COMPREHENSIVE SEMICONDUCTOR SCIENCE AND TECHNOLOGY
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COMPREHENSIVE SEMICONDUCTOR SCIENCE AND TECHNOLOGY Editors-in-Chief Pallab Bhattacharya Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, USA
Roberto Fornari Leibniz Institute for Crystal Growth, Berlin, Germany and Institute of Physics, Humboldt University, Berlin, Germany
Hiroshi Kamimura Research Institute for Science and Technology, Tokyo University of Science, Tokyo, Japan
Volume 1 PHYSICS AND FUNDAMENTAL THEORY Volume Editor Hiroshi Kamimura Research Institute for Science and Technology, Tokyo University of Science, Tokyo, Japan
AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO
Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, UK 30 Corporate Drive, Suite 400, Burlington MA 01803, USA Copyright ª 2011 Elsevier B.V. All rights reserved The following article is a US Government works in the public domain and is not subject to copyright: CHAPTER 4.09 ATOMIC RESOLUTION CHARACTERIZATION OF SEMICONDUCTOR MATERIALS BY ABERRATION-CORRECTED TRANSMISSION ELECTRON MICROSCOPY No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email:
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Contents of Volume 1 Contributors to Volume 1
vii
Preface
ix
Editors in Chief Contents of All Volumes
xiii xv
Physics and Fundamental Theory 1.01
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories J. R. Chelikowsky, University of Texas, Austin, TX, USA
1.02
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors: Bulk Crystals to Nanostructures J. Deslippe and S. G. Louie, University of California at Berkeley and Lawrence Berkeley National Laboratory, Berkeley, CA, USA
1
42
1.03
Impurity Bands in Group-IV Semiconductors M. Eto, Keio University, Yokohama, Japan H. Kamimura, Tokyo University of Science, Tokyo, Japan
77
1.04
Atomic Structures and Electronic Properties of Semiconductor Interfaces T. Nakayama, Chiba University, Chiba, Japan Y. Kangawa, Kyushu University, Fukuoka, Japan K. Shiraishi, University of Tsukuba, Ibaraki, Japan
113
1.05
Integer Quantum Hall Effect H. Aoki, University of Tokyo, Tokyo, Japan
175
1.06
Fractional Quantum Hall Effect and Composite Fermions J. K. Jain, The Pennsylvania State University, University Park, PA, USA
210
1.07
Spin Hall Effect S. Murakami, Tokyo Institute of Technology, Tokyo, Japan N. Nagaosa, University of Tokyo, Tokyo, Japan
222
1.08
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures W. R. Clarke and M. Y. Simmons, University of New South Wales, Sydney, NSW, Australia C.-T. Liang, National Taiwan University, Taipei, China
279
1.09
Thermal Conductivity and Thermoelectric Power of Semiconductors I. Terasaki, Nagoya University, Nagoya, Japan
326
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vi Contents of Volume 1
1.10
Electronic States and Transport Properties of Carbon Crystalline: Graphene, Nanotube, and Graphite Y. Iye, University of Tokyo, Kashiwa, Chiba, Japan
359
1.11
Angle-Resolved Photoemission Spectroscopy of Graphene, Graphite, and Related Compounds T. Sato and T. Takahashi, Tohoku University, Sendai, Japan
383
1.12
Theory of Superconductivity in Graphite Intercalation Compounds Y. Takada, University of Tokyo, Kashiwa, Chiba, Japan
410
Contributors to Volume 1 H. Aoki University of Tokyo, Tokyo, Japan
Chapter 1.05 p. 175
J. R. Chelikowsky University of Texas, Austin, TX, USA
Chapter 1.01 p. 1
W. R. Clarke University of New South Wales, Sydney, NSW, Australia
Chapter 1.08 p. 279
J. Deslippe University of California at Berkeley and Lawrence Berkeley National Laboratory, Berkeley, CA, USA
Chapter 1.02 p. 42
M. Eto Keio University, Yokohama, Japan
Chapter 1.03 p. 77
Y. Iye University of Tokyo, Kashiwa, Chiba, Japan
Chapter 1.10 p. 359
J. K. Jain The Pennsylvania State University, University Park, PA, USA
Chapter 1.06 p. 210
H. Kamimura Tokyo University of Science, Tokyo, Japan
Chapter 1.03 p. 77
Y. Kangawa Kyushu University, Fukuoka, Japan
Chapter 1.04 p. 113
C.-T. Liang National Taiwan University, Taipei, China
Chapter 1.08 p. 279
S. G. Louie University of California at Berkeley and Lawrence Berkeley National Laboratory, Berkeley, CA, USA
Chapter 1.02 p. 42
S. Murakami Tokyo Institute of Technology, Tokyo, Japan
Chapter 1.07 p. 222
N. Nagaosa University of Tokyo, Tokyo, Japan
Chapter 1.07 p. 222
T. Nakayama Chiba University, Chiba, Japan
Chapter 1.04 p. 113
T. Sato Tohoku University, Sendai, Japan
Chapter 1.11 p. 383
K. Shiraishi University of Tsukuba, Ibaraki, Japan
Chapter 1.04 p. 113
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viii Contributors to Volume 1
M. Y. Simmons University of New South Wales, Sydney, NSW, Australia
Chapter 1.08 p. 279
Y. Takada University of Tokyo, Kashiwa, Chiba, Japan
Chapter 1.12 p. 410
T. Takahashi Tohoku University, Sendai, Japan
Chapter 1.11 p. 383
I. Terasaki Nagoya University, Nagoya, Japan
Chapter 1.09 p. 326
Preface Semiconductors are at the heart of modern living. Almost everything we do, be it work, travel, communication, or entertainment, all depend on some feature of semiconductor technology. Though this domination has its roots in developments dating back 60 years, the progress has not been diminished by achievement, and the demands for improvements in performance continue to push manufacturing limits. As we enhance the performance of devices, we also extend the range of applications, exploiting what we have learned about older semiconductors to grow and process newer ones. Whereas once we were content to explore silicon chips for advanced electronics, and then gallium arsenide and other compound semiconductors for optoelectronics, we are now addressing critical issues in growth and processing of group III nitrides and silicon carbide for highpower, high-frequency, and high-temperature electronics and optoelectronics, with the objective of replacing fluorescent tubes and tungsten filaments in our homes and offices. While the numerous applications of semiconductors are evident to most, less is known about the serendipitous evolution of basic semiconductor physics, which often paved the way to new device concepts. For example, in 2000 the Nobel prize for physics was awarded to Zhores Alferov and Herbert Kroemer for the development of the technology used today in satellite communication and cellular phones, and to Jack Kilby for the invention and development of the integrated circuit, the forerunner of the microchip and the pocket calculator. Further in 2009 Willard Boyle and George Smith shared the Nobel prize in physics for the invention of an imaging semiconductor circuit – ‘the CCD sensor’ – with Charles Kao for his groundbreaking research on the transmission of light in fibers for optical communication. From the standpoint of chemistry and materials science, the prediction of semiconductor superlattices by Leo Esaki and Raphael Tsu in 1969 opened the tailor-made materials age, together with the development of the molecular beam epitaxy (MBE) and metal organic vapor phase epitaxy (MOVPE) techniques. These techniques paved the way for the realization of tailor-made semiconductor materials, in particular low-dimensional (two-, one-, and zero-dimensional) heterostructures which immediately became front runners in materials science. This frontier spirit of semiconductor research has led to the recent splendid realization of a one-atom thick layer of graphite, the so-called graphene, for which Andre Geim and Konstantin Novoselov received the 2010 Nobel prize in physics. The discoveries of integer and fractional quantum Hall effects which were awarded the physics Nobel prize in 1985 (Klaus von Klitzing) and 1998 (Robert Laughlin, Horst Sto¨rmer and Daniel Tsui), respectively, are also undoubtedly among the most fascinating phenomena, not only in semiconductor physics, but also in the entire subject of condensed matter physics. The integer quantum Hall effect was originally discovered in SiMOSFET (MOSFET, metal–oxide–semiconductor field-effect transistor), while the fractional quantum Hall effect was observed in very pure and atomically abrupt GaAs–AlGaAs heterostructures. As regards the theory of semiconductors from bulk to nanostructures, the advances in theoretical methodology and computer technology have been tremendous in the last few decades. In particular, predictions of new stable materials and device-worthy materials have contributed to remarkable progress in tailor-made materials and devices. The uniquely strong interplay between theory, experiment, chemistry, materials science, and technology has greatly contributed to the success of semiconductor-related fields. The revolution in semiconductor science and technology did not happen by accident. It is based on a thorough understanding of the diverse science and technology of these fascinating materials, coupled with ix
x Preface
astonishing facilities for controlling the crystallographic structure and purity. Progress has called for research by scientists and engineers from many disciplines, and the results of their efforts are documented in thousands of archival articles, patents, and licensed technologies. Those new to the subject blanch at the thought of assimilating even a small fraction of this immense knowledge base, and turn to the many specialist books that attempt to summarize specific aspects of the field. To date, there is no comprehensive work that covers the complete spectrum from fundamental semiconductor physics to design and manufacture devices which the physics predicted to be possible, and which are now actually fabricated by materials scientists. In this sense, a comprehensive work is different from an encyclopedia and a handbook. Comprehensive Semiconductor Science and Technology consists of much longer chapters, which contain extensive cross-references to other relevant information within the work and references to the vast available literature beyond the work. Each chapter may be partly tutorial so that graduate students can greatly benefit from its reading. This comprehensive comprises six volumes, and has been broadly divided into three main sections: Section 1: Physics/Fundamental Theory – Edited by Hiroshi Kamimura Section 2: Chemistry/Preparation – Edited by Roberto Fornari Section 3: Devices/Application – Edited by Pallab Bhattacharya Each of these complementary sections will provide a complete description of one aspect of the whole, and will fuse together to give a comprehensive picture of the semiconductor world. The first section, which covers Volumes 1 and 2, is concerned with the fundamental physics of semiconductors, showing how the electronic properties and lattice dynamics change drastically when systems vary from bulk to a low-dimensional structure and further to an interface and to a nanometer size such as quantum dots. The section describes all the important characteristics of transport and optical properties. Further, this section includes many new topics such as spin effects, carbon crystalline systems such as graphene and nanotubes, ultrafast coherent optical phenomena, quantum information processing, etc., since the field of semiconductors continues to expand. Throughout Volumes 1 and 2, there is an emphasis on the full understanding of the underlying physics. Section 2, which covers Volumes 3 and 4, generally deals with technology of semiconductors. This includes a description of the main methods employed for the preparation of bulk single crystals and thin layers. It is shown that large single crystals may be conveniently grown from their melt, from solutions, or from the vapor phase, and that the thermodynamical properties of the semiconductors to be grown actually decide about the most appropriate method. The reader will also find some specific examples in the chapters devoted to bulk growth of silicon, wide-bandgap compounds, II–VI and III–V semiconducting compounds. In addition, Section 2 also contains reviews on thin film technology, in particular MBE and MOVPE. In particular, how a basic technology can be suited to preparation of semiconducting layers with tailored properties is discussed. Examples of advanced low-dimensional heterostructures and nanostructures are provided along with a specific chapter on integration of dissimilar materials. Additional chapters highlight the development of technologies for deposition of high-quality ferroelectric and high-K materials to be applied as memories and gate isolators, respectively. In addition to growth technology, Volumes 3 and 4 also include contributions regarding processing and characterization of semiconductors. The reader will find valuable information about the formation of shallow junctions in semiconductors, the fabrication of ohmic and Schottky contacts as well as several chapters describing the sophisticated methods used nowadays for investigating the physical and structural characteristics of substrates, films, and quantum structures. Volumes 5 and 6, which correspond to Section 3, contain chapters on the physics, technology and application of devices, and circuits realized with diverse materials and heterostructure systems. The materials include Si/SiGe, GaAs-, and InP-based heterostructures, antimonides, GaN-, ZnO-, and SiC-based materials, graphene, HgCdTe, and materials for molecular electronics. Also included are carbon nanotubes and nanostructured materials for flexible and stretchable electronics. The chapters describe in detail the properties of high-speed, high-frequency, and high-power bipolar and field-effect transistors made with a variety of heterostructure systems, negative differential resistance devices, single electron transistors, and microelectromechanical system (MEMS)-based sensors. Several chapters describe the principles and properties of a wide variety of optoelectronic devices including photodetectors, solar cells, light emitting diodes, and lasers. Passive
Preface
xi
photonic devices such as waveguides and filters are also included. The operating wavelength of these devices ranges from very short (ultraviolet) to very long (terahertz). The active region of some of these devices includes low-dimensional quantum-confined heterostructures such as quantum wells and quantum dots. Concepts such as the manipulation of slow and fast light and disordering of quantum structures for optoelectronic device integration are described in detail in a couple of chapters. The use of photonic crystals and microcavities in optical devices is elucidated. Electronics with molecules is included with a good description of the underlying physics. Finally, the physics, fabrication, and characteristics of spin-based electronic and optoelectronic devices, more commonly known as semiconductor spintronic devices, are described in a few chapters. Each chapter and topic is uniquely distinct, complete with the appropriate background material and references. Pallab Bhattacharya, Roberto Fornari, and Hiroshi Kamimura
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Editors in Chief Pallab Bhattacharya is the Charles M. Vest Distinguished University Professor of Electrical Engineering and Computer Science and the James R. Mellor Professor of Engineering in the Department of Electrical Engineering and Computer Science at the University of Michigan, Ann Arbor. He received the M. Eng. and Ph.D. degrees from the University of Sheffield, UK, in 1976 and 1978, respectively. Professor Bhattacharya was an Editor of the IEEE Transactions on Electron Devices and is Editor-in-Chief of Journal of Physics D. He has edited Properties of Lattice-Matched and Strained InGaAs (UK: INSPEC, 1993) and Properties of III-V Quantum Wells and Superlattices (UK: INSPEC, 1996). He has also authored the textbook Semiconductor Optoelectronic Devices (Prentice Hall, 2nd edition). His teaching and research interests are in the areas of compound semiconductors, low-dimensional quantum confined systems, nanophotonics and optoelectronic integrated circuits. He is currently working on highspeed quantum dot lasers, quantum dot infrared photodetectors, photonic crystal quantum dot devices, and spin-based heterostructure devices. From 1978 to 1983, he was on the faculty of Oregon State University, Corvallis, and since 1984 he has been with the University of Michigan. He was an Invited Professor at the Ecole Polytechnic Federale de Lausanne, Switzerland, from 1981 to 1982. Professor Bhattacharya is a member of the National Academy of Engineering. He has received the John Simon Guggenheim Fellowship, the IEEE (EDS) Paul Rappaport Award, the Heinrich Welker Prize, the IEEE (LEOS) Engineering Achievement Award, the Optical Society of America (OSA) Nick Holonyak Award, the SPIE Technical Achievement Award, the Quantum Devices Award of the International Symposium on Compound Semiconductors, and the IEEE (Nanotechnology Council) Nanotechnology Pioneer Award. He has also received the S.S. Attwood Award, the Kennedy Family Research Excellence Award, and the Distinguished Faculty Achievement Award from the University of Michigan. He is a Fellow of the IEEE, the American Physical Society, the Institute of Physics (UK), and the Optical Society of America.
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Editors in Chief
Roberto Fornari studied Solid State Physics at the University of Parma, Italy. He is presently director of the Leibniz Institute for Crystal Growth (IKZ) in Berlin and holds the Chair of Crystal Growth at the Physics Dept. of the Humboldt University Berlin (joint appointment). Before moving to Germany in 2003, he worked over twenty years as a research scientist at the Institute for Electronic and Magnetic Materials (MASPEC, later IMEM) of the Italian CNR where he led different research projects on growth and thermal processing of bulk III-V semiconductors, HVPE and MOVPE of Nitrides, characterization of semiconductors by electrical and optical techniques. He has authored/coauthored about 180 scientific papers, eight patents and different book chapters. He has edited books and proceedings on crystal growth and semiconductors physics and was subject editor of the Encyclopedia of Materials published by Pergamon Press in 2001. He is presently a member of the editorial board of the Journal of Crystal Growth, the Crystal Research and Technology, and the Journal of Optoelectronics and Advanced Materials. He has been Chairman of the IUCr Commission for Crystal Growth and Characterization of Materials from 1999 to 2005 and then member till 2008. From 2001 to 2007 he served on the Executive Committee of the International Organization for Crystal Growth and is currently President of this organization.
Hiroshi Kamimura is currently a Senior Advisor to the Tokyo University of Science (TUS), and a guest professor of the Research Institute for Science and Technology at the TUS. He was awarded a Doctor of Science in Physics from the University of Tokyo in 1959. He worked at Bell-Telephone laboratories at Murray Hill, USA as a Member of Technical Staff from 1961 to 1964. In 1965 he became a lecturer, then an associate professor and a professor at the Department of Physics, Faculty of Science in the University of Tokyo. Between 1974 and 1975 he worked with Sir Nevill Mott as a guest scholar at Cavendish Laboratory in Cambridge, UK. In 1991 he retired from the University of Tokyo, and became a professor at the Department of Applied Physics, Faculty of Science at the TUS. His interests are in the theory of condensed matter physics and of materials science, in particular semiconductor physics, high temperature superconductivity and superionic conduction. He was President of the Physical Society of Japan between 1984 and 1985, and Chairman of IUPAP Semiconductor Commission between 1985 and 1990. He is an honorary fellow of the Institute of Physics, UK, a life-fellow of the American Physical Society, an emeritus professor of the University of the Tokyo and an emeritus professor of the Tokyo University of Science.
Contents of All Volumes Volume 1 Physics and Fundamental Theory Edited by Professor Hiroshi Kamimura, Research Institute for Science and Technology, Tokyo University of Science, Tokyo, Japan 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors: Bulk Crystals to Nanostructures Impurity Bands in Group-IV Semiconductors Atomic Structures and Electronic Properties of Semiconductor Interfaces Integer Quantum Hall Effect Fractional Quantum Hall Effect and Composite Fermions Spin Hall Effect Ballistic Transport in 1D GaAs/AlGaAs Heterostructures Thermal Conductivity and Thermoelectric Power of Semiconductors Electronic States and Transport Properties of Carbon Crystalline: Graphene, Nanotube, and Graphite Angle-Resolved Photoemission Spectroscopy of Graphene, Graphite, and Related Compounds Theory of Superconductivity in Graphite Intercalation Compounds
Volume 2 Physics and Fundamental Theory Edited by Professor Hiroshi Kamimura, Research Institute for Science and Technology, Tokyo University of Science, Tokyo, Japan 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10
Electronic States and Transport in Quantum-Dots Control Over Single Electron Spins in Quantum-Dots Contact Hyperfine Interactions in Semiconductor Heterostructures Semimagnetic Semiconductors Optical Properties of Semiconductors Light Emission from Silicon Nanoparticles and Related Materials High-Density Excitons in Semiconductors Magneto-Spectroscopy of Semiconductors Bloch Oscillations and Ultrafast Coherent Optical Phenomena Optically Controlled Semiconductor Spin Qubits and Indistinguishable Single Photons for Quantum Information Processing
xv
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Contents of All Volumes
Volume 3 Materials, Preparation, and Properties Edited by Dr Roberto Fornari, Leibniz Institute for Crystal Growth, Berlin, Germany and Institute of Physics, Humboldt University, Berlin, Germany 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10 3.11 3.12 3.13
Bulk Crystal Growth of Semiconductors: An Overview Bulk Growth of Crystals of III–V Compound Semiconductors Fundamentals and Engineering of the Czochralski Growth of Semiconductor Silicon Crystals Growth of Cd0.9Zn0.1Te Bulk Crystals Sublimation Epitaxial Growth of Hexagonal and Cubic SiC Growth of Bulk GaN Crystals Growth of Bulk AlN Crystals Growth of Bulk ZnO Organometallic Vapor Phase Epitaxial Growth of Group III Nitrides ZnO Epitaxial Growth Nanostructures of Metal Oxides Molecular Beam Epitaxy: An Overview Growth of Low-Dimensional Semiconductors Structures
Volume 4 Materials, Preparation, and Properties Edited by Dr Roberto Fornari, Leibniz Institute for Crystal Growth, Berlin, Germany and Institute of Physics, Humboldt University, Berlin, Germany 4.01 4.02 4.03 4.04 4.05 4.06 4.07 4.08 4.09 4.10 4.11 4.12 4.13
Integration of Dissimilar Materials Ion Implantation in Group III Nitrides Contacts to Wide-Band-Gap Semiconductors Formation of Ultra-Shallow Junctions New High-K Materials for CMOS Applications Ferroelectric Thin Layers Amorphous and Glassy Semiconducting Chalcogenides Scanning Tunneling Microscopy and Spectroscopy of Semiconductor Materials Atomic Resolution Characterization of Semiconductor Materials by Aberration-Corrected Transmission Electron Microscopy Assessment of Semiconductors by Scanning Electron Microscopy Techniques Characterization of Semiconductors by X-Ray Diffraction and Topography Electronic Energy Levels in Group-III Nitrides Organic Semiconductors
Volume 5 Devices and Applications Edited by Professor Pallab Bhattacharya, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, USA 5.01 5.02 5.03 5.04 5.05 5.06 5.07
SiGe/Si Heterojunction Bipolar Transistors and Circuits Silicon MOSFETs for ULSI: Scaling CMOS to Nanoscale GaAs- and InP-Based High-Electron-Mobility Transistors High-Speed InP-Based Heterojunction Bipolar Transistors Negative Differential Resistance Devices and Circuits GaN-Based Transistors for High-Frequency Applications GaN- and SiC-Based Power Devices
Contents of All Volumes
5.08 5.09 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17
Silicon Single Electron Transistors Operating at Room Temperature and Their Applications Electronics with Molecules Electronic and Optoelectronic Properties and Applications of Carbon Nanotubes Micro- and Nanostructured Semiconductor Materials for Flexible and Stretchable Electronics MEMS-Based Sensors III–V Compound Avalanche Photodiodes Disordering of Quantum Structures for Optoelectronic Device Integration Quantum-Well Lasers and Their Applications Quantum Cascade Lasers Slow and Fast Light in Quantum-Well and Quantum-Dot Semiconductor Optical Amplifiers
Volume 6 Devices and Applications Edited by Professor Pallab Bhattacharya, Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI, USA 6.01 6.02 6.03 6.04 6.05 6.06 6.07 6.08 6.09 6.10 6.11 6.12 6.13 6.14 6.15 6.16 Index
III-Nitride-Based Short-Wavelength Ultraviolet Light Sources Nitride-Based LEDs and Superluminescent LEDs Electronic and Optoelectronic Devices Based on Semiconducting Zinc Oxide Molecular Beam Epitaxy of HgCdTe Materials and Detectors Quantum-Well Infrared Photodetectors and Arrays InAs/(In)GaSb Type II Strained Layer Superlattice Detectors Terahertz Detection Devices Amorphous and Nanocrystalline Silicon Solar Cells and Modules Quantum-Dot Lasers: Physics and Applications High-Performance Quantum-Dot Lasers Quantum-Dot Infrared Photodetectors Photonic Crystal Microcavity Light Sources Photonic Crystal Waveguides and Filters Spintronic Devices Based on Semiconductors Spin-Based Semiconductor Heterostructure Devices Spin-Polarized Transport and Spintronic Devices
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1.01 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories J R Chelikowsky, University of Texas, Austin, TX, USA ª 2011 Elsevier B.V. All rights reserved.
1.01.1 1.01.1.1 1.01.1.2 1.01.2 1.01.2.1 1.01.3 1.01.4 1.01.4.1 1.01.4.2 1.01.4.3 1.01.4.4 1.01.4.4.1 1.01.4.4.2 1.01.5 1.01.5.1 1.01.5.2 1.01.5.2.1 1.01.5.2.2 1.01.5.2.3 1.01.6 References
Introduction Early History The Phillips–Kleinman Cancellation Theorem Defining Pseudopotentials Model Potentials Solving the Electronic Structure Problem The Empirical Pseudopotential Method Optical Properties of Semiconductors The Structure and Form Factors of Semiconductors The Role of Nonlocality in Pseudopotentials The EPM Applied to Diamond Structure Semiconductors The electronic structure of silicon The electronic structure of germanium, gallium arsenide, and zinc selenide The Ab Initio Pseudopotential Method Constructing Pseudopotentials from Density Functional Theory Structural Properties of Semiconductor Crystals Total electronic energy from pseudopotential–density functional theory Phase stability of crystals Vibrational properties Summary and Conclusions
1.01.1 Introduction 1.01.1.1
Early History
A long-sought goal of materials physics is to predict the properties of materials solely from a knowledge of the atomic constituents. Before the turn of the twentieth century, such a goal was not in the realm of possibility. Quantum mechanics had not been invented, much less applied to materials. Moreover, as the rudimentary quantum theory evolved, for example, the Bohr theory of the hydrogen atom, the arguments of the time were often focused on the structure of an individual atom and did not dream of addressing problems of materials. However, as quantum mechanics became accepted for atoms, interest in applying quantum theory to materials grew. Indeed, once the predictive power of quantum mechanics for atoms and simple molecules became clear, the notion of predicting materials properties by solving the quantum mechanical behavior of the
1 1 3 4 4 6 6 6 8 10 11 11 15 26 28 31 31 32 34 37 39
system of interest did not seem so outlandish. One of the first visionaries was Dirac (1929). In a very famous quote circa 1930 he stated: The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.
The implication of this statement was clear. A solution of the quantum mechanical problem would yield the ‘whole of chemistry.’ One could predict properties of the chemical bond and the corresponding materials properties of matter without resort to laboratory work. Chemistry and materials properties could be predicted by calculations. Of course,
1
2 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
‘knowing’ and ‘doing’ are two different issues. The equations in question were much too complicated to be solved when Dirac made this statement. In fact, they are much too complicated to be solved today, even with state-of-the-art computational platforms. As Dirac anticipated, the key to progress in this area is to develop ‘approximate practical methods’. Today, we have a number of such practical methods that allow us to access some properties in a manner Dirac suggested. The first successes appeared some 30 or 40 years after Dirac’s statement and were led by the development of the pseudopotential concept. The pseudopotential model of a solid provided a practical model to address the quantum mechanical problem and advances in computers provided the means to overcome the numerical hurdles. A key attribute of a pseudopotential is that it allows us to address directly the electronically or chemically active valence states of an atom and remove from consideration the electronically inert core states. This decomposition between active and inactive states is readily done for most atoms, for example, in a silicon atom the 1s22s22p6 core states are tightly bound compared to the 3s23p2 valence states. In Figure 1, we illustrate the pseudopotential model for a crystalline solid. In this case, the core
Nucleus Core electrons Valence electrons Figure 1 Pseudopotential concept of a solid. The ion cores consisting of the core electrons and valence electrons are inert. The chemically active valence electrons move within this array of ion cores.
electrons and nucleus of an atom are treated as an inert ion core; the valence electrons are treated as an electron sea moving against the periodic background of the ion cores. This pseudopotential picture sets a length and energy scale to the problem that is determined by only the valence states. A profound consequence of this length and energy scale is that all atoms of the periodic table can be treated on an equal footing. Consider the situation without the pseudopotential approximation; in this case, an atom of hydrogen and an atom of cesium would require all the electronic states to be placed on an equal footing. The energy and length scales between the core and valance states would be different by orders of magnitude and would enormously complicate the problem. The pseudopotential idea is not new. Several key elements were recognized after the advent of quantum mechanics. Both Enrico Fermi and Hans Hellmann contributed seminal papers in the mid1930s (Fermi, 1934; Hellman, 1935). Interestingly, they recognized somewhat different, but important elements of pseudopotential theory. Fermi’s focus was on determining the phase shift in wave functions of high-lying alkali atoms subject to perturbations from foreign atoms. He correctly recognized that if one were interested in the long-range behavior of the wave function, it was not necessary to get the details correct near the nucleus. It should be possible to replace the true potential near the nucleus with a simple potential, or pseudopotential, that yields a similar value for the long-range part. Fermi also established the importance of a scattering length within this paper. Hellmann’s work is probably closer to our contemporary picture (Schwarz et al., 1999a, 1999b). Hellmann recognized that the valence orbitals for many-electron atoms contained pronounced oscillations in the region where the atomic core electrons remain. Determining an accurate description of the oscillatory behavior near the core and the long-range behavior in the chemically relevant region would render the problem computationally complex. Hellmann also recognized these core electrons did not play an important role in determining the chemical bond; after all this was clear from the periodic table. Elements such as Si and Ge have different ioncore configurations, but the valence electron configuration is the same, and their chemical properties are similar. In today’s language, we would argue that the valence electrons experience an additional kinetic
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories 3
the Schro¨dinger equation for the atomic orbitals can be written as
V (r )
~½ Bond length
H n ðrÞ ¼ r Core region
energy contribution in the core region because of the orthogonality requirement, that is, the valence state of an atom is orthogonal to a core state by nodal structure in the core region. In this 1935 paper, Hellmann proposed that this kinetic energy term could be quantified using the Thomas–Fermi expression for the kinetic energy of a free electron gas, which depends only on the electron density. When this term was added to the attractive Coulomb part of the valence electron interacting with the ion core, he argued that the resulting potential was ‘weak and constant’. An example of a pseudopotential similar to those Hellmann suggested is schematically illustrated in Figure 2.
v ðrÞ ¼ pv ðrÞ þ
X
av;c c ðrÞ
ð2Þ
c
The sum is over the core states, c, and pv represents the pseudopotential wave function. This form of the wave function recognizes that near the nucleus the valence wave function should appear atomic like and contain elements similar to the core states. Away from the nucleus, the core states will have little amplitude and we expect: v ðrÞ pv ðrÞ: Moreover, we expect pv(r) to be smoothly varying over all space. Outside of the core, the potential will vary slowly. Within the core, we expect its contributions will be small compared to the core-like functions. To determine the admixture of the core with the pseudo-wave-function, we demand the following condition: Z
c ðrÞv ðrÞd3 r ¼ hc j v i ¼ 0
ð3Þ
where we use the ‘bra–ket’ notation. The valence state must be orthogonal to the core state, owing to the nature of the eigenvalue problem. The orthogonality requirement yields av;c ¼ – c j pv
1.01.1.2 The Phillips–Kleinman Cancellation Theorem While the work of Hellmann and Fermi established the elementary ideas associated with pseudopotentials, a rigorous transformation of an all-electron potential to a pseudopotential was lacking. One of the earliest such transformation is based on the ideas of Phillips and Kleinman (Kleinman and Phillips, 1960a, 1960b; Phillips and Kleinman, 1959; Phillips and Kleinman, 1962). Although their conclusions are more far reaching, we will focus on solving the electronic structure problem for an isolated atom. We will make the one-electron approximation, that is,
ð1Þ
where m is the electron mass, h is Planck’s constant divided by 2, V is all-electron potential, and (En, n) are the eigen pairs: the energy levels and corresponding wave function. Here, we assume both core and valence states exist, that is, at this point we do not consider an element such as H that has no core states. We express the wave function for the valence state, v, as
Ion potential
Figure 2 Schematic of an ion-core pseudopotential. The all-electron potential and the pseudopotential ion-core potential are similar outside of the core region.
– h2 r2 þ V ðrÞ n ðrÞ ¼ En n ðrÞ 2m
ð4Þ
This form of the wave function was first proposed by Conyers Herring (Herring, 1940) in a different context. He proposed describing electronic states in crystals using orthogonalized plane waves. In this model, the valence states in a solid were replicated using plane waves that were orthgonalized to the core levels. If we apply the Hamiltonian in Equation (1) to the valence wave function, we obtain H pv þ
X c j pv ðEv – Ec Þc ¼ Ev pv c
ð5Þ
4 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
We have recognized that Hc ¼ Ecc. We can define a pseudopotential Hamiltonian, Hp, as Hp ¼ H þ V R
where G are reciprocal lattice vectors. S(G) is a structure factor given by
ð6Þ
SðGÞ ¼
where we define a potential operator, V R, as VR¼
X hc j . . .iðEv – Ec Þc
ð7Þ
c
The empty ‘ket’ given by j. . . > requires an operation wherein the wave function is projected on c and the functional dependence now corresponds to c. If we define a pseudopotential as Vp ¼ V þ V R , we have now transformed the one-electron Schro¨dinger equation to Hp pv
– h2 r2 þ Vp pv ¼ Ev pv ¼ 2m
ð8Þ
This description of the atom has a number of advantages. The repulsive part of the potential, V R, largely cancels the attractive part of the all-electron potential in the core region. The pseudo-wave-function is smooth as the core oscillations have been removed and the eigen-value, Ev, is identical to the all-electron potential. While the wave function is now amenable to a simple basis, the pseudopotential within this construction is more complex than the all-electron potential. Although the potential is weak and only binds the valence state, this potential is energy dependent, state dependent and involves a nonlocal, non-Hermitian operator. As such, the Phillips– Kleinman potential is rarely, if ever, used for calculating pseudopotentials. However, the cancellation theorem is useful in demonstrating the essential features of a pseudopotential.
One could attempt to form a potential for an elemental crystal by writing V ðrÞ ¼
X
Va ðr – R – tÞ
ð9Þ
R;t
where R is a lattice vector, t a basis vector, and Va a potential that we associate with the atom. For a crystal, we can explicitly incorporate the periodicity of the crystal by expressing the potential in a Fourier series in three dimensions. We write V ðrÞ ¼
X G
Va ðGÞSðGÞexpðiG?rÞ
ð10Þ
ð11Þ
where Na is the number of atoms in the basis. Va(G) is an atomic form factor given by Va ðGÞ ¼
1 a
Z
Va ðr ÞexpðiG?rÞd3 r
ð12Þ
Here a is the volume per atom and Va is the atomic potential that we take to be spherically symmetric. The structure factor contains information on the crystal structure and the form factors contain information on the electronic interactions. The reciprocal lattice vectors are designed to have the property: exp(iG?R) ¼ 1 (Kittel, 2005). This insures that the potential is periodic: V ðr þ RÞ ¼
P
Va ðGÞSðGÞexpðiG?ðr þ RÞÞ
G
¼
P
Va ðGÞSðGÞexpðiG?rÞ ¼ VðrÞ
ð13Þ
G
While this plane-wave expansion of the potential reflects the translational symmetry, the expansion is only helpful if the number of waves (or reciprocal lattice vectors) is manageable. This will not be the case for the all-electron potential owing to the singularity of the Coulomb potential at the nucleus. The pseudopotential avoids this issue by removing this singularity. What remains is to define a procedure to construct the potential.
1.01.2.1
1.01.2 Defining Pseudopotentials
1 X expðiG ? tÞ Na t
Model Potentials
A realistic and powerful approach to find the required form factors is based on experiment. For example, suppose we consider a potential similar to what Hellmann proposed. Let us consider the following model, which only involves one parameter; the core radius, rc: ( Vcp ðr Þ
¼
r rc
0 2
– Zv e =r
r > rc
ð14Þ
This defines a potential where the cancellation occurs within the ion core (r < rc). Zv is the valence charge of the ion core. This is an ion-core pseudopotential. It represents the interaction of the valence electron with the ion core (the nucleus plus core electrons) sans the role of any valence–valence electron interactions.
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories 5
The next step in defining the atomic form factor is to employ Equation (12): Vcp ðGÞ ¼
– 4Zv e 2 cosðGrc Þ a G 2
Vcp ðGÞ TF ðGÞ
ð16Þ
Thomas–Fermi screening is given by Ks2 q2
TF ðqÞ ¼ 1 þ
ð17Þ
where 1/Ks is the Thomas–Fermi screening length: Ks2 ¼ 6ne 2 =EF , where n is the valence electron density (n ¼ Zv/a) and EF is the Fermi energy. The atomic form factor is now given by Vap ðGÞ ¼
– 4e 2 n cosðGrc Þ G 2 þ Ks2
ð18Þ
A common practice is to set the potential to zero outside of the second node to remove the oscillatory or ringing behavior of the cos(Grc) term. An V(q)
Vap ð0Þ ¼
ð15Þ
The long-range nature of the coulomb tail requires special handling in executing this integral (Kittel, 2005). The form factor ‘rings’, that is, the form factor oscillates as a cosine function owing to the discontinuity in the potential at rc. What remains is to screen the ion-core pseudopotential by including a dielectric screening function that replicates the role of the valence electron–electron interactions. Often Thomas–Fermi screening is used for this purpose (Kittel, 2005): Vcp ðGÞ ¼
interesting consequence of Thomas–Fermi screening is that the potential now has a well-defined value when G ¼ 0: – 4e 2 n 2 ¼ – EF Ks2 3
ð19Þ
The G ¼ 0 term corresponds to setting a reference energy for the solid as it corresponds to the average of the potential. This limiting case from Thomas– Fermi screening is not a very good approximation for a semiconductor such as silicon and is more appropriate for a metal. We illustrate a schematic pseudopotential in Figure 3. The Phillips–Kleinman cancellation theorem yields a state-dependent pseudopotential; the potential depends on the nature of the core states. For example, in carbon there is no 1p core state and the valence 2p is not required to be orthogonal to the core. This is not true for silicon, in which both s and p states exist in the core; both the s and p valence states see a repulsive term. Historically, this difference has been used to explain why silicon and carbon chemistry are different. The previous model potential can be modified to reflect the state dependence: ( p Vc;l ðr Þ
¼
r rc;l
Al 2
– Zv e =r
r > rc;l
ð20Þ
This potential is not a simple function of position, but rather acts on the l-component present in the wave function (Cohen and Chelikowsky, 1982, 1989; Chelikowsky and Cohen, 1992). This can be accomplished by using a projection operator as will be discussed later. Setting this technical issue aside,
V(r) ~ ∫V(q) eiqr dq ~ (½ BOND LENGTH)–1 q
V(q = G) for typical G’s – 2/3 EF Screened ion limit for metals Figure 3 Model of an atomic pseudopotential. The required form factors can be extracted from this model. Note the limiting case of G ¼ 0 for a metallic system.
6 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
there are now additional parameters to be determined as the well depth and size must be established for each l-component. A common procedure in the early days of pseudopotential involved fitting the well depths and sizes (Al, rc,l) to replicate optical data of ionized atoms. For example, to construct an ion-core pseudopotential for an Al atom, one needs to consider the optical excitations for an Alþ2 ion. In some cases, this is relatively easy, for example, treating the Na atom does not involve an ionized atom. However, an O atom would require examining a C5+ ion. Experimentally, it is a difficult task to multiply ionize such an atom and measure the atomic levels. Often the potentials were estimated by extrapolation from lighter atoms to heavier ones. Another issue concerns improved dielectric functions for screening the potentials. Both issues were the subject of a number of studies, but without a definitive resolution. By the mid-1960s, an inventory of such model potentials existed (Animalu and Heine, 1965; Cohen and Heine, 1970) with some success, especially for metals.
1.01.3 Solving the Electronic Structure Problem If we are given the crystalline potential, we can solve, the corresponding eigenvalue problem (Equation (1)) using a variety of approaches. Since the pseudopotential is weak compared to the all-electron potential, we can write the wave function in terms of a plane wave basis: n;k ðrÞ
¼
X
n ðk; GÞexpðiðk þ GÞ?r Þ
ð21Þ
G
where n is a band index and k is the wavevector. The sum is over all reciprocal lattice vectors, {G}, but, in principle, the sum is truncated for jk þ Gj > Gmax . This form of the wave function is consistent with the Bloch form of the wave function for a periodic potential. The Bloch form of the wave function allows one to express the corresponding periodicity of the wave function, save a phase factor determined by the wavevector: n;k ðr
þ RÞ ¼ expðiðk?RÞexp
n;k ðrÞ
ð22Þ
Inserting the crystalline potential (Equation (10)) and the plane wave basis (Equation (21)) in the one-electron Schro¨dinger equation yields the following:
( X h2 2 ðk þ GÞ – En ðkÞ GG9 2m G9 ) þVap ðG9 – GÞS ðG9 – GÞ n ðk; G9Þ ¼ 0
ð23Þ
This corresponds to a set of linear equations, one for each G vector in the set. For a nontrivial solution, we require the following: h2 2 det ðk þ GÞ – En ðkÞ GG9 2m p þ Va ðG9 – GÞS ðG9 – GÞ ¼ 0
ð24Þ
The diagonal elements of this determinant contain the kinetic energy contribution; the off-diagonal elements contain the form factors for the pseudopotential and the structure factor. In this example, we consider only elemental crystals so that the form factors can be separated out from the structure factor. This eigenvalue problem is a standard mathematical problem, and is easily handled, save for highly complex systems. Typically, for common semiconductors a few hundred plane waves are required to obtain a converged result. This operation can often be performed on a laptop computer for crystalline semiconductors.
1.01.4 The Empirical Pseudopotential Method One of the most significant scientific advances of the twentieth century was the development of the empirical pseudopotential method (EPM). The intuitive picture of Fermi and Hellmann was greatly advanced by Phillips and Kleinman; however, a practical method of predicting accurate energy band structures was not achieved until the EPM. This method, developed in the late 1960s, was based on fixing an energy band so that an accurate response function, as measured, was reproduced. To accomplish this feat, one had to develop a framework of translating an energy band solution to a response function. 1.01.4.1 Optical Properties of Semiconductors The optical properties of a semiconductor can be characterized by an understanding of response functions such as the complex index of refraction, N(!) ¼ n(!) þ ik(!), or the complex dielectric
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories 7
function, "(!) ¼ "1(!) þ i"2(!) (Cohen and Chelikowsky, 1989). They are related to each other as follows: N 2 ¼ "1 þ i"2 "1 ¼ n2 – k2
ð25Þ
jMvc ðkÞj2 ¼ jhuv ðkÞjrjuc ðkÞij2
ð31Þ
The functions, un(k), are the periodic part of the Bloch wave functions: un ðkÞ ¼
X
n ðk; GÞexpðiG?rÞ
ð32Þ
G
"2 ¼ 2nk
The normal incident reflectivity of a semiconductor is given by the response functions: N – 1 2 ðn – 1Þ2 þ k2 ¼ R¼ N þ 1 ðn þ 1Þ2 þ k2
ð26Þ
Hence, a knowledge of the dielectric function, "(!), is sufficient to yield the complex index of refraction and the reflectivity. In fact, only the real (or imaginary) part of the dielectric function is required owing to the Kramers–Kronig relation: Z
2 "1 ð!Þ ¼ 1 þ P
1
0
!9"2 ð!9Þ d!9 !92 – !2
ð27Þ
The principal part of the integral is required as is a knowledge of "2 over all frequencies, although, in practice, this is not a stringent requirement. There are useful sum rules that can be applied to test the accuracy of response functions. One obvious example is to use the ! ! 0 of the Kramers–Kronig relationship: 2
"1 ð0Þ ¼ 1 þ
Z 0
1
"2 ð!Þ d! !
ð28Þ
Another sum rule is p ! ¼ 2 2
Z
1
!"2 ð!Þd!
ð29Þ
0
where !p is the plasma frequency given by !p 2 ¼ 4ne 2 =m. The connection between the macroscopic response function and the atomistic world can be made by a computation of the imaginary part of the dielectric function. A realistic, but somewhat simplified, expression comes from Ehrenreich and Cohen (EC59). The dielectric function involves transitions between the valence band (v) to the conduction band (c): "2 ð!Þ ¼
4e 2 h X 2 3m2 !2 v;c ð2Þ3 Z ð!vc ðkÞ – !ÞjMvc ðkÞj2 d3 k
where n is the band index. The periodic part obeys un ðk; r þ RÞ ¼ un ðk; rÞ. Implicit in this expression is the existence of a bandgap that delineates the valence and conduction bands. This form also assumes cubic symmetry. The physical content of the dielectric function is clear. The delta function insures energy conservation and the dipole matrix element accounts for symmetry, that is, we assume dipole transitions. Another implicit factor is that the transitions from filled bands to empty bands are direct; the wavevector of the initial and final states are equal. This is appropriate for optical transitions where the wavelength of light is orders of magnitude larger than the atomic length scale. Also, the Ehrenreich–Cohen form does not include excitonic(electron–hole) interactions, local field corrections, or self-energy effects, which can be included (Rohlfing and Louie, 2000) albeit at some considerable computational cost. Moreover, the qualitative features of the optical spectra are correctly obtained from the Ehenreich–Cohen expression. Once the imaginary part of the dielectric function is known, it is straightforward using the Kramers– Kronig transform to find the real part of the dielectric function and the reflectivity, which can be directly compared to experiment. To implement the Ehenreich–Cohen expression, we need to know the energy bands, En(k) and the corresponding wave functions n,k. This can be obtained once the pseudopotential is defined. Usually, the dipole matrix elements in Equation (31) are smoothly varying with k and can be assumed constant, except at high symmetry points. In this situation, it is useful to remove the matrix elements and isolate that part of "2(!) that produces the major structure. The resulting function is called the ‘joint density of states’: Jvc ð!Þ ¼
2 ð2Þ3
Z
½!vc ðkÞ – !d3 k
ð33Þ
BZ
ð30Þ
This integral can be recast as
BZ
The integral is over all states in the Brillouin zone where the dipole matrix element is given by
Jvc ð!Þ ¼
2 ð2Þ3
Z !¼!vc
ds jrk !vc ðkÞj
ð34Þ
8 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
where ds is a surface element in wavevector space defined by ! ¼ !vc(k). This is similar to the density of states, save here we replace the band energy by the transition frequency between the valence and conduction bands. The structure in this joint density of states occurs at critical points, just as in the density of states (Cohen and Chelikowsky, 1989). These critical points occur when rk !vc kcp ¼ 0
where the lattice basis vectors are given by aˆ1 ¼ aðˆy þ zÞ=2; ˆ aˆ2 ¼ aðxˆ þ zÞ=2; ˆ aˆ3 ¼ aðxˆ þ yˆÞ=2
The diamond and zinc–blende crystals possess a primitive cell given by these vectors, with the cell volume given by c ¼ ja1 ?a2 a3 j ¼ a 3 =4
There are two atoms in the basis located at t ¼ ð1;1;1Þa=8, so the atomic volume, a is given by a3/8. The reciprocal lattice vectors are given by
ð35Þ
Physically, this occurs when the group velocity of the hole in the valence band equals that of the electron in the conduction band. Four topologically distinct critical points occur: M0, M1, M2, and M3, which represent a local minimum, two saddle points, and a local maximum. The structure associated with these critical points is presented in Figure 4. Structure in either the imaginary part of the dielectric function or the density states can be identified with these critical points (Cohen and Chelikowsky, 1989).
Gm1 ;m2 ;m3 ¼ m1 bˆ1 þ m2 bˆ2 þ m3 bˆ3
bˆ1 ¼ ð2=aÞð – xˆ þ yˆ þ zÞ=2; ˆ bˆ2 ¼ ð2=aÞðxˆ – yˆ þ zÞ=2 ˆ ˆb3 ¼ ð2=aÞðxˆ þ yˆ – zÞ=2 ˆ
This set of vectors has the desired property that G ? R is an integral multiple of 2. In some disciplines such as crystallography, the factor of 2 is omitted from the definition of reciprocal lattice vectors and G ? R is an integer. For diamond, the structure factor is given by
This section focuses on cubic semiconductors that occur in either the zinc–blende or diamond structures. These two structures are isostructural; they have the same atomic arrangement, but the constituent atoms can be different. Semiconductors such as Si, Ge, GaAs, InSb, ZnSe, and CdTe occur in these structures. Figure 5 illustrates the diamond structure. The diamond structure has cubic symmetry and can be conceptualized by considering two interpenetrating face-centered cubic crystals. The lattice vectors for diamond are given by
SðGÞ ¼ cosðG ? tÞ
ð36Þ
VA ðGÞexpðiG?tÞ þ VB ðGÞexpð – iG?tÞ=2
J(ω)
M2 M0
M3 ω1
ð39Þ
The structure factor for the zinc–blende crystal cannot be separated from the potential terms. For a zinc– blende crystal, A B , where A is the cation and B is the anion, it is easy to show that the product of the form factors and structure factor in Equations (23) and (24) can be replaced by
M1
ω0
ð38Þ
where the reciprocal lattice basis vectors are given by
1.01.4.2 The Structure and Form Factors of Semiconductors
Rn1 ;n2 ;n3 ¼ n1 aˆ1 þ n2 aˆ2 þ n3 aˆ3
ð37Þ
ω
Figure 4 Structure associated with the four types of critical points.
ω2
ω3
ð40Þ
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories 9 Table 1 Structure factors for diamond and zinc–blende crystals
a
Figure 5 Ball and stick model for the diamond crystal structure. The inset shows the tetrahedral coordination. The zinc–blende structure is isostructural, save the atomic species alternate.
It is often convenient to introduce a symmetric form factor, VS, and an antisymmetric form factor, VA, in which case this term can be regrouped as VS ðGÞcosðG?tÞ þ iVA ðGÞsinðG?tÞ
ð41Þ
where VS ðGÞ ¼ ðVA ðGÞ þ VB ðGÞÞ=2 VA ðGÞ ¼ ðVA ðGÞ – VB ðGÞÞ=2
In the case of the diamond structure the antisymmetric form factor vanishes and the symmetric one corresponds to the constituent element of the diamond crystal. The form factor for the zinc–blende crystal is complex because the zinc–blende structure lacks inversion. The plane wave expansion is also complex, but this does not complicate the calculations significantly. The Hamiltonian remains hermitian. There are two important consequences of this formalism. First, owing to the crystal symmetry, the structure factor can vanish for some reciprocal lattice vectors, meaning that the form factor for this vector is irrelevant. Second, the number of form factors required can be quite small if the pseudopotential is weak and converges quickly in reciprocal space. In the case of the diamond crystal, this can be as few as three distinct form factors. In Table 1, we summarize the needed form factors and structure factors. The form factors depend only on the magnitude of the reciprocal vector and are often tabulated by the value of G2 (a/2)2, that is, the form factors are listed for G2 (a/2)2 ¼ 0, 3, 4, 8, . . . and are denoted by V(0), V(3), V(4), V(8), . . . . The structure factors are
G2
cos(G?t)
sin(G?t)
Form factor
0
1
0
3 4 8 11
pffiffiffi 1= 2 0 1 pffiffiffi 1= 2
pffiffiffi 1= 2 1 0 pffiffiffi 1= 2
VS serves as a reference energy VS, VA both required VA required VS required VS, VA both required
given by cos(G ? t) ¼ cos((n1m1 þ n2m2 þ n3m3)/4); sin(G ? t) ¼ sin((n1m1 þ n2m2 þ n3m3)/4). If we focus on the diamond structure only the symmetric form factors (V(0), V(3), V(8), V(11), . . .) are meaningful. V(0) represents the average potential and appears only on the diagonal of the Hamiltonian matrix. It will displace the energy bands uniformly and is not important for spectroscopy. In the remainder of this section, we will omit this term and focus on spectroscopic issues. If the form factors rapidly converge, it is possible to define the potential by only three parameters corresponding to V(G2 11). Since the form factors are not linearly independent, the number of parameters are less than three. Moreover, we can get a rough estimate for these form factors from model potentials (Cohen and Heine, 1970). At a first pass, it might appear that a total of six parameters are needed for zinc–blende structures: VS(3), VA(3), VA(4), VS(8), VS(11), and VA(11). However, we can constrain the symmetric form factors in the following fashion. Suppose we are given the form factors for Ge and wish to find the form factor of GaAs. We can make the following approximation: VS ðGaAsÞ ¼ ðV ðGaÞ þ V ðAsÞÞ=2 V ðGeÞ
We express the symmetric form factors, the average of the Ga and As form factors to be the same as the Ge form factors. With this constraint, we need to fix only three parameters (VA(3), VA(4), and VA(11)) to obtain the GaAs potential. The EPM method was highly successful in establishing realistic form factors and energy band structures for a wide variety of materials. In Figure 6, we illustrate a block diagram for the EPM. Initially a set of form factors are chosen. The pseudopotential for the crystal and the Hamiltonian is constructed from this set of form factors and the structure factor. The one-electron Schro¨dinger is then solved and the band structure (En(k)) and wave functions ( n,k(r)) are obtained. From the band structure
10 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
resulting form factors and the energy band structure are deemed correct, at least within this framework. One of the first successful implementations of the EPM was performed by Cohen and Bergstresser (1965). They examined 14 semi-conductors in the diamond and zinc–blende structures. The Cohen– Bergstresser form factors were fit to band structure features such as the fundamental gap. The resulting form factors and lattice constants are given in Table 2. The overall agreement is quite good; the bandgaps were replicated to within 0.1–0.2 eV. Other features agreed to within 0.5 eV over a 10 eV range.
EPM v(G)
v(r) = Σ v(G) S(G) exp(i G . r) G
H = P2 + V Hψ = Eψ Get E(k), and ψ k
Calculate R or R′/R and N(E)
1.01.4.3 The Role of Nonlocality in Pseudopotentials
Compare with experiment
Alter v (G)
Figure 6 Flow diagram for the empirical pseudopotential method (EPM).
the imaginary part of the dielectric function can be obtained and the corresponding reflectivity, which is then compared to experiment. (It is also possible to obtain the density of states and compare to the photoemission spectrum.) If the agreement between the calculated and measured properties is not good, the form factors can be altered and the calculations repeated. When the agreement is satisfactory, the
The Cohen–Bergstresser approach treats the pseudopotential as a simple function and neglects the role of the state dependence expected from the Phillips–Kleinman cancellation theorem. This is often the case for tetrahedral semiconductors, AB, where the constituent elements are not transition metals (or rare earths) or from the first row of the periodic table. However, important semiconductors such as GaN do contain elements from the first row, and the state dependence or angular momentum components need to be explicitly considered. The nonlocal character of the ion-core pseudopotential for an atom can be expressed as
Table 2 Pseudopotential form factors (Ry) and lattice constants for selected diamond and zinc–blende semiconductors
Si Ge Sn GaP GaAs AlSb lnP GaSb InAs InSb ZnS ZnSe ZnTe CdTe
a(A˚)
VS(3)
VS(8)
VS(11)
VA(3)
VA(4)
VA(11)
5.43 5.66 6.49 5.44 5.66 6.13 5.86 6.12 6.04 6.48 5.41 5.65 6.07 6.41
0.21 0.23 0.20 0.22 0.23 0.21 0.23 0.22 0.22 0.20 0.22 0.23 0.22 0.20
0.04 0.01 0.0 0.03 0.01 0.02 0.02 0.00 0.00 0.00 0.03 0.01 0.00 0.00
0.08 0.06 0.05 0.07 0.06 0.06 0.06 0.05 0.05 0.04 0.07 0.06 0.05 0.04
0.12 0.07 0.06 0.07 0.06 0.08 0.06 0.24 0.18 0.13 0.15
0.07 0.05 0.04 0.05 0.05 0.05 0.05 0.14 0.12 0.10 0.09
0.02 0.01 0.02 0.01 0.01 0.03 0.01 0.04 0.03 0.01 0.04
From Cohen ML and Bergstresser TK (1965) Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and zinc-blende structures. Physical Review 141: 789.
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
Vcp ðrÞ ¼
1 P l¼0
p P yl Vl;c ðr ÞP l ¼ P ys Vs;c ðr ÞP p
s
ð42Þ
þP yp Vp;c ðr ÞP p þ P yd Vd ;c ðr ÞP d þ
p
p
P l is an operator that projects out the lth component. The ion core pseudopotential is semilocal. The potential is radially symmetric, but angular dependent. For many semiconductors only the s, p and d components of the wave functions are significant. This allows us to write P s þP p þP
d
1
ð43Þ
If two of the components are similar, it is often possible to select one component and subsume the other terms into a ‘local potential’. As a specific example, consider the case of GaAs. The energy band structure can be dramatically improved by considering a local potential and altering it with a potential that acts on only d waves: p
Vcp ðrÞ ¼ Vc;local ðr Þ þ P
y d
h i p Vd ðr Þ – Vc;local ðr Þ P
y d
ð44Þ
A plane wave expedites the use of angular projections. It can be decomposed as follows: expðiK?rÞ ¼
1 X ð2l þ 1Þi l Pl ðcosðÞÞjl ðjK jr Þ
ð45Þ
i¼0
where Pl(x) are Legendre polynomials: P0 ¼ 1, P1 ¼ x, P2 ¼ (3x2 1)/2, is the angle between k and r and j(x) are spherical Bessel functions: sinðxÞ x sinðxÞ cosðxÞ j1 ðxÞ ¼ – x2 x
3 sinðxÞ 3cosðxÞ – –1 j2 ðxÞ ¼ x2 x x2
j0 ðxÞ ¼
Using this decomposition allows one to write the required matrix element for a nonlocal potential as D
E 4 p k þ GVc;l k þ G9 ¼ ð2l þ 1ÞPl ðcoskþG;kþG9 Þ a Z 1 p dr r 2 Vc;l ðr Þjl ðjk þ Gjr Þjl ðjk þ G9jr Þ 0
ð46Þ
The off-diagonal matrix elements now depend on the wavevector, k. Typically, a model potential is used for the nonlocal component. For example, one might p consider a Gaussian potential: Vc;l ¼ Al expð – ðr =rl Þ. In this case the matrix elements in Equation (46) can be evaluated analytically (Cohen and Chelikowsky,
11
1989). Nonlocal components for tetrahedral semiconductors are available in the literature (Chelikowsky and Cohen, 1976). The agreement between experiment and nonlocal pseudopotential can be considerably better than for local pseudopotentials, albeit more parameters are utilized, typically two per element. 1.01.4.4 The EPM Applied to Diamond Structure Semiconductors Group IV elements, save Pb, can form in the diamond structure (Figure 5). All elements of this group have a valence electron configuration of s2p2. When bonds are formed, we view this process as the promotion of an electron in the s-state to the p-state to form sp3 hybrids. This explanation accounts for the nature of the diamond structure as sp3 hybrid orbitals form tetrahedral bonding patterns. Moreover, the hybridization energy increases as one descends down the group IV column (Phillips, 1973). As such, we expect the diamond structure to be less stable versus metallic structures such as the face-centered cubic structure of Pb, or the gray on structure for Sn. The general increase in metallicity down the column is also rationalized by the lack of formation of sp3 hybrid orbitals for the heavy elements. 1.01.4.4.1 of silicon
The electronic structure
The band structure of silicon serves as the hydrogen atom of solid-state physics. Virtually any new band structure method is tested on crystalline silicon. There is general accord on the basic energy band features of silicon, but this was not the case until the end of the 1970s. In Figure 7, we present the band structure of silicon as calculated using two different pseudopotentials. One of the potentials contains nonlocal corrections and yields a slightly wider valence band. The energy bands are plotted along directions between high symmetry points in the Brillouin zone as indicated in Figure 8. Silicon is different in a fundamental way from other diamond or zinc–blende semiconductors. The ordering of the conduction bands of silicon is different from that of germanium or tin. In Si, the 15 band is lower than the 29 band. This difference was the source of some controversy in the early days of energy band calculations. For example, some workers claimed that the first pseudopotential calculations were in strong disagreement with less empirical
12 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
6 L3
Γ2´
Γ2´
4 Γ15
Γ15
2 L1 X1
Γ25´
0
Γ25´
Energy (eV)
L3´
–2 X4
–4
–6 Si
–8
X1
L1 L2´
–10 Γ1
Γ
–12 L
Γ
Λ
Δ X Wavevector, k
U,K
Σ
Γ
Figure 7 Band structure for silicon.
L U X Γ
W
K
Figure 8 Brillouin zone labeling high symmetry points.
methods for determining energy bands and with the available experimental data (Kunz, 1971). However, optical measurements of silicon–germanium alloys strongly suggested that silicon and germanium were different (Kline et al., 1968). The issue was decided by the work of Aspnes and Studna (1972). They used low-field electroreflectance to resolve transitions from 25 to 29, and showed unequivocally that the conduction band ordering is in complete agreement with the pseudopotential predictions.
Another feature of the silicon band structure, related to the lowest conduction band configuration, is the nearly parallel dispersion of the lowest conduction band and the uppermost valence band. Specifically, consider the dispersion along the A direction. Grover and Handler (1974) proposed on the basis of their electroreflectance data that the critical point at L is essentially two dimensional, that is, the experimentally measured transverse band mass virtually vanishes. The value they obtained for m1 was approximately 0.02 m. The corresponding mass calculated by pseudopotential methods is closer to 0. 1 m, but even this value results in a nearly two-dimensional M0 critical point at the L point (Cohen and Chelikowsky, 1989). Moreover, the dispersion for the valence and conduction bands is such that the energy difference between these bands is less than 0.01 eV over half the distance from L to . This valence–conduction band configuration favors excitonic behavior. It has been traditionally suggested that the rather large discrepancies between the theoretical and experimental optical constants in this energy range arise from such an effect. To illustrate the point, we have displayed the measured and calculated imaginary part of the dielectric constant in
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
Figure 9. The measured dielectric function is nearly a factor of 2 larger near 3.4 eV than the calculated value for the energy region in question. This discrepancy is also observed in the real part of the dielectric function, which is displayed in Figure 10. While the Ehrenreich–Cohen formalism (Ehrenreich and Cohen, 1959) replicates the essential features of the optical constants, it omits local fields and electron–hole interactions. Local-field corrections were addressed by Adler (1962) and Wiser (1963). Local fields allow for the microscopic
Si 40 Experiment With local-field corrections Without local-field corrections
ε2 (ω)
30
20 ×3 10
0
0
2
4
10
8 6 Energy (eV)
12
Figure 9 Imaginary part of the dielectric function for Si with and without local fields.
13
variation of the field within the unit cell. Unfortunately, the corrections for local fields do little to improve the dielectric function in this region as indicated in Figure 9. The role of the electron–hole interactions in semiconductors was addressed by a number of groups. One of the first realistic attempts is from Hanke and Sham (1974). They examined the optical properties of diamond, including both local fields and exchange interactions. More recent work by Louie and collaborators (Hybertsen and Louie, 1986; Rohlfing and Louie, 1998, 2000) has unequivocally demonstrated the role of excitonic contributions to the optical response functions. In Figure 11, we exhibit the calculated and measured reflectivity spectrum for silicon. As for most solid-state spectra, the spectrum is not highly structured, and on first inspection appears to contain little information. Such an assessment would be incorrect. Details of the spectrum can be enhanced by numerically differentiating the spectrum. In Figure 12, we present the calculated and measured logarithmic derivative of the reflectivity spectrum. The derivative spectrum does not exhibit the three broad peaks of the undifferentiated spectrum but numerous sharp spectral features. These features are classified according to a standard nomenclature. The lowest energy structural feature is denoted by E0 and corresponds to structure from the smallest direct transition. The next higher energy structural feature is denoted by E1. This feature denotes structure associated with the direction. In the unique case of silicon, the E0 and the E1 structural features occur in
50
0.8
Si Theory Experiment
40
Si Theory Experiment
0.7 0.6
30
R (E )
ε1 (E )
0.5
20
0.4
10
0.3
0
0.2 0.1
–10 –20 0
0 0
1
2
3 4 Energy (eV)
5
6
Figure 10 Real part of the dielectric function for Si.
7
1
2
3 4 Energy (eV)
5
6
Figure 11 Measure and calculated normal incident reflectivity for Si.
7
14 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
+1.0 (a)
E1 + E′0
E1′
E2
+0.5
0
Si
1 dR (eV–1) R dE
–0.5
Experiment –1.0 (b) +0.5
0
–0.5 Theory –1.0 2
3
4 Energy (eV)
5
6
Figure 12 Measured and calculated normal incident modulated reflectivity for Si.
the same energy region, that is, the lowest energy direct transitions at L and at have nearly the same energy. In fact, a good deal of work has been peformed to determine precisely where in the Brillouin zone this transition takes place. Piezoelectric experiments (Gobeli and Kane, 1965), chemical shifts in Ge–Si alloys (Tauc and Abraham, 1961), electroreflectance (Pollak and Cardona, 1968), and some wavelength-modulation techniques (Koo et al., 1971) have suggested that the first reflectivity peak arises near the zone center, perhaps along the direction. However, other work (Grover and Handler, 1974) suggested that this transition occurs along the direction. Unfortunately, the pseudopotential work has not resolved this issue since the reflectivity structure in question arises from contributions near and along both the and directions. The dominant transition in the calculations appears to be along the direction, but no firm conclusion can be drawn. What is clear is that the complexity of this structure does suggest a
multiplicity of critical points within this energy region. This conclusion has been drawn by several authors (Welkowsky and Braunstein, 1972). The E2 structure dominates the reflectivity spectrum and occurs at about 4.5 eV. This structure arises from large regions of the Brillouin zone. While optical properties served as a centerpiece for the first EPM calculations, photoemission measurements provide information inaccessible to interband transitions. In particular, photoemission can provide the absolute position of bands relative to the vacuum. The valence band density of states for silicon calculated using pseudopotentials is compared with experiment in Figure 13. Under certain conditions, it is possible to compare directly the theoretical density of states with photoemission measurements. The density of states, D(E), for crystalline matter, can be defined as DðEÞ ¼
2 X ð2Þ3 n
Z
d3 kðE – En ðkÞÞ
ð47Þ
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
disruption of the crystalline bonding environment will introduce states into the regions of reduced state density. For example, in amorphous materials, where there presumably exist bond angle and bond length deviations from the ideal crystalline state, these regions are strongly affected. The dip around 8 eV fills in, and the upper region of the density of states tends to narrow and shift to higher energies. For surface states or vacancy states where broken bonds are clearly involved, the defect localized states tend to fill in these regions, including the fundamental gap.
(a)
XPS spectra (a.u)
Experiment Si
Density of states (states / eV-atom)
(b) Theory
1.0
1.01.4.4.2 The electronic structure of germanium, gallium arsenide, and zinc selenide
0.5
0 –14 –12 –10 –8
15
–6 –4 –2 Energy (eV)
0
2
4
6
Figure 13 Density of states for crystalline silicon. The top panel is from X-ray photoemission spectroscopy (Ley et al., 1972; Cavell et al., 1973).
where is the cell volume. Typically, the density of states, which is an extensive property, is normalized in terms of states per atom. If matrix element effects are ignored, the density of states can be extracted from X-ray photoemission spectroscopy or ultraviolet photoemission measurements. For the testing of empirical pseudopotential band structures, the photoemission spectra have proven invaluable for the determination of accurate potentials. Photoemission measurements have indicated that the local pseudopotentials cannot provide band pictures where both the valence and conduction bands are simultaneously reproduced accurately when compared to experimental band pictures. For diamond-structure semiconductors, the valence band density of states may be divided into three general regions as in Figure 13. Using the top of the valence band as our zero of energy, the region of 13 to 8 eV is predominantly of s-like character stemming from the atomic 3s-states of Si. The region from 8 to 4 eV is a transition region with contributions from both s- and p-states. The region from 5 to 0 eV is predominantly p-like. The feature which delineates these spectral regions is a sharp reduction or ‘dip’ in the density of states compared to the average density. The nonuniformity of the density of states in contrast to simple metals arises from hybridization among the atomic orbitals. Any
The triad of semiconductors Ge, GaAs and ZnSe are special. They are isostructural with a bond length that hardly changes among the three. As such, the difference between the three can be attributed to potential differences. In pseudopotential theory, the symmetric form factors do not significantly change; the antisymmetric form factors (the difference between the cation and anion pseudopotentials) control the band structure, and the optical and photoemission spectra. Germanium The valence electron configurations of germanium and silicon are both s2 p2; however, significant differences exist for the band structures. The chief differences arise from the corecharge configuration. While the nuclear charge increase is exactly balanced by an increase in core electron number, the larger number of core electrons is more effective in screening the nuclear charge. The valence levels in germanium are less tightly bound compared to silicon, and this is reflected in the more metallic character of the germanium band structure. Another significant feature that distinguishes germanium from silicon is the presence of 3d states within the core. While it is clear that these d-states do not significantly participate in the cohesive process, they can still influence the conduction band configuration. Specifically, the unoccupied bands, which may have 4d character, are influenced by an orthogonality requirement not present for the 3d states in silicon. This reasoning is reinforced by empirical evidence that suggests that the presence of d-electrons, that is, nonlocal pseudopotentials, which incorporate d-orbital corrections, appear crucial in describing the germanium conduction bands (Chelikowsky and Cohen, 1973; Phillips and Pandey, 1973). 1.01.4.4.2(i)
16 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
Γ8
2 L6 0
Energy (eV)
Γ8
Γ6 Γ7 Γ8
L4 + L5
X5
Ge
Γ7
–2 L6
0.6
Γ6 Γ7 Γ8
R (E )
L4 + L5 L6
4
0.4 Ge Theory Experiment
Γ7
0.2 X5
–4 –6
0
–8 L6 X5
Γ6 L
Λ
Γ
1
2
3 4 Energy (eV)
5
6
7
Figure 15 Theoretical and experimental reflectivity for crystalline germanium. The experimental data is from Philipp and Ehrenreich, (1963).
–10 L 6 –12
0
Γ6 Δ x U, K Wavevector, k
Σ
Figure 14 Band structure for crystalline germanium. The top of the valence band is considered as the energy zero.
In Figure 14, we present the band structure of germanium as determined from empirical pseudopotentials. This calculation includes spin–orbit interactions, which are most significant for heavier elements (Cohen and Chelikowsky, 1989). Unlike the case of silicon, most band structures for germanium are in reasonably good agreement. The band structure of germanium differs from silicon most significantly in the conduction band arrangement. The 15 and 29 bands, using the notation in Figure 7, reverse their ordering. Moreover, while germanium and silicon are both indirect semiconductors, the conduction band minimum in germanium occurs at the L-point as opposed to a point near X in silicon. As noted earlier, the valence band configuration for diamond semiconductors is insensitivie to the constituents: silicon, germanium, and tin have almost identical valence band configurations, but they have quite different conduction band configurations. In Figure 15, we present the reflectivity spectrum of germanium as calculated from pseudopotential band theory and as determined by experiment (Grover and Handler, 1974). Compared to silicon, germanium exhibits a richer spectrum. The additional structure arises from the larger spin–orbit coupling occurring in germanium. For example, in the E1 region near 2.2 eV, a distinct doublet structure occurs. This doublet arises from the spin–orbit splitting in the L39 level.
In Figure 16, the logarithmic derivative reflectivity spectrum is displayed. The derivative spectrum vividly illustrates the rich reflectivity structure available using modulated techniques. Perhaps the most interesting feature of the germanium reflectivity spectrum is the E2 peak at 4.5 eV. Initially, the origin of this spectral feature appeared to be located at the X-point and to involve transitions between the highest valence band and the lowest conduction band. The energy of this transition is near the observed E2 peak and the dipole matrix elements for these transitions are large. However, the phase space associated with this critical point is too small to yield such a prominent structure. The observation of a well-defined interband reduced mass for the E2 structure from electroreflectance (Aspnes, 1973) suggests a relatively well-localized critical-point origin for this spectral feature. Ruling out transitions near X, the critical point must reside elsewhere within the Brillouin zone. Empirical pseudopotential calculations reinforce this interpretation; these calculations (Chelikowsky and Cohen, 1973) yield a well-defined critical point near the special k point (Chadi and Cohen, 1973). The density of states for germanium is illustrated in Figure 17 and compared to photoemission and inverse photoemission. Photoemission methods such as X-ray photoelectron spectroscopy (XPS) and ultraviolet photoemission spectroscopy (UPS) measure only properties of occupied states. Inverse photoemission spectroscopy allows one to detect the properties of empty states. Electrons with a well-defined energy are directed at a sample. These electrons decay to the lowest unoccupied states,
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
17
2 Ge Theory Experiment
1 ΔR (eV–1) R ΔE
1
0
–1
–2 0
1
2
3
4
5
6
7
Energy (eV) Figure 16 Theoretical and experimental modulated reflectivity for crystalline germanium. The experimental data is from Zucca and Shen (1970).
with some of the decay transitions being radiative. Photons emitted in this process are detected and a photon counts versus incident electron energy spectrum is measured. Like photoemission, the inverse photoemission can be related to the empty state density of states, ignoring any matrix element effects. One can obtain a direct measure of the ‘quasi-particle’ gap, that is, the energy required to obtain a noninteracting electron–hole pair, by a combination of photoemission and inverse photoemission spectra.
Gallium arsenide We can approach the electronic structure of gallium aresenide from the perspective of perturbing the germanium potential. Intuitively, we expect
1.01.4.4.2(ii)
VS ðGÞ ¼ ½VGa ðGÞ þ VAs ðGÞ=2 VGe ðGÞ
ð48Þ
That is, the symmetric part of the potential should resemble the form factors of germanium. The antisymmetric form factor, VA ðGÞ ¼ ½VGa ðGÞ – VAs ðGÞ=2
ð49Þ
which is a measure of the difference between the gallium and arsenic potentials, should contain the essential information which distinguishes germanium from gallium arsenide. Formally, it is the deviation of VA(G) from zero that distinguishes zinc–blende from diamond semiconductors. In this sense, we expect a smooth transition from a purely covalent semiconductor VA(G) ¼ 0, to a highly ionic semiconductor where VA(G) deviates notably from zero.
We expect that many properties of semiconductors, for example, optical gaps, cohesive energies dielectric constants, bond charges, etc., might be determined simply with an ionicity factor based on VA(G) or a similar measure. Actually, this is the case, and although VA(G) is not explicitly used, a more sophisticated measure, based on similar concepts, of ionic versus covalent contributions to crystal binding in diamond and zinc–blende semiconductors has been devised by Phillips (1973). In Figure 18, we illustrate the calculated band structure for gallium arsenide using the EPM. The nonlocal pseudopotential EPM approach has yielded one of the most accurate band structures for GaAs to date. In fact, as a result of the extensive experimental studies in this compound, this band structure may be the most accurate band structure over a large energy range available for any material. By compiling the most recent experimental data, we estimate an error of about 0.1 eV over an energy span of nearly 20 eV. Thus, we have an accuracy of roughly 1% or better; this is almost an order of magnitude better than other band structures. As we introduce antisymmetric form factors for the gallium arsenide crystal potential, several significant alterations from a germanium-like band structure may be observed. First, the bottom valence band, which now has atomic s-like character and is localized on the anion, splits off from the the rest of the valence band to form the antisymmetric gap. This gap grows with increasing ionicity of the crystal or charge transfer between the cation and anion.
18 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
XPS
BIS
n-Ge(111) 2 × 1
L(6,7) L(10,11) L(2)
L(3,4)
L(1) X(1,2) Γ(1)
Γ(12–14)
X(5,6)
X(13,14)
L(8) X(3,4) L(5)
Γ(6–8)
L Λ Γ Δ X W K Σ Γ
–10
–5 0 5 10 Energy relative to EV (eV)
15
20
Figure 17 Experimental X-ray photoemission spectroscopy (XPS) and Bremsstrahlung isochromat spectroscopy (BIS) or inverse photoemission spectra for crystalline germanium (Chelikowsky et al., 1989). The critical point features of the theoretical density of states are also identified with band structure features.
Another smaller gap arises from the antisymmetric part of the potential and is located between the first and second conduction bands along the -direction. This smaller gap is important for transport properties of zinc–blende semiconductors and has been observed to have subtle effects on the reflectivity spectrum of the zinc–blendes. We also note that the band gaps in gallium arsenide tend to be larger than in germanium.
In general, as a semiconductor becomes more ionic, the valence bandwidths narrow and the optical gap grows in size. Part of this effect arises from dehybridization accompanying the change from covalent to ionic bonding. Perhaps the most important conclusion from our experience with GaAs is not simply that the pseudopotential approach can yield accurate results. Consider the parametrized nature of the potential. One might naively expect that the band structure could be adjusted at will with parametric nature of the form factors. This is not the case for two reasons. First, the form factors are not linearly independent, for example, the form factors V(8) and V(11) characterize the size of the ion core and tend to move the bands in similar fashion. In a fitting sense, we have only two or three parameters to fix all the energy bands. Second, the form factors are chosen to be close to those expected from model potentials and are not arbitrary. In GaAs, this issue is important. The band ordering of the conduction band minima was first believed to be an issue with the EPM. In order to explain the Gunn effect (Gunn, 1964), it was thought that the correct ordering is c6 Xc6 Lc6. (The Gunn effect occurs in III–V and results in negative differential resistance.) If it were possible to obtain this ordering by a simple adjustments of the form factors, the bands would have been fit to give ordering suggested by the Gunn effect. In fact, pseudopotential band calculations place the Lc6 band below Xc6, resulting in a c6 Lc6 Xc6. Since experiment is the arbiter of such issues, the pseudopotential method was deemed to be in error. However, in the late 1970s careful experiments showed the ordering employed to explain that the Gunn effect was not correct. In Figure 19, we illustrate the interpretation by Aspnes (1976). Using electroreflectance for core to conduction band transitions, he was able to show the ordering from the EPM was correct. In Figures 20 and 21, we illustrate the computed optical response functions, that is, real and imaginary parts of the dielectric function, for GaAs. We also compare these functions to experiment (Philipp and Ehrenreich, 1963). The information content of the response functions is considerable. It is for this reason that Phillips (1973) based his bonding theories of semiconductors on spectroscopic data rather than following the approach of Pauling (1960), who used thermochemical data to analyze molecular bonding trends. The general shape of the real part of the dielectric function is that expected for a harmonic
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
L4, 5
6
Γ8
Γ8
L6
Γ7
4
19
Γ7 X7
2 L6
GaAs
L4, 5
Γ7
Γ7
L6
–2
Γ6 Γ8
Γ8
0
Energy (eV)
X6
Γ6
X7 X6
–4
–6 L6
X6
L6
X6
–8
–10
Γ6
Γ6
–12 L
Λ
Γ
X Δ Wavevector, k
K
Σ
Γ
Figure 18 Energy bands for crystalline gallium arsenide. The top of the valence band is taken to be the zero energy reference.
oscillator with a resonant frequency at about 5 eV. We can think of this resonant frequency as a fundamental property of GaAs that represents the average bonding–antibonding energy level separation. Phillips divides up this average bonding–antibonding gap into a part that is ionic and a part that is covalent. One can extract from the dielectric function relevant parameters for formulating ionicity scales (Phillips, 1973). Such scales are useful for making predictions of a variety of semiconductor properties. With a knowledge of the real and imaginary parts of the dielectric function, we can derive the reflectivity spectrum and reflectivity derivative spectrum. While the structure of the calculated imaginary part of the dielectric function is similar, the magnitude in the 2–4 eV region is lower by almost a factor of 2. This discrepancy was initially attributed to a failure of the Ehrenreich–Cohen dielectric function (Ehrenreich and Cohen, 1959) to include
electron–hole ‘excitonic’ interactions. Recent work based on a many-body formulation of the Bethe– Salpeter equation confirms this interpretation as indicted in Figure 22. The Bethe–Salpeter equation strongly enhances the magnitude of the imaginary part of the dielectric function, but does not shift the critical points. The EPM-calculated reflectivity is shown in Figure 23. As for the silicon and germanium semiconductors, we may divide up the zinc–blende reflectivity structure into five distinct regions. For GaAs, the E0 region extends from 1 to 2 eV, the E1 region 2-4 eV, the E90 region 4-5 eV, the E2 region 56 eV, and the E92 region 6-7 eV. The lowest energy region E0 is dominated by structure originating from the fundamental gap at . Spin–orbit interactions split the upper valence bands of gallium arsenide by about 0.3 eV; for heavier metal constituents, the splittings can be quite large, for example, 1 eV or more.
20 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
L
Lc6
E vs. k (Lowest C.B.)
Λ
k
Γ
Γc6
Δ X
Xc6
ER spectrum (Ga 3d Core)
2
0
104 ΔR/R
–2
–4
L
X
L
X
–6 20
21
22
Energy (eV) Figure 19 Fine structure in the electroreflectance spectrum for GaAs (Aspnes, 1976). Assignment to the conduction band minima is indicated.
30 Ga As Theory Experiment
25
30
20 15
20
10
ε2 (E )
ε1 (E )
GaAs Theory Experiment
25
5 0
15 10
–5 5
–10 –15 0
1
2
3 4 5 Energy (eV)
6
7
Figure 20 Real part of the dielectric function for gallium arsenide compared to experiment (Philipp and Ehrenreich, 1963).
8
0
0
1
2
3 4 5 Energy (eV)
6
7
Figure 21 Imaginary part of the dielectric function for gallium arsenide compared to experiment (Philipp and Ehrenreich, 1963).
8
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
30 GaAs
⑀2
20
10
0 0
2
4 6 Energy (eV)
8
10
Figure 22 Calculated optical absorption spectrum of GaAs with (solid lines) and without (dashed lines) electron– hole interaction from Rohlfing and Louie (1998).
0.7 GaAs Theory Experiment
0.6
R (E )
0.5 0.4 0.3 0.2 0.1 0 0
1
2
3 4 5 Energy (eV)
6
7
8
Figure 23 Reflectivity spectrum for gallium arsenide. The experiment is from Philipp and Ehrenreich (1963).
Thus, the E0 structure actually has a doublet character. The E1 reflectivity peak originates from transitions near the zone boundary at L. Again, spin–orbit interactions can become important in this region by splitting the upper valence bands. We note that older, local pseudopotential calculations gave a different critical point symmetry for this region. The local potential results put an M0 critical point at L and an M1 critical point along the direction. The nonlocal results move the M1 critical point to the zone boundary and eliminate the M0 point altogether. This latter configuration is in better accord with experiment. The E90 structure has been somewhat controversial. It can arise from either of two regions: near the
21
zone center or along the direction. If it were to occur at the zone center, we would expect spin–orbit splitting to be important in both the valence and conduction bands. On this basis, recent work by Aspnes and Studna (1973) has given support to the zone-center assignment. The traditional argument against this assignment has been the small phase space associated with the point; however, excitonic effects may very well enhance its importance. The E2 peak, which dominates the absorption spectrum, represents the average bonding–antibonding transitions. This peak arises from transitions from the uppermost valence bands to the lowest conduction bands near the points (2/a)(3/4, 1/4, 1/4) in the Brillouin zone. This assignment is similarly found in the diamond structure semiconductors. The E2 peak dominates the spectrum because of the large phase space which contributes to interband transitions in this region and because of large interband dipole matrix elements. At energies above the E2 peak, reflectivity structure from transitions along the direction. The E91 structure arises from transitions from the top valence band to the second lowest conduction band at or near the zone boundary at L. By examining the energy gradients and dipole matrix elements throughout the Brillouin zone, it is possible to determine the origin of structure of the imaginary part of the dielectric function. In such a manner, we analyzed the contribution to the E90 reflectivity structure. This structure is complex and has been somewhat controversial. In Figure 24, we label this structure as A, B, and C. Rehn and Kyser (1972), using electroreflectance, observed only a symmetry for this structure and attributed the structure to the pseudocrossing of the 5 conduction bands. However, Aspnes and Studna (1973) have pointed out that this interpretation conflicts with band structure calculations where some symmetry structure is predicted. They further proposed that the symmetry structure arises from a pair of M1 critical points approximately one-tenth way from to X. EPM calculations agree with their interpretation (Cohen and Chelikowsky, 1989). The relevant transitions are indicated in Figure 25. Aspnes and Studna also noted the possibility of the pseudocrossing producing some very weak structure at 4.4 eV. The 4.4 eV (A) transition corresponds to this type of transition. It should be noted, however, that there also exists a companion M0 critical point owing to the spin–orbit splitting of the 5 valence band. Since this companion critical point occurs
22 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
2
GaAs
1 dR (eV–1) R dE
Theoretical Experimental
A 0
C B
–2 0
1
2
3
4
5
6
Energy (eV) Figure 24 Reflectivity derivative spectrum of gallium arsenide. See the text for a discussion of the structures: A, B, and C. Experiment is from Zucca and Shen (1970).
about 0.1 eV higher in energy, it is nearly degenerate with the E90 structure from and at 4.5 eV. In the calculated derivative spectrum, this structure is masked by the stronger M1 critical points, and this may be the case in the electroreflectance measurements. In Figure 26, we illustrate the calculated valence band density of states for GaAs, and the results of XPS measurements and inverse photoemission (Chelikowsky et al., 1989). As for the diamond structure semiconductors, we may divide the density of states into three general regions. The first region is the most tightly bound energy band. Electron states corresponding to this band are strongly localized on the anion and are descendants of the atomic As 4s states. The next region of note is a peak arising from the onset of the second valence band. This band shows almost no energy variation along the X–U symmetry direction; in fact, it is very flat over the entire square face of the Brillouin zone. This energy band configuration results in a sharp onset of states above the antisymmetric gap. The character of states associated with the second valence band changes from predominantly cation s-like states at the band edge to predominantly anion p-like states at the band maximum. The third region of interest in the
density of states extends from the onset of the third valence band (at about 4 eV below the valence band maximum) to the valence band maximum. This region encompasses the top two valence bands and is predominantly p-like and is associated with anion states.
Zinc selenide The prototypical II– VI semiconductor is zinc selenide. However, unlike the prototypical III–V semiconductor (GaAs) or our prototypical diamond semiconductor (Si or Ge), we do not have as detailed a picture for this class of semiconductors. There are several reasons for this situation. First, extensive experimental information is lacking for ZnSe as compared to GaAs. The larger bandgap in ZnSe requires higher photon energies than GaAs for reflectivity measurements. Photon sources at these higher energies have not been routinely available. Also, some of the powerful optical techniques such as Schottky barrier electroreflectance cannot be easily applied to ZnSe owing to fabrication problems. Second, the Zn 3d-level resides close to the valence band. Traditionally, empirical pseudopotential band calculations do not explicitly
1.01.4.4.2(iii)
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
23
6 Λ6 Δ5 Δ5
Λ4,5 Γ8
GaAs Λ6
Γ7
4
Δ5
E0′ (Γ)
E0′ (Δ)
Energy (eV)
Δ5
Λ6 2
Γ6
Γ8
0 Λ4,5
Δ5 Γ7
Λ6 Λ6 (0.3,0.3,0.3)
Λ
Δ5 Δ5
Γ
Δ
(0.5,0.0,0.0)
Figure 25 Band structure of gallium arsenide near showing the critical point locations for the E90 structure. The indicated transitions give rise to the structure labeled A, B, and C in Figure 24.
include this level in ZnSe. The 3d cation level in gallium arsenide is significantly below the valence band and in Ge the 3d level is more than 30 eV below the top of the valence band. In Figure 27, we illustrate the band structure of ZnSe. Most features of this band structure can be accounted for by extrapolating from Ge to GaAs. Compared to Ge and GaAs, the bandgaps in ZnSe, that is, the optical gap and the antisymmetric gap, have increased considerably in size. Moreover, the valence bands have narrowed and show less dispersion.
In Figure 28, we display the calculated and measured reflectivity spectrum for ZnSe (Freeouf, 1973; Walter et al., 1970). Some work using derivative spectroscopy is available (Theis, 1977). Owing to the increased ionic component of the bonding in ZnSe, the reflectivity spectrum is shifted to higher energy as compared to Ge or GaAs. It also exhibits much more structure than Ge or GaAs because of the larger separation between bands. For the most part, the reflectivity spectrum can be analyzed by analogy with GaAs. The E0 reflectivity
24 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
XPS
BIS
p-GaAs(110) 1 × 1
X(11,12)
X(2) L(9) L(3,4)
L(8)
X(1) X(15) Γ(10,11) X(7,8) L(6,7)
Γ(1)
X(3,4) Γ(5)
Γ(6–8)
L(1) X(6)
L Λ Γ Δ
X
W K Σ Γ
–10
–5
0
5
10
15
20
Energy relative to Ev (eV) Figure 26 Experimental X-ray photoemission spectroscopy (XPS) and Bremsstrahlung isochromat spectroscopy (BIS) or inverse photoemission spectra for crystalline gallium arsenide (Chelikowsky et al., 1989). The critical point features of the theoretical density of states are also identified with band structure features.
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
L4, 5
8
L6
25
Γ8
Γ8 Γ7
Γ7
6 X7 4
Γ6
2
Zn Se
Γ8
0
Energy (eV)
X6
L6
L4, 5 L6
Γ6 Γ8
Γ7
Γ7 X7
–2
X6 –4 L6
X6
–6
–8
–10 L6
X6
Γ6
–12 L
Γ
X
Γ6 k
Γ
Wavevector, k Figure 27 Energy bands for crystalline zinc selenide. The top of the valence band is taken to be the zero energy reference.
structure lies between 2 and 4 eV and corresponds to the minimum energy gap at . The E1 structure at 4–5 eV arises from transitions near to or at the L point. The structure corresponding to the E2 peak arises from a localized region near the special point (2/a)(3/4, 1/4, 1/4). The E90 structure at 7–8 eV may be an exception having no analogy in GaAs. It lies above the E2 structure; thus, the origin of structure is reversed in ZnSe as compared to GaAs. However, the origin of the E90 structure appears to be the same, that from near and along the direction. The highest energy reflectivity structure, E91, occurs at 8–9 eV and corresponds to transitions near the L point. With the exception of the line shape near the E2 peak, the pseudopotential results are consistent with experiment. It is possible that the line-shape discrepancy is an artifact of the sampling scheme used. We note that at energies above 10 eV or so, the structure
in the reflectivity might arise from transitions involving the Zn 3d-states. It should not be surprising that the density of states of ZnSe may also be interpreted in terms of an extrapolation from the Ge and GaAs band structures. In Figure 29, we compare the calculated density of states for ZnSe with photoemission measurements and inverse photoemission (Chelikowsky et al., 1989). Some of the early pseudopotential band structures, which had potentials based on optical data alone, were in mediocre agreement with the photoemission results. Specifically, local pseudopotentials chosen to reproduce optical gaps tend to overestimate the ionicity of the II–VI semiconductors. As a consequence, the valence bands become quite narrow compared to experiment. Nonlocal pseudopotentials do not suffer from this malady; they can be used to fit optical gaps with no corresponding increase in ionicity.
26 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
0.6 ZnSe
Experiment 1 Experiment 2 Theory
0.5
0.4
R 0.3
0.2
0.1
0 0
1
2
3
4
5 6 Energy (eV)
7
8
9
10
Figure 28 Reflectivity spectrum for zinc selenide. Two experimental spectra are presented: the dashed line is from Walter et al. (1970) and the dotted line from Freeouf (1973).
1.01.5 The Ab Initio Pseudopotential Method A deficiency of the empirical pseudopotential method is that the potential is ‘biased’ by the experimental data to which it was fit. For example, suppose we fit the optical properties of GaSb and extract a Ga potential, we must be careful in trying to use this fit Ga potential in another crystal. This potential might work fine in GaAs, which has a similar bonding configuration when compared to GaSb, but the potential may fail to describe GaN, which is considerably more ionic than GaSb. A related problem is that the form factors are fit only to a subset of reciprocal lattice vectors. If the crystal volume or structure changes, one must extrapolate to a new subset. Another problem can occur at a surface. In crystalline GaAs, each Ga atom is surrounded by four As atoms. For the cleavage plane the (110) surface, the Ga atom is bonded to three As atoms. Given the coordination change, the surface Ga atom cannot be expected to retain a ‘bulk-like’ screening potential. There is no reason to be confident that the Ga pseudopotential extracted from the crystalline environment will be very accurate at the surface. Of course, this problem is made worse if one wants to examine a liquid containing Ga atoms where the coordination may continuously change. We can consider the form factors to be composed of two interactions: the ion core–valence electron
interaction and the valence electron–valence electron interaction. A fundamental postulate of the pseudopotential method is that the ion-core pseudopotential is not dependent on the chemical environment. We assume that this part of the potential can be transferred with no loss in accuracy. The key problem is to determine the total pseudopotential by determining the potentials from the valence electrons interacting among themselves. A common procedure is to construct a selfconsistent potential. The wave functions obtained in an electronic structure calculation can be used to construct a new screening potential, which in turn can be used with the ion-core potential to compute a new total potential and new wave functions. When no changes occur in a feedback loop of this kind, the solution is considered to be selfconsistent. Let us assume a one-electron Hamiltonian, which can be based on density functional theory (Hohenberg and Kohn, 1964; Kohn and Sham, 1965):
– h2 r2 p þ Vion ðrÞ þ VH ðrÞ þ Vxc ðrÞ n ðrÞ ¼ En n ðrÞ 2m ð50Þ
The ion-core pseudopotential, Vpion, can be taken as a linear superposition of ion-core atomic potentials. Determining the ionic potential can be accomplished by resorting to atomic structure
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
Zn 3d
XPS
27
Assume initial density: ρ
BIS
÷8
Δ
Solve:
2V H
= –4 πeρ
p
Form: VT = Vion +VH +Vxc n-Zn Se (100) c 2 × 2
Solve:
L (3,4) X(3,4)
2
2m
2
p
Vion +VH +Vxc Ψn = EnΨn
Γ(15)
Γ(6–8)
X(2)
–
Δ
X(1)
L (8)
X(10)
X(6)
Form: ρ = e
Ψn
2
n, occup Γ(1) L(2)
Figure 30 Self-consistent field loop. The loop is repeated until the input and output charge densities are equal to within some specified tolerance.
Γ(10,11)
L(5)
Γ(9)
Γ(5)
L (13,14)
L Λ Γ Δ X W K Σ Γ
–10
–5
0
5
10
15
20
Energy relative to EV (eV)
Figure 29 Experimental X-ray photoemission spectroscopy (XPS) and Bremsstrahlung isochromat spectroscopy (BIS) or inverse photoemission spectra for crystalline zinc selenide (Chelikowsky et al., 1989). The critical point features of the theoretical density of states are also identified with band structure features.
calculations, as discussed in the following section. The potential arising from the valence-electron interactions can be divided into two parts. One part represents the classical electrostatic terms, the Hartree or Coulomb potential: r2 VH ðrÞ ¼ – 4eðrÞ P ðrÞ ¼ e jn ðrÞj2 n;occup
ð51Þ
where is the valence electron charge density. The second part of the screening potential, the exchange-correlation part of the potential, Vxc, is quantum mechanical in nature. A common approximation for this part of the potential arises from the local density approximation, that is, the potential depends only on the charge density at the point of interest, Vxc(r) ¼ Vxc[(r)]. In principle, density functional theory is exact, provided one can obtain an exact functional for Vxc. This is an outstanding problem. It is commonly assumed that the functional extracted for a homogeneous electron gas (Ceperley and Alder, 1980) is universal and can be applied to the inhomogeneous gas problem. The procedure for generating a self-consistent field (SCF) potential is given in Figure 30. The SCF cycle is initiated with a potential constructed by a super-position of atomic densities. These densities are then used to solve a Poisson equation for the Hartree potential, and a density functional is used to obtain the exchange-correlation potential. A screening potential composed of the Hartree and exchange-correlation potentials is then added to the fixed ion-core pseudopotential, after which the one-electron Schro¨dinger equation, or Kohn–Sham equation, is solved. The resulting wave functions from this solution are then employed to construct a new potential and the cycle is repeated. In practice, the output and input potentials are mixed using a scheme that accounts for the history of the previous iterations (Broyden, 1965; Chelikowsky and Cohen, 1992).
28 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
1.01.5.1 Constructing Pseudopotentials from Density Functional Theory The construction of ion-core pseudopotentials has become an active area of electronic structure theory. Methods for constructing such potentials have centered, on ab initio or first-principles pseudopotentials; that is, the informational base on which these potentials are based does not involve any experimental input. The first step in the construction process is to consider an electronic structure calculation for a free atom. For example, in the case of a silicon atom, the Kohn–Sham equation (Kohn and Sham, 1965) can be solved for the eigenvalues and wave functions. Knowing the valence wave functions, that is, 3s2 and 3p2, and corresponding eigenvalues, the pseudowave functions can be constructed. This is an easy numerical calculation as the atomic densities are assumed to possess spherical symmetry and the problem reduces to a one-dimensional radial integration. Once we know the solution for an all-electron potential, we can invert the Kohn–Sham equation and find the total pseudopotential. We can unscreen the total potential and extract the ion-core pseudopotential. This ion-core potential, which arises from tightly bound core electrons and the nuclear charge, is not expected to change from one environment to another. It should be transferable from the atom to a molecular state or to a solid state or liquid state. The issue of this transferability is one which must be addressed according to the system of interest. The immediate issue here is how to define pseudo-wavefunctions which can be used to define the corresponding pseudopotential. Suppose we insist that the pseudo-wave-function be identical to the all-electron wave function outside of the core region. For example, let us consider the 3s state for a silicon atom. We want the pseudo-wavefunction to be identical to the all-electron state outside the core region: p
3s ðr Þ ¼
p3s
3s ðr Þ
r > rc
ð52Þ
where is a pseudo-wave-function and rc defines the core size. This assignment will guarantee that the pseudowave-function will possess properties identical to the all-electron wave function, 3s in the region away from the ion core. For r < rc, we alter the all-electron wave function. We are free to do this as we do not expect the valence wave function within the core region to affect the
chemical properties of the system. We choose to make the pseudo-wave-function smooth and nodeless in the core region. This will provide rapid convergence with simple basis functions. One other criterion is mandated. Namely, the integral of the pseudocharge density within the core should be equal to the integral of the all-electron charge density. Without this condition, the pseudo-wavefunction differs by a scaling factor from the all-electron wave function. Pseudopotentials constructed with this constraint are called norm conserving (Hamann et al., 1979). Since we expect the bonding in a solid to be highly dependent on the tails of the valence wave functions, it is imperative that the normalized pseudo-wave-function be identical to the all-electron wave functions. There are many ways of constructing norm conserving pseudopotentials as within the core the pseudo-wave-function is not unique. One of the most straight-forward construction procedures is from Kerker (1980) and later extended by Troullier and Martins (1991): ( p l ðr Þ
¼
r l expðpðr ÞÞ l ðr Þ
r rc
ð53Þ
r > rc
p(r) is taken to be a polynomial of the form: pðr Þ ¼ c0 þ
6 X
c2n r 2n
ð54Þ
n¼1
This form assures us that the pseudo-wave-function is nodeless and by taking even powers there is no cusp associated with the pseudo-wave-function. The parameters, c2n, are fixed by the following criteria: (1) The all-electron and pseudo-wavefunctions have the same valence eigenvalue. (2) The pseudo-wave-function is nodeless and is identical to the all-electron wave function for r > rc. (3) The pseudo-wave-function must be continuous as well as the first four derivatives of the wave function at rc. (4) The pseudopotential has zero curvature at the origin. This construction is easy to implement and extend to include other constraints. An example of an atomic pseudo-wavefunction for Si is given in Figure 31 where it is compared to an all-electron wave function. Unlike the 3s all-electron wave function, the pseudo-wavefunction has no nodes. The pseudo-wave-function is much easier to express as a Fourier transform or a combination of Gaussian orbitals than the allelectron wave function.
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
ZR d 2 ln – 2 ðr Þ2 ¼ 4 2 r 2 dr ¼ QðRÞ dE dr R
1.0 3s Radial wavefunction of Si
29
ð56Þ
0
0.5 Pseudoatom All-electron 0
–0.5 0
1
2 3 Radial distance (a.u.)
4
5
Figure 31 An all-electron and a pseudo-wave function for the silicon 3s radial wave function.
Once the pseudo-wave-function is constructed, then the Kohn–Sham equation can be inverted to arrive at the ion-core pseudopotential: p
p
Vion;l ðr Þ ¼
h2 r2 l p – En;l – VH ðr Þ – Vxc ½ðr Þ 2m l
ð55Þ
The ion-core pseudopotential is well behaved as p, has no nodes and does not vanish. The ion-core potential appears to be both state dependent and energy dependent. The energy dependence is usually weak. For example, the 4s state in silicon computed by the pseudopotential constructed from the 3s state is usually accurate. Physically, this happens because the 4s state is extended and experiences the potential in a region where the ion-core, potential has assumed a simple Zve2/r behavior. However, the state dependence through l is an issue; the difference between a potential generated via a 3s state and a 3p can be an issue. In particular, for first row elements such as C or O, the nonlocality is quite large as there are no p states within the core region. For the first row transition elements for such as Fe or Cu, this is also an issue as again there are no d-states within the core. This state dependence can be addressed in similar fashion as for a nonlocal empirical pseudopotential. An additional advantage of the norm conserving potential concerns the logarithmic derivative of the pseudo-wave-function (Bachelet et al., 1982). An identity exists:
The energy derivative of the logarithmic derivative of the pseudo-wave-function is fixed by the amount of charge within a radius, R. The radial derivative of the wave function is related to the scattering phase shift from elementary quantum mechanics. For a norm-conserving pseudopotential, the scattering phase shift at R ¼ rc and at the eigenvalue of interest is identical to the all-electron case as Qall elect(rc) ¼ Qpseudo(rc). The scattering properties of the pseudopotential and the all-electron potential have the same energy variation to first order when transferred to other systems. There is some flexibility in constructing pseudopotentials. The nonuniqueness of the pseudowave-function was recognized early in its inception. For example, within the Phillips–Kleinman formulation, one can always add a function, f, to the pseudo-wave-functions without altering the pseudopotential provided f is orthogonal to the core states. Consider the matrix element in Equation (7). If one pv to pv þf, changes p p then c jv þ f ¼ c jv þ hc jf i ¼ c jpv . Nothing is changed in the Phillips–Kleinman pseudopotential by this addition. The nonuniqueness of the pseudopotential can be exploited to optimize the convergence of the pseudopotentials for the basis of interest. Much effort has been made to construct soft pseudopotentials. By soft, one means a rapidly convergent calculation using plane waves as a basis. Typically, soft potentials are characterized by a large core size, that is, a larger value for rc. However, as the core becomes larger, the goodness of the pseudo-wave-function can be compromised as the transferability of the pseudopotential becomes more limited. In Figure 32, the ion-core pseudopotential for carbon is plotted in real space and in reciprocal space. Although the potentials look quite different in the core region, they all give reliable electronic structure properties for carbon. As for the nonlocal terms that occur in empirical potentials (Section 1.01.4.3), we can handle nonlocality using a plane wave basis in a straightforward fashion. If we consider a reference potential such that Vl ¼ Vlocal Vl, then we need to determine matrix elements as in Equation (46) where Vpc,l is replaced by Vl. The reference potential can be chosen to be a local potential. In general, the nonlocal elements are very short ranged in real space as the
30 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
(a)
–5
1
2
(c) 5 1
–5
2
r (a0 )
(d) 5 1
–5
2
r (a0 )
–5
–15
–15
–25
–25
–25
–25
–35
–35
–35
–35
8
8
8
8
0
0
12
6 q(a0–1)
0
12
6
12
6
–8
12
6
q(a0–1)
q(a0–1)
–8
2 r (a0 )
–15
0
1
r (a0 )
–15
q 2Vl (q) (Ry/a02)
Vl (r) (Ry)
(b) 5
5
q(a0–1)
–8
–8
Figure 32 Ion-core pseudopotentials for carbon generated by four different methods in real and reciprocal space; (a) Troullier and Martins (1991), (b) Kerker (1980), (c) Hamann et al. (1979), and (d) Vanderbilt (1985). The dotted and solid lines correspond to the s and p pseudopotentials, respectively.
size of the core is determined by rc. As such, the upper limit of the real space integral is slightly more than rc in practice. A major strength of ab initio pseudopotentials is that they can be employed for systems with many atoms of low symmetry, for example, clusters, liquids, and surfaces, without having to fit any form factors. However, a plane wave basis will require a large cutoff for such systems. In this case, the matrix elements in Equation (46) can be difficult to evaluate as elements depend on both k + G and k + G9. Kleinman and Bylander (1982) suggested an alternate form for the required nonlocal matrix elements: P VlKB ðk þ G; k þ G9Þ ¼ Z
m
Ylm ðkd þ GÞYlm ðkd þ G9Þ
4a
Z
p
p l ðr ÞVl ðr Þjl jk þ G9jr Þr 2 dr Z p l ðr ÞVl ðr Þjl jk þ Gjr Þr 2 dr
p
l ðr ÞVl ðr Þl ðr Þdr
ð57Þ
where pl is the atomic reference pseudo-wave-function and Ylm is a spherical harmonic with the angles determined by the unit vector: k d þ G. This matrix
element form has a great advantage as the integrals are separable in that they are solely a function of k + G or a function of k + G9. The individual integrals can be stored and retrieved as required in setting up the matrix. It is also possible to solve the Kohn–Sham problem directly in real space (Chelikowsky et al., 1994). In this case, the Kleinman–Bylander form can be cast as P p VlkB ðx;y;zÞ p ðx;y;zÞ¼ Glm ulm ðx;y;zÞVl ðx;y;zÞ Z Glm ¼Z
lm
p
ulm Vl p dxdydz p
ð58Þ
p
ulm Vl ulm dxdydz
where uplm are the reference atomic pseudo-wavefunctions. The nonlocal nature of the pseudopotential is apparent from the definition of Glm; the value of these coefficients are dependent on the pseudowave-function, p, acted on by the operator Vl. This is very similar in spirit to the pseudopotential defined by Phillips and Kleinman (1959). While we have focused on a simple plane wave basis, there are other bases that can be employed, for example, one
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
can combine pseudopotentials with Gaussians (Chan et al., 1986) or a uniform grid (Chelikowsky et al., 1994). These real space methods can be easily implemented with real space ab initio pseudopotentials (Kronik et al., 2006). While our focus has been on norm conserving pseudopotentials, it is possible to generalize the pseudopotential method to include systems that are not based on such potentials. It is possible to make the pseudopotentials even weaker by relaxing this condition. Vanderbilt (1990) proposed such a method for constructing ultrasoft pseudopotentials, which are constructed as a generalized eigenvalue problem. The norm conservation constraint is relaxed, and the charge density within the ion core is not explicitly considered as part of the pseudo-wave-function. The relaxation of the norm conservation constraint allows one to consider a much larger core radius and a much softer potential. One issue which is relevant for pseudopotential constructions, regardless of whether the potential is intended for use with a plane wave basis or not, concerns the issue of unbound, or weakly bound, atomic states. If an atom does not bind a state of interest, then the atomic wave function corresponding to this state is clearly not normalizable. Nonetheless, the pseudopotential corresponding to this state might be of some interest, for example, in a crystal such diverging wave functions are captured by the potentials of neighboring atoms. For example, Ba has a strong f-component resonance. However, these f-states are not bound for neutral Ba atom. In order to bind such states, one must consider highly ionized atomic states, which result in very strong pseudopotentials. Sometimes these potentials are so strong as to be useless for a plane wave basis, or so far removed from the chemical environment of interest that their transferability may be suspect. Hamann (1989) has suggested a method for handling such cases by integrating out the Kohn–Sham equation to a large distance and at that point terminating the pseudo-wave-function. The corresponding terminated wave function is then used to generate a pseudopotential for the component of interest. 1.01.5.2 Structural Properties of Semiconductor Crystals 1.01.5.2.1 Total electronic energy from pseudopotential–density functional theory
Empirical pseudopotentials have been one of the most effective tools in understanding the optical and dielectric properties of semiconductor crystals (Cohen and
31
Chelikowsky, 1989); however, to understand structural properties, we need to employ a different approach. Since the form factors in the empirical pseudopotential have been fit to optical transitions, there is no reason to believe that they will be very accurate for structural properties such as the phase stability, the equilibrium bond length, the bulk modulus, and the phonon spectrum of a crystal. Ab initio pseudopotentials can be used for this purpose and, in this section, we illustrate their application to the phase stability of tetrahedral semiconductors such as silicon. To evaluate the total energy of a crystal can be a difficult task. The total energy of the system contains terms that individually diverge, for example, the repulsive Coulomb terms between the ion cores and the electron–electron interactions. These terms can be handled in Fourier or momentum space using a plane wave basis (Ihm et al., 1979). The total energy of the system can be written as ET ¼ Ekin þ Eec þ EH þ Exc þ Ecc
ð59Þ
ET is the total electronic energy, Ekin represents the electronic kinetic energy, the electron–ion-core interaction energy is given by Eec, the electrostatic Coulomb energy is given by the Hartree energy, EH, the nonclassical exchange-correlation energy is given by Exc, and the ion core–ion-core classical term is given by Ecc. The momentum-space expressions for the electronic energy terms are as follows: Ekin ¼
2 h 2 j k þ Gj 2 1 X p n ðk þ GÞ N n;k;G 2m
1 X p ðk þ GÞ pn ðk þ G9Þ N n;k;G n
Z 1 Ze 2 3 p d r Vc ðk þ G; k þ G9Þ þ G;G9 c r
ð60Þ
Eec ¼
EH ¼
ð61Þ
2 c X 4e 2 ðGÞ 2 2 G;G6¼0 G
ð62Þ
c X ðGÞ"xc ðGÞ 2 G
ð63Þ
Exc ¼
" ( 4 X 1 e2 X cosðG?ðtS – tS9 ÞÞ Zv;s Zv;s9 Ecc ¼ c G;G6¼0 G 2 2 S;S9
2 # –G – exp 4 2 c 2 ) ð64Þ X erfcð jR – tS þ tS9 jÞ 2
– pffiffiffi S;S9 þ jR – tS þ tS9 j R;R6¼0
32 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
Each term yields the energy per cell; N is the total number of cells, c is the cell volume, R is a lattice vector, G is a reciprocal lattice vector, Z is the total core charge, which is a sum of the individual core charges Zv, t is the basis vector, k is the crystal momentum, and pn is the wave function for state n. ðGÞVcp ðk þ G; k þ G9Þ, and pn ðk þ GÞ represent the Fourier components of the charge, ion-core potential and the pseudo-wave-function. The sums involving the pseudo-wave-functions run only over the occupied states, is a parameter that controls the convergence of the Ewald summation in real space versus momentum space (Ihm et al., 1979). In practice, it is easier to make use of the eigenvalue explicitly and subtract off the double counting terms, that is, one can write the total energy as ET ¼
1X En ðkÞ – EH þ Exc þ Ecc N n;k
ð65Þ
The sum is over all occupied states. The eigenvalue term contains the exchange-correlation term, whereas the total energy should contain the exchange-correlation energy. This term can be written as Exc ¼ c
X
ðGÞ½"xc ðGÞ – Vxc ðGÞ
ð66Þ
G
where "xc is related to the exchange correlation via a functional derivative: Vxc ¼
ð "xc ½Þ
ð67Þ
where, in the local density approximation, Exc ½ ¼
Z
ðrÞ"xc ½ðrÞd 3 r
ð68Þ
In the simplest formalism, "xc is extracted from a homogeneous electron gas (Kohn and Sham, 1965).
1.01.5.2.2
Phase stability of crystals The total energy, ET, of a crystal structure can be calculated using Equation (65) as a function of the lattice constant and any internal structural parameters. One of the first applications of this work was carried out by Ihm and Cohen (1980), and Yin and Cohen (1980, 1982c, 1982b). This work represents a seminal point in condensed matter physics (Chelikowsky, 2000). It demonstrated a workable scheme for examining the structural properties of solids and its extension led to molecular dynamics using quantum forces (Car and Parrinello, 1985).
