of both. The electrodes on top of the substrate are elongated by increasing the bending of the substrate. The elongation can be reduced again by pulling back the pushing rod and thus reducing the curvature of the substrate. In order to break a junction to the tunneling regime, considerable displacements of the pushing rod and thus important bending of the substrate is required. Therefore, the most common substrates are metals with a relatively high elastic limit like spring steel or bronze. The substrates are covered by an electrically isolating material such as polyimide before the junction can be fi xed on it. The notched-wire MCBJ (Figure 37.6) uses a thin metallic wire (diameter 50–200 μm) which has a short, knife-cut constriction to a diameter of 20–50 μm. The wire is glued at both sides of the L Counter supports Metal wire

u

t

Pushing rod

δx

Elastic substrate Sacrificial layer

FIGURE 37.5 Working principle of the mechanically controllable break-junction (not to scale) with the metal wire, the elastic substrate, the insulating sacrificial layer, the pushing rod, the counter supports and the dimensions used for calculating the reduction ratio (see text).

37-5


vibrations. Assuming homogeneous beam-bending of the substrates results in the so-called reduction ratio r between the length change of the bridge δu and the motion of the pushing rod δx (see Figure 37.5) r=

δu 6tu = δx L2

where t is the thickness of the substrate u is the length of the free-standing bridge arms L is the distance of the counter supports FIGURE 37.6 Optical micrograph of a notched-wire MCBJ made of a 100 nm thin gold wire (top view). The wire is glued with epoxy resin (black) onto the substrate. The electrical contact is made by thin copper wires glued with silver paint. The inset shows a zoom into the notch region between the two black drops of epoxy resin. (Reprinted from Agraït, N. et al., Phys. Rep., 377, 81, 2003. With permission.)

notch to the substrate and both ends. The distance between the glue drops is of the order of 50–200 μm. The thin-fi lm MCBJs are fabricated by the standard techniques of nanofabrication, i.e., electron beam lithography, metal deposition, and partial removal of a sacrificial layer underneath the metal fi lm for suspending a bridge with the typical dimensions of 2 μm in length and 100 × 100 nm2 at the narrowest part of the constriction (Figure 37.7). Both versions of the techniques share the idea of enhanced stability due to the formation of the contact by breaking the very same piece of metal on a single substrate and by the transformation of the motion of the actuator into a much reduced motion of the electrodes perpendicular to it. The small dimensions of the freestanding bridge-arms give rise to high mechanical eigenfrequencies, much higher than the ground frequencies of the setup and thus reduced sensibility to perturbations by

This quantity thus denotes the factor with which any motion that acts on the pushing rod is reduced when it is transferred to the point contact. It has a typical value of 10−3 to 10−2 for the notched-wire MCBJs and 10−6 to 10−4 for the thin fi lm MCBJs. The relatively weak reduction ratio of the notched-wire MCBJs usually requires the use of a piezo drive for controlling and stabilizing single-atom contacts, while the thin-fi lm MCBJs can be controlled with purely mechanical drives, i.e., a dc motor with a combination of gear boxes. The typical motion speeds of the piezo drive lie between 10 nm/s and 10 μm/s, which results in 10 pm/s to 100 nm/s for the electrodes forming the atomic contacts. For thin-fi lm MCBJs, these values are 10 nm/s to 1 μm/s for the pushing rod and 10 fm/s to 10 nm/s for the contact. Also, due to the in-built reduction, the piezo-driven setups are slower than STM systems. On the other hand, the small r values require considerable absolute motion of the pushing rod and thus deformations of the substrate in order to achieve displacements of the electrodes. The high stability allows comprehensive studies on the very same atomic contact at various values of control parameters such as fields and temperature. MCBJ mechanisms have been developed for various environments including ambient conditions, vacuum, very low temperatures (Scheer et al. 1997), or liquid solutions (Grüter et al. 2005). The disadvantages as compared to STM techniques are the low speed, the fact that the surrounding area of the contact cannot easily be scanned, and clean contacts can only be guaranteed when working in good vacuum conditions. The sample preparation itself, however, does not require clean conditions because the atomic contacts are only formed during the measurement by breaking the bulk of the electrodes.

37.3.4 Electro-Migration Technique

FIGURE 37.7 Electron micrograph of a thin-fi lm MCBJ made of cobalt on polyimide taken under an inclination angle of 60° with respect to the normal. The distance between the rectangular shaped electrodes is 2 μm, the thickness of the thin fi lm is 100 nm and the width of the constriction at its narrowest part is approximately 100 nm.

A third method for the formation of atomic-size contacts is controlled burning of a wire by electro-migration caused by high currents. This technique has been optimized for the formation of nanometer-sized gaps for trapping individual molecules or other nano-objects (van der Zant et al. 2006, Wu 2007). Before the wire finally fails and the current drops drastically, atomic size contacts are formed for a rather short time span (Trouwborst et al. 2006, Hoff mann et al. 2008). During the electromigration

37-6


Support

Wire bond

32 nm-thick Au

Si-Chip 500 nm Drain

Source

Contact

Pushing screw 10 mm

SiO2 Au

200 nm

Si (Gate)

100 nm

(a)

Gate (Si)

(b)

FIGURE 37.8 Working principle (a) and electron micrograph of an electromigrated MCBJ (b). (Reprinted from Champagne, A.R. et al., Nano Lett., 5, 305, 2005. With permission.)

process, the transport changes from ohmic behavior, i.e., limited by scattering events to wave-like behavior, which can be described by the Landauer picture. The main drawback of this technique is the fact that it is a single-shot experiment; after burning through the wire, it cannot be closed again. A combination of electromigration with the thin-fi lm MCBJ overcomes this problem (Champagne et al. 2005): a thin-film MCBJ is thinned-out by electromigration to a narrow constriction with a cross-section of less than 10 nm (see Figure 37.8). The substrate is then bent carefully for completely breaking the wire or arranging single-atom contacts. This last step is reversible and repeatable for studying small contacts or trapped nano-objects. Because only the very last part of the breaking requires mechanical deformation of the substrate, it is rather fast and enables the use of more brittle substrates such as silicon.

37.3.5 Recent Developments A new version of the MCBJ technique has been introduced by Waitz et al. (2008). It uses thin-film wires on silicon membranes with a thickness of a few hundred nanometers (Figure 37.9). The membrane is deformed by a fine tip on the rear side. At variance

(a)

to the MCBJ techniques on bulk substrates, the elasticity of the membrane is used rather than the bending. Thus, the deformation is applied locally and it is possible to address particular positions while the rest of the circuit on the substrate remains unaffected.

37.4 Conductance of Atomic-Size Contacts 37.4.1 Opening and Closing Traces The natural characterization method of atomic contacts that also enables us to roughly determine the size of the contacts is the recording of the so-called opening and closing curves, i.e., measurements of the conductance as a function of time when enhancing or reducing with constant speed the distance of the two electrodes forming the contact. A typical opening and closing sequence measured consecutively on a gold thin-fi lm MCBJ is shown in the top panel of Figure 37.10. Equivalent data for Al and Co is given in the central panel and the bottom panel. Both the opening and the closing traces show regions in which the conductance is rather constant (so-called conductance plateaus)

(b)

FIGURE 37.9 MCBJ on silicon membranes. (a) Working principle of the MCBJ (not to scale). The insets illustrate the formation of a point contact as well as an artist’s view of the atomic arrangement of single-atom contact. The thickness of the membrane is in the order of 300 nm, the lateral dimension of the membrane is typically 0.5 mm × 0.5 mm. The dimensions of the suspended metal fi lm are comparable to the ones for thin fi lm MCBJs on massive substrates but with smaller thickness of the sacrificial layer of typically 100 nm. (b) Optical micrograph of a membrane MCBJ with patterned Au electrodes.

37-7


0.3 nm

6 4 2 Au

Conductance [G0]

0 0.1 nm 6 4 2 Al 0 0.1 6

0.2 nm

0.01 1E–3

4 2 0.2 nm

Co 0 Distance

FIGURE 37.10 Conductance as a function of distance measured for thin-fi lm MCBJs made of Au (top), Al (center), and Co (bottom) at low temperature T < 10 K. The distance axes have been calibrated using the exponential distance dependence in the tunneling regime. For Au two examples of opening traces, for Co one opening and one closing trace are shown. The inset displays the tunneling regime in logarithmic conductance scale. The atomic configurations are given to illustrate the typical size of the contacts on the plateaus and the transition to the tunneling regime where both electrodes are broken to form a vacuum gap.

interchanged with pronounced steps. The lengths of the plateaus and the step heights are of comparable size but not as regular as those observed for the 2DEGQPCs. Furthermore, the plateaus are not all horizontal but may have a fine structure such as small steps, curvature, or inclination (Sanchez-Portal 1997, Krans 1993, Agrait 2003). The typical shape depends on the metal under study and also on experimental details (see below). At some point of the opening traces, the series of plateaus and steps comes to an end and the conductance decreases exponentially with the time, as can be seen in the inset of the bottom panel where the conductance is plotted on a logarithmic scale. This is the signature of vacuum tunneling: the two electrodes are broken to form a small tunnel gap. Because vacuum tunneling is a well-studied phenomenon, this region can be used to calibrate the setup and to transform the time axis into a distance axis. Assuming clean surfaces of the electrodes, the conductance G as a function of distance d is ⎛ 2 ⎞ G(d ) ∝ exp ⎜ − 2m * Φ ⋅ d ⎟ ⎝ ⎠

where Φ is the work function m* is the effective mass of the electrons Using bulk values for Φ and m* yields the distance scales given by the arrows in the individual panels of Figure 37.10. Since the work function of most metals is affected by contamination and surface geometry, the distance values determined with this method bear an error in the order of 20% as can be verified in an STM configuration with an independent calibration of the piezo. The plateau lengths are in the order of several tenths of nanometer, i.e., the typical atomic size. The interpretation of the opening and closing traces is that the plateaus correspond to a particular atomic configuration that can elastically be stretched until a reconfiguration occurs. This gives rise to a change of the minimal cross-section of the contact and the conductance. The relation between both of these quantities is not straightforward to deduce but requires a detailed consideration. A reasonable assumption is, however, that the smaller the cross-section. i.e., the smaller the number of atoms forming the contact, the smaller the conductance. Another reasonable assumption is that the last contact before breaking to the tunneling regime is given by a single-atom contact. Experimental support for this assumption will be given in Section 37.4.3. These considerations lead us to the artist’s view of the opening trace given in the central panel of Figure 37.10. Starting from the right, it is well proven that a vacuum gap exists between the two electrodes, when the conductance depends exponentially on the distance. The last plateau before breaking is depicted as a contact formed by a single-atom that forms the apexes of the two pyramidal electrodes. The contacts with higher conductance are most likely formed by more than one atom; the number and arrangements are, however, not easy to guess from the bare conductance since several different arrangements may give rise to similar conductance values even for simple metals like gold (Dreher et al. 2005). We will come back to this point in Sections 37.4.2 and 37.4.3. Additional important information that can be deduced from the examples shown in Figure 37.10 is the fact that the closing traces are not time reversed versions of the opening traces, although the adopted conductance values are similar. For most metals, the opening traces give rise to longer plateaus than the closing traces (Agraït et al. 1993, Böhler et al. 2004). From the area surrounded by the opening and closing trace, quantities like Young’s module and deformation energy may be deduced (Rubio et al. 1996).

37.4.2 Conductance Histograms An in-built problem of the study of atomic point contacts is the fact that each contact is unique in the sense that the exact atomic arrangement of the atoms in the narrowest part varies from contact to contact. Th is results in manifold transport behavior, such as varying conductance, varying noise properties, varying current-voltage characteristics, and conductance fluctuations. Nevertheless, experiments on a large ensemble of metallic contacts—realized with various techniques—have

37-8

Handbook of Nanophysics: Nanotubes and Nanowires 0.5 Potassium, 4.2 K Normalized number of counts

Counts [a.u.]

30

20

10

0.4

0.3

0.2

0.1

0.0

0

0

1

2

3

4

5

Conductance [2e2/h]

FIGURE 37.11 Conductance histogram of a Au thin-fi lm MCBJ calculated from 66 opening traces recorded at T = 10 K in ultra-high vacuum.

demonstrated the statistical tendency of atomic-size contacts to adopt configurations leading to some preferred values of conductance. The usual tool for deducing the statistical behavior is the recording of conductance histograms, i.e., a plot of the probability with which a particular conductance value is adopted as a function of the conductance. An example for Au recorded with a thin-fi lm MCBJ at 10 K in ultrahigh vacuum conditions is given in Figure 37.11. Despite the relatively small number of opening traces (M = 66), it displays a pronounced and narrow maximum at G = 1G 0 and two smaller and wider ones at G = 2G 0, and G = 3G 0. The actual preferred values do not only depend on the metal but also on the experimental conditions and on the way the histograms are built. Obviously, conductance histograms made from opening traces may differ from those obtained from closing curves, as discussed above. For most of the metals, the plateaus are not horizontal. It is thus crucial for the shape of the histogram whether the conductance values recorded all over the plateau or only the last ones–obtained just before break–are used. Both methods are applied, however. A more detailed discussion of this issue can be found in the review of Agraït et al. (2003). Other external parameters that determine the shape of the histograms are, among others, the temperature, the vacuum conditions, the applied voltage, the hardness of the electrode material (Yanson et al. 2005, 2008), and the number and the speed of the formation of the contacts. Luckily, most of these parameters mainly affect the height and the width of the histogram peaks, while the positions, i.e., the preferred conductance values, are rather robust. For many metals, in particular monovalent ones (like Na, Au, etc.) that as bulk material fulfill the free-electron model, the smallest contacts have a conductance G close to G0 (Olesen et al. 1994, Krans et al. 1995, Costa-Krämer 1997). Furthermore, many monovalent metals show several peaks in the histograms that presumably correspond to contacts with more than a single atom in the

0

1

2

3

4

5

6

7

8

Conductance [2e2/h]

FIGURE 37.12 Conductance histogram of potassium measured at 4.2 K. The fact that the peaks at G = 2 and G = 4 are suppressed is typical for free electron systems. (Reprinted from Agraït, N. et al., Phys. Rep., 377, 81, 2003. With permission.)

smallest cross-section. Also, the higher-conductance peaks are often close to integer values of G0, but not all integers are observed, a fact which can be explained with the help of electronic shell effects similar to magic numbers of clusters (Mares and van Ruitenbeek 2005). In particular, for the alkali metals, not all multiples of G0 give rise to the same peak height and the pattern is strongly temperature dependent (Figure 37.12; Yanson et al. 1999). Interestingly, there also exists a few multivalent metals such as Al (Yanson et al. 1997, 2008; see Figure 37.13) or Zn (Scheer et al. 2006) that show several rather well pronounced maxima, while the majority of the electronically more complex metals reveal either a single peak or wide bumps (for examples, please see Nb: Ludoph et al. 2000; Fe, Co, Ni: Untiedt et al. 2004, Sirvent et al. 1996; Figure 37.14) that do not have their maximum at integer multiples of G 0. The spacing between the neighboring maxima is in the order of G 0 as well but is only rarely exactly 1G 0. As shown in Section 37.4.4, the fact that the first peak in the conductance histograms often is close to 1G 0 does not in general mean that the smallest contact (presumably a single-atom contact) corresponds to a single, perfectly transmitted Landauer channel with τi = 1. Summarizing, the origin of the peaks in the conductance histograms is not yet fully understood because it is an interplay of electronic and geometrical effects.

37.4.3 Determination of Conduction Channels The total transmission T = Στi of the contacts can be easily deduced from their measured conductance using the Landauer formula G = G0T. As discussed in Section 37.2, the total ensemble of transmission coefficients {τi} cannot be determined solely by conductance measurements. However, transport quantities exist that do not depend linearly on the τi, which thus may serve to provide additional information on the transmission matrix.

37-9


0.89 10,000

2,000 5

5,000

20,000

6

7

8

9 10 11 12 13 10.99 12.03

5

9.72

4

7.72 8.62

2 3 Peak index

5.50

1

6.69

0

2,500

4.28

Slope = 1.05+/–0.01

1.94

1

3,000 15,000 3.12

40,000

2

0

3,500

20,000

3

Number of counts

Conductance [2e2/h] 2.945

60,000

4

3.965

0.825

80,000 1.885

Number of counts

100,000

25,000 5

0

0 1

2

3

4

5

6

7

8

9

1

Conductance [2e2/h]

2

3

4

5

6

7

8

9 10 11 12 13 14 15

Conductance [2e2/h]

FIGURE 37.13 Influence of the hardness of the electrode material onto the conductance histogram of Al. Left (right): Histogram of annealed (work-hardened) Al measured with a notched-wire MCBJ at 4.2 K. The inset in the left panel displays the conductance values for the positions of the first four peaks. The inset in the right panel is an enlargement of the data for conductance values between 5 and 13G 0. (Reprinted from Yanson, I.K. et al., Phys. Rev. B, 77, 033411, 2008. With permission.)

3000 Fe 2000

1000

0

0

1

2

3

4

5

3000

Counts

Co 2000

1000

0

0

1

2

3

4

5

4000 Ni

3000 2000 1000 0

0

1

2 3 Conductance [2e2/h]

4

5

FIGURE 37.14 Conductance histograms of the ferromagnets Fe. Co, Ni measured with notched-wire MCBJs at 4 K with and without external magnetic field. (Reprinted from Untiedt, C. et al., Phys. Rev. B, 69, 081401, 2004. With permission.)

For example, the power of the shot-noise associated with the current at voltage V is a measure of the second-moment of the transmission distribution {τi} (Büttiker 1990) S = 2eVG0Στi(1 − τi). A perfectly transmitted mode (τi =1) does not contribute to the noise because of electron correlations arising from the Pauli principle. A measurement of this noise power provides a second relation between the matrix elements of t†t. However, this information is still not sufficient to determine the eigenvalues ensemble {τi} if the scatterer has more than two modes. In principle, one needs to measure a number of moments equal to the total number N of channels in order to determine all the transmission coefficients, which is possible by analyzing measurements in the superconducting state of atomic-size contacts. Instead of going through a complete opening trace, one can stop at any point and record a current-voltage characteristic (IVs) in the superconducting state. The first observation is that one finds different IVs for the same conductance i.e., the same total transmission. Examples for three Al contacts realized with a thin-film MCBJ on the same last plateau before breaking, all three with G ≈ 0.8G 0, are given in Figure 37.15. The origin of the well marked current increases at voltage values V = 2Δ/me (with the superconducting gap parameter Δ and m a natural number) are multiple Andreev reflections (MAR) (Averin and Kardas 1995, Cuevas et al. 1996). Each current onset corresponds to a charge transport with a varying number m of coherently transferred elementary charges. For example, the structure at eV = 2Δ is the well-known single-electron transfer also present in superconducting tunnel contacts. The height of the current onset depends nonlinearly on the transmission coefficients and thus allows one to determine this value. The set j(V,τ) of the currentvoltage characteristics of single-channel superconducting QPCs with arbitrary transmission 0 < τ ≤ 1 has been calculated exactly by several groups independently and applying different methods (Averin 1995, Cuevas 1996). Since the channels are independent of each other (see Section 37.2), the current contributions of N channels with an ensemble {τi} give just the total current

37-10


4

4

2 eI/GΔ

3 0

1

2

2

1

0

0

1

2

3

eV/Δ

FIGURE 37.15 Current–voltage characteristics of three single-atom contacts of Al in the superconducting state measured at T = 75 mK with a thin-fi lm MCBJ. The overall conductance of all three contacts is G ≈ 0.8G 0. Symbols represent experimental data, lines are fit to the theory of MAR. Current and voltage are given normalized to the superconducting gap energy eΔ. Inset: Calculated IV characteristics for a single channel superconducting QPC at T = 0 for various values of τ (from bottom to top: 0.1, 0.4, 0.7, 0.9, 0.99, 1). (After Cuevas, J.C. et al., Phys. Rev. B, 54, 7366, 1996; Averin, D. and Bardas, A., Phys. Rev. Lett., 75, 1831, 1995.) N

I (V ) =

∑ j(τ ,V ) i

i

This is used to fit the measured IVs and thus to decompose the total current into the contributions of the individual channels for contacts with more than one channel. It is worth mentioning that superconductivity itself does not alter the channel ensemble: it only serves as a tool for their determination as can be demonstrated for contacts with two channels, the ensemble of which can also be determined by analyzing the shot-noise or other methods that do not rely on superconductivity (van den Brom and van Ruitenbeek 1999). The examples given in Figure 37.15 are each composed of the contributions of three channels with varying transmissions. Although the total conductance is smaller than G = 1G0. In other words, these single-atom contacts that contribute to the first peak in the conductance histogram do accommodate three conductance channels instead of a single one. This observation in spite of the fact that all three contacts are part of the same conductance plateau, immediately yields to the conclusion that the transmissions do depend very sensitively on the precise atomic arrangement.

37.4.4 Atomic Contacts of Conventional Metals The number N = 3 for a single atom contact is not at all unique. Experiments with lead (N = 3), niobium (N = 5), zinc (N = 1 or 2), and gold (N = 2, which has been rendered superconducting by the so-called proximity effect; Scheer et al. 2001) have demonstrated that not only the preferred conductance of single-atom

contacts, but also the number of channels is a function of the chemical valence (Scheer et al. 1998, Konrad et al. 2005, Scheer et al. 2006). Several models have been put forward for explaining this material dependence. One of these is based on a free-electron model for the electrons neglecting the precise atom arrangement (Torres et al. 1994, Stafford et al. 1997; for a review see Grabert 2009). The other one uses a linear combination of atomic orbitals (LCAO) via tight-binding methods (Levy Yeyati et al. 1997, Cuevas et al. 1998; for a review see Agraït et al. 2003) or the so-called ab initio calculations (Bagrets et al. 2006). Tight-binding LCAO models are known to be very suitable for the description of the band structure of insulators and semiconductors because the wave functions are rather localized and only few of them overlap with the wave function of the given atom. This is why tight binding is rather unusual for calculating the electronic properties of metals since the wave functions of the conduction electrons are extended over many atom positions. Therefore, for bulk metals, the free-electron model seems to be more appropriate. However, when aiming at calculating the electronic situation in restricted geometries such as an APC in which only a few atomic waves overlap, the LCAO models have their advantages. The general rule is that the more wave functions overlap, the wider the resulting electronic band in the energy range. According to the wave-guide model that successfully describes the behavior of 2DEGQPCs, the shape of the cross-section as well as the ratio between the size of the contact and the Fermi wave length determine the number of channels. Applying this model to APCs does not straightforwardly account for the observed variability of N. The LCAO model and the ab initio calculations, however, are able to describe the conductance behavior successfully. The latter ones are particularly useful for describing the influence of a finite measuring voltage, which is, however, out of the scope of this chapter. For simplicity, we will describe here the main idea and findings of the LCAO approach. It is not necessary to go into the details of the models to understand how it addresses the most important observations (conductance of a single-atom contact, number of channels, transmissions of the channels, and sensitivity to atomic rearrangements). Due to the fabrication method of the contacts, the exact geometry of the contact on the atomic scale differs from contact to contact. Since the Fermi wavelength is of the same order as the diameter of the contacts, the electrons crossing the contact experience all the imperfections on this scale. Their wave functions will thus be influenced by these imperfections. The next important conclusion is that even for a good free-electron metal, the eigenfunctions of an APC will not be Bloch waves, like the ones of the electrodes. It is thus not taken for granted that the transmission coefficients achieve the value τI = 1, even for a perfectly ordered cut-out of a single crystal of the metal. Rather, the eigenchannels are determined by the overlap of the wave functions of the atoms forming the central cluster with the wave functions of the electrodes and thus vary from contact to contact. The number of wave functions with overlap to the neighbors, i.e., the number of extended electronic


37-11

states or in other words, the number of channels is at most the number of partially fi lled electronic orbitals, i.e., the number of valence orbitals. Since the overlap of the wave functions depends on the exact configuration of the atoms, it varies from contact to contact, and the transmission coefficients of the individual channels do so as well, giving rise to a broad variation of possible {τi} although the sum of the coefficients, i.e., the conductance, may be similar. Because the exact geometry of the contacts at the atomic scale is in general unknown, the channels are calculated for either perfectly ordered model geometries (Cuevas et al. 1998a) or the most likely configurations deduced from molecular dynamics calculations (Dreher et al. 2005). Interestingly, for most of the metals studied so far, it turned out that the number of channels with measurable transmission does not depend on the geometry of single-atom contacts, however, their transmissions vary strongly. By comparing the results obtained for Al and Pb, Figure 37.17 shows how the chemical properties of the element enter into the conductance properties: Al and Pb are sp-like metals, i.e., because of their orbital structure single-atom contacts could give rise to up to four channels (1s orbital, 3p orbitals). The main differences between Al and Pb are that the level spacing between the s and the p orbitals in the isolated atom amounts to approximately 7 eV in the case of Al and 10 eV in the case of Pb. Furthermore, Al has three electrons on the outer shell while Pb has four (see the top of Figure 37.17). When combining these atomic wave functions in order to calculate the band structure of the bulk metal, one uses iterative processes varying the numerical parameters (Papaconstantopoulos 1986) in order to obtain good agreement with experimentally available data for the band structure. The results for Al and Pb are displayed in the two upper panels of Figure 37.17 named “bulk DOS”. As a result of the mentioned element-specific ingredients, the Fermi energy of Al in the bulk metal lies in a region where both s and p electrons have a measurable density of states, while at the position of EF of Pb the p-orbitals dominate by far. In the APC geometry of Figure 37.16, the central atom has only very few neighbors, which results in a weaker broadening of the electronic bands. As a consequence,

the local density of states (DOS) calculated for the central atom interpolates between the spiky energy levels of the isolated atom (see the top of Figure 37.17) and the bulk DOS. Furthermore, the broken isotropy makes the p orbital, which is directed along the transport direction (here: the pz orbital), split-off in energy from the two remaining ones (here: px and py). The new eigenfunctions of the APC structure are two linear combinations of the pz orbital with the s orbital and the two perpendicular px and py orbitals. Due to symmetry reasons, only the symmetric spz hybrid has a measurable weight while the antisymmetric one has negligible DOS throughout the whole energy range interesting for transport. This local DOS now enters into the calculation of the transmission functions, i.e., the energy dependent transmissions. Their values at EF are called transmission coefficients and are entering the linear conductance via the Landauer formula (see Section 37.2), which can be written as

Finite number of atomic layers

Left lead

Right lead

z (Bulk)

(Bulk)

FIGURE 37.16 Schematics of a single-atom contact used for the calculations within the LCAO model. (Adapted from Cuevas, J.C. et al., Phys. Rev. Lett., 80, 1066, 1998a.)

ˆ (EF )tˆt (EF )⎤ G = G 0 Tr ⎡⎣tg ⎦ 12 12 where tˆ(E) = 2 ⎡⎣ Im ∑ L (E)⎤⎦ G1rN (E) ⎡⎣Im ∑ R (E)⎤⎦ are the trans-

mission functions, ΣL,R are the so-called self-energies of the left and right lead, respectively, and G1rN (E) is the so-called Green’s function of the central atom and corresponds to its local DOS. The self energies account for the coupling of the wave functions of the central atom to the leads. As a result, this coupling causes the difference between the energy dependence of the local DOS and the resulting transmission functions shown in the two lower panels. In the linear conductance, which is measured in the experiments when applying a small bias voltage, solely the value of tˆ(E) at the Fermi enters. As can be deduced from Figure 37.17, in the case of Al, three channels contribute with a transmission varying between 0.1 and roughly 0.8, while for Pb three channels with an almost equal transmission of 0.8 are predicted. For the geometry shown in Figure 37.16, i.e., two perfectly ordered pyramids grown in the (111) direction sharing one atom at the apex, the sum of the transmission amounts to roughly 1.2G0 for Al and 2.5G 0 for Pb. For the same geometry but for the transition metal Nb (1s orbital, 5d orbitals), the calculation yields five channels with nonvanishing transmission, adding up to G = 2.8G 0. The situation is more complex for the divalent metal Zn, which crystallizes in a hexagonal structure. Here, the number of channels does indeed depend on the growing direction of the contact (Konrad 2004, Häfner 2004, Scheer et al. 2006). Introducing disorder into the structure does, in general, reduce the transmission values and may lift degeneracies (Cuevas 1998a,b). All multivalent metals have the following in common: even for perfectly ordered single-atom contacts, none of the τi arrives at the maximum possible value τi = 1, reflecting that the eigenstates of the APC are not perfectly matched to the eigenstates of the electrodes. Thus, there is in general no transmission quantization going along with perfect conductance, i.e., perfect sums of transmissions. This is particularly striking for the element Al,

37-12

Handbook of Nanophysics: Nanotubes and Nanowires Aluminum 3s

Lead

3p

6s

~7 eV

6p

~10 eV

Bulk DOS

0.2

0.1

0.4 p

p

s

s

0.2

Transmission

Neck LDOS

0.0

0.0

0.2 spz

px, py

0.4

spz

0.1

0.2

0.0

0.0

1.0

1.0

0.6

spz

spz

0.2

px, py

0.6 0.2

px, py –10 –5

(a)

px, py

0 5 E [eV]

10

–10 (b)

–5 0 E [eV]

5

FIGURE 37.17 LCAO model for single-atom contacts of Al (a) and Pb (b) in the model geometry shown in Figure 37.16. Schematic development of the atomic valence levels into bulk conduction bands. The panels in the second row depict schematically the local density of states (LDOS) in eV−1 at the central atom of the model geometry. The global energy dependence of the transmission coefficients {τi} shown in the bottom most panels. The dotted lines indicate the position of the Fermi level. The spz mode is the best transmitted one for both materials. The px and py modes are degenerate due to the symmetry of the model geometry. The indicated energy differences between the atomic s and p levels are the ones used to fit the bulk band structure. (Data taken from Scheer, E. et al., Nature (London), 394, 154, 1998.)

the histogram of which reveals several pronounced peaks. For instance, the first peak at G = 0.8G 0 is mainly caused by contacts accommodating three channels with varying distribution of transmissions (Scheer et al. 1998, Böhler et al. 2004), although its total conductance would be small enough to be provided by a single channel. According to the LCAO model, single-atom contacts of monovalent metals are expected to accommodate only a single channel, which has been found by theory and by experiment for gold (van den Brom et al. 1999, Scheer et al. 2001) and several alkali metals (Yanson et al. 1999). Whether the transmission of this channel is equal to one that would give rise to conductance quantization (i.e., histogram peaks at multiples of G0) depends on the geometry of the atomic arrangement and the local electronic band structure. This can be visualized in a simple model, assuming that the coupling of the single valence orbital of the central atom is given by a coupling constant to each of the two electrodes (Cuevas et al. 1998b). Two conditions have to be fulfilled for the resulting extended electronic mode to be perfectly matched: The coupling to both electrodes has to be equal, i.e., the central atom is symmetrically bonded to the

neighbor atoms on both sides. The second condition is that the energy of the valence orbital corresponds to the Fermi energy of the electrodes. This is fulfilled for monovalent free electron metals, since the valence band is half filled. The origin of the existence of preferred conductance values in spite of the fact that the electronic wave functions are not quantized is still not completely settled. The situation is complex because, as mentioned earlier, several length scales that are important for conductance—the interatomic distance, the lateral dimension, the Fermi wavelength, momentum scattering length—do coincide. Thus, there is a competition between electronic and structural effects (Yanson et al. 2005, 2008). The electronic effects are closely connected with the physics of nanowires as mentioned in the introduction. We will therefore only briefly mention the most important facts. The electronic effects include effects typical for free electron systems (Yanson et al. 1999, Grabert 2009) as well as interaction effects (Kirchner et al. 2003). For alkali metals, i.e., the best freeelectron metals, the electronic effects dominate giving rise to shell effects that are equivalent to the existence of magic clusters

37-13

Electronic Transport through Atomic-Size Point Contacts Vp [V] 100

150

200

3 2 1 0 (a)

≈ 12 Å

1.0 Conductance [2e2/h]

consisting of a number of atoms for which the electronic shells are closed (see chapters by Grabert (2009) and Agraït (2009) ). For contacts of alkali metals and gold, a transition from the electronic shell effects to the geometry shell effects for larger contacts have been found, which means that those contacts are the most stable and do give rise to pronounced peaks in the histograms, for which the structure is highly favorable, and electronic shells are closed (Yanson et al. 2001). The investigation of atomic contacts with a TEM (see Section 37.3.2) suggests that highly symmetric and well-ordered atomic configurations are preferred. This could explain the appearance of particular conductance values. However, it would go along with preferred transmissions as well, a point that is much less studied so far. The few transmission histograms that are published so far suggest a wider distribution of transmission values (Böhler et al. 2004). It has been demonstrated that the height of the histogram peaks is markedly influenced by the hardness of the electrode material (Yanson et al. 2005, 2008). An outcome of these investigations is that the harder the metal, the more pronounced the shell effects.

0.8 0.6 0.4 0.2 0.0 (b)

≈4 Å

0.06

37.4.5 Atomic Contacts of Semimetals

0.05

The coincidence of Fermi wavelength and contact size is lifted when studying APCs of semimetals. The term semimetal denotes materials for which the electronic bands cross the Fermi level in particular directions only. The two best studied elementary semimetals are Bi and Sb, which fail to be insulators because of a small lattice distortion with respect to a cubic symmetry. This slide distortion makes parts of the bands that would be unoccupied if in cubic symmetry to lie below the Fermi energy. Consequently, the density of conduction electrons is low giving rise to a rather large Fermi wavelength of 10–100 nm, comparable to the one in 2DEGs. It is thus instructive to study point contacts for the electronic properties that resemble 2DEGQPCs but the structure and size is that of APCs. The most obvious question is whether APCs of semimetals adopt preferred conductance values in the order of G 0 with steps in the same order. Such behavior would be expected for an electronic wave-guide similar to what was realized in the 2DEGQPCs. A single channel with perfect transmission would be expected for contacts with lateral sizes in the order of λF/2, much larger than an atom. If, however, atomic configurations dominate the transport behavior, much smaller steps both in height and in length should be observed. When interpreting the data recoded on semimetal APCs, one has to keep in mind that because of the low crystal symmetry electrons and holes in different bands may have different effective mass. This makes the total electronic behavior complex and strongly dependent on external influences like external fields, disorder (Garcia-Mochales 1996, Garcia-Mochales and Serena 1997), and the fabrication method of contacts. It is thus necessary to fabricate the APCs with different methods that result in different contact structures. At first glimpse, the experimental findings appear to some extent to be controversial. For Sb small plateaus with a typical height of 0.02G0 and plateau length in order of the lattice constant

0.04 0.03

Mechanical contact

Vacuum tunneling

0.02 0.01 0.00 (c)

≈2 Å

FIGURE 37.18 Th ree opening traces recorded in different conductance ranges of a Sb notched-wire MCBs recorded at 4.2 K. The horizontal axis is the voltage at the piezo which drives the pushing rod and is proportional to the distance Sb. The step sequence stops around G ≈ 0.01G 0. (From Krans, J.M. and van Ruitenbeek, J.M., Phys. Rev. B, 50, 17659, 1994. With permission.)

have been found (Krans and van Ruitenbeek 1994). These findings are in accordance with the atomistic model (Figure 37.18). The value of conductance corresponds well to the ratio between the lattice constant and the Fermi wavelength a0/λF = 0.02. Several experiments have been performed for Bi with STMs (Costa-Krämer 1997, Rodrigo 2002) as well as with thin-fi lm MCBJs. While the histogram in the article by Costa-Krämer does show rather wide bumps around multiples of G0, individual opening traces depict extremely well marked and horizontal plateaus (Figure 37.19). This seemingly contradictory behavior can partially be explained by the investigations of Rodrigo et al. (2002) who observed both the wave guide as well as the atomistic effects in the same experiment. Depending on the shape of the constriction on the length scale of λF, one or the other dominates, as depicted in Figure 37.20. For long necks, plateaus at multiples of G 0 are found. For short constrictions small steps are observed with a minimum conductance of GBi ≈ 0.2G 0. This finding is in agreement with the results from the thin-fi lm MCBJ measurements, from which GBi ≈ 0.15G 0 for single-atom contacts was deduced.

37-14


100,000

3004 consecutive curves

90,000

1736 consecutive curves

80,000 Counts [a.u.]

Counts

70,000 60,000 50,000 40,000

20

30,000 20,000 10,000 0 1

2

3

4

5

6

0 0.0

7

Conductance [2e2/h]

Conductance (G0 units)

FIGURE 37.19 Conductance histogram of Bi measured with an STM at 4 K. (Reprinted from Costa-Krämer, J.L. et al., Phys. Rev. Lett., 78, 4990, 1997. With permission.)

2 1 1

0

0 0.0

(a)

0.5 1.0 1.5 Elongation [nm]

0.0 0.2 0.4 (b) Elongation [nm]

FIGURE 37.20 Opening and closing traces of Bi contacts realized with a STM at T = 4.2 K. (a) contacts revealing subquantum plateaus and (b) contact showing a last plateau with conductance close to G0. (Reprinted from Rodrigo, J.G. et al., Phys. Rev. Lett., 88, 246801, 2002. With permission.)

The histogram corresponding to the latter investigation reveals peaks at even multiples of GBi, supporting the atomistic model of conductance (Figure 37.21). In summary, in APCs of semimetals, waveguide as well as atomic effects seem to be present. Proof of this interpretation could be delivered by measuring the number of conductance channels: According to the waveguide model contacts with GBi < G ≤ G0 would all have a single channel. Increasing the size would increase the transmission of this channel but not open new channels. In the atomistic model, however, these contacts would have more than one channel. This information could be deduced from shot noise measurements or with the method described in Section 37.4.3.

37.4.6 Atomic Contacts of Magnetic Metals There is a particular interest in APCs of magnetic metals for several reasons. On the one hand, APCs of band magnets (Fe, Co, Ni) are proposed to act as sources of spin-polarized electrons. It has been predicted that the spin polarization, i.e., the relative

0.3

0.6 0.9 Conductance [2e2/h]

1.2

FIGURE 37.21 Histogram calculated from 80 opening traces of a Bi thin-fi lm MCBJ measured at 30 mK.

preference of one of the two spin directions should strongly depend on the atomic configuration. For instance, in the tunnel limit of Ni, the spin polarization could achieve very high values close to 100% (Häfner et al. 2008). On the other hand, the interest was triggered by the observation of very large magnetoresistance values, i.e., changes of the resistance (or conductance) as a function of the applied magnetic field. The latter effect could be used for magnetic storage devices or magnetic field driven switches in nanoelectronic circuits. Although much research effort has been put into this field during the last decade, the mechanism behind the observations is still under debate. According to the Jullière model, the spin polarization is given as P=

D↑ − D↓ D↑ + D↓

where D↑ and D↓ denote the electronic density of states at the Fermi energy for electrons with spin up or spin down, respectively. Th is quantity can, for example, be deduced from spin-resolved photo-emission measurements. In this defi nition, all electrons in all bands crossing the Fermi energy contribute to P. In the electronic conductance, however, not only the energy of the electronic states but also their wave functions are important. In ferromagnetic metals, the density of states at the Fermi energy is dominated by electrons in s-bands and in d-bands, both of which have very different wave functions. In general, the s-bands give rise to more extended wave functions than the d-bands and therefore a stronger contribution to the conductance. In APCs, the different wave function characters result in different values of the transmission coefficients for the channels with dominating s-characters or d-characters, respectively. Therefore, spin polarization values deduced from transport phenomena may differ from those determined from x-ray photoelectron spectroscopy (XPS) or other equilibrium electronic quantities. For the analysis of magneto-transport experiments,

37-15


the definition of an effective transport spin polarization P T is more appropriate:

1200

I

II

III

IV

V 1040 Ω

G↑ − G↓ PT = ↑ G + G↓

MRR =

R↑↓ − R↑↑ R↓↓

where R↑↓ and R↑↑ are the resistance in antiparallel and parallel orientation of the magnetization of the two banks forming the GMR or TMR device. Usually the resistance is highest in antiparallel orientation and lowest in parallel orientation. Since in atomic devices the spin orientation is not always straightforward to deduce, and since additional effects may contribute to the MRR, the usual definition for the MRR of APCs is MRR =

Rmax − Rmin Rmin

These MRR values depending on the actual realization achieve the enormous values of 100.000% (Hua and Chopra 2003) (Figure 37.22). The size of the field that is necessary for switching between the two extremal resistance values usually is in the order of several

Resistance [Ω]

Since in a transport measurement there is no direct access to the spin-orientation of the conducting electrons, yet another model has been established for the analysis of magneto-transport experiments, inspired by the observation of extremely strong magneto-resistance effects of ferromagnetic APCs. Most of the magneto-resistance experiments are carried out in the following way: A single- or few-atom contact between two ferromagnetic electrodes is established using one of the methods described in Section 37.3. Opening and closing traces are recorded with and without applied magnetic fields. From these data histograms are calculated. In addition the conductance is measured as a function of the applied external magnetic field. In a third type of measurement, the conductance is recorded when rotating the sample in a constant external field. MCBJ experiments on Co, Ni, and Fe do not reveal preferred conductance values at multiples of G 0 in accordance with findings at nonmagnetic multivalent metals (see Section 37.4.2). In most of the experiments, the histograms do not vary by much even when applying strong external fields of up to several Tesla (see Figure 37.14) (Untiedt et al. 2004). However, it is possible only in very special realizations to prepare contacts with integer multiples of G 0 that split-up into half-integers in an external field (Ono et al. 1999). The most common findings of the magneto-resistance investigations are the following: Depending on the field orientation, the starting conductance, the material under study, and the technique, the resistance undergoes pronounced changes. In the context of Giant Magneto-Resistance (GMR) or Tunnel Magneto-Resistance (TMR) used for read-write heads, the magneto-resistance ratio (MRR) is defined as