The first application involved the silicon crystal with a number of different phases: face-centered cubic, body-centered cubic, hexagonal close-packed, hexagonal diamond, cubic diamond, white tin and simple cubic. For each structure, the energy was minimized for a given volume by optimizing the internal coordinates, for example, for a hexagonal structure the c/a ratio was optimized. Depending on the system studied the range of volumes considered is varied. In the case of silicon. Cohen, et al. considered a volume of one-half the ambient volume of the known diamond phase of silicon to an upper limit of about 20% larger than the ambient volume. For such a diverse set of crystal structures, it is important to make certain the convergence is similar to all phases. For structures which yield metallic systems, the k-point sampling is sometimes critical because the variations in the Fermi surface make it difficult to assure that occupied states are being sampled. Once the total energy is computed for a finite number of volumes, the energy versus volume points are fit to an equation of state such as the Birch (1952) equation or the Murnaghan (1944) equation. B0 V ET ðV Þ ¼ ET ðV0 Þ þ B90
! ðV0 =V ÞB90 B0 V 0 þ1 – ð69Þ B90 – 1 B90 – 1
where B0 and B90 are the bulk modulus and its pressure derivative at the equilibrium volume V0. Using Equation (69), the fit of the calculated points yields the equilibrium energy ET(V0), the equilibrium volume, and the bulk modulus and its pressure derivative at equilibrium. Representative calculated and measured values for the lattice constant and bulk modulus results are given for Si in Table 1. The results are impressive when one considers that this is an ab initio calculation requiring only the atomic number and crystal structure as input. Generally, the pseudpotential–density functional method yields lattice constants and bulk moduli to an accuracy of about 1% and 5%, respectively. The cohesive energy can be calculated by comparing the total energy at the equilibrium lattice constant with the energy of the isolated atoms. Spin polarization effects (von Barth and Hedin 1972; Gunnarsson et al., 1974) and zero point vibrational energy need to be considered. Early results computed within the local density approximation gave reasonable estimates of the cohesive energies; however, in general the generalized gradient approximation yields more accurate cohesive energies (Becke, 1992).
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
G ¼ U þ PV – TS
ð70Þ
for the phases in question are equal. U is the internal energy, P is the pressure, and S is the entropy. Although some attempts have been made to consider phase transitions at finite temperature (Sugino and
Car, 1995), most phase stability studies are done at zero temperature. In this cases, G ¼ U þ PV, where the internal energy is given by the total electronic energy and the pressure is given by P ¼ dET/dV. –7.82
Ge
–7.84
Estructure (Ry/atom)
Silicon and germanium phases The total energy versus volume curves for seven crystal phases of silicon are shown in Figures 33 and 34. It is encouraging that the lowest energy state corresponds to the diamond structure, as this structure is observed experimentally under ambient conditions. (As with any such calculation, there is no guarantee that there is an untried structure with a lower energy state.) Experimental data for the structural properties of silicon and germanium are listed in Table 3. For small volumes other structural phases are lower in energy than the diamond phase; hence, pressure-induced solid–solid structural phase transitions are predicted. Structural transitions occur between phases when the Gibbs free energy
1.01.5.2.2(i)
33
fcc 4
–7.86
hcp
bcc sc
β-Tin
3
–7.88
Hexagonal diamond
2 –7.83
Diamond
1
Si –7.90 0.6
0.7
1.0
1.1
Volume
–7.85
Energy (Ry/atom)
0.9
0.8
Figure 34 Total electronic energy for seven phases of crystalline silicon as function of the atomic volume. The volume has been normalized to the measured value of diamond. The dashed line is a common tangent of the energy curves for the diamond and -tin phases. The calculations are from Yin and Cohen (1982c). The total energy reference is the energy per pseudo-Ge atom. fcc, face-centered cubic; bcc, body-centered cubic; hcp, hexagonal close packing.
fcc bcc
–7.87
hcp sh
sc
β-Sn –7.89
Table 3 Comparison of calculated and measured static properties of silicon and germanium
Hexagonal diamond
Semiconductor
Lattice constant (A˚)
Cohesive energy (eV/ atom)
Bulk modulus (Mbar)
Si Calculated Measured
5.45 5.43
4.84 4.63
0.98 0.99
Ge Calculated Measured
5.66 5.65
4.26 3.85
0.73 0.77
Diamond –7.91 0.5
0.7
0.9
1.1
Volume
Figure 33 Total electronic energy for seven phases of crystalline silicon as a function of the atomic volume. The volume has been normalized to the measured value of diamond. The dashed line is a common tangent of the energy curves for the diamond and -tin phases. The calculations are from Yin and Cohen (1982c). The total energy reference is the energy per pseudo-Si atom. fcc, face-centered cubic; bcc, body-centered cubic; hcp, hexagonal close packing.
The calculated values are from Yin and Cohen (1982c). Measured values are from Donohue (1974), Brewer (1977) and McSkimin and Andreatch Jr. (1963).
34 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
For T ¼ 0, the pressure-induced phase transformation can be found by a common tangent line between the ET(V) curves of the two phases considered. The negative of the slope of the tangent line is the transition pressure. Examples of common tangent constructions are given in Figures 33 and 34 for the transitions from the diamond structure to the tin phase for Si and Ge. As hydrostatic pressure is applied the path indicated in the figures, 1 ! 2 ! 3 ! 4, illustrates the change from diamond at 1 to -tin at 4, with the transformation occurring along the 2 ! 3 segment of the path where both structures can coexist. The predicted phase transition pressure for the diamond ! -tin phase is 99 kbar for Si and 96 kbar for Ge; the measured values are 125 and 100 kbar, respectively (Yin and Cohen, 1982c). The successful prediction of high-pressure structural phases such as hexagonal forms of Si and Ge and face-centered cubic Si are one of the impressive results of the pseudopotential-density functional theory calculations (Chang and Cohen, 1984; Liu et al., 1988; Vohra et al., 1986). Moreover, these high-pressure phases are metallic and predicted to be superconductors (Chang et al., 1985). Two of the predictions of superconductivity have been verified while the others remain untested. The calculations for the group IV semiconductors serve as prototypes for the application of the totalenergy pseudopotential method to study structural properties of solids. III–V phases The first extensions to other solids were made for the III–V semiconductors: GaAs, AlAs, GaP, AlP, AlSb, GaSb, InP, InAs, and InSb (Froyen and Cohen, 1983a, 1983b; Zhang and Cohen, 1987). Total electronic energy calculations for these III–V semiconductors are displayed in Figures 35 and 36. Although these studies for III–V semiconductors generally give good results for the lattice constants and bulk moduli, the agreement between the calculated and measured properties of the high-pressure phases and for phase transition pressures and volumes is not always satisfactory. As an example of the latter, the calculated transition pressures for the Al compounds are consistently lower by around 50% than the measured values (Zhang and Cohen; 1987). In contrast, the calculated lattice constants are within 0.4, 0.3, and 0.3% of the measured values for AlP, AlAs, and AlSb, respectively (Zhang and Cohen, 1987). It is unclear why
1.01.5.2.2(ii)
the transition pressures of the Al compounds are underestimated.
1.01.5.2.3
Vibrational properties An important application of the ab initio pseudopotential approach is calculating the vibrational properties of solids. These calculations are based on the change in the energy of atom as it moves from an equilibrium position. Once we have established that the energy of atom can be accurately calculated as a function of position, phonon and lattice vibrational modes can be calculated. The input needed for such calculations is minimal. In contrast to phenomenologic force constant models, which often require as many as 15 parameters to achieve reasonable fits to phonon dispersion curves, the total energy approach uses only the masses and atomic numbers of the constituent atoms. To calculate the phonon dispersion curve, !(q), along a specific direction, the frozen-phonon approach is easy to implement (Chadi and Martin, 1976; Heine and Weaire, 1970; Wendel and Martin, 1979; Yin and Cohen, 1980). In this approach, a supercell is chosen, and the atomic cores are displaced to simulate a particular phonon mode. The change in the total energy arising from the distortion depends on the type of distortion assumed and the amplitude. Since a specific phonon wavevector, q, is chosen, this method is limited to calculating !(q) at specific points in the Brillouin zone. The phonon frequencies can also be evaluated by computing the Hellman–Feynman forces on the displaced atoms. Another approach that allows the full phonon dispersion to be computed is based on linear response theory (Baroni et al., 2001). However, the frozen-phonon approach is the simplest method for computing !(q) at symmetry points. Calculations at nonsymmetry points are possible but require larger unit cells. Standard calculations usually limit the computations to three or four q-points along a symmetry direction. As an example of a frozen phonon supercell calculation, we briefly examine the calculations of the phonon properties for the and X points of the Brillouin zone for Si and Ge. The phonon polarizations for these points can be easily determined using group theory as illustrated in Figure 37, and the primitive cell for the distorted lattice contains two atoms for the LTO() mode and four atoms for the phonon modes at X. The phonon-distorted lattice has inversion symmetry, which simplifies the calculation.
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
35
150 AIAs CsCI +
100
Energy - mRy
AIP
Rocksalt
β-Sn
Rocksalt β-Sn
50 NiAs
Zinc–blende
Zinc–blende
0
150 GaAs CsCI
100
Energy - mRy
GaP
Rocksalt
β-Sn
NiAs β-Sn
Rocksalt
50
Zinc–blende
Zinc–blende
0 0.6
0.7
0.8
0.9
1.0
V/V0
1.1 0.6
0.7
0.8
0.9
1.0
1.1
V/V0
Figure 35 Total electronic energy for some representative III–V structures as function of the atomic volume. The calculations are from Froyen and Cohen (1983a, 1983b).
For a given amplitude u, the change in the total energy arising from the distortion is ET(u). The force constant k can be obtained from a second derivative of the ET(u) or a first derivative of the force F(u):
k¼
@ 2 ET @u2
ffi u¼0
2ET ðuÞ u2
ð71Þ
or k¼
@F F ðuÞ ffi – @u u¼0 u
ð72Þ
The phonon frequency is given by ! ¼ 2f ¼
pffiffiffiffiffiffiffiffiffi k=M
ð73Þ
where M is the atomic mass. The calculation of the total energies or forces is done typically for five different amplitudes ranging from 0.01 to 0.1 A˚. The total energies can fit by a quadratic function to about 1%. For some phonon mode calculations such as the LTO() modes in Si and Ge, higherorder fits such as the third-order term are needed. These fits allow the evaluation of anharmonic terms. Table 4 illustrates some typical results. An impressive achievement of these calculations is the good agreement with experiment obtained for the TA(X) modes in Si and Ge. For empirical force constant model calculations, many-neighbor parameters are needed to obtain the correct values for these modes because of the important role played by long-ranged forces.
36 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
(b) 180
(a) 120
(c) 120 InP
GaSb
AISb 150
90
60 A5
30 Zinc–blende
120
Energy (mRy/molecule)
Rocksalt
Energy (mRy/molecule)
Energy (mRy/molecule)
90
Rocksalt 90
60 A5
SH
A5 Rocksalt
60
30
30
Zinc–blende Zinc–blende
0
0
0
–30 0.5
0.6
0.7 0.8 0.9 Normalized volume
1.0
1.1
–30
–30 0.5
0.6
0.7 0.8 0.9 Normalized volume
1.0
1.1
0.5
0.6
0.7 0.8 0.9 Normalized volume
1.0
1.1
(e) 150
(d) 120 InAs
InSb 120 Rocksalt Energy (mRy/molecule)
Energy (mRy/molecule)
90
A5 60
Rocksalt 30
90
60
A5
30
Zinc–blende 0
–30
Zinc–blende 0
0.5
0.6
0.7 0.8 0.9 Normalized volume
1.0
1.1
–30
0.5
0.6
0.7 0.8 0.9 Normalized volume
1.0
1.1
Figure 36 Total electronic energy for some representative III–V structures as function of the atomic volume. The calculations are from Froyen and Cohen (1983a, 1983b).
The use of the total electronic energy or the forces via the Hellmann–Feynman theorem involves different approaches to calculate the phonon frequencies, but the results are essentially the same as indicated in Table 4. Additional information can be obtained from the force calculation. For example, for the LOA(X) mode, the forces on atoms 2 and 3 differ for finite amplitude (see Figure 37). This arises from the effects of anharmonicity and illustrates that it is more difficult to compress a bond than to stretch it. For the phonon calculation, the average of the two force constants is used, and this quantity does not vary by more than 0.5% with the amplitude. If the interlayer force constants are computed, then the full dispersion can be calculated as discussed
earlier. This gives the value of the sound velocity. For example, the q ! 0 TA velocity vTA [110] propagating along the [110] direction with [110] polarization is associated with the shear modulus C11 C12: vTA ½110 ¼
C11 – C12 1=2 2M
ð74Þ
where M is the mass density. The calculated results for the velocity and shear modulus are in good agreement with experiment as shown in Table 5. When interlayer force constants are calculated, the resulting phonon spectrum is in good agreement with experiment. For example, for Si the to X phonon dispersion curve was computed by Yin and Cohen (1982a) using the calculated interlayer
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
force constants. The results are close to the experimentally determined points as indicated in Figure 38. The frozen-phonon total-energy approach can also be used to extract nonharmonic contributions to the phonon couplings. By examining the ET(u) curves terms beyond the quadratic validity, higher order terms can be obtained. For example, Vanderbilt et al. (1986) performed a detailed study on third- and fourth-order anharmonic coupling constants for optical, phonons in C, Si, and Ge. These calculations determined bare phonon– phonon scattering amplitudes. By including virtual processes, renormalized multiphonon vertices and phonon self-energies were calculated. One specific application was the study of the renormalized fourphonon vertices. These were found to be negative, which is the wrong sign for allowing the formation of a proposed two-phonon bound state (Cohen and Ruvalds, 1969).
(110) Plane LTO (Γ) TA(X) TO(X) 1
37
LOA(X)
2
3
1.01.6 Summary and Conclusions
4 Figure 37 Phonon polarization at and X for the diamond structure. Atoms are numbered and denoted by black dots. The solid lines denote an atomic chain in a (110) plane, and the dashed lines denote the projection of an atomic chain a pffiffiffiffiffiffiffiffi distance a=2 away from that plane where a is the lattice constant (Yin and Cohen, 1982b).
The pseudopotential concept has had a profound impact on our understanding of the electronic structure of semiconductors. In this chapter, both empirical and ab initio pseudopotential concepts were outlined and some central applications discussed. Empirical pseudopotentials provided a means for understanding the optical and dielectric properties of semiconductors and the underlying energy band structures. It is sometimes stated that the
Table 4 Comparison of the calculated phonon frequencies, f (in THz), of Si and Ge (Yin and Cohen, 1982b) at and X with experiment (Dolling, 1963; Nilsson and Nelin, 1971, 1972) LTO()
LOA(X)
TO(X)
TA(X)
15.16 ( 2%) 15.14 ( 3%) 15.33
12.16 ( 1%) 11.98 ( 3%) 12.32
13.48 ( 3%) 13.51 ( 3%) 13.90
4.45 ( 1%) 4.37 ( 3%) 4.49
8.90 ( 2%) 8.89 ( 3%) 9.12
7.01 ( 3%) 6.96 ( 3%) 7.21
7.75 ( 6%) 7.78 ( 6%) 8.26
2.44 (2%) 2.45 (2%) 2.40
Si f (energy) f (force) f (expt) Ge f (energy) f (force) f (expt)
The values of the frequencies were determined by energy and by force calculations. The deviation from experimental values are given in parentheses.
38 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories Table 5 Comparison of calculated values (Yin and Cohen, 1982b) of C11–C12 and the TA velocity along the polarization) for Si and Ge with experiment [110] (with [110] (McSkimin and Andreatch Jr., 1963). C11 C12 (Mbar)
vTA [110] (105 cm s 1)
Theory Expt.
1.07 1.03
4.79 4.69
Theory Expt.
0.74 0.82
2.64 2.77
Si
Ge
16
LO
Phonon frequency (THz)
TO 12
LA 8
TA 4
0 0
Γ
0.2
0.4
Δ
0.6
0.8
1.0
X
Figure 38 Phonon branches from to X along the direction from Yin and Cohen (1982a). The dashed lines are calculated from the computed force constants. The solid lines are calculated using the computed thirdnearest layer force constants and the frozen-phonon results at and X. Experimental points are from Dolling (1963), Nilsson and Nelin (1972), and Sinha (1973) and denoted by dots for the transverse modes and triangles for the longitudinal modes.
empirical pseudopotential method (EPM) established the validity of the energy band concept for solids in general. There is much truth to this statement. In the 1950s, there was extended discussions about the validity of the one-electron picture
and whether it could be applied to solid and specifically semiconductors such as silicon. Some workers thought that the electron–hole interactions for a optical excitation would be so strong as to obscure any attempt at understanding such excitations using energy band theory. Given that the EPM allowed adjustments to the one-electron potential, if the EPM had failed to work with the flexibility of choosing a potential, one would have questioned whether it was possible to construct a meaningful one-electron potential and corresponding energy band structure. The consequences of this failure would have been very damaging to prospects for using energy bands to interpret optical, dielectric, photoemission, and transport properties of semiconductors. Fortunately, this was not the case; the EPM provided a simple and effective way to couple band structure theory to experimental work and its impact was immediate. While much of this effort took place in the 1960s and 1970s, the EPM still serves as an effective means to interpret optical data for semiconductors. While more sophisticated methods now exist to examine optical properties, few of the conclusions of the EPM have been overturned by these methods. The coupling of density functional theory with pseudopotentials in the early 1980s resulted in another impressive advance for understanding the electronic structure of materials. The accuracy of density functional theory for predicting electronic and structural energies was problematic at that time. However, the direct numerical application of structural energies to problems involving bond lengths, compressibilities, phonon modes, and phase stabilities showed conclusively the applicability of pseudopotential methods to these problems. Over the last 10 years, more than 10 000 papers have appeared with pseudopotentials in the title or abstract and these papers have been cited over 100 000 times. The vitality and future of the pseudopotential method is without question. The method is now used to examine new and exciting materials systems. These systems include nanoscale systems, amorphous solids, glasses, and liquids. Moreover, owing to strong advances in both hardware and software (algorithms), it is now possible to address systems with thousands of atoms on an almost routine basis. It is for this reason that the pseudopotential concept is said to be the standard model for condensed matter. (See Chapter 1.02).
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
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eleven diamond and zinc-blende semiconductors of semiconductors. Physical Review B 14: 556. Chelikowsky JR and Cohen ML (1992) Ab initio pseudopotentials for semiconductors. In: Moss TS and Landsberg PT (eds.) Handbook of Semiconductors, 2nd edn., p. 59. Amsterdam: Elsevier. Chelikowsky JR, Troullier N, and Saad Y (1994) The finitedifference-pseudopotential method: Electronic structure calculations without a basis. Physical Review Letters 72: 1240. Chelikowsky JR, Wagener TJ, Weaver JH, and Jin A (1989) Valence- and conduction-band densities of states for tetrahedral semiconductors: Theory and experiment. Physical Review B 40: 9644. Cohen M and Chelikowsky J (1982) Pseudopotentials for semiconductors. In: Paul W (ed.) Handbook on Semiconductors, vol. 1, p 219. Amsterdam: North Holland. Cohen MH and Ruvalds J (1969) Two-phonon bound states. Physical Review Letters 23: 1378. Cohen ML and Bergstresser TK (1965) Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and zinc-blende structures. Physical Review 141: 789. Cohen ML and Chelikowsky JR (1989) Electronic Structure and Optical Properties of Semiconductors, 2nd edn.; Berlin: Springer. Cohen ML and Heine V (1970) The fitting of pseudopotentials to experimental data and their subsequent application. In: Ehrenreich H, Seitz F, and Turnbull D (eds.) Solid State Physics, vol. 24, p 37. New York: Academic Press. Dirac PAM (1929) Quantum mechanics of many electron systems. Proceedings of the Royal Society of London A 123: 714. Dolling G (1963) Inelastic Scattering of Neutrons in Solids and Liquids, Vol I and II, Vienna: IAEA. Donohue J (1974) The Structure of the Elements. New York: Wiley. Ehrenreich H and Cohen MH (1959) Self-consistent field approach to the many-electron problem. Physical Review 115: 786. Fermi E (1934) Sullo spostamento per pressione dei termini elevati delle serie spettrali (on the pressure displacement of higher terms in spectral series). Nuovo Cimento 11: 157. Freeouf JL (1973) Far-ultraviolet reflectance of II-VI compounds and correlation with the Penn-Phillips gap. Physical Review B 78: 3810. Froyen S and Cohen ML (1983a) Static and structural properties of III-V zinc blende semiconductors. In: Proceedings of the 16th International Conference on the Physics of Semiconductors. Part I, p. 561. Amsterdam: North-Holland. Froyen S and Cohen ML (1983b) Structural properties of III-V zinc-blende semiconductors under pressure. Physical Review B 28: 3258. Gobeli GW and Kane EO (1965) Dependence of the optical constants of silicon on uniaxial stress. Physical Review Letters 15(4): 142. Grover JW and Handler P (1974) Electroreflectance of silicon. Physical Review B 9(6): 2600. Gunn JB (1964) Instabilities of current in III–V semiconductors. IBM Journal of Research and Development 8: 151. Gunnarsson O, Lundqvist BI, and Wilkins JW (1974) Cohesive energy of simple metals. Spin-dependent effect. Physical Review B 10: 1319. Hamann DR (1989) Generalized norm-conserving pseudopotentials. Physical Review B 49: 2980. Hamann DR, Schlu¨ter M, and Chiang C (1979) Norm-conserving pseudopotentials. Physical Review Letters 43: 1494. Hanke W and Sham LJ (1974) Dielectric response in the wannier representation. Application to the optical spectrum of diamond. Physical Review Letters 33(10): 582.
40 Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories Heine V and Weaire D (1970) Pseudopotential theory of cohesion and structure. Solid State Physics 24: 249. Hellman H (1935) A new approximation method in the problem of many electrons. Journal of Chemical Physics 3: 61. Herring C (1940) A new method for calculating wave functions in crystals. Physical Review 57: 1169. Hohenberg P and Kohn W (1964) Inhomogeneous electron gas. Physical Review 136: B864. Hybertsen MS and Louie SG (1986) Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies. Physical Review B 34: 5390. Ihm J and Cohen ML (1980) Calculation of structurally related properties of bulk and surface Si. Physical Review B 21: 1527. Ihm J, Zunger A, and Cohen ML (1979) Momentum-space formalism for the total energy of solid. Journal of Physics C 12: 4409. Kerker GP (1980) Nonsingular atomic pseudopotentials for solid state applications. Journal of Physics C 13: L189. Kittel C (2005) Introduction to Solid State Physics 8th edn., New York: Wiley. Kleinman L and Bylander DM (1982) Efficacious form for model pseudopotentials. Physical Review Letters 48: 1425. Kleinman L and Phillips JC (1960a) Crystal potential and energy bands of semiconductors. II. Self-consistent calculations for cubic boron nitride. Physical Review 117: 460. Kleinman L and Phillips JC (1960b) Crystal potential and energy bands of semiconductors. III. Self-consistent calculations for silicon. Physical Review 118: 1153. Kline JS, Pollak FH, and Cardona M (1968) Electroreflectance in Ge-Si alloys. Helvetica Physica Acta 41: 968. Kohn W and Sham LJ (1965) Self-consistent equations including exchange and correlation effects. Physical Review 140: A1133. Koo J, Shen YR, and Zucca RRL (1971) Effects of uniaxial stress on E90-peak of silicon. Solid State Communications 9: 2229. Kronik L, Makmal A, Tiago ML, et al. (2006) Parsec – the pseudopotential algorithm for real-space electronic structure calculations: Recent advances and novel applications to nanostructures. Physica Status Solidi (b) 243: 1063. Kunz AB (1971) Energy bands and soft X-rays absorption in Si. Physical Review Letters 27: 567. Ley L, Kowalczyk S, Pollak R, and Shirley DA (1972) X-ray photoemission spectra of crystalline and amorphous Si and Ge valence bands. Physical Review Letters 29: 1088. Liu AY, Chang KJ, and Cohen ML (1988) Theory of electronic, vibrational, and superconducting properties of fcc silicon. Physical Review B 37: 6344. McSkimin HJ and Andreatch P, Jr. (1963) Elastic moduli of germanium versus hydrostatic pressure at 25.0 and 195.8 . Journal of Applied Physics 34: 651. Murnaghan FD (1944) The compressibility of media under extreme pressures. Proceedings of the National Academy of Sciences of the United States of America 30: 244. Nilsson G and Nelin G (1971) Phonon dispersion relations in germanium at 80 K. Physical Review B 3: 364. Nilsson G and Nelin G (1972) Study of the homology between silicon and germanium by thermal-neutron spectrometry. Physical Review B 6: 3777. Pauling L (1960) The Nature of the Chemical Bond. Ithaca, NY: Cornell University Press. Philipp HR and Ehrenreich H (1963) Optical properties of semiconductors. Physical Review 129: 1550. Phillips JC (1973) Bonds and Bands in Semiconductors. New York: Academic Press. Phillips JC and Kleinman L (1959) New method for calculating wave functions in crystals and molecules. Physical Review 116: 287. Phillips JC and Kleinman L (1962) Crystal potential and energy bands of semiconductors. IV. Exchange and correlation. Physical Review 128: 2098.
Phillips JC and Pandey KC (1973) Nonlocal pseudopotential for Ge. Physical Review Letters 30: 787. Pollak FH and Cardona M (1968) Piezo-electroreflectance in Ge. GaAs, and Si. Physical Review 172(3): 816. Rehn V and Kyser D (1972) Symmetry of the 4.5 eV optical interband threshold in gallium arsenide. Physical Review Letters 28: 494. Rohlfing M and Louie S (2000) Electron-hole excitations and optical spectra from first principles. Physical Review B 62: 4927. Rohlfing M and Louie SG (1998) Electron-hole excitations in semiconductors and insulators. Physical Review Letters 81: 2312. Schwarz W, Andraea D, Arnold S, et al. (1999a) Hans G.A. Hellmann (1903–1938): Part I. A pioneer of quantum chemistry. Bunsen - Magazin 1: 10. Schwarz W, Andraea D, Arnold S, et al. (1999b) Hans G.A. Hellmann (1903–1938): Part II. A German pioneer of quantum chemistry in Moscow. Bunsen - Magazin 2: 60. Sinha SK (1973) Phonons in semiconductors. CRC Critical Reviews in Solid State and Materials Sciences 3: 273. Sugino O and Car R (1995) Ab-initio molecular-dynamics study of first-order phase-transitions – melting of silicon. Physical Review Letters 74: 1823. Tauc J and Abraham A (1961) Optical investigation of the band structure of Ge-Si alloys. Journal of Physics and Chemistry of Solids 20: 190. Theis D (1977) Wavelength-modulated reflectivity spectra of ZnSe and ZnS from 2.5 to 8 eV. Physica Status Solidi (b) 79: 125. Troullier N and Martins J (1991) Efficient pseudopotentials for planewave calculations. Physical Review B 43: 1993. Vanderbilt D (1985) Optimally smooth norm-conserving pseudopotentials. Physical Review B 32: 8412. Vanderbilt D (1990) Soft self-consistent pseudopotentials in a generalized eigenvalue formalism. Physical Review B 41: 7892. Vanderbilt D, Louie SG, and Cohen ML (1986) Calculation of an harmonic phonon couplings in carbon, silicon, and germanium. Physical Review B 33: 8740. Vohra YK, Brister KE, Desgreniers S, Ruoff AL, Chang KJ, and Cohen ML (1986) Phase-transition studies of germanium to 1.25 Mbar. Physical Review Letters 56: 19444. von Barth U and Hedin L (1972) Local exchange-correlation potential for the spin-polarized case. I. Journal of Physics C 5: 1629. Walter JP, Cohen ML, Petroff Y, and Balkanski M (1970) Calculated and measured reflectivity of zinc telluride and zinc selenide. Physical Review B 1: 2661. Welkowsky M and Braunstein R (1972) Interband transitions and exciton effects in semiconductors. Physical Review B 5: 497. Wendel H and Martin RM (1979) Theory of structural properties of covalent semiconductors. Physical Review B 19: 5251. Wiser N (1963) Dielectric constant with local field effects included. Physical Review 129: 62. Yin MT and Cohen ML (1980) Microscopic theory of the phase transformation and lattice dynamics of Si. Physical Review Letters 45: 1004. Yin MT and Cohen ML (1982a) Ab initio calculation of the phonon dispersion relation: Application to silicon. Physical Review B 25: 4317. Yin MT and Cohen ML (1982b) Theory of lattice-dynamical properties of solids: Application to Si and Ge. Physical Review B 26: 3259. Yin MT and Cohen ML (1982c) Theory of static structural properties, crystal stability, and phase transformations. Application to Si and Ge. Physical Review B 26: 5668. Zhang SB and Cohen ML (1987) High-pressure phases of III–V zinc-blende semiconductors. Physical Review B 35: 7604. Zucca RRL and Shen YR (1970) Wavelength-modulation spectra of some semiconductors. Physical Review B 1: 2668.
Electrons in Semiconductors: Empirical and ab initio Pseudopotential Theories
Further Reading Cohen ML and Chelikowsky JR (1989) Electronic Structure and Optical Properties of Semiconductors, 2nd edn. Berlin: Springer. Karixas E (2003) Atomic and Electronic Structure of Solids. Cambridge: Cambridge University Press.
41
Kittel C (2004) Introduction to Solid State Physics, 8th edn. New York: Wiley. Marder MP (2000) Condensed Matter Physics. New York: Wiley. Martin RM (2004) Electronic Structure: Basic Theory and Practical Method. Cambridge: Cambridge University Press.
1.02 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors: Bulk Crystals to Nanostructures J Deslippe and S G Louie, University of California at Berkeley and Lawrence Berkeley National Laboratory, Berkeley, CA, USA ª 2011 Elsevier B.V. All rights reserved.
1.02.1 1.02.2 1.02.3 1.02.4 1.02.5 1.02.6 1.02.7 1.02.8 1.02.9 1.02.10 1.02.11 1.02.12 References
Introduction Ground-State Properties and DFT Ab Initio pseudopotentials Selected Applications of DFT to Ground-State Properties Excited States and Spectroscopic Properties Quasiparticle Properties and the Single-Particle Green’s Function The GW Approximation The GW Formalism and Applications Two-Particle Excitations and the Bethe–Salpeter Equation The GW-BSE Formalism and Optical Response Interactions in Nanostructured Semiconducting Materials Summary
1.02.1 Introduction The last few decades have seen dramatic advances in both theoretical methodology and computer technology that have allowed researchers to investigate materials properties in a great variety of systems in a quantitative way from first principles (i.e., without any input parameters other than the constituent atoms making up the material). The advances in theory have been tremendous. Theory has succeeded in reducing a seemingly intractable quantum manybody problem into one that is computationally feasible while retaining the important physics. Today, it is possible to predict the ground- and excited-state properties for a wide range of important materials, in particular semiconductor systems, using first principles (or ab initio) techniques. Such studies have been particularly useful in reduced-dimensional (nano) semiconducting systems characterized by enhanced many-electron interaction effects. The advances in computer technology have been equally tremendous, coming in the form of faster central processing units (CPUs) with larger cache, larger capacity, better performing memory, and greater space. The same period has seen the advent of the massively parallel high-performance computer 42
42 44 47 49 54 55 57 59 63 65 71 73 74
systems. A laptop today has the same computing power as the fastest supercomputer in the world only 15 years ago. In principle, determining the properties of a material from first principles involves the solution of a quantum many-body interacting problem (over both the atomic nuclei and electron coordinates) with the following Hamiltonian (omitting spin and relativistic effects for simplicity): Htot ¼
X P2j
þ
X p2 i
2Mj 2m i X – Zj e 2 : þ jRj – ri j i;j j
þ
X Zj Zj 9 e 2 X e2 þ jRj – Rj 9 j i
ð1Þ
This Hamiltonian dictates the quantum states via the many-body time-independent Schro¨dinger equation H ðr1 ; r2 ; . . . ; rN Þ ¼ Eðr1 ; r2 ; . . . ; rN Þ:
ð2Þ
The exact numerical solution of this problem is, in all but the simplest cases, impractical, due to the exponential increase in size of the Hilbert space for an N-body problem. The exact solution for a typical system of interest would yield an unwieldy many-body
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
wave function, which, in general, would give little or no physical insight. The evaluation of physical quantities of interest with appropriate approximations is then required and desirable. Moreover, the majority of the properties of a condensed matter system can be determined by investigating just the interactions of the valence (outer) electrons with the relatively slower-moving atomic ionic cores (nuclei plus core electrons) and the interactions of the valence electrons among themselves. In the Born–Oppenheimer, or adiabatic, approximation for electronic properties, the atomic positions are assumed to be a set of parameters due to the fact that they are slowly varying compared to motion of the electrons. This approximation allows the elimination of two of the terms in Htot, leaving just a Hamiltonian for the electron wave function: Hel ¼
X p2 i
i
2m
þ
X
X – Zj e 2 e2 þ : jr – ri9 j jRj – ri j i;j i
ð3Þ
Mechanical, structural, and dynamic quantities associated with the motion of the atoms (such as phonon frequencies) can be calculated by evaluating groundstate total energies at different atomic positions. As we will discuss later in this chapter, the explicit dependences of the core electrons can also be removed through the pseudopotential approach, leaving an equation for the valence electrons alone. These interactions determine the electronic structure of the material which, in turn, determines the behavior of the material in the ground state and under external perturbations. We therefore focus the discussion in this chapter on the state-of-the-art theoretical and computational methods for calculating these properties from first principles to further our understanding of the structural, electronic, and optical properties of semiconducting materials from bulk crystals to nanostructures. It is useful to distinguish the ground-state properties of a semiconductor from its electronic excitedstate or spectroscopic properties. Properties such as structure, cohesive energies, charge densities, phase stability, and vibrational properties are ground-state properties because they are given collectively by all the electrons in the many-body ground state. In general, these properties may be determined from knowing the ground-state total energy and related quantities of the system as a function of the atomic positions. Practically, for semiconductors, this is done by applying methods such as those based on the density functional theory (DFT) (Hohenberg and Kohn,
43
1964; Kohn and Sham, 1965). Such approaches have been very successful in the ab initio calculation of many ground-state properties for a wide variety of materials. Spectroscopic properties, such as those probed in photoemission, tunneling or transport experiments, however, involve creating a specific excited particle (electron or hole) above the ground state. Owing to many-particle interactions, the excited particle (or quasiparticle) acquires a self-energy, resulting in a modified dispersion and finite lifetime. Calculating such quasiparticle properties is an N þ 1 particle problem in many-body theory, where N is the number of particles in the ground-state problem, that, in principle, requires a different theoretical treatment from those (such as the standard DFT-based approaches) used for the ground-state problem (Hedin, 1965; Hedin and Lundqvist, 1969; Hybertsen and Louie, 1985, 1986). Understanding the optical properties from first principles is an N þ 2 particle (photo-excitation of an electron and hole pair) problem (Rohlfing and Louie, 1998a, 1998b; Albretch et al., 1998; Benedict et al., 1998). To study optical properties, therefore, it is necessary to include the electron–hole interaction, which can have significant effects on the optical response in many systems from bulk crystals to nanostructures where such effects often qualitatively change the calculated spectra. Spectroscopic properties are best discussed and computed in terms of a Green’s functions approach (Kadanoff and Baym, 1999), where the Green’s functions describe the evolution of the amplitude of a many-body system with the addition of one or more particles at certain time and positions. Because of the reduced dimensionality of nanosystems, many-electron interaction effects have been shown to be extraordinarily important in the spectroscopic properties of these systems including bandgap renormalization, quasiparticle spectral functions such as those measured in photoemission, and optical properties such as those probed in absorption and resonant photon scattering. For example, dominantly strong excitonic effects (resulting from correlations between the photoexcited quasi-electron and the quasi-hole) in the optical response of semiconducting and metallic nanotubes have been predicted by first-principles theory (Spataru et al., 2004a; Deslippe et al., 2007) and have subsequently been confirmed by experiment (Wang et al., 2005, 2007). Experimental advances have also allowed the measurement and characterization of excitation features in the optical response of other nanostructures. In this chapter, we present some of the fundamental
44 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
concepts and theoretical methods for investigating the ground- and excited-state properties of semiconductor systems. We illustrate the use of methods first developed for bulk systems to a wider variety of structures through their application to nanomaterial systems. In particular, we discuss several novel theoretical results that have been obtained from firstprinciples calculations on one-dimensional (1D) and 2D nanostructures. The organization of the remainder of the chapter is as follows. In Sections 1.02.2 and 1.02.3, we discuss the calculation of ground-state properties within the ab initio pseudopotential density functional formalism. Sections 1.02.5–1.02.8 introduce the quasiparticle concept, the electron self-energy, and the GW approximation and method for calculating the quasiparticle properties of interacting electron systems. In Sections 1.02.9 and 1.02.10, we discuss the extension of this formalism to the calculation of optical properties of materials with electron–hole interaction effects included through the Bethe– Salpeter equation (BSE). The formalism for calculating accurate quasiparticle and excitonic properties within the GW–BSE formalism is thus outlined. In Sections 1.02.11 and 1.02.12, we discuss interesting features that arise from the application of this formalism to nanosystems, and provide some future goals and avenues of research in this area.
1.02.2 Ground-State Properties and DFT The major obstacle in solving the quantum manybody problem described by the Hamiltonian in Equation (1), even within the Born–Oppenheimer approximation, is the interaction between the electrons, which are identical Fermionic particles. For the purpose of calculating ground-state properties, this complex many-body problem can be reduced to solving the simpler problem of a single noninteracting electron moving in an effective field. This field is, itself, determined from the single-particle orbitals, and thus the problem must be solved self-consistently using an iterative procedure. Early attempts at such a formalism were devised by Hartree and Fock, where the self-consistent potential (or mean-field potential) takes the form VH ðrÞ ¼
Z X 2 e jjn ðr0 Þj2 0 dr jr – r0 j n
ð4Þ
or VHF ¼ VH ðrÞ þ Vex
ð5Þ
where, jn are the independent electron orbitals, and Vex is the nonlocal exchange operator. The first term on the right side in Equation (5), the Hartree potential, describes the electrostatic interaction of a single electron with the average charge density produced by all the electrons in the system. The second term is the exchange term, resulting from the enforcement of the Pauli exclusion principle. The Hartree and Hartree–Fock (HF) potentials can be derived from minimizing the total ground-state energy if the many-body wave function is assumed to be a simple product (the Hartree approximation) or a single Slater determinant (the HF approprimation) of single-particle orbitals: H ðr1 ; . . . ; rn Þ ¼ ji ðri Þ:
ð6Þ
j1 ðr1 Þ . . . jn ðr1 Þ HF ðr1 ; . . . ; rn Þ ¼ . . . . . . . . . : j ðrn Þ . . . j ðrn Þ 1 n
ð7Þ
i
This leads directly to the HF mean-field, selfconsistent field equation (atomic units are used): –
r2 þ Vion þ VH þ Vx jn ¼ "n jn : 2
ð8Þ
The approximation in the HF approach for the ground-state energy comes from the fact that the true many-body wave function cannot, in general, be described by a single-Slater determinant; rather, it is a superposition of all Slater determinants that can be formed by a complete set of single-particle orbitals. Going beyond the HF formalism leads to corrections that lower the total energy of a manyelectron system. These corrections are collectively referred to as the correlation energy. Techniques that systematically improve the many-body wave function by including increasingly larger number of higher-energy Slater determinants, called multiconfiguration interaction (CI) techniques, generally scale exponentially with the number of singleparticle orbitals included; the formalism becomes quickly intractable for all but the smallest systems such as atoms and small molecules typically studied in quantum chemistry. Unlike mean-field theories such as the Hartree or HF approximations described above, DFT is in principle exact in giving the total energy, electron charge-density distribution, and other related
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
ground-state properties (such as the structural and vibrational properties) of an interacting manyelectron system. DFT does not approximate the ground-state energy by putting restrictions on a trial many-body wave function; instead, DFT rigorously reformulates the total energy in terms of a functional of the charge density of the system. Within the Kohn–Sham scheme, the charge density can be expressed exactly in terms of sums over singleparticle orbitals. As we will see, in practical calculations, the basic approximation lies in the construction of the energy functional itself – no explicit manybody wave function enters the theory. Today, static DFT within the Kohn–Sham formulation is arguably the most popular approach for first-principles investigations of the ground-state properties of condensed matter systems from bulk semiconductors and metals to nanostructures to complex biological molecules. For an interacting many-electron system in an external static potential V(r), Hohenberg and Kohn (1964) demonstrated that, instead of the electronic ground-state energy being a functional of the complicated ground-state many-body wave function (r2,r2,. . .,rN), it can be reformulated as a functional of the electronic ground-state charge density, , alone. This functional obeys a variational principle, that is, it is a minimal value at the physical electron density. Moreover, Kohn and Sham introduced an explicit, practical formulation for the total energy (atomic units are used): Z
1 Ev ½ ¼ V ðrÞðrÞdr þ Ts ½ þ 2 þ Exc ½
R
ðrÞðr9Þ dr dr9 jr – r0 j ð9Þ
Here, V(r) is the ionic potential, Ts is the kinetic energy of a noninteracting electron system with the equivalent ground-state density (r), the third term is
the Hartree energy described above, and Exc is a universal functional, termed the exchange–correlation functional, of the density encapsulating everything not included by the first three terms on the right side of Equation (9). The exchange–correlation functional, Exc[], includes all the effects of many-electron exchange and correlations and the corrections to the kinetic energy of the interacting system not captured by that of the noninteracting system. This formalism is a tremendous conceptual and technical simplification since the unwieldy 3N-dimensional many-electron wave function is, in principle, eliminated from the problem entirely without any approximation to the many-body total energy or charge density. The major drawback of this approach is that although the existence of the universal functional (Exc[]) was shown, its exact form is not known. A number of highly successful approximations to Exc[] have been developed over the years (Dreizler and Gross, 1995; Parr and Yang, 1989), including the local density approximation (LDA) in which the exchange–correlation energy density is taken as that of the homogeneous electron gas at the local density or the generalized gradient approximations (GGAs) in which the effects of both the local density and its gradients are included. These useful approximations to Exc[] have allowed practical and quantitatively accurate calculations for ground-state properties to be made using the DFT approach. Kohn and Sham (1965) showed that the electron charge density, and hence the total energy, may be obtained by solving an associated system of noninteracting electrons with the same charge density as the interacting system. In the Kohn–Sham approach, properties of the original many-body material system of interest with interacting electrons are associated with a fictitious system of noninteracting electrons under the influence of an effective single-particle potential Veff (see Figure 1). The equations for
Kohn–sham DFT formulation Real interacting system: External potential V(r) uij = e 2/rij Ψ(r1, r2, ...), Ev[ρ]
45
Non interacting system: Effective potential Veff(r) uij = 0 {ϕ j(r)} → ρeff(r)
Minimize E V[ρ] with ρeff(r) by varying Veff(r) Figure 1 Schematic of Kohn–Sham DFT. The real interacting system density is obtained in terms of single-particle amplitudes of a fictitious non-interacting system. The energy functional is minimized by varying the effective potential.
46 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
solving for the physical electron density take on the form of those of a self-consistent field problem. The mapping makes use of the fact that the electron density of the noninteracting system is given by the single-particle orbitals. The electron density of the noninteracting system (corresponding to a particular Veff) is then used as an input to evaluate the total energy functional of the real system, making use of the variational principle that the physical density minimizes the total energy functional given by Equation (9). By varying the noninteracting system until the functional is minimized, we arrive at the physical density and energy of the real fully interacting electron system. This variational approach gives rise to a set of Euler–Lagrange equations (the Kohn–Sham equations) that govern the singleparticle orbitals, ji(r), with Lagrange multipliers "i to ensure orthogonal normality, for the fictitious noninteracting system: –
r2 þ Vion þ VH þ Vxc jnk ¼ "nk jnk 2
ð10Þ
with ðrÞ ¼
occ X
jji ðrÞj
ð11Þ
i
and Vxc ðrÞ ¼
Exc : ðrÞ
ð12Þ
The effective, self-consistent potential Veff for the noninteracting system has two terms in addition to the external potential: a Hartree term, VH, which is the same as described above in relation to HF theory, and an exchange–correlation term, Vxc, which is given by the functional derivative of Exc with respect to the electron charge density. In principle, if the exact form of Exc were known, a self-consistent solution to the Kohn–Sham equations would give the exact electron density and, therefore, the exact ground-state energy of the fully interacting system as a function of the atomic coordinates. Here, it is to be noted that unlike the HF approximation, Vxc is simply an energy- or state-independent local operator within the Kohn–Sham framework. However, one should remember that it can have a highly nontrivial dependence as a functional of the whole density, and that it contains the many-electron interaction contributions to the kinetic energy that are not captured by Ts[].
The ground-state total energy as a function of atomic coordinates yields a large number of other properties such as structural and vibrational parameters. However, as mentioned above, Exc is unknown, and approximations must be made. A common approach is to assume that the functional can be written in terms of expectation value of a local operator Exc ¼
Z
ðrÞ"xc ðrÞdr
ð13Þ
where "xc, an exchange–correlation energy density, is assumed to be a function of the density (r) alone in the local density functional approximation (LDA) or a function of (r) and its derivativesr(r) in GGA. These approximations use data from the interacting homogeneous electron gas problem, for which accurate numerical results have been obtained and approximate analytic expressions derived. Such approximations have allowed the very accurate ab initio computation of many ground-state material properties over the past several decades, in particular, those of semiconductors. However, there are well-known cases, such as systems where van der Waals interactions or strong electron correlations are important, that remain outside the scope of such approximate functionals – though, in principle, not outside of an exact DFT ground-state formalism. It is important to note, however, the eigenvalues "i and eigenfunctions, ji(r), of the Kohn–Sham equations (Equations (10)–(12)) do not correspond rigorously to the excitation (or quasiparticle) energies and amplitudes of the electrons/holes. Formally, only the electron charge density and total energy are meaningful quantities within the Kohn–Sham formalism. The Kohn–Sham eigenvalues are only Lagrange multipliers in the Kohn–Sham variational construct, and in general, they are not equal to the electron excitation (or quasiparticle) energies of a system, even if we knew the exact exchange–correlation functional. As an example, let us consider the case of the interacting homogeneous electron gas. Because the density is homogenous, both the Hartree potential plus the external potential and the exact exchange–correlation potential in the Kohn–Sham equation do not depend on the coordinates of the electron, regardless of the exact form of Vxc. Equation (10) then reduces to
p2 þ const: ji ¼ "i ji 2m
ð14Þ
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
The solutions to Equation (14) are always plane waves or free-electron-like states. The dispersion relation of the Kohn–Sham eigenvalues is then identical to that of the free (noninteracting) electron gas system. Therefore, if we were to interpret the Kohn–Sham eigenvalues as electron quasiparticle excitation energies, then, the band structure and therefore the effective mass, m, and occupied bandwidth would be identical to those of free electrons, completely independent of the magnitude of the interaction. In addition, the lifetime of a quasielectron or quasi-hole produced in such a system would be infinite. This is clearly incorrect. It was a similar misuse and misinterpretation of the Kohn–Sham eigenvalues (i.e., using the Kohn–Sham eigenvalue gap as the quasiparticle excitation gap) that led to the famous band-gap problem in semiconductors. Despite this caveat, however, the Kohn–Sham band structure nevertheless does often provide a reasonable starting point for the qualitative understanding of the electronic structure of semiconductor materials.
1.02.3 Ab Initio pseudopotentials Since the valence electrons of the constituent atoms determine most of the properties of a material and the core electrons are often inert from their environment, it is possible to greatly simplify Equation (3) by describing the interaction between these outer electrons with the combined nucleus and core electrons near an atom in terms of pseudopotentials. The concept of pseudopotentials explains the apparent weak interaction between the active electrons and the ionic cores in solids, explaining the many successes of the nearly-freeelectron model in describing simple metals. Further, the use of pseudopotentials greatly improves the efficiency in electronic structure calculations of the properties of real materials. The use of pseudopotentials removes the core electrons from the calculations, reducing the complexity of the problem. A weaker pseudopotential and nodeless pseudo-wave-functions result in smooth eigenfunctions, greatly reducing the size of the basis set required to study the valence electrons. The latter simplification results from the ability to remove the part of the all-electron valence electron wave function that oscillates rapidly in order to remain orthogonal to the core states (Phillips and Kleinman, 1959).
47
The use of pseudopotentials dates back to Fermi’s work in the 1930s (Fermi, 1934). However, they are now best viewed conceptually in terms of the Phillips–Kleinman cancellation theorem (Phillips and Kleinman, 1959). In this view, the valence orbitals (e.g., the Kohn–Sham orbitals in Equation (8)) being orthogonal to the core orbitals may be written in the form of jji ðrÞi ¼ ji ðrÞi þ
X
jfc ihfc ji i
ð15Þ
c
where ji ðr Þi is a smooth function (pseudowavefunction) and jfc i are core electronic states or properly normalized sums of these states. The single-particle Schrodinger equation for the valence electrons can then be constructed within one of the theories described in the previous section as –
p2 þ V ji ¼ "i ji 2m
ð16Þ
where V is the standard independent-particle potential from the Kohn–Sham formalism, for example. Solving this equation is then equivalent to solving the following equation for the pseudo-wave-function ji ðrÞi: –
p2 þ V þ VR i ¼ "i i : 2m
ð17Þ
The additional term, VR, given by VR ¼
X ð" – "c Þjfc ðrÞihfc ðr9Þj
ð18Þ
c
is, in principle, a nonlocal, energy-dependent operator. However, since the core state energies are, in general, significantly lower than those of the states of interest, VR is effectively a repulsive potential with negligible energy dependence. Thus, VR effectively cancels the strong electron-ion interaction potential, V, near the core region, resulting in a net potential seen by the valence electrons that is weaker than the original potential. This weak net potential is what is termed the pseudopotential: Vp ¼ V þ VR :
ð19Þ
The solutions to Equation (17) have the same eigenvalues as the original Equation (16), and have wave functions that are similar to those of the allelectron wave functions outside of the core region but significantly smoother near the core. The pseudopotential concept and methodology justify
48 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
why the interaction between the valence electrons and the crystal potential appears to very weak in many systems, and provides a number of computational advantages such as the removal of the core electronic degrees of freedom in the calculation and a great reduction in the size and complexity of the necessary basis sets to expand the pseudowave-function, since these are smooth functions. The pseudopotentials described above are not unique since a different choice for the second term on the right-hand side of Equation (15) would lead to equally valid but significantly different-looking pseudopotentials. This ambiguity has led to many different construction methods in order to optimize the accuracy and computational efficiency of pseudopotentials (Pickett, 1989). In general, modern practical construction of ab initio pseudopotentials is done by first solving the interacting all-electron problem for a given atom of interest within one of the self-consistent theories described in the previous section, such as DFT in the LDA. The resulting all-electron valence orbitals ji are used to form a set of pseudo-wave-functions, i, using Equation (15) implicitly, by joining j to a properly chosen smooth, nodeless function for small distances from the nucleus (called the cutoff radius), as illustrated in Figure 2 for the case of the 3s and 3p states of sodium. One then inverts the Schro¨dinger equation (for the constructed i and the true "i),
–
p2 þ Vpi i ¼ "i i 2m
ð20Þ
to arrive at the corresponding pseudopotential Vp for the ith state of this particular element. The procedure is usually done for a neutral atom in a configuration that is appropriate in the condensed state. An ab initio ionic pseudopotential describing the intrinsic interaction of an electron with an atom stripped of the other outer valence electrons is obtained by unscreening the neutral pseudopotential, that is, removing the screening Hartree and exchange–correlation potential resulting from the valence electrons themselves. (The interactions with the other valence electrons will be treated explicitly in subsequent calculations on real materials since these effects are environment dependent.) Various other constraints (such as norm conservation; Hamann et al. (1979)) to ensure accuracy and transferability can be implemented in this procedure for the construction of pseudopotentials (Pickett, 1989). The resulting ab initio ionic pseudopotentials are, in general, highly accurate and transferable to different environments where the atoms are not isolated (i.e., where the valence electron distribution differs greatly from the isolated atoms) and have been successfully used in first-principles calculations on materials systems for many different properties in different environments.
2 1
Na 0.4
0 Potential (Ry)
Wave functions
s-Potential
3s
3p 0.2
p-Potential –1 –2 d-Potential –3
0
All electron
–4 –0.2
0
1
2
3 r (a.u.)
4
5
–5
0
1
2 r (a.u.)
3
4
Figure 2 (Left) Constuction of ab initio pseudopotentials. The all-electron wave functions for the 3s and 3p states (dashed lines) oscillate rapidly near the origin because of the orthogonality requirement to the core states. The pseudo-wave-functions are constructed by replacing this oscillatory part of the wave function with a smooth, nodeless function. By inverting Equation 20, one finds the corresponding pseudopotentials shown in the (right) panel. Figure courtesy of J.R. Chelikowsky.
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
1.02.4 Selected Applications of DFT to Ground-State Properties The use of the ab initio pseudopotential approach within the Kohn–Sham formalism of DFT has proven to be a powerful approach for calculating the ground-state properties of many materials (Cohen, 1982) from bulk semiconductors and metals to nanosystems such as molecules, wires, and sheets. There are now a number of DFT computational packages available that vary in the types of exchange–correlation functionals, basis sets, pseudopotentials, and numerical algorithms used. These packages share the general computational flow structure of a typical Kohn–Sham calculation laid out in Figure 3. The input parameters of such a calculation are typically the atom types, initial structural arrangement, and methods for construction of pseudopotentials. Convergence parameters for the basis set such as the plane-wave energy cutoff (maximum energy plane wave to use in the bases) or basis set components in the case of localized basis orbitals are specified from which an initial set of Kohn– Sham orbitals and charge density are constructed using
49
the atomic wave functions. From the charge density, the Hartree and exchange–correlation potentials are constructed, and the Kohn–Sham eigenvector/eigenvalue problem is then solved (typically through an iterative diagonalization method) to get a set of new Kohn–Sham eigenvalues and orbitals. The process is repeated until the change in orbitals or eigenvalues between iterations is lower than a predefined convergence criteria (i.e., when self-consistency is reached). The final products are the self-consistent Kohn–Sham eigenstates of the associated noninteracting system from which the final electron density is obtained so that the electron ground-state energy of the physical system may be evaluated using Equation (7) within a given approximation to the exchange–correlation functional used (e.g., the LDA). In this section, we give several selected examples of DFT calculations of ground-state properties. Figure 4 is taken from the classic work of Yin and Cohen (1980), who first showed that structural energies (as a function atomic positions) may be accurately determined using the ab initio pseudopotential density functional approach. The total energy of the system is given by Etotal ¼ Eel þ Eionion
ð21Þ
Procedure for ab initio DFT calculations Generate pseudopotentials, atomic positions Atomic wavefunctions {ϕ(r)} Create Veff (r)
Construct Heff =
p2 + Veff (r) 2m
Diagonalize Heff Is Veff (r) converged? yes
no
Compute total energy and forces
Figure 3 Components in performing DFT calculations. Initial wave functions are constructed from atomic pseudopotentials. From these wave functions, an initial effective potential (ionic plus Hartree and exchange– correlation) is constructed. The Kohn–Sham Hamiltonian is diagonalized to yield a new set of wave functions and eigenvalues. The process is repeated until the changes in wave functions or energies between cycles are below specified convergence criteria.
for different structural arrangements, where Eel is the DFT electronic ground-state energy from Equation (10) and Eion–ion is the classical electrostatic interaction energy among the ions. By comparing this quantity over several assumed structures (i.e., considering the atomic degrees of freedom or configurations), Yin and Cohen were able to determine with high accuracy the stable atomic configuration and other structural parameters such as the cohesive energies, lattice constants, and bulk and other elastic moduli. With the approximate exchange–correlation functionals described above (i.e., the LDA and GGA), one can now calculate cohesive energies to within a few percent of the experimental values. To achieve a higher level of accuracy, however, further improvement in the treatment of the many-electron interaction effects (such as the use of hybrid or other functionals) is required. Relative energies are more accurate, yielding lattice constants and elastic moduli that are typically within 1% and few percent of experimental values, respectively. In particular, for semiconductor systems, an accurate determination of energy landscapes and phase stability is achievable within implementations of the Kohn–Sham formalism.
50 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
–7.84
–7.82 Si
Ge
–7.86
–7.84 hcp Estructure (Ry/atom)
Estructure (Ry/atom)
tcc bcc –7.88
sc
4
β-TIN 3
–7.90 2
–7.92 0.6
0.7
0.9 0.8 Volume
–7.86
sc
β-TIN
3
2
Hexagonal diamond
Diamond
1
1.0
hcp
bcc
4
–7.88
Hexagonal diamond
Diamond
tcc
1.1
–7.90 0.6
0.7
0.8 0.9 Volume
1
1.0
1.1
Figure 4 Phases of Si and Ge under pressure. The common tangent line between the E vs. V curves show the point where the Gibb’s free energy at zero temperature (EþPV) of the two phases is the same, since P ¼ -dE/dV. The slope of the line gives the transition pressure and the change in volume across the phase transition can be determined from the volume of the two phases at the transition pressure. Modified from Yin MT and Cohen ML (1980) Microscopic theory of the phase-transformation and lattice-dynamics of Si. Physical Review Letters 45: 1004.
Atomic forces can be calculated by derivatives of the total energy with respect to the atomic positions (using, for example, the Feynman–Hellman formula): Fi ¼
@Etotal @Ri
ð22Þ
These forces can be used to evolve a nonequilibrium atomic distribution in space and time, in molecular dynamics simulations (Frenkel and Smit, 2002). New atomic positions and velocities are calculated and evolved in time using _ Þ Rðt þ t Þ ¼ Rðt Þ þ t ? Rðt
ð23Þ
_ þ t Þ ¼ Rðt _ Þ þ t ? Fðt Þ Rðt
ð24Þ
and
where t is an appropriately small time step. In common applications, it is necessary to use thousands and tens of thousands of time steps to evolve the system for even short durations of the order of picoseconds. The ab initio forces can be taken from a standard DFT calculation at a given set of coordinates, but the cost of doing a complete DFT
calculation at thousands of atomic configurations becomes prohibitive. An alternative approach developed by Car and Parrinello (1985) to overcome this limitation is to simultaneously evolve both the atomic positions and electron wave functions. In this approach, a full solution to the Kohn–Sham equations is not obtained at every atomic configuration. Instead, both the atomic positions and the electronic wave functions are evolved in the molecular dynamics simulation, with the latter done through a fictitious dynamics introduced for the degrees of freedom associated with the Kohn–Sham eigenstates (Car, 1984, 2006). The Car–Parrinello and related approaches have extended DFT to the realms of ab initio investigation of the dynamics and thermodynamic properties of solids and liquids, including phase transitions, impurity dynamics, etc. For example, this methodology has been used to examine the dipole moment and paircorrelation function of water under various conditions (Tassaing, 1998; Silvestrelli and Parrinello, 1999; Boero et al., 2000). In addition, computations of phase diagrams at high temperatures and pressures
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
including liquid phases have been done using the ab initio MD approach. Below, we shall consider the simpler case of pressure-induced phase transition obtainable through the application of the standard DFT approach. It is straightforward, utilizing the equation of state for the various structures, to determine the behavior of structural phase transitions under pressure by evaluating the Gibbs’ free energy (Kittel and Kroemer, 1980): G ¼ E þ PV – TS
ð25Þ
In particular, at low temperature, the critical transition pressure between two adjacent structures (as depicted in Figure 4) is given by the negative of the slope of the common tangent (pressure) of the two equations of state, yielding the point where the Gibb’s free energy of the two phases is the same. This leads to the determination of the critical pressure and the volume discontinuity at the phase transition and provides an ab initio understanding and prediction of the high-pressure phases of matter such as those of Si and Ge as shown in Figure 4 (Cohen, 1982). The preferred structure of a material system such as that at a surface or near a defect can be found by minimizing the total energy or the forces on the atoms. This approach has been used with success in determining the structural parameters of complex materials, surfaces, defects, clusters, molecules, and nanostructured systems of interest under imposed boundary conditions. For nonperiodic systems, the standard technique is to carry out the electronic structure calculations employing a supercell scheme (Cohen et al., 1975) in which the surface, defect, or molecule is repeated periodically with sufficient separation between them in order to mimic the isolated structure. As we will discuss more below, for studying isolated nanostructures, a way to minimize the vacuum size needed is to replace the Coulomb interaction by a new interaction that is the same as the Coulomb interaction within a cutoff radius but zero outside of the radius. In addition, from information on the total energy of the system as a function of the atomic positions as well as the forces, we can study the lattice dynamics within the Born–Oppenheimer approximation. There are two common schemes in determining the phonon eigenmodes and dispersions from first principles: (1) the frozen phonon method or (2) the linear response method. Conceptually, in the
51
frozen phonon approach (Yin and Cohen, 1980), one calculates the energy difference in the presence of a finite atomic distortion of the form ui ðkÞ ¼ u0 cosðk?i þ k Þ frozen into the lowest energy structure. k here is an overall phase shift of the distortion. By computing the change in the total energy of the system, E(uo), as a function of the distortion amplitude uo, one obtains the phonon properties via a power series expansion E ¼ K2 uo2 þ K3 uo3 þ K4 uo4 þ
ð26Þ
where the harmonic contribution to the phonon frequency is given by the coefficient K2 and the higher-order coefficients give rise to anharmonic contributions to the phonon frequency and to processes such as phonon lifetime and phonon–phonon interactions. This ability to easily compute the anharmonic contributions is a major advantage of the frozen phonon methodology. However, the size of the supercell required to contain the distortion depends on the wavelength of the distortion; only a finite set of discrete phonons (those of relatively small wavelength, which are commensurate with the lattice) can be calculated. Points away from symmetry points or high symmetry directions, which lead to a very large supercell, may not be studied in practice. Another approach to calculating phonons involves the use of the density functional perturbation theory formalism to create the dynamical matrix. In this scheme, the atomic force constants, Cij ¼
@2E @Ri @Rj
ð27Þ
are evaluated using linear response theory (Baroni et al., 2001; Yin and Cohen, 1982). The resulting dynamical matrix can be diagonalized to obtain the phonon frequencies and eigenmodes. This approach allows the direct computation of phonon properties at arbitrary wave vector, k, in the Brillouin zone, but is limited to the harmonic approximation due to the use of linear response theory. Figure 5 shows the results of such a linear-response calculation for the phonon dispersions and density of states for silicon and germanium. In practice, both frozen phonon and linear response approaches have given very accurate phonon results for a variety of materials ranging from metals to semiconductors to nanosystems and molecules. This shows again the power of the ab initio pseudopotential DFT Kohn–Sham formalism for calculating ground-state properties.
52 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
Frequency (cm–1)
600
400
Si
200
0
Γ
K
X
Γ
L
X
W
L
DOS
Γ
L
X
W
L
DOS
Frequency (cm–1)
400
300 Ge
200
100
0
Γ
K
X
Figure 5 Phonons dispersion relations and density of states (DOS) in Si and Ge calculated within the linear response theory. After Giannozzi P, Degironcoli S, Pavone P, and Baroni S (1991) Ab-initio calculations of phonon dispersions in semiconductors. Physical Review B 43: 7231.
In the study of nanostructures, the predictive power of the ab initio pseudopotential density functional method has led to many important predictions. We give one illustrative example here – the graphene nanoribbons. Graphene, a single atomic layer of graphite with a honeycomb structure, has generated considerable excitement since its isolation and study in 2005 (Novoselov et al., 2005; Zhang et al., 2005). This 2D system exhibits many highly unusual properties and is a promising material for future electronics and other applications (Geim and Novoselov, 2007; Castro Neto et al., 2009). If cut into strips of nanometer widths with the dangling -bonds passivated with adatoms, the resulting homogeneous edge graphene nanoribbons (e.g., with either zigzag shaped or armchair-shaped edges) are predicted to be all semiconductors (Son et al., 2006a), unlike the case of carbon nanotubes (Saito et al., 1998). The semiconducting nature of the graphene nanoribbons has been confirmed by recent experiments (Li et al., 2008). Figure 6 shows the calculated gaps as a function of ribbon width for the armchair nanoribbons. The
importance of DFT over lower-level theories such as a simple -orbital-only tight-binding model is illustrated in the structural dependence of the band gap. The qualitative differences between the two sets of results arise from changes at the boundaries of the structure that are not captured in tight-binding models based on pristine graphene. In carbon nanotubes (i.e., graphene nanoribbons rolled into tubes), somewhat similar corrections to the tight-binding results from relaxing the coordinates with respect to the total energy are found and termed curvature effects (Blase´ et al., 1994; Charlier et al., 1994, 1995; Louie, 1994). As we will see in the following sections, the DFT band structure, however, still needs to be corrected to represent the quasiparticle dispersion. Another interesting result from DFT calculations is that the zigzag graphene nanoribbons have a semiconducting antiferromagnetic ground state that can be made into a half metal by an applied external electric field (Son et al., 2006b). Figure 7 shows the predicted ground-state spin-charge density for a
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
2.5
1.5
LDA Na = 3p Na = 3p + 1 Na = 3p + 2 Δ3p Δ3p + 1 Δ3p + 2
2.0 Δa(eV)
2.0 Δa(eV)
2.5
Tight binding Na = 3p Na = 3p + 1 Na = 3p + 2
1.0 0.5
53
1.5 1.0 0.5
0.0
0.0 0
10
20
30
40
50
0
10
20
30
40
50
Figure 6 The Kohn–Sham electronic gap () of armchair graphene nanoribbons as a function of ribbon width between tightbinding (left) and DFT LDA (right) calculations. The difference stems from the structural relaxation at the edges and a more accurate treatment of electron–electron interaction effects. N is the number of rows of C-C dimers forming the width of the ribbon. Armchair nanoribbons can be broken into three families with N¼3p, 3pþ1, or 3pþ2 for integer p. Note that the LDA predicts a finite gap for the first family which is qualitatively different from the tight-binding result which predicts zero gap. Modified from Son Y-W, Cohen ML, and Louie SG (2006a) Energy gaps in graphene nanoribbons. Physical Review Letters 97: 216803.
(a)
2
(b)
E–EF (eV)
1 0 –1 Electric field –2 –3
1.4 0.0
π
0
–1.4
k(1/a) (c)
2
E–EF (eV)
1 0 –1 –2
Eext = 0.0 (V/Å)
Eext = 0.0 (V/Å)
Eext = 0.0 (V/Å)
π0
0 k(1/a)
π0 k(1/a)
π k(1/a)
Figure 7 Electronic properties of zigzag graphene nanoribbons. (a) Band structure of an 16-ZGNR with spin degree of freedom neglected, showing two very flat edge-state bands at the Fermi energy. (b) The spatial distribution of the charge difference between -spin and -spin ðrÞ – ðrÞ for the ground state when there is no external field. (c) The spin-resolved band structures of a 16-ZGNR at different external electric field strengths. The red and blue lines denote bands of -spin and -spin states, respectively. Inset, the band structure in the range of 50 meV to þ 50 meV from the Fermi level. Modified from Son Y-W, Cohen ML, and Louie SG (2006b) Half-metallic graphene nanoribbons. Nature 444(7117): 347–349.
16-zigzag nanoribbon (16-ZGNR), which has 16 zigzag chains forming the width (Son et al., 2006b; Yang et al., 2007b). The DFT calculations show that, for this system, the bands near the gap are twofold degenerate spin-polarized edge states with
one band localized on one edge and the other band on the opposite edge. The edge-state electrons are ferromagnetically coupled along one edge and antiferromagnetically coupled across the ribbon. By applying a transverse electric field, thus creating a
54 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
While DFT has proven to be an extremely useful and accurate method in calculating the ground-state properties of semiconductors and many other condensed matter systems, the Kohn–Sham formalism within static DFT does not generally give accurate excited-state properties such as band gaps, quasiparticle energies, and optical spectra. Extensions of DFT to time-dependent density functional theory (TDDFT), in principle, would be able to yield certain spectroscopic properties such as the optical absorption spectrum (Reining et al., 2002). In practice, however, approximations to date to the timedependent exchange–correlation functional have restricted these studies to small finite systems such as atoms or molecules. For bulk semiconductors and extended semiconducting nanostructures, current implementations of TDDFT (such as those based on the adiabatic approximation) yield results, which converge to those of independent particle transitions for the optical response and thus do not provide any improvements. This is due to an inadequacy in the description of both quasiparticle self-energy and electron–hole interaction effects within current approximations, although considerable efforts are being expended to make further progress (Reining et al., 2002; Botti et al., 2004). Figure 8 shows a comparison of the calculated electronic band gap with experimental quasiparticle band gaps for a range of semiconductors and insulators. The LDA Kohn–Sham eigenvalues underestimate the gaps by as much as 50% or more. For germanium, even the band topology is qualitatively incorrect; LDA predicts Ge to be a metal instead of a semiconductor with a gap of 0.7 eV. HF theory, which often does well for small molecules and atoms where screening is unimportant, consistently overestimates the band gaps of semiconductors and insulators – the addition of
LIF Fluorite LiCl
10
w-BN
Diamond
5
w-AIN c-BN
Quasiparticle theory
InSb InAs Go GaSb Si InP GaAs CdS, AlSb CdSe, CdTe, BP AlAs SiC, Fluorite, GaP, AIP Znlo, ZnSe c-GaN w-GaN InS
1.02.5 Excited States and Spectroscopic Properties
15
Theoretical band gap (eV)
potential drop across the ribbon, edge states with one spin orientation will be pushed into the band gap and edge states with spin of opposite orientation will be pushed away from the gap. In this a way, it is possible to create a half metal with a 100% spin-polarized carrier by either applying an electric field with strength that would close the gap or by doping or gating the system with carriers in the presence of a weaker applied transverse field.
Many-body corrections
LDA
0 0
5 10 Experimental band gap (eV)
15
Figure 8 Comparison of calculated energy band gaps with those measured by experiments. The diagonal line corresponds to a perfect agreement between experiment and theory. Kohn–Sham band gaps calculated within LDA (crosses) consistently underestimate the experimental bandgap, whereas the GW quasiparticle gaps (diamonds) consistently give good agreement with experiment.
correlation effects systematically reduces the band gaps since correlations lower the energy required to create an excited electron or hole. This band-gap problem and other similar problems that exist in comparing calculated Kohn–Sham eigenvalues to experimental excitation energies do not stem from the approximation of the exchange–correlation functional in DFT, but arise from a fundamental and conceptual difference between the theoretical and measured quantities. As discussed in previous sections and illustrated by the example of the homogeneous interacting electron gas, the Kohn–Sham eigenvalues are merely Lagrange multipliers within a variational derivation of the conditions for minimizing the total energy – that is, they do not physically correspond to the true quasiparticle energies in general. For an interacting system, it is important to distinguish the different kinds of excitations probed in various experiments. Measurements employing photoemission, transport, or tunneling techniques yield, in general, information on individual particlelike excitations or quasiparticles (excited electrons and holes) of a system. For example, photoemission
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
(or inverse photoemission) probes a system with a single additional hole (or electron) of a certain wave vector and energy. Due to many-electron interactions, the excited particle acquires a self-energy which changes its dispersion relation from that of the independent-particle or mean-field picture, and the excitation acquires a finite lifetime. In an optical experiment, on the other hand, one creates neutral correlated electron–hole pair excitations. The electron–hole interaction (excitonic) effects become important and can in fact be dominant in many systems, particularly nanostructures. A treatment of excitonic effects requires an effective two-particle approach on top of the quasiparticle picture. Selfenergy, excitonic, and other many-electron interaction effects exist in all semiconductor systems to various degrees. They are, however, particularly relevant in understanding the spectroscopic properties of many systems of contemporary interest, including clusters, surfaces, polymers, defects, nanotubes, and other lower-dimensional materials since the Coulomb interaction is enhanced in reduced dimensions. The behaviors of the single-particle-like excitations in semiconductor materials are best described in terms of the quasiparticle concept (Hedin, 1965; Hybertsen and Louie, 1985; Kadanoff and Baym, 1999). It is mostly transitions between the quasiparticle states that are probed in spectroscopic measurements. Because of electron–electron and other interactions in a solid, an excited particle of wave vector k is dressed with a screening cloud of electrons and other excitations (e.g., phonons or spin waves) resulting in a different energy E, effective mass m, and a finite lifetime. In this chapter, we focus on the effects of electron–electron interaction since they give rise to eV-scale modifications in the electronic excitations in semiconductors. An accurate treatment of the exchange and dynamical correlation effects, arising from the Pauli exclusion principle and Coulomb repulsion, seen by an electron is essential in calculating the quasiparticle excitation energies and properties. The understanding of optical properties further involves the interaction between the excited quasi-electron and the hole that is left behind, which is mediated through all the other electrons in the system (Strinati, 1988; Rohlfing and Louie, 1998a, 1998b; Albretch et al., 1998; Benedict et al., 1998). As in the case of ground-state properties, tremendous progress has been made in the past two decades toward calculating the electronic excited-state
55
properties of condensed matter systems. It is now possible to accurately calculate the quasiparticle and optical spectra of many systems from first principles, in particular, those with moderately correlated electrons. The approach involves solving for the single-particle and two-particle Green’s function of the interacting-electron system. The advantage of the approach is that one can obtain the spectroscopic properties including the relevant self-energy and electron–hole interaction effects without any empirical parameters and yet still be applicable to real materials. Applications of this approach have explained and predicted the spectroscopic properties of a number of semiconducting systems from bulk crystals to nanostructures.
1.02.6 Quasiparticle Properties and the Single-Particle Green’s Function To predict the spectroscopic properties of semiconductors and other systems, it is convenient to formulate the problem in terms of quasiparticles. A direct way to calculate quasiparticle properties is through a Green’s function formalism of the manybody problem. The single-particle Green’s function describes the propagation of the many-body system after a single particle is added at time t ¼ 0 and location r. For an interacting many-electron system, it is defined as Gðr; r9;t Þ ¼ – ih0jT fcðr; t Þcy ðr9;0Þgj0i
ð28Þ
where the c’s are electron field operators, T the time order operator, and j0i is the many-electron groundstate wave function. A useful alternate form is the diagonal elements of G in an orbital basis representation Gðp; t Þ ¼ – ih0jT fCp ðt ÞCpy ð0Þgj0i
ð29Þ
where p denotes the quantum numbers of a singleparticle state, for example, the band index n and wave vector k. Physically, the Green’s function is related to the probability of finding a particle with quantum number p at time t if one was injected into the system at time t ¼ 0. While the many-body ground-state and excited-state wave functions are extremely complex and impractical to express explicitly in terms of the 3N coordinates of the N electrons in a system, the single-particle Green’s function depends on only two space–time coordinates and is, thus, substantially
56 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
easier to deal with. With some manipulation, for system with periodic translation symmetry (such as crystals), one can arrive at another representation of the Green’s function, the Lehmann representation (described here for the interacting electron gas for simplicity) (Fetter and Walecka, 1971): " X h0 jcð0Þjik ihik jcy ð0Þj0 i h! – – "i ðN þ 1Þ þ i
i y h0 jc ð0Þji – k ihi – k jcð0Þj0 i þ ð30Þ h! – þ "i ðN – 1Þ – i
A(k, ε)
Gðk; !Þ ¼ hV
where, represents the many-body states (0 denotes the N-particle system ground state and ik denotes the ith many-body excited state with wave vector k of the N þ 1/N 1 particle system), is the chemical potential, and "n is the energy of the nth excited state of the N þ 1/N 1 particle system above the N þ 1/N 1 ground-state energy, and the second quantized operator is cðrÞ ¼ e – iP:r=hcð0Þe – iP:r=h (Fetter and Walecka, 1971; Mahan, 1981). For a noninteracting system, this representation illustrates that the poles of the Green’s function occur at the independent electron excitation energies of the N þ 1 and N – 1 particle systems. For such a system, the ik state represents a simple single-particle excitation and each ik state contributes to a single isolated pole in the Green’s function. For a system with relatively moderate electron correlations, G(k, !) along the real ! axis consists of a series of well-defined peaks, similar to the case in the noninteracting spectrum, but each with a finite width corresponding to a pole in the analytical continuation of this quantity off the real axis. In terms of the measurable spectral weight function, A(p, !)¼(1/)|ImG(p, !)|, a dominant pole in the Green’s function at a complex energy Ep corresponds to an A(p, !) of the form Aðp; !Þ ¼
ði=2ÞZp þ c:c: þ correction ! – ½Ep –
ð31Þ
which is sharply peaked as a function of ! (see Figure 9) and gives rise to a form for G in the time domain, for positive t : Gðp; t Þ ¼ – iZp e – iReðEp Þt e – pt þ correction
ð32Þ
where Zp is a renormalization factor describing the relative spectral weight present in the quasiparticle peak and p is the imaginary part of Ep. The physical content of Equation (32) is that G, in this particular single-particle orbital basis denoted by p, describes a propagation amplitude, which is oscillatory with
k < kF
ε
μ
Figure 9 Qualitative figure of the spectral weight function A(k,") of the single-particle Green’s function of an interacting electron system.
characteristic frequency given by the real part of Ep, Re(Ep), and damped by 1/p related to the imaginary of Ep. This leads to the usual interpretation that the peak position in A(p, !) is the quasiparticle energy (real part of Ep) and the width (imaginary part of Ep) gives the lifetime of the quasiparticle (contributing to the linewidth in a photoemission experiment, for example). Finding the quasiparticle properties is then equivalent to solving for the appropriate singleparticle states, which give rise to sharp peaks in A(p,!) and is also equivalent to solving for the position of the poles of G in the complex energy plane. For a many-body system with Hamiltonian given by Equation (1) or equivalently, H¼
X
H0 ðri Þ þ
i
X
Vc ðjri – rj jÞ
ð33Þ
i<j
where H0 ðrÞ ¼
p2 þ Vion ðrÞ 2m
is a one-particle term and Vc the bare Coulomb interaction, the equation of motion for the singleparticle Green’s function can be written as (Mahan, 1981; Hedin, 1965; Hybertsen and Louie, 1985) ðh! – H0 – VH ÞGðr; r9; !Þ –
Z ðr; r0; !ÞGðr0;r9; !Þdr
¼ ðr; r9Þ
ð34Þ
where VH is the usual Hartree potential and is known as the electron self-energy operator, which itself is a functional of G. A formal solution for G can be expressed in terms of single-particle amplitudes as (Hedin, 1965; Hybertsen and Louie, 1985) Gðr; r9; !Þ ¼
X c ðrÞc nk ðrÞ nk ! – Enk – ink nk
ð35Þ
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
where Enk and cnk(r) are the complex solutions to the quasiparticle Dyson equation Z
Enk – H0 ðrÞ – VH ðrÞ cnk ðrÞ – ðr; r9; Enk Þcnk ðr9Þ dr9 ¼ 0: ð36Þ
Here, n and k are again the band and wave vector quantum numbers describing the quasiparticle excitations in a crystal. The positions of the poles of the Green’s function G(!), hence, may be identified with the complex solutions of Equation (36) and, as discussed above, are connected to the excitation energy and lifetime of the quasiparticles. The quasiparticle band structure and properties, thus, may be obtained by solving Equation (36). The quasiparticle equation, Equation (36), is similar in form to the Schro¨dinger equation in oneelectron or self-consistent-field theories (such as the Kohn–Sham or HF formalism). However, in the Dyson’s equation, ðr; r9; !Þ is a nonlocal, nonHermitian and energy-dependent operator, leading to complex eigenvalues, Enk. As seen in Equation (32), the real part of Enk corresponds to the quasiparticle energy and the imaginary part gives the lifetime. We see that this corresponds to a Green’s function with poles off the real axis, giving rise to peaks of finite width in the functions G(k, !) and A(p, !) along the real ! axis. In practice, since Enk is often computed within perturbation theory, it is useful to write the complex quasiparticle energy as a sum of an unperturbed non-interacting energy "nk plus a complex self-energy nk containing the many-electron (exchange–correlation) effects: Enk ¼ "nk þ nk
Σ = 2
ð37Þ
1 + 2
57
This physical description of the single-particle excitations of an interacting electron system relies on the lifetime of the quasiparticles being sufficiently long on the time scale of the relevant experimental probes that is, the widths of the peaks in the Green’s function should be sufficiently small so that there are well-defined peaks in the measured spectra. Within Fermi liquid theory, for excitation energies near the Fermi energy of a metal or the fundamental energy gap region of a semiconductor or insulator, the excitations are long-lived and the quasiparticles are well-defined, allowing us to pursue this description (Mahan, 1981; Figure 9).
1.02.7 The GW Approximation As discussed above, the Dyson’s equation, Equation (36), may be solved to yield the quasiparticle energies, lifetimes, and amplitudes. To make the approach practical for real materials, we need an accurate and computationally tractable method to construct the nonlocal, non-Hermitian self-energy operator ðr; r9;!Þ. It was shown by Hedin that can be systematically expanded in a series in terms of the dressed Green’s function G and the screened Coulomb interaction W (Hedin, 1965; Hybertsen and Louie, 1985): W ðr; r9; !Þ ¼
Z
" – 1 ðr; r0; !ÞVc ðr0; r9Þdr0
ð38Þ
where "ðr; r9; !Þ is the full r; r9-dependent timeordered dielectric matrix of the system (see Figure 10). The advantage of this expansion in terms of the screened Coulomb interaction over an
1
+ 2
1 + 2
1 + 2
1
+ 2
1 + 2
1 + 2
1...
Figure 10 Diagramatic expansion of the self-energy in terms of the screened Coulomb interation.
58 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
expansion in terms of the bare Coulomb interaction Vc is that W is, in general, much weaker than Vc due to the dielectric screening. For example, the static dielectric constant of a typical semiconductor is of the order 10. Thus, a series expansion in W should converge with fewer terms than one in Vc. An alternative view is that, since W and the dressed Green’s function themselves can be expressed as series expansions in the bare quantities, each diagram in Figure 10 is a partial summation over terms in a conventional expansion, leading to more accurate physical results with only low-order terms. In practice, the Dyson’s equation (Equation (36)) is typically solved within a perturbation theory approach. In perturbation theory, it is greatly beneficial to start from a mean-field scenario that is as close as possible to the physical system (Hybertsen and Louie, 1985). The mean-field Hamiltonian that is typically used in quasiparticle studies for semiconductors is the Kohn–Sham DFT Hamiltonian HKS within either the LDA or GGA, which already contains the effects of exchange and correlation in an averaged way. Perturbation theory calculation is then performed on H ¼ H – HKS ¼ - Vxc. It is important to point out, however, that the quasiparticle approach does not depend upon DFT, but only utilizes it as a convenient starting point for many systems. For some systems, such as materials with more correlated electronic states, other starting mean-field Hamiltonians may be used such as those from LDAþU or the hybrid functionals (Miyake et al., 2006); or one may use some selfconsistent scheme to determine the best mean-field (van Schilfgaarde et al., 2006). In the studies of real materials, it is found that, for most cases, only the first term of the series in Figure 10 for the self-energy operator is necessary, yielding what is called the GW approximation (Hedin, 1965), since when expressed to first order in G and W is given by Z d! – i! ðr; r9; EÞ ¼ i Gðr; r9; E – !ÞW ðr; r9; !Þ e 2
ð39Þ
where ¼ 0þ. Within the GW approximation, one computes using Equation (39) and solves the quasiparticle Dyson’s equation for the complex energy Enk and wave functions cnk ðrÞ, which yield the quasiparticle properties of the system. If, instead of expanding in terms of G and W, we replace W with the unscreened (or bare) Coulomb interaction Vc, Equation (39) would reduce to the HF exchange operator. Therefore, much of the physics
in determining the many-electron effects on the quasiparticle properties, within the GW approximation, resides in the dielectric response function or matrix "ðr;r9; !Þ, which is a two-point function in the spatial coordinates and depends on frequency. It has been shown that both the nonlocality and frequency-dependence of the screening matrix play central roles in obtaining accurate quasiparticle properties in semiconductors (Hybertsen and Louie, 1985). The construction of the dielectric response function "ðr; r9; !Þ and the single-particle Green’s function G require the greatest effort in a typical GW calculation. Both have to be treated adequately to obtain quantitative results that may be compared with experiment. As mentioned above, if W is replaced by the bare Coulomb interaction in Equation (39), then Equation (36) reduces to the HF equations. From this point of view, the HF eigenvalues may be considered as a lowest-order approximation to the quasiparticle energies, consistent with Koopman’s theorem. The dielectric function "ðr; r9; !Þ provides the dynamical and spatial-screening response of the electrons that gives rise to correlation effects going beyond bare exchange. Similarly, the eigenvalues from the Kohn–Sham equations in the density functional formalism may be viewed as another set of very approximate quasiparticle energies, with the exchange–correlation potential Vxc(r) approximating the nonlocal, energydependent self-energy operator in Equation (36). However, we must emphasize that, as discussed above, in principle, both the HF and Kohn–Sham eigenvalues are only Euler–Lagrange parameters in minimizing the total energy in ground-state theories, and as illustrated in Figure 8, they are usually not accurate enough as approximations to the quasiparticle energies. It is only in the extent that the exchange–correlation operators within selfconsistent field theories actually do approximate that these theories approximate the solution of the Dyson’s equation. In fact, different formulations of the DFT within the Kohn–Sham schedule can lead to significantly different Kohn–Sham eigenvalues (Seidl et al., 1996). The GW approximation was formulated for the electron gas in the 1960s by Hedin and others. However, the first-principles approach to make computing quasiparticle excitations practical in real materials was not developed until over 20 years later by Hybertsen and Louie (1985, 1986). This approach has since been employed with success in
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
ab initio studies of the quasiparticle properties of semiconductors, insulators, surfaces, nanostructures, and other material systems. A crucial step in the development was the realization of the importance of the off-diagonal elements of the dielectric matrix (called local-field screening effects) in the self-energy as well as efficient ways to calculate and apply the equally important frequency dependence of the dielectric response. The latter is typically done by calculating explicitly the static dielectric matrix within the random-phase approximation and extending it to finite frequencies using schemes such as the generalized plasmon pole (GPP) model (Hybertsen and Louie, 1985). The frequency dependence of the dielectric response has also been computed from analytic continuation from a few points on the imaginary frequency axis or explicitly on the real frequency axis in recent applications (Rojas et al., 1995). Hybertsen and Louie (1985) showed that, with the inclusion of the full dielectric matrix, one can obtain highly accurate quasiparticle energies. Similar to first-principles ground-state studies, the only inputs to the calculations are the atomic numbers of the constituent elements and the geometric structure of the system, which can be determined separately from a total energy calculation. Since the screened Coulomb interaction, W, incorporates the dynamical many-body effects of the electrons, the dielectric response function "ðr; r9; !Þ is key in determining the electron self-energy. Owing to the spatial distribution of atomic positions and electron density inhomogeneity, "ðr; r9; !Þ of a solid is a function of r and r9 separately. In a k-space formulation, the crystalline dielectric function is a matrix "G; G9 ðq; !Þ in the reciprocal lattice vectors G; the off-diagonal elements of this matrix describe the local field effects due to the charge inhomogeneity in real materials (Hybertsen and Louie, 1987). Inclusion of both local field and dynamic effects is essential for an accurate description of the quasiparticle properties of a real material system. Let us return to Figure 8, which compares both the Kohn–Sham gaps and the GW band gaps of a number of insulating materials against the experimentally measured quasiparticle band gaps. A perfect agreement between theory and experiment would place the data points on the diagonal line. As we discussed before, the Kohn–Sham gaps in the LDA (or GGA) significantly underestimate the experimental values, giving rise to the well-known band-gap problem. Some of the LDA Kohn–Sham
59
gaps are even negative. However, the GW quasiparticle energies (which provide an appropriate description of particle-like excitations in an interacting system) result in band gaps that are in excellent agreement with experiments for a range of materials – from the small gap semiconductors such as InSb, to moderate gap materials such as GaN and solid C60 and to the large gap insulators such as LiF. Similarly accurate results have been obtained for surface-state band gaps and for nanostructures.
1.02.8 The GW Formalism and Applications The procedure for an ab initio GW calculation of the quasiparticle energies and wave functions is depicted in Figure 11. In principle, the GW formalism is not a density-functional-based formalism – nor does it depend on a particular choice of basis set or input wave functions so long as the basis is complete and the Dyson’s equation is solved in full (i.e., diagonalized within the basis). However, as discussed above, a good mean-field starting solution is desirable. In practice, GW calculations often take as input the Kohn–Sham wave functions and eigenvalues. For the case of most bulk semiconductors, it is further found that the Dyson’s equation as a matrix equation is nearly diagonal if it is expressed in terms of the basis of the LDA Kohn–Sham eigenstates. This is not
Procedure for ab initio GW calculations Mean Field Solution {ϕ nk (r ), εnk } Construct ε (q, G, G′, ω ) Construct W= ε–1Vc Construct Σ=GW
Diagonalize H =
p2 2m
+ VH + Σ{ϕ nk (r), Enk }
Figure 11 Procedure for carrying out GW calculations as developed by Hybertsen and Louie. From Hybertsen MS and Louie SG (1985) First-principles theory of quasiparticles: Calculation of band gaps in semiconductors and insulators. Physical Review Letters 55: 1418.
60 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
to say that LDA does an accurate job representing the quasiparticle dispersion in the system; it only says that the Schro¨dinger-like equation with either the LDA Vxc or ¼iGW often has similar eigenfunctions. They may have significantly different eigenvalues as illustrated in the band-gap problem discussed above. In the case of silicon, the LDA wave functions differ from the fully diagonalized GW wave functions by less than 0.1% (Hybertsen and Louie, 1985). However, there are important cases where expressing as a matrix within the LDA orbital basis does not yield a diagonal representation and a more preferred basis set can be used. We discuss such cases below. In standard GW studies, the frequency-dependent polarizability and dielectric function are calculated within the random-phase approximation (RPA) using quantities from the mean-field reference system. W is constructed from Equation (38) and G is constructed from Equation (35). In principle, we then diagonalize Equation (36) for the quasiparticle wave functions and eigenvalues with the approximation that ¼ iGW. However, since in many cases, as discussed above, only the diagonal elements are sizable within the mean-field orbital basis, in applications to real materials, the effects of can be treated within first-order perturbation theory. In this approach, the idea is to consider in the form ¼ Vxc þ ( Vxc), where Vxc is some appropriately chosen independent particle mean-field approximation to the exchange– correlation, and then to calculate within perturbation theory the effect of the additional interaction ( Vxc) within the GW approximation. For moderately correlated electron systems, the bestavailable mean-field Hamiltonian may be taken to be the Kohn–Sham Hamiltonian. Thus, for most studies in the literature, the Green’s function is typically constructed using the Kohn–Sham wave functions and eigenvalues. In principle, the process of correcting the eigenfunctions and eigenvalues (which determine W and G) could be repeated until self-consistency is reached or diagonalized in full; however, in practice, it is found that an adequate solution is often obtained within first-order perturbation theory on the Dyson’s equation for a given . Comparison of calculated energies with experiment shows that this level of approximation is very accurate for semiconductors and insulators and for most conventional metals. Also, there is only very limited experience on the importance of including the detailed structures of the spectral function selfconsistently for the quasiparticle energies; and it seems that the effects of self-consistency tend to
cancel those of vertex corrections to (Hedin and Lundqvist, 1969; Holm and von Barth, 1998; Aryasetiawan and Gunnarsson, 1998; Aulbur et al., 2000). However, as in the case of the effect of higher-order terms or vertex corrections to , only an a posteriori experience truly justifies the approximations used. A typical GW calculation using the procedure in Figure 11 has a few technical issues worthy of some discussion. The key ingredient to the calculation (that is not found in HF, for example) is the full two-point dynamic dielectric function within RPA. The calculation of this function is based on the polarizability and has been traditionally computed as a sum over Kohn–Sham empty orbitals. Both the calculations of the empty orbitals and the matrix elements are considerably more demanding than that of a typical ab initio pseudopotential DFT calculation for the ground state. Even requiring a larger number of empty orbitals is the expression for itself. can be broken into two terms: COH þ SEX. SEX, called the screened-exchange term, is simply the exchange, or Fock, term from Hartee– Fock with Vc replaced by W. COH is an entirely new term absent in HF theory describing the interaction of an electron with the Coulomb-hole created around the charge quasiparticle. In the static limit, COH is a local operator but in the dynamic limit, the COH term, like the polarizability, involves a sum over the empty orbitals. This sum converges very slowly with respect to the number of empty orbitals. In practice, the dependence of COH on empty orbitals significantly increases the CPU time needed to generate the large number of initial Kohn–Sham orbitals. However, although each quasiparticle state’s COH converges quite slowly, often needing thousands of empty orbitals, differences between quasiparticle energies converge significantly faster. In bulk systems, where absolute energies are unimportant, it is sufficient to converge the calculation to this lower number of conduction bands, whereas in nanosystems such as molecules, it becomes essential to often use huge number of empty orbitals, up to an energy cutoff of 10s of Rydbergs. Recent research has focused on approximating this sum with various schemes, such as calculating the high-energy state contribution within the static limit or reformulating the problem so that no empty orbitals are involved (Tiago and Chelikowsky, 2006; Giustino et al., 2009; Wilson et al., 2009; Umari et al., 2009a,b; Kohn and Sham, 1965; Bruneval and Gonze, 2008; Reining et al., 1997).
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
10 Ge
8 6 4
L4,5
Γ8
L6
Γ6
2 Energy (eV)
CBM
Γ7 Γ8
0
L4,5
–2
X5
Γ7
L6
X5
–4 –6 L6 –8 X5
–10 L 6 Γ6
–12 –14 L
Λ
Γ Wave vector k
Δ
X
Figure 12 GW band structure of bulk Ge as compared to inverse photoemission experiments (open and full symbols) Modified from Ortega JE and Himpsel FJ (1993) Inverse-photoemission study of Ge(100), Si(100), and GaAs(100) – bulk bands and surface-states. Physical Review B 47: 2130.
The full quasiparticle band structures calculated within the GW approximation for bulk semiconductors agree very well with data from photoemission and inverse photoemission measurements. Figure 12 shows an early calculation of the quasiparticle band structure of Ge (Hybertsen and Louie, 1985; Ortega and Himpsel, 1993) as compared to photoemission data for the occupied states and inverse photoemission data for the unoccupied states. The DFT LDA band structure for Ge exhibits no gap at the point, that is, the system is predicted to be a metal. This incorrect metallic behavior is corrected in the GW calculation. For Ge, the agreement in Figure 12 is within the error bars of experiments. In fact, the conduction band energies of Ge were theoretically predicted (Hybertsen and Louie, 1985) before the inverse photoemission measurement (Ortega and Himpsel, 1993). For bulk Si and diamond, in the original work of Hybertsen and Louie, the indirect gap is corrected from 0.52 eV to 1.29 eV and 3.9 eV to 5.6 eV respectively as shown in Figure 8. Thus, we see that quasiparticle calculations within the GW approximation effectively resolve the so-called band-gap problem discussed above.
61
For isolated nanosystems (molecules, clusters, nanotubes, nanowires, etc.) there are a few more technical issues to consider. Since the calculations are typically done with periodic boundary conditions in a supercell geometry (Cohen et al., 1975), the effects of screening by neighboring systems when a charge quasiparticle is created are often large for practical supercell dimensions. An efficient way to eliminate this spurious interaction is to replace the Coulomb interaction with a truncated Coulomb interaction in both the construction of the dielectric function " ¼ 1 Vtrunc and in W itself. The truncated Coulomb interaction is equal to the true Coulomb interaction, which goes as 1/r, within a region encompassing the entire nanostructure, but is zero outside of this zone (Spataru et al., 2004b; Ismail-Beigi, 2006). Beyond the need for truncating the Coulomb interaction, nanostructures require other considerations when applying the GW formalism. For molecules, it is often the case that H ¼ – Vxc is not diagonal within the Kohn–Sham eigenfunction basis. That is, the Kohn–Sham orbitals are not the same as the quasiparticle states of the molecule for some of the states. In particular, quasiparticle states above or near the vacuum energy can change character dramatically going from a DFT–LDA to a GW description. One such example is the silane (SH4) molecule (Rohlfing and Louie, 1998a). The lowest unoccupied molecular orbital (LUMO) has a quasiparticle energy of 1.1 eV above the vacuum level after applying the GW approximation in the common perturbation theory approach outlined above, but has a quasiparticle energy of only 0.27 eV above the vacuum level if the Dyson’s equation is fully diagonalized (Rohlfing and Louie, 1998a). It is important to note that this is not a limitation of the GW approach but is a limit of treating the selfenergy within perturbation theory starting with LDA eigenfunctions. For such systems, one must either find a better mean-field starting point for the first-order perturbation procedure or fully solve the Dyson’s equation. Semiconducting single-walled carbon nanotubes (SWCNTs) illustrate the importance of self-energy corrections to the Kohn–Sham eigenvalues in lowerdimensional systems. Carbon nanotubes, like graphene nanoribbons, are derived from nanometer wide strips of graphene. In the case of carbon nanotubes, though, the strips are rolled up into 1D tubes. Depending on the rolling (chiral) angle and on the tube diameter, nanotubes can be either metallic or semiconducting with
62 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
(8, 0)
Enk – EnkLDA (eV)
1.5 1 0.5 0 –0.5 –1 –3
–2
–1
1 0 EnkLDA – EF (eV)
2
3
Figure 13 GW corrections to the LDA eigenvalues in the (8,0) SWCNT. Modified from Spataru CD, Ismail-Beigi S, Benedict LX, and Louie SG (2004a) Excitonic effects and optical spectra of single-walled carbon nanotubes. Physical Review Letters 92: 077402.
varying energy gaps. Figure 13 shows the self-energy corrections to the Kohn Sham eigenvalues for an (8,0) semiconducting single-walled carbon nanotube (SWCNT). The LDA Kohn–Sham eigenfunctions do represent a good approximation of the quasiparticle wave functions for this system, yet the eigenvalues are dramatically changed between solutions to the Dyson’s equation and the Kohn–Sham equation. For the (8,0) SWCNT, the electronic energy gap is opened up by over one eV, yielding a quasiparticle gap more than twice the Kohn–Sham gap. Obviously, such a change dramatically impacts researchers’ ability to connect with experiments that directly probe the quasiparticle states, not the fictitious Kohn–Sham states. It also illustrates the role of reduced dimensionality which, in general, enhances electron interaction effects. In addition, a scaling of the band dispersion of approximately 10% occurs for the (8,0) tube, increasing the bandwidths of the carbon nanotube. Although metallic nanotubes seem to be outside of the scope of this volume on semiconducting systems, the quasiparticle properties of metallic nanotubes in many ways are more similar to semiconducting systems than typical bulk metals – as we see in the sections on optical properties. In metallic SWCNT tubes, there is no quasiparticle energy gap at the Fermi energy, but a quantity of central experimental interest is the band velocity, the slope of the quasiparticle energy dispersion, which regulates the carrier dynamics. In the usual nomenclatures, the (n,n) SWCNTs are metallic carbon nanotubes with increasing diameter as the integer n increases
(Saito et al., 1998). GW calculations show that the band velocity of the metallic tubes is renormalized due to electron–electron interactions, with increases by up to 30% from the Kohn–Sham value in the large diameter limit (i.e., for graphene). In the case of graphene, where the band structure about the Fermi energy can be described by the conical dispersion of 2D massless Dirac fermions, this quantity has been directly measured in transport (Zhang et al., 2005; Novoselov et al., 2005) and in angle-resolved photoemission spectroscopy (ARPES) experiments, which shows excellent agreement with the GW value (Siegel et al., in press). Similar to those in carbon nanotubes, the quasiparticle excitations in graphene nanoribbons also show important self-energy effects (Yang et al., 2007b). Figure 14 shows the band gap versus ribbon width for the three families of armchair-edge graphene nanoribbons. In each case, the band gap is dramatically increased with the inclusion of quasiparticle self-energy corrections. Similar results were obtained for the zigzag edge graphene nanoribbons. We note that, in Figure 14, the quasiparticle band gap scales roughly as the inverse of the width and has a value of near 1 eV for graphene nanoribbons with width about a few nanometers, a very desirable range for possible electronics applications.
6 LDA GW 3p + 1 3p + 2 3p
5
Band gap (eV)
2
4 3 2 1 0
4
6
8
10 12 Width (Å)
14
16
Figure 14 GW quasiparticle band gaps (red solid) compared to the corresponding LDA band gaps (blue empty) for armchair edge graphene nanoribbons of varying width for the three families N¼3p, 3pþ1,3pþ2 where p is an integer, as described in Figure 6. From Yang L, Park C-H, Son Y-W, Cohen ML, Louie SG (2007) Quasiparticle energies and band gaps in graphene nanoribbons. Physical Review Letters 99: 186801.
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
From the above examples, it is clear that the correct formalism, including many-electron effects, is extremely important in describing the electronic properties of semiconductors, particularly nanostructured systems, in connection with quasiparticle excitations such as those measured in photoemission, tunneling, and other transport experiments. In many cases, neglecting these effects can lead to errors on the order of magnitude of the quantities being measured or even give qualitatively incorrect behavior –for example, germanium is predicted to be a metal within DFT–LDA while the GW approach correctly captures its semiconducting character and yields quantitatively the value of the band gap.
1.02.9 Two-Particle Excitations and the Bethe–Salpeter Equation
63
Here avþ and bcþ are the second-quantization creation operators for holes and electrons, respectively. For simplicity of notation, we use v and c to represent both the band index and wave vector of the hole and electron state, respectively, and we use jvc i ¼ avþ bcþ jN ;0i to denote a configuration in which a hole with wave vector k is created in the valence band v and a quasi-electron is created in the conduction band c with wave vector k þ q, as illustrated in Figure 15, where the q 0 limit is taken. Electron–electron interactions mix electron–hole configurations of the same center-of-mass q to form the excited (excitonic) state jN ;S i. The electron–hole amplitude or exciton wave function in real space, which describes the spatial correlation of the electron and hole, is then given by S ðr; r9Þ ¼
X
AScv cc ðrÞcv ðr9Þ;
ð41Þ
cv
We now discuss the formalism for the ab initio calculation of optical properties. For an interacting electron system, such as a semiconductor, we must include the interaction between the quasi-electron and the quasi-hole created in the optical excitation process. Thus, the study of the optical response requires knowledge of the correlated electron–hole excitations, jN ;0i ! jN ;S i, that do not change the total number of electrons N in a system. (Here jN ;S i denotes the Sth neutral excited state). That is, we consider excitations for which no particles are added or subtracted from the system. Just as the quasiparticle excitations are given by the one-particle Green’s function G, the electronhole excitations may be obtained by investigating the interacting two-particle Green’s function G2 (which is a two-particle generalization of Equation (24)) and solving its equation of motion. In the above discussion on quasiparticles, we assume the single-particle excitations are long-lived; the theory on optical excitations here also assumes that the electron–hole excitations are long-lived and that the Tamm–Dancoff approximation (Kadanoff and Baym, 1999) is valid. The excited state jN ;S i with energy S (referenced to the ground-state energy) can then be expressed as a linear combination of noninteracting electron–hole configurations plus correction terms that are not important in the evaluation of the optical oscillator strength: j N ; Si ¼
hole X elec X v
c
ASvc avþ bcþ j N ; 0i þ
ð40Þ
with c(r) being the quasiparticle amplitudes (or wave functions), for example, those obtained from a GW calculation. As in the case of the Dyson’s equation for quasiparticles, we use the two-particle Green’s function’s equation of motion to arrive at an effective equation for AS and S. These quantities can be shown to satisfy a Bethe–Salpeter equation of the form (Strinati, 1988; Rohlfing and Louie, 2000)
X EckQP – EvkQP ASvck þ vckK eh v9c9k9 ASv9c9k9 k9v9c9
¼ S ASvck :
ð42Þ
Here the EQPs are the quasiparticle energies and Keh is the electron–hole kernel describing the interaction between the excited electron and hole. Solving this Bethe–Salpeter equation yields the excitation energies and the excited-state wave functions from which one can compute the optical absorption spectrum, exciton binding energies and wave functions, and other related optical quantities. (a)
c
c
c
(b)
υ υ υ Figure 15 (a) Schematic of a photo-excited exciton state as a superposition of independent electron–hole vertical transitions. (b) Feynman diagrams for the BSE consisting of a repulsive exchange term and an attractive direct interaction.
64 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
K ð12; 34Þ ¼
½VH ð1Þð1; 3Þ þ ð1; 3Þ : Gð4; 2Þ
ð43Þ
where the self-energy operator, , is typically treated within the GW theory in practical implementations. With the additional assumption that the functional derivative of W with respect to G can be neglected (Rohlfing and Louie, 2000), K is simplified to K ð12; 34Þ ¼ – ið1; 3Þð2 – ; 4ÞV ð1; 4Þ þ ið1; 4Þð3; 2ÞW ð1þ ; 3Þ
ð44Þ
¼ Kx þ Kd :
Equation (44) has two terms, as illustrated in Figure 15: a bare repulsive exchange interaction Kx and a screened attractive interaction Kd. The methods for the evaluation of the matrix elements of the kernel K and the solution to the Bethe–Salpeter equation have been discussed in detail in the literature (e.g., Rohlfing and Louie, 2000). When spin is considered, the exchange term only affects the singlet excitons. Dynamical (i.e., the frequency dependent) screening can further be neglected in Kd if the excitation energies S are close to the energy of the noninteracting pairs – in other words, the excitonbinding energy is small compared to the optical excitation energy itself. From the solution of the Bethe–Salpeter equation (Equation (42)), the optical absorption spectrum and other optical properties are obtained from the imaginary part of the macroscopic dielectric function, "2 ð!Þ ¼
162 e 2 X jhN ; 0 j e?v j N ; S ij2 ðs – h!Þ ð45Þ !2 s
where e is the normalized polarization vector of light, and v ¼ ð1=hÞ½H ; r is the single-particle velocity operator. The importance of inclusion of the electron–hole interaction comes from the nontrivial correlations of different electron–hole configurations jvc i in the excited state, jN ; S i, leading to coherence effects in the optical transition matrix elements: hN ; 0 j e?v j N ; Si ¼
X cv
AScn hc j e?v j vi:
ð46Þ
Equation (46) is a coherent sum of the interband transition matrix elements of the individual contributing electron–hole pair transitions where the amplitudes and phases of the AS often lead to interesting interference effects. It is absolutely crucial therefore to keep the relative phases of A intact during the calculation and to consider a sufficiently complete basis sets of single-particle transitions. The Bethe–Salpeter formalism for two-particle excitations is therefore a natural extension of the GW formalism for quasiparticle excitations. In firstprinciples implementations, calculations within both formalisms use the same approximations for the selfenergy and Green’s Function G. Figure 16 illustrates the importance of the BSE framework in calculating the imaginary part of the dielectric function in silicon (Rohlfing and Louie, 1998a; Albretch et al., 1998; Benedict et al., 1998). Only after including excitonic effects is quantitative agreement with experiment in the first peak height and also the position of the second peak reached, resolving a long-standing puzzle in the field. This redistribution of oscillator strength does not derive from the existence of bound excitons at the peak position but of the mixing interband excitations from a large k and energy range giving a large enhancement in the first absorption peak. Such rearrangements of optical strength also give rise to an apparent shift of nearly 50
Expt Interacting Nonint.
40
30 ε2
The interaction kernel, Keh, is an operator that describes the scattering of an electron–hole pair going from one configuration to another ðjvc i ! jv9c9iÞ due to the Coulomb interaction among the electrons. Using the notation 1¼(r1, 1, t1), K is given by the functional derivative (Rohlfing and Louie, 1998a; Strinati, 1988)
20
10
00
2
6 4 Energy (eV)
8
10
Figure 16 The imaginary part of the macroscopic dielectric function, "2(!), as a function of incident photon energy. Dashed line: without electron–hole interaction effects. Solid red line: with electron-hole interaction effects. Circles are data from ellipsometry experiment. Modified from Jellison GE, Chisholm MF, and Gorbatkin SM (1993) Optical functions of chemical vapor deposited thin-film silicon determined by spectroscopic ellipsometry. Applied Physics Letters 62: 3348.
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
0.5 eV for the second peak in Figure 16 (Rohlfing and Louie, 1998a). Beyond the oscillator strength redistribution, as in a simple hydrogenic model for excitons in semiconductors, a series of bound exciton states is found below the onset of absorption edge, though their oscillator strength is weak and binding energy is only of the order of 10 meV for Si. As seen below, the GW–BSE approach is capable of obtaining highly accurate optical spectra from first principles and has helped elucidate the optical response of a wide range of systems ranging from nanostructures to bulk semiconductors to surfaces and defects to 1D systems such as polymers and nanotubes.
1.02.10 The GW-BSE Formalism and Optical Response Figure 17 illustrates the procedure for carrying out an ab initio GW–BSE calculation to obtain quasiparticle and optical properties. The BSE calculations take as input the quasiparticle energies and wave functions from a GW calculation. In addition, the BSE calculation takes the full dielectric matrix calculated in the GW step, which is used to screen the attractive direct electron–hole interaction. The direct and exchange part of the electron–hole kernel are calculated, and the electron–hole kernel plus the kinetic energy term as a matrix in the quasiparticle electron–hole pair basis is then diagonalized yielding the exciton wave functions and excitation energies. Exciton binding energies can be inferred from the
65
energy of the correlated exciton states relative to the interband transition continuum edge. With the excitation energies and amplitudes of the electron–hole pairs, A, one can then calculate the macroscopic dielectric function for various light polarizations using Equation (45), and also obtain quantities such as electron–hole correlation functions in real space and higher-order optical effects such as multi-photon absorption and phonon-assisted absorption spectra. When applying this method to isolated nanosystems in supercell calculations, it is important, as it is in the GW calculation, to replace W with the appropriate screened truncated interaction. As will be illustrated in the final section, even with well-separated systems that are considered reasonable in DFT calculations, the nature of the interaction between the electron and hole in an untruncated interaction calculation can be very different from the isolated case owing to the unwanted influence of neighboring replicas. Because of the generally reduced screening and confinement effects, one expects stronger excitonic effects in reduced dimensional systems, which as seen below, is indeed the case. Figure 18 depicts the optical spectrum of bulk GaAs. As seen from this figure and the results for Si in Figure 16, only after the inclusion of electron– hole interaction, there is good agreement between theory and experiment. As already discussed above for Si, there is a redistribution of optical oscillator strength to the lower energy peak, enhancing it by nearly a factor of 2. In addition, the electron–hole interaction shifts and heightens the second main peak, yielding agreement with experiment. This 30
Procedure for GW-BSE calculations
25
Mean Field Solution {ϕnk (r ), εnk }
ε2 GW solution
{ϕQPnk (r
),
εQP
nk }
GaAs
20 15 10 5
Construct Bethe−Salpeter K (k,c,v,k′,c′,v ′)
Diagonalize BSE Hamiltonian A S c,v,k,E S
Figure 17 Procedure for a GW-BSE calculation. The end result of such a procedure is the ab initio calculation of the quasiparticle and exciton states of a material system as well as a description of the optical properties through the expression of the oscillator strength, Equation (45).