1000 800 600 400 200 0 3000

1Ω –3000

3000

–3000

3000

–3000

Applied field [Oe]

FIGURE 37.22 Five magnetoresistance curves of a Ni atomic-size contact as a function of applied magnetic field. The MRR value is in the order of 100.000%. (Reprinted from Hua, S.Z. and Chopra, H.D., Phys. Rev. B, 67, 060401, 2003. With permission.)

hundred millitesla but can arrive at several tesla for particular sample geometries and magneto-crystalline anisotropy (Egle et al. 2010). As mentioned in the introduction, the origin of the enormous MRR effects is still not clear because several microscopic processes may contribute to the resistance and its field dependence. One possible mechanism is the so-called Ballistic MagnetoResistance (BMR). If no spin-degeneracy is given, the Landauer formula has to be modified to distinguish the properties of the transport channels in the two spin orientations: ⎛ e2 G= ⎜ h⎜ ⎝

N↑

N↓

∑ ∑ ↑ i

τ +

i =1

i =1

⎞ τi↓ ⎟ ⎟ ⎠

Here the arrows indicate the spin-up and spin-down channels, respectively. Both the number and the transmissions of the individual channels may differ for spin-up and spin-down electrons. The magnetic field may now act in such a way that the transmissions of the channels in one spin direction would be suppressed, giving rise to strong MRR values depending on the distribution of the transmissions. So far, detailed calculations of the transmission coefficients of APCs only exist for a few model geometries of contacts of the band metals Fe, Co, and Ni (Häfner et al. 2008). The typical behavior is that the spin-up channel ensemble is very different from the spin-down channel ensemble. Both are strongly dependent on the exact geometry. For particular geometries, i.e., the tunnel geometry, only one spin direction would contribute to the transport giving rise to a fully spin-polarized current (Figure 37.23). In other geometries, both spin orientations contribute to the transport with unequal weight. The application of a magnetic field could then act in such a way that it blocks the contribution of one of the spin directions. Depending on the size of its contribution, this could result in a considerable MRR. The calculation of the effect of the magnetic field onto the channels is

37-16


[111]

fcc Ni Ni

1

Transmission

2 1

0.1

0

0.01

2

0.001

(a)

80 60

G

40 20 G

1 0

100 P

0

D

–20

0.0001 –4

–2 0 E [eV]

–40 3

2 (b)

Polarization P [%]

3

3.5

4

4.5 5 D [Å]

5.5

6

–60

FIGURE 37.23 (a) Calculated spin resolved transmission of a single-atom contact of Ni in the geometry shown in the top as a function of energy. The distance between the two central atoms corresponds to the equilibrium distance in bulk. At the Fermi energy one almost perfectly transmitted channel T↑ = 0.86 contributes in spin-up direction, while four channels with a total transmission T↓ = 2.66 in spin down direction adding up to a total conductance of 1.8G 0. (b) Calculated development of the spin-resolved conductance and the resulting spin polarization of a dimer atomic contact of Ni upon stretching to the tunnel regime. When the two apex atoms are in their bulk equilibrium distance the spin polarization is about −51% and arrives at +100% in the far tunnel regime. (Reprinted from Häfner, M. et al., Phys. Rev. B, 77, 104409, 2008. With permission.)

a tricky task because it requires the incorporation of spin-orbit coupling. First results exist for contacts of Ni, predicting that not only complete blocking of channels but also gradual changes of the conductance as a function of field would be expected. These gradual effects have also been observed when performing magneto-resistance measurements both in parallel and in perpendicular magnetic fields (Egle et al. 2009) and when changing the field orientation with respect to the current direction (Viret 2002, Gabureac et al. 2004, Viret et al. 2006, Bolotin 2006). The latter are interpreted in terms of Anisotropic MagnetoResistance (AMR), an effect that is well known from bulk materials and thin magnetic films (Barthelemy et al. 2002). It is caused by the spin orbit coupling and usually amounts to a few percent in bulk materials and wider contacts. The reason why this effect in general is very small is that the orbital angular momentum in extended metals is suppressed (“quenched”) because of the interaction with the neighboring atoms. In atomic-size contacts, however, the quenching mechanism is less active due to the reduced number of neighboring atoms. The spin-orbit coupling and consequently the AMR can thus be strongly enhanced (Atomically-enhanced AMR [AAMR], Viret 2006). Besides these intrinsic origins of magneto-resistance several extrinsic effects exist. These include domain walls in the constriction (Bruno 1999), shape anisotropies, and magneto-striction, i.e., a length change of the electrodes going along with the reorientation of the domains in the external field. This change in length may affect the contact size and therefore the resistance

of the contacts. From the theoretical side, promising tools have been developed that will hopefully lead to an increased understanding within the next years. From the experimental point of view, the determination of the spin polarization PT and the transport channels would be required in order to elucidate the complex behavior of the contacts. In superconductor-ferromagnet heterocontacts in the few nanometer range produced with the needle-anvil method mentioned in Section 37.3.1, PT has been determined by analyzing the suppression of the Andreev reflection (i.e., the contribution of electron pairs n = 2) similarly to the determination of the transport channels of normalconducting APCs (see Section 37.4.3; Soulen et al. 1998, PerezWillard et al. 2004). The analysis is based on the assumption that all spin-up and spin-down electrons can be described by one effective transmission coefficient, each. Using the same method for APCs is therefore not straightforwardly possible because several channels with different transmissions contribute per spin direction. For the same reasons as detailed in the introduction, it is not possible to deduce the full channel set from only measuring the conductance, shot noise, or Andreev reflection if more than two quantities, i.e., spin-resolved transmission coefficients contribute.

37.5 Conclusions and Outlook In summary, atomic-size contacts represent a powerful and versatile tool for the quantitative verification of the fully quantum mechanical description of electronic transport, i.e., the Landauer


37-17

theory. Since their first realization approximately 20 years ago, much progress has been achieved in both fabrication methods and the understanding of their transport properties. On the experimental side, various experimental realizations have been developed that can be applied for different purposes, e.g., depending on whether highest flexibility or highest stability are required. Although several aspects of the understanding of their transport properties are still under debate, they now serve as test beds for advanced transport theories. As an example, we mention the issue of vibrational excitation and their relation to heat dissipation on the atomic level. Furthermore, the techniques used for the investigation of atomic-size contacts are now used to fabricate atomically sharp and well-characterized electrodes for contacting nanoobjects such as individual molecules. The first results of these investigations are covered in the chapter by Devos (Devos 2009).

Champagne, A. R., Pasupathy, A. N., and Ralph, D. C. 2005. Mechanically adjustable and electrically gated single-molecule transistors. Nano Lett. 5: 305–308. Costa-Krämer, J. L. 1997. Conductance quantization at room temperature in magnetic and nonmagnetic metallic nanowires. Phys. Rev. B 55: R4875–R4878. Costa-Krämer, J. L., García-Mochales, N. P., and Serena, P. A. 1995. Nanowire formation in macroscopic metallic contacts: Quantum mechanical conductance tapping a table top. Surf. Sci. 342: L1144–L1149. Costa-Krämer, J. L., Garcia, N., and Olin, H. 1997. Conductance quantization in bismuth nanowires at 4 K. Phys. Rev. Lett. 78: 4990–4993. Cuevas, J. C., Martín-Rodero, A., and Levy Yeyati, A. 1996. Hamiltonian approach to the transport properties of superconducting quantum point contacts. Phys. Rev. B 54: 7366–7379. Cuevas, J. C., Levy Yeyati, A., and Martín-Rodero, A. 1998a. Microscopic origin of conducting channels in metallic atomic-size contacts. Phys. Rev. Lett. 80: 1066–1069 Cuevas, J. C., Levy Yeyati, A., Martin-Rodero, A. et al. 1998b. Evolution of conducting channels in metallic atomic contacts under elastic deformation. Phys. Rev. Lett. 81: 2990–2993. de Jong, M. J. M. and Beenakker, C. W. J. 1997. Shot noise in mesoscopic systems. In: Mesoscopic Electron Transport. Eds. L. L. Sohn, L. P. Kouwenhoven, and G. Schön, NATO-ASI Series E, Appl. Sci., 345: 225–258. Dordrecht, the Netherlands: Kluwer Academic Publishers. Devos, A. 2009. Phonons in nanoscale objects. In: Handbook of Nanophysics. Ed. K. Sattler. Boca Raton, FL: Taylor & Francis. Dreher, M., Heurich, J., Cuevas, J. C., Nielaba, P., and Scheer, E. 2005. Theoretical analysis of the conductance histograms of Au atomic contacts. Phys. Rev. B 72: 075435. Egle, S., Bacca, C., Pernau, H.-F., Hüfner, M., Hinzke, D., Nowak, U., and Scheer, E. 2010. Magneto-resistance of atomic-size contacts realized with mechanically controllable break-junctions. Phys. Rev. B (in press). Gabureac, M., Viret, M., Ott, F., and Fermon, C. 2004. Magnetoresistance in nanocontacts induced by magnetostrictive effect. Phys. Rev. B 69: 100401 (R). García-Mochales, P. and Serena, P. A. 1997. Disorder as origin of residual resistance in nanowires. Phys. Rev. Lett. 79: 2316–2319. García-Mochales, P., Serena, P. A., Garcia N., and Costa-Krämer, J. L. 1996. Conductance in disordered nanowires: Forward and backscattering. Phys. Rev. B 53: 10268–10280. Gimzewski, J. K. and Möller, R. 1987. Transition from the tunnelling regime to point contact studied using scanning tunnelling microscopy. Physica B 36: 1284–1287. Grabert, H. 2009. The nanoscale free-electron model. In: Handbook of Nanophysics. Ed. K. Sattler. Boca Raton, FL: Taylor & Francis. Grüter, L., Gonzalez, M. T., Huber, R. et al. 2005. Electrical conductance of atomic contacts in liquid environments. Small 1: 1067–1070.

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38 Quantum Point Contact in Two-Dimensional Electron Gas 38.1 Introduction ...........................................................................................................................38-1 38.2 Conductance Quantization in the Quantum Point Contact ..........................................38-1 Electrostatic Confinement in Modulation-Doped Heterostructures • Basics of the Conductance Quantization • Breakdown of Quantized Conductance due to Impurities • Visualizing the Electron Flow through QPC • QPC as a Charge Detector • Shot Noise in QPC

38.3 Quantum Point Contact in Magnetic Field .......................................................................38-9 Depopulation of Magnetosubbands: Basics • Interacting Electrons in Quantum Point Contact • Quantum Point Contact in the Quantum Hall Regime

Igor V. Zozoulenko Linköping University

Siarhei Ihnatsenka Simon Fraser University

38.4 0.7 Anomaly and Many-Body Effects in the Quantum Point Contact .......................38-15 Experimental Evidence of 0.7 Anomaly • Theoretical Models for 0.7 Anomaly • 0.7 Anomaly and Kondo Physics

38.5 Conclusion and Outlook.....................................................................................................38-19 References ........................................................................................................................................38-19

38.1 Introduction Semiconductor nanostructures and mesoscopic electronic devices based on a two-dimensional electron gas (2DEG) have been a focus of attention for the semiconductor community during the past few decades. This is due to the new fundamental physics that these structures exhibit, and also due to possible applications in future electronic devices and devices for quantum information processing. A quantum point contact (QPC) represents the cornerstone of the mesoscopic physics. This is not only because the QPC is the simplest mesoscopic device, but also because most of the mesoscopic devices contain the QPC as their integral part. This has apparently motivated a strong interest in the various aspects of the electronic and transport properties of the QPC. In fact, one of the most important discoveries that gave a strong momentum to the whole field of mesoscopic physics was the discovery of the conductance quantization of the QPC in 1988 (van Wees et al., 1988; Wharam et al., 1988). Studies of the conductance quantization in the QPC provided valuable information not only on the fundamentals of the phase-coherent electron motion in low-dimensional structures, but also outlined important material aspects (such as the effect of impurities, potential confinement, etc.). The QPC has also proven to be the key system for studying the various aspects of the quantum Hall physics. This applies both to the pioneering studies in the beginning of the “mesoscopic era” in the early 1990s focusing on the basic aspects of the subband depopulation as well as to more recent studies of the fractional Hall regime revealing the exotic

features of interacting electrons in a high magnetic field such as fractional statistics and charge. It is also important to mention that apart from strong fundamental interest, a QPC found its important practical application as a noninvasive voltage probe and a single-electron charge detector. By now, many features of the electronic and transport properties of the QPC are well understood. However, even 20 years after the discovery of the conductance quantization, some of the important aspects of the QPC conductance still represent topics at the forefront of research and lively debates, where the emphasis is shifted to aspects of the electron interaction, spin, and nonequilibrium effects. In this chapter, we present the basic physics of the QPC in a 2DEG. Section 38.2 outlines the major experimental results of the conductance quantization along with its essential theory. Section 38.3 discusses the transport in the QPC in the presence of a magnetic field. Section 38.4 addresses the many-body effects in the QPC with a particular focus on the so-called 0.7 anomaly in conductance. A brief conclusion and an outline are given in Section 38.5.

38.2 Conductance Quantization in the Quantum Point Contact 38.2.1 Electrostatic Confi nement in Modulation-Doped Heterostructures A QPC represents a constriction defi ned in a 2DEG formed in modulation-doped semiconductor heterostructures. Figure 38.1a 38-1

38-2

Handbook of Nanophysics: Nanotubes and Nanowires Lead ConstrictionLead V(z) ++

++

+

Gates ++

++

z

+++++++++++++

GaAs

2DEG

Charge density

x 12 x

G (2e2/h)

y

10

(b) Potential

y

250 nm

Gates

(a)

1 μm

8 6 4 2

x (c)

y

0 (d)

–2.0 –1.8 –1.6 –1.4 –1.2 –1.0 Gate voltage (V)

FIGURE 38.1 (a) A schematic geometry of a quantum point contact defined by split-gates in a GaAs/AlGaAs heterostructure. An inset to the right shows the band diagram. (b) The electron density and (c) the total self-consisted potential in the vicinity of the quantum point contact. (d) The conductance quantization of the QPC shown in the inset. (Adapted from van Wees, B.J. et al., Phys. Rev. Lett., 60, 848, 1988.)

illustrates a typical GaAs/AlGaAs heterostructure. It has several layers including cap GaAs layers, an n-doped AlGaAs layer, a spacer AlGaAs layer, and a GaAs substrate. The calculated band diagram of the structure is shown in the inset to Figure 38.1a (for a detailed discussion of the 2DEG in modulation-doped heterostructures, see e.g., Davies, 1997). Due to the conduction band discontinuity between GaAs and AlGaAs, a triangular-shaped quantum well is formed at the interface between the spacer layer and the substrate, trapping electrons from the n-doped layer. Typically, only the lowest state in the well is occupied. The electrons trapped in the well form a high mobility two-dimensional electron gas. The electron mobility in the best samples reaches μ ~ 104 m2/Vs, which corresponds to the mean free path of the order of 100 μm. This is to be contrasted with the mobilities μ ~ 1 m2/Vs in Si MOSFET devices where 2DEG is trapped in an inversion layer at the interface between Si and SiO2. There are two major factors that contribute to such high mobilities in the modulation-doped heterostructures. First, the dopants are spatially separated from the quantum well where the 2DEG resides, which eliminates the large-angle scattering due to direct collisions between electrons and donors. Second, the surface scattering on the interface between the spacer AlGaAs layer and the GaAs substrate is strongly suppressed because of an almost perfect match between the lattice constants of GaAs and AlGaAs. The electron phase coherence time in the GaAs/AlGaAs modulation-doped heterostructures at milliKelvin temperatures is of the order of τϕ ~ 0.1−0.01 ns, which for typical sheet electron

densities (ns ~ 3 × 1015 m−2) corresponds to the coherence length of the order of lϕ ~ 20 μm. One of the most common ways to defi ne a lateral confi nement in the 2DEG is to use the split-gate technique (Thornton et al., 1986). By applying a negative gate voltage to the gates, the electrons underneath the gates become depleted such that electron motion is confi ned to the regions defi ned by the lateral patterning of the AlGaAs heterostructure. Other popular techniques for the lateral patterning include an etching technique where depletion is achieved by the removal of a thin layer on top of the heterostructure. This brings the surface of the heterostructure (containing electrons trapped in the surface states) close to the 2DEG, which is sufficient to deplete the electrons below the etched regions. A closely related technique is the local oxidation method where, instead of etching, an oxide line is written with the help of an atomic force microscope (AFM; for a review of the experimental techniques, see e.g., the textbook by Heinzel, 2007). The simplest lateral structure that can be defined in the 2DEG is a constriction, termed as a QPC, see Figure 38.1a. The constrictions defined in metals (known as Sharvin point contacts) have been intensively studied since the mid-1960s. Even thought these studies addressed the ballistic regime when the electron mean free path exceeded the characteristic size of the constriction, the electron transport regime in metals still remained classical because the Fermi wavelength (λ ~ 0.5 nm) was much smaller than any characteristic system size. In contrast, the semiconductor heterostructures offer a unique opportunity to study the quantum regime not only because of the exceptionally large electron mean free path, but primarily because the Fermi wavelength (λ ~ 50 nm) is comparable to a characteristic system size L, and, at the same time, the phase coherence length may significantly exceed it, lϕ >> L. In 1988, two independent experimental groups made the remarkable discovery that the conductance through a narrow constriction in a 2DEG is quantized (van Wees et al., 1988; Wharam et al., 1988), see Figure 38.1b. The essential mechanism basic to this fact was immediately recognized: transverse quantization allows the propagation of a discrete set of modes through a QPC and, as follows from the Landauer–Buttiker formalisms, each such (spin degenerate) channel contributes 2e 2/h to the conductance. A more detailed understanding of the phenomenon, including the shape of the conductance steps, involves the precise geometry of the QPC and coupling to the reservoirs and, in addition, the influence of disorder and of temperature. In this section, we concentrate on the exact and approximate descriptions of transport through QPCs with various geometries including the effect of the impurities.

38.2.2 Basics of the Conductance Quantization The central quantity in transport calculations is the conductance. In the linear response regime, it is given by the Landauer formula G =

∑G σ

σ

38-3

Quantum Point Contact in Two-Dimensional Electron Gas

Gσ = −

e2 ∂f ( E − EF ) dET σ ( E ) , h ∂E

∫

(38.1)

where Tσ(E) is the total transmission coefficient for the spin channel σ = ↑, ↓ f(E − EF) is the Fermi-Dirac distribution function EF is the Fermi energy (for the derivation of the Landauer formula, see e.g., the textbook by Davies, 1995) In this section, we limit ourselves to the spin-degenerate electrons; the electron interaction and spin effects in the QPC will be discussed in the sections that follow. In the linear response regime for spin-degenerate electrons in the limit of zero temperature, the above expression for the conductance is reduced to G=

2e 2 T (EF ). h

(38.2)

In order to calculate the transmission coefficient, let us define the scattering states of the system. For simplicity, let us first consider a hard-wall confining potential. Divide the QPC structure into three regions: two wide semi-infinite leads of the constant width w playing a role of electron reservoirs, and the central region including a constriction, see Figure 38.1a. Because the leads are of a constant width, in the leads regions one can separate the variables in the Schrödinger equation such that its general solution for the energy E reads ψ lead (x , y ) = ( Aα eikα x + Bα e −ikα x )φα ( y ),

(38.3)

where the summation runs over all modes (both propagating and evanescent). The reflection and transmission coefficients from the propagating mode α to the propagating mode β are defined as the ratio of the fluxes associated with each mode: Tβα =

Assuming that all incoming modes are independent (i.e., there is no interference between them), the total transmission and reflection coefficients are obtained by summing the contribution from all the transmitted and reflected states: T=

φα ( y ) = 2/π sin(παy /w) represents the αth transverse mode in the leads E = ( 2k 2 )/2m * , where k 2 = (kα )2 + (kα⊥ )2 with k α and kα⊥ = (πα)/w being respectively the wave vectors of the longitudinal and transverse motions, m* is the effective mass for GaAs, and w is the width of the leads Note that the modes with (kα)2 > 0 are propagating, whereas those with an imaginary k-vector, (kα)2 < 0, are evanescent. Let us inject in the left lead electrons in the propagating transverse mode α. This state will be scattered by the constrictions and thus transmitted and reflected into all available modes of β (both propagating and evanescent) in the right and left leads with the corresponding transmission and reflections amplitudes tβα and r βα . The total wave functions in the left and right leads corresponding to the incoming state α therefore reads ψ left (x , y ) = eikα x +

∑r

βα

e

−ikβ x

φβ ( y ),

∑T

βα

; R=

α, β

∑R

βα

.

α, β

The transmission and reflection coefficients can be calculated exactly using well-established numerical methods such as the scattering matrix or Green’s function techniques (Datta, 1997). In order to get a better insight into the problem at hand, we start with some approximate treatment. Let us expand the wave function for the QPC structure in terms of local transverse eigenfunctions, Ψ(x , y ) = χα (x )ϕα ( x, y) (note that in the lead

∑

α

regions, this form of the wave function reduces to Equation 38.3). Substituting this expansion into the Schrödinger equation, we obtain the following coupled differential equations for the mode coefficients χα(x): −

2 d2χα ( x) + Eα ( x )χ α ( x) 2m * dx 2 = Eχ α ( x) +

where

kβ | t βα |2 ; Rβα =| rβα |2 . kα

2 2m *

∑ ⎡⎢⎣2A

αβ

(x)

β

d ⎤ + Bαβ ( x )⎥ χβ ( x ), dx ⎦

(38.4)

where the transverse eigenfunctions φα(x, y) satisfy the (local in x) transverse Schrödinger equation

−

2 ∂2ϕ α (x , y ) + V (x , y )ϕ α (x , y ) = Eα (x )ϕ α (x , y ), 2m * ∂x 2

with the transverse eigenenergy (local in x) Eα ( x ) = ( 2kα⊥ 2 ( x ))/2m * (with kα⊥ (x ) = (πα)/w(x )) and the coupling matrixes are given by the expressions Aαβ (x ) = ϕ α (x , y )

∫

∂ ϕβ (x , y )dy , ∂x

∫

∂2 ϕβ (x , y )dy. ∂x 2

Bαβ (x ) = ϕ α (x , y )

β

ψ right (x , y ) =

∑ β

ikβ x

tβα e

φ β ( y ),

In a realistic QPC structure, the confi ning potential varies slowly along the x-direction and, therefore, one may expect an adiabatic transport regime when the mode mixing

38-4


Eα(x)

EF

E3 E2

of the adiabatic approximation have been pointed out by Yacoby and Imry (1990). They calculated the leading corrections to the adiabatic approximation and demonstrated that the adiabaticity breaks down (i.e., the mode mixing becomes significant) at some distance from the constriction d when w′(d) ~ 1/Nprop(d), with Nprop(d) = int[kw(d)/π] being the local number of propagating modes (typically Nprop(d) >> 1). At the same time, at this distance the correction to the reflection amplitudes is of the order of −2 rαβ (d ) ∼ N prop (d ) > ω0) it approaches the subband spacing ħωc of LLs of the homogeneous 2DEG. Due to the increase in the subband spacing, the number of occupied subbands Nocc below the Fermi level decreases. Because a number of conductance plateaus are equal to Nopp, the reduction of the plateau number directly reflects the effect of the magnetosubband depopulation. As for the case of the zero magnetic field, the magnetoconductance of a realistic saddle-point parabolic QPC (described by Equation 38.5) can be written in a simple analytic form given by the expression similar to Equation 38.6 (for details, see Buttiker, 1990a,b). This is illustrated in Figure 38.10f for a QPC with a rather narrow constriction, ωx = ωy. Because for this case the tunneling barrier is thin, the zero-field quantization is poor with

n(y)

nB

1 (c)

2

E EF

1 mω20 y2

1

2 yk

y

2

(d)

y

G (2e2/h)

(a)

΄ωc = 0 1

ωc = 5ωx

0 (b)

(e)

(f )

2

4 6 (E – V0)/ћωx

8

FIGURE 38.10 Magnetosubband structure and conductance of a parabolic quantum wire within a one-electron picture. (a) Schematic illustration of the structure of the energy levels for the case of a parabolic confi nement. A thin parabola shows the electrostatic confi nement; flat parabolas represent the magnetic confi nement. Each parabola is shifted by the distance of the guiding center yk that depends on the wave vector k of the longitudinal motion. For the given k the vertex of the magnetic parabola gives a position of the center-of-the-mass of the corresponding wave function. Dashed lines inside the parabolas (marked by 1, 2, 3) show positions of the three lowest states. Filled and empty circles indicate respectively fully occupied and empty states. (b) Classical cyclotron motion for the states inside the wire and classical skipping orbits near the wire boundary. (c) The electron density in a quantum wire for the case when three LLs are occupied in the center of the wire. (d) Adiabatic bending of the Landau levels by the external confi nement as illustrated in (a). (e) Edge states. (Aft er Chklovskii, D.B. et al., Phys. Rev. B, 47, 12605, 1993.) (f) Transition from the conductance quantization in a zero field to the conductance quantization in high magnetic fields for a quantum wire with ωx = ωy. For the case of magnetic field note wide quantization plateaus and transition regions of a negligible width. (Aft er Buttiker, M., Phys. Rev. B, 41, 7906, 1990a.)

38-11


barely defined quantization steps. The effect of the magnetic field on the conductance is two-fold. First, the magnetic field reduces a number of conductance plateaus due to the magnetic subband depopulation as discussed above. Second, the quality of the quantization improves drastically. This is because of the edge-state character of propagating states in the magnetic field. Because left- and right-propagating edge states are spatially separated and localized in the vicinity of different boundaries, the coupling between them can be exponentially small. Th is, in turn, leads to a strongly suppressed backscattering and to almost perfect quantization. The improvement of the conductance quantization by the application of a magnetic field is illustrated in Figure 38.9b. At a magnetic field of zero, the conductance quantization of a QPC is deteriorated (presumably due to the effects of the impurities after several thermal cycling). A relatively small magnetic field (when the cyclotron radius is still smaller than the QPC width) is sufficient to improve the quantization. Note that the drastic improvement of the QPC quantization in the high magnetic field and the remarkable accuracy of the quantum Hall effect have the same origin related to the suppression of the backscattering in the edge-state transport regime.

38.3.2 Interacting Electrons in Quantum Point Contact In the previous section, we presented the basics of the magnetoconductance in the QPC based on a one-electron picture of noninteracting electrons. It reproduces well the essential 3nB

n (y)

3nB

2nB

2nB

nB

nB

(a)

physics related to the depopulation of the magnetosubbands (Berggren et al., 1986). However, there are two important features in the experimental conductance that are not captured by the one-electron description. First, in the one-electron description, the calculated magnetoconductance exhibits wide quantized plateaus separated by transition regions of an essentially negligible width. The experiments, however, show that an extent of these transition regions can be comparable to the width of the plateaus (c.f. Figures 38.9a and 38.10f). Second, at a sufficiently high magnetic field, a spin degeneracy is lifted and odd plateaus are manifested in the experimental conductance (see Figure 38.9a). These two effects (that are due to enhanced screening and exchange interaction, respectively) will be discussed in this section. Within a one-electron picture, the edge states have an essentially zero width and their spatial position is determined by the intersection of a corresponding subband with the Fermi energy as illustrated in Figure 38.10d and e. This naive picture does not account for screening that takes place at the Fermi level where the states are only partially fi lled; hence, the system has a metallic character. Because of the metallic behavior, the electron density can be easily redistributed (compressed) to screen the external potential in order to keep it constant. As a result, the edge states transform to the strips of the finite width called compressible strips. The compressible strips are separated by the incompressible regions where all the levels lie below EF, hence they are completely fi lled, see Figure 38.11a through f. As a result, in these regions, the local electron density is constant

n (y)

(d) E

E

EF

EF

G(2e2/h)

3

(b)

y

(e)

y

2

1

0 (c)

(f )

(g)

1

2

3

Filling factor ν(0)

FIGURE 38.11 Magnetosubband structure and conductance of a parabolic quantum wire for spinless interacting electrons for the case of three occupied subbands in the center of the wire. (a–c) and (d–f) correspond to the cases of respectively incompressible and compressible strips in the middle of the wire. (a, d) The electron density in the quantum wire. In the region of the incompressible strips the electron density is quantized in units of nB. (b, e) Magnetosubbands for interacting electrons. In the compressible regions the subbands are pinned to the Fermi energy. (c, f) The compressible and incompressible strips in the channel (shaded and unshaded regions, respectively). Filled and empty circles indicate respectively fully occupied and empty states. Half fi lled circles indicate partially fi lled states. (g) Magnetoconductance of a representative quantum point contact as a function of the fi lling factor in the center of the channel v(0) = n(0)/nB with n(0) being the electron density in the absence of magnetic field. Note wide transition regions and quantization plateaus a negligible width. (Adapted from Chklovskii, D.B. et al., Phys. Rev. B, 47, 12605, 1993.)

38-12


(incompressible) and is equal to the number of allowed states in each LL, nB = eB/h, times the number of occupied LLs. A quantitative semi-classical treatment of the electrostatic of the edge states including the calculation of the positions and the widths of the compressible and incompressible strips for spinless electrons was proposed by Chklovskii et al. (1993). To calculate the magnetoconductance of the QPC, they used a conjecture that the QPC conductance is given by the filling factor at the saddle point of the electron-density distribution multiplied by 2e2/h. Hence, the plateaus in conductance are identified with the situation when there is an incompressible strip in the center of the channel (see Figure 38.11a through c), whereas the transition regions correspond to the central compressible strip (see Figure 38.11d through f). A conductance of a representative QPC calculated according to Chklovskii et al.’s conjecture is shown in Figure 38.11g. In contrast to the one-electron description, the magnetoconductance of interacting electron was shown to exhibit very narrow quantized plateaus separated by much broader rises where the conductance was not quantized. This conclusion (being opposite to the prediction of the one-electron picture, c.f. Figure 38.10f) is also in apparent disagreement with the experiments that typically show that an extent of the transition regions is smaller than the width of the plateaus. This indicates that even an accurate quantitative description of the magnetoconductance requires quantum mechanical treatment including electron interaction and spin effects. A quantitative quantum mechanical description of the magnetoconductance of split-gate structures focusing on the formation and evolution of the even and odd (spin-resolved) conductance plateaus was given by Ihnatsenka and Zozoulenko (2008) on the basis of the spin density functional theory (DFT) in the local spin density approximation. The starting point is the Hamiltonian within the Kohn–Sham formalism H (r) = H 0 + VKS (r),

(38.13)

where H0 represents the kinetic energy VKS(r) is the mean-field Kohn–Sham potential (Giuliani and Vignale, 2005) The central idea of the Kohn–Sham formalism is that the ground state density of the interacting system is represented by the ground state density of a noninteracting system in some local external potential VKS(r). The quantum mechanical exchange and correlation effects are included in this potential via the local exchange-correlation potential Vxc(r). The exchange-correlation potential Vxc is typically calculated within a so-called local density approximation, when for a system with a spatially varying density, the exchange-correlation energy is locally approximated by that for the corresponding system at a constant density (for details, see e.g., Giuliani and Vignale, 2005). For the system at hand, the mean-field Kohn–Sham potential reads VKS = Vconf + VH + Vxc + VZ ,

(38.14)

where Vconf is the confining potential from metallic gates VH is the Hartree potential VZ is the Zeeman energy The magnetosubband structure and the electron density are calculated self-consistently and the conductance is computed on the basis of the Landauer formula. Figure 38.12a shows the magnetoconductance of a representative wire calculated within the spin DFT and the Hartree approximations (note that the Hartree approximation corresponds to the case of spinless electrons when the exchange interaction is set to zero, Vxc = 0). The Hartree magnetoconductance shows the plateaus quantized in units of 2e2/h separated by transition regions whose width grows as the magnetic field is increased. For large fields, the width of the transition regions is comparable or can even exceed the width of the neighboring plateaus. Analyses of the band structure (Figure 38.12b and c) demonstrate that the formation of the transition regions between the plateaus is related to the development of the compressible strip in the middle of the wire corresponding to the highest occupied subband. The transition between the conductance steps starts when the compressible strip reaches the center of the wire and it ends when the compressible strip disappears and two highest (spin-degenerate) magnetosubband are pushed above EF. Accounting for the exchange and correlation interactions within the spin DFT leads to the lift ing of the spin degeneracy and the formation of the spin-resolved plateaus at odd values of e2/h. The most striking feature of the magnetoconductance is that the width of the odd conductance steps in the spin DFT calculations is equal to the width of the transition intervals between the conductance steps in the Hartree calculations (Figure 38.12a). This is because the transition intervals in the Hartree magnetoconductance correspond to the formation of the compressible strip in the middle of the wire. At the same time, in the compressible strip in the center of the wire, the states are only partially occupied. As a result, the exchange interaction enhances the difference in the spin-up and spin-down population (triggered by the Zeeman interaction VZ), which leads to the lift ing of the subband spin degeneracy and the formation of the odd conductance plateaus, see Figure 38.12b and c. Another striking feature of the magnetoconductance is the effect of the collapse of the odd conductance plateaus for lower fields. This effect is attributed to the reduced screening efficiency in the confined (wire) geometry when the width of the compressible strip in the center becomes much smaller than the extent of the wave function (see the lower panels in Figure 38.12b and c). A detailed comparison of the experimental data (see Figure 38.12d) demonstrates that the spin-DFT calculations reproduce not only qualitatively, but rather quantitatively all the features observed in the magnetoconductance of the QPC. This includes the dependence of the width of the odd and even plateaus on the magnetic field as well as the estimation of the subband index corresponding to the last resolved odd plateau in the magnetoconductance.

38-13

Quantum Point Contact in Two-Dimensional Electron Gas B = 1.7 T + DFT Hartree

+

5 0

16

0 12 (b)

E (meV)

G (e2/h)

B = 1.84 T

v

20

(c)

8

wH comp

H wcomp

–5 V

–10 4

Hartree

0.0 40

0.5

1.0

1.5

Hartree

2.0 0

36

Theory Experiment

32

E (meV)

(a)

8

EF 8 7

7

–5

–10

24

2

20 16 12 8 4 0.0

V

DFT Current (arb. u.)

G (e2/h)

28 DFT

V

Hartree

Hartree

DFT

DFT

1 0 2 1 0

0.2

0.4

0.6

(d)

0.8

1.0

1.2

1.4

1.6

B (T)

0 (b)

100 y (nm)

200 (c)

0

100

200

y (nm)

FIGURE 38.12 (a) Conductance of the quantum wire calculated within the Hartree approximation and the spin DFT (the former is shifted by −2e 2/h for clarity). Note that the transition regions in the Hartree conductance correspond to the odd plateaus in the spin DFT conductance. (b, c) The Hartree and the spin DFT electron densities and the magnetosubband structure calculated for the magnetic fields marked by respectively (b) and (c) in (a) (corresponding to the cases of the incompressible (b) and the compressible (c) strips in the center of the wire). Lower panels show the current densities for the last two subbands (N = 7, 8). (d) Comparison of the calculated and experimental conductances (the effective width of a QPC is 700 nm). Experimental data is by Radu et al. (2008a). (After Ihnatsenka, S. and Zozoulenko, I.V., Phys. Rev. B, 78, 035340-1, 2008.)

38.3.3 Quantum Point Contact in the Quantum Hall Regime In previous sections, we discussed the basics of the magnetoconductance of a QPC in a two terminal geometry. The majority of the magnetotransport measurements are done in the quantum Hall geometry including four or six voltage probes (see Figure 38.13). The resistance of a mesoscopic conductor in a multi-terminal geometry can be calculated on the basis of Landauer–Buttiker formalism (Buttiker, 1986). Consider a QPC in a multi-terminal geometry depicted in Figure 38.13. Each ith terminal (lead) is characterized by the chemical potential μi. Define the transmission probability from the lead i to the lead j as Tj ← i ≡ Tji. (Note that Tji can exceed 1 if there are more than one propagating modes in a lead j; Tii ≡ Ri is defined as a reflection coefficient in the lead i). The net current emitted by the lead i is then given by 2e 2 Ii = h

∑ (T V − T V ) , ji i

j

ij

j

(38.15)

where Vi = μi/e is the voltage in the terminal i. The current is drawn through the source and drain terminals, Is = −Id = I; the voltage probe terminals 1−4 draw no current. We define the multi-terminal resistance in a standard way, Rij = (Vi − Vj)/I. Hence, the longitudinal resistance is R L = R l2 = R34; the Hall resistance is RH = R l3 = R24; and the two-terminal resistance is R 2t = Rsd. For the geometry of Figure 38.13, one can also defi ne the two diagonal resistances R14 and R 23. Assume that in the bulk regions (i.e., in the contact regions), the system has a fi lling factor v = N (which corresponds to N propagating edge states in the leads). The QPC is set to transmit only M states (i.e., T43 = T12 = M). By applying the Buttiker formula (38.15) to the geometry of Figure 38.13 and setting the drain voltage to Vd = 0, we obtain RL =

h ⎛ 1 1⎞ h 1 h 1 − ⎟ , RH = 2 , R2t = 2 , 2 ⎜ ⎝ ⎠ 2e M N 2e N 2e M

h ⎛ 1 2⎞ h 1 R14 = 2 ⎜ . − ⎟ , R23 = 2 ⎝ ⎠ 2e M N 2e M

(38.16)

38-14

Handbook of Nanophysics: Nanotubes and Nanowires R2t 1

RL

2

RH

R14

Drain

Source

QPC R23

3

4

FIGURE 38.13 Multi-terminal measurements of the resistance of a QPC. A current is passed between the source and drain terminals, and the voltage is measured between terminals i and j as schematically indicated in the figure.

(Note that the diagonal resistances R14 and R 23 are interchanged on field reversal.) The above relations predict “fractional” values of the conductance GL = RL−1, as was confirmed by Kouwenhoven et al. (Beenakker and van Houten, 1991). A concept of the edge channels utilized in Equation 38.15 can be generalized to the case of the fractional quantum Hall (FQH) effect (Beenakker, 1990). In the FQH regime, the energy of the homogeneous 2DEG has cusps at densities n = vpBe/h corresponding to certain fractional fi lling factors vp. As a result, the chemical potential has a discontinuity at vp such that in the vicinity of the boundary of the 2DEG the electron density decreases stepwise from its bulk value v bulk to zero with steps at vp. The strips of the constant filling factors vp can be identified as incompressible strips, whereas the compressible strips that separate them play a role of current-currying edge channels. This is illustrated in Figure 38.14, which shows alternating compressible and incompressible strips in the vicinity of a QPC in the FQH regime. A generalization of the Landauer–Buttiker formula for the FQH regime is given by equations similar to Equation 38.15 (Beenakker, 1990) Ii =

e2 h

∑

(TjiVi − TijVj ), Tij =

j

where Tij defi nes the transmission probability from lead j to lead i in terms of a sum over the generalized edge channels in lead j. The contribution from each edge channel p = 1, 2, …, Pj contains the weight factor Δvp = vp − vp−1 and the fraction Tp,ij of the current of the pth edge channel of lead j that reaches lead i. Note that Equation 38.17 describes the spin-resolved edge channels (hence the absence of a factor “2” on the right-hand side). Consider now a QPC in the FQH regime corresponding to the fi lling factor v bulk in the bulk and vQPC in the narrowest part of the constriction, see Figure 38.14 for an illustration. In this case, the fractionally quantized resistance is given by expressions similar to Equation 38.16 ⎞ h 1 h 1 , R2t = 2 , ⎟ , RH = 2 2 e v 2 e vQPC bulk ⎠ (38.18) h ⎛ 1 2 ⎞ h 1 R14 = 2 ⎜ − , R = . 23 ⎟ 2e ⎝ vQPC v bulk ⎠ 2e 2 vQPC RL =

h ⎛ 1 1 − ⎜ 2e 2 ⎝ vQPC v bulk

Pj

∑T

p ,ij

Δv p ,

i =1

(38.17)

Quantization of the QPC resistance in the FQH regime according to Equation 38.18 was first confirmed by Kouwenhoven et al.

2

vBulk

vQPC

vBulk

RD (h/e2)

0.5 1.2 μm 0.8 μm

2⅓ 0.4

5/ 2

2⅔

v

3 (a)

vBulk vQPC

3.0 (b)

(c)

3.5

4.0 B (T)

4.5

5.0

FIGURE 38.14 (a) Schematic drawing of the variation in the fi lling factor ν in a channel. (b) Schematic drawing of the incompressible bands of the fractional fi lling factor v buIk and vQPC , alternating with the edge channels (arrows indicate the direction of electron motion in each channel). (After Beenakker, C.W.J., Phys. Rev. Lett., 64, 216, 1990.) (c) The diagonal resistance R 23 from v = 3 to v = 2, measured concurrently in different QPCs of lithographic size 0.8 and 1.2 μm. Th in horizontal bars indicate fi lling factors for the corresponding plateaus. (Adapted from Miller, J.B. et al., Nat. Phys., 3, 561, 2007.)