0
0
2
4
6
8
10
Energy (eV) Figure 18 Calculated optical absorption of GaAs with (solid line) and without (dashed line) electron–hole interaction, compared to experimental data [Aspnes, Lautenschlager]. Modified from Rohlfing M and Louie SG (1998a) Excitonic effects and the optical absorption spectrum of hydrogenated Si clusters. Physical Review Letters 80: 3320.
66 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
y
z
x
Figure 19 Ball and stick model of a silicon nanowire terminated with hydrogen (small atoms). The wire is periodic in the z-direction but finite in the x and y directions. Modified from Yang L, Spataru CD, Louie SG, and Chou MY (2007c) Enhanced electron–hole interaction and optical absorption in a silicon nanowire. Physical Review B 75: 201304(R).
No e–h interaction (// polarization)
4 2 A
α2(ω) (nm2)
shift of nearly 0.5 eV in the second peak (with position near 5 eV) is not due to a negative shift of the transition energies since the density of photo-excited states remains nearly unchanged by the electron– hole interaction from that of the non-interacting case. The changes in the position and height of the second peak in the optical spectrum in Figure 18 originate from the coupling of states over a large energy range, effectively leading to a constructive coherent superposition of oscillator strengths for states on the lower-energy side of the peak and a destructive superposition for states on the highenergy side. Spin–orbit interaction was not included in the calculated results in Figure 18, and, hence, some of the fine structures associated with the two peaks in the experimental spectrum of GaAs are not reproduced in the theory. Within the GW–BSE approach, in addition to oscillator strength redistribution in the continuum part of the spectrum, discrete bound exciton states below the absorption edge are also obtained. These states come out directly from the BSE calculation without the use of any effective mass or other approximations. The calculated binding energy of the lowest energy exciton states in GaAs (Rohlfing and Louie, 1998a) is 4.0 meV as compared to experimental value of 4.2 meV (Michaelis et al., 1996; Sell, 1972). The theory has reproduced essentially all the observed bound excitonic structures to a good level of accuracy in GaAs. Thus, it is possible to obtain the binding energy of bound excitons very accurately from Equation (42), even if the binding energy is only of the order of few meVs. To achieve such a good description of the optical response of GaAs, a very fine sampling of the Brillouin zone and the inclusion of spin–orbit interaction are required, so that the quasiparticle energy bands are calculated with high accuracy. Electron–hole effects are even more important when applied to silicon nanowires (Yang et al., 2007c). Silicon nanowires are quasi-1D solid wires of silicon terminated by passivating adsorbates on the wire surface. Such nanowires represent a potential building block for nanoscale devices and have recently been synthesized (Cui and Lieber, 2001; Duan et al., 2001; Morales and Lieber, 1998; Holmes et al., 2000). Si nanowires with diameter as small as 1.3 nm have been fabricated (Ma et al., 2003). Figure 19 depicts a model of a Si nanowire passivated with hydrogen of a diameter of 1.2 nm, with the wire axis along the [110] direction.
0 With e–h interaction (// polarization)
4 2
A1
0 With e–h interaction (⊥ polarization)
2
0 2
4
6
8
Energy (eV) Figure 20 Absorption vs. photon energy for the silicon nanowire shown in Figure 19 for light polarized (top two panels) parallel to the wire axis and (bottom panel) perpendicular to the wire axis. The arrows marked by A and A1 indicate the fundamental gap and the lowest-energy exciton locations, respectively. The top panel shows the spectrum without electron–hole interaction effects included, whereas the middle panel contains the full solution to the BSE. The bottom panel illustrates that light polarized perpendicular to the wire axis is absorbed very weakly due to depolarization effects. Modified from Yang L, Spataru CD, Louie SG, and Chou MY (2007c) Enhanced electron–hole interaction and optical absorption in a silicon nanowire. Physical Review B 75: 201304(R).
In Figure 20, we show the calculated optical spectra of the Si nanowire shown in Figure 19 employing the GW-BSE approach, with and without electron–hole interaction effects included. The absorption for light polarized parallel to the wire axis exhibits two new peaks with inclusion of
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
67
(and quasiparticle self-energy) effects (Spataru et al., 2004a, 2004b; Ando, 1997) are shown to be dominantly important in these systems even at room temperature. Figure 21 compares the calculated absorption spectrum of a (8,0) semiconducting SWCNT between the cases with and without electron–hole interactions. As for the Si nanowires, because of depolarization effects (Ajiki and Ando, 1994), the optical response of a tube-like object is the strongest when the polarization of the light is along the tube axis – the polarization used in the calculations. The two resulting spectra are radically different. When electron–hole interaction effects are included, the spectrum is dominated by bound and resonant exciton states. Each of the structures derived from a van Hove singularity in the noninteracting joint density of states gives rise to a series of
electron–hole interaction, giving rise to an appearance of a shift of the spectrum by nearly 2 eV (comparison of top and middle panel in Figure 20). The dominant peaks in the spectrum (middle panel) are due to resonant excitons. In addition, there is a bound exciton state although with very weak optical strength, labeled A1, deep inside the quasiparticle band gap at 2.1 eV with a binding energy of 1 eV. This binding energy is two orders of magnitude larger than the exciton-binding energies in bulk Si. This enhanced binding energy is a result of the reduced dimensionality of the system, and of the unique nature of screening in lower dimensions as discussed in the next section. When the GW-BSE method is applied to semiconducting SWCNTs, the results are extraordinarily. Because of the 1D nature of the nanotubes, excitonic
(a) 250 (8,0)
A′1 200
(b)
With e−h interaction Without e−h interaction
ε2 (a.u.)
150 B′1
C′1
100
Exciton A’1 A′2
50
A′2 0
1
1.5
B′2
A
C
B C′2
2
2.5
3
3.5
4
Photon energy (eV)
(c) 300
(d) 100 Exciton A′1
Exciton C′1 75
|φ|2
|φ|2
200 50
100 25
0 −50
−25
0 Z (Å)
25
50
0 −100
−50
0
50
100
Z (Å)
Figure 21 Excitons in semiconducting (8,0) SWCNTs. (a) Absorption spectra plotted with and without the electron–hole interaction included. The oscillator strength from the continuum is almost entirely moved to the sharp bound exciton transitions. (b) Exciton amplitude square plotted with the hole position fixed (dot in the center). (c,d) Exciton amplitude square plotted along the tube axis with hole position fixed at z¼0.
68 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
exciton states, and these states rob all the optical transition strength from the continuum transitions. Panels (b), (c) and (d) in Figure 21 give the spatial correlation between the excited electron and hole for two of the exciton states, one bound (A’1) and one resonant state (C’1). The extent of the exciton wave function is about 2 nm for both of these states. For the (8,0) nanotube, the lowest-energy bound exciton has a binding energy of nearly 1 eV. Note that the excitonbinding energy for bulk semiconductors of similar size band gap is in general only of the order of tens of meVs, illustrating again the dominance of many-electron Coulomb effects in reduced dimensional systems. Similar results have been obtained for other semiconducting carbon nanotubes (Spataru et al., 2004a, 2004b). These results predicted extraordinarily large exciton binding energies, although first met with skepticism, have been verified by recent experiments (Wang et al., 2005; Ma et al., 2005). Owing to the unique electronic structure of the carbon nanotubes, in addition to the optically active (or bright) excitons shown in Figure 21, there exist also a number of optically inactive (or dark) excitons associated with each of the bright ones. These dark excitons can be observed experimentally with an applied magnetic field or light with different polarization. These dark excitons also play an important role in the optical properties of the nanotubes; for example, they strongly affect the radiative lifetime of the optically created excitons in semiconducting carbon nanotubes (Spataru et al., 2005). For the boron nitride (BN) nanotubes, which are wide band-gap materials, it is found that excitonic effects are even stronger than in SWCNTs. Figure 22 shows the calculated optical spectra of an (8, 0) BN 32 28
With e−h interaction Without e−h interaction
24
I1
(b) 6 E QP (eV)
α2 (ω) (nm2)
(a)
nanotube. Again, the electron–hole interaction leads to a series of sharp lines in the absorption spectrum due to strongly bound excitonic states, and the optical strength as in the case of carbon nanotubes is virtually completely transferred to these states. For the (8, 0) BN nanotube, which has the same diameter as the (8, 0) SWCNT, the exciton binding energy of the lowest energy exciton is over 2 eV (Park et al., 2006). This is consistent with the fact that BN nanotubes are widegap insulators. Unlike the carbon case, however, the lowest-energy exciton is composed of almost equal weight from four sets of nearby interband transitions, which lead to a considerably more localized electron– hole correlation in real space along the circumference direction. A surprising prediction was further made when GW-BSE calculations were carried out for the metallic SWCNTs – bound/resonant exciton states exist despite metallic screening from the carriers (Spataru et al., 2004a; Deslippe et al., 2007). To our knowledge, these are the first bound excitons known to exist in metallic systems, and they have subsequently been observed by measurements (Wang et al., 2007). The existence of excitons in metallic SWCNTs is a result of the increased efficiency of an attractive interaction in forming bound states in one dimension. This can be seen as a result of the fact that, in one dimension, (1) electronic screening is greatly reduced and(2) any Rpotential (other than V ¼ 0) satisfying the constraint V ðx Þdx 0 (i.e., is negative on average) is guaranteed to have at least one bound state (Kocher, 1977; Simon, 1976). Figure 23 shows the LDA band structure for a (10, 10) and a (12, 0) metallic SWCNTs. For incident
20 16 12 8
4 2 0
I′1
K II1
4 5
I2 I3
(x4)
–2
I4
6 7 8 Photon energy (eV)
9
Γ
X
Figure 22 Absorption spectra and band structure of a (8,0) boron-nitride nanotube. (a) The spectrum is plotted with (solid) and without (dashed) electron–hole interaction included. (b) Schematic of the four pairs of bands from which mixing of these interband transitions forms the lowest bound exciton. Modified from Park CH, Spataru CD and Louie SG (2006) Excitons and many-electron effects in the optical response of single-walled boron nitride nanotubes. Physical Review Letters 96: 126105.
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
69
2
Energy (eV)
2
1 0
0
(10,10)
(12,0)
−1 −2
−2
−3 0
0.2
0
0.4
0.2
0.4
k(2π/a) Figure 23 Band structure of metallic nanotubes showing allowed optical transitions (solid arrow) and forbidden optical transitions (dashed arrow).
Without e−h interaction With e−h interaction
α (ω) (a.u.)
light polarized along the tube axis, the optical perturbation operator, A ? p, belongs to the identity representation in the group of the k-vector (Deslippe et al., 2007; Barros et al., 2006a). Therefore, the optical transitions obey well-defined selection rules determined by the representations of this group (Barros et al., 2006a, 2006b). This is a result of the quasi-1D nature of nanotubes where every k-point lies along the same high-symmetry direction causing optical transitions to obey well-defined selection rules across entire subbands instead of just at points of high symmetry. The first optically allowed transitions in the (10,10) tube occur between the second and third valence subbands (the bands are degenerate) and the second and third conduction subbands (shown by the solid red transition line drawn in Figure 23. The facts that (1) there exists a symmetry gap preventing optical transitions between the linear dispersing bands that cross at the Fermi energy, and 2), in achiral tubes, electron–hole pairs formed from these bands are of a different representation of the group of the k-vector (Barros et al., 2006a, 2006b) than those of the first optically active electron–hole pair (preventing mixing between these two types of electron–hole states) are both factors that also contribute to the prominence of the excitons in the metallic carbon nanotubes. Mixing does occur between the continuum of electron–hole states associated with the first active interband transition (denoted by E11) and excitons associated with the second interband transitions (E22); however, owing to the dominance of the joint density of states (JDOS) at the E22 van Hove singularity, resonant excitons are formed with a contribution of over 90% from E22 transitions. Figure 24 presents the optical absorption spectra for the (12,0) and (10,10) carbon nanotubes over the energy range of the first three optically allowed
Without e−h interaction With e−h interaction
1.5
2
2.5 3 3.5 Energy (eV)
4
4.5
Figure 24 Absorption spectra of the (10,10) SWCNT (top) and (12,0) metallic nanotubes (bottom). Although these tubes are metallic, they behave optically as if they were semiconductors. The binding energy for exciton states in each tube is 50 meV, greater than the exciton binding energy in bulk Si.
inter-subband transitions. Each feature in the fully interacting spectrum is dominated by a single symmetric peak followed by a smaller tail derived from the 1D van Hove singularity located at the position of the peak in the noninteracting spectrum (Deslippe et al., 2007). In both metallic tubes, excitonic effects qualitatively change the spectrum due to the existence of the exciton states. In the (10,10) tube, the bound exciton associated with the first interband transition has a binding energy of 50 meV. Figure 25 provides a more detailed view of the lowest exciton in the
70 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
Without e−h Int. With e−h Int.
α 2 (ω)
100
Without e−h Int. (scaled) With e−h Int.
25
80
20
60
15
40
10
20
5
0 2
1.5
0
2.5
1.5
2
2.5
Energy (eV) Figure 25 Spectral shape of absorption spectra in the (10,10) metallic nanotube plotted with (solid) and without (dashed) the electron–hole interaction and with Lorentzian broadening of 20 meV (left panel) and 80 meV (right panel). The right panel plot has the noninteracting spectra shifted and scaled to match the position and height of the interacting spectra for comparison.
(10,10) SWCNT. In the left panel of Figure 25, we see that the optical peak arises almost entirely from the single exciton state, showing a profile that is significantly different from the interband transition case. As a comparison, the right panel in Figure 25 shows the noninteracting spectrum scaled and shifted to match the peak height in the interacting case. We see that the two lineshapes are distinctly different. This predicted change in lineshape was recently confirmed by experiment using optical absorption techniques on isolated metallic SWCNTs (Wang et al., 2007). The measured exciton-binding energy
for the (21,21) metallic SWCNT was 50 meV (Wang et al., 2007), in good agreement with values derived from the ab initio calculations (Deslippe et al., 2009). The large excitonic effects described here are not unique to the systems studied but are characteristic of 1D (as well as 0D and 2D) systems in general. Graphene nanoribbons (see Figure 26), 2D graphene and bilayer graphene all show similar large excitonic effects (Yang et al., 2007a, 2008, 2009). This is largely because of the unique nature of screening in reduced dimensional systems that will be discussed in the next section. (b)
(a) E11
E11
1
1
Optical absorption (a.u.)
W/o e−h interaction With e−h interaction
W/o e−h interaction With e−h interaction
11-AGNR
12-AGNR
E11 E11
0
2
4
0
2
4
Energy (eV) Figure 26 Absorption spectra of armchair graphene nanoribbons with 11 carbon dimer rows (left panel) and 12 carbon dimer rows (right panel) forming the width of the ribbon. Oscillator strength from the continuum is moved almost entirely into the sharp bound exciton peaks. Modified from Yang L, Cohen ML, and Louie SG (2007a) Excitonic effects in the optical spectra of graphene nanoribbons. Nano Letters 7(10): 3112–3115.
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
1.02.11 Interactions in Nanostructured Semiconducting Materials The application of the GW-BSE method to semiconducting nanostructured materials has illuminated and predicted several exciting new physical behaviors of these systems. As discussed above, many-body effects are crucial for even a qualitative understanding of their excited state properties. This can be seen as consequence of two general principles: first, the Coulomb interaction is more effective in lower dimensions, and, second, screening has a unique nature in two- or less-dimensional systems. The first point is illustrated in Figure 27. In the case of three (or two) dimensions, a particle under the influence of a Coulomb potential can avoid the singularity by traveling around the origin of the potential. The expectation value of the 3D potential can be approximately expressed as R 3 ðrÞ d r jr j . If is reasonably well behaved, as it is for the hydrogen model, than the 3D integral about the origin integrates to a finite value with R no divergence. However, in the 1D case, dz jðzzjÞ logarithmically diverges if the origin is included in the integral. In the 1D Coulomb interaction, the singularity is unavoidable for an oscillating particle, leading to large binding energies. In real quasi-1D nanosystems, it is the finite extent of the physical system in the other directions that prevents the interaction between charge particles in the system from diverging. For example, one can define an effective screened electron–electron or electron–hole interaction in a quasi-1D system using an averaging in the nonperiodic directions as (Deslippe et al., 2009)
1D
3D/2D
e e h
h
Figure 27 3D/2D vs. 1D electron–hole interaction. In 1D, the singularity is unavoidable, leading to greatly enhanced exciton binding energies and other electron–electron interaction effects. For example, the 1D bare attractive Coulomb interaction has a bound state of infinite binding energy compared to the value of 13.6 eV of the 3D case.
V ðzÞ ¼
Z
71
dx9dy9dr2 W ðr9 þ r2 ; r2 Þjcc ðr9 þ r2 Þj2 jcv ðr2 Þj2 : ð47Þ
The divergence at the origin is removed due to the spatial extent to which the particles can still be separated when the z separation is zero. In an SWNT, this spatial extent is the tube diameter. While not containing a singularity, the potential still leads to exciton binding energies much greater than those found in higher-dimensional systems due to the fact that the phase space over which the interaction is large does not vanish. In order to truly appreciate all the physics contained in the electron–hole kernel, Equation (43), it is useful to model the dielectric function, which determines the screened interaction W. We can express an effective 1D dielectric function in the case of a nanotube as "(q)¼1 – (q)Vbare (q) and the total screened potential generated by a single ring of charge on a tube is then (Deslippe et al., 2009; Leonard and Tersoff, 2002) V ðqÞ ¼
Vbare ðqÞ : 1 – ðqÞVbare ðqÞ
ð48Þ
For the case of a semiconducting tube with band gap, Eg, and diameter, d, the susceptibility, , can be obtained from a 1D generalization of the Penn model (Penn, 1962). The interesting behavior is that the dielectric function in momentum space approaches unity at both large and small q (Leonard and Tersoff, 2002; van den Brink and Sawatzky, 2000). In a typical bulk semiconductor on the other hand, the dielectric function approaches the bulk dielectric constant at small q. This behavior is a direct result of the fact that, unlike in 3D semiconductors where the induced charge is of the opposite sign as the added charge and is a finite fraction (1 – 1/"0) of the added charge, in 1D the total induced charge seen far away from the added charge particle is zero with regions where charge density of the opposite sign as the external charge is induced, as well as regions where charge density of the same sign is induced. A qualitative explanation for the phenomenon is shown in Figure 28 where the polarizable charge distribution of a semiconducting system is modeled by a simple ball-and-spring dielectric medium (Deslippe et al., 2009). In three dimensions, because the surface area of a spherical shell is proportional to the radius squared and the force generated by a charge at the
72 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
h
V(z) (eV)
0
–0.5
(8,0) (8,0) Vbare
h
–1
0
origin on a spring at distance r is proportional to 1/r2, the total induced charge in a shell of radius r is constant with respect to r. So, at large distances from the external charge, there is a net induced charge of the opposite sign observed surrounding the external charge. In quasi-one-dimension, however, the surface area perpendicular to the tube axis of a box does not change as the box length changes in the z-direction. The total induced charge in larger and larger size boxes drops to zero; so at large length scales, there is effectively no screening (Deslippe et al., 2009). What is even more unusual is that at some intermediate distance, the effect of screening in 1D systems, such in SWCNTs, is to actually enhance rather than weaken the Coulomb interaction. That is, there are ranges of separations over which two particles see an anti-screening effect. This is demonstrated for the case of a (8, 0) SWNCT using Equation (48) in Figure 29. Because the screened Coulomb interaction is very different in 1D, a very distinct excitation spectrum is created for the higherenergy exciton states form from a given pair of interband transitions. The actual spectra are qualitatively and quantitatively different from those of excitons in 3D or in 1D models that neglect the anti-screening effect. The higher states in the excitonic series are considerably more bound due to the presence of the anti-screening region; evidences consistent with this prediction have been observed in recent measurements on semiconducting SWCNT (Lefebvre and Finnie, 2007, 2008).
10 z (nm)
15
20
Figure 29 Screened (solid) and bare (dashed) Coulomb interaction in semiconducting (8,0) SWCNTs. Modified from Deslippe J, Dipoppa M, Prendergast D, Moutinho MVO, Capaz RB, and Louie SG (2009) Electron–hole interaction in carbon nanotubes: Novel screening and exciton excitation spectra. Nano Letters 9(4): 1330–1334.
As discussed above, because essentially every k-point in a quasi-1D system is on a high symmetry direction, the optical absorption to exciton states obey strict symmetry selection rules across entire bands. In the semiconducting SWCNTs, the first allowed optical excitation is just one of many (four in the case of the zigzag tubes mentioned below) nearly degenerate exciton states due to the band or valley degeneracy of the quasiparticle states. These exciton states, and their relative oscillator strengths, come out directly from the GW-BSE calculation and are shown in Figure 30 for the (10, 0) SWCNT. These states can be investigated through symmetry breaking probes such as by applying a magnetic field or through careful analysis of photoluminescence spectra (Shaver et al., 2007).
1.02 (10,0) Dark Bright
1 hΩ(0) (ev)
Figure 28 Screening in a polarizable medium in three dimensions (left) and one dimension (right). In three dimensions, the amount of charge induced inside a sphere of radius r is constant as a function of increasing radius, whereas in one dimension the amount of charge induced inside a box of length z goes to zero for large z.
5
–
0E6 – 0A0
δ1
0.98 0E6 A 0B0
0.96
Dark
δ2
-
0B0
0 0
0.94 Triplet
Singlet
0.92 Figure 30 The first optical transition oA 0 in zigzag (10,0) SWCNT is accompanied by several dark states due to the well-defined symmetry in SWCNTs.
Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
1.02.12 Summary In this chapter, we have presented the basic concepts and methodologies in understanding and calculating both ground- and excited-state properties of semiconductors and semiconductor nanostructures. The development of ab initio pseudopotential and DFT has led to the ability to calculate accurate ground-state properties for many systems from first principles. Through the advent of the GW-BSE methodology, it is now also possible to explain and compute the excited-state and spectroscopic properties from first principles employing the interacting Green’s function formalism. Together these methods form a very versatile set of theoretical techniques that can be used on a wide range of material systems with predictive power. We demonstrate their usefulness in understanding and predicting the structural, electronic, and optical properties through some selected examples, with a particular emphasis on the rapidly developing field of nanoscience. The challenges for first-principles theories are evident on several fronts. One is on investigating complex materials with large numbers of different type of atoms in complex arrangements or on processes/phenomena that span very long timescales. The challenges here are mainly due to the greatly increased Hilbert space needed to study large systems, and that time steps in ab initio molecular dynamics simulations are dictated by electronic frequencies. In general, standard DFT, GW, and BSE calculations all tend to have bottlenecks that scale roughly as N3 where N is the number of atoms in the system. Even with this scaling (which is in general superior to quantum chemistry techniques), current methodologies and algorithms have allowed the study of systems containing thousands of atoms at the DFT level and hundreds of atoms at the GWBSE level. Another frontier is the first-principles study of correlated electron systems such as the transition metal oxides, multiferroics, and high Tc superconducting cuprates and pnictides. At the DFT level, standard approximations such as the local spin density approximations (LSDAs) can sometimes give an incorrect ground state for the highly correlated materials. The construction of better exchange–correlation functionals is needed, and is currently an active and important endeavor in the field. Since the GW-BSE formalism is based on many-body perturbation theory, having a good mean-field starting point is essential for more
73
correlated systems. There have been considerable efforts along this direction to improve upon the methodology (Miyake et al., 2006; Bruneval et al., 2006). Also, attempts are being made to practically rid of the empty orbitals in the GW formalism and to include vertex corrections or higher-order terms (Giustino et al., 2009; Wilson et al., 2009; Umari et al., 2009; Bruneval and Gonze, 2008; Reining et al., 1997). For many semiconducting/insulating systems, upon promotion to an excited state, the system may undergo major changes that would give rise to interesting and technologically important phenomena such as Stokes shifts, self-trapped excitons, molecular or defect confirmational changes, etc. Some recent work has been done to include forces within the GW-BSE framework to study the structural effects through the many-body Green’s function formulation (Ismail-Beigi and Louie, 2003). Finally, for the case of nanostructured materials where symmetry plays a larger role, nonlinear optical spectroscopies (Wang et al., 2005; Maultzsch et al., 2005) have proven particularly efficient in characterizing the excited states of the system. Work toward extending the GW-BSE methodology to calculate nonlinear optical properties such as multi-photon absorption, ultrafast spectroscopy, and phonon-assisted absorption is another avenue attracting considerable attention. Progress in these and other areas are anticipated to greatly expand our ability to study from first-principles increasingly complex materials and novel phenomena. (See Chapters 1.01, 1.04, 1.10 and 2.05).
Acknowledgments This work was supported by National Science Foundation Grant No. DMR07-05941 and by the Director, Office of Science, Office of Basic Energy Sciences, Division of Materials Sciences and Engineering Division, US Department of Energy under Contract No. DE-AC02-05CH11231. Computational resources have been provided by NSF at the San Diego Supercomputing Center (SDSC) and DOE at the National Energy Research Scientific Computing Center (NERSC). J. Deslippe acknowledges funding from the DOE Computational Science Graduate Fellowship (CSGF) under grant number DE-FG02-97ER25308
74 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors
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Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors Kittel C and Kroemer H (1980) Thermal Physics, 2nd edn., Freeman. Kocher CA (1977) Criteria for bound-state solutions in quantum mechanics. American Journal of Physics 45: 71. Kohn W and Sham L (1965) Self-consistent equations including exchange and correlation effects. Physical Review A 140: 1133. Lautenschlager P, Garriga M, Logothetidis S, and Cardona M (1987) Interband critical points of GaAs and their temperature dependence. Physical Review B 35: 9174. Lefebvre J and Finnie P (2007) Polarized photoluminescence excitation spectroscopy of single-walled carbon nanotubes. Physical Review Letters 98: 167406. Lefebvre J and Finnie P (2008) Excited excitonic states in single-walled carbon nanotubes. Nano Letters 8(7): 1890. Leonard F and Tersoff J (2002) Dielectric response of semiconducting carbon nanotubes. Applied Physics Letters 81(25): 4835. Li X, Wang X, Zhang L, Lee S, and Dai H (2008) Chemically derived, ultrasmooth graphene nanoribbon semiconductors. Science 319(5867): 1229–1232. Louie SG (1994) Electronic structure of C60 fullerites and nanatubes. In: Kuzmany H, Fink J, Mehring M, and Roth S (eds.) Progress in Fullerene Research: International Winter School on Electronic Properties of Novel Materials, 303. Singapore: World Scientific. Ma DDD, Lee CS, Au FCK, Tong SY, and Lee ST (2003) Smalldiameter silicon nanowire surfaces. Science 299(5614): 1874–1877. Ma YZ, Valkunas L, Bachilo SM, and Fleming GR (2005) Exciton binding energy in semiconducting single-walled carbon nanotubes. Journal of Physical Chemistry B 109: 15671. Mahan GD (1981) Many-Particle Physics. New York: Plenum. Maultzsch J, Pomraenke R, Reich S, et al. (2005) Exciton binding energies in carbon nanotubes from two-photon photoluminescence. Physical Review B 72: 241402. Michaelis JS, Unterrainer K, Gornik E, and Bauser E (1996) Electric and magnetic dipole two-photon absorption in semiconductors. Physical Review B 54: 7917. Miyake T, Zhang P, Cohen ML, and Louie SG (2006) Quasiparticle energy of semicore d electrons in ZnS: Combined LDA þ U and GW approach. Physical Review B 74: 245213. Morales AM and Lieber CM (1998) A laser ablation method for the synthesis of crystalline semiconductor nanowires. Science 279: 208. Novoselov KS, Geim AK, Morozov SV, et al. (2005) Twodimensional gas of massless Dirac fermions in graphene. Nature 438(7065): 197–200. Ortega JE and Himpsel FJ (1993) Inverse-photoemission study of Ge(100), Si(100), and GaAs(100) – bulk bands and surfacestates. Physical Review B 47: 2130. Park CH, Spataru CD, and Louie SG (2006) Excitons and many-electron effects in the optical response of singlewalled boron nitride nanotube. Physical Review Letters 96: 126105. Parr RG and Yang W (1989) Density-Functional Theory of Atoms and Molecules. Oxford University Press. Penn DR (1962) Wave-number-dependent dielectric function of semiconductors. Physical Review 128(5): 2093. Phillips JC and Kleinman L (1959) New method for calculating wave functions in crystals and molecules. Physical Review 116: 287. Pickett WE (1989) Pseudopotential methods in condensed matter applications. Computer Physics Reports 9: 115. Reining L, Olevano V, Rubio A, and Onida G (2002) Excitonic effects in solids described by time-dependent densityfunctional theory. Physical Review Letters 88: 066404.
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Reining L, Onida G, and Godby RW (1997) Elimination of unoccupied-state summations in ab initio self-energy calculations for large supercells. Physical Review B 56: R4301. Rohlfing M and Louie SG (1998) Excitonic effects and the optical absorption spectrum of hydrogenated Si clusters. Physical Review Letters 80: 3320. Rohlfng M and Louie SG (1998) Electron–hole excitations in semiconductors and insulators. Physical Review Letters 81: 2312. Rohlfing M and Louie SG (2000) Electron–hole excitations and optical spectra from first principles. Physical Review B 62: 4927. Rojas HN, Godby RW, and Needs RJ (1995) Space–time method for ab initio calculations of self-energies and dielectric response functions of solids. Physical Review Letters 74: 1827. Saito R, Dresselhaus G, and Dresselhaus MS (1998) Physical Properties of Carbon Nanotubes. London: Imperial College Press. Seidl A, Goerling A, Vogl P, Majewski JA, and Levy M (1996) Generalized Kohn–Sham schemes and the band-gap problem. Physical Review B 53: 3764. Sell D (1972) Resolved free-exciton transitions in the opticalabsorption spectrum of GaAs. Physical Review B 6: 3750. Shaver J, Kono J, Portugall O, et al. (2007) Magnetic brightening of carbon nanotube photoluminescence through symmetry breaking. Nano Letters 7(7): 1851–1855. Siegel DA, Park CH, Hwang C, et al. (in press) Many Body Interactions in Quasi-Freestanding Graphene. Silvestrelli PL and Parrinello M (1999) Water dipole moment in the gas and the liquid phase. Physical Review Letters 82: 3308. Simon B (1976) The bound state of weakly coupled Schrodinger operators in one and two dimensions. Annals of Physics 97: 279. Son Y-W, Cohen ML, and Louie SG (2006) Energy gaps in graphene nanoribbons. Physical Review Letters 97: 216803. Son Y-W, Cohen ML, and Louie SG (2006) Half-metallic graphene nanoribbons. Nature 444(7117): 347–349. Spataru CD, Ismail-Beigi S, Benedict LX, and Louie SG (2004) Excitonic effects and optical spectra of single-walled carbon nanotubes. Physical Review Letters 92: 077402. Spataru CD, Ismail-Beigi S, Benedict LX, and Louie SG (2004) Quasiparticle energies, excitonic effects and optical absorption spectra of small-diameter single-walled carbon nanotubes. Applied Physics A 78: 1129. Spataru CD, Ismail-Beigi S, Capaz RB, and Louie SG (2005) Theory and ab initio calculation of radiative lifetime of excitons in semiconducting carbon nanotubes. Physical Review Letters 95: 247402. Strinati G (1988) Application of the Green’s-function method to the study of the optical-properties of semiconductors. Rivista del Nuovo Cimento 11(12): 1. Tassaing T (1998) The partial pair correlation functions of dense supercritical water. Europhysics Letters 42: 265. Tiago ML and Chelikowsky JR (2006) Optical excitations in organic molecules, clusters, and defects studied by firstprinciples Green’s function methods. Physical Review B 73: 205334. Umari P, Stenuit G, and Baroni S (2009a) Optimal representation of the polarization propagator for large-scale GW calculations. Physical Review B 79: 201104(R). Umari P, Stenuit G, and Baroni S (2009b) GW quasiparticle spectra from occupied states only. Physical Review B 81: 115104. van den Brink J and Sawatzky GA (2000) Non-conventional screening of the Coulomb interaction in low-dimensional and finite-size systems. Europhysics Journal 50(4): 447. van Schilfgaarde M, Kotani T, and Faleev S (2006) All-electron selfconsistent GW approximation: Application to Si, MnO, and NiO. Physical Review Letters 96: 226402.
76 Ab initio Theories of the Structural, Electronic, and Optical Properties of Semiconductors Wang F, Cho D, Kessler B, et al. (2007) Observation of excitons in one-dimensional metallic single-walled carbon nanotubes. Physical Review Letters 99: 227401. Wang F, Dukovic G, Brus LE, et al. (2005) The optical resonances in carbon nanotubes arise from excitons. Science 308: 838. Wilson HF, Lu D, Gygi F, and Galli G (2009) Iterative calculations of dielectric eigenvalue spectra. Physical Review B 79: 245106. Yang L, Cohen ML, and Louie SG (2007) Excitonic effects in the optical spectra of graphene nanoribbons. Nano Letters 7(10): 3112–3115. Yang L, Cohen ML, and Louie SG (2008) Magnetic edge-state excitons in zigzag graphene nanoribbons. Physical Review Letters 101: 186401. Yang L, Deslippe J, Park C-H, Cohen ML, and Louie SG (2009) Excitonic effects on the optical response of graphene and bilayer graphene. Physical Review Letters 103: 186802. Yang L, Park C-H, Son Y-W, Cohen ML, and Louie SG (2007) Quasiparticle energies and band gaps in graphene nanoribbons. Physical Review Letters 99: 186801. Yang L, Spataru CD, Louie SG, and Chou MY (2007) Enhanced electron–hole interaction and optical absorption in a silicon nanowire. Physical Review B 75: 201304(R). Yin MT and Cohen ML (1980) Microscopic theory of the phasetransformation and lattice-dynamics of Si. Physical Review Letters 45: 1004. Yin MT and Cohen ML (1982) Theory of lattice dynamical properties of solids: Application to Si and Ge. Physical Review B 26: 3529. Zhang Y, Tan Y-W, Stormer HL, and Kim P (2005) Experimental observation of quantum Hall effect and Berry’s phase in graphene. Nature 438(7065): 201–204.
Further Reading Castro Neto AH, Guinea F, Peres NMR, Novoselov KS, and Geim AK (2009) The electronic properties of graphene. Reviews of Modern Physics 81: 109. Fetter AL and Walecka JD (1971) Quantum Theory of ManyBody Systems. San Francisco, CA: McGraw-Hill. Hedin L and Lundqvist S (1969) Effects of electron–electron and electron–phonon interactions on the one-electron states of solids. In: Seiz F, Turnbull D, and Ehrenreich H (eds.) Solid State Physics, vol. 23, pp. 1–181. London: Academic Press. Hybertsen MS and Louie SG (1986) Electron correlation in semiconductors and insulators: Band gaps and quasiparticle energies. Physical Review B 34: 5390. Mahan GD (1981) Many-Particle Physics. New York: Plenum. Martin R (2004) Electronic Structure Basic Theory and Practical Methods. New York: Cambridge University Press. Rohlfng M and Louie SG (1998) Electron–hole excitations in semiconductors and insulators. Physical Review Letters 81: 2312. Saito R, Dresselhaus G, and Dresselhaus MS (1998) Physical Properties of Carbon Nanotubes. London: Imperial College Press. Yu PY and Cardona M (1996) Fundamentals of Semiconductors. New York: Springer.
Relevant Website http://www.top500.org – TOP500 Supercomputing.
1.03 Impurity Bands in Group-IV Semiconductors M Eto, Keio University, Yokohama, Japan H Kamimura, Tokyo University of Science, Tokyo, Japan ª 2011 Elsevier B.V. All rights reserved.
1.03.1 1.03.1.1 1.03.1.2 1.03.2 1.03.2.1 1.03.2.2 1.03.2.3 1.03.2.4 1.03.2.5 1.03.3 1.03.3.1 1.03.3.1.1 1.03.3.1.2 1.03.3.1.3 1.03.3.2 1.03.3.2.1 1.03.3.2.2 1.03.3.3 1.03.3.3.1 1.03.3.3.2 1.03.3.3.3 1.03.3.4 1.03.3.4.1 1.03.3.4.2 1.03.3.4.3 1.03.3.4.4 1.03.4 1.03.4.1 1.03.4.1.1 1.03.4.1.2 1.03.4.2 1.03.4.3 1.03.4.3.1 1.03.4.3.2 1.03.4.3.3 1.03.4.4 1.03.4.5 1.03.5 References
Introduction Isolated Donor Atom in Semiconductors Quantum Computer Using Si:P Localization Theories in Impurity Bands Impurity Conduction Mott Transition Effects of Localization and Anderson Transition Computer Studies of Anderson Model Scaling Theory of Localization Electron–Electron Interactions in Impurity Bands Theoretical Formulation Transfer diagonal representation Anderson-localized states Interaction terms Theory of Intrastate Interaction Spin susceptibility and specific heat Comparison with experiment on Si:P Interstate Interactions Direct and kinetic-type exchanges Spin-pair model Specific heat anomaly and spin susceptibility Numerical Simulation for Interacting Donor Electrons in Si:P System Gaussian model Electron configurations and many-electron states Specific heat and spin susceptibility Comparison with experiment Hopping Conduction and Related Phenomena VRH and Mott’s Law Formulation of hopping conduction Mott’s law Coulomb Gap Formula for Magnetoresistance in the Variable-Range Regime Appearance of spin-dependent mechanisms Calculated results Comparison with experiments Magnetocapacitance in Intermediate-Concentration Regime Spin-Dependent Behavior of Magnetoresistance in Other Systems Conclusions
78 79 80 81 81 82 82 84 84 86 87 87 87 88 88 89 90 91 91 92 93 94 95 96 98 99 101 101 101 102 102 104 104 105 106 107 109 109 110
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78 Impurity Bands in Group-IV Semiconductors
1.03.1 Introduction Research on the electronic structures and physical properties of lightly doped semiconductors is very important, because it forms the basis of present-day semiconductor technology. In this context, research on impurity bands in group-IV semiconductors such as Si and Ge has played an important role not only in the semiconductor physics but also in the device physics. When group-V elements such as P, Sb, and As are doped into Si or Ge, a hydrogen 1s-like shallow impurity state appears just below the conduction band. This impurity state is called a donor state, because it can contribute a free electron to the host crystal. With an increase in impurity concentration, various energy levels are formed by random impurity distributions. The term ‘impurity band’ means the continuous distribution of one-electron energies with a certain probability distribution. In this sense, the impurity band is quite different from an ordinary energy band in periodic systems in which the wavevector is a good quantum number. Similarly, groupIII elements such as B, Al, Ga, and In doped into Si or Ge form an acceptor state, which can contribute a free hole to the host semiconductor. When the impurity concentration is sufficiently low, electronic states are localized in an impurity atom (low-concentration regime). In the intermediate concentration regime, electronic states in the impurity band are still localized in space, but spread over a number of impurity atoms (Anderson localization). The hopping conduction takes place from a localized state to another, which shows an activation-type temperature dependence. When the impurity concentration is larger than a critical value, the electronic states are extended over the whole samples, which give rise to a metallic conductivity. This metal–insulator (MI) transition, the socalled Mott–Anderson transition, in which both the randomness of donor position and electron–electron interaction are important, has been one of the main topics in solid-state physics (Mott, 1990). Although diamond is also the same group-IV element as Si and Ge, pure diamond is an insulator with a wide bandgap of 5.5 eV, so that one has not considered the existence of an impurity band in diamond. Recently, however, Ekimov et al. (2004) reported the discovery of superconductivity in the boron-doped diamond synthesized at high pressure and high temperature. Their measurements showed that boron-doped diamond with a hole-carrier
density of 5 1021 cm3 is a type II superconductor below the superconducting transition temperature Tc 4 K. The doped boron acts as an acceptor with a hole-binding energy of about 0.37 eV. Baskaran (2006) suggested an important role of the electron correlation-driven superconductivity in an impurity band in p-type diamond, so that a topic of the impurity band in diamond suddenly attracted keen interests. Although the observation of superconductivity in boron-doped diamond is interesting, recent experimental results suggested that superconductivity occurs in heavily boron-doped samples (Yokoya et al., 2005). This means that the holes in valence bands in diamond may be responsible for superconductivity. Thus, the topic of superconductivity in boron-doped diamond is not a main theme of the this chapter. In this context, for readers who have interests in the boron-doped superconductivity, we would like to suggest to read a review article on superconductivity in diamond by Takano (2008). In this chapter, we describe various physics of an impurity band, mainly focusing on the localized states and hopping conduction on the insulating side of the MI transition. The MI transition from the metallic side was studied by many theoretical groups, using the scaling theory and perturbation calculations with respect to the random potential (e.g., refer to a review article by Lee and Ramakrishnan (1985)). On the other hand, localized states of electrons on the insulating side were examined by a limited number of researchers. We would like to stress the importance of strong electron correlation as well as the disorder effect in the insulating regimes, which results in very rich physics: unique properties of electronic states and their influence on physical properties, for example, specific heat, spin susceptibility, and hopping conductivity. Recently, the impurity state in semiconductors is attracting a new interest in the field of quantum information processing. In the quantum computation the fundamental unit is a quantum bit (qubit), which is a quantum-mechanical superposition of ‘0’ and ‘1’ (Nielsen and Chuang, 2000). An electron spin S ¼ 1/2, or a nuclear spin I ¼ 1/2, can be used as a qubit. Indeed, if spin states j "i and j #i represent ‘0’ and ‘1’, respectively, a spin state is given by their coherent superpositition j i ¼ C0 j "i þ C1 j #i
(jC0j2 þ jC1j2 ¼ 1). Kane proposed a silicon-based nuclear spin quantum computer utilizing a
Impurity Bands in Group-IV Semiconductors
nanofabrication technique on phosphorus-doped silicon (Si:P) (Kane, 1998). The qubits are nuclear spins I¼1/2 of 31P atoms, which can be controlled by nuclear magnetic resonance (NMR). The nuclear spins interact with electron spins in the phosphorus atoms by the hyperfine interaction. Any qubit operation is achieved by tuning the resonant frequency of NMR and electrical control of electronic states using small metallic gates attached on the silicon surface. One of the key elements in Kane’s quantum computation is the control of an antiferromagnetic coupling between electron spins at adjacent donors (Section 1.03.1.2). This coupling was studied by one of the authors of this chapter with his collaborators in the 1970s for the impurity band of Si:P with intermediate concentration regime, to explain experimental results of the anomalous behavior of specific heat at low temperatures. The spin coupling can be ferromagnetic as well as antiferromagnetic in the case of impurity band (Section 1.03.3.3). Therefore, our study of electronic states in the impurity band should be noteworthy from the modern point of view. The organization of this chapter is as follows. In the rest of the present section, we briefly explain the hydrogen 1s-like electronic state in an isolated impurity doped in semiconductor (Section 1.03.1.1) and the basic idea of Kane’s quantum computer using Si:P (Section 1.03.1.2). In Section 1.03.2, we overview the transport phenomena in the impurity band and the localization theories of electronic states. The historical development of localization theory and MI transition in doped semiconductors are also surveyed. Sections 1.03.3 and 1.03.4 are the main parts in this chapter. These sections are devoted to a theoretical investigation of the electronic states and hopping conduction in the insulating regimes of doped semiconductors. Section 1.03.3 begins with the theoretical formulation to examine the interplay between disorder and electron–electron interaction (Section 1.03.3.1). In Section 1.03.3.2 the important role of the intrastate interaction in localized states, the largest component of the interaction, is described in detail. The description clarifies that the intrastate interaction results in the coexistence of doubly occupied, singly occupied, and unoccupied states. The experimental result of specific heat is compared with the theoretical result. In Section 1.03.3.3, we describe the effects of the interstate interactions, particularly antiferromagnetic or ferromagnetic coupling between electron spins in the singly occupied states. The theoretical and experimental results of
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specific heat at low temperatures and of spin susceptibility are reviewed, and it is shown how the observed strange behaviors of specific heat and spin susceptibility can be explained by the theory described in this subsection. In Section 1.03.3.4, numerical simulation based on a cluster model is reviewed. Then, it is shown how the physics of the localized states with interstate as well as intrastate interactions is verified by the numerical study for real Si:P systems. The hopping conduction is described in Section 1.03.4. In the impurity band the variable-range hopping plays an important role, as was proposed by Mott (1968). The Mott’s law, resistivity (T) _ exp[(T0/T)1/4] as a function of temperature T, is described in Section 1.03.4.1. Then it is shown how the Mott’s law is modified in the lowconcentration regime of compensated samples. In connection with this topic, the so-called Coulomb gap, which is a gap in the density of states at the Fermi level caused by the long-range Coulomb interaction, is discussed in Section 1.03.4.2. In Section 1.03.4.3, we describe the hopping conductivity in the intermediate concentration regime using the theoretical formalism given in Section 1.03.3. In particular, theoretical work on the magnetoresistance developed by Kurobe and Kamimura (1982) is reviewed in Section 1.03.4.3, and that on magnetocapacitance in Section 1.03.4.4. It should be mentioned that their spin-dependent theory has been applied not only to impurity bands in doped semiconductors but also to a wide variety of materials and of systems such as sandwiched semiconductors, amorphous semiconductors, doped polymers, carbon fibers, etc., as briefly mentioned in Section 1.03.4.5. In reviewing theories for impurity bands in doped semiconductors, we should first mention that we are greatly indebted to the textbooks by Mott (1990), Kamimura (1986, 1985), Kamimura and Aoki (1989), and Shklovskii and Efros (1984). 1.03.1.1 Isolated Donor Atom in Semiconductors We briefly explain the electronic state in an isolated donor atom doped in silicon (Si) or germanium (Ge). These elements possess a diamond crystal structure in which (ns)2(np)2 valence electrons form sp3 covalent bonds. When an atom of group-V element is doped into the materials, four valence electrons in the atom participate in forming the sp3 covalent bonds, whereas a remaining electron is bound to the nucleus of the atom by the Coulomb potential:
80 Impurity Bands in Group-IV Semiconductors
U ðrÞ ¼ –
e2 r
ð1Þ
where is the static dielectric constant of the host crystal. A large value of ( 12 for Si and 16 for Ge) reduces the binding energy of the impurity state and enlarges the spatial extension compared with the lattice constant. This enables us to employ the effective mass approximation for the impurity state (Kohn and Luttingel, 1955). If the bottom of the conduction band is nondegenerate in the host crystal, the wave function of the impurity state is expressed as ðrÞ ¼ F ðrÞuðrÞ
ð2Þ
where u(r) is the Bloch state at the bottom of conduction band. The smooth envelope function F(r) follows the effective mass equation –
h2 e2 F ðrÞ ¼ EF ðrÞ – 2m r
ð3Þ
The isotropy of the effective mass m is assumed. Equation (3) yields a hydrogen 1s-like ground state F ðrÞ ¼ ðaB3 Þ– 1=2 expð – r =aB Þ
ð4Þ
with the effective Bohr radius aB ¼ h =ðm e 2 Þ. Here aB 30 A˚ for the case of Si:P, which is indeed much larger than the lattice constant. The binding energy is given by E1s ¼ m e 4 =ð22 h2 Þ X Ry , where Ry is called an effective Rydberg energy and Ry 45 meV in Si:P. The shallow impurity state appears at Ry below the bottom of conduction band. At finite temperatures, electrons are excited from the impurity state to the conduction band. Hence, the impurity atoms of group-V element work as donors. In a real case of Si (Ge), the conduction band has six (eight) equivalent minima, the so-called valleys. The wave function of impurity state is given by 2
ðrÞ ¼
X
i Fi ðrÞui ðrÞ
ð5Þ
i
where ui(r) is the Bloch state at the bottom of valley i, and the coefficients i are determined from the crystal symmetry. In the effective mass equation, we have to take account of anisotropy of m, since an effective mass in the longitudinal direction ml is different from that in the transverse direction mt , where ml ¼ 0:98 and mt ¼ 0:19 (ml ¼ 1:6 and mt ¼ 0:08) for Si (Ge) in units of the electron mass in the vacuum. For the calculations considering the multivalley and anisotropic effective mass, the reader is referred to
Kamimura and Aoki, (1989) and Kohn and Luttinger (1955). In a usual case, the wave function in Equation (4) is a good approximation to describe the impurity state in the length scale much larger than lattice constant. The valley degeneracy at the bottom of conduction band is irrelevant except for quantitative discussion of excited states (Section 1.03.3.4). 1.03.1.2
Quantum Computer Using Si:P
As an example of recent application of the abovementioned impurity state in Si:P, we briefly explain a silicon-based nuclear spin quantum computer proposed by Kane (1998). A phosphorus atom possesses a nuclear spin I¼1/2, which is used for a qubit in the proposed quantum computer. As shown in Figure 1(a), phosphorus atoms are aligned in a one-dimensional array. Two kinds of metallic gates are attached on the surface of the host silicon separated by a barrier of silicon oxide layer: A-gates are located just above phosphorus atoms while J-gates are between neighboring phosphorus atoms. For the NMR, static and alternating magnetic fields are applied on the whole system. A nuclear spin I interacts with an electron spin S trapped by a phosphorus atom through the contact hyperfine interaction Hhyperfine ¼ A I ? S
ð6Þ
with A ¼ (32/3)Bgnnj (0)j2, where B is the Bohr magneton, and gn and n are nuclear g-factor and nuclear magneton, respectively. The (0) is the value of the electron wave function at the position of nucleus. The electronic states can be controlled electrically using metallic gates. An idea of quantum computation is as follows. Single-qubit operation. Nuclear spins I ¼ 1/2 in phosphorus atoms can be controlled by the conventional NMR technique. To rotate one of the qubits selectively, we change the resonant frequency of the qubit using the A-gate. An electric field applied on the A-gate pulls the electron wave function away from the donor and thus the value of (0) is reduced (Figure 1(b)). As a result, the effective magnetic field by the hyperfine interaction is decreased and hence the resonant frequency of the nucleus decreases. Two-qubit operation. Two nuclear spins in adjacent donors interact with each other through electrons trapped at the donors: H ¼ A1 I1 ? S1 þ A2 I2 ? S2 þ J S1 ? S2
ð7Þ
Impurity Bands in Group-IV Semiconductors
(a)
1.03.1.1. By the random distribution of the impurities, they form an impurity band in which the energy levels are continuously distributed. When the impurity concentration is low, the electronic states are still localized in space. Then, the hopping transport is observed between the localized states. By increasing the concentration above the critical value, the insulating phase is changed to metallic phase. In this section, we review the localization theories and the scaling theory of the MI transition from a historical point of view.
A-gate J-gate Barrier 31P
(b) A-gate (V = 0)
e−
31P
(c) A-gate (V > 0)
e−
Si
31P
J-gate (V > 0)
e−
Figure 1 Schematic drawing of silicon-based nuclear spin quantum computer proposed by Kane. (a) Phosphorus atoms are aligned in a one-dimensional array in the host silicon. Each phosphorus atom has a nuclear spin I ¼ 1/2, which is used as a qubit, and weakly binds an electron. Two kinds of metallic gates, A-gates and J-gates, are attached on the surface of silicon. (b) Using A-gates, electron wave functions of the impurity state are modified, which changes the effective magnetic field on the nuclear spin through the hyperfine interaction. (c) Using J-gates, the antiferromagnetic coupling J is tuned between electrons in adjacent phosphorus atoms. Modified from Kane BE (1998) Nature 393: 133.
where Ii and Si are nuclear and electron spins at donor i. An antiferromagnetic interaction J works between the electron spins, the strength of which can be controlled by changing the electron population between the donor atoms using a J-gate. Through the control of electronic states using the J-gate, the energy difference between two states of the nuclear spins I1 and I2, þ and , can be tuned, where 1 ¼ pffiffiffi ðj "i1 j #i2 j #i1 j "i2 Þ 2
81
ð8Þ
Here j"ii, j#ii indicate the states of Ii. In , nuclear spins I1 and I2 are entangled with each other: when spin 1 is j "i, spin 2 is j #i, while when spin 1 is j #i, spin 2 is j "i. The creation and control of the entanglement is one of the key ingredients of the two-qubit operations. The combination of single- and two-qubit operations enables any quantum computation in principle (Nielsen and Chuang, 2000). The fundamental researches for this quantum computer are in progress.
1.03.2 Localization Theories in Impurity Bands When impurities of group-V element (P) are doped in Si or Ge, the shallow impurity states appear just below the conduction band, as described in Section
1.03.2.1
Impurity Conduction
Various transport phenomena observed in the impurity band are called impurity conduction. The phenomenon of the impurity conduction was introduced by Hung and Gliessman (1950) as a new conduction mechanism at low temperatures in doped silicon and germanium. Then, Fritzsche (1958) reported that in n-type-doped semiconductors there are three regimes of donor concentration according to the features of electronic conduction at low temperatures. In the low-concentration regime, each electron is trapped in the hydrogenic 1s state around each donor impurity. The conduction takes place at low temperatures only in compensated samples, where, besides donors, the smaller number of acceptors are doped. In the compensated samples, the empty donor states coexist with the occupied ones since the acceptors capture an extra electron. Then an electron in an occupied state can hop to a vacant state, giving rise to an activation-type conductivity: ¼ 3 expð – "3 =kB T Þ
ð9Þ
(see Section 1.03.4.1.1). As the impurity concentration is increased, there appears another insulating regime called intermediate concentration regime. In this regime, the conduction is of activation type: ¼ 2 expð – "2 =kB T Þ
ð10Þ
The activation energy "2 decreases with increasing donor concentration and finally vanishes at the critical impurity concentration nc, where the MI transition takes place. Above nc, the conduction becomes metallic. Summarizing the above results, the conductivity is expressed in the form ¼ 1 expð – "1 =kB T Þ þ 2 expð – "2 =kB T Þ þ 3 expð – "3 =kB T Þ
ð11Þ
82 Impurity Bands in Group-IV Semiconductors
Here, "1 is the energy required to eject an electron into the conduction band. As regards the origin of the concentrationdependent activation energy, "2, it was first ascribed to the energy required to remove an electron from one donor and transfer it on another donor that already has an electron. The final state is called D ion, that is, a negative donor ion. Figure 2 shows that, with increasing donor concentration, both the donor ground (1s) state and D state broaden to form a band of energy levels. They are called D band and D band, respectively. In the uncompensated case, the D band is completely filled by electrons and the conduction occurs only by exciting electrons from the D band to the D band. Thus, the activation energy, "2, corresponds to the energy gap between these bands. It is seen in Figure 2 that this gap energy, "2, decreases as the donor concentration, nD, increases and finally it vanishes at a certain concentration, nc, at which the MI transition takes place. Since the appearance of the D band is due to the electron-correlation effect in a donor atom, this MI transition was first considered as the Mott transition, which was induced by electron–electron interaction.
1.03.2.2
Mott Transition
Mott pointed out an important role of electron– electron interaction in the electronic states, originally to explain the insulating character of NiO (Mott, 1990). If the disorder effect is irrelevant in the impurity band, the simplest model to consider the electron–electron interaction is the Hubbard model
D − band
D − state
ε2 D state
D band
nc
nD
Figure 2 Schematic illustration of D and D bands as a function of donor concentration (nD). When each donor impurity has one electron, the D band is completely filled by the electrons when nD is smaller than the critical value (nc). Thus, an extra electron is accommodated in the D band, while a hole exists in the D band when an electron is taken out from a donor impurity. In this sense, the D and D bands are the bands for quasi-particles.
in a regular lattice (Hubbard, 1963, 1964a, 1964b; Mott, 1976): H¼
X ij
y tij ci cj þ U
X
ni" ni#
ð12Þ
i
y where ci and ci are creation and annihilation operators of an electron at site i with spin , respectively. The transfer integral tij is a constant V for the nearest neighbors and zero otherwise. Further, U is the Coulomb energy when two electrons occupy the same site, or it corresponds to the energy of D state. Let us consider the half-filling case where the number of electrons is the same as the number of lattice sites (donors), corresponding to the uncompensated case in doped semiconductors. When the distance between the donors is so large that the transfer integral V is much smaller than interaction energy U, it is favorable that each electron stays at one donor owing to the repulsive interaction energy between two electrons. This is the case of insulating phase (Mott insulator). With increasing donor concentration, nD, the donor–donor distance decreases and in consequence the transfer integral, V, increases: the D and D bands in Figure 2, which are called lower and upper Hubbard bands, respectively, in terms of the Hubbard model, are broadened with V. When V U, the electrons move around to gain the kinetic energy, which corresponds to the metallic phase. At a critical value of the concentration, nc, the MI transition takes place. This MI transition due to the electron–electron interaction is called Mott transition. Nowadays, it is known that the disorder effect is relevant to the MI transition in the impurity band (Anderson transition) rather than the electron–electron interaction (Mott transition). On the other hand, the Hubbard model has been intensively studied in the context to investigate the strongly correlated systems, including copper-oxide materials of high-Tc superconductivity (see a review article by Imada et al. (1998)).
1.03.2.3 Effects of Localization and Anderson Transition Anderson (1958) showed that in a random potential one-electron wave functions are localized if the random potential is strong enough. Following Anderson’s theory, Mott (1967) and Cohen et al. (1969) introduced the concept of the mobility edge in such a way that the electronic states caused by the
Impurity Bands in Group-IV Semiconductors
for
E < Ec
ðEÞ 6¼ 0
for
E > Ec
r
ð13Þ
where (E) represents the mobility at energy E. This is why Ec is called as the mobility edge. In the late 1960s, Mott developed an idea of localization by disorder and suggested that the effect of disorder plays an important role in the impurity band (Mott, 1990; Mott and Davis, 1979). According to his idea, at low donor concentration all states in the D band are localized in the Anderson sense, which he called Anderson-localized states. As the donor concentration nD increases, D and D bands broaden and eventually overlap with each other. In this case the states at the upper part of the D band become extended, as in Figure 3, and a mobility edge Ec separates extended states from localized states. This is the situation in the intermediate concentration regime, where the current is carried by electrons excited above the mobility edge Ec. The activation energy, "2, in Equation (10) is thus given by " 2 ¼ Ec – EF
ð14Þ
where EF is the Fermi energy. As nD further increases, Ec EF diminishes and eventually disappears at nD ¼ nc. For nD > nc, the system shows a metallic behavior. In this case, the mechanism of the MI transition is of one-electron nature, namely, at nc the wave function at EF changes from a localized nature illustrated in Figure 4(b) to an extended one in Figure 4(a). This MI transition is called where Ef is the Fermi energy Anderson transition. It was suggested that the MI transition at nc in doped semiconductors is of Anderson type rather than Mott type.
(b) exp(−|r − r0| /ξ)
ψ(r)
ðEÞ ¼ 0
(a) ψ(r)
disorder are localized in one energy range and extended in another, both ranges being separated from each other by a critical energy Ec:
r r0
Figure 4 Wave function of an electron in (a) extended states and (b) Anderson-localized states.
The existence of localized states around EF for nD < nc was supported by experimental results of variable-range hopping conduction in Ge:Sb at very low temperatures in the intermediate concentration regime by Allien and Adkins (1972). Prior to their experiment, the conduction was predicted by Mott (1968), in which the conductivity varies with temperature T as hop ¼ 4 expð – A=T 1=4 Þ
Ec
E
Figure 3 Schematic drawing of the electronic density of states in the intermediate concentration regime. Electronic states are localized at E < Ec (hatched region).
ð15Þ
This hopping conduction is treated in detail in Section 1.03.4. In addition to the concept of the variable-range hopping conduction, Mott (1970, 1972) also proposed another important concept, minimum metallic conductivity. According to his idea, the zero-temperature conductivity (T ¼ 0) jumps at nc from zero in the insulating phase to a minimum metallic value given by min ¼ const e 2 =ðhnc– 1=3 Þ
D(E)
EF
83
ð16Þ
where the constant is 1/20 within a factor of 2. An intuitive derivation of min is as follows. In the metallic regime, the conductivity can be expressed by the Drude formula, ¼ e 2 kF2 l=ð32 hÞ, where kF is the wavevector at the Fermi level and l is the mean free path. With increasing disorder, the mean free path decreases but it cannot go below the wavelength 1/kF where the Drude formula becomes meaningless. Therefore, should be larger than the value which is estimated by at kFl 1. Mott called min as the minimum metallic conductivity, and
84 Impurity Bands in Group-IV Semiconductors
using the Kubo–Greenwood relationship he showed that 2 in Equation (10) is equal to min. Although the scaling theory does not favor the presence of min, as we see later, the concepts of minimum metallic conductivity and variable-range hopping stimulated many researchers in the field of disordered systems.
1.03.2.4 Model
Computer Studies of Anderson
Through a number of attempts to check the validity of Mott’s concepts theoretically and experimentally, much progress was made in the research of disordered systems in the 1970s. As one such attempt, various theoretical groups tried to calculate the electronic structures and metallic conductivity of disordered systems by computer simulation. The computer simulation is based on the Hamiltonian proposed by Anderson (1958), called the Anderson model. The Anderson model is essentially a tightbinding model in a regular lattice, with on-site energies distributed randomly. In the notation of second quantization, the Anderson model is expressed by the Hamiltonian H¼
X i
y "i ci ci þ
X
y tij ci cj
ð17Þ
ij
where i, j represent the sites in a regular lattice. The one-electron energy, "i, which takes a random value, is uniformly distributed over the range from W/2 to W/2. The transfer integral is tij ¼ V for the nearest neighbors and zero otherwise. The computer simulation can be carried out more easily for twodimensional systems than three-dimensional systems. Edwards and Thouless (1972) were probably the pioneers for such a calculation. Yoshino and Okazaki (1976, 1977) performed elaborate calculations on a square lattice of size 100 100 and obtained the eigenstates in this system. They found that for W/V ¼ 8 the wave functions at the center of the band are strongly localized and decay exponentially with a localization length of about 5 atomic spacings. Further, they inferred that the MI transition might take place at W/V 6.5. Although the existence of exponentially localized states in two dimensions was established from their results, there remained a problem concerning the existence of extended states. Licciardello and Thouless (1978) extended their earlier calculations (Licciardello and Thouless, 1975a,b) based on the
scaling idea of Thouless (1974) to the systems of larger size. In the scaling arguments, Thouless related the degree of localization to the sensitivity of the energy levels to the boundary condition. For samples of size L, a dimensionless conductance is defined by gðLÞ ¼ "ðLÞ="ðLÞ
ð18Þ
where "(L) is the mean shift of the energy levels by changing the boundary condition from periodic to antiperiodic and "(L) is the mean spacing of energy levels. If the wave functions at energy E are localized with an exponential envelop, like j ðrÞj exp½ – ðEÞjr – r0 j
ð19Þ
with (E) being the inverse localization length, then "(L) and hence g(L) will behave as gðLÞ ¼ exp½ – 2ðEÞL
ð20Þ
If the wave functions are extended, on the other hand, "(L) should be basically independent of L. Then, g(L) yields the conductance in units of e2/h. The quantity g is often called the Thouless number. Licciardello and Thouless found that, with sufficiently large sample sizes, the states identified as extended previously exhibit the Thouless number that decreases with increasing sample size. They suggested, therefore, that these states are actually localized states with long localization lengths. This also meant that their earlier results, which were consistent with the existence of minimum metallic conductivity, might not be conclusive. This conclusion affected the research on disordered systems since most researchers had assumed the existence of the minimum metallic conductivity. Since the computer simulation was limited to systems of finite size, it was necessary to derive real conductance in a system of infinite size. For this purpose, the scaling theory was most appropriate. This was the dawn of the scaling theory of localization.
1.03.2.5
Scaling Theory of Localization
Abrahams et al. (1979) developed a scaling theory based on the arguments of Thouless (1974). The problem concerns the behavior of the conductance g in Equation (18) when the linear-dimension L of the system is changed. To change the system size, cubes of size L in d dimensions are put together to make a larger cube of size bL. Abrahams et al. (1979, 1980)
Impurity Bands in Group-IV Semiconductors
gðbLÞ ¼ f ½b; gðLÞ
ð21Þ
where f is a universal function depending only on the dimensionality of the system. Taking an infinitesimal increase in the scale size, b ¼ 1 þ ", Abrahams et al. obtained the following scaling equation for g(L): d ln gðLÞ 1 qf ðb; gÞ X ðgÞ ¼ d ln L gðLÞ qb b¼1
ð22Þ
where (g) is called scaling function. The asymptotic forms of (g) are given for g ! 0 and g ! 1 in the following: (1) in the strongly localized regime (g ! 0), we have g(L) _ exp(2L/ ) with being the localization length, so that (g) ! ln g þ constant; and (2) in the opposite limit of g !1, the conductance is given in terms of the conductivity , as g(L) _ Ld 2 in d space dimensions, which yields (g) ! d 2. Assuming that (g) is monotonic in g, Abrahams et al. obtained the behavior of the scaling function as a function of g for one (d ¼ 1), two (d ¼ 2), and three (d ¼ 3) dimensions, as illustrated in Figure 5. Important conclusions deduced from the figure are: 1. For dimensionality d > 2, there exists a mobility edge at which conductivity continuously tends to zero. This implied that, first, there is no minimum metallic conductivity, and, second, d ¼ 2 is the lower critical dimensionality, above which both extended and localized states can exist.
β(g)
1 gc
d=3
2. For d ¼ 2, there is no true metallic conduction. All the states are localized in the limit of infinite system size. The conductance as a function of L always decreases with increasing L in two dimensions, with a logarithmic dependence for small g and an exponential dependence for larger g. The scaling theory gave great impetus to the problem of the impurity conduction. In order to check experimentally whether or not the minimum metallic conductivity exists at the MI transition in doped semiconductors, Rosenbaum et al. (1980) measured the conductivity of uncompensated Si:P samples with donor concentration very close to the MI transition. Their result indicates that the transition is very sharp but the conductivity extrapolated to T ¼ 0 goes below the value of the minimum metallic conductivity, min, predicted by Mott. Paalanen et al. (1982) tried to confirm the above-mentioned experimental results by making a high-resolution, zero-temperature study of the MI transition, applying uniaxial stress to one sample of slightly insulating concentration. Figure 6 shows their experimental results, which can be fitted by the form ð0Þ ¼ c ½ðn=nc Þ – 1 &
ð23Þ
with n being the electron concentration (n ¼ nD in the uncompensated samples), where c ¼ 260 20
103 Insulator
σ (T = 0) (Ω−1cm−1)
asserted that g(L) is the relevant dimensionless ratio that determines the change of energy levels when the hypercubes are fitted together. Thus, when a new g for larger blocks is computed, the only relevant quantity determining the new g(bL) is the old one, namely g(L). Hence,
85
Metal
102
σmin 10
g
0 d=2
1
−1 d=1
Figure 5 Schematic drawing of scaling function (g) in one, two, and three dimensions. Arrows indicate the flow toward larger scales.
0
2
4 nD (1018 cm−3)
6
Figure 6 Experimental results for zero-temperature conductivity (0) shown against the donor concentration nD for Si:P systems obtained by Paalanen et al. (1982) (open circles) and by Rosenbaum et al. (1980) (solid circles). Modified after Rosenbaum TF, Andres K, Thomas GA, and Bhatt RN (1980) Physical Review Letters 45: 1723.
86 Impurity Bands in Group-IV Semiconductors
( cm)1 and ¼ 0.55 0.10. This observed critical behavior is not consistent with the scaling theory by Abrahams et al. that predicted ¼ 1. The inconsistency between the scaling theory and experimental result in Figure 6 may suggest an important role of the electron correlation at the transition in Si:P systems, which is not taken into account in the scaling theory. The experimental result for another three-dimensional disordered system, Si-doped Alx Ga1xAs, is shown in Figure 7 (Katsumoto et al., 1987). In this system, carriers are created when electrons trapped in deep donor states are photoexcited to conductive states, resulting in the persistent photoconductivity. By this method, Katsumoto et al. tuned the electron concentration, n, to a resolution of 0.2% around the critical concentration. Figure 7 clearly shows that the conductivity changes as _ n/nc 1 in the metallic region (n > nc), indicating ¼ 1 in accordance with the scaling theory. Regarding the puzzle of a critical exponent, that is, about a question of whether the value of is explained by the scaling theory or not, there have been a significant amount of experimental work to
n (1016 cm−3) 2.8
2.4
σ (S cm−1)
0.4
2
0.2
0
0 520
determine the critical exponent (Milligan et al., 1985; Lo¨hneysen, 2003). Recently, Itoh et al. (1996, 2004) performed a detailed study for the critical behavior of conductivity on the metallic side and that of dielectric susceptibility on the insulator side in homogeneously doped p-type Ge samples. To examine the vicinity of the MI transition, the preparation of homogeneously doped samples is required. For this purpose, they employed the neutron transmutation doping (NTD) technique to a chemically pure and isotropically enriched 70Ge crystal. When the samples are irradiated with thermal neutrons, some of the 70Ge atoms capture a neutron and change to 71Ga acceptors. This NTD technique enables homogeneous doping to the atomic level and fine-tuning of the carrier concentration. They concluded that 1 for compensated samples and 0.5 for uncompensated samples (Itoh et al., 1996, 2004). Even in uncompensated samples, the critical behavior with 1 was observed in the very vicinity of MI transition. However, this should be attributable to unexpected compensation effect. Their experimental result implies that the scaling theory can describe the MI transition for the compensated samples but cannot for the uncompensated samples, possibly due to strong correlation effect in the latter. In this context, the understanding of the MI transition requires further theoretical studies, taking into account the electron–electron interaction (Kawabata, 1985; Altshuler and Aronov, 1985; Fukuyama, 1985; Kaveh, 1985; Di Castro, 1988; Belitz and Kirkpatrick, 1994).
1.03.3 Electron–Electron Interactions in Impurity Bands 0
0.02 n/nc−1
600 Total exposure time (a.u.)
Figure 7 Experimental results for the conductivity as a function of the concentration of photoexcited electrons in Sidoped AlxGa1xAs. The carrier concentration n is tuned by changing the total exposure time of light. The conductivity changes as _ nD/nc 1 in the metallic region (n > nc) in the vicinity of critical concentration nc (inset). Modified from Katsumoto S, Komori F, Sano N, and Kobayashi S (1987) Journal of the Physical Society of Japan 56: 2259.
There are two main streams of theoretical approach to investigate the MI transition in doped semiconductors, which are complementary to each other. The first stream of approach starts from the freeelectron picture in the metallic phase and investigates how the localized states begin to be formed by the scattering of free electrons with impurities. These studies were prevalent with use of the diagrammatic expansion and renormalization technique for the so-called weakly localized regime of doped semiconductors (Lee and Ramakrishnan, 1985). The second stream of approach covers the insulating phase consisting of the Anderson-localized states and investigates the Anderson localization itself.
Impurity Bands in Group-IV Semiconductors
In this section, we describe the theoretical work along the latter stream (Kurobe and Kamimura, 1982, 1983; Kamimura, 1986, 1985, 1978b, 1978a, 1980, 1982; Kamimura and Yamaguchi, 1978; Yamaguchi et al., 1979, 1980; Kamimura et al., 1982, 1983; Takemori and Kamimura, 1982, 1983a, 1983b; Kurobe et al., 1984). In the preceding section, we have already seen that both the electron correlation and disorder play important roles in the MI transitions in doped semiconductors. Here, we would like to point out remarkable effects of electron–electron interaction in the intermediate concentration regime of doped semiconductors. In this regime, T-linear specific heat was observed by Kobayashi et al. (1977). This fact means that the density of states at the Fermi level D(EF) is finite in spite of the insulating phase. Nevertheless, the spin susceptibility does not obey the Pauli law. Ue and Maekawa (1971) and Quirt and Marko (1972a,b) observed Curie-type susceptibility in the intermediate concentration regime in electron spin resonance (ESR) experiments. The occurrence of the Curie-type susceptibility should be ascribable to the effect of electron correlation, as pointed out by various authors (Kamimura, 1978a, 1978b; Kapian et al., 1971; Kobayashi et al., 1978). Therefore, we have to investigate the electronic states in the intermediate concentration regime, taking into account both effect of the electron–electron interaction and disorder on an equal footing.
1.03.3.1
Theoretical Formulation
1.03.3.1.1 Transfer diagonal representation
H ¼ H0 þ H1 P y P y H0 ¼ "i ci ci þ tij ci cj ij
1XX y y H1 ¼ hij jU jklici cj 9 cl9 ck 2 ijkl 9
ð24Þ
y ci
both random variables, while the second part represents the electron–electron Coulomb interaction U, where hij jU jkl i is its matrix element. If we adopt the eigenstates of H0 as a set of basis functions, called transfer-diagonal representation, the Hamiltonian can be rewritten as H¼
X
" nˆ þ
þ
1X U nˆ nˆ – 2
1 X9 X y y U c c 9 c9 c 2 9
ð25Þ
where ji represents an eigenstate of H0 with eigenenergy ". It is expressed as ji ¼
X
vi jii
ð26Þ
i
with a unitary matrix vi. Here, the one-electron energy " is also a random variable. We have also introduced the notation of y nˆ X c c ;
U X h jU j i;
U X U
In Equation (25) we have separated the electron– electron interactions into those within a state ji and the remaining off-diagonal interactions, denoted by the summation 9 excluding the case of ¼ ¼ ¼ . The former are called intrastate interactions and the latter interstate interactions. The interstate interactions are further classified into direct Coulomb U , direct exchange U , and the remaining interactions as we shall see later. 1.03.3.1.2
Following Kamimura (1985, 1982), the Hamiltonian in the presence of both disorder and electron–electron interaction is written generally in the tight-binding scheme as
i
87
where and ci are the creation and annihilation operators of an electron at the ith site with spin , respectively, ji i is a Wannier function at the ith site with one-electron energy "i, and tij is the transfer integral between ith and jth sites. In this Hamiltonian, the first part represents the singleparticle Hamiltonian with "i and tij, which are
Anderson-localized states If there is stronger disorder, the transfer-diagonal representation bases ji are localized in the Anderson sense although there exist also extended states above the mobility edge. In the strongly localized regime the amplitude of the wave functions decreases with the distance apart from the localization center, on the average, in an exponential envelope function (Figure 3(b)). Thus, the coefficients in the linear combination of impurity wave functions in Equation (26) can be written as vi ¼ v~i expð – jri – r j= Þ
ð27Þ
where v~i represents a random phase factor, the localization length, and r the center of the localized state ji. From the normalization condition, hji ¼ 1, we obtain jvei j2 ¼ 4ðR0 = Þ3 =3
ð28Þ
88 Impurity Bands in Group-IV Semiconductors
where R0 is half the average distance between impurities defined by 4R30 =3 ¼ 1=nD
ð29Þ
In deriving Equation (28), the summation over the impurity sites has been replaced by the integral over the whole space divided by the average volume occupied by each impurity. The quantity is a function of one-electron energy " and considered to have the following critical behavior in the vicinity of the mobility edge Ec:
=aB _ j1 – " =Ec j –
where is the critical Bohr radius.
exponent and aB
ð30Þ
is the effective
1.03.3.1.3
Interaction terms Bearing the form of Equation (27) in mind, we can estimate the magnitudes of interaction terms U and U . If we neglect the overlaps between different Wannier states, we obtain U ¼
X
vi vj vi vj hij jU jij i
ð31Þ
ij
If we further assume that an intra-Wannier-state correlation energy hiijU jiii is a constant U0 and a Coulomb potential between different Wannier states has a functional form of hij jU jiji ¼ e2/ jri rj j, the largest is the intrastate interaction energy U and is estimated to be U
1 4U0 R30 5e 2 þ 8 3 3 8
ð32Þ
The first term dominates in the Anderson-localized regime where the localization length is comparable to the average impurity distance. In such a case, the intrastate interaction energy depends on the state energy in the following way: U =U0 _ ð1 – " =Ec Þ3
ð33Þ
if we adopt the form of Equation (30) as the stateenergy dependence of localization length . Similarly, we can estimate the magnitudes of the other interactions U (Kamimura, 1985; Takemori and Kamimura, 1983). Assuming the random distribution of localization centers, the distribution of nearest-neighbor distance is given by the Poisson distribution: PðRÞ ¼
d exp½ – ðR=R0 Þ3 dR
ð34Þ
For ¼ R0, the most probable values of the spindependent nearest-neighbor interactions are estimated to be U < 0:47U
ð35Þ
U < 0:31U
ð36Þ
and
These values are, however, still overestimates, because if we are to consider only the localized states with state energies close to each other, the average distance between their localization centers r must be larger than the average impurity distance R0, and the interactions between them should be much smaller than the values estimated above. It must be noted that the above estimates of interaction terms are made for the case where is of an order of R0. If the impurity concentration is increased to approach nc, U becomes smaller as localization becomes weaker and all the interaction terms U will be of the same order of magnitude.
1.03.3.2
Theory of Intrastate Interaction
As shown in the previous subsection, in the Anderson-localized regime the interstate interactions are small compared with the intrastate interaction except in the vicinity of the MI transition. Thus, in this regime it is reasonable to consider only the intrastate interaction as the first approximation, and we consider the following Hamiltonian: H¼
X
" nˆ þ
1X U nˆ nˆ – 2
ð37Þ
On the basis of this Hamiltonian, Yamaguchi et al. (1979, 1980) investigated the interplay of disorder and intrastate interaction in the Anderson-localized regime. The effects of interstate interactions, which are neglected in this subsection, are discussed in the following subsection. Since there is no electron transfer in Hamiltonian (37), the eigenstates are given by single configurations for electron occupation of one-electron states. For each state , there are four possible values of energy E, corresponding to the four different electron occupancies of the state: (i) E ¼ 0 when state ji is empty; (ii) E ¼ " þ BH when occupied by a spin-up electron; (iii) E ¼ "BH when occupied by a spindown electron; and (iv) E ¼ 2" þ U when occupied by spin-up and -down electrons, where we have included the Zeeman energy in an external magnetic
Impurity Bands in Group-IV Semiconductors
field H. Denoting the chemical potential by , the partition function is then given by
E
89
E W
Y
" – þ B H 1 þ exp – ¼ kB T " – – B H 2" – 2 þ U þ exp – þ exp – kB T kB T
EF
ð38Þ
so that the free energy is expressed as an independent sum over states F ¼ – kB T ln X " – þ B H ¼ – kB T ln 1 þ exp – kB T " – – B H 2" – 2 þ U þ exp – þ exp – kB T kB T ð39Þ
The average occupation number of state with spin ¼ is calculated from this free energy as hn i ¼ kB T
q F q" kB T
¼
U
e þe 1 þ eþ þ e – þ eU
ð40Þ
where " ¼ " þ BH with ¼ . We have abbreviated as e ¼ exp[(" þ BH)/kBT] and eU ¼ exp[(2" þ U 2)/kBT]. The chemical potential is determined by the condition X hn i ¼ ne
ð41Þ
where ne is the concentration of electrons. According to Hamiltonian (37), an energy " is required for the first electron to occupy state while an energy " þ U is necessary for the second electron to occupy the state which is already occupied by another electron of opposite spin. If the energy for the second electron " þ U is below the Fermi level, the state is occupied by two electrons. If " þ U is above the Fermi level, on the other hand, this state cannot accommodate two electrons even if the oneelectron state energy " is below the Fermi level. In consequence, at and below the Fermi level there are the states that are occupied by only one electron (singly occupied (SO) states) as well as those that are doubly occupied (DO states). The states with energy higher than the Fermi level are unoccupied (UO states). The density of states for the first electron and that for the second electron to occupy a state are denoted by D1(E) and D2(E), respectively, and these are shown schematically in Figure 8, where a
0
N/W
D1(E)
N/W
D2(E)
Figure 8 Density of states for the first electrons, D1(E), 6, where U and that for the second electrons, D2(E). W=U¼ is defined by Equation (46). Regions corresponding to singly and doubly occupied states are shown by hatched and cross-hatched areas, respectively.
constant density of states is assumed for the state energy distribution for 0 " W. Then D1 ðEÞ ¼
8 < 1=W
for 0 E W
: 0
otherwise
ð42Þ
The density of second-electron states D2(E) is related to D1(E) by dU – 1 D2 ð" þ U Þ ¼ D1 ð" Þ 1 þ d"
ð43Þ
This is enhanced over D1(E) because the intrastate correlation energy, U, is a decreasing function of state energy ("), as seen in Equation (33). 1.03.3.2.1 Spin susceptibility and specific heat
The expression for a magnetic moment M(H) is obtained from the free energy (39): MðH Þ ¼
X qF eþ – e – ¼ B þ þ e – þ eU qH 1 þ e
ð44Þ
The spin susceptibility is given by ¼ (qM/ qH)H !0. Similarly, the specific heat C(H) is obtained from CðH Þ ¼ – T
q2 F ðq2 F =qT qÞ2 – qT 2 q2 F =q2
ð45Þ
Yamaguchi et al. (1979, 1980) calculated the temperature dependence of susceptibility and specific heat
90 Impurity Bands in Group-IV Semiconductors
Z
W
U ðEÞ dE
ð46Þ
2B nS =kB T
20
40
80
2B ne =kB T
0.1
1
10 kB T/U
(b) 0.6
10
0.4
6 W/U = 5 0.2
0
ð48Þ
ð49Þ
where ð50Þ
0.05
0.10 kBT/W
0.15
Figure 9 (a) Calculated results of spin susceptibility as a function of temperature T in logarithmic scales, using a model with an energy-dependent intrastate interaction U. The value is indicated in the panel. (b) Calculated results of of W=U electronic specific heat C as a function of temperature T, using the same model as in (a). The dotted line represents the specific heat in the absence of electron–electron interaction. Modified from Yamaguchi E, Aoki H, and Kamimura H (1979) Journal of Physics C: Solid State Physics 12: 4801.
We stress that the T-linear behavior of the electronic specific heat is a result of the continuous distribution of the random energy ". 1.03.3.2.2 on Si:P
1
x2e x dx 1:885 x 2 – 1 ð1 þ 2e Þ
0.1
20
In this case, the total number of electrons ne contributes to . The calculated results of electronic specific heat are shown in Figure 9(b), in the absence of magnetic field (H ¼ 0). At low temperatures, the specific heat is a superposition of Schottky-type specific heat corresponding to the fluctuation of occupation number at each state . The superposition of the specific heat with various excitation energies results in the T-linear behavior, which reflects the density of states at the Fermi level: C ¼ 2kB2 T ½D1 ðEF Þ þ D2 ðEF Þ I
W/U = 5
ð47Þ
where ns is the number of SO states per unit volume. It should be noted that the appearance of this Curie-type susceptibility is the result of the intrastate interaction, because in the absence of this interaction the Paulitype susceptibility is expected from the finite density of states at the Fermi level. As an increase in temperature T, the susceptibility gradually becomes less dependent on T and tends to show a Pauli-type behavior in the temperature range of U kB T W when W is much larger than U . When the temperature is further increased beyond the state-energy distribution width W, electrons are no longer degenerate and the susceptibility becomes again of Curie type:
Z
10 1
0
The numerical results of spin susceptibility and specific heat are presented in Figure 9 for several values of W=U . As seen in Figure 9(a), the spin susceptibility shows Curie-type behavior at low temperatures so that kB T U . This is due to the localized spins of SO states and is expressed as
I ¼
10
C/kB n
1 U ¼ W
(a)
χ (μ 2B n/W )
for the case of half-filled band. They took into account the explicit dependence of U on the state energy " in Equation (33), where the mobility edge was assumed to be located at the top of the band (Ec ¼ W). The only parameters in the theory are the bandwidth W and the averaged magnitude of the intrastate interaction energy:
Comparison with experiment
Let us now compare the calculated results with the experimental data on phosphorus-doped silicon
Impurity Bands in Group-IV Semiconductors
(Si:P) in the intermediate concentration regime, where the phosphorus concentration, nD, is between 1.0 1018 and 3.2 1018 cm3 (see, e.g., a review article by Kamimura (1980)). The spin susceptibility was first measured by Sasaki and Kinoshita (1968) using a static method and then by Ue and Maekawa (1971) and by Quirt and Marko (1972a, 1972b) using the ESR method. The overall behavior of spin susceptibility is of Curie type at low temperatures (see Figure 11 in the next subsection), consistent with the present theory with intrastate interaction. The specific heat of Si:P was measured by Marko et al. (1974) and by Kobayashi et al. (1977, 1979). Figure 10 shows the experimental data of the electronic specific heat taken by Kobayashi et al. (1977). The specific heat is almost T-linear, as is also expected from the present theory, except at very low temperatures below 2 K. The anomaly at low temperatures is discussed in the next subsection. We can determine the values of parameters in the theory by fitting the calculated results to the experimental data. Yamaguchi et al. (1979) found that W ¼ 19 meV and U ¼ 3.4 meV by fitting to the susceptibility and the specific heat observed for nD ¼ 1.7 1018 cm3. The solid line in Figure 10 shows the results calculated for these values of the parameters.
1.03.3.3
Interstate Interactions
1.03.3.3.1 Direct and kinetic-type exchanges
The SO states carry a localized spin of S ¼ 1/2. If there are interactions among these spins, they are expected to affect the spin susceptibility and to give an excess specific heat which is dependent on an external magnetic field. It was seen in Figure 10 that there appears an excess specific heat over the T-linear behavior at temperatures below 2 K. This anomaly in the specific heat of Si:P was first observed by Marko et al. (1974) and was investigated in detail by Kobayashi et al. (1979) for various samples of different donor concentrations in the presence of magnetic field. Kobayashi et al. observed that the hump in the specific heat grows and shifts to higher temperatures with increasing magnetic field. These results suggest that the excess specific heat is due to the freezing of the spin degrees of freedom in the SO states. The spin susceptibility of Si:P at low temperatures, on the other hand, was investigated in detail by Andres et al. (1981) for samples of donor concentration between 1.1 1017 and 4.0 1018 cm3, at temperatures down to 2 mK. Figure 11 shows the Curie– Weiss plot of the molar donor susceptibility for
20 18
n =1.2 × 10 D
χ−1 (mole/e.m.u.)
C (μJK −1 mol−1)
100
50
4
Figure 10 Electronic specific heat of Si:P for nD ¼ 1.7 1018 cm3 as a function of temperature. The full curve represents the theoretical result by Yamaguchi et al. (1979) with W ¼ 19 and U ¼ 3.4 meV, whereas the circles represent the experimental result reported by Kobayashi et al. (1977). The enhanced region of the specific heat above the T-linear line is shown as the hatched area.
18
cm
18
cm
0.7 × 10
0.1 × 10
−3
0 150 18
2 T (K)
−3
cm
10
n =3.7 × 10
100 0
91
D
−3
cm
50
0
2
4 T (K)
Figure 11 Curie–Weiss plots of the molar donor susceptibility for several concentrations of phosphorus reported by Andres et al. (1981). The solid line shows free-spin behavior.
−3
92 Impurity Bands in Group-IV Semiconductors
several samples. The solid straight line in the figure indicates the free-spin behavior. It is seen from the figure that the susceptibility increases monotonically as if it were to diverge with decreasing temperature, in the same temperature range as the excess specific heat is observed. In this subsection, the microscopic origin of these features is discussed by taking into account the interstate interactions U in the last term of Hamiltonian (25). As estimated in Section 1.03.3.1.3, the dominant spin-dependent interstate interactions are U and U . The first type is the direct exchange, which favors the parallel-spin configuration of two SO states and , just like the Hund’s rule in atoms, because the Anderson-localized states ji are orthogonal to each other. Now, we discuss the effect of the second type of interaction, U , which also acts between SO states (Kamimura, 1982; Yamaguchi et al., 1979; Takemori and Kamimura, 1982). Suppose we consider two-electron configurations shown in Figure 12, one in which there are two SO states and with their spins being antiparallel to each other, and the other in which two electrons occupy the same state . We denote the former and latter configurations by 1 ¼ j", #i and 2 ¼ j", #i, respectively. Then, the interstate interaction y y c# c # c" mixes states 2 and 1. This U c" type of configuration interaction always favors energetically the spin-singlet state for a pair of electrons by transferring an electron virtually from state j #i to state j#i. By the secondorder perturbation calculation with respect to U , the change of the energy of state 1, Ekin, can be expressed in the following spin Hamiltonian: kin E
α
β
¼
kin J
1 S ?S – 4
α
ð51Þ
β
Mixed by Uαααβ Figure 12 Two-electron configurations mixed by kinetictype exchange process.
with kin J ¼
jU j2 U þ " – "
ð52Þ
kin Since J is always positive, the energy of the singlet state is lowered. In this view, the interstate interaction resulting from U is called the kinetic-type exchange interaction although in this case the mixing matrix elements arise from the electron–electron interaction, but not from the transfer terms. dir The direct-exchange interaction J ðX U Þ which favors the spin-triplet configuration also exists between SO states and , as already mentioned. Thus, the spin Hamiltonian for the SO states and can be written as
H ¼ – J ðr Þ S ? S
ð53Þ
dir kin where S ¼ S ¼ 1/2 and J ¼ J – J . The dir dependence of J on the distance between the localization centers in states and , r , is the kin same as J in Equation (52) and both decay exponentially with the distance r . We reasonably assume the following form for the exchange interaction J (r ):
J ðr Þ ¼ J0 expð – 2r = Þ
ð54Þ
where represents the localization length, which is assumed here to be independent of states and . The sign of J may be positive or negative, depending on the relative magnitude of the direct and kinetic-type exchanges. 1.03.3.3.2
Spin-pair model As we have a spin system with a network of effective exchange interactions J which take randomly positive or negative values, we are faced with an intractable problem of the random spin glass. In order to calculate the specific heat and magnetization of the spin system of SO states, a simple model of spin pairs was proposed by Takemori and Kamimura (1982) as a first approximation of treating such a system. As we consider the concentration region where the localization length is close to half the average impurity distance R0, there are a few SO states within the distance from a localization center, and we expect that the behavior of each localized spin is most strongly influenced by the spin nearest to it. In fact, if localization centers are randomly distributed with ¼ R0 and 30% of electrons are in the SO states, which is actually the case for Si:P of nD ¼ 1.7 1018 cm3 as seen later, the probability of
Impurity Bands in Group-IV Semiconductors
finding more than two SO states within the distance
from a localization center of SO state is less than 3.7%. Moreover, when the nearest spin is located within /2, the probability that the interaction with the next-nearest spin exceeds 1/e of the interaction with the nearest spin is less than 16%. Therefore, one can assume that each localized spin of SO state forms a pair with its nearest neighbor, so that the whole spin system can be regarded as an ensemble of noninteracting spin pairs. Let us now proceed with the spin-pair model. By the interaction (53), each pair splits into a spin-triplet state (S ¼ 1) and a spin-singlet state (S ¼ 0) with the energy difference J . The partition function of the spin pair with exchange interaction J is given by ZðJ Þ ¼ 1 þ 2coshð2B H =kB T Þ þ expð – J =kB T Þ ð55Þ
and the free energy of the spin system is expressed as F ¼ – kB T
X
lnZðJ Þ
ð56Þ
pairs
If the probability distribution function of the random values of J is denoted by P(J), Equation (56) can be written in the form F¼ –
nS kB T 2
Z
dJPðJ ÞlnZðJ Þ
ð57Þ
where ns is the number of SO states per unit volume. Function P(J) is determined, for example, from Equation (54) with Poisson distribution for the distance between localization centers. For simplicity, we assume an equal magnitude of jJ0 j for both positive and negative J . Then, the parameters we have are jJ0 j, the ratio of the number of spin-triplet pairs to that of spin-singlet pairs, nþ/n, the number of SO states ns ( ¼ nþþ n), and the localization length .
1.03.3.3.3 Specific heat anomaly and spin susceptibility
From free energy in Equation (57), the specific heat and spin susceptibility are calculated as follows: 2
2h ½2 þ coshðhÞ þ e coshðhÞ C¼
1 kB T 2
Z
–4jhe j sinhðhÞþj 2 e j ½1þ2coshðhÞ ½1 þ 2coshðhÞ þ e j 2
PðjkB T Þdj ð58Þ
and ¼ 22B
Z
1 PðjkB T Þdj 3 þ ej
where h¼2BH/kBT and j¼J/kBT. These values are evaluated numerically as functions of temperature T and magnetic field H for various values of the parameters. Let us first consider the cases of spintriplet (S¼1) pairs or spin-singlet (S¼0) pairs only. The calculated results are shown in Figure 13. The spin susceptibility is small for the spin-singlet case, while it is approximately of Curie type for the spin-triplet case but with slight deviation from the Curie law due to the existence of excited states of spin singlet. The specific heat is quite small at H¼0 for the spin-triplet pairs because the ground state is triply degenerate, so that there is residual entropy at T¼0. It then rises sharply with an external magnetic field and the peak shifts to higher temperatures as the field is increased. For the spin-singlet pairs, the specific heat is insensitive to the magnetic field. Although the calculated susceptibility for the spintriplet pairs can explain the Curie-type susceptibility observed in Si:P, the calculated results of specific heat give too sharp rise with increasing magnetic field and are not consistent with the observation. A good fit to the experimental results for both C and is obtained by mixing an equal number of spin-triplet and singlet pairs. The calculated results of specific heat in this case are shown in Figure 14 together with the experimental data taken by Kobayashi et al. (1979). The best fit values are jJ0 j ¼ 5.1 K, nþ/n ¼ 1.0, ns ¼ 0.3nD ¼ 5.6 1017 cm3, and ¼ 81 A˚. Although the spin-pair model is simple, the bestfit values of the parameters seem reasonable. The magnitude jJ0 j is of the order of the intrastate interaction. The localization length and the number of SO states do not contradict the estimates made by the intrastateinteraction theory. The number of SO states, ns/nD ¼ 33%, yields the residual entropy, Sres ¼ kBnsln 2 ¼ 96 J (K?mol)1 for nD ¼ 1.7 1018 cm3. This agrees with the experimental value of 90 J (K?mol)1 obtained from the formula for the residual entropy: Sres ¼
j
ð59Þ
93
Z
C dT T
ð60Þ
where C is the observed excess specific heat over the T-linear term (Kobayashi et al., 1979). It is concluded that the coexistence of the direct and the kinetic-type exchange interactions is important in an Anderson-localized electronic system. Both the specific-heat anomaly and spin susceptibility of Si:P in the intermediate concentration regime can be
94 Impurity Bands in Group-IV Semiconductors
(a) (a)
9.7
6
10
4.9
4
2 ΔC/kB
χ (μ 2B /K)
5
0 (b)
H = 0 kG
0 (b)
10
H = 0 kG
6
9.7 4.9
5
4
2 0
0.2
0.4
0.6
0.8
T (K) 0
0.2
0.4
0.6
0.8
T (K) Figure 13 Spin susceptibility and anomaly in electronic specific heat, C per pair of singly occupied states, as functions of temperature T, calculated in the spin-pair model for (a) spin-triplet pairs and (b) spin-singlet pairs. Modified from Takemori T and Kamimura H (1982) Solid State Communications 41: 885.
explained by the spin-dependent interstate interactions between SO states. 40
ΔC (μ J K–1 mol–1)
H = 9.7 kOe
4.9 20
2.9
0
0
0.4
0.8 T (K)
Figure 14 Calculated results of magnetic-fielddependent anomaly in the electronic specific heat for donor concentration of nD ¼ 1.7 1018 cm3 in Si:P (solid curves). The experimental data taken by Kobayashi et al. (1979) are also shown for various values of magnetic field.
1.03.3.4 Numerical Simulation for Interacting Donor Electrons in Si:P System In order to investigate how the Anderson-localized states are constructed from the impurity states on each donor site, Takemori and Kamimura (1983a,b) performed numerical simulation using a finite cluster model for donors. The cluster model contains six donors located randomly in a space, corresponding to several donor concentrations between 1.0 1018 and 2.4 1018 cm3 of Si:P. The calculation is carried out for an ensemble of 80 samples with different geometrical configurations of donors, for each donor concentration. The electron–electron interaction is fully taken into account in the simulation. Although the cluster size is quite small, this should not cause serious limitations because the localization length is of an order of average donor distance, for the concentration considered here, and does not exceed the cluster diameter. By examining the ground and low-lying excited states, the validity of
Impurity Bands in Group-IV Semiconductors
the physical picture described in the preceding subsections is confirmed. Furthermore, the direct calculations of the specific heat and spin susceptibility reproduce the characteristic features observed in Si:P.
specific heat, and spin susceptibility do not depend on the special choice of the one-electron Hamiltonian. As one choice, the one-electron Hamiltonian H0 is taken in such a way as hijH0 jii ¼ h j jH0 jj i ¼ hijK jii þ hijVi jii
1.03.3.4.1
Gaussian model
The Hamiltonian is composed of the kinetic energy of electrons K, Coulomb potential due to donor impurities V, and electron–electron interaction U: H ¼K þV þU
ð61Þ
where K ¼
V ¼
Ne X 1 2 p 2m i i¼1 Ne X
ð62Þ
Vi
ð63Þ
e2 jri – rj j
ð64Þ
i¼1
U¼
X i<j
Here, ri and pi are the space coordinate and momentum of the ith electron, m is the effective mass, and Vi the Coulomb potential of the i th donor. The number of electrons, Ne, is identical to that of donor impurities in uncompensated samples considered here. No valley degeneracy of the conduction band in Si is assumed. The donor centers are distributed randomly in a sphere of volume Ne/nD, where nD is the donor concentration. A Gaussian wave function i ðr Þ
¼ expð – jr – Ri j2 =2 Þ
ð65Þ
is attached to each donor center Ri and the electronic wave functions are represented by their superpositions. The spatial extent in Equation (65) is determined to be 1.88 times the effective Bohr radius so as to give the largest binding energy (0.82 Ry) for an isolated donor. The Gaussian function is employed instead of the usual exponential 1s function, which enables us to obtain the analytical expression for all the matrix elements for the electron–electron interaction (Boys, 1950). First, the one-electron Hamiltonian is diagonalized to obtain the orthonormal base for the transfer-diagonal representation. On doing so, an appropriate choice of the one-electron Hamiltonian is required in order to facilitate the interpretation of the results although the final calculated results of physical quantities such as energy levels,
95
ð66Þ
for the diagonal elements and hijH0 jj i ¼ hijK j j i þ hijVi j j i þ hijVj j j i
ð67Þ
for the transfer energy. Note that the Coulomb potentials due to impurities other than i (i and j) do not contribute to the ith diagonal element (transfer energy between i and j). This choice of one-electron Hamiltonian effectively introduces the long-range screening effect into the one-electron Hamiltonian. The one-electron Hamiltonian H0 is diagonalized, and the eigenstates and eigenenergies are adopted as the orthonormal basis functions and corresponding state energies for the transfer-diagonal representation. Using this basis set, all the matrix elements of the Hamiltonian are calculated. Next, we calculate the matrix elements of the Hamiltonian in the multielectron configuration space. The total number of the configurations in which six electrons occupy the six transfer-diagonal ! 12 states is ¼ 924 if the spin degrees of freedom 6 are considered. Although the Hamiltonian has no symmetry in the coordinate space, it is invariant against the rotation in the spin space. Each manybody eigenstate of the Hamiltonian is always an eigenstate of total spin S and its z-component Sz and thus either a spin-degenerate multiplet or a nondegenerate singlet. It is, therefore, convenient to confine ourselves to the subspace with Sz ¼ 0. This reduces the number of configurations to 400, while we still retain the whole energy spectrum. The energy levels and many-electron wave functions in a cluster are then obtained by diagonalizing the obtained 400 400 Hamiltonian matrix in the configuration space. Since the Hamiltonian is diagonalized exactly, the calculated results do not depend on the choice of basis set for one-electron states. Advanced methods developed in quantum chemistry yield a more suitable basis set to elucidate many-body states. We adopted the multi-configuration self-consistent field (MCSCF) method to discuss the Anderson-localized states in the uncompensated samples of Si:P (Eto and Kamimura, 1988, 1989) and in the compensated samples (Eto and Kamimura, 1989).
96 Impurity Bands in Group-IV Semiconductors
1.03.3.4.2 Electron configurations and many-electron states
Density of one-electron states (1/Ry*)
We now present numerical results for the ground state and low-lying excited states of the clusters. For the comparison with the preceding subsections, we concentrate on the results for the donor concentration of nD ¼ 1.7 1018 cm3 for Si:P. One-electron states. The distribution of the eigenenergies of transfer-diagonal states is shown in Figure 15. It has a peak near the isolated donor level ("0 ¼ – 0:82Ry ) and extends to both higher and lower energies with a broad width. The distribution width of state energies is found to increase with donor concentration nD. The transfer-diagonal wave functions in one of the clusters are shown in Figure 16 schematically, where the states are numbered in increasing order of the energy. The wave functions extend typically over two donor sites for the donor concentration considered here. Spin-dependent interactions. We can determine the spin degeneracy of the ground state in a cluster by examining mixing coefficients of electron configurations. About one-fourth of the clusters are found to have a spin-triplet ground state. Here, we describe in detail the ground state of the cluster in Figure 16, which actually turns out to be one of the spin-triplet clusters. The ground state of this cluster has an energy 12.4329 Ry, whereas the first excited state has an energy higher by 0.0044 Ry. These many-electron states are shown in Figure 17, in which many configurations appear since the electron–electron interaction is taken into account exactly. Table 1 presents the average occupation numbers by six
z 2, 5
y
x
4
1, 6
3
Figure 16 Schematic drawing of the transfer-diagonal states for one sample. The states are numbered in increasing order of state energy. Solid circles (*) represent donor sites.
electrons in each transfer-diagonal state with its state energy. It is found that the transfer-diagonal states 1 and 2 are doubly occupied, states 3 and 4 are singly occupied, and states 5 and 6 are unoccupied, in consistent with the physical picture given in the preceding subsections. As seen in Figure 17(a), the largest contributions (75%) to the ground state jgrdi come from two configurations where states 1 and 2 are doubly occupied and states 3 and 4 are singly occupied: jgrdij1"2"3";1#2#4#i – j1"2"4";1#2#3#i
1.0
0.5
εo 0
–1.6
–0.8
0.0 E (Ry*)
0.8
1.6
Figure 15 Distribution of transfer-diagonal state energies of the six-donor clusters corresponding to the donor concentration nD ¼ 1.7 1018 cm3 of Si:P.
The mixing coefficients have the opposite signs for the two configurations, indicating that spins are parallel in states 3 and 4 in the present notation so that the ground state is triply degenerate with total spin S ¼ 1. As regards the first excited state, the average occupation numbers of transfer-diagonal states are almost the same as in the ground state, as seen in Table 1. The largest contributions come also from the same two configurations as those in the ground state, but with the mixing coefficients of the same sign. This means that electron spins are antiparallel in states 3 and 4 and hence the first excited state is a spin singlet (nondegenerate) with
Impurity Bands in Group-IV Semiconductors
97
(a)
0.61
C:
–0.24 –0.14
0.13
–0.07 –0.06
–0.61 0.24 0.14 –0.13 0.07 0.06 74.8% 11.4% 3.7% 3.2% 0.9% 0.8%
C: Weight: (b)
C:
0.46
0.42
–0.36 –0.18 –0.16
0.15
0.14
–0.12
0.14 –0.12 –0.18 C: 0.46 Weight: 42.0% 17.5% 13.1% 6.3% 2.6% 2.1% 3.8% 2.8% Figure 17 The electron configurations in terms of transfer-diagonal states for (a) the ground state and (b) the first excited state in a cluster shown in Figure 16. The mixing coefficient C is shown at the bottom of each configuration together with the weight of the spin conjugate configurations C2. The horizontal lines representing the one-electron states merely represent the order of the state energy and the line spacing does not represent the actual difference in energy.
Table 1 The state energies of transfer-diagonal states in a cluster shown in Figure 16 Average occupation number
State
State energy " (Ry)
Ground state
First excited state
1 2 3 4 5 6
1.474 1.104 0.834 0.582 0.424 1.202
1.947 1.742 1.000 1.001 0.289 0.018
1.958 1.755 1.096 0.916 0.276 0.001
The average occupation numbers are also tabulated in the ground and first excited states.
total spin S ¼ 0. The next largest components are the configurations in which the transfer-diagonal state 3 or 4 is doubly occupied. This is due to the kinetic-exchange interaction between the two transfer-diagonal states, discussed in Section 1.03.3.3.1. The effective exchange interaction J is given by the energy difference E between the ground and the first excited states, that is, J ¼ E ¼ 0.0044 Ry. Besides the spin-triplet clusters, there are a large number of clusters in which the transferdiagonal states 3 and 4 are singly occupied with antiparallel spins in the spin-singlet ground state (S ¼ 0) and parallel spins in the first excited states
98 Impurity Bands in Group-IV Semiconductors
(S ¼ 1). In such cases, the energy difference can be regarded as the antiferromagnetic coupling J between the spins in states 3 and 4. This result indicates the coexistence of ferromagnetic and antiferromagnetic interactions between the spins in SO states. 1.03.3.4.3 Specific heat and spin susceptibility
Since all the eigenstates and eigenenergies for each cluster are obtained by the numerical study, we can readily calculate the spin susceptibility and specific heat. In the presence of magnetic field H, the energy level En of each spin multiplet splits into the Zeeman terms: Eðn; Sz Þ ¼ En þ 2B Sz H
ð68Þ
if we neglect the higher-order effects. The specific heat C and spin susceptibility are then calculated by the following equations:
E 2 E 2 – kB T kB T 2 P Eðn; Sz Þ exp½ – Eðn; Sz Þ=ðkB T Þ ¼ kB T Z n;Sz
C=kB ¼
" #2 X Eðn; Sz Þ exp½ – Eðn; Sz Þ=ðkB T Þ
–
n;Sz
kB T
Z
ð69Þ
and ¼ 42B
Sz2 kB T
H ¼0
X S 2 exp½ – Eðn; Sz Þ=ðkB T Þ z ¼ Z k T n;Sz B
H ¼0
ð70Þ
where the partition function Z is given by Z¼
X
exp½ – Eðn; Sz Þ=ðkB T Þ
ð71Þ
n;Sz
and the summation over Sz is taken from S to S when the energy level n has the total spin S. Figure 18(a) shows the calculated results of the specific heat and spin susceptibility of a spin-triplet cluster. The spin susceptibility is of Curie type with a slight deviation from the Curie law owing to the presence of spin-singlet excited states. The Schottky-type hump in the specific heat, which is located at kBT 103 Ry in the absence of magnetic field, is caused by the ferromagnetic exchange interaction J between the SO states. The lowest state is a spin triplet and this state splits into three components of Sz ¼ 1, 0, 1 in magnetic fields. The hump of C shifts to higher temperatures with increasing
magnetic field, reflecting the excitation among the Zeeman-split states. The specific heat increases again at higher temperatures, due to the excitations that change the occupation numbers of transfer-diagonal states. For a spin-singlet cluster (Figure 18(b)), a Schottky-type hump appears in the specific heat, but it is insensitive to the magnetic field. The peak of C is situated at kBT 102Ry, which reflects the excitation energy to the spin-triplet state, E ¼ J. The spin susceptibility is quite small compared with the spin-triplet clusters. The tends to zero at low temperatures, whereas increases with temperatures up to kBT E because the excited state is a spin triplet. In order to simulate the experimental observation; the specific heat and spin susceptibility are averaged over all the clusters for the same donor concentration. The calculated results of specific heat for various magnetic fields are plotted against temperature in Figure 19. At high temperatures, such as kBT > 6 103 Ry, the curves for H ¼ 0 are almost T-linear but do not extrapolate to zero at T ¼ 0, in contrast to the experimental results. As discussed in Section 1.03.3.3.3, the T-linear specific heat is caused by continuous distribution of state energy ". For the clusters of a finite size, the state energies " are discrete and the calculated specific heat retains the structure of the discrete excitations. Therefore, the weak downward bending of the calculated curves of C should be attributable to the limited size of the cluster. On the other hand, there appears a hump in the specific heat for H ¼ 0 that grows and shifts to higher temperatures with increasing magnetic field. As seen above, this is due to the ferromagnetic exchange interaction between the spins of SO states. The hump is smaller relative to the T-linear part for higher donor concentration. This indicates that the ratio of the number of SO states to that of donors, ns/nD, decreases with increasing nD, in agreement with the experimental data on Si:P (Kobayashi, 1979). Figure 20 shows the Curie–Weiss plot of the ensemble-averaged spin susceptibility for several values of donor concentration in Si:P. The curves are almost T-linear (Curie type) with a downward bending at low temperatures. This characteristic behavior is in good agreement with the experiment on Si:P by Andres et al. (1981). The Curie–Weiss plot for six free spins is also plotted in the figure for comparison. The ratio of the number of SO states is estimated from the susceptibility at kBT ¼ 1.4 102 Ry,
Impurity Bands in Group-IV Semiconductors
(a)
(b) 2.0
C/kB
C/kB
0.8
H = 3 × 10−3 (Ry*/μ B)
0.4
1.0
H=0 1 × 10−3 (Ry*/μ B) 2 3
2 1 0 0
0
12
6 χ/μ 2B (10 Ry*–1)
χ/μ 2B (103 Ry*–1)
99
8
4
0
0.8
4
2
0
1.6
kBT (10−2 Ry*)
8
16
kBT (10−2 Ry*)
Figure 18 Electronic specific heat C for various values of magnetic field and spin susceptibility , as functions of temperature T, of (a) a spin-triplet cluster and (b) a spin-singlet cluster consisting of six donors.
nS =nD ¼ cluster =free spins ¼ 28:3% 18
ð72Þ
3
for nD ¼ 1.7 10 cm . The molar donor susceptibility decreases with increasing donor concentration nD, which is also in accordance with the observation in Si:P. This also indicates a decrease in the number of SO states with increasing nD. 1.03.3.4.4
Comparison with experiment In order to compare the calculated results with experimental data on Si:P, it is necessary to take into account multivalley effects in the conduction band of Si, because no valley degeneracy is assumed for the numerical calculations. There are six valleys in the conduction band of Si (see Section 1.03.1.1). The crystal field of tetrahedral symmetry around an impurity atom splits the one-electron energy levels of sixfold degeneracy into a nondegenerate state of A1 symmetry, doubly degenerate state of E symmetry, and triply degenerate state of T2 symmetry. This is called valley–orbit splitting. The effect is large for the first electron to occupy the state, but small for the second electron because the crystal field is screened by the first electron. The optical
experiment on Si:P shows that the binding energy of the orbitally nondegenerate one-electron state with A1 symmetry is larger than that for the other states by 12 meV (Aggarwal and Ramdas, 1965). This implies that, for T < 100 K, the electronic state for the first electron is expressed as a superposition only of A1 states of donors with no appreciable valley effects. When two electrons occupy a donor, the electronic configuration is specified by assigning the first electron to the orbital of A1 symmetry and the second electron to that of A1, E, or T2 symmetry with appropriate spin configuration. Thus, the two-electron energy levels are split due to the valley–orbit interactions and by the Coulomb interaction between the two electrons. The energy splitting can be estimated from the binding energy of the D state, which was evaluated to be 1.01 meV (11.7 K) for Si:P (Natori and Kamimura, 1977; Kamimura 1979). The splitting is expected to be smaller for Anderson-localized states, which are more extended than the isolated-donor state, and we can consider that all the six valleys are available for the second electrons in the temperature range larger than 1 K.