38-15


(1990). Figure 38.14c shows a representative example of a diagonal resistance QPC exhibiting plateaus corresponding to both fractional (vQPC = 5/2, 7/3, 8/3) and integer (vQPC = 2, 3) quantum Hall states. The study of transport through a QPC in a FQH regime represents an active field of current research. These studies are motivated by e.g., a search for Luttinger liquid behavior (Roddaro et al., 2005), an interest in the non-Abelian quasiparticle statistics for certain fi lling factors such as v = 5/2, and the building of topologically protected gates for quantum computing by the manipulation of non-Abelian quasiparticles (Das Sarma et al., 2005; Miller et al., 2007; Dolev et al., 2008; Radu et al., 2008b).

fabricated by the split-gated technique (Thomas et al., 1996, 1998; Reilly, 2001), quantum wires fabricated by shallow etching (Kristensen et al., 2000), GaAs wires patterned by focused ion beams (Tscheuschner and Wieck, 1996), and InP based quantum wires (Ramvall et al., 1997). The structure was observed with a different strength of confinement, different distances from the confining gates to one-dimensional electron gas, and different densities. In this section, we present the main experimental properties of the 0.7 anomaly, and in Section 38.4.2 we review the main theoretical models aimed at the explanation of this effect. A hallmark of the 0.7 anomaly is its highly unusual temperature dependence, see Figure 38.15. At very low temperatures, the 0.7 anomaly is only weakly developed (and in some experiments is not seen at all). As the temperature is raised, the first quantization plateau at 2e2/h becomes gradually thermally smeared, whereas the 0.7 plateau becomes even stronger. The 0.7 structure can be observable even at 4.2 K when all the quantized plateaus have disappeared. Kristensen et al. (2000) have demonstrated that the relative conductance suppression shows an activated Arrhenius-type behavior, G(T)/G 0 = 1 − C exp(−TA/T) (where G 0 is the measured conductance value of the 0.7 feature, C is a constant, and TA is the activation temperature that depends on the density). The 0.7 feature strengthens as the two-dimensional carrier densities n2D decreases, see Figure 38.15. As n2D is decreased from 1.4 to 1.13 × 1015 m−2, the 0.7 anomaly becomes strongly

38.4 0.7 Anomaly and Many-Body Effects in the Quantum Point Contact 38.4.1 Experimental Evidence of 0.7 Anomaly In 1996, the Cambridge group pointed out that in addition to quantization in 2e2/h steps, the zero-field conductance of a QPC exhibits a step-like feature at G ~ 0.7 × (2e 2/h) that was coined as a “0.7 anomaly” (Thomas et al., 1996). Since then this feature has attracted enormous experimental and theoretical attention and it is now widely believed that it is an intrinsic property of clean one-dimensional ballistic constrictions at low electron densities. The 0.7 anomaly has been seen in short and long QPCs 2 2

T = 60 mK

0.07 K 0.46 K 0.93 K 1.5 K

0.5

B = 13 T

1.1 · 1015 m–2 0.0

0 –4.5 (b) 3 0.3

0

–3 (d)

–4.0 Vg (V)

–3.5

–6.8 (c)

1.5

–6.6 Vg (V)

–6.4

–6.2

2.5

5

–0.2

Vsd (mV)

–0.6 –0.4 Vg (V)

B =0 T

0.8

–0.8

G (2e2/h)

1

0 (a)

G (2e2/h)

G (2e2/h)

1.0 1.4 . 1015 m–2 1

0

1

–6.0

2

–5.5

3

–5.0

–4.5

Vg (V)

FIGURE 38.15 Experimental data for the temperature (a) (Adapted from Thomas, K.J. et al., Phys. Rev. Lett., 77, 135, 1996.), electron density (b) (Adapted from Thomas, K.J. et al., Phys. Rev. B, 58, 4846, 1998.), magnetic field (c) (Adapted from Thomas, K.J. et al., Phys. Rev. Lett., 77, 135, 1996.), and bias (d) (Adapted from Thomas, K.J. et al., Phys. Rev. B, 58, 4846, 1998.) dependence of the 0.7 anomaly. In (b), the electron density is subsequently reduced from 1.4 × 1015 to 1.1 × 1015 m−2 from left to right. (c) The evolution of the structure at 0.7(2e 2/h) into a step at e 2/h in a parallel magnetic field B ∙ = 0–13 T, in steps of 1 T. For clarity successive traces have been horizontally offset by 0.015 V. (d) The grayscale plot of the zero-field transconductance dG/dVg as a function of the side gate voltage Vg and the applied source-to-drain bias Vsd is shows for T = 1.2 K. The numbers indicate the plateau conductances in units of 2e 2/h and the 0.7 anomaly is the bright region at Vsd = 0 between G = 0 and G = 2e 2/h.

38-16


pronounced. At the highest density, shown in the left-hand trace, the 0.7 plateau is visible only as a weak knee. A strong in-plane magnetic field B lifts the spin degeneracy of the one-dimensional subbands giving conductance plateaus quantized in units of e 2/h. Figure 38.15c shows that as B increases, the zero-field 0.7 structure evolves continuously into spin-split half-plateaus e2/h. The effect of a source–drain voltage Vsd on the conductance characteristics G = G(Vg) is shown in Figure 38.15d. As Vsd is increased, half-plateaus appear at (N + 1/2) × 2e2/h for G > 2e2/h, whereas Vsd-induced structures appear at 0.85 × 2e2/h and 0.3 × 2e2/h for G < 2e2/h. The gate voltage scale is a smooth measure of the one-dimensional confinement energy, so a grayscale plot of the transconductance dG/dVsd allows one to follow the energy shifts of the subband features. The dark lines show transitions between plateaus and the white regions are the conductance plateaus (where the numbers denote the conductance in units of 2e2/h). Features moving to the right (left) with increasing Vsd do so as the electrochemical potential of the source (drain) crosses a subband edge, and if the subband energies were independent of their occupation, we would expect a linear evolution of the transconductance structures with Vsd. This is clearly not the case for the features associated with the 0.7 structure in the lowest subband, suggesting that the subband configuration is occupation-dependent, for which an interaction effect could be responsible. It should be stressed that the clean quantized conductance plateaus and the absence of additional structures when the channel is moved from side to side by changing the gate voltage demonstrate the lack of potential fluctuations in and around the one-dimensional constriction, and hence rule out the potential fluctuations, resonant, or Coulomb blockade effects as possible origins of the 0.7 structure. Instead, the 0.7 anomaly was linked to the spontaneous lifting of spin degeneracy in the one-dimensional constriction driven by an electron–electron interaction effect related to the exchange interaction (Thomas et al., 1996, 1998). The strengthening of the 0.7 structure as n2D is lowered and is consistent with

an exchange interaction mechanism. Further evidence that an exchange mechanism may be responsible for the 0.7 structure is provided by the source–drain measurements (see Figure 38.15d) where the features in the lowest subband are sensitive to the occupation statistics in the channel, as well as from the enhancement of the g factor for the last few occupied subbands. Valuable independent information supporting the mechanism related to the spontaneous spin splitting is given by the shot noise measurements (Roche et al., 2004). The results for the measured shot noise power clearly indicate that spin-up and spin-down channels do not have the same transmission on the 0.7 structure. The evolution of the noise power with a parallel magnetic field B supports the picture of two channels with different spin orientations. A direct measurement of the spin polarization of the 0.7 structure was performed by Rokhinson et al. (2006). In contrast to a majority of the studies utilizing 2DEG, Rokhinson et al. studied a QPC defined in a two-dimensional hole gas (2DHG). The 0.7 structure in p-type QPCs shows all the essential features reported for n-type QPCs, such as a gradual evolution into the 0.5(2e2/h) plateau at high in-plane magnetic fields, survival at high temperatures, a gradual increase toward (2e2/h) plateau at low temperatures, and the zero bias anomaly, which is suppressed by either temperature increase or the application of a magnetic filed. The similarities between p-type and n-type QPCs suggest that the underlying physics responsible for the appearance of the 0.7 structure should be the same. The spin polarization was measured in a ballistic magnetic focusing geometry consisting of two QPC in parallel, see Figure 38.16. Due to the enhanced spin-orbit interaction in 2DHG, the carriers with opposite spin have different cyclotron orbits even in a small external perpendicular magnetic field B⊥. By injecting current through the first QPC and monitoring the voltage across the detector QPC, the focusing peaks were clearly observed (Figure 38.16). The polarization of the injected current is related to the relative heights of the focusing peaks (similar peak heights correspond to the equal population of spin-up and spin-down

A 1.0 0

B C

B C

–1

D

G (2e2/h)

V (μV)

A

0.5

E

D Detector QPC

Injector QPC

E –2 0.1

(a)

(b)

Injector QPC 0.0 0.10

0.2 B (T)

(c)

0.15 Vg (V)

0.20

FIGURE 38.16 Polarization detection via magnetic focusing. (a) Magnetic focusing geometry and the schematic trajectories of the ballistic holes in a perpendicular magnetic field. (b) The fi rst spin-split focusing peak is measured at different injector conductances. The curves are vertically offset relative to the top one. (c) The gate voltage characteristic of the injector QPC. Vertical arrows (A–E) mark the positions where the curves in (b) are taken. (Adapted from Rokhinson, L.P. et al., Phys. Rev. Lett., 96, 156602-1, 2006.)


subbands). The main finding of Rokhinson et al. is a detection of a spin polarization below the first plateau, which is found to be as high as 40% in samples with a well-defined 0.7 structure. In the fi rst paper (Thomas et al., 1996), the origin of the 0.7 anomaly was already attributed to spontaneous spin polarization. Since then, many dozens (if not hundreds) of various studies have addressed this problem. Despite this, no consensus has been reached concerning the microscopic origin of the 0.7 anomaly. Different viewpoints with confl icting conclusions are reported in the literature, and in this section we briefly present some of these theories.

38.4.2 Theoretical Models for 0.7 Anomaly There are several phenomenological models quantitatively reproducing the main features of the 0.7 anomaly including its unusual temperature dependence (Bruus et al., 2001; Reilly et al., 2002). While differing in details, these models essentially rely on the assumption of the spin gap that opens up between the quasi one-dimensional subbands in the constriction. Such approaches, while providing an important insight for an interpretation of the experiment, are not, however, able to uncover the microscopic origin of the observed effect. The microscopic origin of the 0.7 anomaly was addressed in studies based on mean-field approaches. The spontaneous spin-splitting of the one-dimensional subband in quantum wires was studied by Wang and Berggren (1996) using a spin DFT. Starting with the Hamiltonian of the form of Equation 38.13 and solving the Schrödinger equation within the Kohn–Sham formalism, they demonstrated that the exchange interactions cause a large subband splitting whenever the Fermi energy passes the subband threshold energies and full spin polarization appears at low electron densities, see Figure 38.17. Numerous subsequent mean-field 8

En (meV)

4 EF 0

–4

0

4

8 n1d (105 m–1)

FIGURE 38.17 Sublevels in an infinite, straight quantum wire vs. the one-dimensional electron density n1d. Solid and dotted lines correspond to the spin-up and spin-down electrons. The dashed line shows the position of the Fermi level EF. (Adapted from Wang, C.-K. and Berggren, K.-F., Phys. Rev. B, 54, 14257, 1996.)

38-17

studies addressed spin polarization in a constricted geometry of a QPC. These studies include the spin-DFT approaches (Wang and Berggren, 1998; Berggren and Yakimenko, 2002; Hirose et al., 2003; Havu et al., 2004; Ihnatsenka and Zozoulenko, 2007), Hartree–Fock approaches (Sushkov, 2001), and Hubbard models (Cornaglia et al., 2005), just to name a few representative publications. Various mean-field approaches obviously treat the exchange interaction in different ways; the details of the modeling of a QPC (such as modeling of the confinement, calculation of the electron density, and the conductance.) differ in different works. This sometimes led to somehow different results and conclusions whose detailed review is far beyond the scope of the present chapter. Nevertheless, regardless of the approaches used (spin-DFT, Hartree–Fock, Hubbard) or the details of the modeling of the QPC, all mean-field calculations arrive to the same qualitative conclusion. Namely, as was already shown in 1998 by Wang and Berggren, the spontaneous spin polarization occurs locally in the region of the saddle point as the electron density is lowered. As a consequence, the effective potential barriers become different for spin-up and spin-down electrons. Transport associated with, say, spin-up electrons takes place via tunneling, while spin-down electrons still carry the current via propagating states in a normal way (Wang and Berggren, 1998). The difference of the effective potential might result in a formation of a localized quasi-bound state for one of the spin species (Hirose et al., 2003). The conductance of a representative QPC along with the effective potential and the local density of states for spin-up and spin-down electrons calculated within the spin-DFT approach is illustrated in Figure 38.18. Despite the prediction of the spontaneous spin polarization in the QPC consistent with the initial interpretation of the effect in the first paper of Thomas et al. (1996), the meanfield approaches fail to account for the key experimental features of the 0.7 anomaly. In particular, the mean-field approaches do not reproduce the unusual temperature dependence of the 0.7 anomaly. In contrast to the experimental data, the spin polarization predicted by the mean field theories is maximal for zero temperature and is gradually smeared as the temperature rises. Besides, the calculated conductance predicts a plateau value of 0.5 instead of 0.7. This is related to the above-mentioned fact that one of the spin channels remains blocked and the corresponding tunneling amplitude is exponentially small. Note that it has been argued recently that the failure to reproduce 0.7 anomaly within the spin-DFT approach might be related to the derivative discontinuity problem of the DFT, leading to spurious self-interaction errors not corrected in the standard local density approximation (Ihnatsenka and Zozoulenko, 2007). There are several studies reported in the literature that did not attribute the 0.7 anomaly to spin effects. For example, Seelig and Matveev related the 0.7 anomaly to the enhanced backscattering by the acoustic phonons. For typical GaAs quantum wires, the effect of electron–phonon scattering on transport in quantum wires is strongly suppressed. On the other hand, the electron density in a QPC in the vicinity of the first conductance step is very low. As a result, a minimum energy of a phonon required to backscatter an electron inside a QPC is much

38-18

Handbook of Nanophysics: Nanotubes and Nanowires x (nm) –100

0

x (nm) 100

–100

0

100

100

l = 200 nm Unpolarized +

0

1

–100

Spin-up

Spin-down 0

(b) LDOS

1

0 E (meV)

G (2e2/h)

2

y (nm)

n2D (1015 m–2) 2

12 –1

EF

>1.5 0.75

–4 0

0 –1.0 (a)

–1

(10 J m )

–0.9 Vg (V)

–8

–0.8 (c)

–100

0 x (nm)

100

–100

0 x (nm)

100

FIGURE 38.18 (See color insert following page 20-16.) Conductance of the quantum point contact calculated within the spin DFT approach as a function of the gate voltage Vg. The geometrical width of the constriction is w = 100 nm; the geometrical length is l = 200 nm. Dashed line corresponds to the spin-unpolarized solution. (b, c) Formation of quasibound states in the quantum point contact. (b) The charge density and (c) the local density of states are shown for the regime of one transmitted spin-up and totally blocked spin-down channel. The left and right columns correspond to the spin-up and spin-down electrons. White dashed lines in (c) indicate the self-consistent Kohn–Sham potential in the center of the QPC along the transport direction. The geometrical length and width of the QPC are l = 200 nm and w = 100 nm, respectively; the gate voltage Vg = −0.98 V. (Adapted from Ihnatsenka, S. and Zozoulenko, I.V., Phys. Rev. B, 76, 045338, 2007.)

smaller in comparison with a corresponding bulk value. Seelig and Matveev demonstrated that such a backscattering gives rise to the negative correction to the quantized value 2e 2/h of the conductance of a QPC with activated temperature dependence (consistent with the observation of Kristensen et al., 2000) and also gives rise to a zero-bias anomaly in conductance in agreement with the observation of Cronenwett et al. (2002). Sloggett et al. (2008) studied the effect of inelastic electron scattering on the conductance of a QPC. They demonstrated that at zero temperature, the approach results in the usual Landauer formula and the conductance does not show any structures. However, at nonzero temperature, the electron–electron interaction gives rise to a current of correlated electrons, which scales as squared temperature. The corresponding correction to conductance is negative and strongly enhanced in the region below the first conductance plateau. While the above studies accounting for the acoustic phonon scattering and the inelastic electron–electron scattering seem to reproduce the unusual temperature dependence of the 0.7 anomaly, it is not clear how to reconcile these theories with the experimental evidence of the spin polarization of the current below the fi rst conductance plateau reported in the experiments of Roche et al. (2004) and Rokhinson et al. (2006).

38.4.3 0.7 Anomaly and Kondo Physics A many-body origin of the 0.7 anomaly related to the Kondo physics was advocated in an experimental study by Cronenwett et al. (2002; for an introduction to the Kondo physics see e.g.,

Hewson, 1997). They systematically measured the dependence of the conductance on temperature, magnetic field and applied source–drain voltage and found a number of similarities with the Kondo effect seen in quantum dots (Goldhaber-Gordon et al., 1998). In particular, (1) a narrow conductance peak forms at zero source–drain bias and low temperature, (2) conductance data collapses onto a single Kondo-like function with a single scaling parameter (designated as the Kondo temperature), (3) the width of the zero-bias peak is proportional to the Kondo scaling factor, and (4) the zero-bias peak splits in the magnetic field. A theoretical model of the Kondo effect in a QPC was developed by Meir et al. (2002). Supported by the spin-DFT calculations predicting a formation of the localized state in a QPC, they used the Anderson impurity Hamiltonian, where the localized state inside the QPC (a “magnetic moment”) plays a role of an unpaired spin. Within the Kondo model, this magnetic moment hybridizes with delocalized electrons in the leads forming a spin singlet state. According to the model of Meir et al., the 0.7 plateau is attributed to a high background conductance (giving rise to a 0.5 plateau) plus a Kondo enhancement. The Kondo enhancement of the conductance is suppressed with increasing temperature, which explains the unusual temperature dependence of the 0.7 anomaly that becomes more pronounced with the increase of the temperature. Because of the formation of a singlet state, the Kondo model predicts fully polarized transport below the first conductance plateau. Th is prediction, however, is in obvious disagreement with the experimental fi ndings of Rokhinson et al. (2006) of the partially spin-polarized current for the 0.7 anomaly as well as with the measurements of the shot


noise by Roche et al. (2004) indicating a presence of two channels with different spin orientations. Very recently, Sfigakis et al. (2008) reported a study of the Kondo effect in a QPC structure showing that the Kondo effect and the 0.7 structure are separate and distinct effects. Thus, the role of the Kondo correlations in the “0.7 anomaly” still remains an open question.

38.5 Conclusion and Outlook The discovery of the conductance quantization in the QPC laid the foundation and provided a strong momentum for developments in many subfields of mesoscopic physics and the physics of low-dimensional semiconductor structures. Many aspects of the electron transport through a QPC are well understood by now. This includes, for example, the basics physics of the conductance quantization, effect of impurities, magnetosubband depopulation, and many others. Some features of the QPC have found their important practical application. For example, the QPC is routinely used as a standard tool for noninvasive voltage probing and charge detection. However, some important aspects of the electron transport in the QPC still remain highly controversial. This primarily concerns the 0.7 anomaly whose detailed microscopic origin is still unresolved and is under lively debates in the current literature. The current research is also focused on the many-body, nonlinear, and spin effects; study of a QPC in a two-dimensional hole gas; the search for the Luttinger liquid behavior; and the study of quasiparticles properties in the fractional quantum Hall regime. Despite the 20 years that have passed since the discovery of the conductance quantization, the electronic and transport properties of the QPC still remain an active field of research, and one can expect a number of new exciting discoveries in many years to come!