100 Impurity Bands in Group-IV Semiconductors
The multivalley effect on the specific heat is, therefore, included in the following way. The T-linear specific heat, which reflects the density of states at the Fermi level, is enhanced by a factor of 6. The hump in the specific heat, however, comes from the spin-dependent interactions for the SO states. Since the SO states are constructed by linear combinations of the A1 state of each donor, no effect of valley degeneracy is expected. Thus, the observed specific heat is given by
(a)
μ BH = 3 × 10−3 Ry* 2
0.4 C/kB
1
0.2
(b)
0
0
μ BH = 3 × 10−3 Ry* 0.4
C ¼ 6CL T þ CR
2
C/kB
1 0 0.2
0
0.8 kBT (10−2 Ry*)
1.6
Figure 19 Ensemble-averaged electronic specific heat against temperature with various values of magnetic field for the cluster model of six donors. The donor concentration corresponds to (a) 1.7 1018 cm3 and (b) 2.4 1018 cm3 in Si:P, respectively. Modified from Takemori T and Kamimura H (1983a) Journal of Physics C: Solid State Physics 16: 5167; Takemori T and Kamimura H (1983b) Advances in Physics 32: 715.
where CLT and CR represent the T-linear part and the remainder of the calculated specific heat, respectively. The result is in fair agreement with the experimental results for Si:P of nD ¼ 1.7 1018 cm3 (Kobayashi et al., 1979), as shown in Figure 21. As for the spin susceptibility at T < 1 K, it is determined by the number of SO states and interactions among them. Since the valley–orbit splitting is large for SO states, we expect few valley-degeneracy effects on . The overall behavior of calculated spin susceptibility shown in Figure 21 is in good agreement with the experimental observation on Si:P in Figure 11. For the case of nD ¼ 1.7 1018 cm3, the estimated number of SO states (ns/nD ¼ 28.3%) also agrees with the experimental result of ns/nD ¼ 29% (Kobayashi et al., 1979).
1.2
60
1.7 1.0
0.4
C (μJ K–1 mol–1)
1/χ (10–2 Ry*/μ 2B )
nD = 2.4 × 1018 cm–3 0.8
ð73Þ
40
20 Free spins
0
0.8 kBT (10−2 Ry*)
1.6
Figure 20 Curie–Weiss plot of the ensemble-averaged spin susceptibility for the cluster model of six donors corresponding to various donor concentrations in Si:P. The susceptibility for six free spins is also shown (broken line) for comparison. Modified from Takemori T and Kamimura H (1983a) Journal of Physics C: Solid State Physics 16: 5167; Takemori T and Kamimura H (1983b) Advances in Physics 32: 715.
0
1.0
2.0
T (K) Figure 21 Specific heat calculated for Si:P with impurity concentration nD ¼ 1.7 1018 cm3, as a function of temperature, for H = 0 (full line) and 20 kOe (broken line). The multivalley effect has been taken into account. Experimental data taken by Kobayashi et al. (1976) are also shown for H = 0 (triangles) and H = 19.1 kOe (circles).
Impurity Bands in Group-IV Semiconductors
1.03.4 Hopping Conduction and Related Phenomena
in the case of phonon emission with E < 0. Here, r is the distance between the localization centers and
In this section, the hopping conductivity in doped semiconductors at low temperatures T is discussed. When kBT is much smaller than the width of the impurity band, the variable-range hopping (VRH) ðT Þ ¼ 0 exp½ðT0 =T Þ1=4
ð74Þ
is observed, instead of activation-type conductivity, Equation (11), as predicted by Mott (1968). The Mott’s theory of VRH is briefly reviewed in Section 1.03.4.1. In the low-concentration regime of compensated samples, the VRH shows a different T dependence because of the disappearance of the density of states at the Fermi level caused by the long-range Coulomb interaction. The theory of the so-called Coulomb gap is described in Section 1.03.4.2. In Sections 1.03.4.3 and 1.03.4.4, VRH and related phenomena are studied for the intermediate concentration regime, based on the transfer-diagonal representation formalism presented in the preceding section. The theory successfully explains experimental results of magnetoresistance and magnetocapacitance.
1.03.4.1
VRH and Mott’s Law
Let us begin with the hopping conduction between Anderson-localized states, neglecting the electron– electron interaction. The hopping conductivity is usually evaluated using a model of random resistance network proposed by Miller and Abrahams (1960) (Section 1.03.4.1.1). Based on the model, the Mott’s law of the VRH is derived in Section 1.03.4.1.2. 1.03.4.1.1 Formulation of hopping conduction
Consider an elementary hopping process from ji to j i when the state energies " and " are different. The hopping of an electron is accompanied by the phonon absorption or emission for the energy conservation. The number of electrons making the transition per unit time is given by ¼ ph e – 2r = Nph ðEÞf ð" Þ½1 – f ð" Þ
101
ð75Þ
in the case of phonon absorption with E ¼ " " > 0, and ¼ ph e – 2r = ½Nph ð – EÞ þ 1 f ð" Þ½1 – f ð" Þ ð76Þ
Nph ðEÞ ¼
1 1 ; and f ð"Þ ¼ "=ðk T Þ e E=ðkB T Þ – 1 e B þ1
are boson and fermion distribution functions, respectively. The state energies are measured from the Fermi level. The localization length, , is assumed to be independent of state energy, so that the transfer integral between the two states gives the factor of e – 2r = . The pre-exponential factor, ph, is related to the product of the square of electron–phonon matrix element and the density of states of phonons. (Although ph depends on r , its dependence is much weaker than that of the exponential part. Hence, ph is treated as a constant.) The electron spins are disregarded in Sections 1.03.4.1 and 1.03.4.2. At sufficiently low temperatures, it is usually expected that kBT j" " j, j"j, and j" j. Then, the hopping rate can be written in a unified form of ¼ ph exp½ – 2r = – =kB T
ð77Þ
1 " ¼ ½j" – " j þ j" j þ j" j 2
ð78Þ
where
Here is equal to j" " j when the state energies lie on opposite sides of the Fermi energy and max (j"j, j" j) otherwise. The resistance of the hopping is given by R ¼
kB T kB T ¼ 2 expð2r = þ " =kB T Þ 2 e e ph
ð79Þ
(Shklovskii and Efros, 1984). Miller and Abrahams (1960) proposed a random resistance network in which resistance R connects a pair of vertices, as shown in Figure 22(a). The conductivity of the whole system is determined by a set of {R }. An important feature of the network is its extremely wide spectrum of resistance, R , due to the exponential dependence on r and . When the temperature is not too low, the r dependent part in Equation (79) dominantly determines the conductivity in the system. Then electrons hop from a localized state to its nearest-neighbor state with smallest r , as illustrated by process (i) in Figure 22(b). The temperature dependence in Equation (79) results in an activation-type conductivity: ¼ 3 expð – "3 =kB T Þ
102 Impurity Bands in Group-IV Semiconductors
yields the optimized hopping distance
(a)
r¼
1=4 ð80Þ
and average hopping energy
α Rαβ (b)
9
8Dð0ÞkB T
β
¼
(ii) (i)
EF
kB ðT0 T 3 Þ1=4 4
This gives the optimized hopping rate ¼ ph exp½ – ðT0 =T Þ1=4
Optimized r (T ) Figure 22 (a) Random resistance network proposed by Miller and Abrahams (1960) to evaluate the hopping conductivity in the Anderson-localized regime. The resistance R between sites and is given by Equation (79). (b) Schematic drawing for (i) the nearest-neighbor hopping and (ii) variable-range hopping conduction. In the latter, the hopping length increases with decreasing temperature.
for the nearest-neighbor hopping. The activation energy, "3, is evaluated using the percolation method. The calculation is explained in detail in Section 8 in the textbook by Shklovskii and Efros (1984). 1.03.4.1.2
Mott’s law At low temperatures, the energy term in Equation (79) leads to a strong resistance dispersion as well as the distance term. Then, there are two competing factors in the hopping rate: a larger hopping distance, r , enables us to find a state with smaller " , but a larger r results in a smaller transfer integral at the same time. The average of hopping distance, r, which optimizes the product of the two factors, turns out to be r _ T – 1=4 , which leads to the hopping conductivity in Equation (74). Since the optimized hopping distance varies with T, as illustrated by process (ii) in Figure 22(b), this mechanism of hopping conduction is known as the VRH. Following Mott, we present an intuitive derivation of Equation (74). The problem is to find the optimized hopping distance r which maximizes in Equation (77). Since there are (4/3)r3D(0) states available in a spherical region of radius r, where D(0) is the density of states at the Fermi level per unit volume, the average of " is estimated to be [(4/3)r3D(0)]1. The minimization of the average hopping rate 2r 1 3 – ¼ ph exp –
kB T 4r 3 Dð0Þ
ð81Þ
ð82Þ
3
with kBT0 ¼ 18.1/[D(0) ]. The conductivity is expressed as ¼ e 2 DDð0Þ from the Einstein relation, where the diffusion constant D is written as D ¼ r 2 . Thus, we arrive at ¼ 0 exp½ – ðT0 =T Þ1=4 2
ð83Þ
2
with 0 ¼ e D(0)r ph, where r is given by Equation (80). Generally, in d-dimensional systems, we have _ exp½ – ðT0 =T Þ1=ðdþ1Þ
ð84Þ
kB T0 ¼ =½Dð0Þ 3
ð85Þ
with
where the numerical coefficient depends on d. After the qualitative derivation by Mott, Equation (84) was confirmed numerically by the percolation method (Ambegaokar et al., 1971; Pollak, 1972) and Monte Carlo method (Skal and Shklovskii, 1971). The numerical coefficient in Equation (85) was determined by these numerical calculations. Those are 21.1 for d ¼ 3 and 13.8 for d ¼ 2. As regards the VRH, it should be mentioned that Kamimura and Mott (1976) pointed out a possibility that the VRH is induced by the ESR. 1.03.4.2
Coulomb Gap
In the low-concentration regime of compensated samples, the long-range Coulomb interaction between charged impurities plays an important role in the VRH conduction. The interaction results in the reduction in density of states around the Fermi level. This is called the Coulomb gap. The Coulomb gap affects remarkably the T dependence of the hopping conductivity. In this subsection, we briefly explain the theory of Coulomb gap following Shklovskii and Efros (1984) (also see (Efros and Shklovskii, 1985)). At low temperatures, all the impurities of groupIII elements are occupied by an extra electron and
Impurity Bands in Group-IV Semiconductors
thus negatively charged, resulting in acceptors. Some of the donor levels are occupied by an electron and the others are empty. Here, the former are neutral while the latter are positively charged. The electronic states in the system are identified by the number of electrons ni (¼1 or 0) at donor site i if the electron spins are neglected. This situation is described by the Hamiltonian " don X don ð1 – ni Þð1 – nj Þ e2 1 X H ¼ "D n i þ 2 ri j i i j 6¼i # don X acc acc X acc X X 1 – ni 1 1 þ – 2 k l6¼k rkl rik i k d on P
ð86Þ
where "D is the energy level of the isolated donors, don and acc mean the summations over all donor and acceptor sites, respectively. The second term in the Hamiltonian represents the long-range Coulomb interaction between ionized donors, the third term the Coulomb interaction between ionized donors and acceptors, and the last term the Coulomb interaction between acceptors. The set of occupation numbers {ni} in the ground state is determined by minimizing H with a fixed total number of electrons, or by minimizing H~ ¼ H –
don X
ni
i
The chemical potential is chosen to be zero. As a result, the energy level at donor site i is given by "i ¼
" # don acc X 1 – nj X qH e2 1 ¼ "D þ – þ qni rik rij k j 6¼i
ð87Þ
The determined values of "i’s become random due to the long-range Coulomb interaction from the other impurities. Hence, the width of the impurity band is of an order of e 2 =ð r Þ, where r¼ ð4nD =3Þ – 1=3 is the average distance between donors. For an electron transfer from a filled donor i to an empty donor j in the ground state, the total energy must be increased: ij ¼ "j – "i –
e2 >0 rij
ð88Þ
The increment ij in energy is derived by the following argument. First, an electron is transferred from donor i to infinity, which requires the energy "i. Second, the electron is placed on donor j, which is empty in the ground state. The energy cost in the second stage is "j e2/rij since donor j is empty,
103
whereas the Coulomb energy from donor i is counted in "j because donor i is occupied in the ground state. The condition (88) must hold for any pair of donors with "i < 0 and "j > 0. Consider donors whose energies fall in a narrow band [ – "~=2; "~=2] around the Fermi level. From Equation (88), any two donors in this band must be separated from each other by a distance rij larger than e2/"~ when the energies are on the opposite sides of the Fermi level, "i"j < 0. As a result, the donor concentration in this band n("~) cannot exceed ð3=4Þð"~=e 2 Þ3 . Thus, the density of states Dð"~Þ ¼ dnð "~Þ=d "~ tends to zero when "~! 0, at least as fast as "~2 . (D("~) cannot tend to zero faster than "2 because the decrease in the density of states stems from the strong interaction between the low-energy states. Otherwise, the separation between the donors is too large and hence the interaction is too weak to be responsible for lowering the density of states so much.) We conclude that Dð"~Þ ¼ 3
3 "~2 e6
ð89Þ
where 3 is a numerical coefficient. The reduction in the density of states in the vicinity of the Fermi level is called Coulomb gap by Efros and Shklovskii (1975). The width of the Coulomb gap is estimated from D() D0(0), where D0(0) is the density of states at the Fermi level in the absence of Coulomb interaction. Thus, e3D0(0)1/2/3/2. Similar arguments in the two-dimensional case lead to the law Dð"~Þ ¼ 2
2 j"~j e4
ð90Þ
The numerical coefficients were estimated to be 3 ¼ 3/ and 2 ¼ 2/ (Baranovskii et al., 1978). The width of the Coulomb gap is e4D0(0)/2 in the two-dimensional case. The Coulomb gap in the density of states significantly affects the T dependence of the VRH conduction at temperature T . The hopping conductivity is derived in the same way as that in the preceding subsection. Let us find the optimized hopping distance r which maximizes in Equation (77). From the above-mentioned argument, the hopping energy is evaluated to be " ¼ 9e2/r for a given hopping distance r, with 9 being a numerical factor. Hence, we minimize the hopping rate of "
0
2r e2 – ¼ ph exp –
kB T r
#
104 Impurity Bands in Group-IV Semiconductors
1.03.4.3.1 Appearance of spin-dependent mechanisms
and obtain the optimized hopping distance r¼
e2 kB T
1=2 ð91Þ
apart from a numerical prefactor, and average hopping energy ¼
kB ðT1 T Þ1=2 2
ð92Þ
The optimized hopping rate is ¼ ph exp½ – ðT1 =T Þ
1=2
ð93Þ
with kB T1 ¼ 1
e2
ð94Þ
The numerical coefficient 1 depends on the dimensionality d, whereas the exponent of T dependence of is identical for both dimensions of d ¼ 3 and 2. Finally, the hopping rate in Equation (93) yields the hopping conductivity of _ exp½ – ðT1 =T Þ
1=2
ð95Þ
This VRH conduction is observed in many samples. Here, we have considered only the single-electron transfer from a donor to another. In some cases, a simultaneous transfer of many electrons plays an important role in the hopping conduction. Then, a hopping process of an electron is accompanied by the transfer of many electrons surrounding the initial and final donors of the electron hopping, that is, a polaron cloud is formed around the hopping electron. For a detailed discussion, the reader is refered to section 10 in Shklovskii and Efros (1984). 1.03.4.3 Formula for Magnetoresistance in the Variable-Range Regime As discussed in Section 1.03.3, intrastate and interstate interactions play important roles in the Anderson-localized states. In this subsection, we show that these interactions are important for the magnetoresistance in the VRH conduction in the intermediate concentration regime of uncompensated samples. We assume that the long-range part of the Coulomb interaction is screened out by extended states over several impurity sites. Kurobe and Kamimura (1982) calculated the resistivity due to the VRH between the Anderson-localized states by the percolation method (Pollak, 1972) taking into account the intrastate interaction.
It was mentioned in Section 1.03.3.2 that there exist three types of electronic states in the presence of intrastate interaction: unoccupied (UO), singly occupied (SO), and doubly occupied (DO) states. Corresponding to these three electronic states, there are four kinds of elementary hopping processes, as shown in Figure 23: 1. 2. 3. 4.
from an SO state to an UO state; from an SO to another SO state; from a DO to an UO state; and from a DO to an SO state.
The total hopping rate from a localization state ji to state j i is given by the sum of the hopping rates of the four elementary processes: ¼
4 X
ðlÞ
ð96Þ
l¼1
If we assume that each state is statistically indepenðlÞ dent of each other, each hopping rate ðl ¼ 1 – 4Þ ðlÞ is the product of an intrinsic hopping rate and appropriate occupation probabilities, as follows: ð1Þ
¼ ð2Þ
¼
ð3Þ
¼
ð1Þ
hn ð1 – n – Þihð1 – n Þð1 – n – Þi
¼
P ¼
¼ ð4Þ
P
P ¼
ð2Þ
hn ð1 – n – Þihð1 – n Þn – i
ð3Þ
hn n – ihð1 – n Þð1 – n – Þi
P ¼
ð4Þ
hn n – ihð1 – n Þn – i
where n is the number operator in state ji and spin . Regarding functional forms of the intrinsic hopðlÞ ping rates , we assume that they are similar to those given in Equation (75) and (76): ðlÞ
ðlÞ
¼ ph e – 2r = Nph ðE ðlÞ Þ 0 exp½ – 2r = ðlÞ – E ðlÞ =kB T
εα
εβ
εβ
εα
U
εβ (1) S
U
ð97Þ
(2) S
S
EF U
U
(3) D
εβ
εα
εα U
(4) D
Figure 23 Four kinds of hopping processes in the presence of intrastate interaction U.
S
Impurity Bands in Group-IV Semiconductors
for E(l) > 0, where as ðlÞ
¼ ph e – 2r = ðlÞ ½1 – N ð – E ðlÞ Þ 0 exp½ – 2r = ðlÞ ð98Þ (l)
for E < 0, where r is the distance between the localization centers of states ji and j i. The preexponential factor, 0, is assumed to be independent of l. The E(l) is the difference in the total energies before and after the hopping process of type l: E ðlÞ ¼ " – "
þ exp½ – ð2" þ U Þ=kB T ð3Þ
In the presence of intrastate interaction, the localization length of DO state 2(") is different from that of SO states 1(") by an analogy with the case of a D state (Natori and Kamimura, 1977; Kammimura, 1979). We take the larger of the two as the localization length (l) in the hopping processes l ¼ 2 and 3 in which the localization length is different before and after the hopping, following the treatment by Kirkpatrick (1973). Therefore, the relevant localization lengths in Equation (97) and (98) are (1) ¼ 1 and (l) ¼ 2 (l ¼ 2, 3, 4). 1.03.4.3.2
Calculated results For the random resistance network in the present problem, the percolation calculation gives the following results for the temperature dependence of resistivity in d dimensions (Kurobe and Kamimura, 1982, Kurobe 1986): ð99Þ
where kB T0 ¼
c ð3Þ
13 þ 3 23 3 D1 ðEF Þ ð 1 þ 23 Þ2 þ 4 26
ð100Þ
for d =3 and kB T0 ¼
c ð2Þ
12 þ 3 22 D1 ðEF Þ ð 12 þ 22 Þ2 þ 4 24
ðH Þ ¼ ½Z ðH ÞZ ðH Þ – 1 Xn exp½ – ð" þ B H Þ=kB T ð1Þ þ exp½ – ð" þ " Þ=kB T ð2Þ
ðl ¼ 3Þ
¼ 0 expðT0 =T Þ1=ðd þ1Þ
Now, we discuss the magnetic-field dependence of VRH conduction following Kurobe and Kamimura (Kurobe and Kamimura, 1982; Kamimura et al., 1982, 1983; Kurobe, 1986). By neglecting the change of localization length due to a magnetic field, the total transition rate (H) from localized state to state can be expressed as
¼
ðl ¼ 1; 4Þ
¼ " – " þ U ðl ¼ 2Þ ¼ " – " – U
105
ð101Þ
for d ¼ 2. The coefficients are c(3) ¼ 2268 (3)/(115) and c(2) ¼ 105 (2)/(11) with (d) ¼ 201/(d þ 1). Here, D1(EF) is the density of states for the first electron at the Fermi level defined in Section 1.03.3.2. The energy dependence of localization lengths 1 and 2 is neglected here, and 2 is taken to be larger than 1. This result shows that Mott’s law still holds even in the presence of intrastate interaction, but with a prefactor T0 which now depends on both the localization lengths of SO and DO states, 1 and 2.
þ exp½ – ð2" þ " þ U – B H Þ=kB T ð4Þ
o
ð102Þ
where Z(H) is the partition function in magnetic field H: Z ðH Þ ¼ 1 þ 2coshðB H =kB T Þexpð – " =kB T Þ þ expð – ð2" þ U Þ=kB T Þ
ð103Þ
When a magnetic field is applied, the spins of SO states tend to become parallel to the field. Therefore, the probability to find a pair of SO states whose spins are antiparallel to each other becomes smaller with increasing magnetic field. As a result, the hopping processes from SO to SO states, which take place only when the spins are antiparallel to each other, are suppressed by the magnetic field as long as spinflip processes are neglected. The processes from DO to UO states are connected with those from SO to SO states by the relation of a detailed balance. Therefore, the processes from DO to UO states are also suppressed. This suppression of hopping processes by the magnetic field gives rise to a positive magnetoresistance. When the hopping processes of type 2 and 3 are completely suppressed, only the processes of types 1 and 4 contribute to the hopping conduction. Thus, the magnetoresistance saturates for a magnetic field stronger than a certain value. Figure 24 presents numerical results for the magnetic-field dependence of the resistivity for various temperatures in three dimensions (Kamimura et al., 1982, 1983). The state-energy dependence is neglected for the localization length, , and for the intrastate interaction energy (U). The values of the parameters,
1, 2, D1(EF) and U (X U), correspond to those in Si:P with a donor concentration of nD ¼ 1.7 1018 cm3. The result clearly indicates that the magnetoresistance grows and then saturates with an increase in magnetic field. The saturation value increases when the
106 Impurity Bands in Group-IV Semiconductors
E T = 0.2 K
Ec
3
Ec
Δρ /ρ
0.5 K 2 1.0 K
EF
EF
1 3.0 K 2μ BH 10
0
20 H (kOe)
30
40
Figure 24 Calculated result for the positive component of the magnetoresistance in the variable-hopping conduction regime, as a function of magnetic field for various temperatures, in three dimensions. The parameters used are U ¼ 3.4 meV, D1(EF) ¼ 9.5 1019/cm3 eV, 1 ¼ 49 A˚, and
2 ¼ 98 A˚. Modified from Kurobe A and Kamimura H (1982) Journal of the Physical Society of Japan 51: 1904.
temperature T decreases. The saturation value also increases when the density of states D1(EF) decreases, or when the localization lengths decrease with the ratio of 1/ 2 being fixed. The analytical expression for the saturated value is given by d d 1=ðdþ1Þ 8 1=ðd þ1Þ
1 þ 2 < ðH Þ c ðdÞ 2d 2d ¼exp
1 þ 2 : D1 ðEF ÞkB T ð0Þ sat #) ð104Þ ! 1=ðd þ1Þ d d
1 þ3 2 –1 – ð 1d þ 2d Þ2 þ4 22d
j
in d dimensions (Kurobe and Kamimura, 1982; Kurobe, 1986). 1.03.4.3.3
Comparison with experiments As an effect of the magnetic field on the transition rate, we have so far considered the change in occupation probabilities of UO, SO with spin ¼ , and DO states. In addition, we have to take into account an effect of the magnetic field on the localization length. Since the localization length is a function of state energy, it is changed through the Zeeman shift, as schematically shown in Figure 25. This mechanism was first proposed by Fukuyama and Yosida (1979). The localization length increases to
(" þ BH) for spin-up electrons while it decreases to (" BH) for spin-down electrons if (") is an increasing function of state energy " (we assume that EF is located below the mobility edge Ec). Since the majority carriers have the magnetic moments
D–(E )
D+(E )
Figure 25 Schematic illustration of the negative contribution to the magnetoresistance due to the stateenergy dependence of the localization length.
parallel to the magnetic field, this effect gives rise to a negative magnetoresistance. Combining this effect with the opposite effect discussed above, the total magnetoresistance is written conventionally in the form ðH Þ ðH Þ – ð0Þ ¼ ¼ P ðH Þ þ N ðH Þ ð0Þ ð0Þ
ð105Þ
where P(H) and N(H) represent positive and negative parts of the magnetoresistance, respectively. For comparison with experimental results, we discuss the magnetoresistance in a transition metal dichalcogenide system, 1T-TaS2, which has a layertype sandwich structure with a commensurate charge density wave (CDW) phase below T ¼ 200 K (Williams et al., 1974, 1975; Wilson et al., 1974, 1975). In this phase, the material is an insulator with an energy gap which is interpreted as arising from the CDW modulation. DiSalvo and Graebner (1977) found a T1/3 law of conduction in this phase, which should be ascribed to the VRH conduction among the Anderson-localized states in the energy gap in the quasi-two dimensional system. The magnetoresistance was later measured in this system by Kobayashi and Muto (1979), Onuki et al. (1980), and also by Tanuma (1983). Figure 26 shows their experimental results of the magnetoresistance as a function of magnetic field for various temperatures. Let us apply the present theory to the abovementioned experimental results. We assume that
1(") ¼ 2(")/2 and that the localization lengths depend on the state energy, ", in the following way:
1 ð" Þ ¼ 2 ð" Þ=2 _ ðEc – " Þ – 1
ð106Þ
Impurity Bands in Group-IV Semiconductors
0.2
2
107
T = 0.4 K
T = 1.42 K
Δρ /ρ
0 0.39 K Δρ /ρ
–0.2
1
0.6 K 1.0 K
0 –0.4 0.1 K –1 0
–0.6 0.05 K –0.8
0
20
60 40 H (kOe)
80
Figure 26 Experimental data on magnetoresistance of 1T-TaS2 in the temperature range from 1.42 to 0.05 K. Modified from Kobayashi N and Muto Y (1979) Solid State Communications 30: 337.
where Ec is the mobility edge although there is no mobility edge in two-dimensional systems of infinite size (Abrahams et al., 1979, 1980). Choosing the values of the parameters to be Ec " ¼ 1.0 meV, U ¼ 1.7 meV, D1(EF) ¼ 5.3 1014 cm2eV, and
1(EF) ¼ 100 A˚, the magnetoresistance was calculated. The calculated result is shown in Figure 27. The result shows that the magnetoresistance is positive in low magnetic fields and then changes to negative in high magnetic fields. The magnitude of the positive part decreases with increasing temperature. These features are consistent with the experimental results. However, experimental data show only negative magnetoresistance below about 0.1 K (see Figure 26). This behavior does not agree with the theoretical result. This discrepancy between the theoretical result and experiments can be explained by the spin-dependent interstate interactions between Anderson-localized states, which is given by Equation (53). When the interstate interaction is antiferromagnetic, the spins in the SO states retain spin-singlet states and hardly become parallel to the magnetic field if the magnetic field is weak. Thus, the suppression of hopping processes of types 2 and 3 in Figure 23 may not occur, and in consequence
50 H (kG)
100
Figure 27 Calculated result for the magnetoresistance of a two-dimensional 1T-TaS2 system, in the presence of intrastate interactions. Both positive and negative components are taken into account. The parameters are (EcEF) ¼ 1.0 meV, U ¼ 1.7 meV, D1 (EF) ¼ 5.3 1014 cm2eV and 1(EF) ¼ 100 A˚. Modified from Kamimura H (1982) Progress of Theoretical Physics Supplement 72: 206.
P(H) disappears in Equation (105). When the temperature is lower than a certain temperature corresponding to the order of average interstate interaction, the magnetoresistance becomes negative even in low fields. The observation of an anomaly in the specific heat of this material (Nishio et al., 1978) suggests the existence of spin-dependent interstate interactions, as discussed in Section 1.03.3.3, where the magnitude of the interaction is estimated to be about 1 K. If the average interstate interaction is taken as 0.5 K, the calculated result of magnetoresistance is in good agreement with the experimental results (Kamimura et al., 1982, 1983). 1.03.4.4 Magnetocapacitance in Intermediate-Concentration Regime In this subsection, we consider the magnetic-field dependence of the electrical polarizability for Si:P in the intermediate-concentration regime. The magnetocapacitance reflects elementary processes of the hopping between the Anderson-localized states although it is not a transport property. New et al. (1982) reported an experimental result that the electrical polarizability depends drastically on the magnetic field. Figure 28(a) shows their experimental result of the magnetocapacitance (MC) of Si:P, where the MC is plotted as a function of magnetic field H. The MC is defined as
108 Impurity Bands in Group-IV Semiconductors
(b)
(a) 10–1
10–2
ln Reff (H)
ln Reff (H)
10–1
T = 1.3 K
T=1K 10–2
2 10–3 10–3
1.9 2.9 4.2 10
1
0.1
B (T)
3 4 5
1 B (T)
10
Figure 28 (a) Experimental result for logarithmic of magnetocapacitance, Reff(H), plotted against magnetic field for Si:P at various temperatures. The donor concentration is nD ¼ 1.0 1018 cm3. (b) Calculated result for Reff(H) for Si:P with donor concentration nD ¼ 1.7 1018 cm3 with U ¼ 3.4 meV and W ¼ 19 meV. (a) Modified from New D, Lee NK, Tan HS, and Castner TG (1982) Physical Review Letters 48: 1208. (b) Modified from Kurobe A, Takemori T, and Kamimura H (1984) Physical Review Letters 52: 1457.
Reff ðH Þ ¼
D ð0Þ D ðH Þ
ð107Þ
where the effective polarizability of donors, D, is related to the dielectric constant by – 0 ¼
4nD D 1 – 4nD D =ð30 Þ
Anderson-localized states in the intermediateconcentration regime. The electrical polarizability, , is calculated by the Kubo formula, taking account of four kinds of excitations in Figure 23: ¼ 1 þ 2 þ 3 þ 4
ð108Þ 1 ¼
Here 0 is the dielectric constant of the host material. The electrical polarizability is obtained from the relation of ¼ 0 þ 4 . This experimental result shows that ln Reff(H) _ H2 for low fields and the dependence becomes weaker for higher fields, implying the behavior of spin-dependent electrical polarizability. New et al. (1982) ascribed their experimental result to the Zeeman splitting of excited spin-triplet states in spin-singlet paris of donors. Their model may be suitable for the low-concentration regime of the donors. For the present case, however, we have to consider the Anderson-localized states in the intermediate-concentration regime. Kurobe et al. (1984) adopted the theory in the preceding subsection to calculate in the intermediate-concentration regime. Although there are various excitations, the dominant contribution to electrical polarizability comes from the one-electron excitations among the
ð109Þ
2e 2 X hn ð1 – n – Þihð1 – n Þð1 – n – ÞijhjX j ij2 V " – " ð110Þ
2 ¼
2e 2 X hn ð1 – n – Þihð1 – n Þn – ijhjX j ij2 V ð" þ U Þ – " ð111Þ
3 ¼
2e 2 X hn n – ihð1 – n Þð1 – n – ÞijhjX j ij2 V " – ð" þ U Þ ð112Þ
4 ¼
2e 2 X hn n – ihð1 – n Þn – ijhjX j ij2 ð113Þ V ð" þ U Þ – ð" þ U Þ
where h i represents the thermal average and h jX j i is the matrix element of the x-coordinate which is proportional to the electric dipole moment. In general, there is also contribution to from excitations into states of antibonding characters, which reduces in the low-concentration regime, as discussed by New
Impurity Bands in Group-IV Semiconductors
et al. (1982). In the intermediate concentration, however, the antibonding states have much higher energies. Besides, the states are more extended and hence the matrix elements in Equations (110–113) are small. Hence, the above-mentioned excitations among the Anderson-localized states dominantly contribute to . The calculated result is shown in Figure 28(b) on simplifying assumptions that the matrix element h jX j i does not depend on the kind of process and that the localization length, , and intrastate interaction, U, are independent of the state energies. For low magnetic fields, we have ln½Reff ðH Þ _ ðB H =kB T Þ2
ð114Þ
Physically, spins of SO states tend to be aligned with increasing H. Since an electron cannot be excited to a state which is already occupied by another electron of the same spin, the contribution from an SO state to another SO state in Equation (111) (2) decreases with increasing magnetic field. Therefore, at high fields (H > 5 T), where this type of excitation is almost suppressed, MC becomes less dependent on magnetic field. All these characteristics of MC are in good agreement with experimental results.
1.03.4.5 Spin-Dependent Behavior of Magnetoresistance in Other Systems As we have described in the previous subsections, the theory of magnetoresistance and MC developed by Kurobe and Kamimura (Kurobe and Kamimura, 1982; Kamimura et al., 1982, 1983; Kurobe, 1986) has played an important role in clarifying the spin-dependent effects on the transport properties due to the electron–electron interactions in doped semiconductors (New et al., 1982; Agrinskaya and Kozub, 1998; Agrinskaya et al., 2000; Kozub, 2002; Meir, 2001). Recently, their theory has been widely applied to other systems. For example, two-dimensional hole system in the GaAs/AlGaAs heterostructure in magnetic fields (Yoon et al., 2000; Noh et al., 2004), silicon MOSFETs in magnetic fields (Mertes et al., 2001), amorphous indium oxide films (Frydman and Ovadyahu, 1995; Lee et al., 2002; Orlyanchik and Ovadyahu, 2005), amorphous Bi films (Hernandez et al., 2003), Mn–Fe–Ge antiferromagnets (Du et al., 2007), and icosahedral Al-Pd-Re materials (Srinivas et al., 2001). In particular, it is interesting to see that the theory of Kurobe and Kamimura has been used for elucidating magnetoresistance phenomena in one-dimensional and quasi-one-dimensional systems
109
such as metallic or doped polyacetylene (Kaneko et al., 1994; Ishiguro et al., 1995; Choi et al., 1999), metallic polypyrrole (Singh et al., 2008), entangled carbon-nanotube network (Kim et al., 1998), carbon nanomaterials (Demishev et al., 2008), disordered nanographite network (Takahara et al. 2007), etc.
1.03.5 Conclusions In this chapter, we have described how the impurity band in group-IV semiconductors manifests a rich diversified physics. In Section 1.03.2, we have surveyed the transport phenomena in the impurity band, MI transition, and the historical development of localization theory. In Section 1.03.3, we have described the theoretical formulation to treat the interplay of disorder and electron–electron interactions in the strongly Anderson-localized regime just in the insulating side of the MI transition, developed by Kamimura and co-workers. Their theory successfully explains the anomaly of specific heat and spin susceptibility at low temperatures. The theoretical formulation has been confirmed by numerical simulation for the electronic states in the Anderson-localized regime. Section 1.03.4 has been devoted to the description of the hopping conduction in the Anderson-localized regime. We have reviewed the Mott’s law for the VRH and Coulomb gap theory for the compensated samples. Then, we have described the theory by Kurobe and Kamimura on the magnetoresistance and MC in the VRH regime, considering spin-dependent effects due to electron–electron interactions. Their theory has now been applied to magnetoresistance phenomena observed in a wide variety of systems such as polymers, carbon-nanotubes, and other exotic materials. Recent development of the nanofabrication technology on semiconductors is opening new research fields in solid-state physics. As we have mentioned in Section 1.03.1.2, for example, it is an interesting subject to control single impurity states in doped semiconductors for the quantum information processing. It is very impressive that Kamimura and co-workers predicated the spin-dependent effects on various physical phenomena in the strongly Anderson-localized regime two decades ago before extensive researches on solid-state quantum computers have begun. We believe that the present chapter will be useful for graduate students and young researchers who will work on the physics and technology of semiconductors. (See Chapters 1.05, 1.06, 2.02 and 2.04).
110 Impurity Bands in Group-IV Semiconductors
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1.04 Atomic Structures and Electronic Properties of Semiconductor Interfaces T Nakayama, Chiba University, Chiba, Japan Y Kangawa, Kyushu University, Fukuoka, Japan K Shiraishi, University of Tsukuba, Ibaraki, Japan ª 2011 Elsevier B.V. All rights reserved.
1.04.1 1.04.2 1.04.2.1 1.04.2.1.1 1.04.2.1.2 1.04.2.2 1.04.2.2.1 1.04.2.2.2 1.04.2.2.3 1.04.2.2.4 1.04.2.3 1.04.2.3.1 1.04.2.3.2 1.04.2.4 1.04.2.4.1 1.04.2.4.2 1.04.2.4.3 1.04.2.5 1.04.2.5.1 1.04.2.5.2 1.04.3 1.04.3.1 1.04.3.1.1 1.04.3.1.2 1.04.3.1.3 1.04.3.2 1.04.3.2.1 1.04.3.2.2 1.04.3.2.3 1.04.3.2.4 1.04.3.3 1.04.3.3.1 1.04.3.3.2 1.04.3.3.3 1.04.3.3.4 1.04.4 References
Introduction Interface Formation and Stability Semiconductor Epitaxy Epitaxial growth diagram Competition of growth direction Growth Modes in Heteroepitaxy Growth modes Theory based on macroscopic energetics Mode transition during epitaxy Theory based on atomistic energetics Defect Formation in Heteroepitaxy Heterovalency-induced defects Impurity-induced defects Interface Formed by Oxidation Phenomenological theory Mechanism of oxidation Optical control of oxidation Stability of Interfaces Metal induced inter-diffusion Silicidation Electronic Structures of Interfaces General Features Origins of interface states Semiconductor/semiconductor interface states Metal/semiconductor interface states Band Alignment Schottky barrier Charge neutrality level Band bending Band offset New Features of Schottky Barrier Breakdown of Schottky–Mott limit Generalized charge neutrality level Interface reaction and Fermi-level pinning Correlation between interfaces Future Prospects
114 115 115 115 117 121 121 122 123 124 126 126 128 130 130 131 137 141 141 143 145 145 145 146 149 150 150 151 153 154 157 157 158 161 163 167 170
113
114 Atomic Structures and Electronic Properties of Semiconductor Interfaces
1.04.1 Introduction An interface is a boundary between two different materials A and B (denoted henceforth as A/B). Because the vacuum, a gas, or a liquid system is a material form, the surface of any material is also considered as an interface. Since the number of material combinations, A/B, are infinite, we can expect a variety of physical phenomena to be observed at the interfaces. Moreover, in the last decade, the subject of interfaces has increasingly extended from representative solid/solid interfaces made of inorganic materials to a wide area that includes organic and liquid/solid interfaces. A complete review of the overall features of these interfaces and of the important progress in relation to ordinary solid/solid interfaces is beyond the scope of the present chapter. Thus, we restrict ourselves to the limited subject of atomic and electronic structures of semiconductor interfaces, with which the authors have been closely concerned, and explain fundamental physical concepts revealed by recent researches. To understand the atomic structures of interfaces, it is essential to clarify the formation of an interface and its stability. This is because the interface is sometimes produced in nonequilibrium environments and often becomes a metastable structure. In Section 1.04.2, we consider the formation and stability of interfaces. Two representative phenomena are discussed with respect to the interface formation. One is the epitaxial growth. Semiconductor interfaces are often produced by stacking other materials on the substrate. In this case, we naturally ask what conditions are necessary to realize the epitaxial growth, which interface direction is selected in the epitaxy, what kinds of crystal shapes are produced in the epitaxy, and why the defects are generated at the interface. The answers to these questions are discussed in Sections 1.04.2.1, 1.04.2.2, and 1.04.2.3. The other topic concerning the interface formation is the oxidation. Si/SiO2 interfaces, which are the most important interfaces of present semiconductor technology, are produced by oxidizing the Si surfaces. Recent progress in surface/interface measurements has uncovered a surprising oxidation process; Si/SiO2 interfaces are produced by layerby-layer oxidation. We wonder how the oxidation proceeds, why the layer-by-layer chemical reaction is realized, and how we observe and control such
oxidation. Section 1.04.2.4 is devoted to a discussion of the answers to these questions. On the other hand, the produced interfaces are not stable forever. They sometimes change the shape by the interface reaction of interface atoms. For an example of these reactions, the random atom intermixing and the compound formation like silicidation at metal/ semiconductor interfaces are discussed in Section 1.04.2.5. The most important physical quantity to characterize the electronic structures of an interface A/B is the Schottky barrier in case of metal/ semiconductor interfaces and the band offset in case of semiconductor/semiconductor interfaces. These quantities represent the energy difference of the electronic states between A and B materials. The Schottky barrier height becomes a key quantity to inject the electron and hole carriers from metal electrodes into semiconductors, thus governing the electronic transport properties in semiconductors. Meanwhile, in quantum superstructures made of semiconductors, such as a quantum-well and a superlattice, the band offset values determine the electron and hole confinement into the well-material layers and control band structures, thus becoming the key quantities to control optical properties. In Section 1.04.3, we concentrate on these quantities and explain what produces such energy difference of electronic states. Interfaces have unique electronic states not present in bulk systems having a periodicity. The general features of interface electronic structures and their examples are illustrated in Section 1.04.3.1. Based on these features, in Section 1.04.3.2, we explain the conventional views on the origins of Schottky-barrier and band-offset formation. On the other hand, unusual behaviors of Schottky barriers, such as the breakdown of the Schottky limit and the reaction-induced Fermilevel pinning, which have never been explained by conventional views, have been found very recently in the development of some of the leading Si nanoscale devices. These new features are explained in detail in Section 1.04.3.3. Section 1.04.4 is devoted to some of the prospects for future interface physics, by illustrating a new Fermi-level stabilization by multivalent materials and crossdimensional phenomena at nanocontact interfaces, such as quantum frictions in current and vibrational motion and electron transport between an electrode and the quantum-dot.
Atomic Structures and Electronic Properties of Semiconductor Interfaces
1.04.2 Interface Formation and Stability 1.04.2.1
Semiconductor Epitaxy
1.04.2.1.1
Epitaxial growth diagram Fabrication of semiconductor devices such as highelectron-mobility transistors (HEMTs) and lightemitting diodes (LEDs) are carried out using vapor phase epitaxy (VPE) techniques, for example, molecular beam epitaxy (MBE) and metal organic vapor phase epitaxy (MOVPE). The adsorption– desorption behavior of the constituent atoms on semiconductor surfaces is of fundamental importance in these growth techniques. In particular, the behavior of constituent atoms must be known to understand surface and thin film growth-related phenomena such as the change in surface reconstructions and growth processes. In the present section, we discuss the relationships between adsorption–desorption behavior and growth conditions such as beam equivalent pressure (BEP), p, and temperature, T. We can consider adsorption–desorption behavior from a viewpoint of vapor–solid phase equilibrium. If the free energy of an atom/molecule in the vapor phase is smaller than that on the surface, an impinging atom/molecule is easily desorbed from the surface and stabilized in vapor phase. On the other hand, an atom/molecule is absorbed if the free energy of the atom/molecule in vapor phase is larger than that on the surface. Here, the free energy of the atom/molecule, which is called the chemical potential (), can be computed by using quantum statistical mechanics. The energy gain of an adsorbed atom/molecule on the surface (i.e., the adsorption energy (Ead)) is estimated, for example, by ab initio calculations. By comparing with Ead, we can discuss the adsorption–desorption behavior as shown in Figure 1. Adsorption
Desorption
E
μ Ead
Ead μ
Figure 1 Schematics of the magnitude correlation between chemical potential and adsorption energy Ead and its relationship with the adsorption–desorption behavior.
115
The chemical potential for the ideal gas is given by (Fowler, 1936; Tolman, 1938; Fowler and Guggenheim, 1939; Mayer and Mayer, 1940; Rodebush, 1931; Fowler and Sterne, 1932; Kassel, 1936): ¼ – kB T
lnðgkB T =p trans rot vibr Þ
3=2 trans ¼ 2mkB T =h 2 n on=2 rot ¼ ð1=Þ 83 ðIA IB :::Þ1=n kB T =h 2 vibr ¼ Pi3N – 3 – n f1 – expð – hvi =kB T Þg – 1
ð1Þ ð2Þ ð3Þ ð4Þ
where trans, rot, and vibr are the partition functions for the translational motion, the rotational motion, and the vibrational motion, respectively. Here, kB is Boltzmann’s constant, T the temperature, g the degree of degeneracy of the electron energy level, p the BEP of the particle, m the mass of one particle, h the Planck’s constant, the symmetric factor, II the moment of inertia, n the degree of freedom of the rotation, N the number of atoms in the particle, i the degree of freedom for the vibration, and the frequency. II is written as II ¼ mI r 2
ð5Þ
where mI is the reduced mass and r the radius of gyration. Practical applications of this theory are as follows. During the typical GaAs MBE process, monatomic Ga and As2 molecule are supplied on the substrate surface under an ultrahigh vacuum (UHV) condition. In the UHV condition, the ideal-gas assumption is applicable because impingement of each atom/molecule is negligible. In the case of monatomic Ga, the value of N and I are 1 and 0, respectively. Therefore, all that is required to compute the chemical potential of monatomic molecule is considering the partition function of translational motion. The value of parameter g for monatomic Ga in Equation (1) is 2. The values of g for typical elements are summarized in Table 1 (Glasstone et al., 1964). In the case of As2 molecule, we should consider the partition functions for rotational and vibrational motions. Here, the value of the symmetry number in (3) is 2 for As2. The values of and r for As2 in Equations (4) and (5) can be estimated by ab initio molecular orbital calculations. The values of and r for As2 estimated by the Gaussian 98 program (Frisch et al., 1998) are 446 cm1 and 1.0621 A˚ (Kangawa et al., 2002), respectively. By using these values, we can compute the chemical potentials of monatomic Ga and As2
116 Atomic Structures and Electronic Properties of Semiconductor Interfaces Table 1 Degeneracy of the electron energy level of the various elements g
H, Li, NA, K, Rb, Cs, Cu, Ag, Au Be, Mg, Ca, Sr, Ba, Zn, Cd, Hg B, Al, Ga, In, Tl C, Si, Ge, Sn, Pb N, P, As, Sb, Bi O, S, Se, Te, Po F, Cl, Br, I He, Ne, Ar, Kr, Xe, Rn
2 1 2 3 4 3 2 1
As
[110]
I II III IV V VI VII 0
Ga
Element
B
A C
3
D
E 1
2
[110]
[001]
Reproduced with permission from Kangawa Y, Ito T, Taguchi A, Shiraishi K, and Ohachi T (2001) A new theoretical approach to adsorption–desorption behavior of Ga on GaAs surfaces. Surface Science 493: 178.
molecule as a function of growth parameters p and T. On the other hand, the adsorption energy Ead of Ga or As2 on reconstructed surfaces can be obtained by ab initio calculations. For the results in this section, the first-principles pseudopotential method based on the local-density functional formalism (Hamman et al., 1979) is used, adopting the Kleinman–Bylanders separable pseudopotentials, careful treatment of the cut-off value of local potential (Ohno, 1993), the conventional repeated-slab geometry, and termination of bottom-layer Si by fictitious H atoms. Here, we show the results of the calculated adsorption energy of Ga on GaAs(001)-(4 2)2 surface and desorption energy (Ead) of As2 from c(4 4) surface. First, we discuss about adsorption–desorption behavior of Ga on GaAs(001)-(4 2)2 surface. Figure 2 illustrates the GaAs(001)-(4 2)2 surface. The calculated results show that the Ga adatom is most stable at the site ‘E’ on the surface where two Ga–Ga bonds (E-1 and E-2 bonds) and a Ga–As bond (E-3 bond) are formed around the adatom. The most favorable site for adatom on the semiconductor surface is frequently discussed using an electron counting model (Farrel et al., 1987; Pashley, 1989). This is a model for discussing the structural stability of reconstructed surface structure of compound semiconductors. The lowest-energy structure is obtained by filling all dangling bonds on the anion atom and keeping empty those on the cation atom. This is because energy levels of dangling bonds on anion atom are generally located around the valence bands and those on cation atom are around the conduction bands. The stability of reconstructed structure and the adsorption site of constituent atoms on GaAs surface is well discussed by Ito (1995). The adsorption energies of Ead can be
[110] Figure 2 Schematic of GaAs(001)-(4 2)2 surface. Adsorption sites are indicated by the letters A–E. Reproduced with permission from Kangawa Y, Ito T, Taguchi A, Shiraishi K, and Ohachi T (2001) A new theoretical approach to adsorption–desorption behavior of Ga on GaAs surfaces. Surface Science 493: 178.
calculated as the difference between the total energy when the Ga adatom is located at the most stable site (E-site) and when the Ga atom is in the vacuum region. In the calculation, we gradually pulled the Ga apart from the E site to the vacuum region and confirmed the convergence of the total energy difference when the distance between the position of Ga atom and the E site is larger than 4 A˚. Consequently, the calculated adsorption energy Ead for Ga at the E site on the Ga-rich surface is estimated to be 3.3 eV. This implies that the Ga adsorbed structure is formed when the chemical potential for Ga atom is larger than the calculated adsorption energy (3.3 eV). On the other hand, it is thought that the GaAs(001)-(4 2)2 surface without Ga adatom is stable when the chemical potential for Ga is less than the adsorption energy (3.3 eV). Figure 3 shows the chemical potential for Ga atom as a function of temperature. Considering the typical BEP value (pGa ¼ 1.0 105 torr), the calculated line crosses the line of Ga ¼ 3.3 eV at 700 C; therefore, the critical temperature for Ga adsorption is 700 C at this BEP value. By similar calculations to obtain crossing points at other BEPs, a p–T diagram for the adsorption–desorption transition was obtained as Figure 4. This result agrees well with experiments, that is, GaAs(001)-(4 2) surface becomes unstable and Ga droplets are formed at
Atomic Structures and Electronic Properties of Semiconductor Interfaces
–2.0
700
Chemical potential (eV)
–2.5
Temperature (°C)
1.0 ×10–4 torr 1.0 ×10–5 torr 1.0 ×10–6 torr
–3.0 –3.5
Desorption c(4 × 4) unstable
600 550 500
Adsorption c(4 × 4) stable
450 10–8
μ Ga = –3.3 eV
–4.0 –4.5
600
800 700 Temperature (°C)
900
Figure 3 Chemical potential as a function of temperature. Reproduced with permission from Kangawa Y, Ito T, Taguchi A, Shiraishi K, and Ohachi T (2001) A new theoretical approach to adsorption–desorption behavior of Ga on GaAs surfaces. Surface Science 493: 178.
850 Temperature (°C)
650
117
800
Desorption
750 700 650
Adsorption
600 10–7
10–6
10–5
10–4
10–3
Ga BEP (torr) Figure 4 p–T dependence of adsorption–desorption transition curve for Ga. Reproduced with permission from Kangawa Y, Ito T, Taguchi A, Shiraishi K, and Ohachi T (2001) A new theoretical approach to adsorption– desorption behavior of Ga on GaAs surfaces. Surface Science 493: 178.
600 C during the conventional MBE growth under Ga-rich condition (Yamada et al., 1989), whereas Ga desorption proceeds and reconstructed surface appears at 700 C after turning off the Ga flux (Kojima et al., 1985; Gibson et al., 1990). This suggests that the theoretical approach is feasible for predicting the dependence of the temperature and BEP on adsorption–desorption behavior. Next, we discuss the stability conditions of GaAs(001)-c(4 4) surface. The conditions where the -c(4 4) appears are predicted by comparing the adsorption energy of As-dimer on the surface
10–6 10–5 10–7 As2 BEP (torr)
10–4
Figure 5 p–T dependence of adsorption–desorption transition curve for As2. Reproduced with permission from Kangawa Y, Ito T, Hiraoka YS, Taguchi A, Shiraishi A, and Ohachi T (2002) Theoretical approach to influence of As2 pressure on GaAs growth kinetics. Surface Science 507– 510: 285.
Ead-dimer (¼3.6 eV per dimer at Ga coverage Ga ¼ 0.0 (Shiraishi and Ito, 1998)) with the chemical potential of As2 molecule As2 . That is, desorption of As-dimer proceeds and the c(4 4) reconstructed structure may be decomposed when As2 is less than Ead-dimer, whereas adsorption of As2occurs and Asdimer on c(4 4 ) is stabilized when As2 is larger than Ead-dimer. Figure 5 shows the p–T dependence of adsorption–desorption transition curve of As2 (Asdimer) on GaAs(001)-c(4 4). In Figure 5, we find that c(4 4) surface is stable at higher pressures and lower temperatures. This agrees well with experiments, that is, the c(4 4)-like structure appears at T < 500 C for pAs2 1 107 torr (Bell et al., 1999), while the (2 4)-like structure is observed at T ¼ 580 C for pAs2 1 106 torr (Itoh et al., 1998). The result suggests that the theoretical approach is applicable for predicting the adsorption–desorption behavior of As2 molecule. More detailed discussions are described elsewhere (Ito et al., 2004).
1.04.2.1.2
Competition of growth direction Control of the interface/surface shape is a crucial factor for fabricating semiconductor devices with betterdefined epi-layers. In this section, we discuss, for example, how to control the surface shape of cubic GaN (c-GaN) during growth. If we can control the surface shape, we can also control the interface shape between the layer and the post-growth layer. In case of c-GaN growth, control of surface shape is indispensable to fabricate a better-defined epi-layer as well as to control the grown phase.The following information has been obtained about c-GaN growth: {111} facet formation
118 Atomic Structures and Electronic Properties of Semiconductor Interfaces
(a)
Top view 2nd layer N 1st layer Ga A
B
C E
D Side view A
B
C E
D
1st layer Ga 2nd layer N
Ga
(b)
N
0 –2 –3 –4
–6.4 eV
–1 –2.1 eV
Adsorption energy (eV)
during growth causes hexagonal GaN (h-GaN) mixing, that is, in the regions grown toward the <111> directions, the stacking sequence of . . .ABCABC. . . easily collapses and changes to . . .ABABAB. . . (Kuwano et al., 1994). This implies that two-dimensional (2D) growth without facet formation is important to grow singlephase c-GaN. Here, we apply the theoretical approach described in Section 1.04.2.1.1 on the adsorption process of monatomic Ga and N on GaN(001) and (111) facet planes in order to investigate the 2D growth condition range. The shape of a growing surface can be investigated by comparing the adsorption energy of the constituent atoms on each surface; top surface and facet plane. It is known that GaN(001)-(4 1) reconstructed structure based on a linear Ga tetramer is stable under typical MBE growth conditions for c-GaN (Neugebauer et al., 1998; Al-Brithen et al., 2005). Therefore, we assumed the (4 1) reconstructed surface as the initial substrate and investigated the relationship between adsorption–desorption behavior of monatomic Ga and N on the surface and growth conditions such as BEP and temperature. Here, we consider monatomic N, because the source gas is supplied on the substrate surface in case of radiofrequency (rf) MBE. Figure 6 shows a schematic drawing of the GaN(001)-(4 1) surface and the adsorption energy on the sites denoted by the capital letters (A–E), respectively. In Figure 6, one can see that the most favorable sites for adsorption are ‘E’ site for Ga and ‘A’ site for N. In the case of Ga adsorption, the adsorption energy is 2.1 eV. This value is larger than the formation energy of a Ga droplet (2.8 eV; Honing and Karmer, 1969). Therefore, the formation of a Ga droplet is easier than that of a Ga-adsorbed structure in this case. In the case of N adsorption, the adsorption energy is 6.4 eV. Using these values of adsorption energy, we can obtain phase diagrams related to the surface structures (Figure 7). In Figure 7 shows that no Ga-adsorbed structure would be formed, though Ga droplets seem to be formed on the surface at high Ga BEP and in the low-temperature region. Here, the boundary line satisfies the conditions of Ga ¼ EGadroplet where Ga is the chemical potential of monatomic Ga and EGa-droplet the formation energy of a Ga droplet (2.8 eV). On the other hand, the N-adsorbed structure is stable at typical growth conditions as shown in the phase diagram. These results suggest that a N-adsorbed structure appears instead of a Ga-adsorbed structure in the case of adsorption of the first atom.
–5 –6 A B C D E Site (–)
A B C D E Site (–)
Figure 6 (a) Schematic drawing of GaN(001)-(41) surface. (b) Adsorption energies of Ga and N on sites A–E shown in (a). Reproduced with permission from Kangawa Y, Matsuo Y, Akiyama T, Ito T, Shiraishi K, and Kakimoto K (2007) Theoretical approach to initial growth kinetics of GaN on GaN(001). Journal of Crystal Growth 300: 62.(Kangawa et al. 2007)
In the same way, we can discuss the adsorption of a second atom on the N-adsorbed GaN(001)-(4 1) surface. Figures 8(a) and 8(b) show a schematic drawing of the N-adsorbed GaN(001)-(4 1) surface and the adsorption energy on each site, respectively. In Figure 8(a), it can be seen that the most favorable sites are ‘D’ for Ga adsorption and ‘B’ for N adsorption. However, if a N atom is adsorbed near the N atoms on the surface, that is, the case of adsorption on ‘A’ and ‘D’ sites, N atoms would desorb as a N2 molecule instead of N-dimer formation because a N-dimer is unstable on the surface. Phase diagrams were made using the calculated adsorption energies, that is, 3.7 eV for Ga and 6.0 eV for N, as shown in Figure 9. The results imply that the stable region of the Ga-adsorbed structure appears in the phase diagram after the first N adsorption. That is, a Ga-adsorbing site appears after adsorption of the first N adatom. On the other hand, it is found that N adsorption might occur under the condition shown in the diagram; however, N adatoms seem to desorb
Atomic Structures and Electronic Properties of Semiconductor Interfaces
(b)
(a) Temperature (°C)
119
800
Ga desorption Ead-Ga > μGa
700
N adsorption Ead-N > μN
600
Ga droplet
10–6 10–5 10–4 10–3 10–2 10–6 10–5 10–4 10–3 10–2 Ga BEP (torr)
N BEP (torr)
Figure 7 p–T dependence of adsorption–desorption behavior of (a) Ga and (b) N on the initial surface. Reproduced with permission from Kangawa Y, Matsuo Y, Akiyama T, Ito T, Shiraishi K, and Kakimoto K (2007) Theoretical approach to initial growth kinetics of GaN on GaN(001). Journal of Crystal Growth 300: 62.
(a)
A
B
Temperature (°C)
Top view
(a)
C
D
E
Side view D A B
(b) 800
700
Ga adsorption Ead-Ga < μGa
Ead-N> μN 600
Ga droplet
C E
10–6
–5
10
10–4 10–3 10–2 10–6 10–5 10–4 10–3 10–2
Ga BEP (torr)
Ga 0
–4
–6.0 eV
–3
–3.7 eV
Adsorption energy (eV)
N
–1 –2
–5 –6
N2
A B C D E Site (–)
N BEP (torr)
Figure 9 p–T dependence of adsorption–desorption behavior of (a) Ga and (b) N on a N-adsorbed surface. Reproduced with permission from Kangawa Y, Matsuo Y, Akiyama T, Ito T, Shiraishi K, and Kakimoto K (2007) Theoretical approach to initial growth kinetics of GaN on GaN(001). Journal of Crystal Growth 300: 62.
1st layer Ga 2nd layer N (b)
N adsorption
N2
A B C D E Site (–)
Figure 8 (a) Schematic drawing of N-adsorbed GaN(0 0 1)-(41) surface. (b) Adsorption energies of Ga and N on sites A–E shown in (a). Reproduced with permission from Kangawa Y, Matsuo Y, Akiyama T, Ito T, Shiraishi K, and Kakimoto K (2007) Theoretical approach to initial growth kinetics of GaN on GaN(001). Journal of Crystal Growth 300: 62.
as a N2 molecule if a N adatom migrates to the sites on another N adatom. These results suggest that c-GaN growth on the GaN(001)-(4 1) surface proceeds via the following process during the initial growth stage. First, a N-adsorbed structure is formed instead of a Ga-adsorbed structure.
Subsequently, Ga adsorption on the N adatom occurs. Here, the N adatom is stabilized by being covered with a Ga adatom. This is different than the GaAs case, that is, group-III atoms are incorporated in the crystal by being covered with group-V atoms in the case of GaAs. Though more detailed examinations are required, Ga adsorption might be the rate-limiting process during c-GaN growth and the adsorption energy is 3.7 eV in this case. We have already noted that the vapor–solid equilibrium condition changes during the growth process. That is, change in coverage of adatom/molecule leads to a change in adsorption site for the subsequent adatom/molecule and adsorption energy. Therefore, the adsorption–desorption behavior changes regularly during the growth process. In a similar manner, we have to consider the influence of the spatial dependence of the adsorption–desorption behavior if there are more than one plane on the surface. In the following paragraph, we discuss the
120 Atomic Structures and Electronic Properties of Semiconductor Interfaces
(a)
Top view
(b)
1st layer Ga 2nd layer Ga
Top view
1st layer Ga 2nd layer Ga
Table 2 Desorption energies of Ga at T1 and B sites shown in Figure 10 on Ga-monolayer and bilayer surfaces
Monolayer
Bilayer
Site
T1
B
T1
B
Desorption energy (eV)
4.1
3.8
3.7
3.4
B B
T1
T1
Side view
Side view B
B
Reproduced with permission from Kangawa Y, Akiyama T, Ito T, Shiraishi K, and Kakimoto K (2009) Theoretical approach to structural stability of GaN: How to grow cubic GaN. Journal of Crystal Growth 311: 3106.
T1
T1
Figure 10 Schematic of (a) Ga monolayer and (b) Ga bilayer structure on (111) surface. Reproduced with permission from Kangawa Y, Akiyama T, Ito T, Shiraishi K, and Kakimoto K (2009) Theoretical approach to structural stability of GaN: How to grow cubic GaN. Journal of Crystal Growth 311: 3106.
adsorption–desorption behavior on the faceted surface and the change in growth form. To compare the adsorption energies of constituent atoms on GaN(001)-(4 1) surface with those on {111} facet plane, we studied the adsorption– desorption behavior of monatomic Ga and N on {111} facet plane. Figures 10(a) and 10(b) show schematics of Ga monolayer and bilayer surfaces, respectively, which were observed on GaN(0001) (Adeimann et al., 2002; Northrup et al., 2000). These structure also appear on GaN(111) because the atomic arrangement of the topmost and second monolayers of (111) is the same as that of (0001). Therefore, we discuss the adsorption–desorption behavior on these surfaces henceforth. In Figures 10(a) and 10(b), one can see that there are two kinds of atomic sites, that is, T1 site which is on the Ga in underling layer and B site which is on the bridging site between Ga atoms in the underling layer. Table 2 shows the desorption energies of Ga occupying the T1 and B sites on Ga monolayer and bilayer surfaces. In Table 2, it is found that Ga atoms at the B sites are unstable compared with that at T1 sites. This implies that decomposition of each layer starts by desorption of Ga at B sites. Moreover, the results suggest that
Ga monolayer is stable compared with Ga bilayer surface because desorption energy of Ga on the monolayer surface (3.8 eV) is larger than that on the bilayer surface (3.4 eV). Here, the adsorption energy is equal to the negative of the desorption energy if there is no activation energy during the desorption process. That is, the adsorption energy of Ga at B site on the monolayer and bilayer surface is 3.8 eV and 3.4 eV, respectively. Using these values and the adsorption energy of Ga on (001)-(4 1) surface (3.7 eV), we can classify the growth conditions into three regions: region (I)–(III), as shown in Figure 11. In regions (I) and (II), the chemical potential, Ga , is less than 3.4 eV, which is the adsorption energy of Ga on (111) Ga bilayer surface. Therefore, Ga monolayer surface appears on the faceted surface and the adsorption energy on the surface is 3.8 eV. In case of region (I), Ga is less than 3.7 eV, which is the adsorption energy of Ga on GaN(001)-(4 1). In this region, impinging Ga adsorbs only on the (111) facet plane. In the case of region (II), impinging Ga adsorbs on the facet plane, and then adsorbs on the (001) surface because Ead-Ga(111) < Ead-Ga(001) < Ga , where Ead-Ga(hkl) is the Ga adsorption energy on (hkl) plane. In the case of region (III), Ga is larger than 3.4 eV, which is the Ga adsorption energy on Ga bilayer surface, and, therefore, the bilayer surface appears on the facet plane under the growth conditions. In the region, impinging Ga atoms adsorb on the (001), and then adsorb on the facetted surface because EadGa(001) < Ead-Ga(111) < Ga . These results imply that 2D growth occurs when the growth condition is suitable in regions (I) and (II), whereas facet formation occurs when the growth condition is in region (III). That is, we can control the surface shape and suppress h-GaN mixing by choosing proper growth conditions.
Atomic Structures and Electronic Properties of Semiconductor Interfaces (b) (I) –3.8 < μ Ga < –3.7 (eV)
(a) (I)
(II)
–3.7
(III)
–3.8
Monolayer
800 Temperature (°C)
121
(II) –3.7 < μ Ga < –3.4 (eV) –3.7
700
Monolayer
600 10–6
–3.8
Ga droplet 10–5
10–4
10–3
(III) –3.4 < μ Ga < –2.8 (eV) 10–2
Ga partial pressure (torr)
–3.7
–3.8
Bilayer Figure 11 (a) Phase diagram of growth form and (b) schematic of growth mode in regions (I)–(III). Reproduced with permission from Kangawa Y, Akiyama T, Ito T, Shiraishi K, and Kakimoto K (2009) Theoretical approach to structural stability of GaN: How to grow cubic GaN. Journal of Crystal Growth 311: 3106.
1.04.2.2
Growth Modes in Heteroepitaxy
1.04.2.2.1
Growth modes Heteroepitaxial growth is one of the most important techniques for the fabrication of semiconductor heterostructures which is the basic building block of semiconductor optical and electronic devices. Therefore, an understanding of heteroepitaxial growth mechanism is crucial for both scientific and technological viewpoints (Venables et al., 1984). Among them, it is well known that several growth modes appear in a heteroepitaxial growth with lattice-mismatched systems. In Figure 12, three typical examples of Frank–van der Merwe (FM) mode, Stranski–Krastanov (SK) mode, and Volmer–Weber (VW) mode are schematically illustrated. As shown, a two-dimensional flat epitaxial layer is observed during growth in the FM mode. In the SK mode, however, three-dimensional islands (called SK islands) appear after two-dimensional wetting-layer
(a)
(b)
(c)
Figure 12 Schematic illustration of three typical growth modes: (a) Frank–van der Merwe (FM) mode; (b) Stranski– Krastanov (SK) mode; and the (c) Volmer–Weber (VM) mode.
formation. In the case of the VW mode, three-dimensional growth proceeds without forming a twodimensional wetting layer. It is noted that the FM mode originally means the growth mode in which a two-dimensional growth layer appears after the formation of misfit dislocations at the interface between a substrate and a growth layer, as described in the original paper of van der Merwe (1991). Nowadays, all two-dimensional growths are often categorized into the FM mode, contrary to the original definition. It is well known that growth modes of a semiconductor are quite complex and the physical mechanisms of various growth modes have intensively studied so far (van der Merwe, 1963a,b, 1991; Matthews and Blakeslee, 1975, 1976; Tersoff, 1988, 1998; Nakajima, 1999). However, it is difficult to systematically explain the various and complex behaviors of the growth modes observed experimentally. This is because semiconductor crystal growth involves various and complex phenomena: for example, the formation of misfit dislocations at growth interfaces and of three-dimensional growth islands, as well as the inter-diffusion of atoms at the growth interfaces. In the case of lattice-mismatched epitaxial growths, the most crucial question is about how the strain relaxation occurs. There have been a lot of reports that discuss the relationship between growth processes and each individual strain relaxation mechanism. For example, van der Merwe (1963a,b, 1991) and Matthews and Blakeslee (1975, 1976)
122 Atomic Structures and Electronic Properties of Semiconductor Interfaces
investigated the relationship between the generation of misfit dislocations and growth processes from the viewpoint of the energy competition between the strain in growth layers and the generation of misfit dislocations. However, their approaches focused on the misfit dislocations and three-dimensional island formations were not taken into account. On the other hand, Tersoff (1988, 1998) intensively studied the relationship between the three-dimensional SK island formation and growth mode, although he did not discuss the generation of misfit dislocations. Nakajima (1999) systematically studied the competition between strain energies, surface energies, and interfaced energies, and discussed the growth modes. However, he did not include the effect of misfit dislocations. In this subsection, we show the simple but systematic theoretical studies that discuss the competition between several strain relaxation mechanisms, such as the generation of misfit dislocations and three-dimensional island formations (Okajima et al., 2000). This theory is not only easy to understand but also can naturally include various mechanisms such as misfit dislocation and three-dimensional island formations. Moreover, we comment on the importance of atomistic level information for systematic understanding of various macroscopic growth processes. 1.04.2.2.2 Theory based on macroscopic energetics
In this subsection, we discuss the growth mode of lattice-mismatched epitaxy when two kinds of strain relaxation mechanisms of misfit dislocation and three-dimensional island formations are taken into account (Okajima et al., 2000). First, we discuss a theory that includes misfit dislocation and SK island formations. This theory includes several macroscopic parameters, for example, the elastic constants. In the following, we will discuss a procedure that determines these macroscopic parameters based on the atomistic simulations (e.g., first-principles calculations) showing an example that initial heteroepitaxial growth process of InAs/GaAs(110) can quantitatively be reproduced by the discussion using first-principles calculations. It should be noted that dynamics as well as the energetics play a crucial role in crystal growth. In the present discussion, however, we only take into account energetics in order to simplify the discussions. In the following discussion, we will show that various and complex behaviors of lattice-mismatched
heteroepitaxial growth can be qualitatively described on the basis of energetics. First, we define the ‘optimal growth mode’, which is key to the discussion in lattice-mismatched heteroepitaxy. The optimal growth mode has the minimum free energy at a given film layer thickness h. Here, we determine the optimal growth mode by considering the energetics, instead of the system free energies, for simplicity. To determine the growth mode, we should consider the possible growth modes. If two mechanisms of misfit dislocation and SK island formations are considered, possible growth modes can be categorized as follows: 1. mode with neither SK islands nor misfit dislocations (two-dimensional coherent growth), 2. mode with only SK island formation (SK coherent growth), 3. mode with only misfit dislocation formation (twodimensional growth with misfit dislocation), and 4. mode with both misfit dislocations and SK islands (SK growth with misfit dislocation). These growth modes are schematically illustrated in Figures 13(a)–13(d). Accordingly, the optimal growth mode is one with the lowest energy among the above four growth modes at a given film thickness h. Therefore, by considering the change in the optimal growth mode with the increase of h, we can systematically investigate the epitaxial growth process. Next, we describe the system energies of four types of growth modes. Here, we define the physical parameters that will commonly be used in this subsection as follows:
(a)
(b)
(c)
(d)
Figure 13 Schematic illustrations of four growth modes that appear when misfit dislocation and SK island formation are competed: (a) two-dimensional coherent growth, (b) SK coherent growth, (c) two-dimensional growth with misfit dislocation formation, and (d) SK growth with misfit dislocation formation.
Atomic Structures and Electronic Properties of Semiconductor Interfaces
: surface energy of a growth layer, i: Increase in the surface area by SK island formation,
: decrease in strain energy by SK island formation, M: effective elastic constant, ": lattice mismatch at the hetero-interface, l: average distance of misfit dislocations at the heterointerface, l0: average distance of misfit dislocations at which lattice-mismatched strain at the hetero-interface can perfectly be relaxed, Ed: formation energy of a misfit dislocation, and h: thickness of a growth layer. It is noted that the reason why lattice-mismatch induced strain can effectively be relaxed by the three-dimensional island is the increase in surface regions in which constituent atoms can freely move. This situation is included in the parameter , which gives the energy gain by the above effective strain relaxation. First, we discuss the energy of two-dimensional coherent growth. The system energy can be described as the sum of the surface energy and the strain energy of a growth film, and it given as a function of a film thickness h: 2D Ecoh ðh Þ
1 ¼ þ M"02 h 2
ð6Þ
In the case of SK coherent growth, an increase in surface area and strain relaxation by the formation of SK islands should be added and given as follows: 1 SK Ecoh ðh Þ ¼ ð1 þ Þ þ M ð1 – Þ"02 h 2
ð7Þ
In the case of two-dimensional growth with misfit dislocations, the system energy is a function of an average distance in misfit dislocations l as well as a film thickness h as follows: 1 l0 2 Ed 2D EMD ðh; l Þ ¼ þ M"02 1 – hþ l l 2
ð8Þ
Similarly, the system energy of SK growth with misfit dislocations can also be described as h and l: SK EMD ðh; l Þ
1 l0 Ed 2 ¼ ð1 þ Þ þ M ð1 – Þ"0 1 – hþ ð9Þ 2 l l
In real epitaxial growth, an average distance of misfit dislocations l should be the value lopt(h) at which the system energy becomes minimum. In addition, lopt(h) can be described as follows in two-dimensional growth with misfit dislocations:
–1 h 2D – MD opt l2D – MD ðh Þ ¼ l0 1 – c h
123
ð10Þ
where hc 2D-MDs is the critical thickness of twodimensional growth with misfit dislocations, and hc 2D-MD can be written as hc2D – MD ¼
Ed M"02 l0
ð11Þ
In the case of SK growth with misfit dislocations, lopt(h) can be given as follows: –1 h SK – MD opt lSK – MD ðh Þ ¼ l0 1 – c h
ð12Þ
where hc SK-MDs is the critical thickness of SK growth with misfit dislocations, and it is given as follows: hcSK – MD ¼
Ed M ð1 – Þ"02 l0
ð13Þ
By using Euations (10) and (12), the system energies of two-dimensional and SK growth with misfit dislocations can be given as follows: Ed Ed2 – l0 2M"02 l02 h
ð14Þ
Ed Ed2 – l0 2M ð1 – Þ"02 l02 h
ð15Þ
2D Eopt MD ðh Þ ¼ þ
SK ðh Þ ¼ þ Eopt MD
By comparing the system energies of four possible growth modes given in Equations (6), (7), (14), and (15) at a given film thickness of h, we can determine the optimal growth mode as a function of h. Further more, we can discuss the various growth processes by considering the h dependence of the optimal growth mode. Here, we briefly mention the treatment of VW growth in which a wetting layer does not appear. To describe VW growth, we should set the parameter (surface energy of a growth layer) as negative. By this treatment, we can describe various and complex growth processes, such as the three-dimensional island formation, without wetting layer formation, subsequent formation of misfit dislocations, and coalescence of three-dimensional islands. 1.04.2.2.3
Mode transition during epitaxy Tracing the trajectory of the optimal growth mode, while varying the layer thickness h, should reveal the change in growth mode during heteroepitaxy. By varying Ed, two types of typical growth behavior are demonstrated when other parameters are appropriately chosen, as shown in Figure 14. In the case of
124 Atomic Structures and Electronic Properties of Semiconductor Interfaces
2D-coh
SK-coh SK-MD 2D-MD
Free energy per unit area
Ed = 2.0
P SK
C1
2D
C1 T
2D-MD
Ed = 1.0 2D
C1
2D
SK
hC1 hT
hC1
hP
Layer thickness 2D
C1
2D
hC1 < hT 2D-coh
: FM-mode 2D-MD SK
C1
T
2D
hC1 > hT 2D-coh
SK-coh P SK-MD
: SK-mode 2D-MD
Figure 14 Change in the free energy with layer thickness h. By tracing the trajectory of the optimal growth mode, the change in the growth mode can be described. Two typical growth behaviors (FM and SK) are illustrated. Typical FM and SK modes can well be reproduced by changing the formation energy of misfit dislocation (Ed ¼ 1 and 2 eV A˚1 for FM and SK modes). Reproduced with permission from Shiraishi K, Oyama N, Okajima L, et al. (2002) First principles and macroscopic theories of semiconductor epitaxial growth. Journal of Crystal Growth, 237–239: 206–211.
the small formation energy of a misfit dislocation (Ed ¼ 1.0 eV A˚1), the initial two-dimensional coherent growth mode changes into the two-dimensional growth mode with misfit dislocations, when the layer thickness h exceeds hc 2D-MD . However, no change occurs after h ¼ hc 2D-MD . This growth process corresponds to the typical FM growth that is observed in InAs/GaAs(110) and InAs/GaAs(111) heteroepitaxial systems (Belk et al., 1997; Yamaguchi et al., 1997). On the other hand, the two-dimensional coherent growth mode first changes into the SK-coherent one at h ¼ hT, when Ed is comparatively large (Ed ¼ 2.0 eV A˚1). By increasing the growth layer thickness, this SK-coherent growth mode changes to the SK growth mode with misfit dislocations when h exceeds hc SK-MD . Further increase in the growth layer thickness, however, induces the change of SK growth mode with misfit dislocations into twodimensional growth mode with misfit dislocations. This is a typical SK growth process that is observed in many lattice-mismatched systems. It is well known that dynamics and energetics play an essential role in determining the growth processes
in actual systems. Now, we briefly mention the effect of dynamics. For example, it becomes difficult to generate misfit dislocations at the interface when the epi-layer thickness is quite large. In this case, even though the free energy of the growth mode with misfit dislocation is at its lowest, misfit dislocations are rarely generatd. This is related to the fact that the experimental critical thicknesses of many lattice-mismatched systems are much larger than the theoretical values. 1.04.2.2.4 Theory based on atomistic energetics
We have noted in the above that the formation energies of misfit dislocations play a crucial role in determining epitaxial growth processes. These energies of misfit dislocations are often discussed based on the elastic theory. However, the above approach cannot include the individual and atomistic characteristics of materials. To investigate the formation energies of misfit dislocations, quantum mechanical and atomistic approaches are desirable, although these involve generally need large-scale calculations.
Atomic Structures and Electronic Properties of Semiconductor Interfaces
Here, we show an example of a first-principles investigation of misfit dislocations formed at the heteroepitaxial interface of InAs/GaAs(110) systems (Oyama et al., 1999). Moreover, we discuss the growth processes based on the first-principles results. It has been reported by scanning tunneling microscopy (STM) experiments that misfit dislocations with average distance of 60 A˚ appear at the very initial stage of InAs heteroepitaxial growth on a GaAs(110) surface after 2.8 ML (‘monolayer units’) InAs growth (Belk et al., 1997). In addition, it is also observed that two-dimensional growth continues without forming SK islands. The existence of misfit dislocations can be observed based on STM experiments by estimating the vertical displacement of surface atoms with 0.5–0.7 A˚. The first-principles calculations with norm-conserving pseudopotentials (Hamman et al., 1979; Ihm et al., 1979) were carried out for model systems in which two and four InAs monolayers are coherently grown on GaAs(110) substrates, and are grown with misfit dislocations, respectively. One InAs epi-layer unit was forcibly removed for every 15 (GaAs) units to generate misfit dislocations. After the optimization, we obtained the final structures. The so-obtained structures of four InAs epi-layers are shown in Figure 15. The 90 perfect-misfit dislocations were generated in the [001] direction. The most noticeable points in this structure are the fivefold coordinated In atoms that have covalent bonds with the nearest five As atoms along the dislocation line. Although this fivefold coordinated structure was not confirmed experimentally, it is expected that further experimental developments will observe such brand-new dislocation core structures. Due to the dislocation formation, the epi-layer
125
surface was depressed just above the misfit dislocation line due to the strain field caused by the misfit dislocations. The vertical displacement of the top surface As atoms is about 0.54 A˚ when the InAs epi-layer thickness is 4 ML. This vertical displacement is in good agreement with the reported value (Oyama et al., 1999). The above first-principles results also gave the total energy difference between coherent and dislocated growth modes. The calculated energy difference are DE(2 ML) ¼ 3.15 eV per cell and DE(4 ML) ¼ 0.85 eV per cell, respectively. From these values, we can estimate the critical thickness as well as the phenomenological parameters. The calculated phenomenological parameters (Ed and M) and the critical thickness hc 2DMD are 0.96 eV A1, 1.22 1011 N m2, and 2.35 ML, respectively, for the InAs/GaAs(110) system. The value of elastic constant M is of the same order as the reported bulk modulus of InAs (0.58 1011 N m2). On the other hand, the value of the formation energy of a misfit dislocation is much smaller than the typical value of misfit dislocations in crystals (1.86.9 eV A˚1). This is because this dislocation core structure only contains stable In–As bonds without relatively unstable In–In or As–As bonds (Kobayashi and Nakayama, 2008). Moreover, the critical thickness is obtained as 2.35 ML. Thus, the first isolated misfit dislocation is expected to appear when the epi-layer thickness exceeds 3 ML, in fairly good agreement with the experiments (Yamaguchi et al., 1997). The above consideration indicates that the atomistic information of misfit dislocations is crucial for understanding the macroscopic growth modes appearing in the heteroepitaxial growth. Vertical surface displacement = 0.542 Å
2.52 2.57
2.57
2.64
2.63
3.12
2.86
2.70 2.39
2.52
:In
2.60
2.28 2.30
:Ga
2.36
:As [110]
2.34 2.34
2.40 2.40
[001] [110]
Figure 15 The dislocation core structure obtained for InAs/GaAs(110) heterointerface with an InAs layer is 4 ML. Light, dark, and black circles indicate In, Ga, and As atoms, respectively. The numbers in the figure represent bond lengths (A˚). Reproduced with permission from Shiraishi K, Oyama N, Okajima L, et al. (2002) First principles and macroscopic theories of semiconductor epitaxial growth. Journal of Crystal Growth, 237–239: 206–211.
126 Atomic Structures and Electronic Properties of Semiconductor Interfaces
1.04.2.3
Defect Formation in Heteroepitaxy
1.04.2.3.1
Heterovalency-induced defects In the previous section, we explained how the elastic strain originating from lattice mismatch produces nanostructures such as quantum-dots in heteroepitaxy. In both hetero- and homoepitaxy of semiconductor films, another kind of nanostructure is often generated near the interface originating from other origins. In this section, we concentrate on the stacking-fault tetrahedron defects (SFTs) (Finch et al., 1963; Mendelson, 1964; Kanisawa et al., 2001; Nakayama and Kobayashi, 2005) and show that they are generated by (i) the heterovalency, that is, the valency mismatch of two semiconductors, and (ii) the existence of impurity atoms on the growth surface, by demonstrating the cases of ZnSe/GaAs interface formation and Si homoepitaxy. ZnSe and GaAs have the same zinc-blende crystal structure and similar lattice constants of 5.65 and 5.64 A˚, respectively. Therefore, there is no elastic strain at ZnSe/GaAs interfaces. However, the heteroepitaxy of ZnSe on a (001) GaAs substrate often produces various defects such as vacancies and interstitials at the interface, followed by macroscopic defects like dislocations, partials, and SFTs (Kuo et al., 1996a, 1996b; Yasuda et al., 1996). What is the origin of such extensive defect formation at ZnSe/ GaAs interfaces? As shown in Section 1.04.2.1.1, most semiconductor surface structures satisfy the electron counting condition, where the surfaces stabilize to become semiconducting by decreasing the dangling-bond electrons, that is, by producing cation/ anion dimer bonds on the surface and transferring the excess electrons of surface-cation dangling bonds into the partially occupied dangling bonds of surface anion atoms. In the case of ZnSe/GaAs heteroepitaxy, however, the electron-deficit Zn–As acceptor and electron-excess Ga–Se donor bonds inevitably appear at the interface and the electron transfer occurs among these bonds in addition to the surface dangling bonds (Nakayama, 1992a; Oda and Nakayama, 1992; Murayama et al., 1998; Nakayama and Murayama, 2000a). The simplest atomistic model that includes these charge-transfer effects has the total energy form as (Nakayama and Sano, 2001)
H¼
X
VAB A;BX
þ þ
k0 minðjQAþ j; jQA – jÞ A 3 X X
kn minðjQA j; jQB jÞ
ð16Þ
n¼1 A;B
In this model, all atoms are assumed to be located at zinc-blende tetrahedral site. The first term represents the bonding energies between the first-nearest neighboring atoms, A and B, and between the secondnearest neighboring surface dimer-bond atoms in the same epitaxy layer. The values of VAB are evaluated from the first-principles calculations of cohesive energy for zinc-blende AB crystal and surface energies of various configurations. The second term in the equation represents the energy gains due to the electron transfer between four tetrahedral bonds around an atom, A, where electrons are transferred from the donor bonds such as As–Se to the acceptor bonds like As–Zn. Since this transfer changes the electron occupancy from the antibonding states to the bonding states, the gain energy is proportional to the sum of excess or deficit electron number, QAþ or QA around the A atom. After this intra-site electron transfer, there still exist excess or deficit electrons, QA ¼ QAþ þ QA, around the atom A. We can assume that such electrons are transferred within the first (n ¼ 1) to third-nearest (n ¼ 3) neighboring sites, hence the third term being introduced. The parameters, k0, k1, k2, and k3, are taken to be 2.00, 0.32, 0.08, and 0.02 eV, respectively, which are again obtained by fitting the GaAs and ZnSe surface energies in the first-principles calculations. The growth simulation is performed using the conventional stochastic Monte Carlo method (Ito and Shiraishi, 1998a, 1998b), where the absorption, evaporation, surface-atom diffusion, dimerization, and atom exchange between the surface and substrate are considered as fundamental processes. The occurrence probability of these processes is described by the Boltzman factor, exp(E), where E is the surface energy change between before and after the process and ¼ 1/kBT. The growth simulation starts on GaAs (001) (2 4)2 surfaces with an area of (12 12) and (20 20). Three typical cases are shown here for the initial growth of ZnSe on GaAs: (i) the case when Zn and Se are simultaneously supplied from the beginning, and the case when Zn and Se are simultaneously supplied after some amount of (ii) Zn or (iii) Se exposure.
Atomic Structures and Electronic Properties of Semiconductor Interfaces
(a)
(a)
(b)
As
As
As
As
(001)
z=4 Zn 3
Se
127
Ga
Ga
Ga
Ga
Ga As
(–110)
(001)
2 (b) 1
z = –1 V
Vacancy Zn As
z = –2
0 A
Sc
As
As
As
z = –3 Ga
Ga
Ga
Ga
–1 (c) Se –2
As
As –3 (11
0)
)
10
(–1
Figure 16 Calculated ZnSe atom configurations (a) at the initial stage of growth and (b) after 4 ML ZnSe adsorption on GaAs (001) (2 4)2 substrate. Zn and Se atoms are simultaneously supplied. Reproduced with permission Nakayama T and Sano K (2001) Monte Carlo simulation of defect formation in ZnSe/GaAs heterovalent epitaxy. Journal of Crystal Growth 227/228: 665.
First, we consider case (i). Figures 16(a) and 16(b) show the grown atom configurations at the initial stage and after ZnSe adsorption in the four topmost monolayers, respectively, where the top Asdimer layer of GaAs corresponds to the z ¼ 0 layer. At initial growth in Figure 16(a), Zn and Se atoms are adsorbed at trench (missing Ga) sites in the z ¼ 1 layer, which was also observed in STM experiments (Ohtake et al., 1999). When the growth proceeds further, as seen in Figure 16(b), most of these Se atoms move to other sites by diffusion and exchange processes and most trench sites become occupied by Zn atoms; thus, the cation–anion order is realized in most upper grown layers. However, it is noticed that a few Se antisites, denoted by A, are observed at trench sites (z ¼ 1), and these antisites are accompanied by vacant sites, V, in the upper layers (z > 1). Even if the growth proceeds further, these vacant defect structures remain stationary and never disappear. The appearance of such defect structure is closely related to both the electron transfer between
Ga
Ga
As
Ga
As
Ga
Figure 17 Atom structures of trench sites of GaAs (001) (2 4) 2 surfaces viewed from the (110) direction: (a) before ZnSe adsorption, (b) after Zn adsorption, and (c) after Se adsorption. In (b), the second adsorption occurs at the cation site by Se atom, while, in (c), As evaporation occurs after the Se adsorption. Reproduced with permission Nakayama T and Sano K (2001) Monte Carlo simulation of defect formation in ZnSe/GaAs heterovalent epitaxy. Journal of Crystal Growth 227/228: 665.
heterovalent bonds and the growth process. When the GaAs (2 4)2 surface is exposed to a ZnSe beam, as shown in Figure 17(b), the cation site between the As dimers in the z ¼ 1 layer is first occupied by a Zn atom and the next adsorption occurs at the nearest cation site by breaking the As dimer. The adsorption energies of Zn and Se to this cation site are, respectively, 3.03 and 2.70 eV in the absence of electron transfer, kn ¼ 0, while 3.61 and 3.84 eV in the presence of electron transfer, kn 6¼ 0. The energy gain of Se adsorption occurs in the latter case because the Se adsorption produces As–Se donor bonds, which compensate the electron deficiency in nearest Zn–As acceptor bonds caused by the first Zn adsorption. In this way, the electron transfer prefers to locate a Se atom at this cation site, that is, induces the co-adsorption of Zn and Se, and produces the Se antisite as shown in Figure 17(b). Of course, when the growth proceeds further, most such Se atoms change the position into anion layers by diffusion and atom-exchange
128 Atomic Structures and Electronic Properties of Semiconductor Interfaces
processes. However, once a certain number of antisites are produced nearby, they do not disappear through the diffusion and atom-exchange processes between different layers, and, as seen in Figure 17(b), the vacant sites appear over these antisites. Moreover, it should be noticed here that, as seen in Figure 16(b), the boundary between adsorbed and vacant regions is occupied by antisite atoms of Zn and Se. This indicates the displacement tendency of adsorbed atoms and might induce large defects such as dislocations and SFTs observed in experiments. We now consider the effects of initial treatment of the substrate. Figure 18(a) shows the grown atom configuration after 4 ML of ZnSe are adsorbed in case (ii), while Figure 18(b) shows the surface structure after Se atoms are supplied on GaAs substrate in case (iii). In Figure 18(a), all trench sites in the z ¼ 1 layer are occupied by Zn atoms, and there is no antisite and vacancy, hence no sign of defect production even when further growth proceeds. On the other hand, in Figure 18(b), Se atoms are adsorbed not only on the GaAs surface but also at (a)
(b)
(001)
z=4
3
Zn Se Ga
2
As
the anion sites which had ever been occupied by As atoms. Moreover, one can see that a number of substrate As atoms evaporate and disappear on the surface and even in the inner z ¼ 2 layer. This As evaporation is induced by the Se adsorption as follows: in case (iii), the supplied Se atom first occupies the cation trench site as shown Figure 17(c). Although the Se desorption from this configuration is a major process in the next step, the second-nearest neighboring As atom sometimes evaporates. This is because there are many electrons around the adsorbed Se due to the Se–As donor bonds, and when the neighboring As atom evaporates the large energy gain is obtained by transferring excess electrons into the dangling bond of the nearest As generated by As evaporation. Moreover, since there is no As supply, vacant sites are never occupied by As atoms once As atoms evaporate. Since the resulting surface is so bumpy, there appear more number of defects than those shown in Figure 16(b) when ZnSe is grown on this surface. All the above-mentioned results are qualitatively in agreement with experiments; the defect starts just at the interface and the Zn/Se exposure before ZnSe supply decreases/increases the defect density in grown layers. These results clearly demonstrate that both the heterovalency and the initial treatment of heteroepitaxy are key factors to control the interface crystalline shape. The cases of other interfaces are also shown in the literature (Nakayama et al., 2002a). 1.04.2.3.2
1
0
0 –1 –1 –2
Impurity-induced defects Next, we consider the defect formation in the Si homoepitaxy. During the thin-film growth of most semiconductors, crystallographic defects such as point defects and dislocations are often generated, originating from impurity atoms on the substrate surface. Among various defects, SFTs have unique features; as shown in Figure 19, SFTs have a
–2
(11
) 10
0)
(–1
Figure 18 Calculated ZnSe atom configurations after 4 ML ZnSe adsorption on GaAs (001) (2 4) 2 surfaces. Zn and Se are simultaneously supplied after some amount of (a) Zn and (b) Se exposure. The symbol notations are the same as in Figure 16. Reproduced with permission from Nakayama T and Sano K (2001) Monte Carlo simulation of defect formation in ZnSe/GaAs heterovalent epitaxy. Journal of Crystal Growth 227/228: 665.
Si(111)
–3
–3
Face Ridge Apex
wn ro
e
t tra bs
G
Figure 19 Schematic picture of stacking-fault tetrahedron (SFT) in Si(111) grown film.
film
Su
Atomic Structures and Electronic Properties of Semiconductor Interfaces
diamond structure in their tetrahedron body and are surrounded by (111) and three equivalent faces with stacking-fault layers. Most SFTs are nucleated at the interface between the substrate and the grown film such that one of the four apexes of a tetrahedron is located at the interface, and they become larger, reaching the size of a few tens of micrometers, as the film grows thicker (Mendelson, 1964). However, these are phenomenological characterizations, and it has not been clarified from a microscopic viewpoint how and why the SFTs are nucleated at the interface and what crystal structures are realized around the apex, that is, the starting point of the SFT generation. The standard classical molecular dynamics (MD) simulations were used to simulate the SFT generation (Kobayashi and Nakayama, 2004), by adopting the Nose ´–Hoover thermostat algorithm and the empirical interatomic potentials. Since the Si growth proceeds with the supply of chlorinated Si gases, such as SiH2Cl2, a certain number of Cl atoms always exist on the Si substrate, mainly as SiCl with a single Si–Cl bond (Sakurai and Nakayama, 2002). Thus, the simulations begin with such a Cl-adsorbed Si(111) substrate, with the supply of Si and adoption of the atom diffusing as an equilibrium process at a typical temperature of 900 C. When the Si atoms are supplied on Si substrate, the Si bilayer is first produced, as shown in Figure 20(b), except at the region around the Cl atoms. As the growth proceeds further, however, most Cl atoms leave the substrate by diffusing onto the Si surface, as shown in Figure 20(c), and the grown Si layers produce an ordered diamond structure without any defects. This is because the Cl atom has only a single bond with the nearest neighboring Si due to the monovalency of Cl, thus being pushed out from the substrate onto the grown surface to avoid the energy loss due to promoting Si dangling bonds around Cl atom. With a very small but finite possibility, however, the Cl atoms remain at the initial adsorption sites on the substrate and are buried in the Si film after the Si growth. In this case, after the production of two Si bilayers shown in Figure 21(a), the deposited Si atoms can find stable positions above the Cl atoms by producing covalent bonds with Si atoms in the wall around the Cl atoms, as shown by the arrow in Figure 21(a). Then, as shown in Figure 21(b), the deposited Si atoms can be located at positions about one bilayer above the Cl atoms such that they form a dome over the Cl atoms. Most of these Si atoms in the ceiling of the dome are tetrahedrally coordinated,
129
(a)
(b)
(c)
Figure 20 Side views of surface structures during Si growth. Gray and open circles denote, respectively, Si and Cl atoms. (a) Initial Si substrate for the simulations, where two Cl atoms are adsorbed; (b) surface after initial bilayer growth; and (c) surface after three-bilayer growth. Reproduced with permission from Kobayashi R and Nakayama T (2004) Theoretical study on generation and atomic structures of stacking-fault tetrahedra in Si film growth. Thin Solid Films 464/465: 90.
which is the reason for the stability of this structure. In fact, once this dome structure is produced, it remains intact during the subsequent Si growth even when the Cl atoms are removed. When the Si layers are formed on this dome structure, their crystal structures strongly depend on the number of adsorbed Cl atoms on the substrate. When only one Cl atom is left on the substrate, a small region of Si overlayers shows a disordered structure. As seen in Figure 21(c), however, most Si layers are deposited with the diamond structure and the Cl atom is buried in Si films as a point-like defect, thus no SFT structure appears. On the other hand, when more than one Cl atoms are present in the nearest neighbouring sites on the substrate, as shown in Figure 21(d), the Si layers in a region above the Cl atoms are located at positions slightly higher than the corresponding layers in the other
130 Atomic Structures and Electronic Properties of Semiconductor Interfaces
(a)
(b)
(c)
(d)
Ridge
SF
Figure 21 Side views of surface structures during Si growth. (a) Surface after the growth of two Si bilayers in the case when two Cl atoms are left on the substrate; (b) surface after a few Si atoms are supplied on the surface shown in (a). The dome structure is produced over the Cl atoms. (c) Surface after the growth of three bilayers with one Cl atom left on the substrate. (d) Surface after the growth of three bilayers with two Cl atoms left on the substrate. Dotted lines, labeled SF and Ridge, respectively, show the stacking fault face and the ridge line of SFT. Reproduced with permission from Kobayashi R and Nakayama T (2004) Theoretical study on generation and atomic structures of stacking- fault tetrahedra in Si film growth. Thin Solid Films 464/465: 90.
region and have a diamond structure. Moreover, we can see the stacking-fault faces and the ridge line around this region, indicating the growth of the SFT-like structure. From these results, we can conclude that the SFTs can be produced when more than one Cl atoms are present nearby on the substrate, and the dome structure shown here is one of the candidates for the apex structure of SFTs. This conclusion is consistent with the observation that the SFT density is proportional to the square of the Cl coverage on the Si substrate (Mazumdar et al., 1995; Takakuwa et al., 1997). In fact, after producing such an apex, there appear three partial dislocations as ridge lines (Kobayashi and Nakayama, 2008). 1.04.2.4
Interface Formed by Oxidation
The Si/SiO2 interface is a building block of Si metaloxide field-effect transistor (MOSFET) in largescale integration (LSI) circuits. The realization of MOSFET in modern LSI circuits stems from the high-quality Si/SiO2 interfaces (Kahng, 1960). In
fact, a Si/SiO2 interface can provide us a nearly ideal two-dimensional electron gas (2DEG) that leads to the discovery of quantum Hall effects, etc. (von Klitzing et al., 1980). This is due to the sharp and nearly defect-free characteristics of the Si/SiO2 interface. The typical interface defect density is of the order of 1010 cm2, and the TEM observation really indicates the sharp interface structure. Such sharp interface comes from the layer-by-layer oxidation of Si surfaces and interfaces, which has been observed on Si(111) (Ohishi and Hattori, 1994) and Si(100) surfaces (Watanabe et al., 1998). 1.04.2.4.1
Phenomenological theory The Deal and Grove model of Si oxidationhas been widely accepted for a long time. First, we introduce the conventional understanding of Si thermal oxidation by Deal and Grove (1965). The suggested steps consist of an oxidant diffusion process followed by an interfacial reaction process. When a Si substrate is exposed to an oxygen (O2) atmosphere at around 700–1100 C, the substrate surface changes into Si
Atomic Structures and Electronic Properties of Semiconductor Interfaces
oxide. This is the thermal oxidation process. While this process can be described by a simple chemical reaction formula, Si þ O2 > SiO2, its microscopic picture is much complicated. During the oxidation, O2 molecules from the atmosphere must meet the substrate Si to react, but the surface of the substrate is covered by Si oxide. Therefore, O2 molecules must move through the Si oxide covering the surface. So, Deal and Grove assumed that the O2 molecules move (the diffusion process) and arrive at the interface between the Si oxide and substrate Si and react with the latter (the reaction process). By combining these two processes, they successfully explained the kinetics of oxide thickness growth. Experiments with O isotopes basically supported this scheme (Han and Helms, 1988; Lu et al., 1995): isotope-labeled O atoms are located at Si/SiO2 interfaces, which indicates that O2 molecules diffuse through SiO2 layer and react at the Si/SiO2 interface. Moreover, their model could reproduce the experimental oxidation rate as follows. The time evolution of oxide thickness can be written as X 2 þ AX ¼ B ðt þ Þ, where X and t are oxide thickness and oxide time, respectively, and A and B are, respectively, the constants related to oxidation reaction and O2 diffusion through SiO2. The characteristic nature of their model is that there exist two regions called the reaction- and diffusion-limited oxidation regions that correspond to thin and thick oxide thickness, respectively. In the case of reactionlimited oxidation, oxide thickness X is proportional to the oxidation time, t. On the other hand, pffiffi in the diffusion limited case, X is proportional to t . However, as Deal and Grove also pointed out in their original paper, this picture cannot explain several issues related to the thermal oxidation process (Deal and Grove, 1965; Deal, 1988). In particular, it cannot explain the so-called initial enhanced oxidation, the rapid oxide growth when the oxide thickness is thinner than a few tens of nanometers. One important factor, which is not taken into account in the Deal–Grove scheme, is that the volume of the newly formed oxide is more than 2 times larger than that of the reacted substrate Si. Since the reaction occurs at the interface surrounded by the substrate and the surface oxide, the excess volume should cause a large compressive strain on the newly formed oxide. Such accumulated strain could be released by viscoelastic deformation of the Si oxide. The Si density drastically decreases after Si thermal oxidation due to the volume expansion, indicating that Si species should move from interface region to release accumulated strain. Therefore, it is natural to think
131
that the transport of Si is also important during the thermal oxidation, though the Deal–Grove scheme considers only the transport of O. Thus, the movement of both O and Si should be clarified microscopically in detail to obtain a precise scheme of the thermal oxidation process. 1.04.2.4.2
Mechanism of oxidation Although the Deal and Grove model can describe the important characteristics of thermal oxidation, it is agreed that it fails to explain the oxidation process, especially in the thin-oxide case. Here, we introduce a more realistic scheme for the mechanism of the Si thermal oxidation process based on the knowledge obtained by the computational sciences and related experiments (Kageshima and Shiraishi, 1998; Kageshima et al., 1999; Uematsu et al., 2000, 2001, 2004; Fukatsu et al., 2003; Watanabe et al., 2006; Ming et al., 2006).
•
Oxidation direction Kageshima and Shiraishi clearly show by the first-principles calculations that the preferential growth direction of the oxide nucleus on the surfaces is vertical to the substrate, whereas, at the interfaces, it is lateral. Moreover, they have shown that Si atoms are inevitably emitted from the interface to release the stress induced during Si oxide growth (Kageshima and Shiraishi, 1998) First, the growth directions of an oxide nucleus on Si surfaces and at Si-oxide/Si interfaces are discussed. The surface growth direction is investigated by using the Si(100) surface with buckled dimmers as the initial surface. For such a surface, the most stable adsorption site of the initial O atom is the back-bond of the lower dimer atom (Kato et al., 1998). The first O atom is placed at that site and another is added between the SiSi bonds neighboring the first SiOSi bond. Second, the most stable adsorption site for the second O atom is determined by the firstprinciples results. The possible sites are shown in Figure 22(a). The calculated total energies are 0.10, 0.02, 0.33, and 0.61 eV per unit cell, for the sites B, C, D, and E, respectively, relative to the total energy of the site A. Using the dihydride Si(100) surface model (Northrup, 1991) as the initial surface, Kageshima and Shiraishi also investigated the growth direction for a H-terminated surface. The most stable adsorption site for the initial O atom is the outermost SiSi bond. The first O atom is placed at that site and another one is added between the SiSi bonds neighboring the first SiOSi bond. Then, the most
132 Atomic Structures and Electronic Properties of Semiconductor Interfaces
(a)
(b)
(c) D
C C
C
E
B A
A
D B
B
A
Figure 22 Atomic structures for studying the oxide nucleus growth. (a) Top view for the clean surface, (b) top view for the dihydride surface, and (c) side view for the oxide/Si interface with the less-stressed quartz-like oxide. The filled circles are O atoms, the empty circles are Si atoms, and the small hatched circles are H atoms. Reproduced with permission from Kageshima H and Shiraishi K (1998) First-principles study of oxide growth on Si(100) surfaces and at SiO2/Si(100) interfaces. Physical Review Letters 81: 5936.
stable adsorption site of the second O atom can be determined. The calculated total energies are 0.12, 0.22, and 0.59 eV per unit cell, for the sites B, C, and D, respectively, relative to the total energy of the site A (Figure 22(b)). These calculations indicate that the oxide nucleus on the (100) surface preferentially grows vertically into the substrate, being independent of the surface reconstruction. Next, we discuss the oxide growth direction at Si/ SiO2 interfaces. For the investigation of the growth direction for interfaces, the quartz/Si(100) interface model is used as the initial interface (Figure 22(c)) (Kageshima and Shiraishi, 1998). While the real oxide layer formed by oxidation is amorphous, for simplicity, the oxide is modeled by a crystal of SiO2. However, this crystal model certainly has a perfect bond network without large stress, which is an important feature of the real amorphous oxide interface. The first O atom is added to the interface and the stable structure is determined. Next, the second and third O atoms are introduced to the interface, assuming that all of the SiOSi bonds formed are connected. The calculations show that the structure in which the second O atom is inserted into the site A is energetically more stable (by 0.29 eV per unit cell) than the structure in which the second one is inserted into the site B. Moreover, the structure in which the second and third O atoms are inserted into the sites A and C is more stable (by 0.05 eV per unit cell) than the structure in which the second and third ones are inserted into the sites A and B. These results indicate that the oxide nucleus at the Si-oxide/Si(100) interface preferentially grows laterally, parallel to the interface. The preferential growth direction of the oxide nucleus for the (100) substrate is thus governed
by the difference between ‘on the surface’ and ‘at the interface.’ Since the vertical oxide growth on the surfaces is independent of the surface reconstruction, the stress (rather than the bonding nature or the charge transfer) seems to control the growth direction. Actually, it is easy for SiOSi bonds on the surfaces to expand vertically because the surface atoms in the Si region can move upwards with almost total freedom, while it is not easy for the bonds to expand laterally. Thus, the initial oxide nucleus on the surface should grow vertically in order to minimize the stress. In the case of the interface, the vertical expansion of SiOSi bonds is not easy because their movement is restricted by the covered oxide layer. Therefore, the energy gain due to the stress release by vertical growth is quite restricted. On the other hand, to minimize the interface energy, the initial oxide nucleus at the interface should grow laterally. These results have been confirmed by examining the stress distribution of the calculated atomic structures estimated from the shortening of the Si–Si bond lengths. These findings show the importance of the stress in determining the growth direction. A simple phenomenological model that includes oxidation induced compressive strain also reproduces the lateral oxide growth (Shiraishi et al., 2000). Recently, it has been pointed out that a simple Monte Carlo simulation that includes the diffusion of O species through SiO2 region can reproduce layerby-layer growth qualitatively (Watanabe et al., 2004). The above considerations agree with the experimental results fairly well. A previous STM-based measurement of the oxide growth on a clean Si(111) surface (Ono et al., 1993) showed that oxide islands are formed in the initial stage at 600 C. The depth of the islands reaches several atomic layers at
Atomic Structures and Electronic Properties of Semiconductor Interfaces
the very initial stage. Furthermore, many experiments have clearly shown that the oxide grows atomically layer by layer at the Si-oxide/Si(100) and (111) interfaces (Gibson and Lanzerotti, 1989; Komeda et al., 1998; Ohishi and Hattori, 1994; Watanabe et al., 1998). These are consistent with the findings based on the computational results. The results discussed above indicate that a uniform oxide layer can be obtained with any thickness by thermal oxidation, once a uniform surface oxide layer is formed. Therefore, the preparation of the initial surface oxide is crucial for obtaining a uniform oxide layer with atomically controlled thickness, which gives an important guiding principle for modern Si nanotechnologies. Although first-principles results also indicate that the initial growth direction of the oxide nucleus on the surfaces is vertical into the substrate, this is true only when the thermodynamics govern the oxidation process. Actually, the STM measurement has shown that oxidation does not form islands, but instead forms an atomically thin surface oxide layer from the very first stage at room temperature, where the oxidant cannot diffuse into the substrate easily (Ono et al., 1993). It has been reported that the O2 adsorption in the second layer of the clean Si(100) surface has a nonzero barrier of about 0.3 eV, while the adsorption in the outermost layer is barrierless (Watanabe et al., 1998). Therefore, thermal oxidation at a lower oxidant pressure and
133
lower temperature could result in the formation of a well-controlled atomically thin uniform oxide layer. The efficiency of these oxidation processes is supported by the experiments (Ohishi et al., 1994; Watanabe et al., 1998).
•
Stress release We now turn to the mechanism for the release of the accumulated stress during oxidation. Stress release was first investigated using a dihydrided Si(100) surface as the initial surface. We sequentially insert O atoms between SiSi bonds of the surface, assuming atomical layer-by-layer oxide growth. This assumption simplifies the analysis of the accumulated stress, as will be shown below. When eight O atoms per unit cell are introduced (Figure 23(a)), the formed oxide has a SiOSi network similar to that of the cristobalite of crystal SiO2 (Wyckoff, 1963). However, the structure is highly compressed compared to that of the cristobalite. The a and b axes of the obtained oxidized region, which are parallel to the interface, are 23% shorter than the corresponding axes of the a-cristobalite. Despite the elastic theory, the c axis, which is perpendicular to the interface, is only 20% longer than the corresponding axis of the a-cristobalite. Thus, the structure is largely compressed to about threefourths of the volume of the a-cristobalite. This suggests the existence of strain release mechanism during the oxide growth. One possibility is the
(a)
(b)
(c)
(d)
(e)
(f)
Figure 23 Side views of the atomic structures for studying the accumulation and the release of the stress. (a) The structure after sequential oxidation by two Si atomic layers; (b, c) the structures before and after the Si emission on the dihydride surface; (d) the structure with the Si emission after oxidation by two Si atomic layers; (e, f ) the structures before and after the emission at the oxide/Si interface with the less-stressed quartz-like oxide. The broken circles indicate the position where the Si atom is emitted. Reproduced with permission from Kageshima H and Shiraishi K (1998) First-principles study of oxide growth on Si(100) surfaces and at SiO2/Si(100) interfaces. Physical Review Letters 81: 5936.
134 Atomic Structures and Electronic Properties of Semiconductor Interfaces
breaking, deformation, and rebonding of the formed SiOSi network, which would correspond to a viscous flow of oxide. However, bond breaking and deformation after oxide formation require a lot of energy. Therefore, there must be some other mechanisms that work to release the stress before the compressed oxide is formed.
Energy advantage (eV \ unit cell)
It is found that the atomic structure, when three O atoms per unit cell are introduced, is the key to the stress release (Figure 23(b)) (Kageshima and Shiraishi, 1998). In this structure, an O atom is quite close to a surface Si atom, which has only one SiO bond. Thus, these two atoms can form a bond by breaking the bonds with the second-layer Si atom. Moreover, the second-layer Si atom, whose two bonds were broken, could be emitted from the surface because of laterally compressed stress on it (Figure 23(c)). The calculated total energy of such a Si-emitting structure indicates it to be only 0.04 eV per unit cell higher than that of the nonemitting structure, though there remain two dangling bonds. This structure resembles the well-known A center (or the VO center) in bulk Si crystal (Pajot, 1994; Chadi, 1996). In addition, when we sequentially insert O atoms, the total energies for all of the emitting structures are more stable than those for the corresponding nonemitting structures (Figure 24). The energy advantage is up to 2 eV per unit cell. This is because the two remaining dangling bonds first form a weak bond by laterally compressed stress, and are finally terminated by forming a SiOSi bond. This also indicates that the Si emission scarcely results in the creation of the interfacial gap states.
3 2 Emission preferential 1 0 –1 –2 No-emission preferential –3 1 2 3 4 5 6 7 8 0 Number of O atoms
Figure 24 Energy advantage of Si-emitting structures compared with the nonemitting structures as a function of the number of inserted O atoms per unit cell. The most stable structures for each case are compared, assuming the atomical layer-bilayer oxide growth. Reproduced with permission from Kageshima H and Shiraishi K (1998) First-principles study of oxide growth on Si(100) surfaces and at SiO2/Si(100) interfaces. Physical Review Letters 81: 5936.
Moreover, when six O atoms per unit cell are introduced to the emitting structure (Figure 23(d)), the resulting bond network resembles the quartz structure of SiO2. The a axis of the obtained oxidized region is only 8% longer than the corresponding axis of the b-quartz. The b and c axes are only 1% and 0.2% shorter, respectively, than the corresponding axes of the b-quartz. Thus, the accumulated strain is successfully released by removing Si atoms during Si oxidation. In the real experimental situation, it is expected that the remaining stress in the formed oxide would be completely released after the Si emission during oxide growth. Silicon emission also occurs at the Si-oxide/Si interfaces. Si emission from the interfaces has been considered using the quartz/Si(100) interface model (Kageshima and Shiraishi, 1998). The total energy of the emitting structure (Figure 23(f)) is more stable (by 0.41 eV per unit cell) than that of the nonemitting structure (Figure 23(e)), although two Si dangling bonds are formed after Si emission. This means that, even at the oxide/Si interfaces, Si atoms are preferentially emitted during oxide growth. Moreover, although layer-by-layer oxidation is assumed as mentioned above, further calculations show that the Si emission is independent of the oxide growth mode. Even after the initial vertical oxide growth on the surfaces, the emission can occur again when the oxide islands connect with each other. Since stress accumulation is inevitable in the Si oxidation process, the release of this stress by Si emission should be essential and universal. Si emission As discussed above, the emitted Si atoms should play an important role in the oxidation process. Since the energy advantage of the Si emission (up to 2 eV) is smaller than the formation energy of the Si interstitials (4.9 eV) (Car et al., 1984), which is thought to induce the oxidation-induced stacking faults (OSF) (Thomas, 1963; Ravi and Varker, 1974; Hu, 1975), oxidationenhanced diffusion (OED), and oxidation-reduced diffusion (ORD) (Mizuo and Higuchi, 1982; Tan and Go¨sele, 1985). However, the energetically most stable way for emitted Si is that emitted Si atoms flow back toward the SiO2 region and react with O species as (Si þ O2 ! SiO2). This is because the energy gain of this reaction amounts to 11.0 eV. Accordingly, the microscopic oxidation mechanism given by Deal and Grove (1965), which only considers the O diffusion, should be modified by taking into the backflow diffusion of emitted Si species
•
Atomic Structures and Electronic Properties of Semiconductor Interfaces
135
1019 Si species
Si Si species Figure 25 Schematic illustration of Si oxidation processes. Oxygen molecules should diffuse through a SiO2 layer before reacting with Si at Si/SiO2 interfaces.
from the Si/SiO2 interface. As a result, the oxidation process contains diffusion of both O and Si species, as illustrated in Figure 25. Next, we show the experimental findings of Si species backflow from Si/SiO2 interface to SiO2 region during oxidation. First experimental example is that the B and Si self-diffusion near the Si/SiO2 interface is remarkably enhanced (Uematsu et al. 2004; Fukatsu et al., 2003). This is thought to be the effect of emitted Si species from the Si/SiO2 interfaces. To investigate the B diffusion in SiO2 by secondary-ion mass spectroscopy (SIMS) analysis, the samples are prepared as follows. The isotopically enriched 28Si epi-layer was thermally oxidized in dry O2 to form 28SiO2 with thickness 200, 300, and 650 nm. The samples were implanted with 30Si at 50 keV to a dose of 2 1015 cm2 and capped with a 30-nm-thick silicon nitride layer. Subsequently, the samples were implanted with 11B at 25 keV to a dose of 3 1015 cm2. The final structure is shown in Figure 26. The samples were pre-annealed at 1000 C for 30 min to eliminate implantation damages, and were annealed in the resistively heated annealing furnace at various temperatures in the range of 1100–1250 C. The diffusion profiles of 11B and 30Si were measured by SIMS.
30 nm
Si3N4
200–650 nm 28SiO
2
28Si
Implanted 30Si Implanted 11B
Figure 26 The sample structure employed for considering the enhanced B diffusion near Si/SiO2 interfaces.
B concentration (cm–3)
SiO2
O species
B 5 ×1013 cm–2
1018
1200 °C 24 h 28SiO 2
thickness
200 nm
1017
300 nm 650 nm As-impla 1016
0
50
100 Depth (nm)
150
200
Figure 27 Diffusion profiles of B in SiO2 with various thicknesses. Samples were implanted with B to a dose of 5 1013 cm2 and annealed at 1200 C for 24 h. The nearer the Si/SiO2 interface, the broader the B profiles becomes. Reproduced with permission from Uematsu M, Kageshima H, Takahashi Y, Fukatsu S, Itoh KM, and Shiraishi K (2004) Correlated diffusion of silicon and boron in thermally grown SiO2. Applied Physics Letters 85: 221.
Figure 27 shows the depth profiles of 11B before and after annealing at 1250 C for 6 h. As shown in Figure 27, the profiles of 11B become broader with decreasing thickness of the 28SiO2 layer, that is, B diffusivity increases with decreasing distance from the Si/SiO2 interface. If B diffusion is governed by a single process, the B diffusivity should depend on the distance. However, it is physically unnatural. The distance dependence of diffusivity is also observed in the Si self-diffusion in SiO2 (Fukatsu et al., 2003). In the case of Si self-diffusion, SiO molecules generated at the interface and diffusing into the oxide enhance Si self-diffusion (Fukatsu et al., 2003). These results indicate that SiO molecules also enhance B diffusion, because B diffusivity is higher near the interface, where the SiO concentration is high. Moreover, first-principles calculations show that interstitial BO complexes can diffuse through SiO2 layers with relatively low activation barrier (Otani et al., 2003). Considering that this interstitial BO complex is stoichiometrically equivalent to the BSiSiO complex, these first-principles results may indicate that the existence of SiO species also enhances B diffusion (Uematsu et al., 2006). Taking into account the effect of SiO, coupled diffusion equations that include normal thermal B
136 Atomic Structures and Electronic Properties of Semiconductor Interfaces
diffusion and SiO-assisted B diffusion can readily be constructed (Uematsu et al., 2004). As shown in Figure 27, a numerical simulation that includes the mechanism that SiO species generated from Si/SiO2 interface enhance B diffusion well reproduces the distance dependence of B diffusivity, although constant diffusivities are assumed. Moreover, the time dependence of B diffusivities has also been reported. It is expected that SiO concentration generated from Si/SiO2 interface increases with longer annealing time. Considering the above discussions that SiO species enhance the B diffusion, B diffusivity is expected to be increased with longer annealing time. Actually, the clear enhancement in B diffusivity has been confirmed; 1.5 1016 and 3.0 1016 cm2 s1 B diffusivities are obtained after 8 and 24 h 1200 C anneals, respectively. This time dependence also supports the fact that B diffusion is assisted by SiO (Uematsu et al., 2006). Next, we show much direct proof of Si species emission during Si oxidation (Ming et al., 2006). Experiments use the characteristic material properties of SiO2 in thin HfO2. It is known that SiO2 and HfO2 reveal phase separation when HfO2 is thin enough. Thus, when the HfO2/SiO2/Si stacked sample is oxidized and Si substrate oxidation occurs, it is expected that emitted Si species diffuse through HfO2 and segregate at the surface. Thus, surfacesensitive observations can detect the emitted Si
15
species which segregate at the surface. Ming et al. (2006) performed the high-resolution Rutherford backscattering (HRBS) measurement and confirm the existence of surface Si component around 361 keV in addition to the Si peak at the Si/SiO2 interface near 350 keV, only when interfacial SiO2 growth occurs, as clearly shown in Figure 28. This experiment clearly confirmed that Si emission during Si oxidation, which was predicted by the first-principles calculations (Kageshima and Shiraishi, 1998) is surely observed. Here, we introduce the physical origin of initial enhanced oxidation, which has been a mystery of Si oxidation for a long time. First, we introduce the Si emission model (Kageshima et al., 1999). If the effect of such SiO interstitials as discussed previously under the so-called Si emission model is considered, the initial enhanced oxidation can systematically be reproduced (Kageshima et al., 1999; Uematsu et al., 2000). As per Deal and Grove, the oxide growth rate equation can be derived from the reaction–diffusion equation, while newly considered SiO flow from the interface to the surface should be included besides the O2 flow from the surface to the interface (Figure 29). Due to the SiO flow, the interfacial reaction rate of O2 must be modified. SiO should be much easily oxidized on the surface than in the oxide because the oxidation of SiO should be incorporated with the volume expansion. Then, in the thin-oxide limit,
400 keV He+ HfO2 /SiO2 /Si [111] channeling As-grown
Counts (a.u.)
900 °C × 2 min 10
×5 5
Hf
Si
O 0 320
340
360
380
400
Energy (keV) Figure 28 High-resolution Rutherford backscattering spectra of as-grown and annealed HfO2/SiO2/Si. Reproduced with permission from Ming Z, Nakajima K, Suzuki M, et al. (2006) Si emission from the SiO2/Si interface during the growth of SiO2 in the HfO2/SiO2/Si structure. Applied Physics Letters 88: 153516.
Atomic Structures and Electronic Properties of Semiconductor Interfaces
Gas
137
Oxydant k′
Point (2)
S
C SO
C Si
k: rapid reaction x
k : slow reaction
Point (1) Interfacial reaction rate
DSi Oxide
k
C Si(x)
DO
C ISi
C IO
k = k0 (1–C ISi /C 0Si)
x
k Silicon V Emitted Si Figure 29 Schematic view of the reaction-diffusion kinetics in the Si emission model. From Kageshima H, Uematsu M, Akiyama T, and Ito T (2007) Microscopic mechanism of silicon thermal oxidation process. ECS Transactions 6: 449.
the combined reaction–diffusion equations are analytically solved as ð17Þ
where X is the oxide thickness and t the oxidation time. This is quite similar to the empirical equation for the initial enhanced oxidation proposed by Massoud et al. (1985). The Si emission model also indicates the physical meaning of the parameters A, B, K, and L. For example, L is related to the diffusion length of SiO. By solving numerically the combined equations of Si emission model, the experimental results can well be reproduced including the initial enhanced oxidation as well as the growth rate for thicker oxide, as shown in Figure 30. Recently, it has been reported that the initial enhanced oxidation can be reproduced by taking into account the increase in the diffusion barrier by the accumulated stress near the Si/SiO2 interfaces (Watanabe et al., 2006; Watanabe and Ohdomari, 2007). They suppose a compressively strained oxide layer with thickness L localized in the proximity of the SiO2/Si interface. By considering the diffusion barrier modification, initial enhanced oxidation can also be reproduced (Watanabe et al., 2006; Watanabe and Ohdomari, 2007). 1.04.2.4.3
Our theory Experiment
Optical control of oxidation In the downsizing trends of Si-related nanoscale devices, the in situ monitoring and thickness control of Si oxidation received an increasing demand as one
Oxide thickness (μm)
dX =dt ¼ B=ðA þ 2X Þ þ K expð – X =LÞ
100 1000°C
10–1
1 atm O2 (100) substrate 900°C
10–2
10–3 0 10
800°C
101
102 103 104 Oxidation time (s)
105
Figure 30 Comparison of our theory of the oxide growth kinetics with experimental data. From Uematsu M, Kageshima H, and Shiraishi K (2000) Unified simulation of silicon oxidation based on the interfacial silicon emission model. Japanese Journal of Applied Physics 39: L699.
of the key process technologies. As explained in the above, Watanabe et al. (1998) clearly demonstrated visually by employing the scanning reflection electron spectroscopy (SREM) that the thermal oxidation of Si(001) surface proceeds in a layer-bylayer manner. Since the SREM uses electrons as a probe, however, the observation is limited to the oxidation of ultrathin layers under the ultrahigh
138 Atomic Structures and Electronic Properties of Semiconductor Interfaces
vacuum environment. On the other hand, the reflectance difference spectroscopy (RDS) is a powerful in situ optical measurement to observe the electronic structures of semiconductor surfaces/interfaces and their time evolution (Aspnes and Studna, 1985; Hingerl et al., 1993; Uwai and Kobayashi, 1994; Nakayama, 1997; Murayama et al., 1998). Since the electromagnetic light penetrates deep into the Si substrate, that is, an order of 1000 A˚, the RDS can detect, in principle, the layer-by-layer thermal oxidation. Nakayama and Murayama proposed the use of RDS to control the layer-by-layer thermal oxidation, by theoretically investigating the variation of RDS spectra of buried SiO2/Si interfaces (Nakayama and Murayama, 2000). At first, we briefly explain the RDS. The RDS measures the difference in reflectance of surface/interface between two perpendicular light-polarization directions as shown in Figure 31(a). The inner bulk
ð!Þ ¼
(a) RDS spectra ∝
(001)
ε2,(−110)− ε2,(110) (−110) (010)
Surface is anisotropic
Bulk is isotropic (b)
RDS spectra
layers are isotropic, while the surface/interface layers (ILs) are anisotropic. Thus, since the polarization anisotropy originates from the anisotropy of surface/ interface electronic states, the RDS can teach us the nature of surface/interface electronic structures. In particular, the spectral shape reflects the localized nature of the interface states. The simplest picture of polarization around the semiconductor interface is obtained by the bond polarization model (Nakayama and Murayama, 1999). The bond polarization in the nth semiconductor layer is represented by n ¼ ð – 1Þn b ð!Þ þ sn ð!Þ, where b ð!Þ is the bulk polarization corresponding to the light with h! energy, while sn ð!Þis the deviation of polarization from the bulk values. The (-1) prefactor denotes that the RDS measures the reflectance difference because the odd and even layers contribute to the spectra with alternative signs. By summing up the contributions from all layers, the spectra become
C
A
B Photon energy
Figure 31 (a) Schematic picture of RDS experiment. (b) Typical spectral shapes observed in RDS. Reproduced with permission from Nakayama T and Murayama M (1999) Tight-binding-calculation method and physical origin of refrectance difference spectra. Japanese Journal of Applied Physics 38: 3497.
X
n ?e – 2na ¼ b ð!Þ=2 þ
X
sn ð!Þ
ð18Þ
where – 1 is the decay length of the light in semiconductor. Figure 31(b) shows the typical spectral shapes of the RDS. When there exist interface states strongly localized at the interface, the sn ð!Þ term becomes extremely large because the optical transitions between localized states are large. Thus, the RDS shows the peak-like shape at the transition energy, as shown by A in Figure 31(b). When there are no interface states but the electrons in bulk have localized nature, the first term, b ð!Þ=2, produces spectra having the shape of bulk dielectric function, as shown by B in Figure 31(b). When the electronic states are extended from the interface, we can rewrite the sn ð!Þ term as qb ð!Þ=q!?" because the bond polarization gradually changes the excitation energy from the surface for extended electronic states. As a result, the RDS spectra have the energy-derivative shape for the dielectric function, as shown by C in Figure 31(b). In this way, using the optical properties of surface/interface electronic states, from the RDS spectral shape, one can know the localization nature of the surface/interface electronic states. Then, we explain why the RDS can detect the layer-by-layer oxidation. Figure 32(d) shows the schematic picture of SiO2/Si interfaces viewed from the (001) direction. As explained in the previous section, roughly speaking, the oxidation corresponds to the insertion of the oxygen atoms into SiSi bonds. Whenever the monolayer oxidation is completed in the layer-by-layer process, as seen in this figure, the
(a)
139
(b) O
O
Si
Si
Si
SiO2
[001]
Atomic Structures and Electronic Properties of Semiconductor Interfaces
(c)
(d) O
Si
O
Si
[–110]
Si
O
0 1 2
1)
(00
[110]
3 4 Figure 32 Schematic diagrams of SiO2/Si (001) interfaces. (a) Flat interface A with the desorption of interface Si atoms; (b) flat interface B after 1 ML oxidation of A interface; and (c) flat interface with crystalline SiO2 layers around the interface. (d) Bird’s eyeview of layer-by-layer oxidation of Si(001) surface as noted from the surface along [001]. Reproduced with permission from Nakayama T and Murayama M (2000b) Atom-scale optical determination of Si-oxide layer thickness during layer-by-layer oxidation: Theoretical study. Applied Physics Letters 77: 4286.
edge SiSi bonds terminated at SiO2/Si interface alternatively change the direction between [110] and [110]. Therefore, since the RDS measures the reflectance difference between two perpendicular directions, that is, the anisotropy of interface polarization originating from the interface electronic structure, one can optically detect the change of interface-bond direction and know the advance of monolayer oxidation. Figures 33(a) and 33(b) show the calculated RDS spectra of two SiO2/Si interfaces, A and B, which are displayed in Figures 32(a) and 32(b), respectively (Nakayama and Murayama, 2000). It is noted that the interface B corresponds to the interface when the monolayer oxidation is completed starting from the
interface A, and vice versa. Namely, these two interfaces alternatively appear during the layer-by-layer oxidation. The detailed atomic positions at these interfaces are taken from the first-principles calculations (Kageshima and Shiraishi, 1998). Here, we concentrate on two features in Figures 33: (i) we first notice that the spectra of the interfaces, A and B, have similar shapes and opposite signs. This result indicates that when the A and B interfaces appear alternatively during oxidation, the RDS signals oscillate between positive and negative values, which is similar to the reflectance high-energy electron diffraction (RHEED) oscillation in the epitaxial layer-by-layer growth. Especially, such an observation is apparently effective around 3.5/4.5 eV, where
140 Atomic Structures and Electronic Properties of Semiconductor Interfaces
the RDS signal is large. In this case, the period of oscillation corresponds to the bilayer oxidation and one can know the number of oxidized layers by counting the oscillation; (ii) the other feature observed in Figure 33 is the spectral shape of the large peaks around 3.5 and 4.5 eV, which, respectively, correspond to the E1 and E2 van Hove singularity energies of bulk Si shown in Figure 33(d). These peaks have shapes similar to those of the energy derivatives of "2, which indicates, based on the general theory of spectral shapes
20 Å
(a)
(b)
RD spectra: Δ ε2 •d
ε2 (d)
2ML-Exp. E1
8
E2
E3
6 4 2 0
(c) 5 7 9
1
2
3
1
6 3 4 5 Photon energy (eV)
7
Figure 33 Calculated RDS spectra of SiO2/Si (001) interfaces as a function of photon energy. (a) Interface A shown in Figure 32(a); (b) interface B shown in Figure 32(b); and (c) the crystalline interface shown in Figure 32(c). Observed RDS spectra after 2 ML oxidation are shown in (d), together with the imaginary parts of the dielectric function of bulk Si (dashed line). In (c), the bold line corresponds to the RDS spectra, while the thin lines with numbers n, respectively, correspond to layer contributions to the RDS spectra of the nth layer bonds shown in Figure 32(c). Reproduced with permission from Nakayama T and Murayama M (2000b) Atom-scale optical determination of Si-oxide layer thickness during layer-by-layer oxidation: Theoretical study. Applied Physics Letters 77: 4286.
discussed above, that the anisotropic RDS signals appear due to the modulation of bulk electronic structures that are extended around the interface (Nakayama and Murayama, 1999). In fact, such a modulation can be examined by analyzing the layer contributions to the RDS, which are also shown in Figure 33(c). This theoretical prediction was confirmed by the RDS experiments by Yasuda et al. (2001). Figure 34(a) shows the spectral oscillation measured at around 3.5 eV, while the observed spectra after 2 ML oxidation are shown in Figure 33(d). The spectral oscillation is clearly seen as a function of oxidization time, being in good agreement with the theoretical prediction. We note that the oscillation amplitude decreases as the oxidation proceeds. This is because both interfaces A and B grow to coexist within the spot size of the reflectance light, indicating some disordering of the layer-by-layer process. By analyzing the temperature and pressure dependence of the oxidation time as shown in Figure 34(b), we can know the microscopic mechanism of oxidation, such as the oxidation speed and the activation energy of oxidation (Matsudo et al., 2002; Yasuda et al., 2003). With respect to the spectral shapes, the experiments showed shapes similar to those of the energy derivatives of "2 around 3.5/4.5 eV, again in good agreement with the theoretical prediction in Figure 33(a). However, we note some differences in the magnitude. Figure 29(c) shows the calculated spectra of the interface shown in Figure 33(c), which has crystalline SiO2 structure around the interface. The existence of this kind of ILs was suggested by Ikarashi et al. (2000) and Tu and Tersoff (2002). The agreement of spectral shapes and magnitude is fairly good, indicating the appropriation of SiO2 crystalline structures around the oxidation front interfaces. In this way, the RDS measurement can observe not only the dynamical change of the interfaces but also the detailed electronic/atomic structures of interfaces. Moreover, from these studies, we obtained the in situ method of Si oxidation control. Similarly, the dot formation on the surface was also observed by the optical measurement (Kita et al., 2002). These studies have demonstrated a new direction in semiconductor nanotechnology that the microscopic theoretical investigation is indispensable to the development of new technologies.
Atomic Structures and Electronic Properties of Semiconductor Interfaces
(a)
Δr/r at 2.90 eV
2 × 10–4
Heating on
t2
O2 in
101
(b)
t3
102 Elapsed time (s)
103
The 3rd layer (one-step ox.)
t3 (S)
103
102
pO2 (Pa) 0.060 0.20 0.60 2.0 20
101 0.8
0.9
1
1.1
Figure 34 (a) Observed RDS oscillation during the layer-by-layer oxidation of Si(001) surfaces, as a function of oxidation time. The peaks around 80 and 500 s, respectively, correspond to the completion of 1 and 3 ML oxidation, while the dip around 150 s to that of 2 ML oxidation. Reproduced with permission from Yasuda T, Yamasaki S, Nishizawa M, et al. (2001) Optical anisotropy of oxidized Si(001) surfaces and its oscillation in the layer-by-layer oxidation process. Physical Review Letters 87: 037403. (b) Arrhenius plot of activation energy for the third Si-layer oxidation as a function of inverse temperature. Reproduced with permission from Yasuda T, Kumagai N, Nishizawa M, Yamasaki S, Oheda H, and Yamabe K (2003) Layer-resolved kinetics of Si oxidation investigated using the reflectance difference oscillation method. Physical Review B 67: 195338.
1.04.2.5
gate electrodes and wirings. It has long been known that the intermixing of metal and semiconductor atoms occurs at a number of metal/semiconductor interfaces, even around room temperature (Hiraki, 1980). For example, Au shows intermixing with several semiconductor substrates, such as Si, Ge, InP, GaAs, etc., while the Si substrate exhibits intermixing with many kinds of metals, such as Au, Ag, Cu, Ni, Al, etc. Such intermixing typically ranges from nano- to micrometers and provides serious damages to nanoscale devices such as circuit shortening. Thus, it is important to understand what the motive force of intermixing is and how the intermixing proceeds. On the other hand, in the case of metalSi combinations that show the intermixing at interfaces, it is well known that most transition metals and rare-earth metals, typically Ni metal, produce metal silicides by intermixing, while sp-orbital metals like Al and a few transition metals such as Au and Ag have no silicide phases. In addition, in spite of various bulk silicide phases of NixSiy, only the silicides with specific stoichiometry of (x,y) ¼ (1,2), (1,1), (2,1) are allowed to grow on Si substrate (Hiraki, 1980). What origins distinguish between such differences in silicide formation? In this section, we show one of the answers to these questions, based on the first-principles theoretical calculations. 1.04.2.5.1
1000/T (K1)
Stability of Interfaces
Metal/semiconductor interfaces are essential structures to semiconductor physics and technology because metal layers are often used as source/drain/
141
Metal induced inter-diffusion First, we consider how intermixing proceeds and discuss the motive force of intermixing. Figure 35(a) shows the calculated adiabatic potentials of interface Au and Al atom diffusion movements at Au/Si and Al/Si (111) interfaces, respectively, from the stable initial-interface positions, z ¼ 0, into Si substrate, z < 0 (Nakayama et al., 2002b, 2006a; Murayama et al., 2001). It is clearly seen that both potentials have stable minimum positions at around z ¼ 1.8 A˚. Since the barriers are at most 0.4 eV, the thermal diffusion of metal atoms becomes possible into Si layers at both interfaces. To analyze the origin of such diffusion, we show in Figure 36(a) and 36(b) the valence charge densities of Au/Si and Al/Si systems, respectively, in cases when metal atoms are moved to locate around z ¼ 1.8 A˚. In the case of Au/Si system, it is clearly recognized that the diffused innermost Au atom produces AuSi bonds with the second-top-layer Si as well as the top-layer Si. Note that the charge density of SiSi bond between top and second-top layers is smaller that that of AuSi bonds. Namely, the
142 Atomic Structures and Electronic Properties of Semiconductor Interfaces
1
(b) 3
Adiabatic potential (eV)
Adiabatic potential (eV)
(a) 2
Al/Si
0 Au/Si
−3 −2 −1 0 Metal-atom position z (Å)
2
Si surface w/o Au
1 −0 −1
Au/Si interface
0 1 2 Si-atom position z (Å)
Figure 35 Calculated adiabatic potentials for interfaceatom diffusion. (a) Metal-atom diffusion into Si substrate around Au/Si and Al/Si interfaces; and (b) Si-atom diffusion into metal layers around Au/Si interfaces. z ¼ 0 corresponds to the initial interface, while z < 0 and z > 0 to spaces within Si and within metal layers, respectively. From Murayama et al. (2001); Nakayama et al. (2002b, 2008). Reproduced with permission from Nakayama T, Itaya S, and Murayama D (2006) Nano-scale view of atom intermixing at metal/ semiconductor interfaces. Journal of Physics: Conference Series 38: 216.
charge on Au-Si bonds is mainly supplied from the nearest SiSi bonds. This charge transfer occurs because the electronegativity of Au, 2.5 (Pauling’s value), is much larger than that of Si, 1.9, and the AuSi bonding state has lower energy than the (a)
SiSi one. Since such bond rearrangement apparently stabilizes the system and produces the potential minima, one can say that the rebonding is the motive force of the Au diffusion and thus the intermixing. On the other hand, Figure 35(b) shows the calculated adiabatic diffusion potential of top-layer Si atom into metal layers when Au atom is located around z ¼ 1.8 A˚. For comparison, we also show the diffusion potential (broken line) when Au atom does not diffuse and is located at z ¼ 0. It is clearly seen that, once such rebonding weakens the interface SiSi bonds, the interface Si atoms can easily diffuse into Au layers (z > 0) with a small potential barrier around 0.2 eV. On the other hand, in the case of Al/Si system, we cannot clearly see the creation of AlSi bonds in Figure 36(b). In fact, the charge density between the innermost Al and Si atoms is smaller than that of the SiSi bond between top and second-top layers. However, the charge density of the SiSi bond between top and second-top layers is much smaller than that of the SiSi bonds in inner layers. We note that the charge density has almost constant values in the region surrounded by Al and Si atoms. This is because the electronegativity of Al atom, 1.6, is small compared to that of Si, 1.9, and the interface Al atoms partially present their valence electrons toward Si (b)
Au/Si(111)
Al/Si(111)
Al
Au Al Au Si(1)
Si(1) Au
Si(1) Si(1) Al
Si(2)
Si(2)
Si(2) Si(2)
Si(3)
Si(3) Si
Si(1) Si
Si
Figure 36 Calculated electron-density contour map around (a) Au/Si and (b) Al/Si interfaces. Three ML Au and Al are deposited on Si(111) surface and innermost metal atoms are moved into Si layers by about 1.8 A˚ (z ¼ 1.8 A˚ in Figure 35(a)). From Murayama et al. (2001); Nakayama et al. (2002b, 2008). Reproduced with permission from Nakayama T, Itaya S, and Murayama D (2006) Nano-scale view of atom intermixing at metal/semiconductor interfaces. Journal of Physics: Conference Series 38: 216.
Atomic Structures and Electronic Properties of Semiconductor Interfaces
atoms and spread their valence-electron density widely over the Si substrate. As a result, the electron density increases around Si atoms, and such an increase screens up and decreases the charge density of Si–Si bond (Hiraki, 1980). Once such screening weakens the interface SiSi bonds, the interface Si atoms can diffuse again into Al layers similar to the case of Au/Si interface. In fact, the calculated barrier of Si diffusion in the case of Al/Si is also small, around 0.2 eV. Therefore, one can roughly say that the screening due to the charge extension is the motive force of intermixing for Al/Si interfaces. In this way, the charge redistribution during the diffusion and the motive force of intermixing are somewhat different between Au/Si and Al/Si interfaces. From this result, we can expect the difference of diffusion and intermixing patterns of metal atoms between Au/Si and Al/Si interfaces. Since the production of AuSi bonds is the origin of Au atom diffusion, the Au diffusion has directional tendency and the penetration of Au into Si is expected to proceed in a needle-like line pattern. On the other hand, since the charge extension of Al toward Si atoms in various directions is the origin of Al-Si intermixing, the Al diffusion has no directivity and the intermixing of Al and Si is expected to proceed with a two-dimensional face-like front interface. It should be noted here that, in contrast to the cases shown above, no mixing is observed at most of the metal/(wide-gap-semiconductor) interfaces, such as Au/GaN, although Au shows intermixing at interfaces with several semiconductors. There are two factors to prevent intermixing. One is the elastic energy loss; since the atomic radius of Au, 1.44 A˚, is much larger than the average atomic radii of Ga and N, 1.0 A˚, the diffusion of Au pushes the GaN substrate to induce a simple compression strain of GaN layers and produces the elastic energy loss. For example, Pb metal layers were observed showing no intermixing on Si substrate, because the atomic radius of Pb, 1.8 A˚, is much larger than that of Si, 1.1 A˚. The other factor to prevent the mixing is the chemical-bonding energy loss; since the electronegativities of Ga, Au, and N atoms are 1.6, 2.4, and 3.0, respectively, the Ga atoms prefer locating among N atoms to Au atoms. When the Au atom is located on GaN surface, the top-layer Ga is surrounded by Au and N. On the other hand, when the Au atom moves between Ga and N, the top-layer Ga is surrounded only by Au atoms, thus losing the Ga-N high ionic
143
bonding energy gain. In this way, one can know that the small atomic radius and the larger/smaller electronegativity of metal atoms are relevant factors to realize intermixing.
1.04.2.5.2
Silicidation Next, we consider what factors distinguish the silicide formation between various metals. When MSi2 silicide (M ¼ metal atom) is selected as a representative phase among a variety of stoichiometric phases, the calculated formation energies of TiSi2, NiSi2, AuSi2, and AlSi2 become 0.4, 0.1, þ0.3, and þ0.4 eV, respectively (Nakayama et al., 2008, 2009). This result indicates that Ti and Ni silicides are stable, while Au and Al silicides are unstable, being in good agreement with the observed bulk phase diagrams (Madelung, 1997). The reason for such difference in formation energies is clearly understand by noting the band structures. Figures 37(a) and 37(b) show the calculated band structures of Ni and Au bulks (Nakayama et al., 2008, 2009). In the case of Ni, one can see five flat bands between 4 and þ0.5 eV, which are made of Ni d orbitals. The band that ranges from 9 to 7 eV and crosses these d bands is made of Ni s orbitals. Similar d and s bands are seen for bulk Au between 7.5 and 2 eV and from 11 to 4 eV, respectively. The most important difference between Ni and Au is the electron occupation in d bands; all d bands are occupied by electrons for Au, while about half of the electrons are unoccupied in the top d band of Ni. On the other hand, Figures 37(c) and 37(d) show the band structures of NiSi2 and AuSi2 silicides. One can see the deformation of metaloriginated bands, which is caused by the production of covalent-like bonds between metal and Si atoms due to the hybridization of s þ d orbitals of metal atom and s þ p orbitals of Si, as seen in the case of Au/Si interface shown in Figure 36(a). However, we can roughly identify that the relatively flat d-like bands, from 9 to 3 eV for NiSi2 and from 9 to 6 eV for AuSi2, are located far below the Fermi energy in the case of silicides, compared to the case of bulks. This result indicates that the d bands of Ni, which are partially unoccupied in bulk, become fully occupied in NiSi2., thus indicating the electron charge transfer from Si sp orbitals to Ni d orbitals. However, in the case of AuSi2, the d bands are already fully occupied by
144 Atomic Structures and Electronic Properties of Semiconductor Interfaces
(a)
(b) 25
Ni
Au
15 10
15
Energy (eV)
Energy (eV)
20
10 5
5 EF0
EF0 –5 –5 –10 G
XK
G
–10 G
L KW X
(c)
XK
L KW X
(d) NiSi2
AuSi2 5
Energy (eV)
5
Energy (eV)
G
EF0
–5
–10
–15
EF0
–5
–10
G
XK
G
L KW X
G
XK
G
L KW X
Figure 37 Calculated band structures of (a) Ni, (b) Au, (c) NiSi2, and (d) AuSi2. Reproduced with permission from Nakayama T, Shinji S, and Sotome S (2008) Why and how atom intermixing proceeds at metal/Si interfaces: Silicide formation vs. random mixing. ECS Transactions 16(10): 787–795 and Nakayama T, Sotome S, and Shinji S (2009) Stability and Schottky barrier of silicides: First-principles study. Microelectronic Engineering 86: 1718.
electrons in bulk Au, and such a transfer never occurs. From these results, one can conclude that the stability of metal silicides is realized by the electron transfer from Si to unoccupied d-orbital states of transition-metal atoms. This stabilization scenario of metal silicides well explains why transition metals, except Au and Ag, produce stable silicides at metal/Si interfaces and why Al does not produce silicides. Next, we consider why only the NixSiy silicides with specific stoichiometry, (x,y) ¼ (1,2), (1,1), (2,1), are allowed to grow on Si substrate among various silicide phases. Figures 38(a) and 38(b) show the calculated phase diagrams of NixSiy as a function of chemical potentials of Ni and Si, in the case of the bulk growth and the growth on the Si substrate, respectively (Nakayama et al., 2008, 2009). Chemical potentials represent the supply ratios of Ni and Si and the gray regions correspond to the allowed
regions to grow in thermal equilibrium. It is clearly seen that all NiSi2, NiSi, Ni2Si, and Ni3Si phases are realized in the bulk growth, depending on the growth condition, while the Ni3Si phase becomes difficult to grow on the Si substrate. Such changes in stability are caused by two factors. One is the elastic strain in NixSiy on the Si substrate. As the proportion of Ni increases, the lattice constant of NixSiy increases and the compressed strain in it increases. The other factor is the energy loss of the interface bonding. With increasing Ni, the number of stable Si–Si covalent bonds decreases, which also promotes the instability of Ni3Si. Finally, we point out that the above stabilization mechanism is closely related to the work function at silicide/Si interfaces (Nakayama et al., 2009). Since the p-orbital valence energy states of Si are located above those of Ni, the charge transfer occurs from Si to Ni in NixSiy and lowers the positions of Fermi
Atomic Structures and Electronic Properties of Semiconductor Interfaces
1.04.3 Electronic Structures of Interfaces
(a) –3.8
1.04.3.1
NiSi2 μ of si [Ht]
Ni3Si
–41.6
–41.2
–41.4 μ of Ni [Ht]
(b) –3.8
Ni2Si
NiSi
Ni3Si
NiSi2 –4
–4.2
–41.6
–41.4
–41.2
μ of Ni [Ht] Figure 38 Calculated phase diagrams of NixSiy as functions of Ni and Si chemical potentials, in cases of (a) bulk growth and (b) growth on the Si substrate. Gray regions are regions where NixSiy is allowed to grow in thermal equilibrium. Reproduced with permission from Nakayama T, Shinji S, and Sotome S (2008) Why and how atom intermixing proceeds at metal/Si interfaces: Silicide formation vs. random mixing. ECS Transactions 16(10): 787–795 and Nakayama T, Sotome S, and Shinji S (2009) Stability and Schottky barrier of silicides: First-principles study. Microelectronic Engineering 86: 1718.
Origins of interface states Electronic states at the interface, A/B, are generally categorized into two types: (i) states originated from A and/or B bulks and (ii) those intrinsic to the A/B interface. Figure 40 schematically shows the relation of the complex band structures of A and B bulks to electronic states at the interface. In A and B bulks, because of the crystal periodicity, the electronic states are characterized by the extended Bloch wave functions, n;k ðrÞ, where n is the band index and k the Bloch wave-number real vectors. Reflecting the periodicity, the energy spectra, "n,k, show the band structure having band gaps. However, since the interface breaks this translational symmetry along the z-direction, that is, perpendicular to the interface, the electronic states with the energy within the band gaps of bulks and complex wave-number kz are allowed to exist. Because Im kz 6¼ 0, such states show amplitude decay along the z-direction and are sometimes called as the evanescent wave states. For example, the electronic state having energy "1 in Figure 40 is constructed by the connection of two propagating Bloch states in A and four evanescentwave states in B, which all are described by the intersection points of the band structures and the energy plane. Since this state propagates along the z-direction to z ¼ 1in A (z < 0) but decays sharply in B (z > 0), we can call it the state originating from
Si conduction band
Si conduction band NiSi2
TiSi2
0.55eV 0.42
NiSi
Ni3Si 0.21
Valence band
0.48
E
E
(b)
(a)
General Features
1.04.3.1.1
Ni2Si
NiSi
–4.3
0.27 Ni2Si
145
Ti2Si Ti3Si
Si
ψε1 (r) ε1
0.69 eV
TiSi
Complex band
ψε2 (r)
0.64 0.40
ε2
Valence band
Figure 39 Calculated Fermi-energy positions of NixSiy and TixSiy. Reproduced with permission from Nakayama T, Sotome S, and Shinji S (2009) Stability and Schottky barrier of silicides: First-principles study. Microelectronic Engineering 86: 1718.
energy originating from Si electronic states for Nirich silicides. Thus, as seen in Figure 39, the work function of silicides increases as the proportion of Si decreases in NixSiy.
Real band
Re kz
Re kz Im kz Material A
Im kz Material B
Figure 40 Schematic diagram of complex band structures of A and B materials and connection of wave function at A/B interface.
146 Atomic Structures and Electronic Properties of Semiconductor Interfaces
material A. On the other hand, the electronic state having energy "2 is made of evanescent-wave states of both A and B materials, and is localized near the interface, thus called the interface state. It is apparent that the state that has energy within the band gaps of both A and B bulks becomes the type-(ii) interface state, while the state that has energy within the band spectrum of either A or B becomes the type-(i) state that propagates deep into either A or B. These are necessary conditions of energy for judging the types of interface states. It should be noted here that the number of complex bands that produce the interface states is in general infinite, though the finite ones are displayed in Figure 40. However, their contribution becomes considerably small when they are located far from the real band structures. For the type-(i) bulk-like interface states, the existence of interface is sometimes renormalized to be described as some potential, combined with the effective-mass approximation. This treatment is often called the quantum-well-picture representation. For example, as shown in Figure 41 for the case of GaAs/AlAs superlattice, AlAs layers are treated as potential barriers and GaAs ones as quantumwells. By adopting this picture, many interesting electronic phenomena are analyzed. In such cases, the key quantity that governs the electronic states is the band offset, which is the energy difference of bands between the two semiconductors (represented by Ec and EV in Figure 41). The details of this band offset are discussed in the next subsection. The physical origins of the existence of type-(ii) interface states are also categorized into two groups: (a) extrinsic origins such as dislocations and impurity atoms at interfaces, which are sometimes produced and incorporated in the case of interface formation, as shown in the previous section; and (b) intrinsic E
Ve (z)
Ec (AlAs)
origins for interfaces owing to the breakdown of translational symmetry at the interface and the production of new types of atomic bonds between A and B at the interface. Since, at present, there are no definitive experimental tools, such as STM, for the surface to observe the interface atomic structures, it is sometimes difficult to separate these origins. The details of the intrinsic origins are considered in the next two subsections. Here, we briefly illustrate examples of the extrinsic origins in Figure 42. New atomic environment shown in the number region ‘2’ and the existence of impurity atoms shown in regions ‘6’ and ‘7’ often produce the interface states in the band gap. The step structures of the substrate shown in ‘3’ and ‘4’, which are left in the heteroepitaxy, and the appearance of misfit dislocations shown in ‘5’, which are produced by the lattice mismatch between A and B, also become the structural origins of interface states. 1.04.3.1.2 Semiconductor/semiconductor interface states
We first consider the Ge/GaAs (110) nonpolar interface as an example. Figure 43 shows the atomic structure near the interface (Pickett and Cohen, 1978). Because Ge and GaAs have similar lattice constants, the interface has few dislocations and becomes remarkably sharp. The atomic bonds represented by broken lines, GeGa and GeAs, are not present in tetrahedral bulk semiconductors. GeGa and GeAs bonds have 1.75 and 2.25 electrons, as opposed to the ordinary semiconductors, which have 2.0. Thus, these bonds are often called acceptor and donor bonds, respectively. To judge the existence of interface states, it is convenient to compare the two-dimensional band structure of an interface with the projected spectra E
LB z
Lw ΔEc
Ec (GaAs)
Band gap
Ev (GaAs) z
Ev (AlAs)
ΔEv
Vh(z) AlAs
Conduction band
Valence band kz
kz AlA
GaAs
AlAs
GaAs
AlAs
GaAs
Figure 41 Quantum-well diagram and band offset, in the case of GaAs/GaAs superlattice.
Atomic Structures and Electronic Properties of Semiconductor Interfaces
147
(a)
6 5 7
4 1 (b)
3
2
Figure 42 Schematic examples of interface structures that produce extrinsic interface states.
Ga
Ge
As
Ge-GaAs interface states GaAs
Ge
Projected band structure
2
1st GaAs layer
Interface layer
z
–2 Energy (eV)
1st Ge layer
P2
B2
0
P1
–4 S2
B1
–6 –8
x
–10
y Figure 43 Atomic structure of Ge/GaAs (110) interface. Reproduced with permission from Pickett WE, Louie SG, and Cohen ML (1978) Self-consistent calculations of interface states and electronic structure of the (110) interfaces of Ge–GaAs and AlAs–GaAs. Physical Review B 17: 815.
of bulk band structures into two-dimensional Brillouin zone. Figure 44 shows such a comparison for Ge/GaAs interface, where band structures are calculated using super unit cells and self-consistent pseudopotential method (Pickett et al., 1978). Shaded regions correspond to bulk bands, thus pointing to the existence of only the bulk-like interface states even at the interface. The electronic states, S1B2, appearing outside the shaded regions (gap, pockets), are states intrinsic to interface and are localized at the interface as shown in Figures 45(a)–45(d). S1 and B1 states ((d) and (b)) are made of s and p orbitals and localized at the
–12 Γ
S1 X
M k
X′
Γ
Figure 44 Band structure of Ge/GaAs (110) interface. Reproduced with permission from Pickett WE, Louie SG, and Cohen ML (1978) Self-consistent calculations of interface states and electronic structure of the (110) interfaces of Ge–GaAs and AlAs–GaAs. Physical Review B 17: 815.
GeAs bond, while S2 and B2 ((c) and (a)) are similar states localized at the GeGa bond. In this way, the interface states at nonpolar sharp interfaces appear to originate from new interface bonds. However, there appear no interface states within the fundamental band-gap region, 0.01.0 eV in Figure 44. This is because all dangling bonds that exist on the Ge and GaAs surfaces are terminated at the interface. As a result, the physical
148 Atomic Structures and Electronic Properties of Semiconductor Interfaces
6
(a) 8
Ge
Ge
As 32
Ge 4
4
Ga
Ga
B2 Ge-Ga interface state
(b)
B1 Ge-As interface state
8 Ge
As
As
18
24 14 Ge
Ge
Ga
(c)
4
S2 Ga s-like interface state 6 Ge
Ge
As 4 Ga
Ge
(d)
S1 As s-like interface state
Ge
Ge
Ga
24
As
As
42
8
Ge
Ga
Figure 45 Charge distribution of interface states at Ge/ GaAs (110) interface. Reproduced with permission from Pickett WE, Louie SG, and Cohen ML (1978) Self-consistent calculations of interface states and electronic structure of the (110) interfaces of Ge–GaAs and AlAs–GaAs. Physical Review B 17: 815.
properties such as electron transport are not influenced by these interface states. The above-mentioned features apply to other combinations of semiconductors. Table 3 shows the
chemical tendency of interface states (Pickett and Cohen, 1978). There exist only the bulk-like states when the ionicity of materials is small, while the intrinsic interface states appear and their energy positions leave the bulk band spectra as the material ionicity increases. Since the charge transfer occurs from donor bonds to acceptor ones, the atomic positions at the interface are relaxed. However, even in such cases, there is little change for the energy-level positions of interface states. Next, we consider the polar interfaces. The (001) and (111) interfaces become polar interfaces because either anion or cation atoms are located on these crystal planes. When we consider a Ge/GaAs (001) sharp interface as an example, there exist two kinds of interfaces: the interface with only acceptor bonds, GeGa, and the interface with only donor bonds, GeAs, as shown in Figures 46(a) and 46(b). However, these interfaces are sometimes unstable due to large deficiency or excess of electrons around the interface. Instead, to compensate such unfavorable charge in a short range, the atom intermixing often occurs at these interfaces as shown in Figures 46(c) and 46(d) This kind of interface instability becomes notable for heterovalent interfaces, as discussed in Section 1.04.2.3.1. Even at these interfaces, however, electronic states are produced based on the tetrahedral bonds, thus often appearing around the bulk band edges but not deep in fundamental band gaps (Pollmann and Pantelides, 1980; Konc and Martin, 1981; Oda and Nakayama, 1992). In spite of this, the electron transfer from donor bonds to acceptor ones largely changes the band offset between two semiconductors, as shown in the following subsections. To take advantage of such controllability of band offset, the acceptor and donor bonds are sometimes produced artificially by the insertion of ultrathin layer between A and B semiconductors during the crystal growth.
Table 3 Ionicity and character of inferface state at (110) semiconductor/ semiconductor interfaces Bond charge A/B
Ionicity
Bulk state
Interface state
Acceptor
Donor
GaAs/AlAs GaAs/Ge ZnSe/Ge
Small Middle Large
Yes Yes Yes
No Yes Yes
2.00 1.77 1.54
2.00 2.23 2.46
Reproduced with permission from Pickett WE and Cohen ML (1978) Theoretical trends in the abrupt (110) AlAs–GaAs, Ge–GaAs, and Ge–ZnSe interfaces. Journal of Vacuum Science and Technology 15: 1437.
Atomic Structures and Electronic Properties of Semiconductor Interfaces
(c)
(b)
(d)
(001)
(a)
Ge,
Ga,
As
Figure 46 Schematic atomic structures of Ge/GaAs (001) interfaces. (a) Ge–Ga interface, (b) Ge–As interface, and (c,d) charge-compensated interfaces.
from region III to region IV, which is discussed in the next subsection. Figure 48 shows the local density of states (LDOS) in regions I–VI. The positions of these regions are denoted in Figure 47. In inner layers of Al, that is, in region I, the LDOS shows the square-root energy dependence, indicating a free-electron-like electronic structure of bulk Al. On the other hand, in inner layers of Si, that is, in region VI, the band gap appears around the Fermi energy and the LDOS is similar to that of bulk Si. In the LDOS in region IV, one can see the interface state, Sk, around 8.5 eV. The most important
1.0
1.04.3.1.3 Metal/semiconductor interface states
Al-Si Interface Region I EF
0.5 0 1.0 Region II
EF
0.5 0 1.0 Region III Local density of states (a.u.)
Metal/semiconductor structures are essential in almost all electronic devices to inject carriers from metal electrodes into semiconductors and activate the device operation. Here, we adopt the Al/Si interface as an example and consider the features of metal/semiconductor interfaces. Figure 47 shows the valence electron distribution around the interface, which was calculated by the pseudopotential method using a super unit cell and employing a jellium model for Al layers (Louie and Cohen, 1976). The electron density is almost constant and the small Friedel oscillation is seen in Al sides, while the electrons are accumulated on the covalent bonds between Si atoms in Si side. We can see a small amount of electron transfer
149
0.5
EF
0 1.5 1.0
Region IV SK
EF
0.5 0 1.5
Region V
1.0
EF
0.5 0 1.5 Region VI 1.0
EF
0.5 0 −14 −12 −10 −8 −6 −4 −2 Energy (eV) Figure 47 Valence charge distribution around the Al/Si (111) interface: (a) contour map display and (b) distribution averaged along the interface. Reproduced with permission from Louie SG and Cohen ML (1976) Electronic structure of a metal–semiconductor interface. Physical Review B 13: 2461.
0
2
4
Figure 48 Local DOS around the Al/Si (111) interface. Regions I–VI are displayed in Figure 47. Reproduced with permission from Louie SG and Cohen ML (1976) Electronic structure of a metal–semiconductor interface. Physical Review B 13: 2461.
150 Atomic Structures and Electronic Properties of Semiconductor Interfaces
(a)
Al-Si interface states with E = 0 to 1.2 eV 1.8 1.5 1.8 1.5
2.1 1.8
1.7
0.6
1.0 0.3
1.8
1.5 1.8 1.5
0.3
0.0
1.8 0.9 2.7
(b) 2.0
Al-Si interface
P (z) States with E = 0 to 1.2 eV
1.04.3.2
1.0
0
Figure 49 Charge distribution of metal-induced gap states (MIGSs) at the Al/Si (111) interface: (a) contour map display and (b) display average along the interface. Reproduced with permission from Louie SG and Cohen ML (1976) Electronic structure of a metal–semiconductor interface. Physical Review B 13: 2461.
feature is the appearance of electronic states within the band gap of Si in regions IV and V, which contributes to the formation of Schottky barrier height. The charge density of one of these states is displayed in Figure 49. Though this state has relatively large density on the dangling bond of interface Si, it is apparently different from the dangling-bond state on Si surface that is strongly localized around Si. Instead, this state looks like the free-electron-like state of Al that penetrates into Si layers or connects with the Si localized state. This electronic state, which is also present in semiconductors and has energy within the band gap, is called the metal-induced gap state (MIGS) (Heine, 1965; Louie and Cohen, 1976). The density of states (DOS) is shown in Figure 50 for MIGSs in some 12
DOS (1014 states/eV cm2)
semiconductors. One can see that the number of MIGSs becomes small as the ionicity of semiconductor increases. This is because a semiconductor of large ionicity has larger band gap and thus the DOS is largely decreased because the complex band leaves the real band, has a large imaginary part of wave number, ImKz, and the MIGS is strongly localized at the interface.
(a) Si
(b) GaAs
(c) ZnS
10 8 EF
6
EF EF
4 2 0 0
0.4 0.8
0 0.4 0.8 1.2 0 1.0 2.0 3.0 4.0 Energy (eV)
Figure 50 DOS for MIGSs within the band gap of the semiconductor. Reproduced with permission from Louie SG, Chelikowsky JR, and Cohen ML (1977) Ionicity and the theory of Schottky barriers. Physical Review B 15: 2154.
Band Alignment
The Schottky barrier is the energy difference between the valence (or conduction) band edge of the semiconductor and the Fermi energy of the metal, while the band offset is the energy difference of valence (or conduction) bands of two materials that construct the interface. In this subsection, we review the representative theories and consider how these quantities are determined, that is, how the bands of two materials align at the interface. We start from the Schottky barrier at metal/semiconductor interfaces. 1.04.3.2.1
Schottky barrier We first consider the metal/metal interface. At this interface, the statistical mechanics teaches us that the movable carriers such as free electrons easily move across the interface to produce a dipole potential at the interface and equalize the Fermi energies of both metals with each other. Since there are no electric fields in metals in an equilibrium, the transferred carriers are often localized within a few atomic layers around the interface (Kajita et al., 2007). In the case of a metal/semiconductor interface, if one can define Fermi energies for semiconductors, the same scenario as that of metal/metal interfaces applies. At the metal/semiconductor interface, the semiconductor generally possesses electronic eigenstates that have eigenenergies within the band gap of bulk semiconductor, as shown in Section 1.04.3.1.1. For intrinsic interface states, these states appear due to the breakdown of translational symmetry of semiconductor bulk crystals at the interface perpendicular to the interface direction and are called MIGSs. In fact, as shown in Section 1.04.3.1.3, ab initio calculations demonstrated that these MIGSs really exist in Si at Al/Si interfaces. On the other hand, in the case of extrinsic origins, these states appear due to structural disorders such as defects. In any case, these states are made of eigenstates in complex band structures of bulk materials and have complex wave numbers, and are thus localized around the interface. Such states are schematically shown by short bars in the energy diagram of Figure 51(a).
Atomic Structures and Electronic Properties of Semiconductor Interfaces
(a)
interface to equalize their Fermi energies, CNL , and EF. In the case of Figure 51(a), electrons transfer from the metal to the semiconductor. The excess positive carriers in the metal are accumulated at the interface because the dielectric constant of a metal is infinitely large, while the excess electrons in the semiconductor enter the interface MIGSs and penetrate into the semiconductor at most a few A˚. In this way, the electron transfer between metal and semiconductor produces the dipole at the interface and results in the final band alignment between the two. The energy difference between EF and the lowest conduction-band state of the semiconductor, ECB, or between EF and the highest valence-band one, EVB, acts as a potential barrier for electrons or holes, respectively, and is called the Schottky barrier or ‘contact potential’. In addition, the energy position of EF relative to vacuum is sometimes called as an effective work function (WF) of a metal. The change of the metal Fermi energy (effective WF) with varying metals is simply regulated using the slope parameter (S parameter) (Cowley and Sze, 1965). The S parameter is defined by the derivative as
Conduction band
EF
CB-like MIGS
φCNL
VB-like MIGS Valence band Metal
(b) Schottky barrier
Semiconductor
Conduction band ECB
φCNL
EF
E VB Valence band Metal
Semiconductor
(c)
DCB S
ECB
EF
S ¼ ðeffective WFÞ=ðvacuum WFÞ
(Low-WF metal)
B B
EF
EG
φCNL
S
(High-WF metal) Metals
151
EVB DVB Semiconductor
Figure 51 Schematic band alignment at metal/ semiconductor interface (a) before and (b) after the connection. (c) Normal movement of the metal Fermi energy relative to the electronic structure of semiconductor. In (c), the bold-arrow lines denote that the Fermi energy of the metal tends to move toward the charge neutrality level,
CNL , by transferring electrons around the interface. Dashed-arrow lines S and B, respectively, correspond to the Schottky (S ¼ 1) and Bardeen (S ¼ 0) limits.
As shown in Figure 51(a), some of the MIGSs are occupied from the bottom, corresponding to the number of electrons around the interface and the highest occupied MIGS determines the effective Fermi energy of the semiconductor around the interface. Since this Fermi energy is determined by the electron number to keep the original semiconductor as neutral, it is often called the charge neutrality level (CNL). Hereafter, we denote CNL as CNL . When the semiconductor is in contact with a metal, as in Figure 51(b), electrons in the semiconductor and metal move across the
ð19Þ
where means the variation by the change of metals at the interface (see also Figure 58(a)). The theory of
CNL predicts that the metal Fermi energy moves toward a single position of CNL , as shown in Figure 51(c), thus decreasing the energy differences of effective WF among metals. This indicates that 0 < S < 1. When the densities of MIGSs is small and there is little charge transfer across the interface, effective WF has the same values as vacuum WF, as shown by the S-arrows in Figure 51(c), which realizes the S ¼ 1 limit. On the other hand, when the density of MIGSs is large and the CNL position hardly changes by the interface charge transfer, all effective WFs have the same value (Fermi-level pinning) for all metals, as shown by the B-arrows, which indicates that S ¼ 0. These two limits are called the Schottky–Mott and Bardeen limits, respectively (Schottky, 1938; Mott, 1938; Bardeen, 1947). As long as the Schottky barrier height (SBH) is considered to be produced based on this conventional CNL concept, the S parameter never goes out of the 0 < S < 1 region.
1.04.3.2.2
Charge neutrality level We now consider the location of CNL . Tersoff (1984a,b) noted that the key is the nature of MIGS
152 Atomic Structures and Electronic Properties of Semiconductor Interfaces
and proposed that CNL is determined only by the electronic structure of the bulk semiconductor. As explained above, the MIGS is the eigenstate in complex band structures having energies within the band gap of the bulk system. Since the complex bands connect with real bands at the band edge of real bands, the MIGSs with lower energies possess a valence-band character, while those having higher energies carry a conduction-band character (see Figure 51(a)). The effective Fermi energy, CNL , is defined as the boundary between these valence-bandand conduction-band-like states, and are thus approximately given by the sign-changed boundary energy of the cell-averaged propagating Green function: G ðR; CNL Þ ¼
Z dr unitcell
XZ
d3 k
n
jnk ðr Þjnk ðr þ RÞ ¼0
CNL – "nk ð20Þ
where jnk ðrÞ and "nk are eigen wave function and energy of the electronic states of bulk semiconductor, respectively, with the band index n, and the Bloch wave number k. Here, R is the real lattice vector perpendicular to the interface. In a one-dimensional system, CNL coincides with the branching point of the complex band. Using CNL calculated by this formula, Tersoff succeeded in reproducing SBHs of various metal/semiconductor interfaces. Table 4 shows the calculated SBH values, together with the observed values in experiments. The agreement is quite good, especially for semiconductors with small band-gap energies. This is because the MIGS picture, which assumes the metallic electronic states in semiconductor layers near the interface, is well applicable to such systems because of the large density of MIGSs.
When the interface states appear due to the extrinsic origins such as structural disorders, we have to consider other methods for calculating Based on their systematic XPS
CNL . experiments,Spicer et al. (1980, 1989) were the first to point out the importance of defect-induced interface electronic states to determine the Schottky barrier. Drummond (1999) extended this concept and categorized observations into various defect types such as vacancies and antisite atoms. It is often observed that the DOS of interface states shows a U-like shape in the band gap for a variety of disordered metal/semiconductor interfaces, as shown in Figure 52(a). Hasegawa and Ohno called these states as disorder-induced gap states (DIGS). They found that the Fermi energies of metals match the energy position of the lowest DOS value of DIGS; thus, such energy state acts as a CNL-like Fermi energy of semiconductors (Hasegawa and Ohno, 1986). Figure 52(b) shows the basic concept they used for calculating CNL for disordered interfaces. They considered that the DIGS appears due to the disorder of semiconductor bonds at the metal/semiconductor interfaces and proposed that the CNL-like energy that separates the bonding and antibonding states is obtained as an average of sp3-orbital energies of semiconductors. Here, we would like to point out two important features in these kinds of CNL theories. At first, due to some ambiguity in determining the boundary between valance- and conduction-band-like MIGS or DIGS, there exist several versions of the definition of CNL . For example, using a Penn-like model, Cardona and Christensen (1987) proposed to adopt the dielectric mid-gap energy as CNL . However, most of these theories present similar values of
CNL for many semiconductors. This occurs because
Table 4 Calculated Schottky barrier height, , and localization length, , for various combination of metal and semiconductor f e (eV)
Si Ge GaAs ZnS
Au
Al
Other metals
Theory (eV)
(A˚)
Band gap (eV)
0.83 0.59 0.94 2.00
0.70 0.48 0.78 0.80
0.70–0.82 0.38–0.64 0.71–0.94 0.80–2.00
0.76 0.48 0.74 1.40
3.0 4.0 3.0 1.5
1.12 0.66 1.42 3.60
From Tersoff J (1984a) Schottky barrier heights and the continuum of gap states. Physical Review Letters 52: 465, Tersoff J (1984b) Theory of semiconductor heterojunctions: The role of quantum dipoles. Physical Review B 30: 4874 and Tersoff J and Harrison WA (1987) Transition-metal impurities in semiconductors – their connection with band lineups and Schottky barriers. Physical Review Letters 58: 2367.
Atomic Structures and Electronic Properties of Semiconductor Interfaces
(a)
0
Ec PCVD - SiO2 (annealed at 800 °C)
Energy (eV)
GaAs –04
PCVD - SiO2 Ei
–08 –12
Anodic native oxide
Ev Ec
0 Energy (eV)
InP –04
As-grown
–08
Annealed for 16 h Ei
–12
Optimized Al2O3 /native oxide
Electrolytic Plasma
Ev Ec x = 0.65
0 Energy (eV)
Inx Gax As –04
x = 0.53
–08
x = 0.25
x=0
153
assuming that the interface DOS is considerably large because CNL does not change even if the charge transfer occurs. For example, assuming a simple band structure for semiconductor, where valence and conduction bands have constant DOS, we can derive that CNL ¼ EVB þ EG DVB =ðDVB þ DCB ), where EVB and EG are the valence-band edge and band-gap energies, respectively, while DVB and DCB are DOS of valence and conduction bands, respectively. This formula is derived from the theories of Tersoff and Cardona and Christensen, and indicates that both theories estimate similar values. It should be noted that this formula is described by the physical quantities intrinsic to bulk semiconductors. In case of DIGS interface states, since the interface is randomly disordered, Hasegawa and Ohno assumed the simple average of orbital energies of semiconductors, thus again depending on only the bulk properties.
–12 Al O /native oxide 2 3 1011
(b)
1013 1012 (cm–2 eV–1)
Disordered semiconductor layer
Semiconductor
1014
Ev
E Ec
III
Anti-bonding II I EHO Bonding
Insulator or metal
Nss
Er
Figure 52 (a) DOS of interface states at disordered interfaces. (b) Schematic diagram explaining the appearance of disorder-induced gap states (DIGSs). Reproduced with permission from Hasegawa H and Ohno H (1986) Unified disorder induced gap state model for insulator– semiconductor and metal–semiconductor interfaces. Journal of Vacuum Science and Technology B 4: 1130.
they commonly assume the existence of interfaces states and the charge transfer between metal states and interface states. Since such charge transfer is conceptually similar to that between metallic materials, we can say that the full contact of electronic states is assumed between metals and semiconductors in these conventional CNL theories, details of which will be discussed in the following section. The second important feature is that CNL is defined as intrinsic to not the interface but the bulk properties of semiconductors, which is equivalent to
1.04.3.2.3
Band bending Most semiconductors contain impurity atoms that are introduced unintentionally in the growth process or intentionally by doping. Here, we consider an ntype semiconductor having movable electron carriers as an example. Figures 53(a) and 53(b) show the band alignments before and after the contact. Before contact, in a semiconductor, some of electrons originating from donor atoms transfer into interface states and the positive-impurity sites appear around the interface. Due to this positive charge distribution, the potential (therefore, the conduction and valence bands) bends as shown in Figure 53(a), which is called the band bending. When metal and semiconductor are in contact with each other, electron transfer occurs from the interface states of semiconductor and inner donor states to the electronic states in metal in case of the alignment in Figure 53(a), to equalize the Fermi energies of metal and semiconductor, M and S. The alignment after the contact is (a)
μM
Metal
Depletion layer
μS
(b)
μM
φC
μS
n-Type semiconductor
Figure 53 Schematic view of band bending at metal/ semiconductor interface: (a) before and (b) after contact.
154 Atomic Structures and Electronic Properties of Semiconductor Interfaces
shown in Figure 53(b). A more precise band-bending feature is obtained by solving the Poisson equation in the electromagnetism, considering the charge transfer self-consistently. The region around the interface with no electrons is often called the depletion layer, with typical width around 100 A˚. Here, we briefly comment on the experimental methods to measure SBHs. Figure 54 shows schematically the principal views of three methods. In the current–voltage (I–V) method, one applies a voltage perpendicular to the interface and measures the thermal electrons that pass the Schottky barrier potential from semiconductor to metal. Because the current is proportional to expð – B =kB T Þ, one can determine
B from the temperature dependence. In the capacitance–voltage (C–V) method, one applies the alternating voltage V and measures the capacitance C of inversion layer.pSince the width of inversion layer is ffiffiffiffiffiffiffiffiffiffiffiffiffi proportional to B – V , the capacitance becomes proportional to 1= B – V , from which B is determined. Because the capacitance also includes effects of tunneling currents, resistance in semiconductors, and distribution of impurity atoms, the analysis is sometimes complicated. In the method using internal photoelectric effect, one produces hot electrons by photoelectric excitation and measures the inverse current flowing over the barrier. These methods have an advantage in observing the deeply buried interface. We can also measure the barriers by using XPS and STM when metal layers are thin enough. 1.04.3.2.4
Band offset Next, we consider how the band offset appears at a semiconductor/semiconductor interface. Anderson (1962) proposed that the conduction band offset at A/B interface is given by the electron affinity difference between A and B materials as Ec ¼ B – A . Using the observed band-gap energies, one can also derive the valence-band offset as Thermal electron emission I
φB Conduction band
EF
Inpurity levels Depletion layer Capacitance
C = dQ/dV
Figure 54 Schematic principles to measure the Schottky barrier height (SBH).
EV ¼ EC þ EGB þ EGA . On the other hand, since the valence-band structures of zinc-blende semiconductors are well described by the nearest-neighbor tight-binding approximation using sp3 orbital sets, Harrison and Tersoff (1986) and Monch (1996) proposed that the valence-band offset, EV, is obtained as the energy difference of the valence-band top between A and B; they calculated various EV values by adopting empirical parameters. These results are shown in Table 5 for various combinations of semiconductors. It is seen that the agreement with experiments is largely improved by the latter theory. This is because the affinity is sensitive to the atomic structures of material surfaces and the atomic and electronic structures are generally different for surfaces and interfaces. On the other hand, though we do not consider the charge transfer at the interface in these theories, the agreement is not so poor, indicating the quantity intrinsic to bulk semiconductor is an important factor to determine the offset. In these theories, we use the vacuum level as the reference energy for comparison. Recently, Van der Walle and Neugenbauer (2003) proposed the band alignment assuming that the hydrogen impurity level has the same energy position in most of the semiconductors. This proposal can be conceptually similar to the above-mentioned two theories. To advance a more reliable calculation of the band offset, one has to consider the charge transfer at the interface (Nakayama, 1993). In this case, the scenario based on the CNL theory at metal/semiconductor interfaces applies straightforwardly to semiconductor/ semiconductor interfaces. Figure 55 illustrates how the band offset is determined at the semiconductor/semiconductor interface. At the semiconductor interfaces Table 5 Calculated valence-band offset at various semiconductor/semiconductor (110) nonpolar interfaces, in eV A/B
Exp.
EAR
HAO
TER
LDA
GaAs/AlAs GaSb/InSb GaAs/InAs Si/Ge GaAs/ZnSe Ge/ZnSe Ge/GaAs InP/CdS
0.50 0.51 0.17 0.20 0.96 1.52 0.53 1.63
0.15 1.12 0.24 0.33 1.26 1.97 0.35 1.27
0.04 1.12 0.32 0.38 1.05 1.46 0.41 1.48
0.35 0.43 0.20 0.18
0.37
1.12
1.59 2.17 0.63
Exp: experiments; ERA: electron affinity theory; HAO: tight-binding theory; TER: charge neutrality level theory; LDA: local density functional calculations.
Atomic Structures and Electronic Properties of Semiconductor Interfaces
A
The results in Table 5 demonstrate that not only the electronic structures of bulk semiconductors but also the charge transfer at the interface are important to determine the band offset at the interface. It should be noted here that the charge transfer across the interface is often realized by the orbital hybridization between the constituent interface atoms of semiconductors, thus producing the dipole at the interface. To examine whether such a dipole is really produced at the interface by charge transfer, we had better introduce another freedom at the interface and study the variation of charge transfer. Figure 56 shows the calculated valence-band offset at ZnSe/ZnTe (001) interface as a function of the strain (Nakayama, 1992). The strain can be varied by changing the lattice constant, that is, the composition, of the substrate for ZnSe/ZnTe system. Because by increasing the Te composition, the difference of bulk energy, ", decreases, it reflects the deformation potentials of the bulk system. On the other hand, there is an increase in the charge transfer from ZnTe to ZnSe, , and this increases the dipole potential, V. As a result of compensation between " and V, the valence-band offset gradually decreases. It should be noted that such a correlation between "
Semiconductors A B
B εBF
εAF
εBF
εAF
Figure 55 Schematic view to explain the formation of band offset.
A and B, there exist MIGS-like interface states and these states are occupied by interface electrons from the bottom to the effective Fermi energies (left figure). When the interface is realized, the charge transfer occurs between these states to equalize such Fermi energies (middle figure). As a result, one obtained the band alignment shown in the right. Harrison and Tersoff (1988) were the first to adopt this picture and calculate the band offsets for various combinations of semiconductors; their results are shown in Table 5. It is seen that the CNL theory also succeeded in predicting valence- and conduction-band offsets of a number of semiconductor/ semiconductor interfaces. (a)
3.2
ZnSe/ZnTe
0
(b)
0
Δ ΔEv ,
–2
1
hh
Offset of hh state (eV)
Δρ
0.2
SiC
0.1
GaSb AIAs AlSb AIP, GaP GaAs InSb AIN ZnTe CdSe InP ZnSe ZnS InAs GaN Cds CdTe HgTe HgSe HgS
0 0 0 0
0.5 LznTe X= LZnSe + LZnTe
: IV family : III–V compounds : III–VI compounds
Si
Level-shift difference & offset
ΔV (eV) –1
Δ & ΔEv,hh (eV)
Δρ (10–2e) 1.6
Dipole potential
Charge transfer
C
ΔV
155
2
4
6
Ionicity, ε cp – ε ap (eV)
1
Figure 56 (a) Strain dependence of valence-band offset at ZnSe/ZnTe (001) interface. Strain changes corresponding to the composition of the substrate. (b) Band offset at wurtzite/zinc-blende interface for the heavy-hole states of various semiconductors as a function of ionicity. (a) Reproduced with permission from Nakayama T (1992b) Valence band offset and electronic structures of zinc-compound strained superlattices. Journal of the Physical Society of Japan 61: 2434. (b) Reproduced with permission from Murayama M and Nakayama T (1994) Chemical trend of band offsets at wurtzite/zinc-blende heterocrystalline semiconductor interfaces. Physical Review B 49: 4710.
156 Atomic Structures and Electronic Properties of Semiconductor Interfaces
and V indicates the charge transfer expected by the
CNL theory really occurs. At the heterovalent interfaces such as ZnSe/GaAs (001), the charge transfer governs the band offset itself. Figure 57(a) shows the measured valenceband offset as a function of R ¼ Zn/Se ratio at the interface, which is realized by changing the growth treatment (Nicolini et al., 1994). Zn–As acceptor and Se–Ga donor bonds exist at this interface. The charge transfer of about 101e occurs between these bonds, which produces a dipole potential V. Depending on the growth treatment, the configuration of these bonds changes as shown in Figures 57(b) and 57(c) and the direction of charge transfer is opposite in these two cases. As a result, as seen in Figure 57(a), the offset varies by about 0.7 eV. The above result for ZnSe/GaAs interface demonstrated that the atomic structures at the interface are also important to determine the band offset in some of the interfaces. In fact, by producing the heterovalent bonds by intentionally inserting
foreign-family atoms at the interface, one can sometimes control the modulation of the band offset. Lambrecht and Segall (1990) developed the simple tight-binding model to include such effects in the case of covalent bonding systems. More precise values of band offsets are evaluated using the first-principles calculations. The results using the local density-functional calculations are also shown in Table 5. When the quasiparticle GW calculation is applied, one can obtain the most precise values, for example, 0.5 eV for GaAs/AlAs interface (Zhang et al., 1989). These first-principles calculations can consider the effects of any atomic structures of interface and apply to new systems that have not ever been produced in experiments. For example, the heterocrystalline interfaces, which is the interface between the same semiconductors with different crystal structures, are proposed and produced in experiments (Murayama and Nakayama, 1993, 1994; Hibino and Ogino, 2000). The calculated band offsets at hexagonal/cubic
(a) 1.6
ZnSe/GaAs(001) band offset
R = Zn/Se
1.2
0.8 Substrate types p+
0.4
0 0.5
n+ n nc (4 × 4) n 3×1 R
p pc (4 × 4) p 3×1 No doping 0.6
0.7
0.8 0.9 ΔE (eV)
1.1
1.2
1.3
(c) Se Zn Ga As Small R
Charge transfer direction
(b)
1
Large R
Figure 57 (a) Valence band offset, E, at heterovalent ZnSe/GaAs (001) interface, as a function of the composition ratio of R ¼ Zn/Se. (b,c) Charge transfer at anion and cation mixed interfaces. Reproduced with permission from Nicolini R, Vanzetti L, Mula G, et al. (1994) Local interface composition and band discontinuities in heterovalent heterostructures. Physical Review Letters 72: 294.
Atomic Structures and Electronic Properties of Semiconductor Interfaces
(wurtzite/zinc-blende) homomaterial twin interfaces are shown in Figure 56(b). This result indicates that not only the heteroatoms but also the topology of bond connection are the origins of offsets.
1.04.3.3
New Features of Schottky Barrier
1.04.3.3.1 limit
Breakdown of Schottky–Mott
Koyama et al. (2004) deposited Au, Pt, and Al metals on HfSiO substrate and measured their effective WFs, which results are displayed in Figure 58(a). They found that effective WFs of Au and Pt become large compared to WFs in vacuum, opposite to what happens with Al This result indicates that the S parameter becomes larger than 1, that is, beyond the conventional limit. A similar anomaly was observed in the XPS experiment by the Miyazaki group (Shiraishi et al., 2005). They deposited Au nanoscale dots on HfO2 substrate, measured the
Effective WF on HisiD (eV)
(a) 6
5 4.5
S > 1 !!
Al 4 4
4.5 5 5.5 Vacuum WF (eV)
(b)
6
biding energies of Au, and found that the binding energy increases as the size of dots and thus the interface area increases, as shown in Figure 58(b). This result clearly indicates that at Au/HfO2 interface the electrons transfer from Au to HfO2, which promotes the increase of Au effective WF. Similarly, by using the CV techniques, WF shifts were observed on HfAlOx substrate for Al metal with a decrease of 0.36 eV, and for Ni and Au metals with increases of 0.20 and 0.22 eV, respectively. The change of WFs from in-vacuum to on-high-kdielectrics is schematically summarized by arrows in Figure 58(c), for various metals. The Fermi energies of p-metals such as Ni and Au, which have larger WFs, are shifted toward the valence band of high-k dielectrics, whereas those of n-metals, such as Al, having smaller WFs move toward the conduction band. These WF changes are quite different from expectations deduced by the conventional CNL theories. The conventional theories state that there is a single CNL level in the band gap of a high-k dielectric and the metal Fermi energies are aligned to this CNL , as shown in Figure 51(c). Thus, the variation in Figure 58(c) was beyond our ordinary understanding. In order to clarify what happens at metal/high-k interfaces, Shiraishi et al. (2005) investigated the electronic structures using the first-principles calculations, and found that two important preconditions implicitly assumed in conventional theories are broken at these interfaces. The first is concerned with the penetration length of MIGSs from the interface into insulating materials. Figures 59(a) and 59(b) show the charge densities
(c) Au4f
Intens. (a.u.)
Ps
Au
5.5
157
CB
2 nm 3 nm
Interface (a)
Al Ni
W Ru
4 nm
Metal
Au
Si
EF 88 86 84 82 Binding energy (eV)
HfO2 VB
Figure 58 (a) Effective work functions of Al, Au, and Pt on HfSiON as a function of vacuum work function, reported by Koyama et al. (2004). (b) Observed binding energies of Au nanodots on HfO2 by XPS measurement (Shiraishi et al., 2005). The sizes of dots are described in the figure. (c) Schematic diagram of the observed Fermi-energy movement of metals on Hf-related high-k materials.
Metal (jellium)
HfO2
(b) Hf
O
Interface Figure 59 Contour plots of MIGS wave functions at (a) metal/Si and (b) metal/HfO2 interfaces (Shiraishi et al., 2005).
158 Atomic Structures and Electronic Properties of Semiconductor Interfaces
Energy (eV)
(a)
AI 25 20 15 10 5 0
EF
Γ
XK
(b) Energy (eV)
of typical MIGSs around metal/Si(111) and metal/ HfO2(110) interfaces, respectively. Here, the metal is represented by the jellium model. In the case of metal/Si, the MIGS penetrates deep into Si, about five atomic layers. Thus, the Si side of the interface has movable electrons and looks like a metal. The band alignment is realized by moving electrons in these states. Note here that, since the MIGS penetrates deep and touches a number of Si atoms around the interface, the detailed interface atomic structure does not govern the electronic structure of the interface. In other words, one can say that the full contact of electronic states is realized at this metal/Si interface. On the other hand, in case of metal/HfO2, the penetration of MIGS into HfO2 is seen to be at most one to two atomic layers, thus the full contact of electronic states not being realized at metal/HfO2. This occurs because of the high ionicity and large band gap of HfO2. The present result indicates that the electronic structure at metal/ high-k interface is very sensitive to the interface atomic structures, that is, to which atom contact is realized, and we had better start with the bonding picture of atoms to understand the electronic structure at this interface. The other precondition is related to the individuality of metals. Figures 60(a) and 60(b) show band structures of Al and Au metals around the respective Fermi energies, EF. The schematic pictures of DOS are also shown on the right. As the extended s- and p-orbital electrons are valence electrons, Al has featureless DOS around EF. On the other hand, due to the localized d-orbital electrons, Au has small-dispersion bands below EF. Thus, DOS is extremely large below the Fermi energy, whereas it is comparable to the Al case above the Fermi energy. As shown in the following, this kind of metal character is not considered in the conventional theory. The full contact of electronic states at the interface and the featureless metal DOS are essential conditions in conventional theories. This is apparent because CNL is defined using quantities intrinsic to bulk materials, such as EVB, EG, DVB, and DCB, and does not include the interface information, such as atomic structures and characteristics of metals. The same conditions are also assumed in the case of the theory of DIGSs (Hasegawa and Ohno, 1986); for example, the interface is assumed to have random amorphous-like structures, and some sort of averages that realize the full contact of electronic states are implicitly taken to deduce CNL . However, as shown
Γ
L
DOS
KWX
Au 40 35 30 25 20 15 10 5
DOS EF Γ
XK
Γ
L
KWX
Figure 60 Band structures of (a) Al and (b) Au bulk metals, calculated by the first-principles method. Schematic diagrams of DOS are displayed on the right. From Shiraishi K, Akasaka Y, Miyazaki S, et al. (2005) Universal theory of work functions at metal Hf-based high-k dielectrics interfaces – guiding principles for gate metal selection. In: Technical Digest of IEEE International Electron Devices Meeting, p. 29. Washington, DC, USA, December.
in the above, it is clear that these conditions are not satisfied at metal/high-k interfaces. 1.04.3.3.2 level
Generalized charge neutrality
In order to simulate the electronic structures of metal/high-k dielectrics interfaces, we have to take into account the interface atomic structures and characteristics of metals. To realize such interfaces, Shiraishi et al. (2005) adopted the effective-fourlevel tight-binding model of an interface. The schematic diagram of this model is described in Figure 61(a). They characterize a metal with the Fermi energy, EF, and the effective local DOS at the interface below and above EF, Docc and Dunocc. The high-k dielectric such as HfO2 is represented by the energies of valence-band top and conductionband bottom (EVB and ECB, respectively) and the effective local DOS of valence and conduction bands at the interface (respectively, DVB and DCB). Since HfO2 is an ionic material, the conduction bands are mainly made of Hf d-orbitals, while the valence bands are made of O p-orbitals. tM-Hf is a transfer energy between the occupied metal states and conduction-band states of HfO2, while tM-O is that
Atomic Structures and Electronic Properties of Semiconductor Interfaces
(a)
s. p
EF
Dunocc
ECB φGCNL
s. p. d
Docc
tM–O
Metal (b)
jtM-Hf j2 Docc DCB jtM-O j2 Dunocc DVB ¼0 – ECB þ V – EF EF – EVB – V
Hf 5d
DCB
tM–Hf
O 2p
HfO2 (nonmetal)
G CNL ¼ EVB þ EG
(c) s. p EF
Hf 5d
Hf 5d
s. p
φGCNL
φGCNL d
s. p
O 2p
Al
HfO2
EF O 2p Au, Ni
HfO2
Figure 61 (a) Schematic diagrams of interface hybridization models to derive the generalized neutrality levels at metal/nonmetal interfaces. EF, Doccu., and Dunoccu are Fermi energy and DOS variables for the metal below and above EF, while ECB, EVB, DCB, and DVB are the conduction- and valence-band edges and their DOS for the nonmetal. tM-Hf and tM-O are orbital hybridization energies between electron occupied and unoccupied states, which promote charge transfer across the interface. (b,c) Hybridization-induced charge transfer at Au/HfO2 and Al/HfO2 interfaces. Bold arrows denote major charge transfer.
between the unoccupied metal states and valenceband states of HfO2. It should be noted here that the charge transfer between a metal and HfO2 is generally realized only by such orbital hybridization between unoccupied and occupied states. Transfer energies between occupied states and those between unoccupied states never induce the charge transfer between the metal and HfO2, thus being not relevant to determine the band alignment and being excluded in the present model. By applying the second-order perturbation theory of quantum mechanics, the charge transfer from a metal to HfO2 is written as (Nakayama, 1993; Nakayama et al., 2006b) _
jtM-Hf j2 Docc DCB jtM-O j2 Dunocc DVB – ECB – EF EF – EVB
ð22Þ
By solving this equation, the generalized charge neutrality level of HfO2 that should match the Fermi energy of metal is obtained by G CNL ¼ EF – V as
EVB
DVB
159
ð21Þ
This charge transfer produces the dipole potential at the interface and increases the energies of EVB and ECB in HfO2. Since the charge transfer should be completed by inducing a final dipole potential of V, the self-consistent equation of V becomes as
jtM-O j2 Dunocc DVB jtM-O j Dunocc DVB þ jtM-Hf j2 Docc DCB ð23Þ 2
where EG ¼ ECB EVB is the band-gap energy of HfO2. In the case of ordinary metal/semiconductor interfaces, we can expect the full contact of electronic states, tM-Hf ¼ tM-O, and the DOS of the metal is featureless, Docc ¼ Dunocc. Thus, we can reproduce the conventional CNL as EVB þ EG DVB =ðDVB þ DCB ) (Cardona and Christensen, 1987), which is the quantity intrinsic to a bulk semiconductor. This is why we call the present G the generalized charge CNL neutrality level. Next, we explain how the new G theory CNL explains the unusual behavior of work functions at metal/high-k interfaces. As representative interfaces made of n and p metals, the Al/HfO2(110) and Ni/ HfO2(110) interfaces were investigated by the firstprinciples calculations. Figures 62(a) and 62(b), respectively, display the most stable adsorption positions of Al and Ni atoms on the HfO2 substrate, while Figure 62(c) shows the adsorption energies of Al and Au on HfO2 as a function of the adsorption position (Nakayama et al., 2006b). Adsorption positions, 1–7, are denoted on the right. It is clearly seen that Al atoms prefer to locate on oxygen atoms and produce the connection only with oxygen atoms. This occurs because the reactivity of Al with oxygen is high. As a result, one can reasonably approximate tM-Hf << tM-O. Meanwhile, the variation of Al DOS is monotonously continuous and featureless as shown in Figure 61(b), indicating that Docc ¼ Dunocc. Therefore, the charge transfer occurs from O to Al, as shown in Figure 61(b). As a result, G approaches the bottom CNL of HfO2 conduction bands, ECB, and the effective WF of Al decreases as shown in Figure 58(c). With respect to Au/HfO2, the Au atom appears to have no preference for the adsorption position and is expected to locate between Hf and O atoms on average on the HfO2 surface, thus indicating tM-Hf ¼ tM-O. However, as seen in Figure 60(b), the Au DOS below the Fermi energy is much larger than that above the Fermi energy. Thus, we can approximate
160 Atomic Structures and Electronic Properties of Semiconductor Interfaces
(b) 1.89Å
3.56Å
38 1.95Å
1.97Å
2.64Å
36 Energy (eV)
(a)
2.84Å
Al-adsorbed structure
CB
34 Al Au
32
Ni-adsorbed structure
VB 30
HfO2
Adsorption energy [eV]
(c) 0
2
Au
–0.5
4
–1 Al
1
–1.5 –2
1
top view
1
2
3 4 5 Atom position
6
2 7
3 6 5
0
Hf
7
Figure 62 Stable adsorption positions of metal atoms on HfO2 (110) substrate: (a) Al and (b) Ni atoms. (c) Calculated adsorption energies of Al (solid line) and Au (broken line) atoms on HfO2 (110) substrate, as a function of adsorption position. Adsorption positions are displayed on the right. Reproduced with permission from Nakayama T, Shiraishi K, Miyazaki S, et al. (2006) Physics of metal/high-k interfaces. ECS Transactions 3: 129–140.
Docc >> Dunocc. Therefore, the charge transfer occurs mainly from Au to HfO2 as shown in Figure 61(c), approaches the top of HfO2 valence bands, EVB,
G CNL and the effective WF of Au increases as shown in Figure 58(c). It might seem quite strange that the new G CNL theory predicts the charge transfer from O to Al at Al/HaO2 interface and from Au to Hf at Au/HfO2 interface because electronegativities of O and Au atoms are, respectively, much larger compared to Al and Hf. However, this really does occur! The most important point is the fact that O and Hf are not atoms but elements in bulk HfO2, thus the O and Hf atoms are fully ionized in HfO2 as O2 and Hf4þ. Thus, there is no space to receive additional electrons around O and holes around Hf. Instead, once the Al/HfO2 and Au/HfO2 interfaces are grown and the Al-O and Au-Hf connections are produced, O and Au atoms partially present electrons to Al and Hf atoms, respectively. The strictest examination of the generality of new G
CNL theory is to consider the extreme artificial cases; what happens when the Al atoms contact with not O but Hf atoms at Al/HfO2 interface. This is because the new theory argues the importance of interface atom coupling, thus the theory is justified by
4 5 6 3 Metal-atom position
7
∞
Figure 63 Calculated Fermi-energy positions of Al and Au monolayer metals relative to the conduction- and valenceband edges of HfO2, as a function of interface atomic position on the HfO2 (110) substrate. Atomic positions are displayed in Figure 62(c). 1 shows the Fermi-energy position realized at real metal/HfO2 interfaces, which are obtained for thicker metals and as an average of stable interface configurations in Figure 62(c).
studying the relation between the atom coupling and the WF value at the interface. Figure 63 shows the Fermi-energy positions of Al and Au calculated by the first-principles method when monolayer metal atoms are located at various positions of HfO2 interfaces (Nakayama et al., 2006b). It is clearly recognized that the Fermi-energy position depends on not only the metal kinds but also the metal-atom positions Especially, Fermi energies of both Al and Au are low when the metal atoms are located on Hf (position 1), while they become high when located over O (position 3). This common variation to both Al and theory. By Au clearly justifies the generalized G CNL analyzing the charge distribution, one can also check that such a variation of Fermi energy occurs, reflecting the charge transfer caused by the orbital hybridization at the interface. Other aspects of the universality of the new G CNL theory are observed by considering the band offsets, that is, band-edge discontinuity, at semiconductor/ semiconductor (S/S9) interfaces (Nakayama et al., 2006b). In the case of (S/S9) interface, the band alignment is determined by equalizing the following equation: V þ EVB þ EG
B A ¼ E9VB þ E9G AþB AþB
ð24Þ
where A ¼ |t9CB-VB|2DCBD9VB and B ¼ |t9VB2 CB| DVBD9CB are effective couplings at the interface, while V is the interface dipole. Various D’s are DOS of S and S9, while various E’s and t’s are, respectively, band-edge energies and electron transfer energies at
Atomic Structures and Electronic Properties of Semiconductor Interfaces
the interface, which all are defined as in Figure 61(a). This equation represents the balance of two charge neutrality levels (Nakayama, 1993). Finally, we remark on the universality of the new
G concept. G is constructed by using the repreCNL CNL sentative physical quantities of the interface. The energies EF, EVB, and ECB and DOS values Docc, Dunocc, DVB, and DCB represent properties intrinsic to bulk materials originating from bulk band structures, while the transfer energies, tM-Hf and tM-O, are introduced to describe the microscopic atomic structures of interfaces, that is, the interface bonds. This fact indicates that both band and bond pictures, which, respectively, correspond to itinerant and localized characters of electrons, are necessary to describe electronic structures at the interface. In this way, the first-principles theoretical approaches have played key roles in creating the new science concepts such as G in recent Si nanotechnology, CNL and they are further leading not only the Si nanotechnology but also the frontier fields of nanoscience such as organic semiconductors and nano-bio devices (Oda and Nakayama, 2008). 1.04.3.3.3 Interface reaction and Fermi-level pinning
We first introduce an example where the thermodynamics of interface reaction determine the SBH. An unexpected SBH behavior at the interface between heavily B doped Si (pþpoly-Si) and HfO2. has been reported. Hobbs et al. (2003) reported that the difference in SBH between nþpoly-Si and pþpoly-Si becomes only 0.2 eV, although their intrinsic Fermilevel difference amount to 1 eV reflecting the Si band gap (Figure 64). This is called ‘Fermi level pinning’ (FLP) in LSI jargon. Since HfO2 is a typical wide-gap
0.2 eV Ferml level of n + poly Si
0.2 eV
Ferml level of p + poly Si
0.6 eV
Si conduction band
Si valence band On SiO2
On HfO2
Figure 64 Schematic illustration of the unusual behaviors of the relative Schottky barrier heights (SBHs) at the pþpolySi/HfO2 interface compared with that at the nþpoly-Si/HfO2 interface. For comparison, the well-known SBH behavior at the poly-Si/SiO2 interfaces is also shown on the left.
161
insulator (band gap is about 5.6 eV), SBH behavior is expected to be similar to the Schottky limit according to the conventional CNL concepts. However, the observed behavior rather resembles the Bardeen limit. The physical origin of this unusual behavior is in the mechanism of the SBH formation being much different from the conventional mechanism. This is governed by the thermodynamics of the interface reaction. The physical mechanism is as follows. The relatively higher energy level of oxygen vacancy (Vo) in HfO2 causes a notable thermodynamic behavior of interfaces, when HfO2 is in contact with Si. Recent experiments indicate that the Vo level is located about 0.4 eV above the bottom of Si conduction band (about 1.2 eV below the bottom of HfO2 conduction band) (Takeuchi et al., 2004). It is well known that Hf atoms can bind much more strongly than Si atoms to O atoms. Actually, the formation enthalpy of HfO2 is larger than that of SiO2 by about 2.2 eV. Further more, recent firstprinciples calculations show that the Vo formation energy in bulk HfO2 is larger than that in bulk SiO2 by about 1.2 eV (Scopel et al., 2004). At first glance, the fact that Hf–O bonds are stronger than SiO bonds indicates that the partial oxidation of poly-Si gates by pulling an O atom out of the HfO2 dielectrics is an endothermic reaction with 1.2 eV energy loss, and this reaction occurs with difficulty. However, the situation changes completely if we take into account the electron behavior. FLP in Hf-related high-k gate stacks with pþpoly Si gates can naturally be explained by taking into account the electron behavior as follows. In Figures 65(a)– 65(c), the mechanism of Vo formation in HfO2 and subsequent electron transfer across the poly-Si/HfO2 interface is schematically illustrated. First, let us assume that the poly-Si is partially oxidized by the formation of SiOSi bonds by O atoms being pulled out of HfO2. As a result, SiOSi bonds in the poly-Si and Vos in HfO2 are formed (Figure 65(a)). An O atom in HfO2 takes an O2 ion form, but an O atom in a SiOSi bond is neutral. Accordingly, two additional electrons are generated after one O atom is pulled out, and if these electrons remain inside HfO2, they occupy the Vo level in HfO2 (Figure 65(b)). The assumption that two additional electrons remain in HfO2 corresponds to the same situation as the bulk calculations that give a 1.2 eV energy loss. However, since HfO2 is in contact with the poly-Si gate, electrons have to transfer into the gate. This is because the Vo level is located above the poly-Si Fermi level (Figure 65(c)). Now, we estimate roughly the two ultimate cases of energy loss (gain)
162 Atomic Structures and Electronic Properties of Semiconductor Interfaces
(a)
(b)
0.4 eV Ec
E(Vo)
(c)
0.4 eV Ec
1.1 eV
E(Vo)
0.4 eV Ec
1.1 eV
Ev
1.1 eV HfO2
HfO2 Ev Poly-Sigate
Poly-Sigate
Vo
SiO2
Vo
SiO2
+1.2 eV
E(Vo)
Ev
HfO2 Poly-Sigate
Vo
SiO2
+0.4 eV = (1.2–2×0.4)
–1.8 eV = (1.2–2×1.5)
Figure 65 Schematic illustrations of Vo formation in HfO2 with partial oxidation of poly-Si gate and subsequent electron transfer into the gate electrodes. (a) Partial poly-Si oxidation by pulling out an O atom from HfO2. (b) Energy loss in an nþpolySi gate. (c) Large energy gain in a pþpoly-Si gate. Reproduced with permission from Shiraishi K, Yamada K, Torii K, et al. (2006) Oxygen-vacancy-induced threshold voltage shifts in Hf-related high-k gate stacks. Thin Solid Films 508: 305.
when HfO2 is in contact with the poly-Si gate: one is with an nþpoly-Si gate and the other is with a pþpolySi gate. For an nþpoly-Si gate, the two-electron transfer results in an energy gain of 0.8 eV (2 0.4 eV), and the total energy loss is reduced from 1.2 eV (bulk value) to 0.4 eV. Despite the energy reduction due to the electron transfer, the reaction for an nþpoly-Si gate is still endothermic. For a pþpoly-Si gate, on the other hand, the situation is quite different. The Fermi level position of a pþpoly-Si gate is located about 1.5 eV below the Vo level in HfO2. As a result, the twoelectron transfer results in a total energy gain of 1.8 eV (2 1.5 eV 1.2 eV). Surprisingly, the interface reaction, accompanied by Vo formation and subsequent electron transfer, becomes exothermic with an energy gain of 1.8 eV, when the HfO2 is in contact with a pþpoly-Si gate. Actually, TEM observations show that some interfacial reaction layers are observed in a pþpoly-Si gate HfAlOx MISFET, as shown in Figure 66, and such reaction layers have not been observed in nþpoly-Si gate MISFETs. These results
p + poly -Si
Si substrate
HfAIOx
IL 20 nm
Figure 66 Cross section of replacemnt pþgate HfAlOx MISFET observed by TEM. Reproduced with permission from Shiraishi K, Yamada K, Torii K, et al. (2006) Oxygenvacancy-induced threshold voltage shifts in Hf-related highk gate stacks. Thin Solid Films 508: 305.
completely corroborate the above discussion based on the ‘oxygen vacancy model’. Now, we move on the Vfb shift originating from the formation of Vo in HfO2. For an nþpoly-Si gate, the interface reaction that induces the electron transfer occurs with difficulty, since it is endothermic. For a pþpoly-Si gate, however, the interface reaction accompanied by the formation of Vo in HfO2 occurs easily, since this reaction has a large energy gain of 1.8 eV. At the same time, electrons transfer from HfO2 into the poly Si occurs. As a result, an interface dipole is formed as illustrated in Figure 67. This dipole formation raises the position of the Fermi level of the pþpoly-Si gate and flat band voltage (Vfb) decreases. It is noticeable that the energy gain of the interface reaction decreases when the Fermi level is elevated. A simple consideration indicates that the position at which the Fermi level is pinned corresponds to the energy level that makes the energy gain of the interfacial reaction zero. The final Fermi level position satisfies the equation 1.2 (0.4 þ x) ¼ 0, where x is the final pinning position measured from the nþpoly-Si Fermi level. The Fermi level position of a pþpoly-Si gate obtained from this is about 0.2 eV below the nþpoly-Si gate Fermi level, which is in fairly good agreement with the experimental results (Hobbs et al., 2003). As discussed above, ‘oxygen vacancy model’ can quantitatively reproduce the Fermi level pining position of pþpoly-Si gate/HfO2 interfaces. Next, we comment on the effect of inserting cap insulator between pþpoly-Si and HfO2 gate dielectrics. Recent experiments show that the FLP of pþpoly-Si gates cannot essentially be improved by inserting SiO2 or SiN cap layers between pþpoly-Si gates and high-k Hf-related dielectrics (Cartier et al.,
Atomic Structures and Electronic Properties of Semiconductor Interfaces
(a)
163
(b)
Si CB 1.1 eV
0.4 eV
Vo
EC
0.4 eV
Si VB (EF)
x eV 0.2 eV
Interface dipole
– –
EV
1.1–x eV
Pinning level HfO2
Poly-Si gate Poly-Si gate
Vo
HfO2
SiO2
Cap layer
Vo + +
SiO2 Figure 67 (a) Schematic illustration of interface dipole formation and subsequent Fermi-level elevation toward the pinning level. (b) Schematic illustration of the cap layer effect for Fermi-level pinning (FLP). Reproduced with permission from Shiraishi K, Yamada K, Torii K, et al. (2006) Oxygen-vacancy-induced threshold voltage shifts in Hf-related high-k gate stacks. Thin Solid Films 508: 305.
2004). The oxygen vacancy model naturally explains the results of these experiments. The schematic illustrations are shown in Figure 67. As shown in this figure, the pinning position is governed only by the energy position that balances the Vo formation energy loss and the electron transfer energy gain. The final pinning position measured from the nþpoly-Si Fermi level (x) satisfies the equation 1.2˜ (0.4 þ x) ¼ 0, which is the same equation without a cap layer. Accordingly, if O atoms can penetrate through the cap layer until the system reaches thermal equilibrium, the final pinning position remains the same, regardless of the existence of a cap layer. Systematic experiments have been reported by Kamimuta et al. (2005). They examined three geometries of poly-Si/HfO2 gate stacks. The schematic illustrations are shown in Figure 68. The usual gate stack structures in which FLP is observed in pþpolySi gates are structures with no barrier layer and thin IL as shown in Figure 68(a). At first, the effect of boron segregated near the Si/HfO2 interfaces (Takayanagi et al., 2003) or interfacial Hf–Si bonds (Hobbs et al., 2003) were thought to be the cause of FLP of p-poly-Si gates. However, FLP cannot be avoided even if a barrier layer is inserted between a pþpoly-Si gate and HfO2, as described in Figure 68(b), indicating that neither B effect nor Hf–Si bonds can be the cause of FLP at pþpoly-Si/ HfO2 interfaces. They have found that FLP relaxation can be achieved only when the gate stack structure contains both thick barrier layer and thick
IL, as illustrated in Figure 68(c). Their experimental finding indicates that interaction between HfO2 and a Si substrate is very important for FLP as well as that between poly-Si gate and HfO2. In other words, FLP disappears only when the interaction between Si and HfO2 is weak enough. Further, it is expected that indirect interaction between Si and HfO2 can induce FLP. In short, the thermal equilibrium between Vo formation and annihilation reaction at the Si/HfO2 interface given by the following equation determines the SBH (FLP position) at pþpoly-Si/HfO2 interface, as shown in Figure 69: ðHfO2 Þ þ Si
1.04.3.3.4
! SiO2 þ HfO2 þ Vo 2þ þ 2e
ð25Þ
Correlation between interfaces It has been reported that SBHs of p-metals decrease remarkably, revealing FLP behavior when IL is thin, after high-temperature annealing. However, FLP does not appear when IL is thick enough (Lee et al., 2006). This is called ‘Vfb roll-off’. The noticeable fact is that the energy position of FLP is similar to the pinning position of pþpoly-Si gates (Hobbs et al., 2003), as mentioned in previous section. This experimental fact indicates that FLP of p-metal is also governed by a mechanism similar to FLP of pþpoly-Si gates mentioned above (Akasaka et al., 2006). Now, we consider the mechanism of FLP of p-metal gates subjected to a high-temperature treatment. Since usual p-metals are nonreactive
164 Atomic Structures and Electronic Properties of Semiconductor Interfaces
e– transfer
O transfer
(a) Occur
Occur FLP
Vo
2+
Poly-Si
Occur Very small p+ poly-Si2
HfO
Si sub.
(b) FLP
Not occur
Occur Vo2+
Occur
Poly-Si with barrier layer
Very small p+ poly
Barrier
HfO2
Si sub.
(c) FLP relaxation
Not occur
Not occur
Vo2+
Occur
Poly-Si with barrier layer and thick IL
Very small Barrier
p+ poly
Si sub.
HfO2
Figure 68 Schematic illustration of three poly-Si/high-k gate stack structures examined by Kamimuta et al. (2005). (a) Usual pþpoly-Si gate stack structure. (b) Gate stack structure with barrier layer between a pþpoly-Si gate and a high-k dielectric. (c) Gate stack structure in which barrier layers are inserted both between a pþpoly-Si gate and a high-k dielectric and between a Si substrate and a high-k dielectric. Reproduced with permission from Akasaka Y, Nakamura G, Shiraishi K, et al. (2006) Modified oxygen vacancy induced fermi level pinning model extendable to P-metal pinning. Japanese Journal of Applied Physics 45: L1289.
(a) EC
EV
(b)
Electron transfer
EC
E(Vo) + +
E(p+)
p+polySi-gate SiO2 O transfer
E(Vo) E(p+)
EV p+polySi-gate
HfO2 Interface dipole
Interface dipole
Thermal equilibrium
Vo – –
SiO2 Vo annihilation EV(HfO2)
Pinned (EF = EV = E(p+))
HfO2 Electron transfer Vo – –
Pinned (another aspect)
Figure 69 Schematic illustration of another understanding of Fermi level pinning (FLP). (a) Vo generation and (b) Vo annihilation are balanced, and the system reaches thermal equilibrium.
materials, the situation of p-metal/high-k gate stack can be schematically illustrated as in Figure 70. O transfer hardly occurs from high-k dielectrics to p-metals due to low reactivity of p-metals. However, O transfer is still possible if IL is thin (Figure 70(a)). Transfer of O from high-k
dielectrics is inhibited when IL is thick enough (Figure 70(b)). As discussed in Figure 68, the interface reaction between high-k dielectric and Si substrate still occurs when IL is thin. Furthermore, reaction with the Si substrate induces electron transfer from Vos to p-metal gates. A
Atomic Structures and Electronic Properties of Semiconductor Interfaces
e– transfer
(a)
165
O transfer
Not occur
Occur
FLP
2+
Vo Occur
p-Metal with thin IL
Very small p-metal
HfO2
Si sub.
(b)
FLP relaxation
Not occur
Occur p-Metal with thick IL
Vo2+ Occur Very small p-metal
HfO2
Si sub.
Figure 70 Schematic illustration of two typical p-metal/high-k gate stack structures. (a) A p-metal/high-k gate stack structure with thin IL. (b) A p-metal/high-k gate stack structure with thick IL. Reproduced with permission from Akasaka Y, Nakamura G, Shiraishi K, et al. (2006) Modified oxygen vacancy induced fermi level pinning model extendable to P-metal pinning. Japanese Journal of Applied Physics 45: L1289.
Figure 71, the pinning position corresponds to the energy at which the energy loss (G1) and energy gain (G2) by electron transfer from Vo in high-k to a gate metal are canceled by each other (i.e., G1 G2 ¼ 0). This means that the FLP positions of p-metal gates are the same as that of the pþpolySi gate, irrespective of the metal species. In fact, our observed EWFs of p-metals are almost independent of metal species, as shown by the C–V curves in Figure 72. It is better to note that the Vo-related mechanism is not the only cause of Vfb shift. The Vo-related
schematic illustration of this situation is given in Figure 71. It is noticeable that the net reaction between high-k dielectrics and the Si substrates is the same as that between poly-Si gates and high-k dielectrics, although electron transfer and O transfer directions are opposite to each other. In fact, the reaction equation with the Si substrate can be described as ðHfO2 Þ þ Si
! SiO2 þ HfO2 þ Vo 2þ þ 2e
ð26Þ
which is the same as in the case of pþpoly-Si gates described in the previous subsection. According to
(a) Vo EF Metal
Semiconductor (Si)
HfO2
Reaction with Si sub. – G1
O
SiO2
(b) EF
EF elevation
Vo Energy gain by electron transfer G2
Metal HfO2
Semiconductor (Si) Reaction with Si sub. – G1
SiO2 Figure 71 Schematic illustration of interface reaction with the Si substrate and subsequent electron transfer from Vo to gate metals in p-metal/high-k gate stacks. (a) O transfer into the Si substrate through thin IL. (b) Subsequent electron transfer from a Vo level to a p-metal gate which induces gate Fermi-level elevation.
166 Atomic Structures and Electronic Properties of Semiconductor Interfaces
C (F cm–2)
2×10–6
WF(Ru) = 4.7 eV WF(Ir) = 4.63 eV
WF(TiN) = 4.72 eV
1×10–6 Ru Ir TiN 0 –3
–2
1000 °C spike –1
0 Vg (V)
1
3
2
Figure 72 Observed C–V curves of Ru, Ir, and TiN. The estimated effective work functions are similar to those of Fermi-level pinning (FLP) position of the pþpoly-Si gate. Reproduced with permission from Akasaka Y, Nakamura G, Shiraishi K, et al. (2006) Modified oxygen vacancy induced fermi level pinning model extendable to P-metal pinning. Japanese Journal of Applied Physics 45: L1289.
mechanism determines the final position of FLP. For example, if other factors such as surface strain (Ikeda et al., 2006) or MIGSs lower the EWFs of gate metals, relatively less Vo generation is sufficient to reach the FLP position. The thermodynamics of the interfacial reaction between high-k dielectrics and the Si substrates that generates the Vo and Vo-induced interface dipole determines the final position of FLP, when the system can reach thermal equilibrium. Accordingly, the pinning position is independent of the process condition and the film quality. This is the main concept of ‘oxygen vacancy model’. As discussed above, FLP of p-metals naturally occurs if IL is thin, since the reaction between high-k dielectrics and the Si substrate is inevitable. However,
(a) EF
EF elevation
Vo Energy gain by electron transfer G2
Metal HfSiON
Semiconductor (Si)
Reaction with Si sub.
(b) EF
FLP can be avoided if the reaction between high-k dielectrics and the Si substrate is suppressed. In order to suppress the corresponding reaction, there are two possibilities. One is insertion of thick IL, and the other is the low temperature process. The former one leads to a remarkable increase in the effective oxide thickness (EOT). Thus, it is not suitable for use in future LSI technologies. It is naturally expected that the latter (a low-temperature process) is a promising solution to avoid FLP if we use Hf-based high-k gate dielectrics. It is also noted that the interface dipole modulation between high-k dielectrics and Si substrates is also effective, since this modulation does not change the thermodynamics of interface reactions that generate O vacancies. In other words, the relative energy difference between the neutral Vo level and the FLP position of a gate metal does not change as a result of this dipole modulation. F incorporation (Inoue et al., 2005) or counter-doping effects are categorized in this recipe which modulates the dipole at IL/Si interfaces. It has also been proposed that Al and La incorporation into Hfrelated oxides can modulate the dipole at high-k/ IL interfaces; this is also included in this recipe (Iwamoto et al., 2007). Now, we discuss the experiments that confirm the validity of the oxygen-vacancy model. After the high-temperature treatment, which causes FLP of p-metals, the Si substrate is removed. Next, oxygen atoms are injected from the substrate side into Hfbased high-k dielectrics at room-temperature ozone treatment (Ohta et al., 2006). The schematic illustration of the experiments is given in Figure 73. The results obtained are shown in Figure 74. As clearly shown in this figure, EWF of TiN increases with the
– G1
Vo EF elevation HfSiON
O injection by ozone at RT
Metal
Figure 73 Schematic illustration of the experimental procedure that confirms our Vo model. (a) FLP occurs after a hightemperature treatment. (b) After the removal of Si substrates, ozone is injected into Hf-based high-k dielectrics at room temperature.
Atomic Structures and Electronic Properties of Semiconductor Interfaces
1.04.4 Future Prospects
4.5
TiN work function (eV)
4.6 Ralative Hf4f position
0.1
4.7
On SiO2
0.2
4.8
Relative Hf 4f position (eV)
0
On HfSiON 0.3 4.9
167
0
400 200 UV-O3 oxidation time (s)
600
Figure 74 Observed TiN effective work function and relative Hf4f position as functions of UV-ozone oxidation times. Experiments are based on XPS measurements. From Ohta A, Miyazaki S, Akasaka Y, et al. (2006) Extended Abstracts of 2006 International Workshop on Dielectric Thin Films for Future ULSI Devices – Science and Technology, p. 61. Kawasaki, Japan, November.
ozone injection time, and it reaches the original position releasing FLP. Thus, these experimental results clearly indicate that O deficiency is the main cause of FLP of TiN. It is also noticeable that the quantity of the Hf core level shift does not match the total increase in EWF of TiN. This means that Vo generation in Hf-related high-k dielectrics is not the only cause of Fermi level shift of TiN gates. Other factors also contribute to the shift of EWF. It is consistent with the oxygenvacancy model that the thermodynamics of Vo generation determines the final position of FLP, although other factors also contribute to the Vfb shift (Ohta et al., 2006). As discussed above, the oxygen-vacancy model can naturally explain the FLP of p-metals observed when IL is very thin, and our model is experimentally confirmed. Finally, we mention a new finding of physics of Schottky barriers. In the above mechanism of Schottky barrier formation, SBH at a metal/HfO2 interface is determined by the thermodynamics of the reaction at the other interface at HfO2/Si, instead of the corresponding metal/HfO2 interfaces. This is completely different from the conventional understanding of SBH that it is determined by the dipole at the corresponding interfaces.