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Heinzel, T. 2007. Mesoscopic Electronics in Solid State Nanostructures. Weinheim, Germany: Wiley-VCH. Hewson, A. C. 1997. The Kondo Problem to Heavy Fermions. Cambridge, U.K.: Cambridge University Press. Hirose, K., Yigal Meir, Y., and Wingreen, N. S. 2003. Local moment formation in quantum point contacts. Phys. Rev. Lett. 90: 026804-1–026804-4. Ihnatsenka, S. and Zozoulenko, I. V. 2007. Conductance of a quantum point contact based on spin-density-functional theory. Phys. Rev. B 76: 045338-1–045338-9. Ihnatsenka, S. and Zozoulenko, I. V. 2008. Magnetoconductance of interacting electrons in quantum wires: Spin density functional theory study. Phys. Rev. B 78: 0353401–035340-10. Johnson, A. C., Petta, J. R., Taylor, J. M., Yacoby, A., Lukin, M. D., Marcus, C. M., Hanson, M. P., and Gossard, A. C. 2005. Triplet–singlet spin relaxation via nuclei in a double quantum dot. Nature 430: 431–435. Kataoka, M., Ford, C. J. B., Faini, G., Mailly, D., Simmons, M. Y., Mace, D. R., Liang, C.-T., and Ritchie, D. A. 1999. Detection of Coulomb charging around an antidot in the quantum Hall regime. Phys. Rev. Lett. 80: 160–163. Kouwenhoven, L. P., van Wees, B. J., van de Vaart, N. C., Harmans, C. J. P. M., Timmering, C. E., and Foxon, C. T. 1988. Selective population and detection of edge channels in the fractional quantum Hall regime. Phys. Rev. Lett. 64: 685–688. Kristensen, A., Bruus, H., Hansen, A. E., Jensen, J. B., Lindelof, P. E., Marckmann, C. J., Nygard, J., Sorensen C. B., Beuscher, F., Forchel, A., and Michel, M. 2000. Bias and temperature dependence of the 0.7 conductance anomaly in quantum point contacts. Phys. Rev. B 62: 10950–10957. Lesovik, G. B. 1989. Excess quantum noise in 2D ballistic point contacts. Pis’ma Zh. Eksp. Teor. Fiz. 49: 515–517. [JETP Lett. 49: 592–594.] Maaø, F. A., Zozulenko, I. V., and Hauge, E. H. 1994. Quantum point contacts with smooth geometries: Exact versus approximate results. Phys. Rev. B 50: 17320–17327. Meir, Y., Hirose, K., and Wingreen, N. S. 2002. Kondo model for the “0.7 anomaly” in transport through a quantum point contact. Phys. Rev. Lett. 89: 196802-1–196802-4. Miller, J. B., Radu, I. P., Zumbuhl, D. M., Levenson-Falk, E., Kastner, M. A., Marcus, C. M., Pfeiffer, L. N., and West, K. W. 2007. Fractional quantum Hall effect in a quantum point contact at filling fraction 5/2. Nat. Phys. 3. 561–565. Nixon, J. A., Davies, J. H., and Baranger, H. U. 1991. Breakdown of quantized conductance in point contacts calculated using realistic potentials. Phys. Rev. B 43: 12638–12641. Radu, I. P., Miller, J. B., Amasha, S., Levenson-Falk, E., Zumbuhl, D. M., Kastner, M. A., Marcus, C. M., Pfeiffer, L. N., and West, K. W. 2008a. Suppression of spin-splitting in narrow channels of a high mobility electron gas, unpublished. (The layout and geometry of the devices and heterostructures are similar to those studied by Miller et al. 2007).


Radu, I. P., Miller, J. B., Marcus, C. M., Kastner, M. A., Pfeiffer, L. N., and West, K. W. 2008b. Quasi-particle properties from tunneling in the ν = 5/2 fractional quantum Hall state. Science 320: 899–902. Ramvall, P., Carlsson, N., Maximov, I., Omling, P., Samuelson, L., Seifert, W., Wang, Q., and Lourdudoss, S. 1997. Quantized conductance in a heterostructurally defined Ga0.25In0.75As/InP quantum wire. Appl. Phys. Lett. 71: 918–921. Reilly, D. J., Facer, G. R., Dzurak, A. S., Kane, B. E., Clark, R. G., Stiles, P. J., Hamilton, A. R., O’Brien, J. L., Lumpkin, N. E., Pfeiffer, L. N., and West, K. W. 2001. Many-body spin-related phenomena in ultra low-disorder quantum wires. Phys. Rev. B 63: 121311-1–121311-4. Reilly, D. J., Buehler, T. M., O’Brien, J. L., Hamilton, A. R., Dzurak, A. S., Clark, R. G., Kane, B. E., Pfeiffer, L. N., and West, K. W. 2002. Density-dependent spin polarization in ultra-low-disorder quantum wires. Phys. Rev. Lett. 89: 246801-1–246801-4. Reznikov, M., Heiblum, M., Shtrikman, H., and Mahalu, D. 1995. Temporal correlation of electrons: Suppression of shot noise in a ballistic quantum point contact. Phys. Rev. Lett. 75: 3340–3343. Roche, P., Segala, J., Glattli, D. C., Nicholls, J. T., Pepper, M., Graham, A. C., Thomas, K. J., Simmons, M. Y., and Ritchie, D. A. 2004. Fano factor reduction on the 0.7 conductance structure of a ballistic one-dimensional wire. Phys. Rev. Lett. 93: 116602-1–116602-4. Roddaro, S., Pellegrini, V., Beltram, F., Pfeiffer, L. N., and West, K. W. 2005. Particle-hole symmetric Luttinger liquids in a quantum Hall circuit. Phys. Rev. Lett. 95: 1568041–15680-4. Rokhinson, L. P., Pfeiffer, L. N., and West, K.W. 2006. Spontaneous spin polarization in quantum point contacts. Phys. Rev. Lett. 96: 156602-1–156602-4. Sfigakis, F., Ford, C. J. B., Pepper, M., Kataoka, M., Ritchie, D. A., and Simmons, M. Y. 2008. Kondo effect from a tunable bound state within a quantum wire. Phys. Rev. Lett. 62: 026807-1–026807-4. Sloggett, C., Milstein, A. I., and Sushkov, O. P. 2008. Correlated electron current and temperature dependence of the conductance of a quantum point contact. Eur. Phys. J. B 61: 427–432. Sushkov, O. P. 2001. Conductance anomalies in a one-dimensional quantum contact. Phys. Rev. B 64: 155319-1–155319-8. Szafer, A. and Stone A. D. 1989. Theory of quantum conduction through a constriction. Phys. Rev. Lett. 62: 300–303. Tarucha, S., Honda, T., and Saku, T. 1995. Reduction of quantized conductance at low temperatures observed in 2 to 10 μm-long quantum wires. Solid State Commun. 94: 413–418. Thomas, K. J., Nicholls, J. T., Simmons, M. Y., Pepper, M., Mace, D. R., and Ritchie, D. A. 1996. Possible spin polarization in a one-dimensional electron gas. Phys. Rev. Lett. 77: 135–138.


Thomas, K. J., Nicholls, J. T., Appleyard, N. J., Simmons, M. Y., Pepper, M., Mace, D. R., Tribe, W. R., and Ritchie, D. A. 1998. Interaction effects in a one-dimensional constriction. Phys. Rev. B 58: 4846–4852. Thornton, T. J., Pepper, M., Ahmed, H., Andrews, D., and Davies, G. J. 1986. One-dimensional conduction in the 2D electron gas of a GaAs: AlGaAs heterojunction. Phys. Rev. Lett. 56: 1198–1201. Topinka, M. A., LeRoy, B. J., Shaw, S. E. J., Heller, E. J., Westervelt, R. M., Maranowski, K. D., and Gossard, A. C. 2000. Imaging coherent electron flow from a quantum point contact. Science 289: 2323–2326. Topinka, M. A., LeRoy, B. J., Westervelt, R. M., Shaw, S. E. J., Fleischmann, R., Heller, E. J., Maranowski, K. D., and Gossard, A. C. 2001. Coherent branched flow in a twodimensional electron gas. Nature 410: 183–186. Tscheuschner, R. D. and Wieck, A. D. 1996. Quantum ballistic transport in in-plane-gate transistors showing onset of a novel ferromagnetic phase transition. Superlattices Microstruct. 20: 615–622. van Wees, B. J., van Houten, H., Beenakker, C. W. J., Williamson, J. G., Kouwenhoven, L. P., van der Marel, D., and Foxon, C. T. 1988. Quantized conductance of point contacts in a two-dimensional electron gas. Phys. Rev. Lett. 60: 848–851.

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VII Nanoscale Rings 39 Nanorings

Katla Sai Krishna and Muthusamy Eswaramoorthy............................................................................ 39-1

Introduction • Why Is Ring Shape Important? • Synthetic Methods • Types of Nanorings • Conclusions • Acknowledgment • References

40 Superconducting Nanowires and Nanorings

Andrei D. Zaikin .......................................................................... 40-1

Introduction • Background • Thermal and Quantum Fluctuations in Superconducting Nanowires • Persistent Currents in Superconducting Nanorings • Summary • References

41 Switching Mechanism in Ferromagnetic Nanorings Wen Zhang and Stephan Haas ..........................................41-1 Introduction • Background • Magnetic States • Switching Processes • Applications • Conclusions • Acknowledgments • References

42 Quantum Dot Nanorings Ioan Bâldea and Lorenz S. Cederbaum ....................................................................... 42-1 Introduction • Model of Quantum Dot Nanorings • Tunable Electron Correlations • Optical Absorption • Avoided Crossings • Hidden Quasi-Symmetry • Photoionization • High Harmonic Generation by QD Nanorings • Quantum Phase Transitions • Related Systems: Kondo Effect • Conclusion • Acknowledgments • References

VII -1

39 Nanorings 39.1 Introduction ...........................................................................................................................39-1 39.2 Why Is Ring Shape Important? ...........................................................................................39-1 39.3 Synthetic Methods .................................................................................................................39-3

Katla Sai Krishna Jawaharlal Nehru Centre for Advanced Scientific Research

Muthusamy Eswaramoorthy Jawaharlal Nehru Centre for Advanced Scientific Research

Ring Formation by Evaporation-Induced Self-Assembly • Ring Formation Induced by Chemical Reaction • Ring Formation Induced by Electrostatic Interactions • Ring Formation by Template-Based Approach • Ring Formation by Lithographic Technique

39.4 Types of Nanorings................................................................................................................39-7 Carbon Nanorings • Metal Nanorings • Metal Oxide/Sulfide Nanorings • Organic Nanorings

39.5 Conclusions...........................................................................................................................39-15 Acknowledgment.............................................................................................................................39-15 References.........................................................................................................................................39-15

39.1 Introduction The exciting size- and shape-dependent properties associated with many materials at the nanometer scale evoked a great deal of interest in synthesizing materials with different lengths and shapes [1–9]. Spheres, wires, rods, and tubes are common shapes that are usually prevalent in material synthesis as against uncommon shapes like rings, bowls, and other complex morphologies. Nanorings, as the name implies, are nanoscale entities with ring-shaped geometry (i.e. the inner diameter of the ring is larger than the width and thickness of the rim) and with one of their dimensions (either the width or the thickness of the rim) in the nanometer range (Figure 39.1). Molecular nanorings are entities in which the size of the ring-shaped molecule is in the nanometer scale. A nanoring can be a single entity with an end-to-end closed (a perfect tori structure) or coiled structure, or a superstructure formed by the self-assembly of smaller entities such as molecules and nanoparticles (Figure 39.2). In the case of superstructures, the resulting symmetry is often determined by the shape of the smaller entities used to build the structure. Since the prevalent shapes of the smaller units are mostly spherical, disk/tile, and rod-like, it would be difficult to pack them in the form of a ring-like superstructure. For example, opaline structures commonly result from the packing of (monodispersed) spherical entities (Figure 39.3). Stacked pancake structures (discotic phase) are obtained from the packing of disk-shaped entities. Nematic, semectic, or hexagonal phase superstructures are built from the packing of rod-like entities. These types of superstructures are usually observed in controlling the mesophases in colloidal

suspensions and the superlattices in nanocrystal dispersions [10,11]. Nevertheless, the organization of sphere-, disk- and rod-shaped entities into ring-like superstructures [12] was also reported under certain conditions, which will be discussed later in this chapter.

39.2 Why Is Ring Shape Important? Metal nanoparticles such as Ag, Au, and Cu are known to absorb and emit certain wavelengths of light depending upon their size and shape [13]. Among various shapes, ring-shaped nano objects of similar size exhibit several novel properties due to their unique structural features. For example, gold nanorings are shown to display tunable plasmon resonance in the near infrared, which is not possible with solid gold particles of similar size [14,15]. Further, the optical and electromagnetic properties of gold nanorings can be varied by varying the ratio of the ring radius to the wall thickness. When light falls on a gold nanoring, the electrons get excited and oscillate collectively as a wave (plasmon) that can be tuned by tuning the wavelength of light and the geometry of the ring. By tuning the wavelength of the incoming light, the pool of electrons in the ring can be made to resonate in the same wavelength. Th is resonance generates a strong and uniform electromagnetic field that will oscillate within the ring cavity. If the field inside the ring cavity is optimized to near infrared, it (cavity) can be used as a container for holding and probing smaller nanostructures in sensing and spectroscopy applications. Conducting the experiments inside the nanoring will give amplified infrared signals and better results that would be of interest to the drug industry and to biochemical researchers [16]. 39-1

39-2

Handbook of Nanophysics: Nanotubes and Nanowires Electrons

Ring

Disk

FIGURE 39.1

Cylinder

Path A

Path B

Illustration of a ring, a disk, and a cylinder. B

Interference Observation plane/screen Single ring

Coiled ring

Self-assembled ring

FIGURE 39.2 Schematic showing the single ring, the coiled ring, and the self-assembled ring. Sphere

Tile

FIGURE 39.4 Schematic of double-slit experiment in which Aharonov–Bohm effect can be observed: electrons pass through two slits, interfering at an observation screen. The interference pattern will shift when a magnetic field, B, is turned on in the cylindrical solenoid. Path A

Tube

Electron B

Close-packed structures

FIGURE 39.3 tubes.

Scheme illustrating the packing of spheres, tiles, and

Another interesting phenomenon observed in the metal nanorings is called the Aharanov–Bohm effect [17–19], which cannot be explained by classical mechanics. It is a quantum mechanical phenomenon. It predicts that if a coherent electron beam splits into two parts and the path encloses a fi nite magnetic flux (magnetic field B on the path of the electrons is zero), a phase difference will occur in the electron wave packets traveling along the two parts, which in turn would manifest in an interference pattern at the other end due to the nonzero magnetic vector potential A (Figure 39.4). Similarly, the phase coherent electrons traveling around the magnetic field, B, through the arms of the metal nanorings (remember the magnetic field in the conductor regions, the arms of the rings, is zero) should show oscillations in the magnetoresistance because of the interference between the wave packets of electrons of the two arms (Figure 39.5). The phase shift of electrons occurs in both arms of the ring and can be tuned by changing the magnetic flux encircled by the ring.

Path B

FIGURE 39.5

Schematic of Aharanov–Bohm effect in a nanoring.

Nanorings made up of metallic carbon nanotubes are expected to possess giant paramagnetic moments in the presence of an applied magnetic field (perpendicular to the plane of the ring), owing to the effective interplay between the ring geometry and the ballistic motion of π-electrons in the nanotube [20]. Theoretical studies predict that at 0 K and at applied magnetic fields of ~0.1 T, the rings of metallic carbon nanotubes can exhibit very large paramagnetic moments, which are three orders of magnitude higher than the diamagnetic moment of graphite. Magnetic nanorings have been proposed for applications in high-density magnetic storage and vertical magnetic random access memory (V-MRAM) [21,22] because of their ability to retain the vortex states. Larger magnetic elements such as microdisks have rich domain structures determined by their geometry. On the other hand, very small nano entities generally have a single domain state, with all spins pointing to one direction (Figure 39.6). Further, the microdisks form a vortex state, whose magnetic moments form a closed structure from which

39-3

Nanorings

39.3.1 Ring Formation by EvaporationInduced Self-Assembly

(a)

(b)

(c)

FIGURE 39.6 (a) Vortex state in single circular domain (all spins pointing upward), (b) vortex state in a circular disk (with a vortex core where spins are out of planarity and (c) vortex state of a nanoring (with a missing vortex core). In both the circular disk and nanoring, the left side spins are pointing downward and right side spins are pointing upward.

no stray field is leaked out. The center of the vortex, called vortex core, has the magnetic moments pointing out of the plane, either up or down. As the size of the disks shrinks laterally, the vortex core becomes unstable and fi nally transforms to a single domain state below a certain critical size. The competition between the exchange energy (dominant at small scale) and the magneto static energy (dominant at large scale) basically decides such transitions. Since there is no vortex core, a stable vortex state can be achieved in the case of magnetic nanorings. All moments are completely confi ned within the plane and form a closure (Figure 39.6). The vortex chirality (clockwise and anticlockwise) in the magnetic nanorings can be used for information storage. Nanorings made up of porphyrin entities through π–π stacking and supramolecular self-assembly [23,24] are reported to show excellent florescence properties and are being explored for optical applications [25]. All these applications clearly demonstrate the significance of ring morphology at the nanoscale.

39.3 Synthetic Methods As we discussed earlier, nanorings can be made either from a single entity or from the self-assembly of multiple entities like molecules, nanoparticles, etc. Various approaches are being adopted to make nanoring structures of different components. Among these, evaporation induced self-assembly, templatebased synthesis, and chemical reactions and/or electrostatic interactions induced coiling are the most common approaches to make nanorings and are highlighted below.

In this approach, nanoring formation occurs when a solution containing nanocrystals/polymers/nanowires is allowed to evaporate on a substrate. Physical processes such as dewetting, surface tension, and solvent and solute dynamics play a major role in the ring formation. Two types of mechanism have been put forward in the literature to explain the formation of rings from the evaporating solutions. 39.3.1.1 Coffee-Stain Mechanism This was introduced by Deegan et al. [26] to explain the formation of ring-like stains from the solution drop on evaporation. Enrichment of solute at the edges of the droplet will occur during the evaporation of the solvent (from the solution), if the contact line of the solution with the solid substrate is pinned. The solvent at the edge of a solution droplet evaporates faster than the bulk. This evaporation loss at the perimeter will be offset by an outward flow of fluid from the core, which draws the solute (dispersed material) from the interior to the edge of the drop and deposits it as a solid ring (Figure 39.7). If the solute transfer is not complete, a fraction of the material remains inside the resulting ring. Ring formation can occur over a wide variety of substrates, solutes, and solvents subject to the following conditions: 1. The solvent meets the surface at a nonzero contact angle 2. The contact line is pinned to its initial position 3. The solvent evaporates Further, wetting of the substrate surface by the solvent is an important factor, which can be tuned by modifying the substitution pattern of the solute molecules used, the coating of the surface and the conditions under which the evaporation takes place. Several other factors such as solute concentration, solute– surface interaction, and surface tension gradients (Marangoni effect) also modify the ring pattern. 39.3.1.1.1 Marangoni Effect During the process of evaporation, the solvent movement is determined by the forces of surface tension acting upon them. The phenomenon of liquid flowing along a gas–liquid or a liquid– liquid interface from areas having low surface tension to areas having higher surface tension is called Marangoni effect [27]. Marangoni convection is the flow caused by surface tension gradients originating from concentration gradients. The famous Evaporation

Droplet deposition on substrate

Pinned contact line

Side view

Ring formation

Coffee-stain mechanism

FIGURE 39.7 permission.)

Schematic showing the coffee-stain mechanism. (Adapted from Lensen, M.C. et al., Chem. Eur. J., 10, 831, 2004. With

39-4


Ring formation

Film thinning

Droplet deposition on substrate

Solvent evaporation Solute flow

Pin-hole mechanism

FIGURE 39.8 Schematic showing the pin-hole mechanism. (Adapted from Lensen, M.C. et al., Chem. Eur. J., 10, 831, 2004. With permission.)

“tears of wine” is a result of this effect in which one can observe a ring of clear liquid, near the top of a glass of wine, from which droplets form and flow back into the wine. Th is effect becomes evident when an interface contains traces of surface-active substances. If an interface expands locally, these surface-active solutes are swept outward with the movement, creating a gradient in concentration of these surface-active substances. Th is concentration gradient implies a surface tension gradient, which acts opposite to the movement. The interfacial movement is therefore damped and this effect is labeled the Plateau-Marangoni–Gibbs effect. 39.3.1.2 Pin-Hole Mechanism In the pin-hole mechanism, formation of holes in the liquid films during evaporation is responsible for ring-formation [23]. Holes nucleate when the evaporating solution film on a wetted substrate reaches a critical thickness, where there is a balance between thinning of liquid and wetting of the surface. The holes then open up and expand outward (in order to retain the optimum film thickness while evaporation continues) and push the solute particles toward the growing inner perimeter. As the evaporation continues, the solution around the rim becomes more and more concentrated. Finally, the resulting ring of particles gets stuck when the friction between the particles and the substrate can no longer be overcome by the force acting radially outward to thicken the film. Further, the evaporation rate at the edge of the pin-hole

is higher than in the bulk of the solution film, an inward flow of solute occurs to compensate this loss that will enrich the concentration of the solute at the inner edge of the ring. This process will continue till complete evaporation of solvent occurs, leaving the dispersed materials in the form of rings (Figure 39.8). In the case of wetting solvents, the holes open up when the film thickness reaches the nanometer level, so that the aggregation of dispersed material into rings is controlled by friction forces involving the substrate. For the nonwetting case, hole nucleation starts when the film thickness reaches micrometer level leading to the threedimensional solution/precipitation induced aggregation.

39.3.2 Ring Formation Induced by Chemical Reaction The functional groups present over the nanostructures would also induce the formation of ring-like superstructures around the water or organic droplet due to hydrophilic and hydrophobic interactions [28,29]. For example, the carbon nanofibers in CCl4 derived from the carbonization of polymer followed by acid etching will fuse and roll around the water droplet present in carbon tetrachloride (Figure 39.9a). Sometimes, coiling can be facilitated by a chemical reaction. Carbon nanotubes functionalized at both the ends with acid groups curl to form the nanoring structures by an end-to-end chemical reaction using 1,3-dicyclohexylcarbodiimide (DCC) [30] (Figure 39.9b).