To conclude this chapter, we mention some of our self-opinionated prospects for future studies of interfaces. Owing to the rapid progress in controlling the fabrication of organic systems and our keen interest in biosystems such as proteins (Oda et al., 2008), understanding organic interfaces acquires intensive investigations. The cohesion mechanisms in organic semiconductors are quite different from those in inorganic semiconductors. They mainly consist of P-conjugated highly covalent bonds, fluctuating hydrogen bonding, and very weak van der Waals interactions. Therefore, when we study the formation, stability, and electronic properties of organic interfaces, we can expect the need to apply quite different new physical concepts. For example, it has been well known that the Schottky barriers at metal/ organic semiconductor interfaces often show time evolution due to the quasi-equilibrium nature of interfaces (Ishii et al., 1999). It rapidly becomes indispensable to investigate what really happens. Other advances are expected in the field of electric chemistry, where the liquid/solid interfaces are the main stages for chemical reactions such as material synthesis and decomposition, and catalyst phenomena. Wetting under the nonequilibrium conditions is also included (Kajita et al., 2007). The effects of electric field in liquids and the behavior of water molecules are also keys to understand the interface (Otani and Sugino, 2006; Akagi, et al., 2004). Not only the microscopic elucidation of these interfaces but also the development of universal pictures of interface reactions are greatly expected. Next, we discuss a new technique to improve the Schottky-barrier stability at oxide interfaces. The key is to use multivalent materials such as Ce. As discussed previously, excess or deficiency of oxygen atoms results in an unexpected interface behavior, such as the instability of SBHs, which can sensitively depend on the ambient O chemical potential. Therefore, stabilizing the chemical potential of oxygen throughout the process is the key technology for obtaining reliable interface properties. It is known that Ce can take multivalent states of 3þ and 4þ, with The corresponding Ce oxides Ce2O3 and CeO2, respectively. If Ce2O3 and CeO2 coexist in a capping Ce-oxide layer, as shown in Figure 75, the chemical potential of O can be fixed to the intrinsic value determined by the following equation, irrespective of process conditions (Kouda et al., 2009):
168 Atomic Structures and Electronic Properties of Semiconductor Interfaces
Oxygen chemical potential
Reducing process CeO2 Ce2O3 CeO2
Oxidizing process CeO2 Ce2O3
CeO2 Ce2O3
Ce2O3 Release
Vo2+ High-k
O
Absorb Vo2+
O 2– i
O
High-k
O 2– i
μo Stable flat and voltage
CeO2 → Ce2O3 + O V2– o +O
μo is kept constant with Ce-oxide capping Ce2O3 + O → CeO2 – I2– o –2e → O
Figure 75 Schematic illustrations of the recipe for fixing the chemical potential of oxygen during reduction and oxidation processes by using a capping layer of multivalent oxides. In the reduction process, CeO2 supplies O atoms into the oxide layer. On the other hand, Ce2O3 absorbs O atoms from the oxide layer in oxidizing ambient. From Kouda M, Umezawa N, Kakushima K, et al. (2009) Charged defects reduction in gate insulator with multivalent materials. In: Digest of Technical, 2009 Symposium of VLSI Technology, p. 200. Kyoto, Japan, 18 June 2009.
ðOÞ ¼ 2ðCeO2 Þ – ðCe2 O3 Þ
ð27Þ
Actually, by using the multivalent oxide capping layer technique, we can experimentally succeed in reducing the O-related charged defects, such as the O vacancy and interstitials. Multivalent materials have been studied only as strongly correlated electron systems in pure physics for a long time. However, these materials are now expected as key materials to synthesize stable and reliable oxide interfaces in the technological world. Finally, we discuss other examples of our interest: the interfaces made of two different-dimensional systems. Electron tunneling through such interfaces, for example, would provide new interface physics. Although electron tunneling between differentdimensional systems commonly occurs in real situations, there are few reports on this subject. Tunneling from two-dimensional to zero-dimensional systems is a typical phenomenon. For example, tunneling currents through a dielectric barrier via zero-dimensional quantum-dots (QDs) or defects have been studied extensively from both scientific and technological viewpoints (Torii et al., 2004; Meirav et al., 1990; Takahashi et al., 1996; Austing et al., 1996; Sasaki et al., 2000). However, the geometrical matching of wave functions between initial and final states has not been considered so far, although it should be a very important factor in tunneling phenomena between different-dimensional systems, as
schematically illustrated in Figure 76(a) (Sakurai et al., 2010). As shown in the figure, electron tunneling occurs only when the electron wave functions are sufficiently wave-packet-like just below the zerodimensional sites caused by the thermal fluctuation. Thus, unexpected temperature dependence might occur in the tunneling phenomena between different-dimensional systems, although conventional direct-tunneling phenomena should lack temperature dependence. Sakurai et al. prepared a sample in which sub-10nm-diameter Si QDs were weakly coupled to a twodimensional electron gas through a 3.5-nm-thick SiO2 barrier layer. The electron injection currents from the 2DEG to Si QDs were measured as functions of the gate voltage (VG). Figure 76(b) shows the observed displacement electron currents as functions of VG and temperature (T). Surprisingly, the gate voltages necessary for the electron injection from the 2DEG to Si QDs had a clear temperature dependence, although the electron injection currents through a sufficiently thin 3.5-nm SiO2 barrier layer have conventionally been described using a temperature-independent direct-tunneling scheme. The observed gate voltages necessary for the electron injection markedly changed from 3.7 to 2.2 V as the temperature increased from 120 to 240 K. This unexpected temperature dependence can be qualitatively reproduced on the basis of the phenomenological
Atomic Structures and Electronic Properties of Semiconductor Interfaces
y x
(b) 100
Temp
eratur
e (K)
A) Current (p
80 60 40 20 0 120 140 160
180
200
220 240 1
2 3 Gate voltage (V)
4
Figure 76 (a) Schematic illustration of electron tunneling from a two-dimensional system to a zero-dimensional system. If the geometrical matching of the electron wave functions is not satisfied, electron tunneling from a twodimensional to a zero-dimensional system does not occur (left). Electron tunneling can occur only when geometrical matching of the wave functions is achieved (right). (b) Obtained displacement currents (I) measured at different temperatures and the temperature dependence of the gate voltage necessary for electron injection obtained by experiments. At each temperature, the gate voltage was swept with a 10 mV step between 4 and 4 V, corresponding to the effective gate voltage between 2 and 6 V, at a sweep rate of 63 mV s1. Reproduced with permission from Sakurai Y, Iwata J, Muraguchi M, et al. (2010) Temperature dependence of electron tunneling between two dimensional electron gas and Si quantum dots. Japanese Journal of Applied Physics 49(1): 014001.
Molecular bridge
Step-pulse voltage
20 1 10
Electron number
z
quantum-point contacts (QPCs), and QDs, has recently attracted intensive investigations. From these studies, interesting physics, such as the quantization of conductance, the Coulomb blockade, and Kondo effects, has been revealed. However, most of these are concerned with steady-state properties and physical properties of nanoscale systems themselves. On the other hand, these nanoscale systems are also interesting from the viewpoint of the interface. In the case of molecular bridges, for example, the molecule is sandwiched and connected to metal electrodes by a limited number of atomic bonds. The molecule is sometimes a zero- or one-dimensional system and has a small number of freedoms, while the electrode is a two- or three-dimensional system and has a larger number of freedoms. Therefore, the contact between molecule and electrodes is also a typical interface between two systems with different dimensions. Figure 77 shows the transient current behavior of a molecular bridge (or QPC or QD) system. Such an experiment has become possible recently (Naser et al., 2006). One can see the relaxation of current, which reflects the dissipation of energy and entropy (information entropy) from low- to high-dimensional
Left-contact current (a.u.)
Electron density
(a)
169
v = 0.2 Electrodes 0.1
0
v = 0.2
assumption that sufficiently wave-packet-like wave functions, which satisfy ‘geometrical matching’ between different-dimensional systems, can contribute to electron tunneling of these systems. Other interest is concerned with the dynamics at the interface made of two different-dimensional systems because these systems have different degree of freedom. As discussed in other chapters in this publication, electronic transport through nanoscale systems, such as molecular bridges, semiconductor
0.1 0 0
20
40 Time, t (a.u.)
60
Figure 77 Transient current behavior at nanocontact molecular bridge (quantum point contact) system. Reproduced with permission from Tomita Y, Ishii H, and Nakayama T (2009) Transient current behavior through molecular bridge systems: Effects of intra-molecule current on quantum relaxation and oscillation. e-Journal of Surface Science and Nanotechnology 7: 606.
170 Atomic Structures and Electronic Properties of Semiconductor Interfaces
systems. This occurs because the energy and entropy have the general tendency to move from small-freedom system to large-freedom system, which phenomena are often called the friction and the loss of information in physics. In this way, such interfaces provide a new stage for studying quantum friction (Ishii et al., 2008; Tomita et al., 2009), in addition to the conventional representative friction systems (Caldeira and Leggett, 1985). On the other hand, we can also see the oscillation of current in Figure 77. This occurs due to the quantum motion of electrons between the molecule and electrode, the period of which reflects the difference of Fermi energy in the electrode and the energy level in the molecule. In other words, we can observe the electronic properties of the electrode in the current through the molecule. In this way, not only the electronic properties of the molecule (or QPC or QD) but also the interaction between the molecule and the electrode through the interface becomes important in the dynamics of nanoscale objects. Another example of such quantum friction is seen in the attenuation behavior of microscopic molecular vibration (Shigeno and Nakayama, 2007). (See Chapter 1.01).
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Yamaguchi H, Belk JG, Zhang XM, et al. (1997) Atomic-scale imaging of strain relaxation via misfit dislocations in highly mismatched semiconductor heteroepitaxy: InAs/ GaAs(111)A. Physical Review B 55: 1337. Yasuda T, Kuo LH, Kimura K, et al. (1996) In situ Characterization of ZuSe/GaAs(100) interfaces by reflectance difference spectroscopy. Journal of Vacuum Science and Technology B 14: 3052. Yasuda T, Kumagai N, Nishizawa M, Yamasaki S, Oheda H, and Yamabe K (2003) Layer-resolved kinetics of Si oxidation investigated using the reflectance difference oscillation method. Physical Review B 67: 195338. Yasuda T, Yamasaki S, Nishizawa M, et al. (2001) Optical anisotropy of oxidized Si(001) surfaces and its oscillation in the layer-by-layer oxidation process. Physical Review Letters 87: 037403. Zhang SB, Hybertsen MS, Cohen ML, Louie SG, and Tomanek D (1989) Quasiparticle band gaps for ultrathin GaAs/AlAs(001) superlattices. Physical Review Letters 63: 1495.
Further Reading Breuer H-P and Petruccione F (2002) The Theory of Open Quantum Systems. Oxford University Press. Cappaso F and Marrgaritondo G (1987) Henetrojunction Band Discontinuities: Physics and Device Applications. North-Holland. Edward TY, Mccaldin JO, and Mcgill TC (1992) Band offset an in semiconductor heterojunctions. In: Ehrenreich H and Turnbull D (eds.) Solid State Physics, vol. 46, pp. 2–147. San Diego, CA: Academic Press. Gao D and Wei SH (eds.) (1993) II–VI Semiconductor Compounds. World Scientific. Glastone S, Laidler KJ, and Eyring H (1964) The Theory of Rate Processes. New York: McGraw-Hill. Kamimura H and Toyozawa Y (eds.) (1983) Recent Topics in Semiconductor Physics. Singapore: World Scientific. Kangawa Y, Ito T, Taguchi A, Shiraishi K, Irisawa T, and Ohachi T (2002) Applied Surface Science 190: 517–520. Moench W (2001) Semiconductor Surfaces and Interfaces. Berlin: Springer. Ogawa T and Kanemitsu Y (eds.) (1995) Optical Properties of Low-Dimensional Materials. Singapore: World Scientific. Salaneck WR, Seki K, Kahn A, and Pireaux J-J (eds.) (2002) Conjugated Polymer and Molecular Interfaces. Marcel Dekker. Shiraishi K, Yamada K, Torii K, et al. (2004) Oxygen vacancy induced substantial threshold voltage shifts in the Hf-based high-K MISFET with pþpoly-Si gates – a theoretical approach. Japanese Journal of Applied Physics 43: L1413. Sze SM and Kwok KNg (2007) Physics of Semiconductor Devices. Wiley. Tersoff J and Harrison WA (1987) Transition-metal impurities in semiconductors – their connection with band lineups and Schottky barriers. Physical Review Letters 58: 2367. Weiss U (2008) Quantum Dissipative Systems. World Scientific. You JH and Johnson HT (2009) Effect of dislocations on electrical and optical properties in GaAs and GaN. In: Ehrenreich H and Spaepen F (eds.) Solid State Physics, vol. 61, pp. 144–261. New York: Academic Press.
1.05 Integer Quantum Hall Effect H Aoki, University of Tokyo, Tokyo, Japan ª 2011 Elsevier B.V. All rights reserved.
1.05.1 1.05.1.1 1.05.1.2 1.05.1.3 1.05.2 1.05.2.1 1.05.2.2 1.05.2.3 1.05.3 1.05.3.1 1.05.3.2 1.05.3.3 1.05.3.4 1.05.3.5 1.05.4 1.05.4.1 1.05.4.2 1.05.4.3 1.05.5 1.05.6 1.05.7 1.05.8 1.05.9 1.05.9.1 1.05.9.2 1.05.9.3 1.05.10 1.05.11 1.05.11.1 1.05.11.2 1.05.11.3 References
Introduction 2D Electron Gas 2DEG in Strong Magnetic Fields – Classical Mechanics 2DEG in Strong Magnetic Fields – Quantum Mechanics Integer QHE – Experiments Materials and Sample Geometry Optical Properties Other Properties IQHE – Theories Localization in Landau Levels Linear-Response Theory Strˇeda–Widom Formula Gauge Argument Topological Arguments Localization problem Scaling Theory of Localization in 2D Systems Quantum Criticality and xx xy Diagram Fractal Wave Functions and Dynamical Scaling QHE Edge States and Edge Transport Real-Space Imaging QHE Resistance Standard and the Fine-Structure Constant Breakdown of QHE Quantum-Dot and Periodically Modulated Systems in Strong Magnetic Fields Quantum-Dots in Magnetic Fields Hofstadter Spectrum QHE in Three Dimensions Integer versus Fractional QHEs Recent Developments and Related Phenomena QHE in Oxides QHE in Graphene Anomalous Hall Effect and Spin Hall Effect
1.05.1 Introduction Quantum Hall effect (QHE) is undoubtedly one of the most fascinating and important phenomena not only in the semiconductor physics, but, more generally, in the condensed matter physics. This can be immediately realized if one notes that the quantum Hall physics encompasses fundamental physics – topological phenomenon in terms of the quantum field theory – down to applicational physics – as exemplified by the QHE as the resistance standard.
175 176 177 178 181 181 183 183 184 184 186 188 188 189 189 189 190 192 193 195 196 197 197 197 199 199 200 202 202 203 206 208
To start with, QHE comes from quantum mechanical physics in two spatial dimensions (2D) as opposed to the 3D space in which electrons usually dwell. Surprisingly, 2D is a special dimension in which there exist phenomena specific to 2D, as known in field theories, and QHE is the most remarkable one. In this sense 2D is definitely not just a reduced dimensionality. This accounts for the remarkable width and depth of the physics of QHE, which has now become as large a field as those for superconductivity/superfluidity. 175
176 Integer Quantum Hall Effect
1. Energy spectrum. A completely discrete, line spectrum (i.e., Landau levels) arises in the clean limit, which is most unusual, since the system is a bulk. This gives a starting point for the integer QHE. When an integer number of Landau levels are fully filled, then we can regard the system as a giant closed shell. 2. Transport properties. The closed shell is not an ordinary one, since the quantized Hall conductivity (integer times e2/h) is given entirely in terms of physical constants (e : elementary charge, h : Planck’s constant).
(a)
D
M
2D Electron Gas
As a background we should start with describing the two-dimensional electron gas (2DEG). The usual electron gas is a system of electrons that move more or less freely in a continuous space, as typically realized in simple metals. In semiconductor physics, we can realize 2DEG in metal-oxide-semiconductor field-effect transistors (MOSFETs; Figure 1(a)) mainly before c. 1970s, and subsequently QHE is observed primarily in semiconductor heterostructures (Figure 1(b)) (Ando et al., 1982). In these structures, electrons are confined to a 2D plane, due to a Schottky barrier between the metal and the oxide in an MOSFET, or between different semiconductors in a heterostructure. There, an electron moves in the 2D plane, with the electronic structure of the constituent materials entering only through the effective mass in the effective-mass approximation. The wave function has a finite thickness in the direction perpendicular to the plane, but the motion along this direction is quantized, so that
S
O
A
S VG c.b.
EF v.b.
Distance from interface(z) (b)
In all these, noncommutative x and y coordinates are relevant. Namely, in magnetic fields, the position coordinate in real space and the wave number k for a charged particle are mixed. On top of this, there is a fortuitous coincidence of the Hall conductivity with a topological invariant. 1.05.1.1
p - Si
Metal SiO2
Energy
QHE consists basically of the integer QHE discovered in 1980, which is essentially a one-body problem (but see Section 1.05.10), and the fractional QHE discovered in 1983, which is a many-body effect. Here, we concentrate on the integer QHE. Even so, the field is so vast that here we shall describe the bare essentials. If we just summarize the peculiar properties of the QHE system,
Alx Ga1–x As
GaAs
Si doped
Electrode
AlGaAs c.b.
GaAs c.b. EF
~100Å Figure 1 Structures of MOSFET (a) and semiconductor heterostructure (b). In each frame, the upper panel shows the sample structure, while the lower panel the electronic band structure around the interface. Typical wave functions are plotted in black against the direction perpendicular to the interface.
the band structure comprises 2D electronic bands associated with the quantized levels in the normal direction, which are called subbands. When only the lowest subband is occupied by electrons (or, more precisely, if the transitions between adjacent subbands
Integer Quantum Hall Effect
can be neglected), we can regard the motion genuinly 2D. The Schro¨dinger equation reads H
¼
1 2 p þ U ðzÞ 2m
¼E
ð1Þ
where m is the effective mass of the electron, p is the momentum in the 2D plane (x, y), and U(z) is the confining (Schottky) potential. If we ignore disorder (interface roughness, impurities, etc.), the wave function is expressed as ¼ exp iðkx x þ ky yÞ fn ðzÞ
1.05.1.2 2DEG in Strong Magnetic Fields – Classical Mechanics If we apply a magnetic field, B, normal to a 2DEG, classically an electron undergoes a circular motion (called Larmor’s motion) due to the Lorentz force, ev B (e: elementary charge, v: velocity of the electron). When there is an external electric field, E, the classical orbit becomes a trochoid (Figure 2(a)), where the center of the circular motion drifts in a direction perpendicular to E with a drift velocity cE B/B 2 (c: speed of light). This is because the electron is accelerated (decelerated) when it moves along (against) E, while the Lorentz force (_ v) is always balanced with the centrifugal force (_ the radius of the circular motion), so that the trajectory is elongated in the accelerated part while contracted in the decelerated (a)
z
part. The resulting drift of the center coordinate accounts for the classical Hall effect. When the system is disordered, due to, for example, impurities, then an electron is scattered, and the drift velocity acquires a component along E (Figure 2(b)). If we define the conductivity, the quantity is a tensor in the presence of a magnetic field, where the current, j, and E are related as 0 1 0 10 1 xx xy Ex jx @ A¼@ A@ A jy yx yy Ey 0 1 0 10 1 jx Ex xx xy @ A¼@ A@ A Ey yx yy jy
ð2Þ
up to a normalization constant, where the 2D motion reduces to plane waves and fn is the nth quantized wave function along z. In semiconductor heterostructures, typically GaAs/AlGaAs grown with the molecular beam epitaxy (MBE), the Fermi energy is EF 10 meV, so that the electron system is a degenerate Fermi gas at liquid He temperature (0.4 meV).
B
ð3Þ
where m is the conductivity tensor and m the resistivity tensor. They are inverse matrices with each other, so that we have, with xx = yy, yx = -xy, 0
0 1 xx – xy 1 @ A¼@ A ¼ @ A 2xx þ 2xy yx yy – xy xx xy xx 0 1 0 1 xx xy xx – xy 1 @ A¼ @ A 2 þ 2 xx xy yx yy xy xx xx xy
1
0
xx xy
1–1
ð4Þ
If we introduce a phenomenological relaxation time, 0, in zero magnetic field to describe the scattering in a classical transport theory with the equation of motion given by m(d/dt þ 1/ 0)v ¼ e(E þ v B), then we have 0 1 þ !2c 02 0 !c 0 nec xx þ xy ¼ ¼ – B 1 þ !2c 02 !c 0 xx ¼
ð5Þ
where 0 ¼ ne2 0/m is the conductivity in the absence of magnetic fields, and !c ¼ eB=m c (b)
y
177
z
y
B (X, Y ) Ex
x
x Ex
Figure 2 Classical orbits for a charged particle in a magnetic field (k z) in an applied electric field (k x) for a clean system (a) and in a disordered system with scatterers (crosses) (b).
178 Integer Quantum Hall Effect
(a)
(b)
Rxy
(c)
H′
S
S
B
D H
Rxx
S
2D
EG
S
D
D
y D x
Figure 3 (a) Sample geometry for measuring the QHE. (b) Top views of a Hall bar and a Corbino sample, with source (S) and drain (D) electrodes. (c) Equipotential lines in the QHE condition.
the cyclotron frequency. When !c 0 >>» 1 (which is required for an electron to accomplish the Larmor motion between scattering events), the leading term, xy nec/B, on the right-hand side of Equation (5) is the main term (the classical Hall conductivity in the clean case) while the second term a small correction. Thus, we have a voltage (Hall voltage) in the direction perpendicular to the electric field when the sample has open boundaries in that direction, or we can measure the Hall current if electrodes are attached. Figure 3 depicts the sample geometry.
1.05.1.3 2DEG in Strong Magnetic Fields – Quantum Mechanics If we go to quantum mechanics, an electron is still subject to a circular motion in the correspondence principle, but now with a quantum mechanical uncertainty. We should start from a Hamiltonian, H
0
¼
1 2 2m
ð6Þ
where the momentum p is now replaced with the canonical momentum p ¼ p þ (e/c)A(r) with A being the vector potential representing the magnetic field B ¼ rot A. If we now decompose the cyclotron motion into the center R X (X, Y) of the circular motion (guiding center) and the relative coordinate x X (, ), the latter is related to the velocity as v ¼ !ceˆz x where eˆz stands for a unit vector normal to the 2D plane. The presence of a magnetic field renders a skew (i.e., vector product) relation between x and v. We can now apply the correspondence principle, where we adopt a quantum mechanical expression, v ¼ ði=hÞ½H 0 ; r ¼ p=m , for the velocity. This means we have x ¼ (c/eB)eˆz p, that is,
ð; Þ ¼
,2 y ; – x h
ð7Þ
Here, rffiffiffiffiffi ch ,X eB
ð8Þ
is the length scale of the cyclotron motion, called the magnetic length, which does not depend on material parameters and has a typical value of 81 A for the magnetic field of 10 T. Thus, the relative coordinate, (, ), is a quantum mechanical operator, which implies that the center coordinate, ðX ;Y Þ ¼ ðx – ð,2 =hÞy ;y þ ð,2 =hÞx Þ; is a quantum mechanical operator as well. From the standard commutation relation, ½x; px ¼ y; py ¼ ih, we have commutation relations, ½; ¼ – i,2 ;
½X ; Y ¼ i,2
ð9Þ
namely, x coordinate does not commute with y coordinate, which implies an uncertainty , between the components of the relative coordinate. The same applies to the center coordinate. This is rather unusual, since ordinarily it is the momentum with which the real-space coordinate does not commute. Quantum mechanical states can be derived algebraically. For this we first note that the commutation relations for (X, Y) and (, ) enable us to introduce two sets of harmonic-oscillator operators, , 1 a ¼ pffiffiffi x – iy ¼ – pffiffiffi ð þ iÞ 2h 2, 1 b ¼ pffiffiffi ðX þ iY Þ 2,
ð10Þ
which have bosonic commutation relations, y y a; a ¼ b; b ¼ 1
ð11Þ
Integer Quantum Hall Effect
Then the one-particle Hamiltonian, H 0 for the clean system, which is quadratic in p, can be expressed as H
0
1 ¼ h!c a y a þ 2
ð12Þ
in terms of the operator a only, which is natural since b involving (X, Y) should not appear in a translationally invariant system. Since the Hamiltonian has the same form as a linear harmonic oscillator, the energy eigenvalues are
1 EN ¼ h!c N þ ; N ¼ 0; 1; 2; . . . 2
ð13Þ
where N is called the Landau index. So we have here a truly abnormal situation where the energy spectrum, despite the system being a bulk, is completely discrete (Figure 4(a)). In 3D systems we do have Landau’s quantization, but the extra motion along B makes the density of states a continuum (Figure 4(a)). In 2D each level, called the Landau level as labeled by Landau index, has then a macroscopic degeneracy. The degeneracy can be estimated by noting that the density of states of a 2DEG, which is a constant, DðE Þ ¼ m =2h2 per unit area, integrated over an interval h!c should correspond to the degeneracy per unit area: n ¼ h!c DðEÞ ¼ 1=ð2,2 Þ
ð14Þ
This number can be expressed as n ¼ B= 0
179
where 0 X ch/e ¼ 4 107G cm2 is the flux quantum, so that n amounts to the number of flux quanta penetrating the unit area. Alternatively, we can say that the total degeneracy, S2/2,2, is of the order of the number of cyclotron orbits that cover the sample area S. To see this, we can look at the wave functions. We first note that the operator b is related
with the orbit 1 center as X 2 þ Y 2 ¼ 2,2 b y b þ . The formula2 tion so far does not depend on the gauge (i.e., how we fix the vector potential A which has an ambiguity related with the gauge transformation). The wave function does depend on the gauge. Let us adopt 1 the symmetric gauge, A ¼ B r. This amounts to 2 taking, out of the degenerate wave functions, the set that diagonalizes, simultaneously with the energy, the angular momentum, L ¼ r p, which is along z when the motion is within the (x,y) plane. We can show that Lˆz ¼ ½r ðp – ðe=cÞAz ¼ a y a – b y b
ð16Þ
with h = 1. For the N ¼ 0 Landau level we have a harmonic-oscillator form for Lˆz ¼ byb, so that the eigenfunction having the eigenvalue of Lz ¼ m is given by y m b jmi ¼ pffiffiffi j0i; m ¼ 0;1;2; . . . m!
where j0i is the vacuum of boson b. If we go to the first-quantized form the wave function is expressed, in polar coordinates (r, ), as Nm ðrÞ_
ð15Þ
(a)
jmj exp – im – r 2 =4,2 r m LN ðr 2 =2,2 Þ
ð17Þ
(b)
Density of states
3D 2
0
0 2D
1 N=0 1
2
hωc
0
0
1 hωc 2
3 hωc 2
5 hωc 2
7 hωc 2
9 hωc 2
N=0
E Figure 4 (a) Density of states for a clean system in the absence (dashed lines) and in the presence (solid lines) of a magnetic field in two-dimensional (lower panel) or in three-dimensional (upper) systems. (b) Wave functions for a 2D system for various values of the Landau index N.
180 Integer Quantum Hall Effect ðmÞ
where LN ðzÞ is associated Laguerre polynomial. They are depicted in Figure 4(b). By restricting the radius R of these wave functions within a radius of a disk, we recover the degeneracy of a Landau level. In terms of the degeneracy, we can now define the Landau level filling factor, that is, the fraction of the occupied states for a given Landau level. If we denote the density of electrons by ne, the Landau level filling factor is X ne =n ¼ 2,2 ne
ð18Þ
In other words, 1/ is the number of flux quanta per electron. This is the essence of Landau’s quantization, formulated by Landau in 1930, and the discovery of the QHE in 1980 coincided with its half centenary. When only the lowest (or lowest few) Landau level(s) are occupied (i.e., . 1) for strong enough magnetic fields, the situation is called the quantum limit. For GaAs with a typical density of electrons n 1011 cm2, we have ¼
4 n 1011 cm – 2 B ðTÞ
where B is in units of tesla and ne in 1011 cm2, so that the quantum limit is realized for B & 4 T. So far we have considered a clean system. In the presence of disorder, such as a random potential V(r) arising from impurities and interface roughness, cyclotron orbits that have different center coordinates (X, Y) are no longer degenerate, but are subject to scattering. Then the equation of motion for (X, Y) may be obtained from its commutator with H as
i ,2 qV ½V ;X ¼ X_ ¼ h h qy
i ,2 qV Y_ ¼ ½V ;Y ¼ – h qx h
1 y N þ h!c cNX cNX 2 NM X y þ cN 9X 9 hNX jV jN 9X 9icNX
ð19Þ
X
ð20Þ
NXN 9X 9
cyNX
=ðh!c Þ 1=ð!c 0 Þ1=2
ð21Þ
ðni V02 m =h3 Þ – 1
The fact that R_ _ eˆ z rV implies that R X (X,Y) moves, classically, along equipotential contours with a velocity proportional to jrVj. Quantum mechanically, however, the Hamiltonian, with the Landau wave functions as a basis, reads H ¼
mechanical hopping between cyclotron orbits at different positions. The hopping matrix element hNXjVjN9X9i becomes large when the random potential varies rapidly in space (on the magnetic length scale, Equation (8), since each function jNXi has a spatial extension ,). In the presence of the hopping, each Landau level is broadened from the line spectrum. Electronic structure and transport properties of the disordered, Landau-quantized systems in 2D were theoretically studied with the self-consistent Born approximation in the 1970s by Uemura, Ando and co-workers (Ando et al., 1974, 1974b, 1982). In this approximation, the lifetime of an electronic state due to disorder is taken into account in a self-consistent way, which is imperative, since the unperturbed system has anomalous, delta-function spectra. Let us assume that the random potential V(r) ¼ i 2,2V0 (r ri) is expressed as a sum of the contributions from shortrange (delta-function like) impurity potentials at position ri, with the average number of impurities per unit area ni. For dense impurities (to be more precise, for the dimensionless concentration ci X 2 ,2ni 1), we can adopt the Born approximation. There, the self-energy due to the impurity scattering is ðEÞ ¼ ci V02 GðEÞ, where the self-consistency demands that Green’s function, G(E), contains the effect of (E). Then each Landau level is broadened pffiffiffi with a width, ¼ 2 ci V0 : In other words, the ratio between the Landaulevel broadening and the cyclotron energy is
where creates an electron in the Landau’s wave function jNXi (here in the Landau gauge (A ¼ (0, Bx)) rather than in the symmetric gauge), and the second term on the right-hand side represents the quantum
where 0 ¼ is the scattering relaxation time due to the disorder in zero magnetic field. Intuitively, the quantity !c 0 gives a measure of how many times an electron can rotate on a cyclotron orbit between scattering events on average. For sufficiently larger magnetic field and/or smaller disorder we have =h!c < 1 ðor !c 0 > 1Þ, for which Landau levels are separated, while Landau levels are merged in the opposite condition. To give an idea about the magnitudes of relevant quantities, we have h!c ¼ 0:12 B ðTÞðm0 =m ÞmeV pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:15 B ðTÞ=
where B is measured in units of tesla, the effective mass m ¼ 0.067m0 (m0: bare mass of an electron) for GaAs and ¼ e 0/m is the carrier mobility here measured in units of 104 cm2 V1 s1.
Integer Quantum Hall Effect
1.05.2 Integer QHE – Experiments The Landau quantization in 2DEG described above is reflected in various properties, especially in transport properties. There are prehistories for the the QHE physics that are precursors of the discovery. As early as in the 1960s, Shubnikov de Haas effects were observed in Si-MOSFETs by Landwehr’s group, and by groups in US and in Japan. Shubnikov de Haas effect is an oscillatory (in B) transport phenomenon that generally occurs in magnetic fields, but the effect in 2DEG is peculiar, since the oscillations in xx and xy reflect the line-like spectrum Landau levels. In the 1970s more elaborate experimental studies did exhibit oscillatory behaviors indicative of Landau levels in 2D. Figure 5 shows a typical example (Igarashi et al., 1975; Kawaji and Wakabayashi, 1981; Kawaji, 2008), along with a theoretical result. It was also recognized that there are regions of vanishing xx and flat xy between Landau levels, which were called plateaux. In 1980, von Klitzing, Dorda, and Pepper found an astonishing behavior in the Hall conductivity as shown in Figure 6 (von Klitzing et al., 1980). In a Si-MOSFET in strong magnetic fields, the heights of
–σxy
nec H
10
Theory
σxx , –σxy (10–4Ω –1)
8
6 7 6 4
5 3
2
σxx
4
N=2 1 0
0 0
20
40
60
the Hall resistance Rxy are quantized very accurately into integer multiples of h/e2 ¼ 25 813, or when translated to the Hall conductivity as xy ¼ – N
80
VG(V) Figure 5 A typical experimental result (solid lines) along with a theoretical result (dashed lines) for xx and xy against VG (see Figure 1) which is roughly _ n in a MOSFET sample. From T. Igarashi, J. Wakabayashi and S. Kawaji, Suppl. Progr. Theoret. phys. 57, 176 (1975).
e2 ; N ¼ integer h
ð22Þ
Astonishing, because (1) the quantized value only contains the fundamental physical constants, the elementary charge e and Planck’s constant h (whereas ordinarily transport properties are naturally affected by various material parameters, etc.), and (2) this occurs in disordered systems. This has become known as the quantum Hall effect. A few years after this discovery the fractional QHE was discovered, so the original effect is sometimes called the integer quantum Hall effect (IQHE). Remarkably, the accuracy of the quantization is experimentally confirmed to be better than 107. Hall effect was discovered by Edwin Hall in 1879, so it was almost exactly a century later when the remarkable Hall effect was discovered. Subsequently, experimental data were refined, where the quantum Hall steps are almost a series of step functions, as typically shown in Figure 7, which were then interpreted in terms of the localization, as we shall see in the section on localization.
1.05.2.1
Experiment
181
Materials and Sample Geometry
Historically, the IQHE was discovered in SiMOSFETs, but subsequently studied for 2DEGs in semiconductor heterostructures, typically GaAs/ AlGaAs (Figure 1). Integer QHE has also been observed in heterostructures other than GaAs/ AlGaAs, which include 2D hole (as opposed to electron) gas systems in p-type GaAs, type III heterostructures such as GaSb–InAs where 2D electron and 2D hole gases coexist side by side, and Si/ Si1xGex strained heterostructures with different band structure and higher effective mass than those in Si MOSFETs. There are basically two sample geometries – Hall bar geometry and Corbino geometry (Figure 3(b)). The former is usually adopted with a multiterminal geometry. In the latter, where electrodes are attached to the inner and outer perimeters of an annular sample, an advantage is that we do not have to worry about the sample edges and edge transports, nor do we need to worry about hot spots (Figure 3(c)); see the Section 1.05.6. Namely, in the QHE condition (xy : integer e2/h, xx ¼ 0 with the
Rxx /Ω
182 Integer Quantum Hall Effect 400 200 0
23.0
23.5
24.0
24.5
VG/V
Uxx/mV RH /Ω
UH/mV
6500
25 2.5 6400 B = 13.0 T T = 1.8 K 6300
20 2.0 6200
23.0
23.5
24.0
24.5
VG/V
15 1.5 Uxx 10 1.0
5 0.5
0
0
UH
5 N= 0
10
15
20 N= 2 VG/V
N= 1
25
Figure 6 The original QHE result. For the Hall (UH) and longitudinal (Uxx) voltages against the gate voltage VG. Inset shows the detail around the fully occupied N = 0 Landau level. Reprinted figure with permission from K. von Klitzing, G. Dorda and M. Pepper, Physical Review Letters 45, 494 (1980). Copyright 2010 by the American Physical Society. For information, see: http://link.aps.org/
14 000 12 000
GaAs-AlxGa1–xAs T = 50 mK
ρxy (Ω)
10 000 8000 6000 4000 2000 0
ρxx (Ω)
300 200
5 6 3 7 4
2
100 2 0 0
20
40 B (kG)
1
N=1 60
80
Figure 7 A typical result for IQHE in GaAs-AlxGa1 xAs. Reprinted figure with permission from M. A. Paalanen et al., Physical Review B 25, 5566 (1982). Copyright 2010 by the American Physical Society. For information, see: http://link.aps.org/
Integer Quantum Hall Effect
1.05.2.2
Cyclotron absorption
EF Energy
Hall angle (¼ tan1 xy/xx) of 90 ), the Hall current flows around the annulus. QHE has also been observed in Corbino geometry. In fact, some of the early Shubnikov de Haas experiments were done for this geometry. An electron has a spin 1/2, and the associated Zeeman energy in a magnetic field. The Zeeman energy is much smaller than the cyclotron energy in usual experimental situations (although quantitatively the spin splitting is enhanced due to exchange interactions), so that every Landau level is almost twofold spin-degenerated. Later, the effects of the spin splitting on QHE have been elaborated with various experimental techniques. In the case of Si-MOSFET, a valley degeneracy also exists, since the bulk Si has multiple valleys in the band structure.
Inelastic light scattering
Optical Properties
QHE systems exhibit characteristic optical properties as well. Optical measurements have an advantage that (1) they are a contactless method and (2) we can probe both luminescence and absorption. In the QHE regime luminescence spectra have been obtained for Si MOSFETs and semiconductor heterostructures. The luminescence spectra, which measure the radiative recombination of 2D electrons with photoexcited holes, basically reflect the Landau-level structure. One feature is that the Landau-level width as deduced from the width of the luminescence spectra oscillates with magnetic field with a maximum every time a Landau level is filled. This is associated with the screening which becomes effective when the Landau levels are partially filled (i.e., an open shell). There are other properties such as spin-dependent relaxation. Cyclotron absorption is another important experimental technique in the QHE regime. The technique can probe lower carrier density regions than in transport measurements. Landau-level structures have been obtained, where oscillatory linewidths, spin splitting, nonparabolicity effects, etc, have been observed. Inelastic light scattering is another powerful optical method that compliments luminescence. These methods are schematically illustrated in Figure 8. Most recently, the optical Hall conductivity xy(!) is theoretically predicted to have a Hall plateau structure in the THz regime in quantum Hall systems. Although the plateau height is no longer
183
~ ~
Luminescence
Density of states Figure 8 Various optical processes are schematically shown.
quantized in ac, the structure remains robut against disorder reflecting the localization in QHE regime. The effect has subsequently been experimentally detected with Faraday rotation, with its magnitude characterized by the fine-structure constant (Ikebe et al, 2010).
1.05.2.3
Other Properties
There are host of other properties that have been measured. One is the electronic specific heat, which probes the Landau quantization through the density of states (that include both localized and delocalized states), while transport measurements mainly probe the delocalized states. Another method to probe the density of states is the magnetocapacitance (Figure 9). Magnetization has also been measured, with torque magnetometers or micromechanical cantilevers, where jumps are observed as EF traverses the Landau levels (Figure 10). Various other properties have been experimentally studied, among which is the thermoelectric
184 Integer Quantum Hall Effect
Filling factor
Capacitance (pF)
14
12
10
8
200
180
1.4
1.0
1.8
2.2
Magnetic field (T) Figure 9 Experimental (solid line) and theoretical (dashed) results for the capacitance against magnetic field. Landau-level filling is indicated on the upper axis. Reprinted figure with permission from T. P. Smith et al., Physical Review B 32, 2696 (1985). Copyright 2010 by the American Physical Society. For information, see: http://link.aps.org/
ΔT, Hall resistivity (a.u.)
87 6 5
Filling factor 4 3
flow (or a temperature gradient) that is perpendicular to both the electric current and the magnetic field, known as the Nernst–Ettingshausen effect. Namely, the electric field E and the temperature gradient rT are related as E ¼ S rT. Here, S is the thermopower tensor, where Sxx corresponds to the thermopower, while Sxy to the Nernst–Ettingshausen coefficient (Figure 11) (Fletcher, 1999). The Ettingshausen effect has been studied both theoretically and experimentally in conjunction with the breakdown of QHE as probed by the temperature change rT, in the heat-pulse method.
2
ρxy
1.05.3 IQHE – Theories 1.05.3.1
0
1
2
3 4 5 Magnetic field (T)
6
7
8
Figure 10 Experimental (thick line) and theoretical (thin) results for the magnetization, along with xy against magnetic field. From E. Gornik et al., Physical Review Letters 54, 1820 (1985).
effect, which probes how the electrons sustain temperature gradients in 2DEGs, where the dominant mechanism for the thermopower is phonon drag. Another magneto-thermoelectric effect is a heat
Localization in Landau Levels
Let us start with the localization problem in the QHE system, since this has to do with a first essential question about the integer QHE – the presence of plateaux in xy plotted against the density of electrons n (or vs. B _ 1/ for a fixed n). Experimentally, the plateau in xy goes hand in hand with a region of vanishing xx. This is an interesting situation, since, usually xy should be an increasing function of n and xx as a function of n should not be zero (even when there is a gap in the density of states). This has been explained from the Anderson localization of the wave functions in the Landau levels that arises from disorder in the system. The Anderson localization was proposed back in 1958 for general
Integer Quantum Hall Effect
185
50
Thermopower (μVK–1)
40
–Sxx
30 20 10 0 –10 –Sxy
–20 –30 –40 0.0
1.0
2.0
3.0
4.0
Chemical potential Figure 11 A typical theoretical result for the diagonal (Sxx) and off-diagonal (Sxy) components of the thermopower tensor against chemical potential. Reprinted figure with permission from M. Jonson and S. M. Girvin, Physical Review B 29, 1939 (1984). Copyright 2010 by the American Physical Society. For information, see: http://link.aps.org/
electron systems in the presence of disorder, and was subsequently culminated as the scaling theory of localization in 1979. In a clean crystal, every wave function is a Bloch state that extends over the entire sample. In a disordered system, wave functions can be spatially localized, whose spatial extension is characterized by the localization length that depends on energy. In the presence of magnetic fields, the Anderson localization was suggested to occur as well for wave functions in the Landau levels in the 1970s (Figure 12). For random potentials slowly varying in space, an electron follows, semiclassically, an orbit that is an
equipotential contour as we have shown above. Quantum mechanically, however, an electron tunnel between these orbits even for a slowly varying potential, while the quantum mechanical hopping becomes more frequent for rapidly varying (i.e., short-ranged) random potentials (see Figure 13). So the localization is a quantum mechanical effect. Then plateaux in xy and regions of vanishing xx have been suggested to come from the localization. A peculiarity in the localization in the QHE system is that the disorder has a dual role: disorder causes the localization on the one hand, but it gives rise to quantum mechanical hopping
(a) ψ 2 y | | x (b) Localization length (c) D(E ) (d) E a
bc
d
e
(e)
Figure 12 Typical wave functions in a disordered QHE system obtained in a computer simulation for various eigenenergies as indicated on the right panel, which shows the density of states and the localization length against energy. From H. Aoki, Journal of Physics C 10, 2583 (1977).
186 Integer Quantum Hall Effect
(a)
(b)
l
(c)
Figure 13 A random potential rapidly varying as compared with the magnetic length l (left panel) and a slowly varying one (rigth) are schematically shown. (c) A typical contour plot with shaded areas indicating negative energies and thick lines E ¼ 0 contour. From H. Aoki, Reports on Progress in Physics 50, 655 (1987).
of cyclotron guiding centers to contribute to transport at the same time, as pointed out by Aoki and Kamimura (1989).
1.05.3.2
Linear-Response Theory
The most standard method to treat conductivities microscopically is the linear-response theory, or the Kubo formula. Aoki and Ando (1981, 1993) have applied this to the QHE problem. While the plateaux in xy and vanishing regions in xx are explained by the localization, the puzzle remains as to why the value of the plateaux in disordered systems is quantized into the universal e2/h, which is originally the value for the clean 2DEG with just an integer Landau-level filling. In other words, if a disorder makes xx zero, why does the disorder allow xy to stick to the quantized value. In the linear-response theory, the conductivity is given by a current–current correlation function. In the QHE system, the current has to do with the dynamics of the cyclotron guiding center (X, Y), which is subject to quantum mechanical hopping. The conductivities, both longitudinal and Hall, are given as correlation functions of X_ and Y_ as xx ¼ e 2 X_ X_ ne xy ¼ – þ xy B e _ _ _ _ YX – XY 2
ð23Þ
ðBÞ ¼ ð – B Þ
so that we can decompose into the symmetric part s and the antisymmetric part a. Then the current j is expressed as j ¼
X
s E þ axy ðE eˆz Þ
ð25Þ
where the part of the current (i.e., the Hall current) that is induced by the magnetic field and is perpendicular to the applied electric field E is related to a (Landau et al., 1984). The linear-response formula may be written in terms of Green’s function as xy ¼
e 2 h iL2 *
Z
1
dEf ðEÞ –1
"
q ReGðEÞY_ ImGðEÞ – X_ $ Y_ Tr X_ qE
#+
ð26Þ –1
2
xy ¼
Z Z 1 1 dt expð – t Þ dhAð – ihÞ L2 0 0 Bðt Þi; where hi is the canonical ensemble average plus the average over disorder, ¼ 1/kBT, a positive infinitesimal, and A(t) the Heisenberg representation of A. xy is expressed as a combination hhY_ X_ iihhX_ Y_ ii for the following reason. For the conductivity tensor in a magnetic field B, Onsager’s reciprocal theorem dictates that Here hhAB ii X
ð24Þ
where G ðE Þ ¼ ðE – H þ i Þ is the Green’s function, f(E) ¼ [e(E m) þ 1]1 is the Fermi’s
Integer Quantum Hall Effect
distribution function, and $ denotes the term with X_ and Y_ exchanged. In this expression, the Hall conductivity xy is contributed by all the states below EF, unlike xx that is related to an energy-dissipating process around EF. In terms of the eigenstates of the Hamiltonian, the Hall conductivity is expressed as
ihe 2 X f ð" Þ – f ð" Þ jx jy – jy jx xy ð!Þ ¼ 2 " – " L 6¼ " – " þ i
! ð27Þ
implies that, (ii) at least in the limit of strong magnetic fields all the states cannot be localized, otherwise xy would be zero over the whole region. This is important, since, according to the scaling theory of localization all the states are localized in two dimensions. The only way to go around this universal argument is to go to another universality class, either to the unitary class (e.g., systems in magnetic fields where the time-reversal symmetry is broken) or to the symplectic class (e.g., systems that have spin–orbit interactions). QHE system belongs to the unitary class, which is why delocalized states are allowed to exist, which in turn makes the QHE to be realized. The theoretical picture explained here for broadend Landau levels, xx and xy are summarized in Figure 15. As we shall see below, the quantization of xy into e2/h can be shown for finite magnetic fields, where the required condition is EF being in a gap in the density of states or in a localized regime (mobility gap). Even in the treatment above, however, different Landau levels are not treated as being completely independent. As seen from the fact that the Landau’s quantization is formally equivalent to a 1D harmonic oscillator, the current operator has matrix elements
D(E)
Density of states
where " is -th eigenenergy, and j the current matrix elements between the eigenstates. From this formalism, we can show, in the presence of localization arising from disorder, that (i) the Hall conductivity xy should be rigorously flat (at T ¼ 0) as a function of the density n of electrons in the region where the states are localized. This accounts for the flatness of plateaux in the QHE, as depicted in Figure 14. We can also show that the quantized xy ¼ Ne2/h in a plateau should hold in the limit of strong magnetic fields where adjacent Landau levels are well separated and when EF is in the gap between them. This in turn
N=0 N=1 Inverse loc length
N=2
σxx
σxx
0
187
0
0
–σxy
nec H
0
–σxy(e 2/h)
3
e 2 = ntotalec h H
2 1 0 0
1
0
2πl 2n Figure 14 Theoretical density of states D(E) (with shaded areas indicating the localized states), the longitudinal conductivity xx, and the Hall conductivity xy are schematically plotted against the Landau-level filling 2,2n. The horizontal dashed line indicates how the quantized value is achieved despite the presence of localization.
1
2
3
E/hωc Figure 15 Theoretical density of states (with shaded areas indicating the localized states) along with the inverse localization length, xx, and xy are schematically plotted against energy for a series of Landau levels. Each white region representing the delocalized states actually has zero width in the thermodynamic limit at T ¼ 0. From H. Aoki and T. Ando, Solid State Communications 38, 1079 (1981).
188 Integer Quantum Hall Effect
between N and N 1 levels, so the mixing between them is implicitly included (in fact, the treatment in terms of the guiding center (X, Y) corresponds to taking this mixing to the leading order). It may first seem counterintuitive that xy attains the quantized value despite the presence of localized states. Physically, we can say the following. From the equation of motion for the guiding center the Hall current jx in an electric field Ey is given as jx ¼ – ð1=B Þ
delocalized X
h 9jqV =qy j 9i þ eEy
9
P There is a kind of sum rule, all
9 h 9j V =qy j 9i ¼ 0; which means that delocalized states move faster to just compensate the localized states, which has been confirmed numerically and field theoretically as well. 1.05.3.3
Strˇeda–Widom Formula
As another formalism for the Hall conductivity, Strˇeda’s formula (Strˇeda, 1982) is also often evoked. He showed that the Kubo formula for the Hall conductivity can be decomposed into a form, xy ¼ Ixy þ IIxy ; where Ixy is the Drude-like part that tends to Ixy ! – !c 0 xx in the relaxation time picture. The second term is expressed as IIxy ¼ ec
qN ðEÞ qB
ð28Þ E¼EF
where N(E) is the integrated density of states. When the Fermi energy EF is in an energy gap, Ixy vanishes, while we can show that IIxy ¼ – ðe 2=hÞN when EF is in a gap between the Nth and (N þ 1)th Landau levels. Thermodynamically, we have the electric field, E ¼ r(m/e) with being the electrochemical potential, so that the Hall current jH ¼ cr M with M being the magnetization can be expressed as jH ¼ ecE (qM/q ). If we use, following Widom, the thermodynamic Maxwell’s relation, qM/q ¼ qN/qB, which comes straight from M ¼ q/qB with the grand potential , we recover the formula for IIxy (Widom, 1982). Note that the Strˇeda–Widom formula is applicable to the situation where the spectrum has an energy gap. 1.05.3.4
in which a QHE system is wound into a cylinder (but the magnetic field is still applied perpendicular to every point on the cylinder), as in Figure 16. To this we add a magnetic flux (due to, say, a solenoid) that pierces the cylinder. Through its vector potential, A ¼ /2r (in cylindrical coordinates), the flux exerts an Aharonov–Bohm effect on the electrons. We can eliminate the vector potential with a gauge transformation at the cost of the boundary condition twisted to ( ) ! exp[i(/ 0) ], on the wave function , where 0 ¼ eh/e is the flux quantum. If we increase with time, we recover the original periodic boundary condition every time increases by 0 in a time interval t. When the Fermi energy EF is in a gap in the density of states or in the energy region for localized states (as described in the previous section), the occupation of electrons must be the same when we make ! þ 0. The only change, then, should be a transfer of an integer (M) number of electrons from one electrode to another, which we assume to be attached to either edge of the cylinder as Hall probes. From the Maxwell equation, an electric field LEy ¼ (1/c)q/qt is induced along the cylinder of circumference L, which should result in a Hall current, jx ¼ (e)M/(Lt) ¼ (Me2/h)Ey, and we have xy ¼ Me2/h. So the derivation applies arbitrary strength of the magnetic field as far as the Fermi energy EF is in a gap in the density of states, or in the energy region for localized states.
Φ
B Hall current
Gauge Argument
As a transparent approach to QHE, Laughlin has proposed an argument based on a gauge transformation (Laughlin, 1981). Consider a Gedanken-experiment,
V Figure 16 Geometry considered by Laughlin.
Integer Quantum Hall Effect
1.05.3.5
Topological Arguments
In Laughlin’s argument we have introduced a magnetic flux, but subsequently it has been shown that we can go even further to express the Hall conductivity as a topological invariant, which guarantees the quantization in a mathematically rigorous manner. Namely, Thouless et al. (1982) have considered, for a QHE system periodic in both x and y directions, the twisted boundary condition for both x and y directions, which corresponds to introducing a vector potential, A ¼ (Ax, Ay), describing fictitious fluxes. Then the Hall conductivity averaged over (Ax, Ay) reads
occup Z Z xy 1 X qu qu qu qu ¼ – j j dAx dAy 2 qAy qAx qAx qAy 2i e =h ¼ –i
¼
e2 L2
Z
dk X f ð" k Þ½rk h kjrk j kiz ð2Þ2
e2 X C h
(29) ð29Þ
In the first line u is the th eigenfunction and the summation is over the occupied states, while in the second line L the sample size, f the Fermi distribution function, " k the energy of the Bloch wave function j ki in the th band, and rk is the gradient with respect to k. The expression coincides with a topological invariant C (that always takes integer values) known as the first Chern character in the differential geometry. R This is seen if we rewrite the formula as xy _ dkrk A ðkÞ, where A ðkÞX – i h kjrk j ki is a fictitious gauge potential. So the Hall conductivity (in units of e2/h) is just the topological invariant. The reason why a differential geometry is relevant is that the wave function in 2DEG in a magnetic field has such a property that different wave functions arise between the cases when we adiabatically change (Ax, Ay) as ! (Ax þ Ax, Ay) ! (Ax þ Ax, Ay þ Ay) and as ! (Ax, Ay þ Ay) ! (Ax þ Ax, Ay þ Ay). The difference is related to the phase of the wave function, which is a kind of Berry’s phase that generally arises when a quantum system is subject to an adiabatic change. When a vector (to which a wave function belongs in a Hilbert space) ends up with different vectors for different parallel transports, we say the space in which the vectors reside is curved. The integrand in the expression for the Hall conductivity happens to represent the curvature (Berry’s curvature here), and Euler’s theorem dictates that the curvature integrated over the surface is an integer.
189
Alternatively, we can express the linear-response Hall conductivity in terms of Green’s function G as Z ZZ 0 =L xy 1 ¼ dz dAx Ay 82 C e 2 =h 0 " # qG – 1 qG – 1 qG – 1 – ðx $ yÞ Tr G G G qAx qAy qz
ð30Þ
When EF is in a localized regime, we can close the contour C on the plane of complex energy z, and the above expression coincides with another topological invariant (Pontrjagin number). This expression reduces, for a fixed number of electrons, to the formula by Thouless et al. if we note j ¼ qH =qA. For clean systems, the topological expression indicates that the Hall conductivity, for arbitrary strength of B, is quantized topologically when EF is in a gap of the density of states. For disordered systems, the topological expression indicates that the Hall conductivity is quantized when EF is in the energy region of localized states, since in that case A-averaged xy is equal to the unaveraged one. In reality, xy is a smooth function (rather than a step function) of energy for finite systems or at a finite T, and this may seem contradictory to the topological argument. However, we can show that, even though xy for each of disordered samples is individually quantized, the quantity becomes a smooth function of energy after the ensemble average over randomness (which corresponds to the observable quantity), and the latter tends to the xy (not integrated over A) in the thermodynamic limit (Aoki and Ando, 1986). Field theoretically, we can introduce a topological term (as in the term in the Yang–Mills field theory) in the Lagrangian in the nonlinear model for the QHE system (Pruisken, 1990). Topological phenomena abound in condensed-matter physics, as exemplified by vortices in superfluids and fluxoids in superconductors, but QHE thus forms an important and distinct class in the topological phenomena, which has spinoffs in the physics of topological insulator, etc.
1.05.4 Localization problem 1.05.4.1 Scaling Theory of Localization in 2D Systems As we have seen, the integer QHE is intimately related with the localization problem. We have also mentioned that the system falls upon an interesting universality class of the unitary class in 2D. Details of the
190 Integer Quantum Hall Effect
localization (which is a nonperturbative problem) have been extensively studied with numerical methods and field theoretical methods. Figure 12 shows typical behaviour, obtained numerically, of the wave function along with the localization length versus energy, which shows that the localization is strong toward the edges of a Landau level and the weakest at the center. Classically, this corresponds to the fact that contours of a random potential, along which the cyclotron guiding center drifts in the semiclassical picture, are valleys (hills) for energies well below (above) the level center, while the contours tend to percolating paths for energies close to the center. However, the localization is quantum mechanical phenomenon related with interference of wave functions. If we look at the behavior of the localization length in a numerical method (finite-size scaling for the Thouless number), the localization length, although a smooth function of E, exhibits a rather singular behavior, in which the inverse localization length becomes zero only at the level center (i.e., extended states only occur there), as schematically shown in Figure 15. The delocalized states, marginally allowed to appear in the presence of magnetic fields in two dimensions, thus coalesce into a single point E0 (the center of each Landau level) on the energy axis. The localization length, denoted here as , behaves around this point as 1=jE – E0 js
ð31Þ
where s (numerically shown to be & 2) is the localization critical exponent (Aoki and Ando, 1985; Ando and Aoki, 1985). This value may be compared with the exponent s ¼ 7/3 for the percolation problem. The effect of quantum mechanical tunneling has also been extensively studied in terms of a network model due to Chalker and Coddington, who have modeled the quantum mechanical tunneling across the paths (see Figure 13(c)), (Chalker and Coddington, 1988; Kramer et al., 2005). The critical exponent s has also been shown to depend on the Landau level (e.g., larger for N ¼ 1 than for N ¼ 0). A mapping onto the nonlinear model gives an estimate , exp 2SCBA , with SCBA (N þ 1/2)e2/h being xx in the self-consistent Born approximation (Ando and Uemura, 1974; Ando 1974a, 1974b; Ando et al., 1975) and , the magnetic length. So the localization length increases with N, but so does the critical exponent. The localization length also depends strongly on whether the randomness is rapidly varying in real space or slowly varying as compared with the magnetic length.
1.05.4.2 Quantum Criticality and xx xy Diagram One way to display the critical behavior is the xx–xy diagram. Namely, while at T ¼ 0 xy is a step function and xx nonzero at discrete points, at finite temperatures not only the Fermi distribution is smeared by kBT, but we have also a finite inelastic scattering length L. This can be summarized as the (xx, xy) flow diagram when T is varied for various values of EF, as originally evoked in the nonlinear model (Figure 17(a)). Namely, we can map the QHE system onto a field-theoretic model called the nonlinear model, for which the renormalization into larger sample sizes can be discussed (Huckestein, 1995). We can also vary the sample size for numerical works on the QHE system to look at the renormalization flow. Alternatively, we can change the temperature, which changes the inelastic scattering length, hence effectively changes the sample size (Figure 17(b)). This implies that experimental (xx, xy) flow lines can be obtained by varying the temperature (Figure 17(c)). We can also discuss the so-called plateauto-plateau transition, which is the energy interval over which one plateau crosses over to another. In terms of the localization, the states that satisfy L" < (E) are effectively delocalized, where L" is the inelastic scattering length and the localization length (Figure 18(a)). If we assume the inelastic scattering time behaves like "(Tp) with p ¼ 2 in the diffusive regime, the energy region whose width is Tp/2s behaves as delocalized, since the localization length has a critical behavior jE E0js while L" _ 1/Tp/2. If the magnetic field is varied instead, the plateau transition width against B has the width B Tp/2s with the same exponent. Figure 18(b) displays an experimental result. One important question is how the QHE plateaux vanish as the degree of disorder is increased, which should take place since for a large-enough disorder the Landau quantization, along with QHE, should go away. Kivelson et al. (1992) have considered the problem in terms of what is called the global phase diagram on a xyxx plane. In this diagram xx is regarded as a measure of the strength of disorder, while xy the Landau-level filling. As shown in Figure 19(a), the midpoints, xy ¼ (N þ 1/2)e2/h, between the plateaux in the Hall conductivity are bifurcation points in the xxxy flow lines, so that we can take the points, xy ¼ 1= 2xx þ 2xy ¼ ðN þ 1=2Þe 2=h, as boundaries
Integer Quantum Hall Effect
(b)
(a)
191
(c)
3 1.5K
T = 0.32K 1.78K 10.0K
σxx /σSCBA
0.35K (1 –)
n=1
2
2.0
(0 –) 0.35K
n=0 1 T = 1.5K
0 0
1 2
1
3 2
2
5 2
0
1
2
0 3.0
3.5
5.0
5.5
6.0
–σxy / (e 2/h)
Figure 17 Results for the xxxy diagram: (a) Renormalization flow lines obtained in the nonlinear model [Reprinted figure with permission from A. M. M. Pruisken, Physical Review Letters 61, 1297 (1988) Copyright 2010 by the American Physical Society. For information, see: http://link.aps.org/]. (b) A numerical result [From H. Aoki and T. Ando. Surface Science 170, 249 (1986)]. (c) An experimental result [From M. Yamane et al., Journal of the Physical Society of Japan 58, 1899 (1989)].
Inverse localization length ∝ E s
(b)
10 μm 1
T p/2s
1 Lε
18 μm 32 μm 64 μm
ΔB (T)
(a)
0.3 ~ T p/2
E0 E
0.1
30
100 300 T (mK)
1000
Figure 18 (a) In a plot for the inverse localization length against energy, the effectively extended states (white region) is shown for the inelastic scattering length L" at a given temperature T [after H. Aoki and T. Ando, Surface Science 170, 249 (1986)]. (b) An experimental result for the plateau-to-plateau transition width on the B-axis plotted against T for various values of the sample size. Reprinted figure with permission from S. Koch et al., Physical Review Letters 67, 883 (1991). Copyright 2010 by the American Physical Society. For information, see: http://link.aps.org/
up to which the QHE effect survives when the strength of disorder is increased (Figure 19(b)). The global phase diagram has also been experimentally examined for IQHE systems. It is also important to consider how the situation crosses over to the weak magnetic field case, since all the states should become localized in the limit of
B ! 0, so that the fate of the delocalized states is quite nontrivial. It has been suggested that the energies corresponding to the delocalized states go up in energy, which is called the floating. Another question of heuristic interest is what is the classical limit (i.e., Planck’s constant h ! 0). In this limit, the Landau-level filling 2,2n ¼ (h/eB)n
192 Integer Quantum Hall Effect
(a)
σ
2
σxx
σxy
1
σxx
N=0
1 2
n
σxy /(e 2/h)
3 2
ρxx0
(b)
σxy /(e 2/h) = 1
2 4 3 0
1 ρxy 0/(h/e 2)
2
Figure 19 (a) How a xxxy diagram is obtained from their n-dependence is schematically shown, with the flow lines bifurcating at xy /(e2/h) ¼ half integer indicated. (b) The global phase diagram.
vanishes for a fixed n. The measure of the Landaulevel mixing, on the other hand, is the Landau-level broadening divided by the cyclotron energy, pffiffiffi pffiffiffiffiffiffiffiffiffi =h!c 1= !c 0 . Since =h!c diverges like 1= h for h ! 0 with fixed B and 0, the classical limit amounts to a regime of dilute (_ h) in weak magnetic fields. 1.05.4.3 Fractal Wave Functions and Dynamical Scaling It is an intriguing question to ask whether a transition between localized and extended states, in general, is similar to phase transitions in statistical mechanics. A suggestion came in 1983 by Aoki to the effect that the wave function at the Anderson transition, which has no characteristic length scale with the diverging localization length, should be self-similar (fractal) (Aoki, 1983), just as the critical point in a phase transition is characterized by scale-invariant states with a diverging correlation length. For the QHE system in particular, the transition point occurs at the center of each Landau level, so that the delocalized states at the center of a Landau level are just not the usual extended states, but fractal (sometimes called critical) wave functions (r) that have a fractal dimensionality, d < 2. In other words, the density autocorrelation decays with a power law, hj (r) (r þ R)j2i 1/R2 d . Typical wave functions in Figure 20, which are obtained numerically for
larger sample sizes than in Figure 12, show that the wave function is very scattered both in amplitude and spatial extent. To be more precise, the self-similarity in critical wave functions extends beyond a single scale transformation, so the idea has been subsequently developed into the multifractal analysis, with which we can analyze the delocalized states around the Landau level center (Huckestein, 1995). As described in the section below, real-space imaging such as STM begins to visualize the fractal states. Among the physical quantities that are affected by the fractality of the wave functions is the anomalous diffusion in transport. Namely, the dc conductivity obeys, in ordinary systems, Einstein’s relation, ¼ limq;!!0 ¼ e 2 DðEF ÞD 0 , where D 0 is the diffusion constant. At a critical point, the conductivity becomes q, !-dependent. In the dynamical scaling ansatz, the ac conductivity should depend on !, where the relaxation rate goes to zero like 1/ 1/z around the transition. Here, z is the dynamical critical exponent, which is usually 2 in noninteracting systems. If we combine this with the sample size scaling in L/, q, !-dependence should take a form ðq; !Þe 2 DðEF ÞD q=!1=z
ð32Þ
At the criticality, the offfiffiffithe pffiffiffi fractality p states is shown to lead to D q= ! D 0 = q= ! with ¼ 2 d.
Integer Quantum Hall Effect
193
(a) N=0 d=0
(c)
N=1 d = 0.71l
(b) N=1 d=0
Figure 20 Typical wave functions (represented as their squared amplitude) around the center of each Landau level obtained in a computer simulation for a disordered QHE system with a system size of 300l 300l for (a) N ¼ 0 and (b) N ¼ 1 Landau levels with short-range scatteres, and for (c) N ¼ 1 with long-range (d 6¼ 0) scatteres. Reprinted figure with permission from T. Terao et al, Physical Review B 54, 10350 (1996). Copyright 2010 by the American Physical Society. For information, see: http://link.aps.org/
1.05.5 QHE Edge States and Edge Transport The role of sample edges in the QHE transport has been an issue of interest from an early stage when the problem was raised by Halperin in the 1980s (Halperin, 1982). Classically, there are edge currents that correspond to cyclotron motions skipping along the edges (Figure 21(b)). Quantum mechanically, edge states also exist (Figure 21(a)). The energy diagram plotted against the real-space position across the sample width will look like Figure 22(a). Edge states are rather ubiquitous in quantum mechanics, but the peculiarity in the QHE is that the edge currents flow, with no backscattering, in a definite direction dictated by the direction of the applied magnetic field. When one takes the electron–electron interaction into account, the edge states may be regarded as special, incompressible electronic states called the chiral Tomonaga– Luttinger liquid, usually discussed in the context of the fractional QHE. In terms of the topological nature of the QHE, the appearance of edge states is a prime example of a much wider concept of the bulk-edge correspondence, that is, the nature of edge states are dictated by the nature of bulk states, which corresponds to the problem
of boundary states in field theories. In a finite QHE sample, the edge states exist, which appear on the energy spectrum as the edge modes that cross from one Landau level to another in a Landau gap. These modes have to exist, and sometimes called topologically protected in QHE systems. More importantly, the topological nature of the QHE edge states manifests itself, as shown by Hatsugai, in the properties that (1) the edge Hall conductivity is also expressed as a topological invariant (Chern number), and (2) this quantity exactly coincides with the topological Chern number for the bulk Hall conductivity (Hatsugai, 1997). Alternatively, we can visualize, in Figure 22(b), that we can continuously change the situation from the edge-picture limit (where currents exist only at edges) to the bulk-picture (where a bulk potential gradient exists) by deforming the current distribution, but there is always the total current preservation, I bulk ¼ I left edge I right edge, which implies edge bulk xy ¼ xy (Koshino et al., 2001, 2002a, 2002b). One way to describe the edge transport is Bu¨ttiker’s formula (Bu¨ttiker, 1992) which is an extension of Landauer’s formula for transport processes in terms of an S matrix for transmission and reflection channels in ballistic transports. Bu¨ttiker’s formula can explain some of the experimental results, including the quantization of the Hall conduction. However,
194 Integer Quantum Hall Effect
Edge (a)
(b)
Figure 21 Typical edge states in a QHE system in quantum mechanics (a) and classical mechanics (b) From H. Aoki, in G. Landwehr (ed.): Application of High Magnetic Fields in Semiconductor Physics (Springer, 1983), p. 11.
(a)
(b) μ0 μ0
eV
Bulk picture μ2 E
μ2
μ1
Bulk
Edge
μ1 Edge picture Position along sample width l2
Edge
l1
Edge
Figure 22 (a) Energy spectrum against the position along the width is schematically shown for a sample with edges with the chemical potential at the left edge ( 1) and at the right ( 2). (b) How the edge picture (in which all the currents are carried by edge states) continuously crosses over to the bulk picture (where a bulk contribution exists) is schematically shown. From M. Koshino et al., Physical Review B 66, 081301(R) (2002).
this does not mean that all the QHE currents are carried by edges. In general, there exist both bulk and edge Hall currents. Bulk and edge states can even be hybridized. The details have been studied both experimentally and theoretically, which largely indicate that there are contributions from both bulk and edges, whose proportions depend on the width of the sample, T-dependent xx, etc. Real-space imaging results also indicate that there are bulk potential gradients in QHE samples. Physically, a macroscopic system may be regarded as comprising subsystems each having a linear
dimension of the phase coherence length L . The transport is described by the Kubo formula xx, xy if the sample size is much greater than L where the Hall field and Hall current distributed over the sample, while the edge transport in the manner of Bu¨ttiker starts to dominate when the size is comparable with L . One of the characteristic features in the edge transport is that highly nonlocal phenomena can arise that extend over length scales far exceeding the bulk mean free path. This can be probed in Hall-bar samples by selectively injecting currents
Integer Quantum Hall Effect
from the contacts affects the conduction, where the conduction can strongly depend on which terminals to employ in multiterminal measurements. The width of edge states has experimentally been measured with various methods, including transport (e.g., QHE breakdown), magnetocapacitance, and magnetoplasmon measurements. One recent method utilizes the nuclear spins. Usually, the nuclear spins are irrelevant to electron systems, since the nuclear Zeeman energy is as small as 1/2000 of the electronic counterpart. When there are almost degenerate electronic states, however, nuclear spins can affect the electron system via the hyperfine interaction. This occurs not only in the fractional QHE systems close to a spin-polarization transition, but also in the integer QHE where left and right edge states are degenerate in energy, so that the nuclear spin is probed via the scattering between the left and right edge channels. In semiconductor superlattices, which realize a stack of 2DEGs, an application of a strong magnetic field along the growth direction makes the system a stack of IQHE systems. There, the edge states are also stacked along the edge surfaces, which is called sheath currents and have been experimentally observed.
1.05.6 Real-Space Imaging Experimentally, there have been a body of studies for real-space imaging of the QHE systems, including the Hall field. Fontein et al. have studied a potential profile imaging with the electro-optical effect (which utilizes the birefringence (Pockels effect) with the phase
195
difference between different polarizations probing the potential) to show that an electric field exists over the whole sample, although the field becomes stronger toward the edges. An example of the imaging is shown in Figure 23. This is followed by various other methods, including scanning capacitive, force, and polarization optical microscopies (Morgenstern, 2007). One such probing, which is particularly suited to examine nonequilibrium carriers around the hot spots, is the imaging of cyclotron emission due to Komiyama et al. (2004). In a Hall bar sample in the QHE condition (with xx ¼ 0, xy 6¼ 0), the electric lines of force are forced to be distorted to make the rectangular sample geometry compatible with the Hall angle of 90 , resulting in singularities which have to appear at two positions around the electrodes (Figure 24(a)). These are called the hot spots. In the imaging of cyclotron emission (Figure 24(b)), the QHE detector, which is itself an IQHE system having by nature sensitive photoresponses at the cyclotron resonance frequency, is used to scan the QHE system. In the result the hot spots are clearly seen, with edge channel also visible for the Landau-level filling 6¼ integer. The imaging of cyclotron emission has also been used to examine the breakdown of QHE (see section 1.05.8). Recently, STM and scanning tunneling spectroscopy (STS) begin to be performed for a special kind of 2DEG, that is, the adsorbate-induce 2DEG, where for example, Cs atoms are deposited on cleaved n-InSb. The STS imaging is also used to directly observe the quantum Hall transition from one localized regime to another via the delocalized states that have a fractal character as shown in Figure 25 (Hashimoto et al., 2008).
Figure 23 A typical real-space image for the potential distribution obtained with the electro-optic imaging for B ¼ 10.8 T corresponding to the center of a plateau. From R. Knott et al., Semiconductor Science and Technology 10, 117 (1995).
196 Integer Quantum Hall Effect
(a)
S
D
(b)
ν = 2.22
–
B
+
XY-stage 2DEG Hall-bar (mitter) Si-SIL
Y (μm)
1800 Condenser lens
1400
14.6mm
1000 1000
2000 X (μm)
3000
4000
QHE etector
Figure 24 (a) The electric lines of force are schematically shown for the QHE condition, with S: source electrode and D: drain, (b) A real-space imaging obtained with the cyclotron emission detected by the QHE device (inset). Reprinted figure with permission from K. Ikushima et al., Physical Review Letters 93, 146804 (2004). Copyright 2010 by the American Physical Society. For information, see: http://link.aps.org/
50 nm (a)
(b)
(c)
b a
High
c
dl/dV (a.u.)
dl/dV (a.u.)
Lowest spin-LLs
(d)
d
–120 –110 –100 –90 –80 Sample voltage (mV)
Low
Figure 25 A real-space image obtained by STS for 0.01 monolayer of Cs on a cleaved n-InSb(110) at T ¼ 0.3 K for various value of the Landau-level filling, (a)-(d), as indicated in inset. Reprinted figure with permission from K. Hashimoto et al., Physical Review Letters 101, 256802 (2008). Copyright 2010 by the American Physical Society. For information, see: http://link.aps.org/
1.05.7 QHE Resistance Standard and the Fine-Structure Constant
et Mesures (CIPM), that the QHE be adopted as the electrical resistance standard, with which ohm is defined via the QHE resistance,
After the recognition that the accuracy of the QHE quantization is experimentally better than 106 (nowadays around 107 or even better), it was decided, in 1990 by Comite´ International des Poids
RK ¼ h=e 2 ¼ 25 812:807
where RK is called the von Klitzing constant.
ð33Þ
Integer Quantum Hall Effect
(a)
(b) H
10–8α
10–6h RK NPL-88 RK LNE-01 RK NMI-97
KJ PTB-91 KJ NMI-89
RK NIST-97
KJ2RK NPL-90 KJ2RK NIST-98 KJ2RK NIST-07 CODATA-02 CODATA-06
ae U Washington-87 ae Harvard-06 CODATA-02 CODATA-06 597
197
598
599
600
601
602
603
5 604
6
7 8 [h/(10–34Js) – 6.6260] × 105
9
10
(α–1 – 137.03) × 105
Figure 26 (a) Values of the fine-structure constant obtained with various methods. Those marked with RK are from the QHE measurements at various institutes, ae from the electron magnetic moment anomaly, (b) A similar plot for the Planck constant h. CODATA-recommended values are also indicated. From P. J. Mohr et al., Reviews of Modern Physics 80, 633 (2008).
The constant, now adopted as a resistance standard, has a profound significance related with a fundamental physical constant. Namely, the fine-structure constant
¼
e2 .1=137:036 hc
ð34Þ
which is the coupling constant in the quantum electrodynamics (QED) (Kinoshita, 2007), one of the most basic physical constants, is directly related with RK via ¼ 2/RKc. Here, c ¼ 2.99 792. . . 108 m s1, the speed of light in vacuum, is, in SI, a defined value. Figure 26(a) shows the values with error bars of obtained by various methods including that from RK obtained by various institutes. If we combine RK with the Josephson constant, KJ ¼ 2e/h, we can deduce the value of Planck’s constant via h ¼ 4/K2JRK (Figure 26(b)). For a recent review, see Mohr et al. (2008).
this phenomenon, known from an early stage of the QHE studies, is called the breakdown of QHE. The breakdown is important from both applicational aspects (since it affects the accuracy of the QHE) and fundamental aspects since this is an interesting nonequilibrium phenomena. Several factors can be involved, among which are tunneling between different Landau orbits or Landau levels, and/or electron heating effect. Also relevant is how the breakdown is related with the current profile in real space in the Hall bar. Although there are some prevailing experimental and theoretical results, we seem to be some way from a unified picture (Nachtwei, 1999). Sometimes metastable or bistable states are observed around the breakdown regime, which is also of interest. As for nonequilibrium phenomena, we can mention that the microwave-induced magnetoresistance oscillation is one remarkable nonequilibrium phenomenon, although this occurs in a weak magnetic field regime rather than in the QHE regime.
1.05.8 Breakdown of QHE When we increase the source–drain voltage to increase the source–drain current in Hall effect measurements in the Hall-bar geometry, the longitudinal resistance Rxx, which is close to zero in the QHE plateau region, abruptly increases for sufficiently large voltages, as typically shown in Figure 27. This is accompanied by disrupted plateau structures in Rxy. Thus, the Hall current, which is originally dissipationless in the QHE condition, becomes dissipative, and
1.05.9 Quantum-Dot and Periodically Modulated Systems in Strong Magnetic Fields 1.05.9.1
Quantum-Dots in Magnetic Fields
Semiconductor quantum-dot is a nanostructure that confines a few electrons. The physics of quantumdots is a very large area of investigation, as reviewed in the chapter on this. Here, we only mention its relevance to the QHE system. quantum-dots are
198 Integer Quantum Hall Effect
Ey (Vcm–1) –80
–40
0
40
80
4
ρxx
6
Ex (Vcm–1)
2 0
B(T)
5
0 –2 –4 –6 –0.8
–0.4
0 jx (Am–1)
0.4
0.8
Figure 27 A typical current–voltage characteristics at ¼ 2 (the arrow in the inset). From Nachtwei, Physica E 4, 79 (1999).
generally fabricated by applying a lateral confining potential to a 2DEG system. This can be realized by an electrode that exerts an electrostatic confining potential, or alternatively we can mesa-etch the system to have a finite system (Figure 28(a)). The confining potential is usually cylindrically parabolic to a good approximation.
Quantum-dots are also investigated in strong magnetic fields. In a magnetic field, B = rot A, applied perpendicular to the dot plane the Hamiltonian, in the one-body problem here, reads H ¼
2 1 1 pi þ ðe=c ÞAðrÞ þ m !20 r 2 2m 2
ð35Þ
(b) 50
40 (a)
Drain AlGaAs InGaAs AlGaAs Source
Side Gate
E (meV)
A 30
20
10
0
0
2
4
6
8
10
B (T) Figure 28 (a) A typical mesa-etched quantum-dot structure. From P. A. Maksym et al, Physical Review B 79, 115314 (2009). (b) Energy levels of the Fock–Darwin states against magnetic field for a harmonic confinment. From P. A. Maksym et al, Journal of Physics: Condensed Matter 12, R299 (2000).
Integer Quantum Hall Effect
where h!0 is the confinement energy and we have ignored the Zeeman energy. The exact eigenstates of this Hamiltonian are known as the Fock–Darwin states, since they were first investigated by Fock and Darwin in the 1920s. They are given, up to the normalization constant, by nm ðrÞ
2 2 – im
2 2 ¼ r jmj Ljmj n r =2 exp – r =4 e
ð36Þ
in 2D polar coordinates with eigenenergies given by Enm ¼ ð2n þ 1 þ jmjÞh – mh!c =2. Here, m is the angular momentum, n a radial quantum number, Ljmj n the associated Laguerre polynomial, 2 ¼ !20 þ !2c =4 with the cyclotron frequency !C ¼ eB/m. Thus, the wave function is almost the usual Landau’s wave function as far as the parabolic confinement is concerned, where pthe difference is in the length ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi parameter, ¼ h=ð2m Þ. In Figure 28(b) depicting the energy levels we can indeed see that a crossover from the states of a 2D harmonic oscillator in the zero field limit to the Landau states of a free electron in the strong field limit. When we consider the electron–electron interaction, a variety of states emerge, which include the maximum-density droplet that corresponds to the Landau-level filling ¼ 1, and the electron-molecule states (Maksym et al., 2000). So the dots in strong mangetic fields are a kind of confined QHE systems, or artificial atoms in magnetic fields. Their properties have been extensively studied with various methods. For instance, dot wave functions have been probed experimentally with techniques such as magneto-tunneling spectroscopy. Other properties include magnetocapacitance, electron addition spectra, and transport spectra in magnetic fields, from which the groundstate quantum numbers are deduced. Dot arrays have also been studied in the QHE regime, where resonant scattering effects, etc., have been reported.
1.05.9.2
Hofstadter Spectrum
What happens to QHE when we have periodic systems rather than a translationally invariant 2DEG? Condensed-matter physics tells us that in a periodic system we have Bloch’s theorem, which dictates that electronic states for periodic systems are dominated by Bragg’s reflection, which results in band structures and Bloch states. So here we are talking about Landau’s quantization in the presence of Bragg’s reflection. We can realize that Bragg’s reflection
199
and Landau’s quantization interfere with each other, since the application of a magnetic field, B ¼ rot A, gives rise to, semiclassically, the Peierls phase,
Z e r exp – i Aðr9Þ ? dr9 ðrÞ, in the wave function h . Landau’s quantization in periodic and lattice systems was first considered by Wannier and by Hofstadter (1976). It has been shown that the energy spectrum plotted against magnetic field is a curious fractal (sometimes called Hofstadter’s butterfly); fractal, because each Landau level splits into p levels when the magnetic flux within a unit cell in units of flux quantum equals to q/p (Figure 29). When EF is in one of these gaps, we should have IQHE, for which xy in units of e2/h, the Chern number, has been calculated with the topological formula due to Thouless et al. Recent advances in the electron-beam lithography has made it possible to fabricate 2DEGs with 2D periodic modulations, and the butterfly begins to be observed (Geisler et al., 2004).
1.05.9.3
QHE in Three Dimensions
While usually the QHE is an effect specific to 2D systems, can we conceive a similar effect in 3D systems, and, if so, how? If we go back to the topological argument, we do not use the fact that the system is 2D except for the presence of inter-Landau-level gaps in the energy spectrum. This implies that, if there exist, for some reason, energy gaps in 3D systems in magnetic fields, we should have a quantization (for the Hall conductance Rxy– 1 in 3D), when EF is in a gap. Note that in d-dimensional systems of size L the conductance R-1 and conductivity are related as Rxy– 1 ¼ Ld – 2 xy , so that only in 2D do they happen to coincide with each other. Usual wisdom, however, is that gaps do not tend to appear in 3D. One possibility is to use 3D systems that have periodic structures or potentials. Koshino et al. have shown that 3D Hofstadter spectra can appear in periodically modulated structures in the energy spectrum against the tilting angle in tilted magnetic fields, where an interference of Landau’s quantizations due respectively to the components By and Bz of the magnetic field is responsible (as compared with 2D where Hofstadter’s butterfly comes from an interference between Bragg’s reflection and Landau’s quantization). Then each of xy and zx are quantized when EF is in each gap with the current
200 Integer Quantum Hall Effect
B
0
1/4 3/8 –σxy /(e2/h)
1/2 3 2 1 0 –1 –2 –3 –4
–2
0 E
2
4
–2 –3
E/tx
1
–2
–1
–3
1
3 2
0 3
2
2
–1
1
1
0,3
1,02,–20,2
0,4 0,3
–3 –2
2,0
0,1
–2,2 3,–1
–1,3
1,–1
2
2
–1
1
0,4 1, –1
1,–2 1,–3 1,–4
2
1,
1,
0
3,0
2,
0
2
2
2
0,
3
0,2
3
E/tx
Figure 29 Hofstadter butterfly (energy spectrum against magnetic field) for a square lattice (upper panel), and the corresponding QHE at the value of magnetic field for which the magnetic flux penetrating a unit cell of the periodic system is 3/8 in units of the flux quantum [after H. Aoki, in G. Landwehr (ed.): Application of High Magnetic Fields in Semiconductor Physics III (Springer, 1991), p.17].
–3,4 –2,3
0,1 –1,2
–2
4,–3 3,–2 2,–1
4,0
,1 –1
–2,1 –3,1 –4,1
2,
4,0 3,0
,1 –1
1,0
0
3,–3 3,–3 –1,2
–3
–3
0
0.2
–2
0.4
0 φ
2
0.6
0,0
3
0.8
1
0
2,–2 1,–1
–2,2
2,–1 –1,1
60 30 θ (degree)
0,0 90
Figure 30 (Right) Energy spectrum against the tilting angle, , of the magnetic filed applied to a 3D, periodic system. The QHE values (in units of e2/ah with a: lattice constant) for (xy and zx) are indicated by a pair of numbers attached to each gap in the spectrum. The region indicated by a dashed line has a one-to-one correspondence to the Hofstadter butterfly in 2D (left panel). [Reprinted figure with permission from M. Koshino et al., Physical Review Letters 86, 1062 (2001) Copyright 2010 by the American Physical Society. For information, see: http://link.aps.org/].
j ¼ s E where s = (yz, zx, xy) (Figure 30). Experimental realization is yet to come. A kind of Landau quantization has been known to occur in 3D systems, typically in a quasi-2D organic conductor (TMTSF in the Bechgaard salt family). This is the field-induced spin-density wave (SDW) in strong magnetic fields with a many-body origin. Namely, the Landau quantization takes place within the pockets formed by incompletely-nested Fermi surfaces, which gives rise to a series of gaps around the main SDW gap.
1.05.10 Integer versus Fractional QHEs The integer QHE has subsequently been developed into the fractional QHE (FQHE) as is described in the present volume (see Chapter 1.06). In FQHE the Hall conductivity is quantized into xy ¼ e2/h for fractional Landau-level fillings ¼ 1/3, 2/3, 3/5, . . . as opposed to the IQHE for ¼ integer. Let us compare the IQHE with FQEH in this section. Physically, the integer QHE is primarily understood
Integer Quantum Hall Effect
in terms of a one-body problem as described here, while FQHE is inherently a many-body effect, and this is a customary way to distinguish the IQHE and FQHE. It is then natural that the IQHE was originally discovered in Si-MOSFET, while the FQHE effect in GaAs–AlGaAs heterostructures, where the latter system is atomically much cleaner with typical mobility exceeding 106cm2 V1 ? s1 against the former’s 104cm2 V1 ? s1. The FQHE has in fact been observed in clean-enough Si-MOSFETs. One clear way to realize that the FQHE, a manybody effect, emerges as the degree of disorder is lowered is to look at the historical developments from IQHE to FQHE in Figure 31, which shows how the fractional effect appears as the sample quality (as characterized by the carrier mobility) becomes higher. In the clean limit, the system is indeed in the limit of strong electron correlation in that the kinetic energy is quenched due to Landau’s quantization so that the ratio of the interaction energy to the kinetic energy is infinite (although more rigorously we have to consider the inter-Landau-level matrix elements).
2 3
1 3
ρxx
1982 0
1984 0 2 3
3 3 7 5 6 4 45 9 5 11 79 11 8 7 13 13
2 5
1987 0
4 5 5 7
1
1/2 Landau level filling
1/3
Figure 31 The developments of the fractional structure in samples of progressively higher quality From D. C. Tsui et al., Physical Review Letters 48, 1559 (1982), A. M. Chang et al., ibid 53, 997 (1984), R. Willett et al., ibid 59, 1776 (1987).
201
Although the FQHE as a new class of many-body states is associated with fractional Landau-level fillings, the electron–electron interaction exists for integer fillings as well. In fact, an integer is a kind of fraction in that Laughlin’s wave function for the FQHE liquid for clean systems, allowed for odd fractions ( ¼ 1/m with m an odd integer), also accommodates m ¼ 1. So the real question is the nature of the energy gap: in the integer QHE the excitation gap is primarily the one-body gap between the adjacent Landau levels (or between the mobility edges in disordered systems), while the excitation gap in the FQHE has a many-body origin. The relative magnitude of the one-body and many-body gaps depends on the degree of disorder (and the g-factor in the case in which the adjacent Landau levels are Zeeman-split ones). To quantify this, we can examine various energy scales. Figure 32 plots 1. h!C (the cyclotron energy), 2. e2/", (typical size of the electron–electron Coulomb interaction), where e is the elemental charge, "(¼ 13 for GaAs) the dielectric constant of the material and , the magnetic length, and 3. g BB (the Zeeman energy), where g is Lande´’s g-factor, B ¼ he=ð2mc Þ the Bohr magneton. We can see that for B few tesla the Zeeman energy is relatively negligible, while the cyclotron energy and the Coulomb energy are comparable. For disordered systems we have to compare these with the Landau-level broadening , which is roughly estimated in the self-consistent Born approximation as =ðh!c Þ1=ð!c 0 Þ1=2 , where 0 is the scattering relaxation time in zero magnetic field. So the Landau-level broadening is comparable with the cyclotron energy for !c 0 1. While the two-dimensionality in IQHE appears in the fact that the Hall conductivity represents a topological quantum number (Chern number) in terms of the Berry’s curvature, in FQHE the spatial dimensionality of two is essential in allowing the composite fermion picture (i.e., a Chern–Simons gauge field theoretic treatment) of the many-body quantum liquid. This is the reason why the FQHE accommodates such novel concepts as anyon quasi-particles. The relevant chapter should be referred to for the FQHE, so suffice it to mention that FQHE is regarded as an IQHE of composite fermions in the
202 Integer Quantum Hall Effect
35 30
N=1
25
N=0
hωc
Energy (meV)
hωc Zeeman splitting
20 15
e2/εl
10 5 0
gμBB 0
5
15
10
20
B (T) Figure 32 Various energy scales (cyclotron, Coulomb, and Zeeman) against magnetic field, here plotted for GaAs. Inset schematically shows the Landau levels with Zeeman splitting.
composite fermion picture. As explained in that chapter, the Landau-level filling is expressed as ¼ Ne/N , for Ne electrons in N flux quanta, so that an odd fraction, say, ¼ 1/3 implies that there are three flux quanta per each electron on average. In the composite fermion picture we attach two flux quanta to each electron (in a kind of gauge transformation), and we are left with one flux quantum. So the ¼ 1/3 state can be mapped to a ¼ 1 state of composite fermions in a mean-field sense. One application of this correspondence is the global phase diagram: we have explained the xx xy diagram above for the IQHE. If we combine this with the composite particle transformation, we can, again in a mean-field sense, a phase diagram for the integer and fractional QHE phases against the Landau-level filling and the degree of disorder.
When heterostructures such as ZnO/MgxZn1 xO are grown with MBE, MgZnO layer acts as a potential barrier for the 2DEG in ZnO layer in realizing a 2DEG. The carrier density (typically 1012cm2) systematically depends on x and the growth temperature, where spontaneous and piezoelectric polarization effects work to accumulate carriers at the heterointerface. The condition necessary to have QHE (i.e., !c 0 > 1) is met, and a typical result (Figure 33) exhibits a clear IQHE. Since oxides have various possibilities, broad applications are expected (Tsukazaki et al., 2008). 10
1.0 T = 0.3 K n = 8.7 × 1011 cm–2 μ = 20 000 cm2 V–1s–1
0.8
ν=3 8
There are a multitude of recent developments in the integer QHE and closely related physics. Extensions are being made both for various classes of materials, and various novel phenomena. Let us here briefly mention them.
6 0.4
4
0.2
2
0.0
1.05.11.1 QHE in Oxides A notable breakthrough in the attempts to widen the classes of materials is the realization of QHE in oxide heterostructures. The system is zinc oxide (ZnO), which is an insulator, or a wide-gap semiconductor.
6 5
0
5
10
0 15
B (T) Figure 33 QHE in an oxide heterostructure ZnO/ MgxZn1 xO. Reprinted figure with permission from A. Tsukazaki et al., Physical Review B 78, 233308 (2008). Copyright 2010 by the American Physical Society. For information, see: http://link.aps.org/
ρxy (kΩ)
1.05.11 Recent Developments and Related Phenomena
ρxx (kΩ/ )
4 0.6
Integer Quantum Hall Effect
1.05.11.2 QHE in Graphene One fascinating aspect of the condensed-matter physics is that we can have various field theories effectively realized on low-energy scales. Recent emergence of the physics of massless Dirac particles (or Weyl particles in the language of field theoretic texbooks) in graphene is a good example. While the 3D graphite has long been studied extensively, experimental fabrication of graphene had to wait for the accomplishment by Geim’s group around 2004. IQHE then received a strong impetus when seminal series of works on graphene, in particular the IQHE, were launched for graphene after around 2005 (Novoselov et al., 2005; Geim and Novoselof, 2007; Castro Neto et al., 2009). Graphene is a monolayer graphite with a honeycomb array of carbon atoms (Figure 34(a)), while graphite is a stack of graphene sheets (in a staggered manner called Bernal stacking). Electrons on a single layer of honeycomb lattice, despite its simplicity, provide interesting problems in condensed matter physics. Specifically, it has long been known that its band dispersion (for orbitals) is composed of a pair of k-linear conical electron and hole dispersions that touch with each other at E ¼ 0 (Figure 34(b)), so that the graphene is a condensed-matter realization of massless Dirac fermions around E ¼ 0 (at which EF usually resides). This was noted by Wallece as early as in 1947. Subsequently, the reason why honeycomb symmetry implies the massless Dirac dispersion was revealed group-theoretically by Lomer and by Coulson in the 1950s. Band analysis in terms of the k ? p perturbation was also done by Slonczewski and Weiss. Graphene is to be contrasted with graphite, whose band structure is a semimetal with Fermi surface comprising small electron and hole pockets.
(a)
203
To be more precise, graphene has two, inequivalent Dirac cones at K and K9 in the first Brillouin zone (Figure 34(b)). To see this, we should go back to the honeycomb lattice, which is a non-Bravais lattice that contains two (A and B) sublattice sites in a unit cell. The lattice is bipartite, so that the energy spectrum is electron–hole symmetric about E ¼ 0 if we only consider nearest-neighbour hopping. The tight-binding model (for electrons) on a honeycomb lattice reads, in k-space, 0 P H ¼t @ k
0
ðkÞ
ð k Þ
0
1 A;
ð37Þ
ðkÞ ¼ 1 þ e – ik1 þ e – ik2 ;
where the 2 2 matrix represents the AB sublattices, k1, k2 are the wave vectors along the primitive vectors in k-space. Around K and K9 points in the Brillouin zone, the k?p perturbation shows that the Hamiltonian linearized in p reduces to H ¼ vF x z px þ y py
ð38Þ
where vF . 108 cm s1 is the Fermi velocity, and m Pauli matrix in the 2 2 matrix representation, while z is another Pauli matrix with z ¼ 1(1) for K (K9) point. So the equation (for each value of z) is just the Dirac equation for a massless particle in two spatial dimensions. The Hamiltonian anticommutes with z (which plays the role of 5 in (3 þ 1) dimensions), and this symmetry is called the chiral symmetry. In the presence of a magnetic field, we can replace p in the above equation with p ¼ p þ (e/c)A as far as the effective, k ? p band structure is concerned. Then the Landau levelspbecome, as ffiffiffiffi shown by McClure in 1956, E ¼ N h ! with N c pffiffiffi pffiffiffi 2=, vF ¼ ð2evF =chÞ B , the magnetic !c ¼ (b)
E
ky
Figure 34 Crystal structure (a) and the band dispersion (b) of graphene.
K′
K
kx
204 Integer Quantum Hall Effect
length as usually defined (Equation (8)), and N ¼ 1, 2,. . . (N ¼ 1, 2,. . .) correspond to p electron (hole) ffiffiffiffi Landau levels. The Landau levels _ N is not uniformly spaced in sharp contrast to the usual, uniformly spaced Landau levels, and, in particular, there is the N ¼ 0 Landau level right pffiffiffiat the Dirac point E ¼ 0. The cyclotron energy _ B is also unusual. The N ¼ 0 Landau level is quite peculiar, which is brought home by noting that the level is completely outside the Onsager’s semiclassical quantization scheme in magnetic fields (because the Landau tube, which is the set of cylinders of varying radii for the semiclassical quantization, cannot be defined around the Dirac point). In fact, this level is an outcome of a topological property of the massless Dirac cone, so that its presence is topologically protected. If we go back to the original honeycomb lattice in magnetic fields, the problem becomes a Hofstadter butterfly for the honeycomb lattice (Figure 35), which was obtained by Rammal in 1985. Incidentally, it is heuristic to look at the Landau’s quantization in the (massive) relativistic particles. From the Dirac equation for a Dirac particle in a magnetic field, the energy levels are EN ¼ [m2c4 þ mc2(h!c(2N þ 1 1))]1/2 (MacDonald, 1983), (which contains the nonrelativistic limit, EN ¼ mc2 þ h!c(N þ 1 1/2)), as the leading term in h!c/mc2 expansion). In the massless limit, m ! 0, we have to note that !c ¼ eH/mc also contains the mass to recover the graphene Landau levels.
E 0
B Figure 35 Energy spectrum against magnetic field for a honeycomb lattice.
Soon after the fabrication of graphene samples, an anomalous IQHE was observed by Geim’s group, and by Kim’s group (Novoselov et al., 2005; Geim and Novoselof, 2007; Castro Neto et al., 2009). As Figure 36 shows, the IQHE in graphene has xy ¼ – 2
e2 ð2N þ 1Þ h
ð39Þ 2
e as contrasted with the usual xy ¼ – 2 N when the h carriers are filled up to the Nth Landau level. The prefactor of 2 in these equations is for the spin degeneracy. The peculiarity of the Dirac cone appears as the factor (2N þ 1) replacing the ordinary N as shown by Zheng and Ando, 2002; Gusynin and Sharapov, 2005. There are K and K9 points that contribute equally to the Hall conduction, so that if we take out the factor of 2 along with the spin e2 degeneracy, we can write xy ¼ – 4 ðN þ 1=2Þ, h where, remarkably, a fraction (1/2) appears. The situation around the N ¼ 0 Landau level is indeed unusual. Let us compare the situation with the ordinary one where we have a (massive) conduction band and a (massive) valence band in Figure 37. In the latter case, we have an ordinary IQHE sequence, 0 ! 1 ! 2 ! . . ., for electrons (in the conduction band) and another, 0 ! 1 ! 2 ! . . ., for holes (in the valence band). For a Dirac cone, the IQHE step across N ¼ 0 at the Dirac point (i.e., an electron– hole symmetric point) cannot be 0 ! þ 1 nor 1 ! 0, which would be incompatible with the electron–hole symmetry. Instead, the step changes from 1/2 to þ1/2 at N ¼ 0. Another peculiarity is that the quantum Hall step remains robust even at room temperature (Figure 38), which is quite unusual. We have mentioned that the quantum Hall number in usual systems is identified as a topological quantum number according to Thouless et al. How is the topological number identified in the graphene QHE? Hatsugai et al. (2006) have shown that we can indeed algebraically calculate the Chern number in the manner of Thouless et al. extended to the QHE on the honeycomb lattice, which precisely gives e2 xy ¼ – 2 ð2N þ 1Þ. The topological analysis also h leads to another topological character, namely edge states (Figure 39) for a finite graphene in storng mangetic fields appear (which should not be confused with the graphene edge states in zero magnetic field, an interesting issue in its own right). We have mentioned above that the bulk and edge QHE topological bulk numbers are equal, that is, edge xy ¼ xy , which can
Integer Quantum Hall Effect
B = 14 T and T = 4K
205
7/2 5/2
10
3/2
ρxx (kΩ)
1/2 0 –1/2
ρxx
5
σxy (4e2/h)
σxy = (2N + 1) (e 2/h)
–3/2 –5/2 –7/2 0 –4
–2
2 0 n (1012 cm–2)
4
σxy /(–e2/h)
σxy /(–e2/h)
Figure 36 QHE observed in graphene. From K. S. Novoselov et al., Nature 438, 197 (2005).
2 1
(a)
0 –1
3/2
(b)
1/2 –1/2
–2 vb
k
–3/2
cb
k
E
E –1 N = 0
1
Figure 37 Band dispersion and Landau levels (lower panels) and the QHE steps (upper) are compared between the ordinary 2D system with valence and conduction bands (a) and graphene with k-linear, zero-gap dispersion (b).
30
σxy
B = 29 T T = 300 K
4
ρxx (kΩ)
2 20
0 –2
10
0 –60
ρxx
σxy (e2/h)
be identified by connecting the topological integers for the bulk and for the edge states. The same applies to the graphene IQHE. Energy spectrum against real-space position also differs in graphene from the usual QHE system as depicted in Figure 40. Edge states in graphene in magnetic fields are beginning to be observed with STM. As for the effect of disorder and its effect on localization, graphene samples themselves are atomically clean (although there are extrinsic source of disorder such as charged impurities), so that graphene can have a very high mobility. One intrinsic disorder is corrugations of the graphene sheet, called ripples. As for the Landau quantization in magnetic fields, ripples, when their effect is represented as random hopping energies or random compontents in the magnetic field (which respect the chiral symmetry),
–4 –6
–30
0 Vg (V)
30
60
Figure 38 An experimental result for the QHE in graphene at room temperature. From K. S. Novoselov et al., Science 315, 1379 (2007).
206 Integer Quantum Hall Effect
the other hand, dominates the scattering between K and K9 points, where the longer-ranged the disorder the K–K9 scattering becomes less effective. Physics of graphene is now extended to various properties other than transport (such as optical properties). There is also a proposal for the Landau level laser that exploits the transitions between the Landau levels, which was originally proposed for the ordinary 2DEGs (Aoki, 1986; Morimoto et al., 2008). Extensions are also being made for bilayer graphene, where the particle becomes massive, and for manybody physics including FQHE. Another material with a Dirac cone has also been discovered, which is an organic metal -(BEDT-TTF)2I3, although the cone is tilted and does not sit at the corner of the Brillouin zone in this material. 1.05.11.3 Anomalous Hall Effect and Spin Hall Effect QHE has other Hall effects, the anomalous Hall effect and the spin Hall effect (Figure 41), as close relatives, so let us briefly describe them in relation to the IQHE, while details are described in the chapter on the spin Hall effect. While the QHE is a topological phenomenon intrinsic to the nature of the generic 2DEG, the anomalous Hall effect (Hall effect in ferromagnetic materials) and the spin Hall effect (Hall effect for the spin degrees of freedom in materials that have strong spin–orbit interactions) arise
Figure 39 A typical edge states in graphene in a strong magnetic field obtained numerically from H. Aoki et al., International Journal of Modern Physics B 21, 1133 (2007).
are shown to exert anomalously small effect on the broadening of the N ¼ 0 Landau level. This is also a topological effect, related with Atiyah–Singer’s index theorem in field theory. The range of the disorder, on
(b)
(a)
E
E
N=1 N=0
N=1 N=0
x
x
Figure 40 Energy spectrum against the real-space position along the width for a finite sample is schematically compared between the ordinary QHE system (a) and graphene QHE system (b).
B
+q
E
M
–q
+q
–q
Quantum Hall effect Anomalous Hall effect
Spin Hall effect
Figure 41 Quantum Hall effect, anomalous Hall effect, and spin Hall effect are schematically shown.
Integer Quantum Hall Effect
from system’s magnetic structure, band structure and interactions, so they occur in 3D systems as well. The anomalous Hall effect, in which the Hall resistivity is RH ¼ Rnormal B þ Ranomalous M
ð40Þ
with Rnormal being the normal Hall resistivity, Ranomalous the anomalous Hall resistivity, and M the magnetization of the material, was discovered in metallic ferromagnets in the nineteenth century, where the Hall current flows even in zero external magnetic fields. Theoretical analysis was initiated in the 1950s by Karplus and Luttinger as a (multi-) band effect in the presence of spin–orbit interaction. To be more precise, two mechanisms have been identified, one (called extrinsic) is the impurity scattering in the presence of spin–orbit interaction and magnetization (skew scattering þ side jump) and the other (called intrinsic) is Berry’s curvature contribution to the Hall conductivity as in the QHE, which is included in the linear-response formula. Experimentally, the anomalous Hall effect has been observed in various materials, typically Nd2Mo2O7 with noncollinear spin configurations and (Sr, Ca)RuO compounds. In the spin Hall effect, which also occurs in zero magnetic field, " spins and # spins flow in the opposite directions in an electric field (as opposed to the ordinary Hall effect in which opposite charges flow in the opposite directions in a magnetic field). This effect, which also comes from the spin–orbit interaction, is another topological effect. The effect, predicted in the 1970s, is recently being experimentally observed with, for example, Kerr rotation microscopy in both n- and p-type GaAs-based semiconductor heterostructures. QHE, anomalous Hall effect and spin Hall effect are related in that they are manifestations of the phase of the wave functions in magneto-transport properties. In a wide context, electrical polarization is also formulated recently in terms of Berry’s phase. An intimate relation between QHE and the spin Hall effect has been brought home by a more recent quantum spin Hall effect (QSHE) (Ko¨nig et al., 2008; Murakami, 2008) which was predicted originally for graphene. The idea starts as follows. Graphene is described, as mentioned above, by a Dirac equation for two spatial dimensions, where K and K9 points in the Brillouin zone are distinguished by a pseudospin z. In the presence of a spin–orbit interaction, we
207
have an extra term that couples the real spin and the pseudospin, and this gives rise to an energy gap (a mass gap in the language of the Dirac theory). Since the gapped state cannot be reached by an adiabatic change of the system, the insulator is called a topological insulator. Associated with the gap in the bulk are the edge states, which are also dictated to exist and called topological edge states. In this respect, the QSHE is distinct from the spin Hall effect, and the whole situation is rather similar to the IQHE, where an essential difference is that the edge states carry charges in IQHE while edge states carry spins in the QSHE. The energy dispersion (Figure 42) of the edge states in a QSHE system, which may be thought of as arising from Kramers’ doublets, is gapless with the two dispersion crossing at a point in the Brillouin zone. The conductivity calculated with the linear-response theory for each of the spin-up and spin-down electrons gives a quantized (Chern number) conductivity, where the directions of the current are opposite between the up and down spins, with a quantized spin Hall conductivity shown to be spin xy ¼ e=2. While the original proposal, due to Kane and Mele (2005) was made for graphene, the system has turned out to have too small a spin–orbit interaction. Subsequently, the effect was shown to be realized in quantum-wells with narrow-gap semiconductors, HgTe/CdTe. While both of HgTe and CdTe have zincblende crystal structure, their band structures are affected by the spin–orbit interactions which are significant in these materials. Specifically, the bulk
1 E/t 0
I
–1 0
π/a
kx
2π/a
Figure 42 Energy spectrum for a system that exhibits quantum spin Hall effect. Bunch of lines represent bulk states, while a pair of isolated lines the edge states. Inset shows how the spin-up and spin-down electrons flow in real space. Reprinted figure with permission from C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). Copyright 2010 by the American Physical Society. For information, see: http://link.aps.org/
208 Integer Quantum Hall Effect
0.4 T = 1.6K
ρxx ρxy
0.35
ν=3
0.3
0.8
ρxx (kΩ)
ν=4 0.6
0.25
ν=5
0.2
ν=6 ν=7
0.4
ρxy (h /e2)
1.0
0.15 0.1
0.2 0.05 0.0
0
1
2
4
3
5
6
7
0.0 8
B (T) Figure 43 QHE in a HgTe quantum-well [after A. Pfeuffer-Jeschke et al., Physica B 256–258, 486 (1998).]
HgTe has a band structure called inverted, since a band that usually forms the valence band lies above another band that usually forms the conduction band, where the former consists of two (heavy- and lightmass) bands that touch with each other at a point in the Brillouin zone with opposite curvatures (i.e., the bulk system is a semimetal). While IQHE itself (Figure 43) became observable in HgTe/CdTe quantum-wells and superlattices grown by MBE, a recent breakthrough is the observation of the QSHE. The band structure behind this is, when HgTe is sandwiched between CdTe to make a quantumwell, the bands in HgTe well remain inverted for wide wells, while the bands become normal for thin enough wells. So the well thickness can act to control the presence or otherwise of a mass gap, and the system should be a quantum spin Hall insulator above a critical thickness. With the spin Hall effect, we can electronically manipulate the spin degrees of freedom, so that applications to the spintronics is anticipated. (See Chapters 1.03, 1.06 and 1.07).
References Ando T (1974) Journal of the Physical Society of Japan 36: 1521. Ando T (1974) Journal of the Physical Society of Japan 37: 1233. Ando T and Aoki H (1985) Journal of the Physical Society of Japan 54: 2238. Ando T, Fowler AB, and Stern F (1982) Reviews of Modern Physics, 54: 437.
Ando T, Matsumoto Y, and Uemura Y (1975) Journal of the Physical Society of Japan 39: 279. Ando T and Uemura Y (1974b) Journal of the Physical Society of Japan 36: 959. Aoki H (1983) Journal of Physics C 16: L205. Aoki H (1986) Applied Physics Letters 48: 559. Aoki H and Ando T (1981) Solid State Communications 38: 1079 (reprinted in Aoki H and Ando T (1993) Solid State Communications 88: 951). Aoki H and Ando T (1985) Physical Review Letters 54: 831. Aoki H and Ando T (1986) Physical Review Letters 57: 3093. Bu¨ttiker M (1992) In: Reed M (ed.) Nanostructured Systems, 192. New York: Academic Press. Castro Neto AH, et al. (2009) Reviews of Modern Physics, 81: 109. Chalker JT and Coddington PD (1988) Journal of Physics C 21: 2665. Fletcher R (1999) Semiconductor Science and Technology 14: R1. Geim AK and Novoselof KS (2007) Nature Materials 6: 183. Geisler MC, et al. (2004) Physical Review Letters 92: 256801. Gusynin VP and Sharapov S (2005) Physical Review Letters 95: 146801. Halperin BI (1982) Physical Review B 25: 2185. Hashimoto K, et al. (2008) Physical Review Letters 101: 256802. Hatsugai Y (1997) Journal of Physics: Condensed Matter 9: 2507. Hatsugai Y, et al. (2006) Physical Review B 74: 205414. Hofstadter DR (1976) Physical Review B 14: 2239. Huckestein B (1995) Reviews of Modern Physics, 67: 357. Igarashi T, Wakabayashi J, and Kawaji S (1975) Journal of the Physical Society of Japan 38: 1549. Ikebe Y, et al. (2010) Physical Review Letters 104: 256802. Ikushima K, et al. (2004) Physical Review Letters 93: 146804. Kamimura H and Aoki H (1989) Physics of Interacting Electrons in Disordered Systems. Oxford: Oxford University Press. Kane CL and Mele EJ (2005) Physical Review Letters 95: 226801. Kawaji S (2008) Proceedings of the Japan Academy, Series B 84: 199.
Integer Quantum Hall Effect Kawaji S and Wakabayashi J (1981) In: Physics in High Magnetic Fields, Chikazumi S and Miura N (eds.) p. 284. Springer. Kinoshita T (2007) Progress of Theoretical Physics Supplement 167: 62. Kivelson S, et al. (1992) Physical Review B 46: 2223. Ko¨nig M, Buhmann H, Molenkamp LW, et al. (2008) Journal of the Physical Society of Japan 77: 031007. Koshino M, et al. (2001) Physical Review Letters 86: 1062. Koshino M, et al. (2002) Physical Review B 65: 045310. Kramer B, Ohtsuki T, and Kettemann S (2005) Physics Reports 417: 211. Landau LD, Lifshitz EM, and Pitaevskii LP (1984) Electrodynamics of Continuous Media, 2nd edn., Pergamon. Laughlin RB (1981) Physical Review B 23: 5632. MacDonald AH (1983) Physical Review B 28: 2235. Maksym PA, et al. (2000) Journal of Physics: Condensed Matter 12: R299. Mohr PJ, Taylor BN, and Newell DB (2008) Reviews of Modern Physics 80: 633. Morgenstern M (2007) In: Scanning Probe Microscopy, pp. 349–371. New York: Springer. Morimoto T, et al. (2008) Physical Review B 78: 073406. Murakami S (2008) Progress of Theoretical Physics Supplement 176: 279. Nachtwei G (1999) Physica E 4: 79. Novoselov KS, et al. (2005) Nature 438: 197.
209
Pruisken AMM (1990) In: Prange RE and Girvin SM (eds.) The Quantum Hall Effect, 2nd edn., 117. New York: Springer. Strˇeda P (1982) Journal of Physical: Condensed Matter 15: L718. Thouless DJ, Kohmoto M, Nightingale P, and den Nijs M (1982) Physical Review Letters 49: 405. Tsukazaki A, et al. (2008) Physical Review B 78: 233308. von Klitzing K, Dorda G, and Pepper M (1980) Physical Review Letters 45: 494. Widom A (1982) Physics Letters A 90: 474. Zheng Y and Ando T (2002) Physical Review B 65: 245420.
Further Reading Aoki H (1987) Reports on Progress in Physics 50: 655. Chakraborty T and Pietila¨inen P (1988) The Fractional Quantum Hall Effect. Springer. Das Sarma S and Pinczuk A (eds.) (1997) Perspective in Quantum Hall Effects. New York: Wiley. Landwehr G (1986) Festko¨rperprobleme 26: 17. Prange RE and Girvin SM (eds.) (1990) The Quantum Hall Effect, 2nd edn. New York: Springer. Yoshioka D (2002) The Quantum Hall Effect. Springer.
1.06 Fractional Quantum Hall Effect and Composite Fermions J K Jain, The Pennsylvania State University, University Park, PA, USA ª 2011 Elsevier B.V. All rights reserved.
1.06.1 1.06.2 1.06.2.1 1.06.2.2 1.06.2.3 1.06.3 1.06.3.1 1.06.3.2 1.06.4 1.06.5 1.06.6 1.06.7 1.06.7.1 1.06.7.2 1.06.7.3 1.06.7.4 1.06.7.5 1.06.7.6 1.06.7.6.1 1.06.7.6.2 1.06.7.7 1.06.8 References
Introduction Quantum Hall Effect Phenomenology The Hall Effect Integral Quantum Hall Effect Fractional Quantum Hall Effect Electrons in a Magnetic Field: Landau Levels Landau-Level Degeneracy Filling Factor Integral Quantum Hall Effect Theory The Fractional Quantum Hall Effect Problem Composite Fermions: Basic Foundations Consequences Theory of Fractional Quantum Hall Effect Quantitative Tests against Exact Results Composite Fermion Fermi Sea Effective Magnetic Field Spin Physics Interacting Composite Fermions: New Fractions FQHE of composite fermions Pairing Fractional Charge Open Issues
211 211 211 211 212 212 213 214 214 214 214 216 216 217 218 218 218 219 219 219 220 220 220
Glossary Composite fermions Bound states of electrons and quantized vortices. Degeneracy Multiplicity of an energy level. Fermi sea The state in which fermions occupy all states below certain energy. Filling factor Number of filled Landau levels. Fractional quantum Hall effect Observation of plateaus in the Hall resistance at quantized values characterized by simple fractions.
Hall effect Observation of a voltage transverse to the direction of the current flow in the presence of a magnetic field. Integral quantum Hall effect Observation of plateaus in the Hall resistance at quantized values characterized by integers. Landau levels Quantized kinetic energy levels of electrons in a magnetic field.
Nomenclature
h I N Pf R RH
B B e E
210
magnetic field magnetic field experienced by composite fermions electron charge electric field
Planck’s constant current number of electrons Pfaffian resistance Hall resistance
Fractional Quantum Hall Effect and Composite Fermions
1.06.1 Introduction
B
Collective quantum behavior is of fundamental interest in condensed matter physics, where a collection of interacting particles behaves in a surprising manner that would be difficult to guess from the knowledge of the properties of single particles. Superconductivity and superfluidity are two such well-known examples. During the last three decades, a new collective quantum fluid state of matter has been discovered and studied, which occurs when electrons are confined to two dimensions, cooled to near absolute zero temperature, and subjected to a strong magnetic field. The most dramatic, and unexpected, manifestation of this phase is the ‘fractional quantum Hall effect’ (Tsui et al., 1982). The fractional quantum Hall effect (FQHE) and many other properties of this state result from the formation of a new class of topological particles, called ‘composite fermions’, which are bound states of electrons and quantized microscopic vortices. The compositefermion quantum fluid provides a new paradigm for collective behavior. This chapter describes the essential phenomenology of the fractional quantum Hall effect, how the composite fermion theory resolves it, and other consequences of the existence of composite fermions.
1.06.2 Quantum Hall Effect Phenomenology 1.06.2.1
I I VL
The phenomenon has a classical origin. The Lorentz equation of electrodynamics 1 F¼q Eþ vB c
ð1Þ
where is the conductivity, and J ¼ qv is the current density for particles of charge q and density moving with a velocity v. The more familiar I ¼ V/R can be obtained from this local relation. In the presence of a magnetic field, the electron current flows in a direction perpendicular to the plane containing the electric and the magnetic fields. Alternatively, passage of a current induces a voltage perpendicular to the direction of the current flow, called the Hall voltage, VH, as seen in Figure 1. In addition to the usual (longitudinal) resistance, R ¼ VL/I, a new resistance, called the Hall resistance, is defined as VH I
ð2Þ
ð3Þ
gives the force on a particle of charge q, moving with a velocity v, in the presence of electric and magnetic fields. A consequence of this equation is that for crossed electric and magnetic fields, say E ¼ Eyˆ and B ¼ Bzˆ , the charged particle drifts in a direction perpendicular to the plane containing the two fields, with a velocity v ¼ c(E/B)xˆ . Current density is given by J ¼ qv, where is the (three-dimensional) density of particles. That produces the Hall resistivity H ¼
The Ohm’s law is given by
RH ¼
VH
Figure 1 Schematics of magnetotransport measurements. I, VL, and VH are the current, longitudinal voltage, and the Hall voltage, respectively. The longitudinal and Hall resistances are defined as RL X VL/I and RH X VH/I.
The Hall Effect
J ¼ E
211
Ey B ¼ Jx qc
ð4Þ
The Hall measurement is used routinely to measure the density of the mobile charges, as well as the sign of the charge carriers (i.e., whether they are electrons or holes). 1.06.2.2
Integral Quantum Hall Effect
When the Hall experiments are performed in a system in which electrons are confined to two dimensions (such confinement is achieved, for example, by constructing AlGaAs–GaAs quantum-wells to confined motion in one dimension), the behavior changes in a qualitative manner, as shown in Figure 2, taken from Stormer (1999). Instead of being proportional to B, now Hall resistance shows plateaus. Most prominent are plateaus on which the Hall resistance is precisely quantized at
212 Fractional Quantum Hall Effect and Composite Fermions
Hall resistance, RH (h/e2)
3
2/5 3/7 4/9 1/2
2 3/5 4/7 2/3 4/5
RH 1
Resistance, R
0
more precisely), and therefore the Hall resistance measurements also provide an accurate estimate of the fine structure constant. Concurrent with the quantized plateaus is a dissipation-less current flow in the limit of zero temperature.
1/3
1 4/3 5/3 2 3 4/3 4 5/3
2/5
1/3
3/7
1.06.2.3 2/3 3/5 4/5 4/7
At lower temperatures, higher magnetic fields, and with better quality samples, plateaus are observed on which the Hall resistance is precisely quantized at
4/9
R 43 2
1
RH ¼ 1/2
00
20 10 Magnetic field (T)
30
Figure 2 Hall and the longitudinal resistances for a twodimensional electron system. The Hall resistance shows precisely quantized plateaus whereas the longitudinal resistance exhibits minima. Stormer HL (1999) Nobel lecture: The fractional quantum Hall effect. Reviews of Modern Physics 71: 875–889.
RH ¼
h ne 2
ð5Þ
where n is an integer, h is the Planck’s constant, and e is the electron charge. This is referred to as the integral quantum Hall effect (IQHE), or the ‘von Klitzing effect’ (von Klitzing et al., 1980). The quantization of the Hall resistance is independent of materials details, sample type, or geometry; it is also robust to variation in temperature and disorder, provided they are sufficiently small. The precision of the quantization, that is the correctness of the preceding equation, has been verified to one part in 107 for absolute accuracy, and to a few parts in 1011 for relative accuracy. The constant RK ¼
Fractional Quantum Hall Effect
h ¼ 25813:807 . . . e2
ð6Þ
is called the von Klitzing constant, and has been adopted as the unit of resistance. It should be noted that the fine structure constant ¼
e2 hc
h fe 2
ð8Þ
where f is a fraction. This is the ‘fractional quantum Hall effect’ (FQHE) also known as the Tsur– Stormer–Gossard effect (Tsui et al., 1982). More than 75 fractions have been observed to date, and more are being observed as the experimental conditions and the sample quality are improved. Most of the observed fractions have odd denominators, and they occur in certain sequences, with some examples given later in Equations (50)–(52). One exception to the ‘odd denominator rule’ is the FQHE state with f ¼ 5/2. The theory of FQHE must explain: the origin of gaps in a partially filled Landau level; the dominance of odd denominator fractions; the origin of sequences; the order of stability of fractions; the nature of state at even denominator fractions; the origin of 5/2; the role of spin; and the nature of neutral and charged excitations. We shall see below that the composite fermion theory provides a natural explanation of these facts, besides giving a microscopic theory that allows detailed quantitative comparisons with exact numerical results as well as experiment. At the same time, the composite fermion theory makes verifiable predictions (such as the existence of composite fermions; the composite fermion Fermi sea; and effective magnetic field) and unifies the fractional and the integral quantum Hall effects.
1.06.3 Electrons in a Magnetic Field: Landau Levels
ð7Þ
also involves the same combination of the Planck’s constant and the electron charge as the von Klitzing constant (the speed of light c being known much
The Hamiltonian for a nonrelativistic electron (relativistic effects are neglected throughout) moving in two dimensions in a perpendicular magnetic field is given by
Fractional Quantum Hall Effect and Composite Fermions
H¼
1 eA 2 pþ 2mb c
Here, e is defined to be a positive quantity, the electron’s charge being e, and mb is the band mass of the electron. The canonical momentum is defined as p ¼ ihÑ, and A satisfies Ñ A ¼ B zˆ
ð10Þ
The Schro¨dinger equation H ¼ E
H ¼ ay a þ
ð9Þ
ð11Þ
L ¼ – h b y b – ay a
1=2 hc eB
H jn; mi ¼ En jn; mi 1 En ¼ n þ 2
ð12Þ
ð13Þ
and the unit of energy as the cyclotron energy h!c ¼ h
eB mb c
ð14Þ
1 q2 1 q q –4 z – z þ z þ z 2 qz q z qzq z 4
a; ay ¼ 1; b; b y ¼ 1
ð27Þ
which satisfies aj0; 0i ¼ b j0; 0i ¼ 0
ð28Þ
The single-particle states are especially simple in the lowest Landau level (n ¼ 0): ð16Þ
ð17Þ ð18Þ ð19Þ
1
ðb y Þm zm e – 4z z 0;m ¼ hrj0; mi ¼ pffiffiffiffiffi 0;0 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi m! 22m m !
ð29Þ
where we have used Equations (18) and (27). Aside from the ubiquitous Gaussian factor, a general state in the lowest Landau level is simply given by a polynomial of z. In other words, apart from the Gaussian factor, the lowest Landau level wave functions are analytic functions of z.
ð20Þ
1.06.3.1
ð21Þ
States with a given n but different m are degenerate. To obtain the degeneracy of a Landau level, consider a disk of radius R centered at the origin, and ask how many states lie inside it. The degeneracy can be shown to be independent of the Landau level. Taking, for simplicity, the lowest
which have the property that
ð26Þ
where n is called the Landau level index, and m ¼ n, n þ 1,. . . is the angular momentum quantum number. The single-particle orbital at the bottom of the two ladders defined by the two sets of raising and lowering operators is
ð15Þ
We further define the following sets of ladder operators: 1 z q þ2 b ¼ pffiffiffi qz 2 2 1 z q –2 b y ¼ pffiffiffi q z 2 2 1 z q y –2 a ¼ pffiffiffi qz 2 2 1 z q þ2 a ¼ pffiffiffi q z 2 2
ð25Þ
1 1 hrj0; 0i X 0;0 ðrÞ ¼ pffiffiffiffiffi e – 4zz 2
Here we have defined new variables z and z: z ¼ x – iy ¼ r e – i ; z ¼ x þ iy ¼ r ei
ð24Þ
ðb y Þmþn ðay Þn jn; mi ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffi j0; 0i ðm þ nÞ! n !
the Hamiltonian can be expressed as H¼
ð23Þ
Exploiting the property [H,L] ¼ 0, the eigenfunctions are chosen to diagonalize H and L simultaneously. The analogy to the Harmonic oscillator problem immediately gives the solution
Choosing the unit of length as the magnetic length ,¼
ð22Þ
This is the familiar Hamiltonian of a simple harmonic oscillator. The angular momentum operator is defined as
can be solved in several gauges, but the most convenient for the fractional quantum Hall effect is the symmetric gauge: Br B ¼ ð – y; x; 0Þ A¼ 2 2
1 2
213
and all the other commutators are zero. In terms of these operators, the Hamiltonian can be written as
Landau-Level Degeneracy
214 Fractional Quantum Hall Effect and Composite Fermions
Landau level, the eigenstate j0, mi has itsffi weight pffiffiffiffiffi located at the circle of radius r ¼ 2ml. Thus, the largest value of m for which the state falls inside the disk is given by m ¼ R2/2l2, which is also the total number of eigenstates in the lowest Landau level that fall inside the disk (ignoring order one corrections). Thus, the degeneracy per unit area is G¼
1 eB B ¼ ¼ 2l 2 hc 0
ð30Þ
Here we have defined the flux quantum as 0 ¼ hc/e; the degeneracy for each Landau level is equal to the flux quanta penetrating the sample.
1.06.3.2
Filling Factor
The filling factor is the number of occupied Landau levels, which depends on the density and the magnetic field. It is given by 0 ¼ B
ð31Þ
where is the two-dimensional (2D) density of electrons. The filling factor is inversely proportional to the magnetic field. As the magnetic field is increased, each Landau level can accommodate more and more electrons, and, as a result, fewer and fewer Landau levels are occupied. Only the lowest Landau levels is occupied at sufficient high magnetic fields.
1.06.4 Integral Quantum Hall Effect Theory The integral quantum Hall effect can be explained (Laughlin, 1981) in a model that neglects interactions between electrons. It occurs because the state of electrons at an integral filling factor is very simple: it contains a unique ground state containing an integral number of filled Landau levels, separated from excitations by the cyclotron or the Zeeman energy gap. (In other words, the state is incompressible, because to compress the ground state creates finite energy excitations.) It should be noted that the detailed explanation of the existence of the plateaus also requires a consideration of disorder-induced Anderson localization of some states.
1.06.5 The Fractional Quantum Hall Effect Problem The phenomenon of FQHE indicates that gaps open not only at integral fillings (where their physics is straightforward), but also at many fractional filling factors. The essential goal for the theory of the FQHE is to explain the origin of these gaps. This requires a consideration of interelectron interactions, because gaps occur only at integral fillings for noninteracting electrons. At high magnetic fields all electrons occupy the lowest Landau level. Their kinetic energy is then constant, hence irrelevant. With proper units for energy and length scales, interacting electrons in a high magnetic field are mathematically described by the Hamiltonian H ¼P
LLL
! X1 P r j
LLL
þ Ve – bg þ Vbg – bg
ð32Þ
Here rjk ¼ jzj zkj is the distance between the electrons j and k, P LLL denotes projection into the lowest Landau-level (LLL) subspace, and the last two terms denote interaction between electrons and a uniform positive background. There are no parameters in the Hamiltonian; all energies are to be expressed in units of the Coulomb energy: VC ¼
e2 ",
ð33Þ
where " is the dielectric constant of the background material (" 13 for GaAs). For some condensed matter systems, a nontrivial collective phenomenon can be understood as an instability of a normal state, which is the state obtained when the interaction is switched off. For the FQHE problem, switching off the interaction produces a large number of degenerate ground states; for example, for 109 electrons at ¼ 2/5 the number of degenerate ground states in the absence of inter8 action is 10710 . Further, there is no small parameter in the theory, which prevents a meaningful perturbation theory.
1.06.6 Composite Fermions: Basic Foundations With the observation of many fractions, an analogy between the FQHE and the IQHE could be identified, which led to the postulation of composite
Fractional Quantum Hall Effect and Composite Fermions
fermions, and an explanation of the FQHE as the IQHE of composite fermions (Jain, 1989). The most important accomplishment of the composite fermion (CF) theory is the identification of the particles of the FQHE which can be considered as weakly interacting to a good approximation. The CF theory can be motivated by a Chern–Simons mean field theory (Jain, 1989; Lopez and Fradkin, 1991; Halperin et al., 1993), but its final outcome can be stated succinctly. The eigenfunctions and eigenenergies for the ground and (low-energy) excited states of strongly interacting electrons at an arbitrary lowest Landau level filling are expressed in terms of the known solutions of the noninteracting electron problem at the Landau level filling as follows: ¼ P
LLL
Y
zj – zk
2p
ð34Þ
215
argued that these correlations survive projection into the lowest Landau level. Q 2p attaches 2p The Jastrow factor j < k(zj zk) quantized vortices to each electron in . The bound state of an electron and 2p vortices is interpreted as a particle called the composite fermion. As composite fermions move about, the vortices bound to them produce Berry phases, which cancel part of the Aharonov Bohm phases originating from the external magnetic field. The effective magnetic field can be determined in a Berry-phase calculation. When a composite fermion, that is an electron along with its vortices, is taken in a closed loop enclosing an area A, it acquires a Berry phase ¼ – 2
BA þ 22pNenc 0
ð38Þ
j
and E ¼
E D P j
h j i
þ Vel – bg þ Vbg – bg
ð35Þ
where ¼
2p 1
B ¼ B – 2p0
ð36Þ ð37Þ
Here, are the eigenfunctions of noninteracting electrons at , P LLL projects the wave function to its right into the lowest Landau level, p is an integer, and B is an effective magnetic field. These equations define a one-to-one correspondence between the ground and excited states at filling factor (or magnetic field B) and (magnetic field B). The problem is often much simpler at , and for ¼ n (where n is an integer), we know the exact solution for the ground and low energy states. The wave function for the ground state at ¼ 1/(2p þ 1) was known previously (Laughlin, 1983). Electrons capture vortices to turn into composite fermions because this is the most effective way for them to stay away from one another. A typical wave function satisfying the Pauli principle vanishes as r when two particles approach one another, r being the distance separating Q them, but the unprojected wave functions j < k(zj zk)2p vanish as r2p þ 1, with the Jastrow factor contributing 2p to the exponent and contributing the rest. The Jastrow factor is very effective in keeping particles apart from one another, producing favorable correlations. It can be
where Nenc is the number of composite fermions inside the loop. The first term is the familiar Aharonov Bohm (AB) phase due to a charge going around in a loop. The second is the Berry phase due to the 2p vortices going around Nenc particles, with each particle contributing a phase of 2. For uniform density states, we replace Nenc in Equation (38) by its average value A and equate the entire phase to the AB phase due to an effective magnetic field, B, to obtain Equation (37). Composite fermions thus sense a magnetic field B that is much smaller than the applied magnetic field, and can even be zero or negative. This property of composite fermions distinguishes them from electrons, and lies at the root of most of the phenomenology. Composite fermions form Landau-like levels in the reduced magnetic field, which are called levels, and occupy of them. Composite fermions represent a new class of particles realized in nature. A vortex is a topological object, because the quantum mechanical phase associated with a closed loop around a vortex is exactly 2, independent of the shape and the size of the loop. (The topological character of vortices is implicit in the fact that we count them.) As a result, composite fermions are collective, topological particles. All fluids of composite fermions are thus topological quantum fluids. The most direct consequence of the topological quantization of the vorticity of composite fermions is the effective magnetic field, which is responsible for the FQHE and several other phenomena. The Chern–Simons field theoretic formulation of composite fermions (Jain, 1989; Lopez and Fradkin, 1991; Halperin et al., 1993) proceeds through a singular gauge transformation defined by
216 Fractional Quantum Hall Effect and Composite Fermions
¼
Y j
z –z j k z j – z k
!2p
1.06.7 Consequences ð39Þ
9
under which the eigenvalue problem transforms into H 99 ¼ E9 " H9 ¼
ð40Þ
2 1 X e e pi þ Aðri Þ – aðri Þ þV 2mb i c c aðri Þ ¼
#
9 2p X 0 Ñi ij 2 j
ð41Þ
ð42Þ
zj – zk is the relative angle between zj – zk the particles j and k. The prime denotes that j ¼ i is to be excluded from the sum. The magnetic field corresponding to a(ri) is given by where jk ¼ iln
bi ¼ Ñi aðri Þ ¼ 2p0
9 X
2 ðri – rl Þ
ð43Þ
l
The above transformation thus amounts to attaching a point flux of strength 2p0 to each electron, which is how the composite fermion is modeled in this approach. To proceed further, one makes a mean-field approximation, which amounts to spreading the point flux on each electron into a uniform magnetic field. Formally, one writes A – a X A þ A
ð44Þ
ð45Þ
Ñ A ¼ B ˆz
The transformed Hamiltonian then becomes
2 1 X e H9 ¼ pi þ A ðri Þ þV þ V 9 ¼ H0 þ V þ V 9 2mb i c ð46Þ
where V is the Coulomb interaction and V 9 denotes the terms containing A. The solution to H0 describes free composite fermions in an effective magnetic field B , and V þ V 9 is the effective interaction between them, which is to be treated perturbatively. When one transforms this problem into the field theoretical Langrangian, it is seen to be equivalent to the familiar Chern–Simons theory. This approach is believed to capture the topological properties of composite fermions, but does not lend itself to a systematic perturbative treatment because of the lack of a small parameter.
1.06.7.1 Effect
Theory of Fractional Quantum Hall
The FQHE of electrons is a manifestation of the integral QHE for composite fermions. The latter occurs because a gap opens when composite fermions fill an integral number of levels, that is, when ¼ n. These fillings correspond to electron filling factors given by the sequences: ¼
n 2pn 1
ð47Þ
A gap here results in an FQHE plateau at RH ¼ h/fe2, with f ¼
n 2pn 1
ð48Þ
FQHE at f also implies an FQHE at the hole partner: f ¼ 1–
n 2pn 1
ð49Þ
These fractions can be obtained by defining the original problem in terms of holes – rather than electrons – in the lowest Landau level, and making composite fermions by binding vortices to holes. The composite fermion (CF) theory provides a natural explanation of many experimental facts. The fractions f given by Equations (48) and (49) are precisely the prominently observed fractions. Furthermore, the fractions appear in sequences because they are all derived from the integer sequence of the IQHE. Some of the experimental sequences are: f ¼ 1=3; 2=5; 3=7; 4=9; . . . ; 10=21
ð50Þ
f ¼ 2=3; 3=5; 4=7; . . . ; 10=19
ð51Þ
f ¼ 1=5; 2=9; . . . ; 6=25
ð52Þ
The fractions have odd denominators because the vortex quantum number 2p is an even integer. Different flavors of composite fermions (i.e., composite fermions carrying different numbers of vortices) occur in different filling factor regions. The filling factor range 1/2 1/3 is described in terms of composite fermions carrying two vortices; the range 1/3 > 1/5 in terms of composite fermions carrying four vortices (with B antiparallel to B for 1/3 > > 1/4); and so on. The region 1 > 1/2 of electrons maps into 0 < 1/2 of holes as a result of particle–hole symmetry in the lowest Landau level, and can be understood in terms of composite fermions made of holes.
Fractional Quantum Hall Effect and Composite Fermions
1.06.7.2 Results
Quantitative Tests against Exact
217
and 3/7 (the first, second, and third rows, respectively) are shown in Figure 3 (dashes). This figure also shows the predicted energies from the CF theory, without any adjustable parameters, as dots. The ground state corresponds to one, two, or three filled levels of composite fermions, the the lowest energy branch of excitations is a particle–hole pair of composite fermions. The predicted energies agree to within 0.1%, and the overlaps between the exact and the CF wave functions are greater than 99% for the numerical systems. Furthermore, the states in between the special fractions n/(2pn 1) are well described in terms of composite fermions at a nonintegral filling factor. Monte Carlo methods allow determination of the thermodynamic limits of various experimentally measurable quantities, such as excitation gaps. These are often in 20–50% agreement with experiment, with the discrepancy caused primarily by disorder. An important quantity is collective modes, 31 42 101 111 168 175 227 230 277
Exact results can be obtained for the Hamiltonian in Equation (32) for a small number of particles by a brute force diagonalization, because the dimension of the Fock space in the lowest Landau level is finite for a finite system. Typically, depending on the filling factor, 10–16 electrons can thus be studied. The exact solution gives the eigenenergies and eigenfunctions for all eigenstates. The CF explanation of the FQHE is fully confirmed by comparison to exact results (Dev and Jain, 1992; Wu et al., 1993; Jain and Kamilla, 1998) A convenient geometry is the spherical geometry, in which electrons move on the surface of a sphere, with a radial magnetic field produced by a magnetic monopole of strength Q at the center, which produces a total magnetic flux of 2Q0. The eigenstates are conveniently labeled by the total orbital angular momentum L. The exact spectra at ¼ 1/3, 2/5,
–0.41
–0.43 –0.44
–0.44
N = 8 , v = 1/3
52 83 179 212 304 328 416
–0.45
N = 10, v = 1/3
8 8 21 22 35 33
E (e2/εl)
–0.42
–0.46 –0.47 N = 10, v = 2/5
–0.49 –0.46
127 263 493 621 952 744 1182
–0.48
N = 8 , v = 2/5 8 8 21 22 35 33 45
E (e2/εl)
–0.45
E (e2/εl)
–0.47 –0.48 –0.49 N = 12, v = 3/7
–0.50 –0.51
0
3
6 L
9
12
N = 9 , v = 3/7
0
3
6
9
12
L
Figure 3 Comparison of spectra obtained from exact diagonalization (dashes) and CF theory (dots). The spectra in the three rows are for ¼ 1/3, 2/5, and 3/7, respectively. The x-axis label L is the total orbital angular momentum of the state. From Jain JK and Kamilla RK (1998) Composite fermions: Particles of the lowest Landan level. In: Heinonen O (ed.) Composite Fermions, ch. 1. New York: World Scientific.
218 Fractional Quantum Hall Effect and Composite Fermions
which are understood as excitons of composite fermions; the experimental measurements of the energies and the dispersions (Pinczuk et al., 1993; Kukushkin et al., 2009) at various fractions such as 1/3, 2/5, 3/7, and 4/9 are in excellent agreement with the predictions of the composite fermion theory (Scarola et al., 2000). 1.06.7.3
Composite Fermion Fermi Sea
No general principle excludes FQHE at evendenominator fractions. Such FQHE has been observed, for example, at f ¼ 5/2. The CF theory provides a natural explanation for why even denominator FQHE is rare: the model of noninteracting composite fermions produces only odd-denominator fractions; any even-denominator fraction must necessarily owe its existence to weak residual interactions between composite fermions, and, therefore, can be expected to be much weaker. The nature of state at ¼ 1/2, the simplest fraction, has been of interest. It is obtained as the n ! 1 limit of the sequence f ¼ n/(2n þ 1). If the model of noninteracting composite fermions continues to be valid in this limit, their effective magnetic field B vanishes and they form a Fermi sea, called the CF Fermi sea (Halperin et al., 1993; Kalmeyer and Zhang, 1992). The lack of FQHE follows because the Fermi sea has no gap to excitations. Several experiments have confirmed the formation of composite fermions in the vicinity of ¼ 1/2 and of the CF Fermi sea at ¼ 1/2. These include. Shubnikov de Haas oscillations of composite fermions (Du et al., 1994), linear opening of the CF gap (Du et al., 1993); measurement of the cyclotron resonance of composite fermions (Kukushkin et al., 2007); and measurements of the CF Fermi wave vector through the semiclassical cyclotron orbit of composite fermions (Willet et al., 1993; Kang et al., 1993; Goldman et al., 1994; Smet et al., 1996)
found between the low energy spectra of the exact solutions of interacting electrons at B (from numerical diagonalization) and the exact solutions of noninteracting electrons at B. In addition, several experiments have measured the radius of the cyclotron orbit of the current carrying entities in the vicinity of ¼ 1/2, and confirmed that it is determined by B rather than the applied magnetic field (Willet et al., 1993; Kang et al., 1993; Goldman et al., 1994; Smet et al., 1996).
1.06.7.5
Spin Physics
The spin degree of freedom is frozen when the Zeeman splitting is large compared to the interaction energy. In that limit, all FQHE states in the lowest Landau level are fully spin polarized. One might expect, by application of the Hund’s maximum spin rule to electrons in the lowest Landau level, that the state would be fully polarized at all Zeeman energies. That is not the case, however. The actual state is determined by application of the Hund’s rule to composite fermions. In a model that assumes that composite fermions can be taken as nearly independent particles with an effective mass, their physics is straightforward. The IQHE of composite fermions occurs at ¼ n ¼ n" þ n#, where n" is the number of occupied spin-up levels and n# is the number of occupied spin-down levels. This fraction corresponds to electron fillings ¼
n n" þ n# ¼ 2pn 1 2pðn" þ n# Þ 1
The spin polarization of the state is given by e ¼
n" – n# n" þ n#
LLL n" ;n#
Y
2p zj – zk
ð55Þ
j
Effective Magnetic Field
The reduced effective magnetic field, which is a direct evidence of binding of vortices to electrons and the formation of composite fermions, has been confirmed directly by numerous means. The appearance of fractional sequences that correspond to the integer sequence of the noninteracting fermions and the formation of a Fermi sea at 1/2 filled Landau level are experimental proofs of the effective magnetic field, as is the one-to-one correspondence
ð54Þ
and wave function for the FQHE state is n 2pnþ1 ¼P
1.06.7.4
ð53Þ
or n 2pnþ1 ¼P
LLL n" ;n#
Y
2p zj – zk
ð56Þ
j
where n" ;n# ¼ n" n#
ð57Þ
are wave functions of IQHE states with n" spin-up and n# spin-down Landau levels occupied.
Fractional Quantum Hall Effect and Composite Fermions
Thus, while inclusion of spin does not give new fractions, it produces, in general, many states at a given fraction, whose spin polarizations are predicted by the CF theory. Take the example of ¼ 4/9, which corresponds to n ¼ 4 of composite fermions. There are three possibilities: (n", n#) ¼ (2,2), (3,1), and (4,0). The unpolarized state (2,2) is obtained at the lowest Zeeman energies (as application of Hund’s rule to composite fermions would predict), the partially polarized state (3,1) at intermediate Zeeman energies, and the fully polarized (4,0) at large Zeeman energies. Upon increasing Ez, the groundstate spin changes discontinuously when the levels of up and down spins cross one another. A quantitative determination of the energies of these states leads to a theoretical determination of the actual phase diagram of the FQHE states as a function of the Zeeman energy (Wu et al., 1993; Park and Jain, 1998) The Zeeman energy can be varied experimentally by application of a magnetic field parallel to the two-dimensional layer (tilted field experiment), or by changing the density so the FQHE state occurs at different B. The above physics has been fully confirmed by extensive experimentation (Du et al., 1995; Kukushkin et al., 1999).
1.06.7.6 Interacting Composite Fermions: New Fractions 1.06.7.6.1
FQHE of composite fermions
n are obtained most 2pn 1 immediately in the CF theory, other fractions are not ruled out. To see the physics of the next generation, fractions consider electrons in the filling factor range While the fractions ¼
1=3 < < 2=5
1 2 and f ¼ 1 þ 3 3
f ¼1þ
219
ð60Þ
which produce new electron fractions f ¼
4 11
and
f ¼
5 13
ð61Þ
between the familiar fractions 1/3 and 2/5, consistent with experimental observations (Pan et al., 2003). Here, composite fermions in the partially filled second level capture two more vortices to turn into composite fermions of a different flavor, which condense into their own levels, thereby opening a gap and producing quantum Hall effect. This physics has been confirmed by exact diagonalization studies (Chang and Jain, 2004; Wo´js et al., 2004), as shown in Table 1. Many more FQHE states can similarly be constructed. 1.06.7.6.2
Pairing As noted above, FQHE has been observed (Willet et al., 1987) at ¼ 5/2. Writing 5 1 ¼2þ 2 2
ð62Þ
and treating the lowest filled Landau level as inert (which, counting the spin degree of freedom, accounts for the 2 on the right hand side) shows that 5/2 corresponds to a filling of 1/2 in the second Landau level. Thus, half-filled second Landau level behaves qualitatively differently from the half-filled lowest Landau level. The currently most promising scenario for the explanation of the 5/2 FQHE is that composite fermions form a p-wave paired state, described by a Pfaffian wave function (Moore and Read, 1991): Pf 1=2
ð58Þ
1 ¼ Pf zi – zj
Y
zi – zj
i<j
2
"
1X 2 exp – jzk j 4 k
# ð63Þ
which map into composite fermions in the range 1 < < 2
ð59Þ
The lowest level is fully occupied and the second one partially occupied. (We take composite fermions to be fully spin-polarized.) No FQHE would result in this region for noninteracting composite fermions, just as non-interacting electrons in the partially filled second Landau level do not exhibit any FQHE. The weak residual interaction between composite fermions, however, can possibly cause a gap to open at certain filling factors. A natural expectation is that the strongest new CF fractions are
Table 1 Energies of two wave functions for 4/11: the exact Coulomb state ex 4=11 and the trial wave function tr4=11 ¼ P LLL 4=3 21 Eex
Etr
Overlap
0.441214
0.44088(4)
0.99
Results are for N ¼ 12 particles. The overlap is defined as
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi htr jð0Þ i= hð0Þ jð0Þ ihtr jtr i, where (0) is a state that is very close to the exact ground state (obtained by a method called CF diagonalization). From Chang C-C and Jain JK (2004) Microscopic origin of the next-generation fractional quantum Hall effect. Physical Review Letters 92: 196806.
220 Fractional Quantum Hall Effect and Composite Fermions
where Pf stands for Pfaffian. The Pfaffian of an antisymmetric matrix M is defined, apart from an overall normalization factor, as Pf Mij ¼ AðM12 M34 MN – 1;N Þ
ð64Þ
where A is the antisymmetrization operator. The usual Bardeen–Cooper–Schrieffer wave function for fully polarized electrons can be written as BCS ¼ A½0 ðr1 ; r2 Þ0 ðr3 ; r4 Þ 0 ðrN – 1 ; rN Þ
ð65Þ
which is a Pfaffian. (The fully symmetric spin part is 1 not shown explicitly.) Analogously, Pf ¼ zi – zj describes a p-wave pairing of electrons (p-wave because the system is fully spin-polarized), and Pf 1=2 is interpreted as the p-wave paired state of composite fermions carrying two vortices. The Pfaffian wave function was originally motivated in a conformal field theory approach, which also suggests that its quasiparticles obey so-called nonabelian braiding statistics (Moore and Read, 1991).
1.06.7.7
(Laughlin, 1983), and has been measured in shot noise experiments (de Picciotto et al., 1997).
1.06.8 Open Issues The FQHE and other phenomena in the lowest Landau level are well described by the CF theory. However, the CF theory in its simplest form is not fully satisfactory for the second Landau-level physics. One difference between the two Landau levels is indicated by the observation that while there is no FQHE at ¼ 1/2, an FQHE occurs at ¼ 5/2. Further, far fewer fractions are observed in the second Landau level, and they are much more delicate than those in the lowest Landau level. While other fractions in the second Landau level also have odd denominators (such as 2 þ 1/3 and 2 þ 2/5), their wave functions are rather different than those of the corresponding fractions in the lowest Landau level. Several imaginative approaches are currently being pursued, which have excitations with exotic character (e.g., nonabelian braiding properties). (See Chapters 1.03, 1.05 and 1.07).
Fractional Charge
The excitations of an FQHE state are excited composite fermions. A sole composite fermion in an otherwise empty level is often called a CF quasiparticle, and a missing composite fermion from an otherwise filled level is called a CF quasihole. This description has been confirmed extensively in exact diagonalization studies. In a localized representation, these represent a localized excess or deficiency of charge relative to the uniform ground state. The charge excess associated with a CF quasiparticle is the sum of the charge of an electron (e) and the charge of 2p vortices: – e ¼ – e þ 2pev
ð66Þ
where ev is the charge of a single vortex. The charge of the vortex can be shown to be ev ¼ ve
ð67Þ
The local charge of a CF-quasiparticle at ¼ n/(2pn þ 1) therefore has the fractional value – e ¼ –
e 2pn þ 1
ð68Þ
The topological quantization of the vorticity implies that this charge is precisely quantized. The fractional charge was originally predicted theoretically
References Chang C-C and Jain JK (2004) Microscopic origin of the nextgeneration fractional quantum Hall effect. Physical Review Letters 92: 196806. de Picciotto R, Rezhnikov M, Heiblum M, Umansky V, Bunin G, and Mahalu D (1997) Direct observation of a fractional charge. Nature 389: 162. Dev G and Jain JK (1992) Band structure of the fractional quantum Hall effect. Physical Review Letters 69: 2843. Du RR, Stormer HL, Tsui DC, Pfeiffer LN, and West KW (1993) Experimental evidence for new particles in the fractional quantum Hall effect. Physical Review Letters 70: 2944. Du RR, Stormer HL, Tsui DC, Pfeiffer LN, and West KW (1994) Shubnikov –de Haas oscillations around ¼ 1/2 Landau level filling factor. Solid State Communications 90: 71. Du RR, Yeh AS, Stormer HL, Tsui DC, Pfeiffer LN, and West KW (1995) Fractional quantum Hall effect around ¼ 3/2: Composite fermions with a spin. Physical Review Letters 75: 3926. Goldman VJ, Su B, and Jain JK (1994) Detection of composite fermions by magnetic focusing. Physical Review Letters 72: 2065. Halperin BI, Lee PA, and Read N (1993) Theory of the half-filled Landau level. Physical Review B 47: 7312. Jain JK (1989) Composite-fermion approach for the fractional quantum Hall effect. Physical Review Letters 63: 199. Jain JK and Kamilla RK (1998) Composite fermions: Particles of the lowest Landan level. In: Heinonen O (ed.) Composite Fermions, ch. 1, pp. 1–90. New York: World Scientific. Kalmeyer V and Zhang SC (1992) Metallic phase of the quantum Hall system at even-denominator filling fractions. Physical Review B 46: 9889.
Fractional Quantum Hall Effect and Composite Fermions Kang W, Stormer HL, Pfeiffer LN, Baldwin KW, and West KW (1993) How real are composite fermions? Physical Review Letters 71: 3850. Kukushkin, IV, Smet JH, Scarola VW, Umansky V, and von Klitzing K (2009) Dispersion of the excitations of fractional quantum Hall states. Science 324: 1044. Kukushkin IV, Smet JH, Schuh D, Wegscheider W, and von Klitzing K (2007) Dispersion of the composite-fermion cyclotron resonance mode. Physical Review Letters 98: 066403. Kukushkin, IV, von Klitzing K, and Eberl K (1999) Spin polarization of composite fermions: Measurements of the fermi energy. Physical Review Letters 82: 3665. Laughlin RB (1981) Quantized Hall conductivity in two dimensions. Physical Review B 23: 5632. Laughlin RB (1983) Anomalous quantum Hall effect: An incompressible quantum fluid with fractionally charged excitations. Physical Review Letters 50: 1395. Lopez A and Fradkin E (1991) Fractional quantum Hall effect and Chern–Simons gauge theories. Physical Review B 44: 5246. Moore G and Read N (1991) Nonabelions in the fractional quantum Hall effect. Nuclear Physics B 360: 362. Pan W, Stormer HL, Tsui DC, Pfeiffer LN, Baldwin KW, and West KW (2003) Fractional quantum Hall effect of composite fermions. Physical Review Letters 90: 016801. Park K and Jain JK (1998) Phase diagram of the spin polarization of composite fermions and a new effective mass. Physical Review Letters 80: 4237. Pinczuk A, Dennis BS, Pfeiffer LN, and West KW (1993) Observation of collective excitations in the fractional quantum Hall effect. Physical Review Letters 70: 3983. Scarola VW, Park K, and Jain JK (2000) Rotons of composite fermions: Comparison between theory and experiment. Physical Review B 61: 13064. Smet JH, Weiss D, Blick RH, et al. (1996) Magnetic focusing of composite fermions through arrays of cavities. Physical Review Letters 77: 2272.
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Stormer HL (1999) Nobel lecture: The fractional quantum Hall effect. Reviews of Modern Physics 71: 875–889. Tsui DC, Stormer HL, and Gossard AC (1982) Two-dimensional magnetotransport in the extreme quantum limit. Physical Review Letters 48: 1559. von Klitzing K, Dorda G, and Pepper M (1980) New method for high-accuracy determination of the fine-structure constant based on quantized Hall resistance. Physical Review Letters 45: 494. Willett RL, Eiscnstein JP, Stormer HL, Tsui DC, Gossard AC, and English JH (1987) Observation of an even-denominator quantum number in the fractional quantum Hall effect. Physical Review Letters 59: 1776. Willett RL, Ruel RR, West KW, and Pfeiffer LN (1993) Experimental demonstration of a Fermi surface at one-half filling of the lowest Landau level. Physical Review Letters 71: 3846. Wo´js A, Yi K-S, and Quinn JJ (2004) Fractional quantum Hall states of clustered composite fermions. Physical Review B 69: 205322. Wu XG, Dev G, and Jain JK (1993) Mixed-spin incompressible states in the fractional quantum Hall effect. Physical Review Letters 71: 153.
Further Reading Das Sarma S and Pinczuk A (eds.) (1996) Perspectives in Quantum Hall Effects. New York: Wiley. Heinonen O (ed.) (1998) Composite Fermions. New York: World Scientific. Jain JK (2007) Composite Fermions. London: Cambridge University Press.
1.07 Spin Hall Effect S Murakami, Tokyo Institute of Technology, Tokyo, Japan N Nagaosa, University of Tokyo, Tokyo, Japan ª 2011 Elsevier B.V. All rights reserved.
1.07.1 1.07.2 1.07.2.1 1.07.2.2 1.07.2.3 1.07.2.3.1 1.07.2.3.2 1.07.2.3.3 1.07.2.4 1.07.2.5 1.07.2.5.1 1.07.2.5.2 1.07.2.6 1.07.2.6.1 1.07.2.6.2 1.07.3 1.07.3.1 1.07.3.2 1.07.4 1.07.4.1 1.07.4.2 1.07.4.3 1.07.4.4 1.07.4.5 1.07.4.6 1.07.4.6.1 1.07.4.6.2 1.07.4.6.3 1.07.4.6.4 1.07.4.6.5 1.07.5 1.07.5.1 1.07.5.2 1.07.5.3 1.07.5.3.1 1.07.5.3.2 1.07.5.3.3 References
Introduction – Early History and Background of Spin Hall Effect (SHE) Intrinsic and Extrinsic Mechanism of SHE Intrinsic SHE in p-Type Semiconductors Rashba Model for n-Type Semiconductors Impurity Scattering Effect Spin-independent impurities SO coupling modified by impurities SHE in the hopping regime Mesoscopic SHE Experimental Observation of SHE in Semiconductors Measurement of SHE in semiconductors by KR Measurement of SHE in semiconductors by spin LED Various Issues on the SHE Definition of the spin current Ab initio calculations SHE in Metals Experiments of SHE in Metals Theories of SHE in Metals Quantum Spin Hall Effect Inverted Band Structure and SHI QSHE and Z2 Topological Invariants in 2D and 3D Helical Edge Modes in QSH systems Models and Candidate Materials for QSHE Experimental Observation of QSHE Further Developments in the QSHE Spin-charge separation and charge fractionalization Effective theory and quantized magneto-electric effect Superconducting QSHSs Localization problem related to the QSHS SHE in strongly correlated systems Summary and Perspectives on SHE Comparison between AHE and SHE Comparison between Integer QHE and QSHE Generalization of SHE and Future Directions Optical SHE Multiferroics Future directions
224 227 228 230 231 231 235 236 237 239 239 241 243 243 245 245 245 251 253 253 254 257 258 259 261 261 262 262 263 264 264 264 265 266 266 267 268 269
Glossary Aharonov–Bohm effect The wave function of a charged quantum mechanical particle acquires a phase given by the contour integral of the vector
222
potential along the path, leading to the change in the interferenceofthedifferencepathsduetothemagnetic flux. This effect is called Aharonov–Bohm effect.
Spin Hall Effect
Aharonov–Casher effect Analogously to the Aharonov–Bohm effect, the wave function of a quantum mechanical particle with spin acquires a phase factor which is given by the contour integral of the vector product of the electric field and the spin. This effect is called Aharanov–Casher effect. Angle-resolved photoemission spectroscopy (ARPES) The momentum- and energy-resolved spectroscopy of the emitted electrons from the sample kicked by the high-energy incident light. This can detect the spectral function of the Green’s function for occupied states. Berry phase In 1984, M.V. Berry formulated the quantum phase associated with the adiabatic change of the Hamiltonian. Through the adiabatic change, the wave function is confined within the subspace of the Hilbert space, which leads to the gauge structure and is represented by this Berry phase. BW phase One of the p-wave pairing states for superfluid with the isotropic energy gap. It is now established that the B-phase of 3He is described by this state. Chiral edge channel In quantum Hall system, there appear edge channels at the edges of the sample, which have the definite propagation direction and are called chiral edge channels. The backward scattering is forbidden in the chiral edge channels, which explains the quantized Hall conductance in quantum Hall system. Coulomb blockade In the tunnel junction with small capacitance, the Coulomb charging energy can be larger than the thermal energy, and the tunneling process is blocked. This Coulomb blockade depends on the charge q at the junction periodically, leading to the characteristic oscillation of the tunneling current as a function of gate voltage. Dirac fermion P.A.M. Dirac introduced the relativistic equation of quantum particles with four-component spinor wave function starting from a few fundamental requirements. This is called Dirac equation, and the fermions obeying Dirac equation are the Dirac fermions. In condensed matter systems, it often occurs that the band structure is described by four- or two-component Dirac equations locally in momentum space. eg and t2g orbitals Under the crystal field of Oh symmetry, the five d-orbitals are split into two groups, that is, eg-orbitals (x2 y2, 3z2 r2) and t2g-orbitals (xy,yz,zx).
223
Hanle effect In 1924, Hanle showed the variation of the polarization of the resonant fluorescent light in atoms, which is called the Hanle effect. This effect is now used to measure the lifetime and g-factor of atoms and molecules. Keldysh formalism L. Keldysh developed a Green’s function method which is applicable even to states far away from the equilibrium. This method is called the Keldysh formalism, and is now applied in various transport problems. Kondo effect When a magnetic impurity is inserted into a metallic host, its magnetic moment is screened by the spins of the conduction electrons to form a spin singlet. This phenomenon is called the Kondo effect. Kramers theorem In the system with the odd number electrons, it is proved that the electronic states are energetically (at least doubly) degenerate when the Hamiltonian has the time-reversal symmetry. The doublet related by the time-reversal symmetry operation is called Kramers doublet. Kubo formula In 1957, R. Kubo formulated the linear response theory based on the density matrix formalism. This provides the most fundamental formula of the response functions, and basis to study the physical properties of systems. Light-emitting diode (LED) The light is emitted through the recombination of the electrons and holes when the current is applied into the p–n junction. Magnetic Kerr rotation The rotation of the polarization occurs when the light is reflected from the magnetic system due to the relativistic spin–orbit interaction. Magneto-electric effect The phenomenon where the magnetization is induced by the electric field or the electric polarization is induced by the magnetic field. Majorana fermion In 1937, E. Majorana proposed the real fermions satisfying the relation f† = f called the Majorana fermion (f† (f) is the creation (annihilation) operator of the fermion). Roughly speaking, Majorana fermion is half of the usual fermion. Majorana fermion appears in the exact solution to two-dimensional Ising model, and also in the theory of pairing state of spinless fermions. Mott scattering N.F. Mott derived the formula for the cross section of the Coulomb scattering between the two identical fermions with spin 1/2. By using this formula, the Mott detector to measure the spin state of electrons has been developed.
224 Spin Hall Effect
Multiferroics Materials showing multiple symmetry breakings. Usually, it means those showing the magnetic order and the ferroelectricity simultaneously. Noether’s theorem The theorem connecting the invariance with respect to some transformation and the conservation law in field theory. By using this theorem, one can define the current from the invariance of the Lagrangian. Noncentrosymmetric superconductor Superconductors without the inversion symmetry. In the presence of the spin–orbit interaction, novel properties such as the mixture of singlet and triplet pairing emerge. Nonlinear sigma model A model in field theory describing the field with constraint. This is used to analyze the localization problem in disordered systems.
Seebeck effect The effect where the voltage drop is induced by the temperature gradient. Soliton In field theory, the order parameter field can have the spatial dependence creating a localized kink structure. This localized structure is called soliton. Umklapp scattering In crystals, the conservation of the momentum is relaxed by the reciprocal lattice vectors G’s, and scattering process using this G’s is called Umklapp scattering. Virtual bound state The d-orbitals in metals are almost localized near the atomic nucleus. However, due to the hybridization with the conduction electrons, the d-levels have the finite lifetime and energy broadening, and are called the virtual bound states. Z2 group The group whose elements are (1,1) only.
Nomenclature
si (i = x,y,z) V(x)
H H(k) jj jij k k kb
Hamiltonian Hamiltonian matrix in the Bloch wave number k charge current flowing along jˆ direction spin current with +iˆ spin moving to +jˆ and iˆ spin moving to jˆ wave number the size of the wave number (=k) unit vector parallel to k, defined as kb= k/jkj
1.07.1 Introduction – Early History and Background of Spin Hall Effect (SHE) Hall effect, which is one of the most important effects in condensed matter physics, has been discovered by Hall in 1893 (Hurd, 1973). When the charge current j flows with an applied magnetic field perpendicular to it, a voltage drop occurs in the direction transverse both to the current and magnetic field. This voltage drop VH between the two edges of the sample, called Hall voltage, divided by the current j defines the Hall resistance. This Hall effect originates from the
x * ? (as in x_ ) s i
spin operators with s = 1/2 impurity potential at the position x measured from the impurity position time derivative spin Hall conductivity time-reversal invariant momentum (TRIM). There are four TRIMs in two-dimensional (2D) Brillouin zone and eight in 3D.
Lorentz force acting on the orbital motion of the electrons under the magnetic field; hence, the sign of Hall coefficient is determined by the sign of the charge carriers while its magnitude by the carrier density. A few years after. Hall himself discovered the anomalous Hall effect (AHE) (Hurd, 1973), in which the transverse Hall voltage originates from the spontaneous magnetization even without the external magnetic field in metallic ferromagnets. This effect is driven by the relativistic spin–orbit (SO) interaction, while there are both intrinsic (Karplus and Luttinger, 1954) and extrinsic (Smit, 1955, 1958; Berger, 1970) mechanisms. The intrinsic mechanism
Spin Hall Effect
is due to the anomalous velocity of the Bloch wave function, and is purely the band structure effect. Recently, this intrinsic contribution is reexamined from the modern topological viewpoint, that is, the Berry curvature in the momentum space (Nagaosa, 2006). The extrinsic ones, on the other hand, are due to the impurity scattering influenced by the SO interaction and are classified into (1) the skew scattering (SS) mechanism (Smit, 1955, 1958), where the scattering probability Pk ! k9 from k to k9 is different from Pk9 ! k, and (2) the side jump SJ mechanism (Berger, 1970), where the wavepacket experiences the transverse shift at the scattering. In nonmagnetic metals, one can imagine that each of the up- and down-spin electrons show the AHE in the opposite directions due to the SO interaction. Then, one expects the spin current flow or the spin voltage drop in the transverse direction both in the direction of the charge current and the spin-polarization direction. This effect, called the spin Hall effect (SHE), is the subject of this chapter, which constitutes a rapidly growing field in the spintronics. The SHE was first proposed by D’yakonov and Perel (1971a,b). They considered the extrinsic SS as the origin of the SHE predicting the spin accumulation at the edge of the sample due to the spin current. Here, one remark is that the spin current is time-reversal even, in sharp contrast to the spin accumulation or charge current which are time-reversal odd. This means that the conversion of the spin current into the spin accumulation requires the irreversibility due to the spin relaxation. Later Hirsch (1999) proposed an experimental method to detect the SHE by the measurement of the voltage. With the charge current flowing in the x-direction in the spin Hall system, the spin Hall voltage is produced along the y-direction. When the two edges at y = L/2, with L being the width of the sample, are connected to each other with a conductor, the spin current circulates along the y-direction. This circulating spin current induces the voltage at the conductor along the x-direction. In Hirsch (1999), the voltage signal has been estimated to be of the order of 60 mV for the sample size l ¼ 100 mm for a usual metal. Zhang (2000) also discussed the SHE with a semiclassical Boltzmann equation taking into account the spin diffusion, since SO interaction is the origin for both the SHE and the spin relaxation through the spin-flip processes. As mentioned above, this spin relaxation is also indispensable for the spin accumulation at the edges of the sample.
225
In these works, the SHE has been considered as an extrinsic effect, due to impurities in the presence of SO coupling. Nevertheless, there were no detailed discussions on the size of the effect, and this extrinsic effect is not easily controllable because it is caused by impurities. On the other hand, two groups independently proposed an intrinsic SHE in different systems. Murakami et al. (2003) proposed the SHE in p-type semiconductors such as p-GaAs. Sinova et al. (2004) independently proposed the SHE in n-type semiconductors in two-dimensional (2D) heterostructures. These proposals are called the intrinsic SHE, because they do not rely on impurities but they stem from the band structure itself, as we discuss in this review. These two proposals triggered extensive research, both theoretical and experimental. After 5 years from the theoretical predictions of the intrinsic SHE, more than 500 papers have been published. In p-type semiconductors, for example, the SHE in the linear response is expressed as jji ¼ s "ijk Ek
ð1Þ
where jij is the current of the ith component of the spin along the direction j and "ijk is the totally antisymmetric tensor in 3D (see Figure 1(a)). The spin current jijroughly corresponds to a product of velocity vj and the spin Si. It describes a spin current with spins along +iˆdirection going to the +jˆdirection, and spins along iˆ direction going to the jˆ direction. In Figure 1(a), the electric field is applied along the z-direction (k = z), and it produces the spin currents jy x (S xvy) and jx y (S yvx) with jy x = jx y . For example, the Sx-spin current flows in the y-direction, and S y-spin current flows in the x-direction. As an example, we consider a thin semiconductor slab and apply an electric field as shown in Figure 1(b). The electrons with opposite spins are deflected toward opposite directions, resulting in a spin current perpendicular to the applied field. In the context of SHE, we often restrict ourselves to a situation with time-reversal symmetry, which means the absence of magnetism and magnetic field. We can consider the SHE in systems without the time-reversal symmetry. In such case the charge Hall effect arises (or in ferromagnet AHE arises), and the Hall current is a mixture of charge and spin current. Only when the system respects the time-reversal symmetry, the Hall current is a pure spin current. The spin current will induce a spin accumulation near the sample boundary. The amount of accumulated spins is roughly estimated as a product of spin current and spin relaxation time Ts
226 Spin Hall Effect
(a)
y
(b) x
Spin current
z E
Up-spin
p-GaAs Current
Down-spin
Figure 1 Schematic of the spin current induced by an electric field.
(Murakami et al., 2003). The spin accumulation also affects the spin current itself near the boundaries (Ma et al., 2004; Hu et al., 2004). Because spin current is even under time reversal, it can be induced even when the time-reversal symmetry is preserved. It also implies that the spin Hall current is dissipationless. In doped semiconductors, however, the longitudinal conductivity is finite and the system undergoes Joule heating. Nevertheless, there exist spin Hall insulators (SHIs), which are band insulators with nonzero SHE; in such systems the longitudinal conductivity is zero, and the SHE accompanies no dissipation (Murakami et al., 2004a). Because the predicted effect in p-type semiconductors and in n-type heterostructures is large enough to be measured, this intrinsic SHE attracted much attention, and one of the important issues was the disorder effect. Among many works on disorder effect, Inoue et al. (2003, 2004) considered dilutely distributed impurities with short-ranged potentials in the n-type semiconductor in heterostructure, and calculated the SHE, incorporating the vertex correction in the ladder approximation. Remarkably, the resulting spin hall conductivity (SHC) is exactly zero in the clean limit. After this work, many people thought the SHE to be fragile to impurities; namely, only a small amount of impurities might completely kill the intrinsic SHE. However, this turned out to be a special case, and is not true in general. In fact, the SHC is, in general, nonzero even in the presence of disorder, as we see later. In such circumstances, two seminal experiments on the SHE have been done. Kato et al. (2004) observed spin accumulation in n-type GaAs by means of Kerr rotation (KR). The observed spin accumulation at the two edges was interpreted as that caused by a bulk spin current. Both unstrained and strained samples, with various orientations of the sample, are used to measure the SHE, and the resulting spin accumulation agreed well with that expected from the extrinsic SHE.
Wunderlich et al. (2005) observed a circularly polarized light emitted from a spin light-emitting diode (LED) structure, using spin current from the p-type semiconductor. This circularly polarized light is attributed to the SHE in the p-type semiconductor. After those works, there followed a number of experimental papers. In particular, the SHE has been observed in various metals such as Al, Pt, and Au. As the Hall effect has a quantized variant, that is, the quantum Hall (QH) effect, the SHE also has an analogous effect: quantum spin Hall effect QSHE. (Note that the spin Hall conductance is not quantized, so it cannot be called quantized SHE.) The QSH effect is a spin analog of the QH effect, and they share many fundamental aspects in common. Yet, the QSH effect is realized in zero magnetic field, which is in strong contrast with the QH effect. This effect has been experimentally observed recently (Ko¨nig et al., 2007). Although there have been many works concerning the SHE in these few years, there are many issues to be resolved. In this review, we also introduce these remaining issues and open questions. There are already several articles on the recent developments of this field (Alderson and Hurd, 1971; Awschalom and Flatte, 2007; Bratkovsky, 2008; Chazalviel, 1975; Fabian et al., 2007; Fertig, 2003; Nagaosa, 2008; Wolf et al., 2001). One may be referred also to other introductory articles on the recent developments and perspectives of the SHE (Bauer, 2004; Day, 2005; Inoue and Ohno, 2005). Readers can also be referred to earlier short reviews (Murakami, 2005), and more recent ones (Schliemann, 2006; Engel et al., 2006). This chapter is organized as follows. In Section 1.07.2, we introduce a basic framework and discuss the mechanism of the SHE in semiconductors. In Section 1.07.3, we discuss the SHE in metals. Section 1.07.4 is devoted to the QSH effect. We give concluding remarks and discuss miscellaneous topics in Section 1.07.5.
Spin Hall Effect
1.07.2 Intrinsic and Extrinsic Mechanism of SHE The intrinsic SHE can be described from various viewpoints. In this section, we start with the Berry phase theory of the intrinsic SHE. According to this theory, the SHE is driven by the ‘‘Berry phase curvature in momentum-space.’’ This occurs because of the wave nature of the electrons. The Berry phase (Berry, 1984; Shapere and Wilczek, 1989; Bohm et al., 2003) is a change of a phase of a quantum state caused by an adiabatic cyclic change of some parameters characterizing the Hamiltonian. As this phase is determined only by a path in the parameter space, it is sometimes referred to as a geometric phase. Various physical phenomena involve this Berry phase, depending on which parameters are regarded to change adiabatically, and are studied and reviewed in Shapere and Wilczek (1989), Bohm et al. (2003), Be´rard and Mohrbach (2004, 2006), Bliokh (2005), Bliokh and Bliokh (2005), Culcer et al. (2004), Culcer and Winkler (2007a,b, Culcer et al. 2005a), Dai and Zhang (2007), Erlingsson et al. (2005), Hu et al. (2003a), Imura and Shindou (2005), Raimondi et al. (2007, 2006), Shindou and Imura (2005), Sinova et al. (2006), Sun et al. (2004), Lu et al. (2006), and Strˇeda (2006). As for the electronic states, this Berry phase is known to appear in various kinds of Hall effects, such as the QH effect (Thouless et al., 1982; Kohmoto, 1985; Sundaram and Niu, 1999), the AHE, or the SHE. To demonstrate the effect of the Berry phase in k space, we here consider a clean system with crystal translational symmetry, where the Bloch wave number k is a good quantum number. We regard k as parameters which are adiabatically changed. Then the notion of the Berry phase can be expressed in a gauge field in k-space (Berry, 1984). It plays an important role in AHE (Matl et al., 1998; Ye et al., 1999; Chun et al., 2000; Ohgushi et al., 2000; Taguchi et al., 2001; Jungwirth et al., 2002; Fang et al., 2003) and in magnetic superconductors (Murakami and Nagaosa, 2003). We define the U(1) gauge field An(k) and corresponding field strength Bn(k) as Z q y qunk Ani ðkÞ ¼ – i nk nk X – i unk dx qki qki unit cell Bn ðkÞ ¼ rk An ðkÞ
ð2Þ ð3Þ
where unk(x) is the periodic part of the Bloch wave function nk(x) = eik ? xunk(x) and n is the band index.
227
Here An(k) is a gauge field, and is subject to a gauge transformation accompanied by a change of phase for the wave function of unk: u9nk = ei(k)unk. The corresponding field strength Bn(k) = rk An(k) remains unchanged, that is, gauge invariant. Because a contour integral of A(k) along a loop in k space is the Berry phase, An(k) and Bn(k) are called Berry connection and Berry curvature, respectively. We consider a wavepacket as a superposition of the Bloch states. It is localized both in the real space and in the momentum space. The width x of the wavepacket in the real space and the width k in momentum space is restricted by the uncertainty principle: xk & 2. We consider k is much smaller than the size of the Brillouin zone k 2=a; where a is the lattice constant, and consequently we have x a. This wavepacket acquires a phase when the wave number k is changed. The motion of a wavepacket made of the Bloch states then undergoes a shift because of the interference of Bloch states. By incorporating the effect of Bn(k), the Boltzmann-type semiclassical equation of motion (EOM) acquires an additional term due to the Berry phase (Adams and Blount, 1959; Blount, 1962; Sundaram and Niu, 1999): x_ ¼
1 qEn ðkÞ _ þ k Bn ðkÞ h qk
hk_ ¼ – e ðE þ x_ BðxÞÞ
ð4Þ ð5Þ
_ The term kB n ðxÞ represents the effect of Berry phase, and is called an anomalous velocity. When we apply an electric field, the wave number will change in time. The second term on the right-hand side of Equation (4) then gives rise to an additional velocity perpendicular to the electric field, namely the Hall effect. This theory has already been proposed in the 1950s in a discussion of AHE (Karplus and Luttinger, 1954; Nagaosa, 2006), and has renewed interest from the QHE (Thouless et al., 1982) and the Berry phase (Berry, 1984). The SHE is among the phenomena due to this anomalous velocity. This intrinsic Hall conductivity (7) was first recognized by Karplus and Luttinger (1954). This Berry phase in momentum space has been studied in the recent works on AHE (Ye et al. 1999; Ohgushi et al. 2000; Taguchi et al. 2001; Jungwirth et al., 2002; Fang et al. 2003; Yao et al. 2004), as well as those on the SHE. These equations were originally proposed by Adams and Blount (1959) and Blount (1962). They are rederived in Chang and Niu (1995, 1996) and Sundaram
228 Spin Hall Effect
and Niu (1999) using time-dependent variational principle and investigated in the context of the Berry phase. We note that when there is a band degeneracy for every wavevector k, for example, as a result of symmetry such as Kramers degeneracy, the above formalism is modified accordingly (Murakami et al., 2003; Culcer et al., 2005b). The wavepacket consists of the degenerate wave functions jw i ¼
Z
ddkaðk;t Þ
X
i ðk;t Þji i
ð6Þ
i
where jii (i = 1,. . .,N) denote the degenerate Bloch wave functions and N is the degeneracy of the considered states. The Berry connection and curvature should then be replaced from U(1) to SU(N). The resulting EOMs for x and k are similar to Equations (4) and (5), with Bn(k) replaced by the matrix element of the SU(N) Berry curvature sandwiched by the vector i. In the course of propagation, the components i can change, and this propagation also involves the Berry connection. These EOMs, Equations (4) and (5), can be used for calculating the Hall conductivity. For 2D systems, for example, the Hall conductivity xy in a clean system is calculated from Equations (4) and (5) as xy ¼ –
e2 X 2h n
Z
d2 k nF ð"n ðkÞÞBnz ðkÞ
ð7Þ
BZ
where nF("n(k)) is the Fermi distribution function for the nth band, and the integral is over the entire Brillouin zone. This formula agrees with the result from the Kubo formula in a bare vertex diagram. In this sense, this formula represents the intrinsic contribution, which does not rely upon impurity scattering. In reality, there are also extrinsic contributions coming from impurity scattering, which should be evaluated separately. Given the Hamiltonian, the vector field Bn(k) is calculable, and we can get the intrinsic Hall conductivity, as in the ab initio calculation of the AHE (Fang et al., 2003; Yao et al., 2004). This Hall effect comes not only from the states on the Fermi level but also from all the occupied states. We note that this formula can be recast into a sum of a quantized part and a part which comes from the states on the Fermi energy (Haldane, 2004). When the Fermi energy lies in a gap, the extrinsic contribution vanishes and the intrinsic one (7) can be shown to be quantized in a unit of e2/h, which explains the integer QHE (Thouless et al., 1982; Kohmoto, 1985). This formula was also applied to the AHE in ferromagnetic metals (Matl et al., 1998; Ye et al., 1999; Chun et al., 2000; Ohgushi
et al., 2000; Taguchi et al., 2001; Jungwirth et al., 2002) including first-principles calculations for SrRuO3 (Fang et al., 2003), thin film Sr1 xCaxRuO3 (Mathieu et al., 2004), bcc Fe (Yao et al., 2004), thin film Mn5Ge3 (Zeng et al., 2006), and CuCr2Se4 xBrx (Yao et al., 2007). In some cases, a quantitative agreement has been reached as a function of parameters such as composition x, the magnetization M, or the temperature T. Meanwhile, apart from such intrinsic terms calculated from the first-principles calculation, an extrinsic term coming from impurities always exists, and it depends on whether the intrinsic terms dominate. To see what the field Bn(k) is like and how we can enhance this field, we rewrite the formula as follows: Bn ðkÞ ¼ i
X nkqH mk mkqH nk qk qk mð6¼nÞ
ðEn ðkÞ – Em ðkÞÞ2
ð8Þ
where we assumed that the energy spectrum has no degeneracy. This formula shows that the Berry phase is a multiband effect, and it vanishes when there is only one band. We note that it is straightforward to extend this formula to a spectrum with degeneracy; the gauge field becomes non-Abelian (Wilczek and Zee, 1984). Because this formula has a square of energy difference in its denominator, Bn(k) becomes large, when two energy spectra come close. In particular, when two bands touch (i.e., are degenerate) at a certain wavevector k, B(k) can diverge. We note that degeneracy does not necessarily mean divergence of Bn(k). For example, when the HamiltonianPnear the degeneracy point is expressed as H ðkÞ ¼ a¼x;y;z a ka , like the massless Dirac fermions, Bn(k) diverges and has a monopole or antimonopole, while it does not when P only two components are involved as in H ðkÞ ¼ a¼x;y a ka . 1.07.2.1 Intrinsic SHE in p-Type Semiconductors In cubic p-type semiconductors with diamond structure (e.g., Si, Ge) or zincblende structure (e.g., GaAs, InSb), the valence band consists of two doublydegenerate bands called heavy-hole (HH) and light-hole (LH) bands (Kittel, 1987). These two bands are degenerate at the -point (k = 0). They consist of the three p-orbitals, px, py, pz, with spin up and down. In the presence of the relativistic SO coupling, these six states are split into fourfold degenerate S = 3/2 states and twofold degenerate S = 1/2 states. Here, S denotes the total angular momentum of the atomic orbital, obtained through
Spin Hall Effect
the coupling of the orbital angular momentum l and the spin angular momentum s. The second-order perturbation in the k ? p results in the effective Hamiltonian near k = 0: Hˆ0 ¼
X
y c;k H ðkÞ c;k
k
h2 H ðkÞ ¼ 2m
"
# X 5 2 2 i 2 1 þ 2 k – 2 2 ki ðS Þ 2 i
– 2 3
X
ð9Þ
ki kj ðS i S j þ S j S i Þ
i6¼j
where k = (kx, ky, kz), S = (Sx, Sy, Sz), and k = jkj. We changed the sign of the energy in order that the energy of the hole be positive. 1, 2, and 3 are called Luttinger parameters, for example, GaAs has the values 1 = 6.98, 2 = 2.06, and 3 = 2.93 (Vurgaftman et al., 2001). S = (Sx, Sy, Sz) are the spin-3/2 matrices. Strictly speaking, the zincblende structure has also odd-order terms in k, due to structural inversion-symmetry breaking (Kittel, 1987). These terms are usually very small and often neglected. In many semiconductors 2 and 3 are of similar order, and are often considered to be equal ( 2 = 3) for simplicity. With this approximation, the Hamiltonian becomes H ðkÞ ¼
h2 2m
5 1 þ 2 k2 – 2 2 ðk?SÞ2 2
x˙ ¼
1 qE ðkÞ ˙ þ k B ðkÞ; hk˙ ¼ eE h qk
229
ð11Þ
Because we are considering holes, the sign of the charge has been changed. By straightforward calculation, we get B ðkÞ ¼ ð2 2 – 7=2Þk=k3 . This has a form of the field radiated isotropically from a monopole at k = 0. The reason why the monopole is at k = 0 is because the HH and the LH bands are degenerate at k = 0. When we apply a uniform electric field E(x) = E independent of time, we can write down the EOM for k and x representing the time evolution. When we solve the equations analytically, the trajectory in the x-space is shown as in Figure 3. Due to the anomalous velocity, the motion of the holes is deflected from an otherwise straight motion along k (dashed line). Here, we only show the trajectory in the direction perpendicular to E. The shift of the motion is opposite for the opposite signs of the helicity , referring to whether the spin S and the wavevector k are parallel or antiparallel. This means that the shift is the opposite for the opposite spin (Figure 3). This shift amounts to the SHE. By summing up this shift over all the occupied states of holes, one obtains the spin current as (Murakami et al., 2003) jji ¼ s "ijl El
ð12Þ
ð10Þ 1.5 CB Helicity: 1.4 0 –EF Energy, E (eV)
This approximation is called spherical approximation, because it restores spherical symmetry, while a calculation without it is also possible (Bernevig et al., 2004). This Hamiltonian is theoretically easy to k?S handle, because the helicity defined as ¼ k becomes a good quantum number. This amounts to taking the spin quantization axis to be kb= k/k. The eigenvalues of are H ¼ 3=2 and L ¼ 1=2 for HH and LH bands, respectively. The corresponding 1 – 2 2 2 2 h k (HH band) and eigenenergies are EH ¼ 2m 1 þ 2 2 2 2 h k (LH band). The band structure is EL ¼ 2m schematically shown in Figure 2. Each eigenvalue is doubly degenerate, due to the Kramers theorem based on the time-reversal symmetry. We begin with a semiclassical description of the SHE, and apply it to the p-type semiconductors (Murakami et al., 2003). The semiclassical EOM for the position x involves an anomalous velocity. The semiclassical EOM reads as
HH
k FL
kH F
λ = ± 3/2 λ = ± 1/2
–0.1 LH –0.2 –0.3 SO –0.4 –1 –0.5 0 0.5 1 Wave number k(nm–1)
Figure 2 Schematic band structure for GaAs. CB, HH, LH, and SO represent the conduction, heavy-hole, lighthole, and split-off bands, respectively. This figure is an approximate one, and in reality the cubic structure induces an anisotropy for the band dispersions depending on the direction of k. In addition, the absence of the inversion symmetry in the zincblende structure induces k-linear terms, which are small and neglected here.
230 Spin Hall Effect λ >0
of the s-orbitals. On the other hand, in high-mobility 2D electron gas (2DEG), structural inversionasymmetry (SIA) due to the heterostructure induces the so-called Rashba SO coupling. The Hamiltonian is approximated as
λ <0
Spin
H ðkÞ ¼ H0 ðkÞ þ HR ðkÞ k
Figure 3 Trajectory of holes from the semiclassical equation of motion with Berry-phase terms. This is a projection on the plane perpendicular to the electric field E. The direction of the transverse shift of the trajectory is the opposite to each other depending on the sign of the helicity. The transverse shift of the trajectory is to the opposite direction, depending on the sign of the helicity = b k?S. The bold gray arrows represent the direction of spin S.
with s ¼
ð14Þ
E
e H L 3kF – kF 122
ð13Þ
L which is of the form of (1). Here kH F and kF are the Fermi wave numbers for the HH and LH bands, respectively. Nominal values of the SHC for p-GaAs are of the similar order of magnitude as the conductivity at room temperature (Murakami et al., 2003). In GaAs, the energy difference of the two bands is larger than the room temperature, and the effect can, in principle, survive even at room temperature. This intrinsic mechanism of the SHE and other related phenomena have been studied (Bernevig et al., 2004; Dai et al., 2006; Jiang et al., 2005; Kleinert and Bryksin, 2006; Luttinger, 1956; Ma and Liu, 2006a,b; Zarea and Ulloa, 2006; Kravchenko and Tsoi, 2007, 2008; Lou and Xiang, 2005; Raichev, 2007; Wang and Zhang, 2005). We note that this calculation is based on the semiclassical theory. In quantum mechanics the velocity and the spin are not commutable, and thus cannot be determined simultaneously. Therefore, the above picture that a hole with definite spin is traveling with a definite velocity is not valid. Moreover, in reality, impurities will scatter the holes traveling along the trajectory. These effects should be taken into account, and will be discussed in the subsequent sections.
1.07.2.2 Rashba Model for n-Type Semiconductors In n-type semiconductors with diamond or zincblende structure, the SO coupling is small. This is because the conduction bands (CBs) consist mainly
H0 ðkÞ ¼
h2 k2 2m
ð15Þ
HR ðkÞ ¼ R ð kÞz ¼ R ðx ky – y kx Þ
ð16Þ
i
where is the Pauli matrix. The second term is called the Rashba term (Rashba, 1960; Bychkov and Rashba, 1984), representing the SO coupling. The coupling constant can be experimentally determined, and can be controlled by the gate voltage (Nitta et al., 1997). The Rashba coupling is of the order (0.5–1) 1011 eVm for InGaAs/InAlAs heterostructure (Nitta et al., 1997). Its eigenenergies are given by "s ¼
h2 k2 þ s R k 2m
ðs ¼ Þ
ð17Þ
with eigenstates 1 ik?x ks ðxÞ ¼ pffiffiffiffiffiffi e 2V
isk – =k
!
0
ð18Þ
where V is the area of the system, and k = kx iky. The dispersion and the spin directions for the two bands are shown in Figure 4. In n-type semiconductors with zincblende structure, there is another type of the SO coupling in 2D. The bulk inversion asymmetry (BIA) induces Dresselhaus coupling of the form (Dresselhaus, 1955) HD ¼ D ðkx x – ky y Þ
ð19Þ
The ratio between Rashba and Dresselhaus terms can be experimentally determined through photocurrent measurements of the spin-galvanic effect (Ganichev et al., 2002, 2004). For an InAs quantum-well (QW), the ratio R/ D = 2.15 was obtained (Ganichev et al., 2004). For an InGaAs QW, in a weak antilocalization experiment, the ratio R/ D = 1.5–1.7 is obtained (Knap et al., 1996), while k ? p calculation suggests
R/ D = 1.85 (Pfeffer and Zawadzki, 1999). The total Hamiltonian reads as H0 þ HR þ HD ¼
h2 k2 þ R x ky – y kx 2m þ D x kx – y ky
ð20Þ
Sinova et al. (2004) applied the Kubo formula to this Rashba Hamiltonian. For this procedure, they
Spin Hall Effect
E(p) py
px py
231
2006; Li and Shen, 2007; Shen, 2004; Shen et al., 2005, 2004; Sinitsyn et al., 2004; Wang et al., 2006a; Wang and Vasilopoulos, 2007; Winkler, 2003; Zhou et al., 2008; Li et al., 2005; Lipparini and Barranco, 2007; Lucignano et al., 2008; Miah, 2006; Rashba, 2004a, 2005; Schulz and Trimper, 2008; Sensharma and Mandal, 2006; Shen, 2005; Song et al., 2006b; Wang et al., 2008, 2006d; Zhang and Shen, 2008; Dimitrova, 2005).
px Figure 4 Dispersion of the Rashba model and the spin directions of the eigenstates Sinova et al. (2004).
defined the spin current jzy to be a symmetrized 1 qH product of the spin sz and the velocity vy ¼ , h qky namely jyz ¼
h2 ky z 1 h v y ; z ¼ 2 2 2m
ð21Þ
To calculate the SHC, we calculate the correlation function between the spin current jzy and the charge current jx defined as e qH hkx R y ¼ –e – h qkx m h
jx ¼ – evx ¼ –
ð22Þ
where the electron charge is taken to be e = jej. By using the Kubo formula for the bare bubble diagram in Figure 5 representing the intrinsic SHE, the resulting is zyx = zxy = e/8, which is independent of the Rashba coupling . The Rashba term in Equation (16) can be regarded as a k-dependent effective Zeeman field Beff = (zˆ k). In equilibrium the spins are pointing either parallel or antiparallel to Beff for the lower and upper bands, respectively. An external electric field Ejjxˆ changes the wavevectors k of Bloch wave functions, and Beff also changes accordingly. The spins will then precess around Beff, and tilt to the zdirection, depending on the sign of ky. This appears as the SHE, and the SHC is calculated to be e/8, in agreement with the Kubo formula. We also note that the same result was obtained by the functional integral method (Rebei and Heinonen, 2006). In addition, various aspects of the SHE in Rashba-type models have been studied (Bryksin and Kleinert, 2006; Bernevig, 2005; Chang, 2005; Cheche, 2006; Cheche and Barna, i
jj
jk
1.07.2.3
Impurity Scattering Effect
In this section, we explain various studies on the effects of impurity scatterings on SHE. The extrinsic SHE has been studied in several methods, partially in order to examine to what extent the experimental reports can be accounted for by the extrinsic mechanism. Before going into details, we note that the impurity scattering effect on the SHE is similar to that on the AHE (Cre´pieux and Bruno, 2001). What has been established in the AHE is as follows. The impurity effects are classified into the skew scattering (SS) and the side jump (SJ). First, the SO coupling at the impurity gives rise to a left–right asymmetry at the scattering, as is known as Mott scattering. This mechanism of the AHE is called as the SS (Figure 6(a)). Second, the impurity potential in the Hamiltonian gives rise to an additional anomalous term to the velocity. It gives an SJ of the electron trajectory at the scattering event, without changing the direction of the particle away from the scattering center (Berger, 1970; Figure 6(b)). These two contributions, SS and SJ, sum up to give an extrinsic AHE. In the Kubo formula, the SS and SJ correspond to Figures 7(d) and 7(c), respectively, whereas the self-energy correction (Figure 7(b)) may be regarded as the correction to the intrinsic SHE (Figure 7(a)). The situation for the SHE well resembles that for the AHE, and we use the same terminology as in the AHE. The readers are also referred to the papers focusing on some new aspects of disorder effects (Hankiewicz and Vignale, 2006; Tserkovnyak et al., 2007). 1.07.2.3.1
Spin-independent impurities There are many works on the calculation of the SHC in the presence of impurities. The most studied case is the system with randomly distributed impurities with -function potential: V ðxÞ ¼ V
Figure 5 Diagram for calculating the intrinsic SHC.
X i
ðx – xi Þ
ð23Þ
232 Spin Hall Effect
(a)
Impurity
(b)
Impurity
Spin
Spin
Figure 6 Schematic of (a) side-jump and (b) skew scattering at an impurity.
(b)
(a)
jji
jk
(c)
(d)
jji
jk
jji
jk
jji
jk
jji
jk
jj i
jk
jji
jk
Figure 7 The calculation of the SHC. (a) bare vertex, (b) self-energy corrections, (c) vertex corrections (side-jump), and (d) skew scattering.
One can include an effect of the self-energy broadening 1/ by disorder (Schliemann and Loss, 2004, 2005) (Figure 7(b)). In both Rashba and Luttinger models, the intrinsic SHE is gradually reduced as is decreased from 1. In the clean limit, the SHC reproduces its intrinsic value. In the Rashba model, the characteristic energy scales are the Rashba energy "R ¼ m R =h2 and the Fermi energy " ; the SHC is reduced when h= exceeds pffiffiffiffiffiffiffiffiffiffi F "R "F . In the Luttinger model, the SHC is reduced when h= exceeds "SO, which is the energy splitting between the LH and HH bands at the Fermi energy. Subsequently, Inoue et al. (2003, 2004) found an important result. They assumed dilutely distributed impurities with a -function potential and took the clean limit. In a clean limit, they obtained a vertexcorrection contribution (Figure 7(c)) e/8 to the SHC, exactly canceling the intrinsic value e/8. In addition, subsequent papers found that the SHC vanishes even for finite and for long-range impurity potential (Mishchenko et al., 2004; Dimitrova, 2004b; Chalaev and Loss, 2005; Khaetskii, 2006; Raimondi and Schwab, 2005; Rashba, 2004b; Mal’shukov and Chao, 2005; Liu and Lei, 2004, 2005c,a,b; Sugimoto et al., 2006; Krotkov and Das Sarma, 2006). In Raimondi and Schwab (2005), the SHC is calculated for weak disorder h= R pF "F , with general form of impurity potential. For weak disorder, we can evaluate the SHC within the Born approximation and ladder summation for vertex corrections. It is found that the SHC is zero.
In Krotkov and Das Sarma (2006), the Rashba model with an arbitrary form of (spin independent) kinetic term is considered, and it is found that the cancelation of the SHC is specific to the quadratic 2 2 h k form of the kinetic term. In Dimitrova 2m (2004b) and Chalaev and Loss (2005), in particular, it is argued that because the conventional definition of spin current (21) is proportional to S_ y , its expectation value is physically expected to vanish in the steady state: hjyz i hS_ y i ¼ 0: In an explicit form, the spin current in the Rashba model is written as jyz ¼
hky z 1 1 y ½H ;y ¼ – ¼ s_ 2m 4im R 2m
ð24Þ
which is roughly because the SO term is linear in k (Dimitrova, 2004b; Chalaev and Loss, 2005). The vanishing of the SHC can be shown rigorously using the Keldysh formalism (Sugimoto et al., 2006). Using the Keldysh formalism, one can show that the time derivative O_ of an arbitrary operator O, which is independent of k, x, time t, and frequency !, has a vanishing expectation value in the steady state, _ = 0 even with a general form of the impurity hOi (Sugimoto et al., 2006). We note that a system with Dresselhaus and Rashba terms also satisfies the similar relationship jyz ¼
1 d ð R s y – D s x Þ 2mð 2D – 2R Þ dt
ð25Þ
Spin Hall Effect
and its SHC vanishes for general impurity potential for finite (Liu and Lei, 2005c). In the calculation of linear response such as the SHC, the Keldysh formalism is equivalent to the Kobo formula calculation. Nevertheless, the Keldysh formalism can sometimes give us a clearer physical picture for transport properties. In addition, the Keldysh formalism is suitable for calculating time- and space-dependent properties. For example, in Mishchenko et al. (2004), the Keldysh formalism is used to derive diffusion equation for the spin and charge in the two-probe geometry. The impurity potential is assumed to be of the -function form. The resulting diffusion equation describes the SHE, the spin relaxation, and diffusion in a unified way. It is found that the spin Hall current appears only near the electrodes, whereas the bulk spin Hall current vanishes irrespective of the lifetime (Mishchenko et al., 2004), in accordance with the vanishing SHC by the Kubo formula. In the calculation, some approximation is used based on an assumption that EF is much larger than the SO coupling and the selfenergy broadening 1: EF , EF 1. To analyze the origin of the vanishing SHC in the Rashba model, several works have been done without such approximation. In the Keldysh formalism, the density matrix is calculated using the quantum Boltzmann equation in the electric field E. The resulting density matrix contains several terms linear in the electric field E. Some of such linear terms are proportional to the Fermi distribution function nF("), while the others are proportional to dnF/d". The former terms stem from interband matrix elements of the (spin) current operators, and correspond to intrinsic contributions, while the latter terms represent intraband matrix elements, leading to extrinsic contributions. In the study by Liu and Lei (2004), the range of the impurity potential is finite, and the selfenergy broadening is ignored, and the vanishing SHC has been concluded. Later, this result is generalized by considering also the self-energy broadening (Sugimoto et al., 2006; Liu et al., 2006a). In Liu and Lei (2005c) the system with Dresselhaus and Rashba terms is analyzed on the same assumptions, and again zero SHC is concluded. On the other hand, the cubic Rashba model is also investigated on the same assumptions, which gives nonvanishing SHC (Liu and Lei, 2005a,b). The extrinsic term has the same sign and its magnitude is somewhat smaller than the intrinsic value 9e/8. In the case of the Luttinger model, on the other hand, the -function impurities will have no effect on the SHC (Murakami, 2004).
233
Long-range impurity potential will give an extrinsic term, which has an opposite sign and a bit smaller compared with the intrinsic one; thus, is the extrinsic piece due to long-range impurity potential makes the SHC smaller (Liu and Lei, 2005a,b). The Keldysh formalism is used for more generalized cases in the Rashba model and related models (Liu and Lei, 2004, 2005a,b,c; Sugimoto et al., 2006). One may wonder whether the SHE vanishes in other systems, and there remain some controversies in this respect. After debates on the vanishing of the SHC for finite , has now been reached consensus that the SHE vanishes in the Rashba model with -function impurities in the clean limit (Inoue et al., 2003, 2004) and even for finite (Mishchenko et al., 2004; Dimitrova, 2004b; Chalaev and Loss, 2005; Khaetskii, 2006; Raimondi and Schwab, 2005; Liu and Lei, 2004; Sugimoto et al., 2006; Nomura et al., 2005a; Sheng et al., 2005a) or for finite-ranged impurities (Inoue et al., 2004; Khaetskii, 2006; Raimondi and Schwab, 2005; Liu and Lei, 2004). Here, we note that the Rashba model is exceptional, in that the SHE vanishes rather accidentally, namely because the spin current operator jzy is proportional to s_ y ¼ i½H ;s y (Shekhter et al., 2005; Dimitrova, 2004b; Chalaev and Loss, 2005; Rashba, 2004b; Liu and Lei, 2005c). In fact, one can check that the SHE does not vanish for other models in general (Murakami, 2004; Bernevig and Zhang, 2004; Liu and Lei, 2005a,b; Krotkov and Das Sarma, 2006); for example, when the Rashba model is generalized, to have a higherorder term in k in the Rashba coupling (Murakami, 2004) or to deviate the band dispersion from parabolic to a more general one (Krotkov and Das Sarma, 2006), the SHC no longer vanishes. In addition, there are some models where the vertex correction does not cancel the intrinsic value (Liu and Lei, 2005a,b), or even vanishes by symmetry (Murakami, 2004; Bernevig and Zhang, 2004). Magnetic impurities also make the SHC to be nonvanishing, where the exchange coupling between magnetic impurities and itinerant electrons can be isotropic (Inoue et al., 2006) or anisotropic (Wang et al., 2007). The SHC depends on the anisotropy of the magnetic interaction (Wang et al., 2007) or on the amount of magnetic impurities when both magnetic and nonmagnetic impurities coexist (Inoue et al., 2006). We note that in this case the SHC does not vanish even in the clean limit. In the 2D hole gas (2DHG) models such as the cubic Rashba model, the vertex correction vanishes for -function impurities (Bernevig and Zhang, 2005).
234 Spin Hall Effect
A Boltzmann-type equation can also be used to study impurity scattering with arbitrary angular dependence (Shytov et al., 2006). We consider a 2D system 1 H ¼ "k – bðkÞ? þ V ðxÞ 2
ð26Þ
where k = (kx, ky) and b(k) represents the SO coupling. We note that b(k) = b(k) required by time-reversal symmetry. It is assumed that bz ¼ 0; bx þ iby ¼ b0 ðkÞeiN
ð27Þ
i
where kx + iky = ke . Since the result is also affected by the behavior of b0(k) and the energy dispersion "k near the Fermi energy, we write ˆ
jb0 j _ kN ; vk ¼
q" 1þ _k qk
ð28Þ
near the Fermi energy. These three exponents, N, N˜, and , characterize the intrinsic SHE. For 2D electron system on a (100) surface of a III–V semiconductor, we have = 0 and N˜ = 1. Furthermore, N = 1 corresponds to the pure Rashba term, and N = 1 to the pure (linear-k) Dresselhaus term. The 2D hole system with the cubic Rashba term is represented as N = N˜ = 3 and
= 0. Here, the direct SO term ? (k rV) is neglected because the constants and are, in principle, independent. This gives vanishing result for the pure Rashba system or the pure Dresselhaus system (N = 1). For 2D hole system, for example, the SHC is different from that for the isotropic scattering potential (Shytov et al., 2006). The order of magnitude for the SHC observed in the spinLED experiment by Wunderlich et al. (2005) seems consistent with the value predicted here. Numerical studies based on the Kubo formula have also been carried out. An earlier numerical calculation for the Rashba model (Nomura et al., 2005a) suggested that the SHC is finite for weak disorder for finite systems. Nevertheless, it has been superseded by a later paper (Nomura et al., 2005b, 2006). In the latter paper (Nomura et al., 2005b) three models are studied – the (k-linear) Rashba model, the k-cubic Rashba model, and a modified Rashba model: H¼
h2 k2 þ k2 ðx ky – y kx Þ 2m
ð29Þ
The calculation is based on the Kubo formula with ji ¼ ðe=hÞqH =qki ; jyz ¼ h– 1 qH =qky ;sz =2, where sz ¼ ðh=2Þz . The disorder potential is assumed to
P i ðx – xi Þ, where Ni be short ranged: V ðrÞ ¼ V Ni¼1 is the number of impurities. For each disorder configuration, they calculate the eigenstates numerically and put them into the Kubo formula. The treatment of the small imaginary part of the frequency in the Kubo formula requires care. In the linear response theory of the Kubo formula, the controls the speed of adiabatic switching of the external electric field. In a calculation of longitudinal charge conductivity, in order to absorb energy from the external electromagnetic field, the level spacing of the system, , should be much smaller than (Imry, 1997): . We can expect that similar argument applies to the SHE. In Nomura et al. (2005b), the SHC is calculated for various system sizes, where the relationship is retained. Bulk quantity is obtained by extrapolating to ! + 0. For the k-linear Rashba model, the extrapolation seems to suggest s 0 within numerical error. On the other hand, for the cubic Rashba model and the modified Rashba model, s in the bulk is extrapolated to be nonzero. It is also found that the variation of the s by changing disorder configuration is much larger than the average value of s. The tight-binding counterpart of the Rashba model is numerically studied (Sheng et al., 2005a). The model is represented by a tight-binding Hamiltonian with Rashba SO coupling on a square lattice: H ¼ H TB þ
X
y "i ci ci
ð30Þ
i;
where HTB represents the nearest-neighbor tightbinding Hamiltonian with the Rashba coupling (Ando model (Ando, 1989)), and "i is set to be a random variable within [W/2, W/2] when disorder effect is considered. In Sheng et al. (2005a), the authors apply the Laughlin’s gauge argument (Laughlin, 1981) to relate the SHC to the carried spins during insertion of a flux quantum into a hole of the system, which is periodic only along one direction (see also Figure 27). This flux insertion corresponds to the twisting of the phase in the boundary condition. The energy spectrum, shown as a function of boundary phases, shows anticrossings between energy levels, from which the authors conclude that there is no spin transfer across the sample during flux insertion, resulting in the vanishing SHC. Nevertheless, it is not clear whether the gapless system like the Rashba model can be treated within the Laughlin’s gauge argument to calculate the SHC.
Spin Hall Effect
Numerical calculation for Luttinger model using Kubo formula has been performed as well (Chen et al., 2005b). Eigenstates for each disorder configuration are calculated, and the SHC is calculated through the Kubo formula with i = 0. The spin 1 current operator jxy ¼ fvx ; s y g is defined as an 2 anticommutator between the spin and the velocity, as adopted before. Even in the pure system without disorder, the resultant SHC largely fluctuates as a function of EF when the system is not sufficiently large, and the fluctuation becomes much larger when the disorder is introduced. Instead of increasing the system size to suppress the fluctuation, in Chen et al. (2005b), averaging over the twisted boundary condition reduces the fluctuation effectively, leading to quantitative estimate for the SHC and its width due to disorder. By increasing EF from the band bottom (EF = 3.5), the SHC increases from zero to 4.2e/8. By fixing EF in 2.27 EF 2.07 as an example, the SHC is around 3:5 8e for the pure system, and it is almost unchanged for weak disorder up to W 2t. For stronger disorder, the SHC gradually decreases but remains nonzero up to around W Wc, the critical strength of the Anderson localization. This behavior is different from that of the Rashba model, where the SHC vanishes for the disorder strength much weaker than the Anderson localization. On the other hand, the width s has a prominent e at around W t, and it gradually peak s 15 8 decreases when the disorder becomes stronger. System-size dependence becomes much weaker for W 2t, suggesting that there is a characteristic length scale which decreases with increasing W (Chen et al., 2005b). Various other aspects of the impurity effects are studied (Arii et al., 2007; Bleibaum, 2005, 2006; Engel et al., 2007; Hu and Huang, 2006; Lin et al., 2006; Liu et al., 2007b; Liu and Lei, 2006; Liu et al., 2006b; Maytorena et al., 2006a,b; Rashba, 2004b; Sinitsyn et al., 2006; Wang and Li, 2008; Xing et al., 2008; Xiong et al., 2006; Mal’shukov and Chu, 2007; Dimitrova, 2004b). 1.07.2.3.2 SO coupling modified by impurities
Because the SO coupling is of the form ( rV) ? k, the impurity potential gives an additional term. The Hamiltonian is the simple one-band Hamiltonian: H¼
h2 k2 – ð rV ðxÞÞ?k þ V ðxÞ 2m
ð31Þ
235
where represents the strength of the SO coupling and V represents the impurity potential. In a vacuum, the free-electron value for is = 2c/4, where
c = h/(mec) = 3.8 103A˚, the Compton wavelength. The key issue here is a large enhancement (106) of j j in a solid like GaAs (Berger, 1970; Engel et al., 2005; Tse and Das Sarma, 2006), and the sign is changed (Engel et al., 2005). Because the intrinsic SHE is a multiband effect, the SHE of this single-band Hamiltonian is only due to the extrinsic one. This Hamiltonian is studied using the Boltzmann equation (Engel et al., 2005). The scattering cross section d/d is spin dependent due to the SO coupling. By omitting the details of the calculation, the solution of the Boltzmann equation for the 2 2 distribution function ˆf(k) in the spin space is (Engel et al., 2005) h i fˆðkÞ ¼ f0 ðkÞ1ˆ þ k? E þ ð EÞ 2
ð32Þ
where X kF is the transport skewness at the Fermi level. The spin Hall current due to SS is calculated from this fˆ(k), and the corresponding SHC is SS s ¼ –
h xx 4e
ð33Þ
The SJ term is SJ s ¼ n e
ð34Þ
Thus, the SS becomes more dominant over the SJ as the system becomes cleaner. For the n-type GaAs in experiments by Kato et al. (2004), the transport skewness is evaluated to be 1/900. The calculated SHC 1 1 1 1 is SJ m and SJ m . Here, we yx = 0.8 yx = 1.7 take the unit of SHC as that of the charge conductivity 1 by multiplying e/h. The total SHC is total yx = 0.9 , which reproduces well the order of magnitude for the 1 experimental value total (Kato et al., 2004). yx = 0.5 We note that the sign of the above experimental value is somewhat uncertain. Considering that the two contributions have opposite signs, together with a number of approximations made in this theoretical evaluation, the definite sign of the total SHC cannot be determined within this theory at the present stage. The extrinsic SHE is also studied in the Green’s function method (Tse and Das Sarma, 2006). The velocity operator is i 1 e v ¼ ½H ; x ¼ hk – A – ð rV ðxÞÞ h m c
ð35Þ
while the spin current operator is given as 1 1 e js ¼ fsz ; vg ¼ hk – A z – ðzˆ rV ðxÞÞ ð36Þ 2 2m c
236 Spin Hall Effect
The diagrams contributing to the SHE are shown in Figure 7. These diagrams are classified into the SJ and the SS in the similar way as in the AHE (Cre´pieux and Bruno, 2001). The spin current operator has a term involving V(x), which gives rise to the SJ (Figure 7(c)). The other contribution comes from the impurity term V(x) itself in the Hamiltonian, corresponding to the SS (Figure 7(d)). The resulting formula is SJ s ¼ e n
ð37Þ
for the SJ and ( SS s
¼ – 4e "F =h
1=3 :3D 1=2 :2D
ð38Þ
for the SS. The ratio between the two contributions is SJ =SS ðh= Þ="F which is smaller for cleaner samples. For 1013–1012s and "F 1–10 meV, for example, they are of the same order. For 3D n-type semiconductors, the resulting value of 1 1 1 1 the SHC is SJ m andpffiffiffiSJ m s = 0.75 s 6 ˚ with nominal values of parameters: 2 ¼ 4:7A, n = 3 1016 cm3, and xx = 3 103 1m 1. The sum is s = 5.2 1 m1. It reproduces the order of magnitude for the experimental value 0.7 1m 1, obtained in an experiment in Kato et al. (2004), although we took the simplest model and picked up only the lowest-order diagrams. For 2D p-type semiconductors with the parameters p = 2 1012 cm2 and xx 1.09 1013 1, as in the experiments by Wunderlich et al. (2005), 8 1 the SHC is evaluated as SJ s = 3.10 10 6 1 SS and s = 2.72 10 . The total SHC is SS 2.68 106 1. In this case the ratio jSJ s /s j is as 2 small as 2 10 , because the sample is relatively clean. The results of Engel et al. (2005) and those in Tse and Das Sarma (2006) qualitatively agree with each other. When both types of the SO coupling, namely that of the bulk and that at the impurities, coexist, the problem becomes much more complicated. This problem is studied in Tse and Sarma (2006) for Rashba and cubic Rashba models, and it was revealed that the Rashba model shows a peculiar interplay between the intrinsic and the extrinsic parts, whereas in the cubic Rashba model extrinsic and intrinsic parts are additive in the lowest order. The Rashba Hamiltonian with SO-coupled impurities
H¼
h2 k2 þ R ðx ky – y kx Þ þ V ðxÞ – ð rV ðxÞÞ?k 2m ð39Þ
is studied with the Kubo formula. By including vertex corrections, the SS vanishes and the SJ contribution is reduced to SJ s ¼ e n=2
ð40Þ
which is a half of the value (37) where the SO coupling of the impurities is neglected. This remains true when R ! 0, meaning that the resulting SHC for R ! 0 is a half of the value at R = 0. Thus, the SHC is nonanalytic at R = 0 (Tse and Sarma, 2006). The cubic Rashba model is treated in the same way. In accordance with the previous results on the intrinsic SHE, the vertex correction vanishes, and the result is additive in the lowest order for the intrinsic and extrinsic parts (Tse and Sarma, 2006). This analysis shows a rather peculiar feature of the (linear) Rashba model. We note that the same problem is studied in Hu et al. (2006) with the Boltzmann transport theory. The results look somewhat different from the Kubo formula results (Tse and Sarma 2006). It is not usually easy to trace the reason for the difference in the results with Kubo formula and Boltzmann transport theory. Nevertheless, one of the reasons may lie in the fact that in Hu et al. (2006) the distribution function is considered to be diagonal in the chirality basis jk i, while it may not be always true when the external field is present, as studied in Shytov et al. (2006). The readers are referred also to Hankiewiez and Vignale (2006, 2008), Tse et al. (2005), Cheng and Wu (2008), Chudnovsky (2007), Grimaldi et al. (2006a), and Sherman et al. (2006) for related subjects. 1.07.2.3.3
SHE in the hopping regime The SHE in strongly disordered systems is governed by hopping conduction. In Beckmann et al. (2005), Damker et al. (2004), and Entin-Wohlman et al. (2005), the SO coupling is included in the hopping amplitudes, and interference effect between hopping paths is studied. The effect of the SO coupling appears in the third order in hopping amplitudes; for three sites i, l, and m, the interference occurs between the direct path i ! l and the indirect path i ! m ! l. Because of this interference, when an external electric field is applied, the transverse spin current calculated as a time derivative of the spin polarization becomes nonzero. In Beckmann et al. (2005) and Damker et al. (2004), only the Rashba SO coupling is studied in terms of the rate equations in the Markovian limit. In Entin-Wohlman et al. (2005), the Rashba and Dresselhaus SO coupling are
Spin Hall Effect
included in the hopping amplitudes, and this SHE is proportional to !2, where ! is the frequency of the external electric field (Entin-Wohlman et al., 2005). We note that the way to calculate the spin current as a time derivative of the spin polarization was adopted in several works (Entin-Wohlman et al., 2005; Shi et al., 2006) as discussed in Section 1.07.2.6.1 1.07.2.4
Mesoscopic SHE
In this subsection, we discuss various calculations on SHE in small systems. This includes (1) the calculations based on Kubo formula and (2) the calculations by the Landauer–Bu¨ttiker formalism. We discuss these two types of calculations separately. The Kubo formula calculation for small-size systems (<100 sites) of 2DEG has been done (Sheng et al., 2005a; Moca and Marinescu, 2007a). The resulting SHC strongly fluctuates as a function of the Fermi energy EF. Nevertheless, by averaging over the twisting angle in the boundary conditions, the SHC becomes a smooth function of EF. We note that this trick has been adopted for numerical calculation of Hall conductivity in the QHE. As a function of the system size, the SHC is s 0.48(e/4) for a clean system, while s for disordered systems exponentially decays as a function of the system size hsi _ exp(L/s), with a characteristic length s (Sheng et al., 2005a; Moca and Marinescu, 2007a). This is consistent with the above-mentioned result that the SHC in disordered Rashba models is zero. In Chen et al. (2005a), the 2DHG is investigated numerically with the Kubo formula. The model is a discretized version of the Luttinger–Rashba model. The SHC is calculated with the Kubo formula in a finite-size sample, with the imaginary part set to zero. In a clean system, the SHC is of the order of 11e/8, as long as the EF is near the band bottom. When the system size is not large (around L = 20–30), zxy largely fluctuates by changing disorder configuration. This fluctuation is strongly suppressed when an average over the twisting angle in the boundary condition is performed, as in the electron systems. The resulting value agrees quite well with a result in a large system L = 1000; hence, one can regard that this averaging over the boundary conditions removes a finite-size effect efficiently. As one introduces the disorder gradually by changing W, the SHC is monotonically suppressed. The SHC almost vanishes when W 5t. The Landauer–Bu¨ttiker formalism has been applied to many types of systems. In fact, the SHE in the
237
Rashba model for a four-terminal geometry was first numerically studied by Bulgakov et al. (1999), prior to the theoretical proposals of the intrinsic SHE (Murakami et al., 2003; Sinova et al., 2004). In Bulgakov et al. (1999), the scattering-wave formalism is applied to the four-terminal (continuum) Rashba model without disorder. For example, without the SO coupling whose eigenstates are plane waves are attached to the system. One can calculate the transition probabilities that an incoming wave goes into an outgoing wave in other leads. It was numerically obtained that a left–right asymmetry occurs in the transmission probability into the transverse channels, because of the Rashba SO coupling. When the incoming electrons are not polarized, the outgoing electrons in the transverse channel are spin polarized, which we can call the SHE. On the other hand, polarized incoming electron gives rise to a transverse voltage, associated as an inverse SHE (ISHE) (Bulgakov et al., 1999). This problem was then studied in the four-terminal Landauer–Bu¨ttiker formalism (Nikolic´ et al., 2005b; Sheng et al., 2005b). In this formalism, the authors used a tight-binding Hamiltonian with Rashba SO coupling on a square lattice. The random on-site energy, "i, is set to be a random variable within [W/2, W/2] when disorder effect is considered. When we consider a square sample with the size L L, and attach the leads with no SO coupling at the four edges, we can apply the four-terminal Landauer–Bu¨ttiker formalism. The current is driven through the lead 0 to 1 in the inset of Figure 8. Nikolic´ et al. (2005b) and Sheng et al. (2005b) studied systems with size up to 100 100. The SHE without any disorder is studied by changing the system size and the SO coupling. The SHC with spin along the z-axis, Gz, is of the order of e/4, but not exactly equal to the universal value e/8 calculated from the Kubo formula in the clean system. This SHC Gz depends on the system size L or the SO interaction VRSO. As the SO coupling VRSO increases, the SHC increases (Sheng et al., 2005b). The SHC with spins along the x-direction, Gx, is also nonzero, which is not the case for the Kubo formula calculation in a system with infinite size. The spin Hall current for both edges of the sample has the opposite sign for the x- and z-direction. When the size of disorder W is increased, the SHC is gradually suppressed, and when W 5t the SHC becomes vanishingly small. With the system size L 100, the size dependence of Gx and Gz depends on the SO interaction. There also arises a spin component along the y-direction, and it has the same sign for both edges of the sample. It corresponds to the
238 Spin Hall Effect
1.2
0.0050 0.0025 0.0000 –0.0025 –0.0050
W=0 W = 1t
0.8
W = 2t
0.0 3 L
–0.4 0 L –0.8
1
y 2 x
–1.2 –4
tSO = 0.01to
0.2 Spin conductance (e/4π)
GsH (e/4π)
0.4
G xsH
–2
0 E/t
2
4
Figure 8 Landauer–Bu¨ttiker calculation. From Sheng L, Sheng DN, and Ting CS (2005) Spin-Hall effect in twodimensional electron systems with Rashba spni–orbit coupling and disorder. Physical Review Letters 94: 016602.
current-induced spin polarization (CISP) studied theoretically (Edelstein, 1990; Aronov et al., 1991; Inoue et al., 2004) and experimentally. A similar system with the Rashba coupling also in the leads is studied by Wang and Chan (2005). Qualitatively similar results with the previous cases of leads without SO coupling are obtained, whereas the existence of the SO coupling in the leads makes the definition of the spin current somewhat ambiguous (Figure 9). The reciprocal effect of the SHE can be studied by the Landauer–Bu¨ttiker formalism (Hankiewicz et al., 2005a). In the Landauer–Bu¨ttiker formalism, the lead has no SO coupling, and the spin current is uniquely defined, thanks to the spin conservation. This conservation of spin automatically ensures the Onsager relation xy yx GSC ¼ – GCS
ð41Þ
This relationship holds for disorder-averaged conductances, because its proof uses the inversion symmetry on the x–y plane (Hankiewicz et al., 2005a). This relationship can be directly checked numerically. The 2DEG as well as the 2DHG are studied. The 2DEG is described by the Rashba model, and the 2DHG by a cubic Rashba model p2 i 3 þ 3 – H¼ þ ðp – pþ Þ, where = x iy. 2m 2 – The resulting SHC (which is equal to the Gyx CS) is
60 × 60
0.1 0.0 –0.1
G ysp
–0.2 0.08 0.04 0.00 –0.04 –0.08
G zsH EF = –3.8to
–4 –3 –2 –1 0 1 2 Fermi energy
3
4
Figure 9 Landauer–Bu¨ttiker calculation. From Nikolic´ BK, Zaˆrbo LP, and Souma S (2005b) Mesoscopic spin Hall effect in multiprobe ballistic spin–orbit-coupled semiconductor bridges. Physical Review B 72: 075361.
larger for the 2DHG than that for the 2DEG in general. Nonetheless, they are of the order e/8 and depend unmonotonically on and the system size L. An effect of impurity SO coupling is studied by Pareek (2004) in a Y-shaped geometry. Here a charge current is passed between two terminals, while the third terminal works as a voltage probe without any charge current. A pure spin current is then produced in the third probe, and it can be regarded as a result of an extrinsic SHE. Nonequilibrium spin accumulation has been studied in a two-terminal geometry, namely in a quantum wire geometry (Nikolic´ et al., 2005a; Wang et al., 2006b). The Keldysh nonequilibrium Green’s function is combined with the Landauer formalism to study numerically the spin accumulation. The tightbinding model studied by Ando (1989) is used for the calculation. The spins accumulate at both the edges, with their directions along the z-axis and the x-axis opposite for the two edges, in accordance with the calculation of the spin current by the Landauer– Bu¨ttiker formalism in the previous section (Nikolic´ et al., 2005b; Sheng et al., 2005b). In Nikolic´ et al. (2005a), the leads are without the SO coupling, whereas in Wang et al. (2006b) the leads include the SO coupling in the leads as well, which changes the spin-dependent scattering between the leads and the sample. In Wang et al. (2006b), the spin accumulation
Spin Hall Effect
oscillates along the wire, while this feature seems absent in Nikolic´ et al. (2005a). This might be because in Wang et al. (2006b) the nonlinear region (strong bias voltage eV EF) is studied whereas the system in Nikolic´ et al. (2005a) is closer to the linear region (weak bias voltage eV EF). It is also shown that the spin Hall current can really inject spin into the leads (Nikolic´ et al., 2005a). In contrast, the spin accumulation along the y-axis is the same for the two edges, resulting from the CISP (Nikolic´ et al., 2005a; Wang et al., 2006b). In Hankiewicz et al. (2004), an H-shaped structure is proposed for a measurement of the SHE through detransport properties. This structure follows the same spirit as the structure proposed by Hirsch (1999); in order to measure the spin current due to the SHE, one uses the inverse effect of the SHE to convert the spin current to a voltage signal. In Hankiewicz et al. (2004), the device size is set to be of the order of 100 nm, and one passes the current 10 nA into the lower arm as shown in Figure 10. When the Rashba coupling R is changed from 0 to 80 meV nm, the voltage change between the voltage probes (10) becomes 2–8 mV as shown in Figure 11 (Hankiewicz et al., 2004). By using the Landauer–Bu¨ttiker formalism, the SHC in a ring geometry is calculated with realistic parameters. In Souma and Nikolic´ (2005), the Landauer–Bu¨ttiker formalism is applied to a mesoscopic ring with Rashba SO coupling with four terminals attached. If the ring is smaller than the dephasing length, the SHC is affected by an interference of two paths, due to the Aharanov–Casher phase around the ring. Indeed, the SHC has a quasi-oscillatory behavior as functions of the Fermi energy or the Rashba SO coupling. A 1D mesoscopic ring with the Rashba and Dresselhaus SO coupling is studied (Moca and Marinescu, 2005), and similar oscillatory behavior of the SHC is found as a function of the SO coupling. Disorder reduces the size of the SHC and the amplitude of the oscillation (Moca and Marinescu, 2005). Nonequilibrium spin accumulation for a 1D mesoscopic ring with Rashba coupling is studied (Zhang, 2006), and a similar quasi-oscillatory behavior is observed, depending on whether the beating pattern of the spin accumulation matches with the ring circumference. There are many other publications on the various aspects of the mesoscopic SHE to which the readers are referred (Adagideli and Bauer, 2005; Bardarson et al., 2007; Bellucci and Onorato, 2007a,b, 2008; Brusheim and Xu, 2006, 2007; Chen et al., 2007; Erlingsson and Loss, 2005, 2006; Galitski et al., 2006; Hankiewicz et al., 2005b;
(a)
239
V5 Itot = 0 V4 Itot = 0
V3 Itot = 0
V2 = 0 Itot = –1
V1 Itot = 1 V6 Itot = 0 (b)
a0
Figure 10 (a,b) Schematic picture for the H-shaped device for detection of the SHE. From Hankiewicz EM, Molenkamp LW, Jungwirth T, and Sinova J (2004) Manifestation of the spin Hall effect through charge-transport in the mesoscopic regime. Physical Review B 70: 241301.