Water droplet CCl4 solvent

Mixed with CCl4

Evaporation

(a) Carbon fibers

O C OH OH

O HO C HO C O

(b)

Amorphous carbon nanorings

Carbon nanotube

N C N

DCC

O O C O C C O O

Carbon nanoring

FIGURE 39.9 (a) Schematic of amorphous carbon nanoring formation. (b) Schematic showing the chemical fusion of carbon nanotube into nanoring using DCC. (Adapted from Sai Krishna, K. and Eswaramoorthy, M., Chem. Phys. Lett., 431, 327, 2007; Sano, M. et al., Science, 293, 1299, 2001. With permission.)

39-5

Nanorings

(a) Toroidal cylindrical micelles

(b) Polarity-induced ring formation

FIGURE 39.10 (a) Schematic of toroidal cylindrical micelles formed from triblock copolymers. (b) Schematic of polarity-induced ring formation through self-coiling ZnO nanobelt. (Reprinted from Pochan, D.J. et al., Science, 306, 94, 2004; Kong, X.Y. et al., Science, 303, 1348, 2004. With permission.)

39.3.3 Ring Formation Induced by Electrostatic Interactions The electrostatic interaction between the charged surfaces of the nanostructures has been reported to give ring-shaped morphology in some cases. For example, the cylindrical micelles obtained with a negatively charged amphiphilic polymer undergo a shape change from a cylindrical to toroid shape on interaction with the positively charged divalent counterion [31] (Figure 39.10a). Rings of ZnO are known to form from polarity-induced self-coiling of ZnO nanobelts synthesized at high temperatures [32] (Figure 39.10b). Here again, electrostatic interaction of the charges over the nanobelt edges are responsible for the self-coiling process.

(a) PS monolayer

39.3.4 Ring Formation by Template-Based Approach

(c) Ar+ ion etching

(b) Sputter Co

39.3.4.1 Colloidal Crystal Template Colloidal crystal templating is a well-known method to produce ordered porous structures [33–35]. This method could also be tuned to obtain nanorings [21]. The process uses monodisperse polystyrene (PS) or silica spheres (of micron or submicron size) as templates. The deposition of nanostructure around the colloidal template can be carried out either through a physical or chemical method. In a physical method, a monolayer of PS nanospheres is first formed on a substrate (Figure 39.11a) by manipulating its surface chemistry. A thin film of a desired metal of required thickness is then sputter-deposited onto the substrate (Figure 39.11b). Finally, Ar+ ion-beam etching is used to remove all the deposited metal, except that under the nanospheres. This results in an array of metal nanorings as shown in Figure 39.11c. The PS nanospheres are then chemically removed using calcinations or a solvent extraction process. The size of the nanorings can be controlled by the size of the PS template, whereas the compositions and thickness of the nanorings can be controlled by the deposition process. In the case of a chemical method, the packed colloidal spheres are functionalized before the desired metal oxide or organic precursors are allowed to deposit around the contact points of two spheres. Subsequent washing and chemical or thermal treatment removes the template, leaving the ring-like nanostructures of the desired materials [36,37] (Figure 39.12).

(d) Sputter Au or Cu

(e)

FIGURE 39.11 Schematic of the fabricating process for arrays of nanorings shown from a side view (left column) and top view (right column). (a) A monolayer of polystyrene (PS) nanospheres is deposited onto the substrate. (b) A thin fi lm of Co is sputter-deposited over the surface of PS nanospheres. (c) An Ar + ion beam is used, in normal incidence, to etch away the sputtered Co fi lm and thus leaving only the nanorings protected under the PS nanospheres. (d) A capping layer of Au or Cu is deposited over the entire surface to prevent oxidation. (e) Schematic of the tapered cross section of a nanoring. (Adapted from Zhu, F.Q. et al., Adv. Mater., 16, 2155, 2004. With permission.)

In a different approach, the polymeric hollow array membranes were used as templates to make the nanorings [38]. The hollow scaffold can be obtained by polymerization of the organic components within the spaces of packed colloidal silica spheres,

39-6


Fill with precursor solution

Colloidal crystal

300 nm (b) Remove the solvent, solidify the precursor

Remove colloids Ring-shaped particles (a)

200 nm (c)

FIGURE 39.12 (a) Schematic outline of the formation procedure of rings. (b) SEM images of the positions of the rings. (c) Polystyrene rings obtained by the colloidal crystal template made from SiO2 particles 1100 nm in diameter after the removal of the SiO2 particles. (Reproduced from Yan, F. and Goedel, W.A., Angew. Chem. Int. Ed., 44, 2084, 2005. With permission.)

Remove the particles

1.00 μm

(b) Fill the pores with HAuCl4 solution

Calcination

500 nm

(a)

(c)

FIGURE 39.13 (a) Scheme of the preparation of gold rings on a mica sheet. (b) SEM images of a porous membrane prepared by using 330 nm silica particles as templates, transferred onto a mica sheet (partially removed by a scotch tape to allow a side view of the pores). (c) SEM images of gold rings on mica prepared using molds of porous membranes shown in (b). (Reprinted from Yan, F. and Goedel, W.A., Nano Lett., 4, 1193, 2004. With permission.)

39-7

Nanorings

–

–

–

– AAO

SPM

–

– –

–

SiO2

–

Ni (a)

–

(b)

(c)

(a)

39.3.4.2 Porous Membrane Template The extensive use of porous alumina membranes as templates to prepare an ordered array of nanotubes and nanorods is well known. On the contrary, the use of porous alumina membrane as the template for nanoring formation is very limited. Nevertheless, several metal nanorings like Au, Ni, Fe, and Si were prepared using this method. The process involves Ar+ sputtering of the metal substrates through the pores of the anodic alumina membrane (AAO), which is used to mask the substrate [39,40]. Figure 39.14 shows the schematic diagram of the process. A porous alumina membrane is placed over a nickel substrate that is covered with a 20 nm thick SiO2 layer. Ion etching of the nickel substrate through the holes of porous alumina membrane leaves the redeposition of Ni around the membrane wall as a ring. Subsequent removal of alumina membrane by chemical etching leaves the nickel nanorings on the SiO2 layer. The purpose of the SiO2 layer is to avoid the substrate interference while measuring the properties of resulting nanorings. 39.3.4.3 DNA Template The ability of DNA to condense into well-defi ned toroids provides a unique opportunity to use them as templates to create toroidal nanostructures (nanorings) of controlled shape and dimensions. For example, the negatively charged condensed DNA rings will be an ideal surface for the silver ions to interact, which, on reduction, give silver nanorings [41] (Figure 39.15).

39.3.5 Ring Formation by Lithographic Technique 39.3.5.1 Soft Lithography Soft-lithographic technique has been used to synthesize many novel nanostructures that are otherwise difficult to form. Single crystalline silicon nanorings of uniform diameter were

–

(b) Ag+

FIGURE 39.14 Schematic showing the (a) AAO template on 20 nm of SiO2 on the desired ring material, (b) sputter redeposited material around the pore walls after sputter etching, and (c) the rings after AAO mask is removed (with a plan-view added to guide the eye). (Reprinted from Hobbs, K.L. et al., Nano Lett., 4, 167, 2004. With permission.)

followed by the removal of silica. These polymeric porous scaffolds are partially fi lled with the metal or metal oxide precursors (e.g., HAuCl4 for Au rings and Ti(OEt)4 for TiO2 rings). Subsequent reduction/calcination removes the organic template and retains the metal or metal oxide rings intact (Figure 39.13).

–

Ag+

Ag NaBH4

Ag+

Ag+

Ag+ Ag+

Ag+

Ag+ Ag+

(c)

(d)

FIGURE 39.15 Schematic representation of the silver nanoring synthesis. (a) Unfolded DNA chain. (b) DNA toroidal condensate formed after the addition of a condensing agent (in this case, spermine, noted SPM4+, a tetravalent polycation). (c) After the addition of AgNO3, silver ions bound to the surface of the DNA toroid. (d) DNA-templated silver nanorings formed after Ag+ reduced by NaBH4. (Reproduced from Zinchenko, A.A. et al., Adv. Mater., 17, 2820, 2005. With permission.)

synthesized by Xia and coworkers [42] using this process (Figure 39.16a). The first step in this process involves the creation of nanorings in a thin fi lm of photoresist (placed on a Si/SiO2 substrate) by exposing it to a UV light source through a polydimethylsiloxane (PDMS) mask, patterned with ring-shaped features. These patterned features in the photoresist fi lm were then subsequently transferred on to the underlying silicon layer (in the Si/SiO2 substrate) by reactive ion etching (RIE) process. Further oxidation and wet-etching with hydrofluoric acid removes the underlying silica layer, leaving behind the individual silicon nanorings (Figure 39.16b). 39.3.5.2 Interference Lithography Nanoring arrays of Au and Ni were fabricated based on interference lithography. It involves first the creation of Si3N4 nanohole arrays on silicon wafers by laser interference lithography (LIL), followed by selective electrochemical deposition on the step edge of periodic Si3N4 patterns. Large-scale fabrication of well-ordered metallic nanoring arrays is possible by this method [43].

39.4 Types of Nanorings Depending on the composition, nanorings can be divided into different types namely 1. Carbon nanorings a. Molecular carbon nanorings b. Graphitic carbon nanorings c. Amorphous carbon nanorings

39-8

Handbook of Nanophysics: Nanotubes and Nanowires Photoresist line Photoresist Si Exposure to UV light and developing

SiO2

RIE and removal of photoresist Silicon line

Si nanorings Lift-off in HF

Si

Si

1. Oxidation 2. Cooling down 3. Oxidation

SiO2

(a)

–1 μm

Si

Si SiO2

–300 nm

(b)

FIGURE 39.16 (a) Schematic of the soft lithography process. (b) Silicon nanorings formed through the lithographic route. (Reproduced from Yin, Y. et al., Adv. Mater., 12, 1426, 2000. With permission.)

2. Metal nanorings 3. Metal oxide/sulfide nanorings 4. Organic nanorings a. Porphyrin nanorings b. DNA nanorings c. Protein nanorings d. Amphiphilic molecular nanorings n

39.4.1 Carbon Nanorings 39.4.1.1 Molecular Carbon Nanorings Molecular nanorings are rings that are formed from the molecules as the building blocks. For example, 6-cycloparaphenilacetylene molecule is a ring-shaped molecule that is formed from six repeating units of phenilacetylene in a cyclic fashion [44–46]. The ring size and the diameter can be varied by varying the number of phenilacetylene groups in the molecule. Figure 39.17 depicts a typical structure of a 6-cycloparaphenilacetylene molecule, where n denotes the number of repeating units of phenilacetylene in the molecule. By placing the n value with 1, 2, 3, 4,

FIGURE 39.17 Molecular structure of a 6-cycloparaphenilacetylene molecule, where n = 1, 2, 3, 4. (Reproduced from Kawase, T. et al., Angew. Chem. Int. Ed., 42, 1621, 2003. With permission.)

etc., different sizes of molecular nanorings can be obtained with specific ring diameters. These rings are similar to the crown ether type of molecules, where the inside cavity can be utilized for inclusion of guest molecules [45]. Like crown ethers, these molecules also hold the guest molecules within the ring due to the intermolecular interactions of host–guest chemistry, and hence molecules such as

39-9

Nanorings

1.53 nm (a)

(b)

FIGURE 39.18 Structure of the inclusion complex of (a) fullerene C70 molecule and (b) hexa methyl benzene in cycloparaphenilacetylene molecule. (Reproduced from Kawase, T. et al., Angew. Chem. Int. Ed., 42, 5597, 2003; Cuesta, I.G. et al., ChemPhysChem., 7, 2503, 2006. With permission.)

FIGURE 39.19 Naphthalene units substituted 6- CPPA molecule. (Reproduced from Kawase, T. et al., Angew. Chem. Int. Ed., 42, 5597, 2003. With permission.)

fullerenes (Figure 39.18a) can be held inside a single molecule of 6-cycloparaphenilacetylene. Hexamethyl benzene (HMB) can form inclusion complexes with these molecular nanorings (Figure 39.18b) [46]. Different variants of these nanorings were also formed by replacing the phenilacetylene molecules in the ring with naphthalene units (Figure 39.19) [45]. These molecular nanorings fi nd their application in the supramolecular chemistry wherein similar molecules like crown ethers, cryptands, cyclodextrin-type molecules play a major role. 39.4.1.2 Graphitic Carbon Nanorings The carbon allotropes, graphite, diamond, fullerene, and nanotube in the form of sheets, spheres, rods, and tubes have been well known for a long time. However, the graphitic carbon in the form of nanoring was not known until Smalley and coworkers discovered it during the laser-assisted synthesis of single-walled carbon nanotubes (SWNTs) in the year 1997 [47]. The rings formed in the process have a perfect tori structure with no beginning or no end (Figure 39.20). To explain the formation of these toroidal carbon nanostructures, they proposed a Kekulean image of a growing nanotube eating its own tail. As the two ends of the bending nanotube come close together, the van der Waal’s interaction makes them align and the bending strain makes them slide over one another. In that high-temperature process, the closed hemifullerene end of the nanotube often gets stuck to the metal nanoparticles (which is used as a catalyst to grow the nanotube) of the growing end of the nanotube, resulting in end-to-end fusion. Avouris and coworkers have reported the formation of SWNT rings by the curling of nanotube ropes (no end-to-end fusion) [48]. Rings were produced by ultrasonicating a solution containing SWNTs in a mixture of concentrated sulfuric

(a)

(b)

FIGURE 39.20 (a) Scanning force micrograph (SFM) of nanoring formed from SWNTs by laser-assisted growth of SWNT (reported by Smalley and coworkers). Scale bar: 100 nm. (b) TEM of a ring showing the higher magnification of the rim. The ring has an apparent height of 1.0–1.2 nm (actual height is closer to 1.5 nm) and width of 4–8 nm. Scale bars: 15 and 5 nm (inset). (Reprinted from Liu, J. et al., Nature, 385, 780, 1997. With permission.)

acid and hydrogen peroxide (Figure 39.21). This treatment not only disperses the nanotubes but also shortens the nanotube ropes. A high yield requires shortening the raw nanotubes to a length of 2–4 mm. The ring formation is a kinetic process, with the activation energy for the curling provided by the ultrasonic irradiation. Further, the strain energy caused by the increased curvature while coiling is compensated by the energy gained from the van der Waal’s interactions. It was suggested that the hydrophobic nanotubes that act as nucleation centers for bubble formation bent mechanically at the bubble–liquid interface as a result of the bubbles collapsing during cavitation. Carbon nanorings are also known to form from SWNTs by chemical modification and end-to-end functionalization [30] (Figure 39.22). Functionalization of SWNT were carried out by ultrasonicating them in a concentrated H2SO4/HNO3 solution, followed by etching with H2SO4/H2O2 solution to obtain oxygencontaining groups, such as carboxylic acid groups at both ends. The end-to-end ring closure of the nanotubes was then achieved by using 1,3-dicyclohexylcarbodiimide (DCC) as the coupling agent (Figure 39.9b). 39.4.1.3 Amorphous Carbon Nanorings Formation of amorphous carbon nanorings, different from the tubular and graphitic SWNT nanorings, has been reported by Eswaramoorthy and coworkers [28] through the fusion

39-10


3 μm

(a)

FIGURE 39.22 AFM images of carbon nanotube rings cast on mica. A reaction mixture after washing and fi ltering through 0.8 mm Teflon fi lter. Rings appear to stick to each other after these processes. Scale bar, 2 mm. (Reprinted from Sano, M. et al., Science, 293, 1299, 2001. With permission.)

26 nm

Number of rings

50

30

nanorings are amorphous in nature and of diameter ranging from 0.5 to 20 μm with a rim thickness of about 50 nm. Figure 39.23 shows a distribution of rings with different sizes (with maximum of 2 μm diameter rings).

20 10 0

31 nm (b)

40

200 (c)

300 400 500 Radius (nm)

FIGURE 39.21 Nanoring formed by self-coiling of SWNTs prepared using laser ablation method followed by acid treatment and ultrasonication (Avouris and coworkers). (a) Scanning electron micrograph of a SWNT sample dispersed on a hydrogen-passivated silicon substrate, with rings clearly visible. (b) TEM image of a section of a ring wall. (c) Histogram showing the distribution of ring radii. (Reprinted from Martel, R. et al., Nature, 398, 299, 1999. With permission.)

and evaporation induced self-assembly of carbon nanofibers around water droplets dispersed in a carbon tetrachloride solution. Carbon nanofibers were generated within the pores of mesoporous silica (by carbonizing the polymer template used to synthesize mesoporous silica, SBA-15) prepared within the AAO membranes. These carbon fibers get functionalized during the acid etching treatment to remove the silica and alumina. The dried carbon fibers were then extracted with CCl4. Drop casting of this solution on a copper grid followed by drying leaves a lot of amorphous carbon nanorings on the grid. The surface of the copper grid initially has a lot of adsorbed tiny water droplets. These droplets are replaced by the high density CCl4 solvent containing the carbon nanofibers dispersed. During this process, the hydrophilic nature of the nanofibers self-assemble and fuse to form a ring around the upward moving water droplet (see the scheme, Figure 39.9a). These

39.4.2 Metal Nanorings Metal nanorings like Au, Ag, Ni, and Co were prepared following one of the techniques or multiple techniques discussed in the earlier section. For example, Au and Co nanorings were made through a template-based process using polystyrene spheres (PS) as the template [14,21]. The metals were deposited over the spheres by a sputtering technique followed by Ar+ ion beam etching. Subsequent removal of the template resulted in the formation of the nanorings of these metals (Figure 39.24). Co nanorings were also prepared using evaporation induced, dipole-directed self-assembly [49]. The charged surface of DNA nanorings was used as a template to prepare the Ag nanorings by adsorption of silver ions by electrostatic interaction, followed by reduction [41]. Ni and Au nanorings were also reported by using laser interference lithography [43].

39.4.3 Metal Oxide/Sulfide Nanorings Nanorings of metal oxides/nitrides/sulfides are unique as they show properties that are different from regular nanostructures. Wang and colleagues [32] have shown the formation of freestanding single-crystalline nanorings of ZnO by the epitaxial self-coiling of polar nanobelts synthesized at high temperatures. The synthesis was done by a solid-vapor technique, starting with powders of zinc oxide, indium oxide, and lithium carbonate in a

39-11

Nanorings

(a)

500 nm

(b) 20 μm

(a)

500 nm

200 nm

0.5 μm (b)

(c)

FIGURE 39.23 Amorphous carbon nanorings reported by Eswaramoorthy and coworkers. (a) FESEM image showing the carbon nanorings of different sizes. (b) TEM of a nanoring with nonuniform rim thickness. (c) FESEM image of a single amorphous carbon nanoring. (Reprinted from Sai Krishna, K. and Eswaramoorthy, M., Chem. Phys. Lett. 433, 327, 2007. With permission.)

1 μm (a)

(b)

FIGURE 39.24 (a) SEM image of the Co nanorings made from PS spheres as template. (b) SEM image of gold nanorings synthesized using PS spheres as template. (Reproduced from Zhu, F.Q. et al., Adv. Mater., 16, 2155, 2004; Reprinted from Aizpurua, J. et al., Phys. Rev. Lett., 90, 057401, 2003. With permission.)

FIGURE 39.25 SEM images of nanorings formed through self-coiling of nanobelts: (a) Ag 2V4O11 and (b) K 2Ti6O13 nanorings. (Reprinted from Shen, G. and Chen, D., J. Am. Chem. Soc., 128, 11762, 2006; Xu, C.Y. et al., J. Phys. Chem. C, 112, 7547, 2008. With permission.)

horizontal tube furnace. Heating the materials to 1400°C in argon causes ZnO material to deposit as rings (on a silicon substrate), with the ring diameter about 1–4 μm and shell thickness around 10–30 nm by a spontaneous self-coiling growth process induced by the polar surfaces of ZnO nanobelts. Nanorings of SnO2 [50], Ag2V4O11 (Figure 39.25a) [51], and K2Ti6O13 (Figure 39.25b) [52] were also reported to form from the self-coiling of nanobelts. For example, Shen and coworkers [51] have reported the formation of Ag2V4O11 nanorings through self-coiling under hydrothermal reaction conditions between AgNO3 and V2O5 powders at 170°C. Ag2V4O11 nanorings thus obtained have perfect circular shape and flat surfaces. Typical nanorings have diameters of 3–5 μm and thin and wide shells with a thickness of 30–50 nm. Three dimensional macroporous scaffold obtained by sintering and selective dissolution of a colloidal crystal was used as a template to make crystalline titania nanorings [53] upon infiltration and calcination of titanium alkoxide precursor within the porous scaffold (Figure 39.26). The conventional infiltration step that introduces a polymer precursor prior to selective dissolution of colloidal crystal was not followed in this case. The resulting titania nanorings exhibit a robust, undisrupted rutile phase. In a different approach, Li and coworkers [54] have synthesized α-Fe2O3 nanorings by a rapid microwave-assisted hydrothermal (MAH) method without the use of any template or organic surfactant. Their synthesis involves hydrolysis of an iron precursor (FeCl3) in presence of NH4H2PO4 under

39-12


shaped α-Fe2O3 nanoparticles in a double anion-assisted hydrothermal process [55]. Nanorings made up of semiconducting cadmium sulfide nanoparticles were prepared by the self-assembly of organically protected cadmium sulfide nanoparticles in aqueous phase [56] (Figure 39.28). The evolved structure was based on morphological controls of primary cadmium sulfide nanocrystals of intrinsic hexagonal symmetry. Since many semiconducting materials have similar hexagonal (or cubic) crystal symmetries, it is possible to extend this self-assembling processes to other semiconductor nanocrystals as well.

TiO2

39.4.4 Organic Nanorings 39.4.4.1 Porphyrin Nanorings FIGURE 39.26 SEM images of 2D interconnected titania rings fabricated on the honeycomb template. Scale bar is 1 μm. (Reprinted from Yi, D.K. and Kim, D.Y., Nano Lett., 3, 207, 2003. With permission.)

MAH conditions. It is believed that MAH process facilitates the formation of single crystal nanorings through nucleation aggregation and coordination-assisted dissolution of α-Fe2O3 (by the ligand of phosphate ions) crystals. The superheating and nonthermal effects induced by the MAH process gradually generate a hole at the center of each primarily formed hematite nanodisk, resulting in the formation of nanorings (Figure 39.27). α-Fe2O3 nanorings were also prepared from the preferential dissolution along the long dimension of the elongated capsule-

Porphyrin molecular systems are well known to form ordered stacks due to π–π interactions between the porphyrin entities. Nolte and coworkers have done extensive studies over the formation of porphyrin nanorings and explored their unusual properties [23–25]. The nanorings of porphyrin derivatives formed through evaporation-induced self-assembly process is shown in Figure 39.29. The ring formation is influenced by the properties of both the solvent and the substrate and together they determine the dewetting process. The ease by which the solution flow carries the porphyrin aggregates to the inner edge of the growing hole is determined by the specific interaction between solvent and substrate. Fe3+

(Aggregation)

Primary nanocrystals

H2O

(Aggregation)

Microwave

(Dissolution)

(a)

(d)

[H2PO4]–

Nucleation clusters

(Dissolution)

(b)

(e)

(c)

(f )

FIGURE 39.27 Schematic illustration of a-Fe2O3 nanorings formation. (a), (b), and (c) show the disk, hole form, and ring form, respectively. (d), (e), and (f) show the TEM images of the respective shapes in (a), (b), and (c). The TEM images of (d) are after 10 s, (e) after 50 s, and (f) after 25 min of synthesis time. Scale bar is 20 nm. (Reproduced from Hu, X. et al., Adv. Mater., 19, 2324, 2007. With permission.)

39-13

Nanorings

200 nm

100 nm (a)

(b)

FIGURE 39.28 TEM images of (a) circular CdS ring and (b) hexagonal CdS ring. (Reprinted from Liu, B. and Zeng, H.C., J. Am. Chem. Soc., 127, 18262, 2006. With permission.) RO

N NM RO

N

OR

N N M N N

RO

N NM N N

N

OR

O

O RO

OR

RO OR

O

O N NM N N RO OR

RO

RO N NM N N

1a M = H2, R = C16H33 1b M = Zn, R = C16H33 2

N N M N N

O O

RO

OR

OR

OR

Self-assembly of porphyrin hexamers into columnar stacks

OR

3.5 nm

M = H2, R =

Top view: Proposed alignment of stacks within the ring

AFM: 15 nm

FIGURE 39.29 Chemical structure and schematic showing the stacking of individual porphyrin hexamer units to form a nanoring. (Reproduced from Lensen, M.C. et al., Chem. Eur. J., 10, 831, 2004. With permission.)

The fluorescence microscopy of these rings using polarized excitation light and polarized detection of the emission reveals that only the small rings (up to 5 mm diameter) showed fluorescence anisotropy. The polarized fluorescence microscopy images of the rings on hydrophilic carbon-coated glass are depicted in the figure below (Figure 39.30a). When the excitation and the detection polarization are both oriented vertically, the left and right parts of the rings display higher fluorescence intensity than

the upper and lower parts, respectively. Turning both the excitation and the detection polarization 90° (Figure 39.30b) results in higher fluorescence intensity of the upper and lower parts of the rings. The expression of polarization is directly related to the molecular ordering and these strong polarization effects indicate that the molecules within a ring are ordered. These rings can be designed and constructed with desired morphology and internal

39-14


(a)

Exc.

(b)

Det.

(c)

Exc.

Det.

(d)

5 μm

5 μm

FIGURE 39.30 Polarized fluorescence microscopy images of porphyrin rings (a) excitation and the detection polarization are both oriented vertically (b) excitation and the detection polarization are both oriented horizontally. (Reproduced from Lensen, M.C. et al., Chem. Eur. J., 10, 831, 2004. With permission.)

16 nm

(a)

R1 O

OR2 O

O

R1 O

R1 =

OR2 O O

O OOO O OO O O OO O O OO O

O (CH2)n CH3 R2 =

O (CH2)n O (CH2)n

CH3

50 nm

CH3

(b)

(c)

FIGURE 39.31 (a) Schematic representation of nanorings from amphiphilic molecular dumbbells. (b) Molecular structure of the dumbbell-shaped molecule. (c) Cryo-TEM image of the amphiphilic dumbbell-shaped molecules formed into toroidal structure. (Reprinted from Kim, J.K. et al., J. Am. Chem. Soc., 128, 14022, 2006. With permission.)

molecular order, and with different porphyrin derivatives, such as porphyrins containing catalytic transition metals. Also, with respect to the orientation of the molecules within these rings, they are expected to show interesting properties that may fi nd application in molecular electronics.

39.4.4.2 DNA Nanorings The exploitation of biological macromolecules, such as nucleic acids, for the fabrication of advanced materials is a promising area of research. DNA nanorings are probably the simplest rigid objects with a nanometer size [57]. Small DNA circles were fi rst

39-15

Nanorings

prepared by designing two 21-mer DNA precursor sequences, which, upon hybridization and ligation, resulted in a statistical distribution of DNA nanorings containing 105, 126, 147, and 168, base pairs (bp). Atomic force microscopy (AFM) analysis of 168bp nanorings confirmed their smooth circular structure without any ring deformation or supercoiling. These features predestine them as building blocks for the assembly of objects on the nanometer scale. However, a major drawback of DNA nanorings in the construction of higher ordered DNA architectures is their unbranched, continuously double stranded (ds) nature which prevents the guided aggregation of multiple rings. Furthermore, the statistical assembly of the short oligonucleotides that were applied in the known strategies for the synthesis of DNA nanorings prevents the controlled introduction of customized sequences into the circle that can serve as defined handles for the self-assembly of multiple rings.

interaction between negatively charged hydrophilic block of an amphiphilic triblock copolymer and a positively charged divalent organic counterion.

39.4.4.3 Protein Nanorings

Acknowledgment

Greater variety of structural and functional uses can be envisioned for protein-based nanorings, given the functional specificity of proteins. In the presence of dimeric methotrexate (bisMTX), wild-type Escherichia coli dihydrofolate reductase (DHFR) molecules tethered together by a flexible peptide linker (ecDHFR2) are capable of spontaneously forming highly stable cyclic structures with diameters ranging from 8 to 20 nm [58]. The nanoring size is dependent on the length and composition of the peptide linker, on the affinity and conformational state of the dimerizer, and on induced protein–protein interactions. Control over protein assembly by chemical induction provides an avenue to the future design of protein-based materials and nanostructures. 39.4.4.4 Amphiphilic Molecular Nanorings The exploitation of aromatic rod-like building blocks for the engineering of synthetic nanostructures is a promising area of research [59]. Incorporation of a rigid rod segment into amphiphilic molecular architectures leads to a number of well-defined nanostructures, including bundles, barrels, tubules, ribbons, and vesicles, in solution state [60]. Recently, researchers have demonstrated that dumbbell-shaped molecules consisting of an aromatic stem segment and hydrophilic dendritic branches, in aqueous solution, self-assemble into nanorings in which the rod segments stack on top of each other with mutual rotation (Figure 39.31). The primary driving force responsible for the helical arrangement of the aromatic rods was proposed to be the energy balance between repulsive interactions among the adjacent hydrophilic dendritic segments and attractive π–π stacking interactions. These results imply that incorporation of hydrophobic branches at one end of a molecular dumbbell further extends the supramolecular organization capabilities of stiff rod-like segments due to enhanced hydrophobic interactions. Triblock copolymer poly(acrylic acid-b-methyl acrylateb-styrene) (PAA99-PMA73-PS66) nanorings [31] were reported to form by self-attraction of cylindrical micelles due to the

39.5 Conclusions The search for existing and novel properties in nanomaterials has been an unending quest for researchers. The size- and shape-dependent properties associated with the materials at the nanoscale are fascinating. Nanorings are the new class of nanostructures that will play a prominent role in the future owing to their unusual properties. Their applications in VMRAM devices, nanocontainers for spectroscopy, and nanosized sensors have made nanorings an interesting area of research. Th is chapter provides an overview of the properties, applications, as well as various methods of fabrication of nanorings.

The authors thank Dr. N.S. Vidhyadhiraja for helpful discussions on Aharanov–Bohm effect in nanorings.

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40 Superconducting Nanowires and Nanorings 40.1 Introduction ...........................................................................................................................40-1 40.2 Background.............................................................................................................................40-1 40.3 Thermal and Quantum Fluctuations in Superconducting Nanowires ........................ 40-4

Andrei D. Zaikin Karlsruhe Institute of Technology and P.N. Lebedev Physics Institute

General Theory • Gaussian Fluctuations • Thermally Activated Phase Slips • Quantum Phase Slips

40.4 Persistent Currents in Superconducting Nanorings ......................................................40-15 Persistent Currents and Quantum Phase Slips • Parity Effect and Persistent Currents

40.5 Summary .............................................................................................................................. 40-22 References........................................................................................................................................ 40-23

40.1 Introduction The phenomenon of superconductivity was discovered (Kamerlingh Onnes 1911) as a sudden drop of resistance to an immeasurably small value. With the development of the topic it was realized that the superconducting phase transition is frequently not at all “sudden” and the measured dependence of the sample resistance, R(T), in the vicinity of the critical temperature, Tc, may have a finite width. One possible reason for this behavior—and frequently the dominating factor—is the sample inhomogeneity, i.e., the sample might simply consist of regions with different local critical temperatures. However, with improving fabrication technologies it became clear that even for highly homogeneous samples the superconducting phase transition may remain broadened. This effect is usually very small in bulk samples and becomes more pronounced in systems with reduced dimensions. A fundamental physical reason behind such smearing of the transition is superconducting fluctuations. The important role of fluctuations in reduced dimensions is well known. Above Tc, such fluctuations yield an enhanced conductivity of metallic systems (Larkin and Varlamov 2005). For instance, the so-called Aslamazov–Larkin fluctuation correction to conductivity, δσal ~ (T − Tc)−(2−D/2), becomes large in the vicinity of Tc, and this effect increases with decreasing dimensionality D. Below Tc, according to the general Mermin– Wagner–Hohenberg theorem (Mermin and Wagner 1966, Hohenberg 1967), fluctuations should destroy the long-range order in low-dimensional superconductors. Thus, it could naively be concluded that low-dimensional conductors cannot exhibit superconducting properties because of strong phase fluctuation effects.

This conclusion, however, turns out to be premature. For instance, 2D structures undergo Berezinskii–Kosterlitz–Thouless phase transition (Berezinskii 1971, Kosterlitz and Thouless 1973, Kosterlitz 1974) as a result of which the decay of correlations in space changes from exponential at high enough T to power law at low temperatures. This result implies that at low T, long-range phase coherence essentially survives in samples of a finite size and, hence, 2D films can well exhibit superconducting properties. Can superconductivity survive also in quasi-1D systems or do fluctuations suppress phase coherence, thus disrupting any supercurrent? The answer to this question would clearly be of both fundamental interest and practical importance. On one hand, investigations of this subject definitely help to encover novel physics and shed more light on the crucial role of superconducting fluctuations in 1D wires. On the other hand, rapidly progressing miniaturization of nanodevices opens new horizons for applications of superconducting nanocircuits and requires a better understanding of the fundamental limitations of the phenomenon of superconductivity in reduced dimensions. This chapter is devoted to a detailed discussion of the nontrivial interplay between superconductivity and the fluctuations in metallic nanowires and nanorings.

40.2 Background It was first pointed out by Little (1967) that quasi-1D wires made of a superconducting material can acquire a finite resistance below Tc of a bulk material due to the mechanism of thermally activated phase slips (TAPS). Within the Ginzburg–Landau theory one can describe a superconducting wire by means of a complex order parameter Ψ(x) = |Ψ(x)|eiφ(x). Thermal fluctuations 40-1

40-2


(a)

X

2π(n + 3) 2π(n + 2)

2π(n + 3) |Ψ|

2π(n + 2)

|Ψ|

|Ψ|

2π(n + 3)

2π(n + 2)

2π(n + 1)

2π(n + 1)

2π(n + 1)

2πn

2πn

2πn

(b)

X

(c)

X

FIGURE 40.1 Schematics of the phase slip process. Spatial variation of the amplitude of the order parameter |Ψ| (left axis, dashed line) and phase φ (right axis, solid line) at various moments of time: (a) before, (b) during, and (c) after the phase slippage. (From Arutyunov, K. Yu. et al., Phys. Rep., 464, 1, 2008.)

cause deviations of both the modulus and the phase of this order parameter from their equilibrium values. A nontrivial fluctuation corresponds to a temporal suppression of |Ψ(x)| down to zero at some point (e.g., x = 0) inside the wire, see Figure 40.1. As soon as the modulus of the order parameter |Ψ(0)| vanishes, the phase φ(0) becomes unrestricted and can jump by the value 2πn, where n is any integer number. After this process the modulus |Ψ(0)| gets restored, the phase becomes single valued again and the system returns to its initial state, accumulating the net phase shift 2πn. Provided such phase slip events are sufficiently rare, one can restrict n by n = ±1 and totally disregard fluctuations with |n| ≥ 2. According to the Josephson relation V = ћϕ /2e, each such phase slip event causes a nonzero voltage drop V across the wire. In the absence of any bias current, the net average numbers of “positive” (n = +1) and “negative” (n = −1) phase slips are equal, thus the net voltage drop remains zero. Applying the current I ∝ |Ψ|2 ∇φ one creates a nonzero phase gradient along the wire and makes “positive” phase slips more likely than “negative” ones. Hence, the net voltage drop V due to TAPS differs from zero, i.e., thermal fluctuations cause a nonzero resistance R = V/I of superconducting wires even below Tc. We would also like to emphasize that, in contrast to the so-called phase slip centers (Tidecks 1990, Kopnin 2001) produced by a large current above the critical one I > Ic, here we are dealing with fluctuation-induced phase slips, which can occur at arbitrarily small values of I. A quantitative theory of the TAPS phenomenon was first proposed by Langer and Ambegaokar (1967) and then extended by McCumber and Halperin (1970). The LAMH theory predicts that the TAPS creation rate and, hence, the resistance of a superconducting wire R below Tc is determined by the activation exponent: N Δ 2 (T ) ⎛ −δU ⎞ R(T ) ∝ exp ⎜ , δU ~ 0 0 sξ(T ), ⎟ ⎝ kT ⎠ 2

(40.1)

where δU(T) is the effective potential barrier for TAPS. Th is potential barrier is determined by the superconducting condensation energy per unit volume N0Δ0(T)2/2 (here and below N0 is the metallic density of states at the Fermi energy and Δ0(T) is the BCS order parameter) lost during a phase slip event in a part of the wire where the order parameter gets temporarily suppressed by thermal fluctuations. This volume is given by the wire cross

section s times the typical TAPS size which—according to the LAMH theory—is of the order of the superconducting coherence length ξ(T). At temperatures very close to Tc, Equation 40.1 yields appreciable resistivity, which was indeed detected experimentally (Lukens et al. 1970, Newbower et al. 1972). Close to Tc, the experimental results fully confirm the activation behavior of R(T) expected from Equation 40.1. However, as the temperature is lowered further below Tc, the number of TAPS inside the wire decreases exponentially and no measurable wire resistance is predicted by the LAMH theory except in the immediate vicinity of the critical temperature. Experiments by Lukens et al. and by Newbower et al. have been performed with small whiskers of typical diameters ~0.5 μm. Recent progress in the nanolithographic technique made it possible to fabricate samples with much smaller diameters down to— and even below—10 nm. In such systems, one can consider a possibility for phase slips to occur not only due to thermal, but also due to quantum fluctuations of the superconducting order parameter. The physical picture of quantum phase slippage is qualitatively similar to that of TAPS (see Figure 40.1), except the order parameter |Ψ(x)| gets virtually suppressed due to the process of quantum tunneling. Following the standard quantum mechanical arguments, one can expect that the probability of such a tunneling process should be controlled by the exponent ~exp(−U/ћω0), i.e., instead of temperature in the activation exponent (Equation 40.1) one should just substitute ћω0, where ω0 is an effective attempt frequency. This is because the order parameter field Ψ(x) now tunnels under the barrier U rather than overcomes it by thermal activation. Since such a tunneling process should obviously persist down to T = 0, one arrives at the fundamentally important conclusion that in nanowires superconductivity can be destroyed by quantum fluctuations at any temperature including T = 0. Accordingly, such nanowires should demonstrate a nonvanishing resistivity down to zero temperature. Assuming that ћω0 ~ Δ0(T), one would expect that at Δ0(T) < T < Tc the TAPS

dependence (Equation 40.1) applies, while at lower T < Δ0(T)

quantum phase slips (QPS) take over, eventually leading to the saturation of the temperature dependence R(T) to a nonzero value in the limit T > Δ0 the Drude kernel χD reduces to the standard form χD

σ , | ω | + Dq 2

where σ = 2e2N0D is the wire Drude conductivity in the normal state N0 is the electron density of state at the Fermi level D = vFl/3 is the diff usion coefficient l is the electron elastic mean free path In the opposite limit ω, q → 0 and at low T > Δ0(T), we find ⎡ T ⎛ 1 ω + Dq2 ⎞ ⎛ 1⎞ ⎤ χ Δ = 2N 0 ⎢ ln + Ψ ⎜ + ⎟ − Ψ⎜ ⎟ ⎥ , ⎜⎝ 2 ⎝ 2⎠ ⎥ 4πT ⎟⎠ ⎢ Tc ⎣ ⎦ where Ψ(x) is the digamma function. As the action S (40.13) is quadratic both in the voltage V and the vector potential A, these variables can be integrated out exactly. Performing this integration one arrives at the effective action that only depends on φ and δΔ. We obtain S=

{

}

2 2 s dωdq F (ω, q) ϕ + χ Δ δΔ . 2 2 (2π)

∫

(40.14)

40-6


Since typically metallic nanowires (with the diameter of the order of superconducting coherence length ξ = D / Δ 0 or thinner) are quite strongly disordered, for generic values of the electron elastic mean free path l one has 1/L >> πσΔ0 s/c 2 and C > C/s + χDq2, which is obeyed in the limit of low frequencies and wave vectors ω/Δ0 1 can be translated into that for the wire cross section: s rc ~ λ F

ξ , l

(40.24)

where ξ >> l. This equation defines the critical wire radius rc below which fluctuations completely wipe out superconductivity even at T = 0. For typical wire parameters we have rc ~ 1 ÷ 2 nm. We conclude that wires with thicknesses in this range (or smaller) cannot become superconducting at any temperature down to zero.

40-7

Superconducting Nanowires and Nanorings

Turning to higher temperatures, we observe that at T ~ Tc it is necessary to retain only the contribution from zero Matsubara frequency in Equation 40.21. The terms originating from all nonzero frequencies are small in the parameter Δ0(T)/Tc . (40.28) (1 − T / Tc )3/4 Only provided this condition is satisfied, Gaussian fluctuations of the order parameter remain insignificant at temperatures in the vicinity of Tc.

Let us remain at temperatures sufficiently close to Tc and restrict our attention to superconducting wires in which the condition (40.28) is well satisfied and, hence, the effect of Gaussian fluctuations on the order parameter Δ0(T) can be safely neglected. This condition requires the wire to be not too thin and/or the temperature to be not too close to Tc, i.e., (Tc − T)/Tc >> Gi1D. At the same time we assume that the temperature is still not far from Tc, i.e., Tc − T

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