Hattori and Okamoto, 2006; Hu et al., 2003c; Huh et al., 2007; Jiang and Hu, 2007; Levitov et al., 1985; Liu et al., 2007a; Mal’shukov et al., 2007, 2005b; Moca and Marinescu, 2005, 2007b; Moca et al., 2008; Nikolic´ and Zaˆrbo, 2006; Nikolic´ et al., 2006, 2005c; Ohe et al., 2008; Onoda and Nagaosa, 2005a; Pareek, 2007; Qiao et al., 2007, 2008; Ren et al., 2006; Reynoso et al., 2006; Schliemann, 2007; Sheng and Ting, 2006a, b; Souma and Nikolic´, 2005; Wu and Zhou, 2005; Xing et al., 2006a, b; Zyuzin et al., 2007; Mal’shukov and Chu, 2006; Mal’shukov et al., 2005a; Pershin and Di Ventra, 2008; Sun et al., 2005; Xing et al., 2007; Zhou et al., 2007). 1.07.2.5 Experimental Observation of SHE in Semiconductors The SHE in semiconductors was recently observed experimentally (Day, 2005). There are two types of measurements, that is, the KR and the LED, each of which is reviewed below. 1.07.2.5.1 Measurement of SHE in semiconductors by KR
When the SHE occurs in a semiconductor sample, the induced spin current will eventually reach the sample edge, and they will accumulate, balancing with the spin diffusion and spin relaxation. Such spin accumulation is measured by KR in a series of experiments by Awschalom et al. (Kato and Awschalom, 2008; Sih and Awschalom, 2007). In their experiments, laser beam with linear
240 Spin Hall Effect
L = 90 nm, N1= 36, 4 leads L = 90 nm, N1= 42, 4 leads L = 120 nm, N1= 42, 4 leads L = 120 nm, N1= 48, 4 leads
0 100
–2
|ΔV (λ) – ΔV(0)|/|ΔV (0)|
ΔV34(λ) – ΔV34(0) (μV)
2
–4
–6
1 0.1 0.01 0.001
0.0001
–8 0
|ΔV12(λ) – ΔV12(0)|/|ΔV12(0)| |ΔV34(λ) – ΔV34(0)|/|ΔV34(0)|
10
0
20
20
40 λ (me V nm)
60
40 λ (meV nm)
80
60
80
Figure 11 Calculated voltage drop between terminals 3 and 4 in the H-shaped device. From Hankiewicz EM, Molenkamp LW, Jungwirth T, and Sinova J (2004) Manifestation of the spin Hall effect through charge-transport in the mesoscopic regime. Physical Review B 70: 241301.
polarization, focused to 1 mm size, is incident onto the sample, and they measured the rotation angle of the polarization plane upon reflection. The rotation angle is proportional to the spin polarization perpendicular to the plane. When a static magnetic field is applied, the magnetization will precess around the magnetic field, and decays as A?ð1=½ð!L s Þ2 þ 1Þ by the Hanle effect, where !L ¼ gB B=h is the Larmor frequency and s is the spin coherence time. By measuring the field dependence of the angle, one can determine s. Kato et al. (2004) observed the SHE in n-type semiconductors by measuring spin accumulation at the edges of the sample by KR (Bauer, 2004). The spin accumulation is uniformly distributed along both the edges, while away from the edges there is no spin accumulation, as can be seen from Figure 12. From experimental data they evaluated the amount of spin accumulation and spin lifetime as a function of an external magnetic field. The measured SHC is 5 103 1 cm1. They concluded the observed SHE to be extrinsic for the following reasons: (1) spin splitting is negligibly small in the sample and (2) the effect has no dependence on crystal orientation. Nevertheless, Bernevig and Zhang (2004) argued that the observed SHE can be intrinsic, coming from the dresselhaus term representing bulk inversionsymmetry breaking. They showed that even if the spin splitting due to the Dresselhaus term is negligibly small, the SHE can be as large as the
experimental data. It can also account for the absence of dependence on crystal orientation. Thus, the quantitative explanation of the observed SHE is still to be resolved. Further experiments using the spin mapping by the KR have been done by the same group. Sih et al. (2005) investigated the SHE and CISP in (110) AlGaAs QWs with various direction of electric field. In the (110) QW, the Dresselhaus field is perpendicular to the plane, while the Rashba field is in-plane; thereby, the separation between the two fields is possible. For Ejj[001], the CISP due to the Dresselhaus field vanishes, and the total CISP is in-plane. Thus, the CISP is absent in the KR signal, because the KR detects the spin component perpendicular to the plane. Therefore, the out-of-plane spin is accumulated only at the two edges of the sample. On the other hand, for other orientations of the electric field such as ½110 and ½112, the CISP has an out-of-plane component, and appears in the whole sample, in addition to the SHE signal at the edges. Sih et al. (2006) also investigated the device with side arms, in order to see whether the observed spin accumulation at the edges is due to a spin current, or due to other edge effect. The measurement was done for 3D n-GaAs (2 mm thick) with a Si doping with n = 3 1016 cm3 at T = 30 K. The observed profile of spin distribution is well fitted by assuming that the spins are generated through the SHE in the main channel, and flow into the side arm.
Spin Hall Effect
ns (a.u.) –2 –1 0 1
2
150
100
Position (μm)
50
0
–50
–100
–150 –40 –20 0 20 40 Position (μm) Figure 12 Experimentally observed spatial profile of the spin accumulation. The electric field is in the vertical direction. Red and blue represent the opposite directions of the out-of-plane spin polarizations. From Kato YK, Myers RC, Gossard AC, Awschalom DD (2004) Observation of the spin Hall effect in semiconductors. Science 306: 1910.
Stern et al. (2006) reported the SHE in ZnSe which persists even at room temperature. With the sample of the n-type Cl-doped ZnSe with the carrier concentration n = 9 1018 cm3, they used the KR spectroscopy, where the the pump-probe technique with the normally incident beam focused to a 15 mm spot. The time-resolved KR measurement determined the spin coherence time of 0.5 ns for n = 9 1018 cm3. At T = 20 K, the spin density peak n0 at the edge of the sample is found to be n0 > 16 spins/mm3 under the electric field of 3 mV mm1. With the fitting to solution of the drift-diffusion equation, the spin diffusion length is estimated as Ls = 1.9 0.2 mm at T = 20 K and the estimated SHC is s = 3 1.5 1 m1/jej at T = 20. The temperature dependence of s has also been measured and even at room temperature s > 0.5 1 m1/jej. The measured sign and magnitude of s is consistent with the extrinsic mechanism (Engel et al., 2005). This is a surprising result considering
241
that the SO interaction is weaker in ZnSe compared with GaAs, and is encouraging for the applications. The in-plane CISP in n-type ZnSe is also measured at room temperature (Stern et al., 2006). The spin accumulation due to the CISP obtained from the KR is 12 spins mm3 at 20 K, which is comparable to that in n-GaAs, although the SO coupling is considered to be much weaker in n-ZnSe than in n-GaAs. Another interesting recent development is the time-resolved dynamics of the SHE observed in GaAs. By using the electrically pumped time- and space-resolved KR spectroscopy, they could see the time–space dependence of the spin accumulation after the voltage pulse (Stern et al., 2008). With the voltage applied along x-direction, we expect the spin current to flow along y-direction with the spin polarization along z-direction, and the spin accumulation of that polarization near the edge at y = L with L being the half of the dimension of the sample measured from the center. The dynamics of the spin density s(y, t) is described by the following equation: qjyi qs i s i ðy; t Þ ðy; t Þ ¼ – ðy; t Þ – þ ðgB =hÞ½B sðy; t Þi qt qt ð42Þ
where is the spin decoherence time, B is the magnetic field, and jiy is the spin current given by jiy = Dqysi sE i,z. This equation can be solved by imposing the appropriate boundary condition relevant to the experimental situation, and excellent agreements between the data and the calculation have been obtained as shown in Figure 13, which established the SHE as the origin of the spin accumulation near the edge, and also gave a reliable estimate of various physical quantities as = 4.2 ns and the spin diffusion length Ls = 3.9 mm. 1.07.2.5.2 Measurement of SHE in semiconductors by spin LED
Wunderlich et al. (2005; Kaestner et al., 2006) observed the SHE in a 2D p-type system, using a p–n junction LED. They applied an electric field across the hole channel, and observed a circular polarization of the emitted light, whose sign is opposite for the two edges of the channel as shown in Figure 14. The circular polarization is 1% at maximum. This circular polarization is attributed to the recombination of electrons and spin-polarized holes produced at the p-type channel through the SHE. They argued that it is near to the clean limit, and the
242 Spin Hall Effect
(a) 0.2
(b) 0.2 Model
B (T)
B (T)
Data
0
0 θK (μrad) –4
–0.2
0
10 t (ns)
–0.2
20
0
10 t (ns)
8 20
Figure 13 Experimental setup for the time-resolved measurement of SHE. From Stern NP, Steuerman DW, Mack S, Gossard AC, and Awschalom DD (2008) Time-resolved dynamics of the spin Hall effect. Nature Physics 4: 843.
obtained SHE is mostly intrinsic. More refined argument, by showing vanishing vertex correction, also supports this conclusion (Bernevig and Zhang, 2005). Nomura et al. (2005c) studied numerically the SHEinduced edge-spin accumulation in a 2DHG with
(a)
Ip
strong SOs (Figure 15). They found it is independent of the strength of disorder scattering suggesting the intrinsic nature of the effect, and obtained the similar phenomenological aspects as compared with the experiment (Wunderlich et al., 2005).
LED 1 +Ip
1
LED 1 p
0
n
x
n
z LED 2
(b)
9
(meV)
(d)
–1
–Ip
2.5
+Ip
LED 1
1.5
CP (%)
1.5-mm channel
y
1
0
5 LED 2
0.5 (c)
–1
1 10 P2D
20
30
(1011
cm–2)
1.505
1.510
1.515 E (eV)
1.520
Figure 14 (a) The device setup for the spin LED, (b) circular polarizations measured for the opposite directions of current, (c) circular polarizations measured at LED1 and LED2, and (d) fit with the theory. From Wunderlich J, Kaestner B, Sinova J, and Jungwirth T (2005) Experimental observation of the spin-Hall effect in a two-dimensional spin–orbit coupled semiconductor system. Physical Review Letters 94: 047204.
Spin Hall Effect
243
(a) Spin-density profile
(c) Edge-spin density
4 0
10
5
–4 0
–8 (b) Spin-current profile
(b) Bulk-spin current 10
6 4
5
2
jxzbulk/E (e /8π)
jxz(x)/E (e /8π)
8
0
Szedge/E (e /4πvF)
Sz (x)/E (e /4πvF)
8
0 0
10
20 30 X (kF–1)
40
50
0
2
4
6 8 10 12 14 EF (h/τ)
Figure 15 Spatial dependence of the spin accumulation and spin current in a model for the 2D hole gas. From Nomura K, Wunderlich J, Sinova J, Kaestner B, MacDonald AH, and Jungwirth T (2005c) Edge-spin accumulation in semiconductor twodimensional hole gases. Physical Review B 72: 245330.
A theoretical study on the spin accumulation by the electric field can also be found in Kleinert et al. (2005).
1.07.2.6
Various Issues on the SHE
1.07.2.6.1
Definition of the spin current In the presence of the SO coupling, the total spin is not conserved. Hence, there is no unique way to define a spin current. Naively we expect that the spin current js should satisfy the equation of continuqs i ity þ r?Jsi ¼ 0; this relationship requires the qt conservation of total spin, namely 0¼
q qt
Z
s i dd x ¼ – i
Z
s i dd x; H
ð43Þ
In the cases relevant for the SHE, the SO coupling violates this conservation of total spin. In other words, due to the nonconservation of spin, Noether’s theorem is not applicable for a definition of spin current. The conventional definition of the spin current, (21) or (36), is usually adopted. However, its mathematical meaning as a current is illdefined. A current is always associated with a corresponding conserved quantity. The conventionally defined spin current is not conserved for the Luttinger Hamiltonian due to the SO coupling.
One can adopt the symmetrized product 1 ðvi s j þ s j vi Þ between the velocity v and the spin s 2 as a definition of the spin current as in Sinova et al. (2004). The result given by the Kubo formula with this definition is in general different from that by the semiclassical theory described above (Murakami et al., 2004b). This difference comes from noncommutability between the spin S and the velocity v. In other words, this comes from the nonuniqueness of the definition of spin. One can modify the semiclassical theory to give the same result as the Kubo formula by adding three contributions: spin dipole, torque moment, and change of wavepacket spins due to electric field (Culcer et al., 2004). An alternative way is to separate the spin s into conserved (intraband) part s(c) and nonconserved (interband) part s(n) (Murakami et al., 2004b). As [s(c), H] = 0, spin current can be uniquely defined for s(c). The resulting spin current is jij ¼
e ðkH – kL Þ"ijl El 62 F F
ð44Þ
which is different from Equation (13), and this difference is considered as a quantum correction to Equation (13). Another attempt for defining conserved spin current is done by introducing torque dipole moment (Entin-Wohlman et al., 2005; Shi et al., 2006). A continuity equation for the spin can be written as
244 Spin Hall Effect qs z þ r?Js ¼ Tz qt
ð45Þ
where Tz X dsz/dt is the torque density. This torque density describes a spin precession due to local magnetic field of the SO coupling. If this torque density is zero, it describes the spin conservation that the total increase of the spin in a certain region of the system is equal to a net in-going spin current js; hence, the spin current is measurable as a spin accumulation. On the other hand, if the torque density Tz is nonzero, the spin current is not directly connected with a spin accumulation. If the average of the torque over the entire system vanishes due to some symmetry reasons, that is, 1 V
Z
dV Tz ¼ 0
ð46Þ
one can define another conserved spin current, which is directly related with the spin current. Provided Equation (46) holds, the torque density term can be written as Tz ¼ – r?P
ð47Þ
where P is called a torque dipole density. Hence, we get ð48Þ
J s ¼ Js þ P
ð49Þ
where is called a conserved spin current. In order to calculate the torque dipole moment, P , in the bulk in response to an external electric field E, it is convenient to consider an electric field at a finite wavevector q. The torque dipole density at finite q, Tz(q) = (q) ? E(q) is calculated, and from Equation (47) we get ð50Þ
from which the SHC can be calculated. Let us write ð0Þ ð Þ ðcÞ s ¼ s þ s
(c) s ,
) ( s ,
ð51Þ
(0) s
where and are the SHC for the conserved spin current Js, the conventional spin current js, and the torque dipole density P . In Shi et al. (2006), the SHC for the conserved spin current is calculated for Rashba model, cubic Rashba model, and Luttinger model. For Rashba model ð0Þ s ¼
e e e ; ð Þ ¼ – ; ðsÞ ¼ – 8 s 4 s 8
ð0Þ s ¼
9e 9e 9e ; ð Þ ; ðcÞ s ¼ – s ¼ – 8 4 8
ð52Þ
ð53Þ
The results for the conventional and the conserved spin currents have different sign; hence, the sign of the SHC in experiments or numerical simulations might be a simple test to distinguish between the conserved spin current and the conventional one. This definition ensures the Onsager relation between the SHE and its reciprocal effect. In the reciprocal effect, a spin force Fs, which is a gradient of a spin-dependent chemical potential or that of the Zeeman field, induces a charge current transverse to the gradient (Zhang and Niu, 2004). This reciprocal effect can be evaluated in the semiclassical theory where the Berry phase in a mixed position-momentum space, xk, plays a central role for the effect (Zhang and Niu, 2004). The total response with SO coupling is written as Js jc
! ¼
ss sc
!
cs ss
Fs
!
E
ð54Þ
The Onsager reciprocity is then written as cs sc ¼ –
qs z þ r?Js ¼ 0 qt
P ¼ i½rq ðqÞq¼0 ?E
and for the cubic Rashba model
ð55Þ
which is explicitly shown in the semiclassical theory (Zhang and Niu, 2004). This reciprocity manifests that the conserved spin current Js is a time derivative of an operator xsz, which is conjugate to the spin force Fs. This reciprocal effect was discussed earlier in a context of extrinsic effect (Hirsch, 1999). Because there is no unique definition for the spin current, we have to choose one definition which matches the considered experimental setup to measure the spin current. One possible test to see which definition of spin current is relevant is to measure the spin accumulation. To this purpose, spin accumulation as well as conventional spin current are calculated numerically, by varying disorder. The calculation uses the Kubo formula both in the strip geometry for the 2DHG and 2DEG (Nomura et al., 2005c). The resulting spin accumulation can be fitted by the conventional spin current rather well. The readers are referred to the further references on the spin current issues (Chen et al., 2006; Fu et al., 2007b; Jin and Li, 2008, 2006; Jin et al., 2006; Leurs et al., 2008; Shen et al., 2006; Sun and Xie, 2005; Wang et al., 2006c; Wang and Zhang, 2007; Yang and Chang, 2006; Zhang et al., 2008b; Li and Tao, 2007; Wong et al., 2008).
Spin Hall Effect
1.07.2.6.2
Ab initio calculations On the other hand, the intrinsic SHE is calculated for Si, GaAs, W, and Au for various values of the Fermi energy (Guo et al., 2005). For p-type GaAs the predicted value is maximally 300 1 cm1, which is of the same order of magnitude as that from the Luttinger model (Zhang and Yang, 2005). The p-type Si has somewhat smaller value, 50 1 cm1, which is due to smaller SO coupling than GaAs. For n-type GaAs, the size of the SHC is 50 1 cm1 and undergoes a sign change as a function of carrier density, and n-type Si is found to have a negative SHC ( 50 1 cm1). It is found that even without doping, GaAs and Si show a small but finite SHE (43 and 7 1 cm1, respectively). It is due to a small hybridization. In this sense, these undoped semiconductors are SHI. In the metallic systems such as W and Au, the SHC is calculated to be 1390 and 731 1 cm1, respectively, and is rather robust against disorder (Yao and Fang, 2005). These ab initio calculations are done by calculating only the bare vertex diagram (Figure 5) in the Kubo formula. Nevertheless, analytic calculations revealed that the vertex correction can be of the same order as the intrinsic SHE, and an effect of vertex correction on such ab initio calculations remains to be unsettled. For applications to spintronies (Zˇutic´ et al., 2004), the SHE may potentially be used as an effective means to inject spins into semiconductor spintronics devices. There have been many proposals for semiconductor spintronics devices such as Datta–Das spin transistor (Datta and Das, 1990). Nevertheless, an effective spin injection into semiconductors is one of the elusive issues, because spin injection from a ferromagnetic metal suffers from conductance mismatch (Schmidt et al., 2000). There have been many kinds of attempts to overcome this difficulty, and the SHE might be one of the way out toward effective spin injection into semiconductors. In particular, one of the merits of the SHE is that it does not require magnetic field or magnetism. The main obstacle for the SHE toward application is its smallness. Ab initio calculations would be important for searching systems with large enough SHC.
1.07.3 SHE in Metals 1.07.3.1
Experiments of SHE in Metals
Up to now we have focused mainly on the semiconductor systems with small carrier concentration. The local band structure in momentum bfk-space is well
245
characterized by, for example, k ? p theory, and the theory can be rather well controlled. However, it has the following disadvantages. (1) due to the small number of carriers, the conductivity itself is small, and correspondingly the SHC is even smaller; (2) the interface with magnetic materials, which are mostly metals, suffers from the impedance mismatch (Schmidt et al., 2000); and (3) the systems are rather fragile against disorder and thermal agitation. These disadvantages are overcome by the metallic systems. For example, (1) the SHC is much larger than that of doped semiconductors typically by the factor of 102– 105; (2) the junction with ferromagnetic metals is simple and techniques in metallic spintronics such as the spin injection (Johnson, 1993; Johnson and Silsbee, 1985), spin diffusion (Shchelushkin and Brataas, 2005a,b), and nonlocal effect can be used; and (3) the quantum coherence is much more robust against disorder and thermal agitation due to the large Fermi energy, and hence the operation at room temperature is possible. These advantages make the metallic systems more promising candidates for the real applications of the SHE in spintronies. In semiconductors, the optical detection of the spin accumulation such as the KR has been used to detect SHE, since the spin diffusion length is much larger than the spot size of the laser beam of the order of 1 mm. In metals, on the other hand, the spin diffusion length is much shorter, and the electrical detection is used (Huh et al., 2008). There are two ways to study the SHE in metals. One is to use the nonlocal spin diffusion due to the injection of the spin-polarized current from the ferromagnet (Takahashi and Maekawa, 2003, 2007, 2008a,b). An early experiment on the SHE in Al (Valenzuela and Tinkham, 2006) uses CoFe as the ferromagnet. A thin Al Hall cross is oxidized and contacted with the ferromagnetic electrodes as shown in Figure 16. When the current flows from the ferromagnetic electrode to Al Hall bar, the spin diffusion occurs to the other side of the bar, where the Hall voltage is measured. This is the reciprocal effect to the SHE in which the charge current induces the spin current. This is called the ISHE, and is now widely used to measure the SHC in metals. As shown in Figure 16 the spin transistor configuration with two ferromagnetic electrodes FM1 and FM2 has been used to characterize the spin diffusion length sf, the spin polarization P, and the angle between the magnetization m and the electrode axis. The chemical potential difference between up- and down-spins decays as the distance x from the distance from the
246 Spin Hall Effect
(a) B⊥ M
2
FM
θ M1
F
AI
500 nm B⊥
(b)
(d) +
I
BI
+ V –
VSH
– FM1
FM2
(c)
(e)
μ
μ
Js
Js
0
0
–1
0
FM1 LFM
LSH
1 x/λ sf
2
2
1 0 x/λ sf
–1
Figure 16 (a–e) Experimental setup for the measurement of SHE in Al. From Valenzuela SO and Tinkham M (2006) Direct electronic measurement of the spin Hall effect. Nature 442: 176.
ferromagnet, and the spin diffusion length sf is estimated from the exponential decay of the resistance change due to the spin transistor action as a function of x. The measurement has been done at low temperature (4.2 K), and it turned out that sf depends on the Al film thickness tAl; that is, sf = 705 nm for tAl = 25 nm and sf = 455 nm for tAl = 12 nm from the data shown in Figure 17(b). With the configuration in Figure 16(b), the spin Hall resistance RSH = VSH/I is expressed as RSH ¼
RSH sin 2
ð56Þ
with RSH ¼
Ps expð – LSH = sf Þ tAl 2c
ð57Þ
where s is the SHC, c is the conductivity of Al, and LSH is the distance between the FM1 and the voltage terminal. By plotting RSH as a function of LSH, they estimated the sf and s as sf = 735 nm for tAl = 25 nm and sf = 490 nm for tAl = 12 nm in good agreement with the above estimate by the spin transistor configuration. The SHC is obtained as s = (2.7 0.6) 101 1 cm1 for tAl = 25 nm, and s = (3.4 0.6) 101 1 cm1 for tAl = 12 nm. This corresponds to the spin Hall angle SH = s/ c = (1–3) 104, which is a bit smaller than the typical value of the anomalous Hall angle 103. Later, Kimura et al. (2007) observed the SHE in Pt metal at room temperature. They used the device structure as shown in Figure 18 where the Cu metal is bridging the ferromagnet (Py) and Pt. With this configuration, they could observe both the SHE and ISHE. For example, when the spin-dependent electrochemical potential is induced in Cu and also Pt by the spin injection from Py, the spin current is introduced and the charge current in Pt is induced by the ISHE. When the voltage is applied to Pt and the charge current flows, on the other hand, the SHE induces the spin current in Cu and Py, producing the voltage change in Py. These two effects are related by the Onsager’s reciprocal relation, which is quantitatively confirmed in this experiment. The obtained value for the SHC s in Pt at room temperature is 2.4 102 1 cm1, and the spin Hall angle is SH = s/c = 3.7 103. Recently, even larger SHE has been reported in FePt/Au devices at room temperature (Seki et al., 2008). As shown in Figure 19, the Au Hall cross is attached to the FePt injector in their devices. The resistivities () of Au and FePt at room temperature are 2.7 m cm and 36 m cm, respectively. With this small value of for Au, the main mechanism of the SHE/ISHE is considered to be the extrinsic SS. The spin diffusion length, Au, of Au is estimated as Au = 86 10 nm at room temperature, which is much larger than that of Pt. They also measured both SHE and ISHE, and the obtained value of the spin Hall angle SH is 0.113 from the data in Figure 19, which is very large compared with the values obtained so far. Another method to inject the spin current is to use the ferromagnetic resonance. This method is the so-called spin battery or spin pumping, and is shown schematically in Figure 20. The ferromagnetic metal is attached to the nonmagnetic metal, and the microwave magnetic field is applied to the ferromagnetic metal. The magnetization precesses
Spin Hall Effect
(a)
(b) FM2
FM1
ΔR (mΩ)
I– I+
(c)
1
λsf = 455 nm
0.1
tAI = 12 nm tAI = 25 nm
V+
1 (d)
LFM = 2 μm 2
λsf = 705 nm
10
V–
I–
2 3 LFM (μm)
–2
2
1
0
B⊥
M
0
θ
–2
0
θ
• sin
sin θ
0
4
1
B⊥ (T) –0.4 –0.2 0.0 0.2 0.4
V/I (mΩ)
V/I (mΩ)
247
–1 2 B⊥ (T)
3
–3
0 B⊥ (T)
3
Figure 17 (a–d) Experimental data for the SHE in Al. From Valenzuela SO and Tinkham M (2006) Direct electronic measurement of the spin Hall effect. Nature 442: 176.
Pt
y
Cu
z
y Cu
x Cu
Py
1
Cu
1 2 3
x
2
3
Pt
Py
Cu
(a)
Is (b)
(c)
Ie s Is Cu
Py 1
Is
Pt 2
Ie
3
Is (e)
(d)
Is Cu
Py 1
Pt 2
3
Is
Ie s
Ie
Figure 18 (a–e) Experimental setup for the measurement of SHE and ISHE in Pt. From Kimura T, Otani Y, Sato T, Takahashi S, and Maekawa S (2007) Room-temperature reversible spin Hall effect. Physical Review Letters 98: 156601.
z
(a)
(b) F
y
200 nm FePt injector
150 nm
x d
M
C E
Au
B V Au hall cross
D Fept A
(c) –31.0
(d)
300 ⏐ΔV⏐ (μV)
I = +0.5 mA
–31.5
+
200 –19.5 0
0.5
100
I(mA) ISHE
0
LHE
V
ΔV
– e–
V (μV)
0
RLHE (mΩ)
RISHE (mΩ)
M
–19.0
0.5
–32.0
V–
I = +0.5 mA
V+ V+
I = –0.5 mA M
20.0
–32.5
+ –100
e–
ΔV
19.5
V –
–33.0 –200 V– –10
–5
0
5 H (k0e)
10
15
–8
–4
0
4
8
H (k0e)
Figure 19 (a–d) Experimental setup and results for the measurement of ISHE in Au. From Seki T, Hasegawa Y, Mitani S, et al. (2008) Giant spin Hall effect in perpendicularly spin-polarized FePt/Au devices. Nature Materials 7: 125.
Spin Hall Effect
(a)
Microwave V H
Ni81Fe19
θ
Pt Magnetization Magnetization Ni81Fe19 (b) Spin pumping Pt
σ
Jc Js
σ
+
Jc Jc
–
Figure 20 (a,b) Experimental setup for the spin pumping and the measurement of ISHE in Pt. From Saitoh E, Ueda M, Miyajima H, and Tatara G (2006) Conversion of spin current into charge current at room temperature: Inverse spin-Hall effect. Applied Physics Letters 88: 182509.
around the magnetic field with the Gilbert damping. More explicitly, the EOM for the magnetization reads dm dm ¼ m H eff þ m dt dt
ð58Þ
where the last term on the right-hand side represents the Gilbert damping, and is typically 103–101. This Gilbert damping drives the magnetization precession amplitude smaller and smaller and the magnetization eventually turns the direction of the effective magnetic field Heff. From the viewpoint of the conservation of the angular momentum, the spin angular momentum dissipates to the environment from the ferromagnet, that is, the flow of the spin current is produced from the ferromagnet to the reservoir. In the present configuration, the reservoir is the nonmagnetic metal. To summarize, the spin current with the polarization direction parallel to the averaged magnetization is injected to the nonmagnetic metal from the ferromagnet by the ferromagnetic resonance. This injected spin current will produce the charge current jc in the nonmagnetic metal through the ISHE as expressed by jc ¼ DISHE js
ð59Þ
where DISHE is a coefficient for ISHE efficiency. In the original experiment by Saitoh et al. (2006) and Inoue et al. (2007), Ni81Fe19/Pt sample with the
249
thickness of 10 nm for NiFe layer and 7 nm for Pt layer, respectively, was used to detect the ISHE. The strength of the resonance microwave magnetic field is HFMR = 130 mT, and the resonance line shape is broadened when Pt is attached compared with the case of NiFe only. This is the evidence for spin pumping. They observed the voltage drop at the edge of Pt which is consistent with the angle dependence expected for the ISHE. The thickness of Pt layer is of the order of 1 nm, which is within the decay length of the spin or spin current due to the strong SO interaction in Pt. Figure 21 shows the dV(H)/dH curves for various angle between the applied static magnetic field (the direction of the magnetization) and the direction of the voltage drop V(H) measured for Pt. The line shape of V(H) is fitted by V ðH Þ ¼ IISHE
2
ðH – HFMR Þ2 þ 2 – 2ðH – HFMR Þ þ IAHE ðH – HFMR Þ2 þ 2
ð60Þ
where is the relaxation rate. The second term represents the contribution from the AHE in NiFe system. Taking the derivative dV(H)/dH, the ISHE contribution is odd with respect to H HFMR while AHE contribution is even. Therefore, it is clear from the data in Figure 21 that the ISHE contribution is dominant. Another test of ISHE is the angel dependence. From (58), the signal is expected to be proportional to sin ( is the angle between the external magnetic field and the charge current), which is also consistent with the data. These experimental results provide rather firm evidence for the ISHE in Pt metal, but the magnitude of the SHC could not be derived (Saitoh et al., 2006). Recently, an extension of this experiment has been done by the same group using similar devices (Ando et al., 2008a,b). In this experiment, the magnetization relaxation in NiFe is controlled by the spin current injection from Pt by the SHE. They have carefully studied the change in the ferromagnetic resonance signal, dI(H)/dH, due to the charge current Jc in Pt layer. The idea here is just opposite to that in the above experiment. For example, the injected spin current to NiFe will modify the Gilbert damping, which depends on the direction of the spin current polarization. They studied the resonance lineshape for different values of the current in Pt. Note that the change in the spectra is
250 Spin Hall Effect
H
Ni81Fe19/Pt
0
10–4V/T
0
Ni81Fe19
ISHE dV/dH
Ni81Fe19/Pt
H (mT)
150
100
(d) HFMR H
Ni81Fe19
10–5V/T
H (mT)
150
(e) Ni Fe /Pt 81 19 10–4V/T
calc. exp.
dV (H)/dH
100 (c)
θ = 90°
(b)
θ = 90°
FMR
dV (H )/dH
d/(H)/dH (a.u.)
(a)
AHE dV/dH
H
0 0 = 90°
HFMR 100 150 H (mT)
120 140 H (mT)
Figure 21 (a–e) Experimental data for the ISHE in Pt. From Saitoh E, Ueda M, Miyajima H, and Tatara G (2006) Conversion of spin current into charge current at room temperature: Inverse spin-Hall effect. Applied Physics Letters 88: 182509.
symmetric for positive and negative direction of the current for = 0 ( is the angle between the external magnetic field and the charge current) suggesting that the heating effect is the dominant contribution. When = 90 , on the other hand, the spectral change is clearly different for the different direction of the charge current. By examining the asymmetric part, that is, dI(H,jc)/dH dI(H,jc)/ dH, they could estimate the change of the Gilbert damping coefficient due to the charge current jc through the SHE. Considering the values of spin diffusion lengths Pt = 7 nm, Py = 3 nm, and 1 the conductivities Pt and c = 0.064(m cm) 1 Py c = 0.065(mcm) , they obtained the spin Hall angle for Pt as 0.08, which is much larger than 0.0037 obtained by Kimura et al. (2007). The ISHE has been recently applied to the experimental observation of the spin Seebeck effect (Uchida et al., 2008). With the magnetic field applied also along the x-direction, the spin current is produced by the difference of the chemical potentials " and # for up- and down-spins, respectively. In the attached Pt metal, where the decay of the chemical potential difference, = " #, occurs along the z-direction, the ISHE occurs and results in the electric field ESHE or the voltage drop along the y-direction. This voltage
drop due to ESHE is measured as a function of the temperature difference, T, between the two ends of the sample, the applied magnetic field H, and also the position xp of the attached Pt metal measured from the center of the sample. Results showed that r is almost constant along the xdirection, which suggests that the temperature gradient term, that is, the second term, gives the dominant contribution to r in the expression r = (qc/qn)rn + (qc/qT)rT er (c: spin-dependent chemical potential for spin , : scalar potential). It has been argued that the contribution from rn decays quickly within the length scale of the spin diffusion length (5 nm) much shorter than the size of the sample (6 mm), and does not contribute to the bulk of the sample. From this measurement, the estimated spin Seebeck coefficient for Ni81Fe19 is S spin > 2 nV K1 at T = 300 K, which is orders of magnitude smaller than the usual Seebeck coefficient S > 20 mV K1. In addition to the studies mentioned above, the increasing number of publications address the metallic SHE (Fan and Eom, 2008; Harii et al., 2008; Kent, 2006; Ma et al., 2008; Otani and Kimura, 2008; Song et al., 2006a; Stern et al., 2007, 2008; Valenzuela and Tinkham, 2007; Vila et al., 2007).
Spin Hall Effect
1.07.3.2
Theories of SHE in Metals
Theoretically, both the intrinsic and extrinsic mechanisms can contribute to the SHE in metals as in the case of semiconductors. From the studies on the AHE, the microscopic mechanism depends strongly on the degree of disorder characterized by the longitudinal resistivity or conductivity c (Onoda et al., 2006b; Miyasato et al., 2007). For very clean system where c is larger than 106 1 cm1, the extrinsic SS is expected to be dominant where the spin Hall angle is the proper quantity. In the intermediate region where c lies between 104 and 106 1 cm1, the intrinsic contribution is the dominant one, where s is of the order of 103 1 cm1. This value of 103 1 cm1 corresponds to e2/ha with a > 5 A˚ as the lattice constant, representing the QH conductivity in 3D. Therefore, it is reasonable to assume the intrinsic mechanism for Pt metal since the conductivity of Pt at room temperature is of the order of 105 1 cm1. The first-principles band structure calculation for the intrinsic contribution has been done (Guo et al., 2008). Note that the SHE does not simply scale with the SO interaction strength, which suggests that the perturbative treatment of SO interaction is not valid. Pt shows particularly large SHE surviving even up to room temperature among the 5d elements with the similar SO interaction strength. The band structure of Pt is calculated using a fully relativistic
extension of the all-electron linear muffin-tin orbital method based on the density functional theory with local density approximation. Figure 22 shows the relativistic band structure of Pt, and also the SHC s, as a function of EF. The 6s bands are extending broadly from 10 eV to higher energy, while the 5d bands are within the range of 7 eV < " < 1 eV. Therefore, the electronic states near EF = 0 are the mixture of the 6s and 5d bands. The situation is different for Au, where one extra electron is added compared with Pt, and the Fermi energy is shifted upward. The electronic states near the Fermi energy are mostly 6s and 6p bands in this case. Therefore, we need to look at the electronic structure in more detail to understand the mechanism of SHE. As shown in Figure 22 the SHC s peaks at the true Fermi level (0 eV), with a large value of 2200 1 cm1. This gigantic value of the SHC is orders of magnitude larger than the corresponding value in p-type semiconductors such as Si, Ge, GaAs, and AlAs. Furthermore, the calculated SHC in simple metal Al is only 17 1 cm1, being two orders of magnitude smaller than that of Pt. One important point is the degenerate band structure at L- and X-points near EF in the scalarrelativistic band structure (i.e., without the SO coupling). These double degeneracies are lifted by the SO coupling, with large SO splittings (0.66 and 4
4 3
Energy (eV)
3
SOC noSOC
2
2
1
1
0
0
–1
–1
–2
–2
–3
–3
–4
–4
–5
–5
–6
–6
fcc Pt
–7
–7
–8
–8 –9
–9 (a)
–10 –11
251
W
(b) L
Γ
X W σxyz (102 Ω–1cm–1)
Γ –20
–10 0
20
–11
Figure 22 Band structure and SHC as a function of Fermi energy for Pt. From Guo GY, Murakami S, Chen TW, and Nagaosa N (2008) Intrinsic spin Hall effect in platinum: First-principles calculations. Physical Review Letters 100: 096401.
252 Spin Hall Effect
and hence the SHC shows the maximum when EF lies within the gap. One can also argue from this effective Hamiltonian that the intrinsic SHE is robust against impurity scattering since the vertex corrections for the SHC from the short-ranged impurity scattering vanish in the clean limit due to the symmetry H(k) = H(k). Therefore, the intrinsic contribution in Pt is expected to be robust against the disorder potential, and the observed SHE is most probably of the intrinsic origin. A related work has been done by Kontani et al. (2008) using the tight-binding Hamiltonian for the t2g orbitals. Sr2MO4 (M = Ru,Rh,Mo) has been explicitly considered, and the large SHC of the order of 700 1 cm1 was obtained for Sr2RuO4 (Kontani et al., 2007, 2008). They also calculated the orbital Hall conductivity corresponding to the orbital current. Tight-binding model is convenient to draw an intuitive picture in the real space. The transfer integrals among the t2g orbitals become complex in the presence of the SO interaction as has been discussed in the context of the AHE by Onoda and Nagaosa (2002). This phase of the transfer integral can be regarded as the Peierls phase for the fictitious magnetic field, and induces the Aharonov–Bohm effect in the crystal. A related paper by Tanaka et al. (2008) considered the SHE and the orbital Hall effect in 3D 4d and 5d transition metals. The vertex correction relevant to the SS contribution has not been considered in this work. Therefore, the obtained results are for the intrinsic mechanism only. As for the giant SHE observed in Au by Seki et al. (2008), the role of local electron correlation has been considered by Guo et al. (2009). They considered the vacancy, Fe, and Pt as the possible defects in Au and
σ (103 Ω-cm–1)
Ωz(k) (atomic units)
0.93 eV, respectively). As has been discussed for p-type GaAs, the magnetic monopoles contribute resonantly to the SHC, and this structure is realized at L-and X-points. This is confirmed by the momentum-resolved contribution to the SHC as shown in Figure 23 where the contribution from the region near L- and X-points are dominant. It has been also found that the SHC decreases monotonically as the temperature T is raised, as shown in the inset of Figure 23. This rather strong temperature dependence is also due to the near degeneracies since the small energy scale is relevant to the SHC. Nevertheless, the SHC s > 240 1 cm1 at T = 300K is still large and is close to the measured value (>200 1 cm1). The calculated SHC for Al at T = 4 and 300 K is 17 and 6 1 cm1, respectively. The former value is similar to the experimental values (27, 34) at 4.2 K. To understand the microscopic mechanism of the SHE, the effective Hamiltonians H(k) for the two doubly degenerate bands at X- and L-points have been constructed. This 4 4 matrix can be written as the linear combination of matrices, which is similar to the Luttinger model. The previous analysis for the p-type semiconductors can be equally applied to the present case. An important issue is whether the contributions from conduction band and valence bands cancel or not. For example, the contribution to the SHC becomes zero or finite when the Fermi energy is within the gap between the two bands. This issue has been discussed in the context of SHI, and the inverted band structure as in the case of HgTe has the finite SHC even if the Fermi energy is within the gap, while it vanishes in the usual band structure as in the case of GaAs. The present case of Pt corresponds to the former one,
300 200
2
1
0
100
0
150 300 T (K)
0 W
L
Γ
X
W
Γ
Figure 23 Momentum-resolved contribution to the SHC. Inset: The temperature dependence of the SHC. From Guo GY, Murakami S, Chen TW, and Nagaosa N (2008) Intrinsic spin Hall effect in platinum: First-principles calculations. Physical Review Letters 100: 096401.
Spin Hall Effect
calculated the local density of states for each case. Only in the case of Fe in Au, there appears the resonant electronic states at the Fermi energy which can contribute to the enhanced SS. Fe in Au has been known as a representative Kondo magnetic impurity with a low Kondo temperature TK = 0.4 K. Therefore, naively, it is not relevant to the giant SHE observed at room temperature. However, the detailed first-principles calculation has shown that the Kondo effect in Fe is orbital-dependent in nature. Previously, the crystal field splitting between eg and t2g orbitals is around 0.1 eV, which is much smaller than the hybridization energy > 1 eV, and has been neglected. On the other hand, the LDA + U calculation shows that the electron correlation induces the large orbital polarization and the energy splitting between eg and t2g orbitals is of the order of 2 eV. This means that the spins in eg orbitals are in the Kondo limit, while the electrons in t2g orbitals are in the mixed-valence region. The former ones lead to the Kondo effect at low temperature, while the latter can contribute to the large SHE since the SO interaction is active in the t2g orbitals while it is quenched in eg orbitals. Another scenario for the Kondo effect of Fe in Au has been recently proposed by Costi et al. (2009). Actually, taking into account the SO interaction in the first-principles calculation, the energy splitting between the effective m = 1 and m = 1 states occurs (m: in the z-component of the orbital angular momentum). According to Engel et al., (2005), the spin Hall angle S = s/xx due to the SS is given by the formula Z S ¼ Z
dI ðÞSðÞsin ð61Þ dI ðÞð1 – cosÞ
where is the angle between the incident and scattered waves, is the solid angle, S() is the skewness function, and I() is the strength of the scattering. S() and I() are given by the spin-dependent phase shifts of the scattering, and the large energy splitting between m = 1 states suggests that S can be as large as 0.1. The reason why S cannot be larger than 0.1 is that the phase shift 1 for the p-wave scattering is needed as first noted by Fert and Jaoul (1972). The SHE due to the SS by Ce and Yb impurities has also been considered recently (Tanaka and Kontani, 2008), and it is concluded that the spin Hall angle S is given by 8 sin( 2/7), expecting the giant SHE.
253
1.07.4 Quantum Spin Hall Effect The QSHE and the topological insulator have been theoretically proposed and experimentally confirmed. In the present section, we review these recent developments.
1.07.4.1
Inverted Band Structure and SHI
From the systems discussed so far, we can extend our discussion to insulators. We can see that the SHE can be nonzero in some classes of insulators, which we call SHIs in Murakami et al. (2004a). The reason why the SHI arises can be understood in terms of band structure. The SHE is caused by the SO coupling. In terms of the band structure, the respective bands are classified into multiplets of the angular momentum. SHE appears if the fillings of the bands within the same multiplet are different. Instead, the bands within the same multiplet are all empty or all occupied, and their contribution to the SHE cancels each other. In the Luttinger model representing the valence bands of cubic semiconductors, the J = 3/2 multiplet contains the LH and HH bands; the SHE appears because the fillings of the HH band and the LH band are different. In the semiconductors such as -Sn, HgTe, HgSe, and -HgS, the band structure is inverted from the cubic semiconductors such as GaAs (Figure 24(a)). In the inverted band structure (Figure 24(b1)), the order of the bands is inverted from LH–HH–CB to CB–HH–LH. The mathematical structure of the Hamiltonian is essentially the same with the cubic semiconductors. The only change is that the energy difference at the point between 6 (CB) and 8 (LH and HH) bands has a different sign. Thus, the original LH band and the CB become a CB and the valence band in the inverted semiconductors, respectively. Hence, in the hole picture the LH band is fully occupied and the HH band is empty, and the resulting SHC is nonzero even for the band insulators. In these semiconductors with inverted band structure, the bandgap is zero, which comes from the degeneracy at the point due to the cubic symmetry. When one breaks the cubic symmetry by uniaxial strain, for example, there opens a finite gap, while the SHC remains nonzero (Figure 24(b2)). This nonzero SHC is a result of the nature of the gap; the gap is opened by the SO coupling.
254 Spin Hall Effect
(b1) Zero-gap (HgTe)
(a) Conventional (GaAs)
CB
(b2) Zero-gap + Uniaxial strain (c) Narrow-gap (PbTe) (HgTe)
LH
LH
EF
EF
EF HH
EF HH
HH LH k=0
CB k
CB k=0
k
k=0
k
π (111) k= a
k
Figure 24 Band structures of (a) Cubic semiconductors such as GaAs, zero-gap semiconductors (b1) without strain and (b2) with uniaxial strain along z-axis, and (c) narrow-gap semiconductors such as PbTe. The LH, HH, and CB in (a) stand for the light-hole, heavy-hole, and conduction bands. The LH, HH, and CB correspond to the LH, HH, and CB bands in (a).
Narrow-gap semiconductors such as PbTe, PbSe, and PbS are also SHI. The bandgap in these narrowgap semiconductors is also caused by the SO coupling, and the SHC is nonzero. The crystal structure is that of the rocksalt. The direct gap of the size 0.15–0.3 eV is at the four L-points (Figure 24(c)). The SHC for these compounds is around 0.04e/a, where a is the lattice spacing (Murakami et al., 2004a). In the SHI described above, as in contrast with the QSH system discussed later, there are no gapless edge channels. It is then a question of how the spin current flows in the SHI. The nature of the spin current in the SHI was numerically studied using the Keldysh formalism (Onoda and Nagaosa, 2005b). It is revealed that the spin current flows along the edge only when an electrode is attached along the edge. This may imply that by providing the edge channel from the electrode, the SHI can carry the spin current. 1.07.4.2 QSHE and Z2 Topological Invariants in 2D and 3D The QSHE is theoretically proposed as the spin analog of the QHE (Kane and Mele, 2005a,b; Bernevig and Zhang, 2006; Fu et al., 2007a; Fukui and Hatsugai, 2007a; Hatsugai et al., 2006; Sheng et al., 2005c). The QSH phase is possible both in 2D and 3D. We first consider the 2D QSH phase. In the phase, the bulk is gapped while the edge is gapless carrying a spin current. The simplest version of the 2D QSH can be constructed as a superposition of two QH subsystems with opposite spins (see Figure 25). For the 2D up-spin subsystem, we apply an external perpendicular magnetic field (jj + zˆ ), realizing the QH state with
"xy = e2/h. Meanwhile, for the 2D down-spin subsystem, the magnetic field is opposite, and #xy = e2/h. They have gapless chiral edge states having opposite sense of rotations. Superposition of these two subsystems then results in two gapless edge states which have opposite spins and flow directions, thus carrying a spin current. This phase realizes topological order, as is similar to the QH phase. This QSH phase is characterized by the Z2 topological number, as we discuss below. The whole system preserves the time-reversal symmetry. This state requires a field, which acts as +zˆ magnetic field for up-spin and zˆ magnetic field for down-spin. It is not achievable by an external magnetic field, but instead it is effectively achieved by the SO coupling, which preserves time-reversal symmetry. In some materials the SO coupling is strong enough to realize this topological phase, as we discuss subsequently. There is one important aspect of the QSH system QSHS, which is absent in the QH system; the above example with two QH subsystems conserves the spin sz, while the SO coupling does not conserve sz in general, hybridizing the two subsystems together. The question is whether there is still some interesting physics even in the absence of sz conservation. Remarkably, the answer is positive; the edge states are robust against perturbations which do not break the time-reversal symmetry, such as nonmagnetic impurities or interaction (Wu et al., 2006; Xu and Moore, 2006). This shows the peculiarity of the edge states. The key concept of the QSHS is the time-reversal symmetry, which gives rise to the Kramers degeneracy between k and k. In the band theory, the wave numbers satisfying k X k (mod G) play an important role. Such momenta are called the time-reversalinvariant momenta (TRIM), and are expressed in 2D
Spin Hall Effect
255
B
B
Quantum Hall system
Quantum Hall system
(B > 0 for up-spin)
(B < 0 for down-spin)
Spin Quantum spin Hall system Figure 25 Realization of the QSH system as a superposition of two QH subsystems.
1 as k = i where i¼ðn1 n2 Þ ¼ ðn1 b1 þ n2 b2 Þ with n1, 2 n2 = 0, 1 and b1, b2 are reciprocal lattice vectors (Kane and Mele, 2005a; Fu and Kane, 2006; Fu et al., 2007a). The degeneracy is different depending on the presence or absence of the spatial inversion symmetry I. In I-asymmetric systems the Kramers theorem implies double degeneracy only at the TRIM, and at other wave numbers k the eigenenergies are free from degeneracies. On the other hand, in I-symmetric system the eigenenergies are degenerate at all k. The QSH phase is characterized by the Z2 topological number taking only two values = 0 and = 1. In some literature they are called as = even and = odd, respectively, or X (1) = 1. When = odd (or = 1), the system is in the QSH phase, and when = even (or = 0), the system is an ordinary insulator. It is worth noting that is calculated from the Bloch wave function in the bulk as explained in the next section. The Z2 topological number in noninteracting clean systems is defined in the following way (Fu and Kane, 2006; Fu et al., 2007a). Let N denote the number of Kramers pairs below EF. First, we define a (2N) (2N) matrix w, defined as wmn ðkÞ ¼ u – k;m h j U juk;n
ð62Þ
where U is the time-reversal operator represented as U = iyK, with K being complex conjugation. uk,m is the periodic part of the mth Bloch wave function (m = 1, 2, . . ., 2N). This matrix w(k) is unitary at any k, and is also antisymmetric at k = i. Then, for each TRIM we define the index i as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi det wði Þ i X Pf wði Þ
ð63Þ
for I-asymmetric systems (Fu and Kane, 2006). Here, w depends on the U(1) phase of the Bloch wave function; the phase should be chosen to be smooth over the whole Brillouin zone. On the other hand, for I-symmetric systems, due to double degeneracy of each state, the calculation of i involves the SU(2) gauge choice of juk,mi. In fact, however, for I-symmetric systems the formula for the index i is even simpler (Fu et al., 2007a): i X
N Y
2m ði Þ
ð64Þ
m¼1
where 2m(i)(=1) is the parity eigenvalue of the Kramers pairs at each of these points for I-symmetric systems. This shows a crucial role of the I-symmetry in the theory of the QSHSs. This is because the I symmetry relates between k and k, as does the time-reversal symmetry. The Z2 topological number in 2D is defined as ð – 1Þ ¼
Y
i
ð65Þ
i
where the product over i runs over the four TRIMs in the 2D Brillouin zone. The Z2 topological number allows various equivalent expressions. For example, in the I-asymmetric systems, the Z2 topological number can define how the wave functions are glued together at the TRIMs (Fu and Kane, 2006; Moore and Balents, 2007; Roy, 2006a). We classify the wave functions below EF into two classes juk,2j 1i (j = 1, 2, . . ., N), and juk,2ji (j = 1, 2, . . ., N). When the Z2 topological number is trivial ( = 0), by a judicious unitary transformation, one can make these states to satisfy uk;2j ¼ Uuk;2j – 1 ; uk;2j – 1 ¼ – Uuk;2j
ð66Þ
256 Spin Hall Effect
We note that U2 = 1. When = 1, in contrast, it is not possible for (66) to hold, but there should necessarily appear additional phase factors. Notice that the wave functions are degenerate only at the TRIMs. The nontrivial Z2 topological number works as an obstruction for the wave functions to be glued at the TRIMs in the simple way given in (66) (Fu and Kane, 2006; Moore and Balents, 2007; Roy, 2006b). because the topological order survives perturbations, such as nonmagnetic disorder and interactions, the definition of the Z2 topological number can be further extended (Qi and Zhang, 2008) to systems with nonmagnetic disorder (Kane and Mele, 2005b) and those with interactions (Lee and Ryu, 2008). To illustrate the relationship between Z2 topological number and the edge states, we calculate the bandstructure for geometries with edges in the Kane–Mele model (Kane and Mele, 2005a,b), which is the 2D tight-binding model showing the QSH and insulator phases by changing some parameters (for details, see Section 1.07.4.4). Let us consider a ribbon geometry, which is finite in one direction and is infinite in the other. The phase diagram is in the inset of Figure 26, where R, SO, and v are model parameters. The QSH and I in the phase diagram represent the QSH and insulator phases, respectively. In the QSH phase (Figure 26(a)), there exists gapless edge states, irrespective of the geometry. In contrast, for the insulator phase (Figure 26(b)) there are no gapless edge states. In fact, there are edge states, but they do not go across the gap. In some cases, in general, these edge states in the insulator state may cross the Fermi energy; nevertheless, even if they cross the Fermi energy, it is not an intrinsic property, and the crossing can disappear by perturbations preserving time-reversal symmetry. On the other hand, the bulk bandstructure
is gapped for both cases and looks similar. The topological order, which is not evident in the bulk bandstructure, appears as the existence of the robust edge states. The correspondence between the bulk and the edge is as follows:
• •
bulk: Z2 topological number is = 1 ( = 0) and edge: there are odd (even) numbers of Kramers pairs of gapless edge states.
Because an odd number cannot be zero, the system with = 1 should have at least one Kramers pair of gapless edge states. This bulk–edge correspondence can be shown by the argument similar to the wellknown Laughlin’s gedanken experiment (Fu and Kane, 2006). Suppose that we consider the system on a ribbon, with two opposite ends attached (Figure 27). In one direction the system is periodic, while in the other there are edges. Then, when we increase the flux penetrating the hole from zero to half the flux quantum, the change of the system characterized by the time-reversal polarization (Fu and Kane, 2006) depends on the Z2 topological number, and it is also related with the number of Kramers pair of edge states. The analogous phase is also possible in 3D (Moore and Balents, 2007; Fu et al., 2007a; Roy, 2006b). In this phase in 3D, the system is an insulator in the bulk and supports gapless surface states carrying spin currents. In 3D, the Z2 topological numbers are defined in the following way. The TRIMs are i¼ðn1 n2 n3 Þ ¼ 1=2ðn1 b1 þ n2 b2 þ n3 b3 Þ with n1, n2, n3 = 0, 1 in 3D. There are four Z2 topological numbers j (j = 0,1,2,3) (Moore and Balents, 2007; Fu et al., 2007a; Teo et al., 2008), given by ð – 1Þ0 ¼
8 Y
Y
i ; ð – 1Þk ¼
i¼ðn1 n2 n3 Þ ð67Þ
nk ¼1;nj 6¼k ¼0;1
i¼1
E/t
1 5 λ R / λ so
0
I
QSH
0
λ V / λ so
–5 –5
(a) –1
0
π
ka
2π
0
5
0
(b) π
ka
2π
Figure 26 Band structures of the Kane–Mele model on a ribbon geometry, for parameters in (a) the QSH phase and (b) the insulator phase. Inset: phase diagram of the Kane–Mele model. From Kane CL and Mele EJ (2005) Z2 topological order and the quantum spin Hall effect. Physical Review Letters 95: 146802.
Spin Hall Effect
y
Φ∼ky x
Figure 27 Laughlin’s gedanken experiment. The flux plays the role of the wave number ky.
Each phase is expressed as 0; ( 1 2 3), which distinguishes 16 phases. Among i, 0 is the only topological number robust against nonmagnetic disorder, and hence the phases are mainly classified by 0. When 0 is odd, the phase is called the strong topological insulator (STI), while it is called the weak topological insulator (WTI) for the even case. The other indices 1, 2, and 3 are used to distinguish various phases in the STI or WTI phases, and each phase can be associated with a mod 2 reciprocal lattice vector G 1 23 = 1b1 + 2b2 + 3b3, as was proposed in Fu et al. (2007a). These three topological numbers ( k(k = 1,2,3)) are meaningful only for a relatively clean sample (Fu et al., 2007a). (However, a recent paper proposes that the 1D helical edge channels along dislocation are protected by k (k = 1, 2, 3), and hence the WTI is also topologically nontrivial (Ran et al., 2008b).) These topological numbers in 3D determine the topology of the Fermi curve of the surface states for arbitrary crystal directions (Fu et al., 2007a; Teo et al., 2008). It is an example of the bulk– surface correspondence. Further discussion on the correspondence between bulk topological numbers in 3D and surface states can be found in Teo et al. (2008). The nature of the Z2 topological numbers can be revealed through the study of phase transitions between the QSH and the insulator phases, involving a change of the Z2 topological number (Murakami et al., 2007; Murakami, 2007; Murakami and Kuga, 2008). The Z2 topological number is preserved as long as the gap remains open. Suppose that the system transforms from the insulator to the QSH phase by changing some parameter of the system. Then, the gap should close somewhere in between. Whether or not the gap closes as the parameter is varied reflects the topological properties of the system. The criterion for the occurrence of the phase transition has been studied by Murakami et al. (2007), Murakami (2007), and Murakami and Kuga (2008). The idea is that the specification of each phase requires information over the entire first Brillouin zone, whereas the change of the Z2 number across the transition can be discussed in terms of the local information near the gap closing point in the momentum space.
257
We consider a time-reversal symmetric system with a bulk gap, and assume that it depends on an external parameter m. We assume here that the Hamiltonian itself is generic, and all the terms allowed by symmetry are nonzero in general. When the parameter m is changed, the gap can close in some cases, whereas in other cases the gap closing is impossible because of level repulsion. The condition for the gap closing by varying a single parameter is obtained as follows. In I-symmetric systems, all the bands are doubly degenerate and the level repulsion is strong. In this case, the gap can close only at one of the TRIM k = i , and only when the valence and conduction bands have opposite parities. In I-asymmetric systems, the gap can close at a generic k in 2D at a single value of m. In 3D I-asymmetric systems, there necessarily appears a gapless phase between the QSH and the insulator phases, when m is changed. This gapless phase occurs because of the topological nature of the gapless points in 3D (but not in 2D). To summarize, the universal phase diagrams involving the QSH and I phases in 2D and in 3D are given in Figure 28 (Murakami, 2007; Murakami et al., 2007; Murakami and Kuga, 2008). 1.07.4.3 Helical Edge Modes in QSH systems As we assume the time-reversal symmetry in the QSH phase, the edge states consist of pairs, in each of which the states are related by the time-reversal symmetry. Such pairs are sometimes called Kramers pairs. The gapless helical edge states consist of Kramers pairs of states with opposite spins, propagating in the opposite directions (see Figure 29). As stated previously, the edge states remain gapless even with nonmagnetic impurities or interaction (Wu et al., 2006; Xu and Moore, 2006). However, the two-particle scattering terms such as H9 ¼ g
Z dx
y y R" R" L# L#
ð68Þ
are allowed. These terms give rise to the Tomonaga– Luttinger liquid effect and also the Umklapp term among them leads to the spontaneous symmetry breaking, most probably the magnetic ones. The readers are referred to the original papers (Xu and Moore, 2006; Wu et al., 2006). We note that another proposal to obtain the similar helical edge modes in graphene has been made (Abanin et al., 2006). On the other hand, if we break the time-reversal symmetry by a magnetic field, the helical edge states
258 Spin Hall Effect
(a)
(b) Gapless
Inversion symmetric
Inversion symmetric
QSH
I
QSH
I
δ
δ Gapless
m
m
Figure 28 Universal phase diagram between the QSH and the insulator (I) phases in (a) 3D and (b) 2D (Murakami, 2007; Murakami et al., 2007; Murakami and Kuga, 2008).
V
I
0
Figure 29 Schematic of the helical edge states. From Kane CL and Mele EJ (2005a) Quantum spin Hall effect in graphene. Physical Review Letters 95: 226801.
open a gap. It is demonstrated experimentally in CdTe/HgTe/CdTe QW that the two-terminal conductance decreases rapidly when the magnetic field is increased (Ko¨nig et al., 2007, 2008). A fractional charge appears when the time-reversal symmetry is broken by magnetic domains put on the edge, as discussed in Section 1.07.4.6.1. It is proposed theoretically that the proximity effect to the ferromagnets with the helical edge channels introduces the magnetic domain wall and a fractional charge e/2, which can be detected as the shift of the Coulomb blockade periodicity (Qi et al., 2008b). 1.07.4.4 Models and Candidate Materials for QSHE The QSHE was first discovered in a model for graphene by Kane and Mele (2005a,b). This model is given as HKM ¼ t
X
ciy cj þ i SO
ði;j Þ
þ i R
X hi;j i
X
ij ciy sz cj
hhi;j ii
ciy ðs b kij Þz cj þ v
X
i ciy ci
ð69Þ
i
where hi,ji runs through the nearest-neighbor pairs (i,j), and hhi,jii runs through the next nearest-neighbor pairs (i,j). b kij is the unit vector along the nearest-
neighbor bond from i to j. ij takes the value 1 depending on whether the next-nearest-neighbor hopping from j to i is clockwise or counterclockwise. i takes the value 1 for the A and B sublattices, respectively. The phase diagram is as shown in the inset of Figure 26. We note that there are other tight-binding models showing the QSH phase (Onoda and Nagaosa, 2005b; Qi et al., 2006a,b). This model shows the QSH and the insulator phases, by changing the model parameters. The phase diagram is shown in the inset of Figure 28. A theoretical proposal of the 2D QSH on the cubic semiconductor with strain gradient has been made (Bernevig and Zhang, 2006). In this setup, the strain gradient plays a role of a spin-dependent effective magnetic field through the SO coupling, and gives rise to the spin-dependent QH effect, that is, the QSH effect. In this proposal, the Landau levels are formed, and in the presence of interaction, there might be a chance to have exotic states similar to the fractional QH states. The 3D QSH phase is realized in the model proposed by Fu et al. (2007a). It is the tight-binding model on a diamond lattice: H ¼t
X hij i
ciy cj þ ið8 SO =a2 Þ
X ðhij iÞ
ciy s?ðd1ij d2ij Þcj
ð70Þ
Spin Hall Effect
where t represents the nearest-neighbor spin-independent hopping, and SO represents the SO coupling, exemplified as a next-nearest-neighbor hopping, and a is the size of the cubic unit cell. The vectors d1ij and d2ij denote those for the two nearest-neighbor bonds involved in the next-nearest-neighbor hopping. This four-band model is I-symmetric, and we assume the Fermi energy to be between the upper and lower doubly degenerate bands. The gap between the doubly degenerate CB and valence bands vanishes at three X-points. In order to realize the QSH phase, we need to open the gap. It is achieved by making the hopping amplitudes for the four different bond directions to be different. For example, by making one bond stronger than the other three, it becomes the STI, while by making it weaker, it becomes the WTI. This phase change can be attributed to different signs of masses at the direct gaps at the X-points. Because the QSH phase is caused by the SO coupling, one may wonder if there are real materials which have a sufficiently strong SO coupling to realize this phase. In order to be in the QSH phase, the system should satisfy the following two necessary conditions: (1) nonmagnetic insulator and (2) odd Z2 topological number. These conditions mean that the gap is opened by the SO coupling. One criterion to see whether the gap is opened by the SO coupling is to focus on the magnetic susceptibility (Murakami, 2006). In insulators and semimetals, when the SO coupling has a large matrix element between the valence and the conduction bands, the magnetic susceptibility is enhanced. This has been established through the study of bismuth and bismuth–antimony alloy. Therefore, bismuth and bismuth–antimony alloy might be good candidates for the QSH phase. For 2D QSH phase, we consider a thin-film bismuth. By making a ultrathin film, like (111) 1-bilayer film, it becomes an insulator (Koroteev et al., 2008). This lattice structure is inversion symmetric, and we can easily calculate the Z2 topological number from the parity eigenvalues. Indeed, it is theoretically proposed to be the 2D QSH phase (Murakami, 2006). Another system is CdTe/HgTe/CdTe QW (Bernevig et al., 2006; Ko¨nig et al., 2007, 2008 (Figure 30)). This idea is based on the fact that HgTe has an inverted bandstructure from the usual cubic semiconductors such as CdTe. Thus, by increasing the QW width, the electronic states change from the conventional ones to the inverted ones. This change of character in the electronic states correspond to the transition from the insulator to the QSH phase. Details
259
(a) E (eV) 0.04 0.02
E1
dC 50
–0.02
60
70
80 d (Å)
H1
–0.04 (b)
Figure 30 Change of the bandstructure of CdTe/HgTe/ f0150 CdTe quantum-well as a function of well width d. From Bernevig BA, Hughes TL, and Zhang S-C (2006) Quantum spin Hall effect and topological phase transition in HgTe quantum wells. Science 314: 1757.
are described in the next subsection, in conjunction with the experimental results (Ko¨nig et al., 2007). Next, we consider the 3D QSH phase. Because the bulk bismuth is a semimetal, one possibility to make it nonmagnetic insulator is to dope antimony. Indeed, the alloy Bi1 xSbx with 0.07 < x < 0.22 becomes an insulator. The Z2 topological numbers are calculated from the parity eigenvalues, and it is proposed to be the 3D QSH phase (Fu and Kane, 2007; Fu et al., 2007a). Related subjects are discussed in Fukui and Hatsugai (2007), Fukuyama (2006), Kane (2007), Kane and Mele (2006), Liu et al. (2008), Min et al. (2006), Nagaosa (2007), and Yang et al. (2008).
1.07.4.5 QSHE
Experimental Observation of
So far, there are several experimental reports for the QSH phase in 2D (Ko¨nig et al., 2007, 2008) and in 3D (Hsieh et al., 2008; Xia et al., 2008). One is on the (Cd,Hg)Te/HgTe/(Cd,Hg)Te QW, realizing the 2D QSH phase. By changing the width d of the HgTe QW, the phase changes from the ordinary insulator (d < dc 64 nm) to the QSH phase (d > dc). For the narrow QW (da < dc) the bandstructure is like that of CdTe, where the CB is 6 and is s-like. For the wider QW (d > dc), on the other hand, the bandstructure is more like that of HgTe, where the CB is 8 and is p-like. When the thickness is larger than the critical value, dc, the p-like and s-like bands
260 Spin Hall Effect
touch at k = 0. Thereby the parity of the valence band and the Z2 topological number changes have been predicted theoretically in advance (Bernevig et al., 2006). In the experiments (Ko¨nig et al., 2007, 2008) the charge conductance, G, is measured in a two-terminal setup, and found to be G 2e2/h when d > dc and G 0 when d < dc. An effect of the external magnetic field has also been studied. As the QSHE is fragile to perturbations which breaks time-reversal symmetry, the charge conductance G shows a rapid decrease when the magnetic field is applied. This means that in the presence of magnetic field, there appears a gap in the helical edge channel to suppress the conductance (Figures 31 and 32). The 3D QSH phase has also been observed experimentally in the alloy Bi0.9Sb0.1 (Hsieh et al., 2008). This
alloy is proposed to be the 3D QSH phase, that is, the STI (Fu and Kane, 2007; Fu et al., 2007a). The angleresolved photoemission spectroscopy (ARPES) is used for this alloy, and the Fermi surface of the surface state – and crosses the M line five times (see Figure 33). From the bandstructure calculation, the indices i at – and M are opposite in sign as predicted theoretically (Fu and Kane, 2007). This means that there should be – and odd number of Fermi points between M . Thus, – and indicate that it is the five crossings between M indeed the 3D QSH phase (Hsieh et al., 2008). Further relevant references in this regard are Dai et al. (2008), Day (2008b), Hirahara et al. (2006), Ko¨nig et al. (2008), and Koroteev et al. (2008). Another experiment on the 3D QSH is carried out on Bi2Se3 (Xia et al., 2008). The ARPES measurement
16 14
R = h/(2e2)
12 Rxx / kΩ
10 8 6 4 1 μm × 1 μm, 1.8 K 1 μm × 0.5 μm, 1.8 K 1 μm × 1 μm, 4.2 K
2 0 –1
0
1 (Vg – Vth) / V
2
Figure 31 Conductance as a function of gate voltage. From Ko¨nig M, Buhmann H, Molenkamp LW, et al. (2008) The quantum spin Hall effect: Theory and experiment. Journal of the Physical Society of Japan 77: 031007.
0.14
0°
0° (B|| x) 15° 30° 45° 60° 75° 90° (B|| z)
0.12
G (e 2/h)
0.10 0.08 0.06 0.04 x
0.02 0.00 –0.10
I y
90° –0.05
0.00 B (T)
z
0.05
0.10
Figure 32 Conductance as a function of external magnetic field. From Ko¨nig M, Buhmann H, Molenkamp LW, et al. (2008) The quantum spin Hall effect: Theory and experiment. Journal of the Physical Society of Japan 77: 031007.
Spin Hall Effect
0.1
Topological Hall insulator
1 EB (eV)
261
2
3
4, 5
0.0
–0.1
0.0
0.4
0.2
Γ
0.6
0.8
–kx (Å–1)
1.0
M
Figure 33 ARPES data for Bi0.9Sb0.1. From Hsieh D, Qian D, Wray L, et al. (2008) A topological Dirac insulator in a quantum spin Hall phase. Nature 452: 970.
showed that in Bi2Se3 the surface states traverse the bulk gap. The dispersion of the surface states forms a single Dirac fermion at the point, meaning that Bi2Se3 is the STI if it is insulating in the bulk. Nevertheless, in reality the material is n-doped in nature, and it is not even insulating in the bulk. If the carriers are compensated and the material becomes insulating in the bulk, it will become the STI. Firstprinciples calculations have been done on this and related materials (Zhang et al., 2009), and it was predicted that Bi2Se3 is really the STI. In addition, Sb2Te3 and Bi2Te3 are also predicted to be STI, while they have a smaller gap than Bi2Se3. 1.07.4.6
Further Developments in the QSHE
QSHS and topological insulator are the subject of intensive studies, and constitute a growing field. We mention here some of the interesting recent developments and future directions. 1.07.4.6.1 Spin-charge separation and charge fractionalization
The simplest example of the spin-charge separation is the soliton excitations in the conjugate polymer model such as polyacetylene. Suppose we have the tight-binding model (Niemi and Semenoff, 1986) H¼ –
X ðt þ ð – 1Þj dj Þðcjy cj þ1 þ h:c:Þ
ð71Þ
j
where there is one orbital at each atom j, coupled to the atom at j + 1 by the transfer integral t + (1)jdj, with dj being the dimerization. We assume that the average electron number is 1/2 per atom, that is,
half-filled. In the ground state, dj = d is uniform, which opens the gap. In the real space, this corresponds to the periodic array of dimers. For each dimer, there are bonding and antibonding orbitals and an electron occupies the bonding orbital. Therefore, the electron number per atom is 1/2. There is an energy gap corresponding to the energy splitting between the bonding and antibonding states. Now, we consider an isolated atom separated from the dimers. This can be regarded as the kink or domain wall between the two degenerate dimerization patterns, and called a soliton. Because the isolated atom has one orbital decoupled from the other atoms, it has the energy between those of the bonding and antibonding orbitals in the dimers. Let us define here the charge of the soliton. An important remark here is that the charge is measured from that of the ground state. For example, we count the charge of the soliton from that of the ground-state configuration. As noted above, there is half electron per atom in the ground state, that is, e/2 charge. Therefore, the charge of the soliton q is defined as q = (e) (e/2) = e/2 if an electron occupies the soliton site, while q = 0 (e/2) = e/2 if the soliton site is empty. This means that the soliton charge is a fraction (half) of the unit charge e. This corresponds to the case of jdjj = t, but the physics remains qualitatively the same also in the limit of jdjj << t. In this case, the field theoretical methods can be applied (Niemi and Semenoff, 1986). In the continuum approximation, the Hamiltonian reads H¼
Z dx
y
ðxÞ½z qx þ x mðxÞ ðxÞ
ð72Þ
262 Spin Hall Effect
where (x) = t( 1(x), 2(x)) is the two-component field operator for the right-going and left-going electrons, and m(x) is the mass term. With the sign change of the mass m(x) between the two limits x ! 1, a mid-gap state is topologically guaranteed (Niemi and Semenoff, 1986). This is the simplest example of the fractionalization but can be generalized to many situations in a straightforward way. Now, we consider the spinfull electrons. Each atom can accommodate two electrons, that is, upspin and down-spin electrons. Then, there is one electron per atom in the ground state. There are four possibilities of the soliton in this case: (1) vacancy, (2) double occupancy, (3) up-spin electron, and (4) down-spin electron. Let us count the charge q of the soliton for each of (1)–(4) states. It reads (1): q = 0 (e) = e, (2): q = (2e) (e) = e, (3): q = (e) (e) = 0, and (4): q = (e) (e) = 0. Therefore, the soliton with (1) or (2) configuration is positively and negatively charged, respectively, and is called charge soliton. On the other hand, the soliton with (3) or (4) configuration is neutral. When we consider the spin quantum number S, it is 1/2 for the configuration (3) and (4), while (1) and (2) and also the ground state are the spin singlet. Therefore, the (3) and (4) forms the spin-doublet pair and is called the spin soliton. Summarizing the above consideration, the charge and spin are carried by the different kinds of soliton, and this phenomenon can be called spin-charge separation. We have shown that even though the constituent particle have both unit charge and spin, the excitation in the many-body state could have fractional quantum numbers and also the spin-charge separation. This idea has been applied to the QSHS. As already noted in Section 1.07.4.3, the helical edge modes in QSHS separate the right-going and leftgoing branches of the dispersion at the two edges of the sample for up- and down-spin electrons, respectively. Therefore, the degrees of freedom are reduced to half at each edge, leading to the fractionalized charge. The effective Hamiltonian for the helical edge modes at one edge can be written as (72) with 1(x) ( 2(x)) corresponding to the right-going upspin electron (left-going down-spin electron), while the mass m(x) corresponds to the magnetic order parameter. Therefore, at the magnetic domain wall where m(x) changes sign, one expects the fractional charge e/2 as discussed in Section 1.07.4.3 (Qi et al., 2008b). In the bulk of the QSHS, there are both up- and down-spin states, and the situation is similar to the
spinfull electron model discussed above. The only difference is that the system is 2D and the soliton is replaced by the vortex with the flux . The phase winding around this vortex guarantees the existence of the mid-gap state in the QSHS, and the occupancy of that mid-gap state determines the charge and spin quantum numbers quite analogous to that in the spin and charge solitons (Ran et al., 2008a,b; Qi et al., 2008b; Moore, 2008). Therefore, QSHS offers the first explicit model showing the spin-charge separation in 2D, which has been examined for many years in the context of the theory of high temperature superconductors (Lee et al., 2006).
1.07.4.6.2 Effective theory and quantized magneto-electric effect
In the case of QH systems, the effective theory, that is, the Chern–Simons theory, describes the lowenergy phenomena. Therefore, it is desirable to develop an effective field theory for the QSHS also. Recently, an interesting work has been done in this direction (Qi and Zhang, 2008). The idea is to define the fundamental time-reversal invariant (TRI) insulator in (4 + 1)-dimensions, described by the (4 + 1)dimensional Chern–Simons theory and characterized by the second Chern number. Due to the time-reversal symmetry, the coefficient of the Chern–Simons term is restricted to be 0 or , which corresponds to the Z2 topological number discussed in Section 1.07.4.2. By dimensional reduction, this state is related to the QSH system in (2 + 1)-dimensions and the topological insulator in (3 + 1)-dimensions. As an explicit application of this effective theory, the quantized magneto-electric effect was predicted, where an electric field produces the magnetization in the same direction, with a universal constant of proportionality quantized in odd multiples of the fine-structure constant ¼ e 2 =hc.
1.07.4.6.3
Superconducting QSHSs The topological concept related to the time-reversal symmetry has been generalized to the superconductors. In particular, the superconducting proximity effect between the surface of the (3 + 1)-dimensional STI and the usual s-wave superconductor has been discussed (Fu and Kane, 2008). The Hamiltonian reads H¼
Z dx½
y
ð – iv?r – Þ þ
y y " #
þ h:c:
ð73Þ
Spin Hall Effect
where = t( " #) is the two-component spinor corresponding to the surface Dirac fermion dictated by the topology of the bulk state, while is the superconducting order parameter induced by the proximity effect. This Hamiltonian can be diagonalized to result in a superconducting state resembling the chiral p + ip spinless pairing state. Therefore, a novel vortex satisfying the non-Abelian statistics (Read and Green, 2000; Ivanov, 2001), and the chiral edge modes (Buchholtz and Zwicknagl, 1981; Matsumoto and Sigrist, 1999; Read and Green, 2000) carrying Majorana fermions are expected. Fu and Kane (2008) considered the geometry of the two superconductors with the order parameter 0 and 0ei attached to the topological insulator with the separation W. Then, the energy dispersion of the Andreev bound state is given by E(q) = [v2q2 + 0 2 cos2(/2)]1/2, with q being the wave number along the edge. The fermion operator for this bound state constitutes the Majorana wire, and the circuit of this wire can be used to create, manipulate, and fuse Majorana bound state (Fu and Kane, 2008). On the other hand, one can consider the (p ip)pairing state as the superconducting analog of the QSH state in parallel to the spinless chiral (p + ip)pairing state which can be regarded as the superconducting QH system (Qi et al., 2008a,c; Goryo et al., 2008; Grover and Senthil, 2008; Roy, 2006c; Sengupta et al., 2006). In the (p ip)-pairing state, the helical edge modes appear as the Andreev bound states. However, because the negative and positive energy states are redundant in the Bogoliubov–de Gennes Hamiltonian, only the half of the fermion degrees of freedom should be taken. This means that the Bogoliubov quasi-particles at the helical edge modes are Majorana fermions. In the case of a vortex, a pair of the Majorana fermion states at the core of the vortex appears. On this low-energy state, the time-reversal symmetry operation acts as the supersymmmetry operation changing the even/odd of the fermion number. This leads to a nontrivial nonlocal correlation of some order parameter. They also propose the BW-phase of 3He as the (3 + 1)-dimensional superfluid analog of the topological insulator (Qi et al., 2008c). Recently Tanaka and Kontani (2008) have proposed that the noncentrosymmetric superconductors can be a candidate for the superconducting QSHS. In the presence of the SO interaction, the singlet and triplet pairing states are mixed, and one cannot define the spin Chern number. However, the Z2
263
number can still be defined, and the helical edge modes as the Andreev bound states appear when the order parameter p for the p-wave pairing is larger than that for the s-wave pairing s. An interesting spin-transport property is found in the Andreev reflection at the junction of the normal metal and the noncentrosymmetric superconductor. Another recent development of this field is the mathematical classification of the possible topological classes including both the insulators and superconductors. It is concluded that there are five classes according to the time-reversal symmetry and chiral symmetry. The readers are referred to the recent papers (Roy, 2006a,b; Schnyder et al., 2008). 1.07.4.6.4 Localization problem related to the QSHS
The effect of disorder on the electronic states is an important issue in condensed matter physics. In particular, it is known that the dimensionality, symmetry, and topological properties of the system determine the localization/delocalization behavior of the wave functions (Lee and Ramakrishnan, 1985). There are basically three universality classes, that is, (1) orthogonal, (2) symplectic, and (3) unitary classes, which corresponds to the (1) spinless timereversal system, (2) time-reversal system with SO interaction, and (3) system with broken time-reversal symmetry. It is remarkable that all the states are localized in 2D however weak the disorder strength is in (1) and (3), while the metal–insulator transition occurs in (2). The integer QH system, which corresponds to the case (3) but also is characterized by the topological number called Chern number related to the quantized Hall conductance, has distinct localization properties. For example, there remain extended states at the center of each Landau level (Girvin and Prange, 1987). Therefore, it is an intriguing problem if the similar modification of the localization properties due to the Z2 topological numner occurs also in the QSHS, which belongs to the symplectic class (2). Concerning this problem, Onoda et al. (2007) numerically studied the localization length and its scaling behavior of the Kane–Mele model (Kane and Mele, 2005a,b). The exponent describes the divergence of the localization length as a function of energy difference from the mobility edge, _ jE Ecj . By numerical calculation, it is found to be 1.6, which is different from the symplectic value 2.7 (Onoda et al., 2007). On the other hand, the QSH phase is studied by extending the network model (Obuse et al., 2007).
264 Spin Hall Effect
The result is consistent with the symplectic value (Obuse et al., 2008a,b). For the 3D QSHS, the localization of the (2 + 1)dimensional Dirac fermion at the surface is an interesting issue. A model of a single 2D Dirac fermion is studied by constructing an effective theory including a nontrivial topological term, which is responsible for the protection of the extended states (Ostrovsky et al., 2007; Ryu et al., 2007). Therefore, the localization is prohibited by the topology in the case of the surface states of the STI. From the viewpoint of the nonlinear sigma model approach, the topological terms are classified by the homotopy consideration of the target space, which is determined by the symmetry of the problem, and are restricted to the two cases: (1) there is no topological term and (2) the topological sectors are classified by Z2 number. The former corresponds to the usual symplectic class and also the 2D topological insulator, while the latter to the surface Dirac fermion of the 3D STI. The readers are referred to some further references on this issue (Sheng et al., 2006; Essin and Moore, 2007; Obuse et al., 2008a,b; Shindou and Murakami, 2009). 1.07.4.6.5 systems
SHE in strongly correlated
One of the most important issues to be studied in the future is the role of electron correlation in SHE (Raghu et al., 2008). Concerning this issue, an interesting material has been reported, that is, Sr2IrO4 (Kim et al., 2008). As the atomic number increases from 3d, 4d to 5d, SO interaction increases while the electron correlation decreases since the extent of the d-orbitals gets larger and larger. In the case of 5d orbitals, the SO interaction and the correlation energy U are of the order of 0.5 eV, both of which play important roles, while the crystal field splitting between eg and t2g orbitals is much larger. In Sr2IrO4, Ir4+ ion has d5 configuration, and one hole occupies the t2g orbitals. Because the orbital angular moment is active in t2g orbitals, the strong SO interaction mixes the spin and orbital states, leading to the effective Jeff = 1/2 states and Jeff = 3/2 states. The single hole occupies one of the doubly degenerate Jeff = 1/2 states, and the Coulomb interaction U leads to the Mott insulating state (Kim et al., 2008). This experiment opens a promising route of the SHE in oxides because the effective Hubbard model for Jeff = 1/2 states contains the spin-dependent complex transfer integrals due to the SO interaction. Actually, a tightbinding analysis on a related material, Na2IrO3, with
the honeycomb lattice structure concludes that the system is a QSHS at room temperature (Shitade et al., 2009). Due to the electron correlation, the novel antiferromagnetic state is expected at lower temperature. Further studies on the interplay between the electron correlation and SHE are extremely important especially in correlated oxides, which have many advantageous features such as the systematic control of the band width, carrier doping, and the high energy scales.
1.07.5 Summary and Perspectives on SHE 1.07.5.1 SHE
Comparison between AHE and
SHE is closely related to the AHE in metallic ferromagnets as discussed in Section 1.07.1. Both are driven by the relativistic SO interaction, and intrinsic and extrinsic mechanisms are considered. More explicitly, one may regard the SHE as the two copies of AHE for up- and down-spins. This analogy is correct in the zeroth order approximation, but there are several differences as described below. One important difference is that the charge is a conserved quantity and the charge current in AHE is well defined as the Noether current associated with the U(1) gauge symmetry, while the spin is not conserved in the presence of the SO interaction and correspondingly the continuity equation for the spin and spin current cannot be derived in generic situations. Therefore, the definition of the spin current is somewhat arbitrary, and the direct observation of the spin current is much more difficult than that of charge current, as discussed earlier. This is why the experimental observation of the SHE has been rather indirect, by measuring the consequent spin accumulation near the edge of the sample, for example. In that case, the analysis of the spin diffusion equation including the decay of the spin density due to the SO interaction, which is phenomenologically introduced, should be performed. More direct detection of the spin current is in terms of the voltage drop perpendicular to it. This effect is the so-called Aharonov–Casher effect in the vacuum, but is very small because the magnitude of the effect in the vacuum contains the rest mass of the electron mc2 in the denominator. In solids, the effective SO interaction can be tremendously enhanced by the factor of mc2/E with E being the bandgap. This factor can be as large as 106, and hence the
Spin Hall Effect
detection of the spin current in solids is much easier. This idea has been already pursued in the ISHE as discussed in Section 1.07.3 for metallic SHE. In the case of AHE, the theoretical studies on the cross-over between the extrinsic and intrinsic dominated regions have been developed, taking into account the band crossing near the Fermi energy which resonantly enhances the AHE (Miyasato et al., 2007; Onodaet al., 2006b). The representative model system is the Rashba Hamiltonian with the spin polarization. For example, the cross-over from the very clean metal, where the SS is dominant, to the usual metallic region, where the intrinsic contribution determines H, occurs at h= > where is the transport lifetime, and is the SO energy. At this cross-over, the longitudinal conductivity xx is much larger than e2/h, that is, xx > (e2/h)("F/). If one increases the disorder strength furthermore, the second cross-over occurs at h=ð"F Þ 0:1 to the region where a new scaling law H _ (xx) with the exponent > 1.6 holds. This theoretical prediction (Onoda et al., 2006b), when translated to the 3D systems by replacing e2/h by e2/(ha) > 103 (Ohm cm)1 (where a is the lattice constant assumed to be around 4 A˚), is in agreement fairly well with the recent experimental studies over the many decades of the disorder strength (Miyasato et al., 2007). Therefore, h=ð Þ and hence the absolute value of the longitudinal conductivity is a key parameter to control the behavior of the AHE, which resolves the long-standing controversy. From the viewpoint of the above results, the SHE in semiconductors is all in the strongly disordered region where the 1.6-power law is expected, but detailed study of the xx-dependence of s has never been done. In the case of metals, on the other hand, the vital role of the band crossings is common which leads to the enhanced intrinsic SHE in the usual metallic systems as discussed in Section 1.07.3.2. If the disorder is further reduced, the extrinsic SS contribution is dominant, and the SHC s is proportional to the diagonal charge conductivity xx, and the spin Hall angle S = s/xx characterizes the spin Hall response. The anomalous Hall angle is also defined in a similar way, that is, = s/xx. The typical value of is of the order of 103, corresponding to the ratio of the SOI and the Fermi energy. When the resonant scattering by the virtual bound state of d-orbitals is active, is of the order of ( /) 1 where is the SOI energy, is the hybridization energy between the d-orbitals and the s-bands, that is, the width of the virtual bound state,
265
and 1 is the phase shift for the p-wave scattering (Fert and Jaoul, 1972). This can be of the order of 102 since / > 0.1 and 1 > 0.1 are possible but not larger. Therefore, the giant SHE observed in Au (Seki et al., 2008) suggests an essential difference between the AHE and SHE. For example, the SHE is not a simple two copies of AHE for up- and downspins. This is natural since the x and y components of the spin operator play some role in the quantum fluctuation, which leads to the singlet formation in the Kondo effect as discussed in Section 1.07.3.2 (Guo et al., 2009; Tanaka et al., 2008). This can be the mechanism of the enhanced SHE compared with AHE. In any case, the role of the electron correlation and the quantum fluctuation of the spins will be an important issue in the future. 1.07.5.2 Comparison between Integer QHE and QSHE The QHE and the QSHE have many aspects in common. They come from topological orders in gapped systems. The QH system is represented by the topological number (the Chern number) taking any integers: = . . ., 2, 1, 0, 1, 2,. . . . This is because the underlying symmetry is the continuous U(1) symmetry associated with the charge conservation. This Chern number is equal to the number of chiral edge states, and also appears as the quantized Hall conductance xy = e2/h. In the 2DEG in a strong magnetic field, this number is equal to the number of Landau levels below the Fermi energy. On the other hand, QSHE is related to the discrete symmetry, that is, the time-reversal symmetry, which is not related to the conservation of a physical observable. Correspondingly, QSHS is characterized by the Z2 topological number taking only two values, = even and = odd, that is, (1) = 1. As stated earlier, the simplest example of the QSHS is realized as a superposition of two QH systems of opposite effective magnetic fields for opposite spins. Only for such kinds of the QSHSs, the corresponding topological number is the spin Chern number: s = " # = . . ., 4, 2, 0, 2, 4, . . . . In this criterion, Z2 even (odd) corresponds to s = 0(2)(mod 4). Nevertheless, generic QSHSs allow terms which break spin conservation preserving the time-reversal symmetry. With such spin-nonconserving terms, the set of topological numbers is reduced to only two classes, odd and even. This is the Z2 topological number, which characterizes the QSHS.
266 Spin Hall Effect
The edge states are also similar, whereas the spin-nonconserving terms make the QSH different from a mere superposition of two QH systems. The QH system has chiral edge states whose number is equal to the Chern number. They are immune to backscattering from impurities, since the modes with different chiralities are separated by the sample width. The QSHS has helical edge states, which consist of pairs of Kramers-degenerate edge states. When the spin, sz, is conserved, the number of edge states is equal to the spin Chern number of the original QH system. When the spin, sz, is no longer conserved (but time-reversal symmetry is preserved), some edge states may open a gap; nevertheless, if the Z2 topological number is odd, at least one Kramers pair of edge states remains gapless. An important difference between the QH and QSHSs is that the QHE usually requires a strong magnetic field, while the QSHE is realized in the absence of the magnetic field. In fact, the QHE requires breaking of the time-reversal symmetry, but not the uniform magnetic field; the QHE can arise from external staggered magnetic field (Haldane, 1988). However, in real systems thus far, QHE arises only in external uniform magnetic field. In 2D systems it can be realized because of the cyclotron motion in the plane perpendicular to the magnetic field. Due to the Landau level formation by the strong magnetic field, the motion perpendicular to the magnetic field opens a gap. In 3D, on the other hand, the motion along the magnetic field usually remains gapless; this means that the system cannot be the QH phase in most 3D systems. On the contrary, the QSHS is realized in the absence of the magnetic field; there is no fixed direction for the magnetic field, and the QSH phase can be realized in 3D as well as in 2D. More deeply, the QSHS or the topological insulator in 3D has no correspondence to the QH system.
2007; Wang and Li, 2007; Yao and Niu, 2008; Zhang et al., 2007). In this case the spin of light is nothing but the circular polarized states of light. One can derive the EOM for the center of mass position rc and momentum kc of the wavepacket of light including the lowest-order correction due to the finite wavelength. It reads as dxc kc ¼ vðxc Þ þ k˙ c ðzc jkc jzc Þ dt kc dk c ¼ – ½rvc ðxc Þkc dt
ð74Þ ð75Þ
d jzc Þ ¼ – k˙ c ?kc jzc Þ dt
ð76Þ
where v(x) = 1/n(x) (n(x): refractive index) is the velocity of light, and jz) = [z+, z] represents the polarization state of light. kc and kc are the SU(2) connection and curvature matrices, which correspond respectively to An(k) and Bn(k) in the Berry-phase theory of electron transport. The second term on the right-hand side of (74) is the anomalous velocity due to the Berry curvature. This term will induce the shift of the trajectory of light once the force, that is, the spatial gradient of the refractive index rn(x) is applied. As shown in Figure 34 this leads to the Hall effect of light at the refraction at the spatial change of n (Onoda et al., 2004, 2006a). If we make the spatial change of n to be abrupt, it reduces to the case of interface refraction/ reflection, which is described by the famous Snell’s law. The new finding is that once the light beam is injected to the interface, the shift of the refracted and reflected beams occurs perpendicular to the plane of incident light typically of the order of a fraction of the wavelength. This phenomenon has
Anomalous velocity
k
(z c | Ω kc|z c )
Small n Berry curvature
1.07.5.3 Generalization of SHE and Future Directions Optical SHE The concept of the SHE can be generalized in many directions. One example is the optical SHE, that is, the SHE of light (Fedorov, 1955; Pancharatnam, 1956; Imbert, 1972; Chiao and Wu, 1986; Tomita and Chiao, 1986; Berry, 1987a,b; Onoda et al., 2004, 2006a; Kavokin et al., 2005; Bliokh and Bliokh, 2006; Day, 2008a; Duval et al., 2006; Glazov and Kavokin,
(z c | Ω kc|z c ) k
1.07.5.3.1
Large n
Figure 34 Schematic of the optical SHE at refraction. The polarization is left circular. The gradient of the refractive index induces a transverse anomalous velocity due to Berry curvature.
Spin Hall Effect
been known as the Imbert shift (Fedorov, 1955; Imbert, 1972) since long ago, whereas it is surprising that it shares the similar mechanism with the SHE in electrons. This transverse shift has been reconsidered by Bliokh in the case of Gaussian beam (Bliokh and Bliokh, 2006). Because the semiclassical EOMs (74)–(76) assumes slow spatial variation of n, in a strict sense they cannot be applied for the abrupt interface; as a result, in the abrupt interface the shift depends on the detail of the beam, for example, whether it is a Gaussian beam (Bliokh and Bliokh, 2006; Onoda et al., 2006a). Recently, this transverse shift has been observed experimentally by using the weak measurement (Hosten and Kwiat, 2008). With the setup shown in Figure 35 the transverse shift at the interface refraction is measured to the accuracy of 1 nm, and the result shows good agreement with the theory, as shown in Figure 36. The readers are referred to the original papers for more details. Optical SHE in exciton-polariton system due to the precession of the spin state by the k-dependent Zeeman field has been discussed and observed experimentally (Kavokin et al., 2005; Leyder et al., 2007). This mechanism is distinct from that due to the Berry curvature discussed above. Note that the SHE of excitons due to the Berry curvature has been proposed by Yao and Niu (2008) and Kuga et al. (2008) recently.
267
due to the spatial variation of the spin directions (Meier and Loss, 2003). This is more promising when one wants to suppress the dissipation, compared with the metals and semiconductors. The design of the QSHE has been developed along this line as discussed in Section 1.07.1 but Mott insulating magnetic systems are more common and usual. In the presence of the SO interaction, one expects the coupling between the spin current and the electric polarization/electric field as a natural generalization of SHE. Let us start with the commutation relationship among the components of the spin operator, that is, ½s a ; s b ¼ ih"abc s c
ð77Þ z
which can be translated into that of s and the angle within the xy-plane (s+ = sx + isy ei) as ½; s z ¼ ih
ð78Þ
This relation is analogous to the commutation relation between the phase j and the particle number n of the bosons. In this language, the magnetic ordering can be regarded as the superfluidity of the spin current, and once difference of the phases i and j occurs between the neighboring sites i and j, the Josephson super spin current is induced, which then produces the electric polarization. In this scenario, the electric polarization P produced by the two spins si and sj is given by (Katsura et al., 2005)
1.07.5.3.2
Multiferroics Another generalization of SHE is to the magnetic insulators. The charge transport is forbidden in the insulator, but the spin current can be nonzero once the magnetism is there. For example, the spin current is associated with the spin wave propagation
P ¼ eij ðsi sj Þ
ð79Þ
with being a constant proportional to the SO interaction. Here, the spin current is related to the vector spin chirality si sj.
zI
VAP
θw θI
θT
xI P1
HWP
y
P2
PS L2
VAP
L1
Figure 35 Experimental setup for the measurement of optical SHE. From Hosten O and Kwiat P (2008) Observation of the spin Hall effect of light via weak measurements. Science 319: 787.
268 Spin Hall Effect
46
70
θI = 64°
44 42
y-Displacement (nm)
60
40
δH
38 50
36 34
40
0
20
40
60
80
γI(degrees) 30
δV
20 10
λ = 633 nm 0
20
40
60
80
θI(degrees) Figure 36 Experimental results of optical SHE. From Hosten O and Kwiat P (2008) Observation of the spin Hall effect of light via weak measurements. Science 319: 787.
According to this consideration, one expects the ferroelectric state for the cycloidal helimagnets, that is, when the spin rotation plane includes the direction of the helical wavevector. Experimentally, the multiferroic behavior of RMnO3 has been discovered earlier by Kimura et al. (2003) independently of the theoretical works mentioned above, and the magnetic structure in the ferroelectric phase was shown to be the incommensurate cycloid later (Kenzelmann et al., 2005; Arima et al., 2006) supporting the theoretical proposal. The origin of the ferroelectricity in several other multiferroic materials is now shown to be the spin current mechanism (Katsura et al., 2005). In any case, the generalization of the spin Hall idea to the insulating magnetic insulating materials offers an interesting route to the dissipationless spintronics. 1.07.5.3.3
Future directions The future of the studies on SHE is briefly discussed here. One can imagine two directions for the future developments. One direction is to consider the various kinds of current instead of the spin current such as the heat current and orbital current. In particular, the heat current is an important issue also from the viewpoint of applications, and the interplay between this current and the spin current is an interesting problem. Temperature gradient, magnetic field
gradient, and the voltage drop are the external forces to drive the heat, spin, and charge currents, respectively, and one can consider various kinds of offdiagonal responses among these. In particular, the Hall responses have the topological aspects as described above, which needs to be explored in the future. Orbital current, on the other hand, is rather subtle since the information of the shape of the wave function is lost once the current is taken out of the sample to the leads. However, one can develop various analogies between the spin current and orbital current. Similar ideas have been discussed in the physics of graphene recently, where the valley degrees of freedom is analogous to those of spin, and valleytronics might be an interesting subject (Rycerz et al., 2007). Another direction is to consider the electron– electron and electron–phonon interactions in SHE. As we discussed above, the d-electrons can be a promising candidate for future spintronics, where the electron correlation plays a crucial role. The enhanced SHE by local correlation by Fe impurity has been discussed in Section 1.07.3. SHE with 5d electrons was mentioned in Section 1.07.6.5. The many-body effects combined with the nontrivial topology is a vast field of research as represented by fractional QHE. Whether the analogous fractionalized state exists or not for QSHE is an open
Spin Hall Effect
question, although the quenched kinetic energy by Landau level formation is missing in the case of QSHE. From the viewpoint of the materials, it started with the semiconductors and now extends to metals, band insulators, Mott insulators, and even superconductors. Accordingly, the concept of the spin current and SHE is generalized. The condition is that the SO interaction is effective, which generally requires heavy elements. It is very important to consider the possible mechanisms to enhance the SO interaction. An interesting proposal has been made (HuertasHernando et al., 2006): the curvature in graphene, fullerenes, nanotubes, and nanotube caps induces the new term of SO interaction. This kind of theoretical design will be useful to extend the horizon of the SHE. Various other issues such as the role of correlation (Hu et al., 2003b; Kou et al., 2005; Dimitrova, 2004a), electron–phonon coupling (Grimaldi et al., 2006b), the enhancement by bilayer effect (Jin and Li, 2007), and SHE in atoms (Liu et al., 2007c; Zhu et al., 2006) are left for future studies. (See Chapters 1.01, 1.02, 1.05 and 1.06).
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Zhang S and Yang Z (2005) Intrinsic spin and orbital angular momentum Hall effect. Physical Review Letters 94: 066602. Zhang ZY (2006) Spin accumulation on a one-dimensional mesoscopic Rashba ring. Journal of Physics – Condensed Matter 18: 4101. Zhou B, Liu CX, and Shen SQ (2007) Topological quantum phase transition and the Berry phase near the Fermi surface in holedoped quantum wells. Europhysics Letters 79: 47010. Zhou L, Ma ZS, and Zhang C (2008) Temperature dependence of the intrinsic spin Hall effect in Rashba spin–orbit coupled systems. Europhysics Letters 82: 67003. Zhu S-L, Fu H, Wu CJ, Zhang SC, and Duan LM (2006) Spin Hall effects for cold atoms in a light-induced gauge potential. Physical Review Letters 97: 240401. Zˇutic´ I, Fabian J, and Das Sarma S (2004) Spintronics: Fundamentals and applications. Reviews of Modern Physics 76: 323. Zyuzin VA, Silvestrov PG, and Mishchenko EG (2007) Spin Hall edge spin polarization in a ballistic 2D electron system. Physical Review Letters 99: 106601.
Further Reading Engel H-A, Rashba EI, and Halperin BI (eds.) (2006) Theory of Spin Hall Effects. Handbook of Magnetism and Advanced Magnetic Materials, Vol. 5, Wiley. Ko¨nig M, Buhmann H, Molenkamp LW, et al. (2008) The quantum spin Hall effect: Theory and experiment. Journal of the Physical Society of Japan 77: 031007. Murakami S (2005) Intrinsic Spin Hall Effect. Vol. 45 of Advances in Solid State Physics. Berlin: Springer. Nagaosa N (2008) Spin currents in semiconductors, metals, and insulators. Journal of the Physical Society of Japan 77: 031010. Schliemann J (2006) Spin Hall effect. International Journal of Modern Physics B 20: 1015.
1.08 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures W R Clarke and M Y Simmons, University of New South Wales, Sydney, NSW, Australia C-T Liang, National Taiwan University, Taipei, China ª 2011 Elsevier B.V. All rights reserved.
1.08.1 1.08.1.1 1.08.1.1.1 1.08.1.1.2 1.08.1.2 1.08.1.3 1.08.1.3.1 1.08.1.3.2 1.08.1.4 1.08.2 1.08.2.1 1.08.2.2 1.08.2.3 1.08.2.3.1 1.08.2.3.2 1.08.2.3.3 1.08.2.3.4 1.08.2.4 1.08.2.4.1 1.08.2.4.2 1.08.2.4.3 1.08.2.5 1.08.2.5.1 1.08.2.5.2 1.08.2.5.3 1.08.2.5.4 1.08.2.5.5 1.08.2.5.6 1.08.2.6 1.08.2.6.1 1.08.2.6.2 1.08.3 1.08.3.1 1.08.3.2 1.08.3.3 1.08.3.3.1 1.08.3.3.2 1.08.3.3.3 1.08.3.3.4 1.08.3.3.5 1.08.3.3.6 1.08.3.4 1.08.3.4.1 1.08.3.4.2
Introduction Important Length Scales and Mesoscopic Systems Important length scales Mesoscopic transport 2D Systems 1D Systems Quantum transport in 1D systems QPCs and saddle point potentials Observing Conductance Quantization Ballistic Conduction in n-type GaAs-Based 1D Systems Introduction The SG Technique and the First Observations of Quantized Conductance Other Methods to Fabricate 1DEGs Shallow etching techniques Cleaved-edge overgrowth wire V-groove quantum wires Gated, undoped (induced) heterostructures Energy-Level Splitting and the 1D g-Factor Zeeman splitting in 1D electron systems Source–drain bias spectroscopy Determination of the g-factor in 1D Many-Body Physics in 1D – The 0.7 Structure Identifying the 0.7 structure In-plane magnetic field behavior of the 0.7 feature Temperature dependence of the 0.7 structure Density dependence of the 0.7 feature Source–drain bias dependence of the 0.7 feature Thermopower, thermal conductance, and shot noise dependence of the 0.7 feature The Kondo Model and Zero-Bias Anomaly Overview Experimental and theoretical evidence for a Kondo model Ballistic Transport in p-Type GaAs-Based 1D Systems Introduction The Hole Band Structure Fabricating Stable 1D Hole Systems Etched quantum wires Surface gate-depleted quantum wires Local anodic oxidation Bilayer quantum wires Electrostatically induced quantum wires Cleaved-edge overgrowth hole wires Ballistic Transport in Hole Quantum Wires Source–drain bias spectroscopy Zeeman splitting and the g-factor anisotropy
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280 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures 1.08.3.5 1.08.3.5.1 1.08.3.5.2 1.08.3.5.3 1.08.4 References
The 0.7 Structure in 1D Hole Systems Characterizing the 0.7 structure g-Factor anisotropy of the 0.7 structure and zero-bias anomaly Spin polarization near the 0.7 structure Summary
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Glossary Band-gap engineering The process of controlling or altering the band gap of a material by controlling the growth and composition of layers of different semiconductors, such as GaAs, AlGaAs, InGaAs, and InAlAs. Chemical potential The chemical potential of a thermodynamic system is the amount by which the energy of the system would change if an additional particle was introduced, with the entropy and volume held fixed. If the chemical potential of a source lead of a quantum wire is greater than the chemical potential of the drain lead, then the electrons will flow from source to drain. Density of states The number of states at each energy level of a system. Drude conductivity The conductivity of an electronic system at zero temperature predicted by the classical Drude model. Effective mass A value of the mass assigned to an electron traveling in a periodic lattice that allows the electron to be treated as a particle traveling in free space. Einstein’s relation Also known as Einstein– Smoluchowski relation, it states that the mobility of charges in a semiconductor is proportional to a diffusion coefficient D and inversely proportional to the product of the Boltzmann constant and the absolute temperature. Elastic scattering A form of scattering event in which the energy of the incident particles is conserved, only their direction of propagation is modified. Fermi energy The energy in a solid below which all energy states are occupied at zero temperature and above which all states are unoccupied at zero temperature. Fermi velocity The velocity of particles having kinetic energy equal to the Fermi energy. Fermi wavelength The de Broglie wavelength of a quantum wave packet with kinetic energy equal to the Fermi energy.
Four-terminal configuration A measurement configuration for measuring the resistance of a device in which the current at the source and drain and the voltage at two points along the device are measured simultaneously. With this information, it is possible to determine the resistance of the device between the two voltage probes and eliminate contact resistance. g-Factor A dimensionless quantity which characterizes the magnetic moment and gyromagnetic ratio of a particle. The g-factor determines how strongly the particle will couple to an external magnetic field. g-Factor anisotropy The extent to which the measured g-factor of carriers changes as the direction of the magnetic field changes. Group velocity The velocity of the envelope of a group of interfering waves having slightly different frequencies and phase velocities. Heterostructure A single crystal made up of layers of different crystalline materials such as GaAs, AlGaAs, and AlAs. Inelastic scattering A form of scattering event in which the energy of the incident particles is not conserved. Low-dimensional system Systems of electrons or holes that are confined to approximately one Fermi wavelength in one or more directions. Many-body phenomena Phenomena that can only be described by including the interactions between particles and cannot be described by a single-particle model. Mean free path The average distance an electron travels before being scattered. Mesoscopic Pertaining to a size regime, intermediate between the microscopic and the macroscopic, that is characteristic of a region where a large number of particles can interact in a quantum-mechanically correlated fashion. Mobility The average drift velocity of carriers per unit electric field in a homogeneous semiconductor. Mobility is determined by the
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
number of scattering events and therefore is often quoted as a measure of purity of the system. Modulation doping A technique used in molecular beam epitaxy in which successive heterostructure layers are grown with different types and amounts of dopants. Momentum relaxation time The characteristic time for the momentum distribution of electrons in a solid to approach or relax to equilibrium after an external influence is removed. Phonon The quantum of acoustic or vibrational energy, considered a discrete particle and used especially in models to calculate thermal and vibrational properties of solids. Quantum point contact A quasi-1D system in which the length of the channel approaches zero. Scattering time The time between scattering events of an electron. Spin–orbit coupling In an atom, it is the interaction between a particle’s spin and its orbital angular momentum. In a semiconductor, the spin–orbit coupling refers to the coupling of the spins to the orbital angular momentum of the bands in which they travel. The orbital angular momentum of the band derives from the atomic orbitals from which it is formed. In GaAs, the conduction band derives from s-orbitals and therefore has l ¼ 0. The valence band derives from p-orbitals and therefore l ¼ 1. Transconductance The derivative of the current through a device as a function of source–drain voltage.
1.08.1 Introduction For the past four decades, the semiconductor industry has consistently managed to double the number of transistors on a silicon chip roughly every 18 months to 2 years. While this effort is to satisfy the everincreasing demand for faster and smaller computers, it has also opened up completely new areas of research in condensed matter physics. In particular, the continued miniaturization of devices and increased purity of semiconductor materials has allowed the study of novel quantum phenomena in systems where charge carriers are confined spatially to lower dimensions. This has resulted in important fundamental discoveries in quantum physics, including the integer and fractional quantum Hall effects in
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Weak localization A quantum mechanical property of 2D systems characterized by an enhanced resistance at zero magnetic field that rapidly falls away with the application of a perpendicular field. The enhanced resistance at zero field arises due to time-reversal symmetry. For each particle with a probability to travel a closed loop, there is an equal probability that the particle will travel the same closed loop in the opposite direction. These two probability pathways constructively interfere, increasing the probability that the particle will be trapped in this closed loop and unable to take part in conduction. The size of these localizing pathways is dependent on the phase coherence length and therefore weak localization is most apparent at low temperature where the phase coherence is enhanced. Applying a magnetic field breaks the time-reversal symmetry and thereby causes a decrease in the resistance. Zeeman splitting The breaking of spin degeneracy by the application of an external magnetic field. Zero-bias anomaly In 1D systems, the zero-bias anomaly refers to an enhanced conductance at zero source–drain bias that falls away rapidly as the source–drain bias is increased. The effect draws its name from an analogous effect seen in quantumdots due to the Kondo effect in 0D.
two-dimensional (2D) systems, conductance quantization in 1D quantum wires and a host of spin and charge phenomena in 0D quantum-dots. In this context, gallium arsenide (GaAs) has emerged as an important material system for studying quantum phenomena in low-dimensional systems. This is because it contains high-quality, epitaxial interfaces in combination with modulation doping, which provide an ultralow disorder environment. The very high purity of GaAs heterostructures allows low-dimensional GaAs devices to easily access the ballistic transport regime in which the mean free path of the carriers is longer than the device itself. As a consequence, carriers can pass through the device without scattering from impurities. Electron transport in the ballistic regime is very different from
282 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
conventional diffusive transport, where the electrons are scattered many times by impurities as they make their way through the device. Interestingly, fundamental research in these high-purity GaAs-based devices has been incorporated back into the semiconductor industry, which now uses them in highspeed applications such as monolithic microwave integrated circuits (MMICs). In this chapter, we study the unique properties of ballistic transport in GaAs devices. In particular, we examine the intriguing effects that appear in ballistic, 1D quantum wires. The truly quantum nature of ballistic 1D wires was first demonstrated by two groups independently. In 1988, van Wees et al. (1988) and Wharam et al. (1988) both showed that the conductance of a ballistic 1D channel was quantized in units of G ¼ 2e2/h. This seminal discovery could be explained by a simple single-particle model. However, subsequent experiments revealed the appearance of new features with no single-particle analog. Most notable has been the discovery of a small conductance plateau at the oddly unquantized value of G ¼ 0.7(2e2/h). This so-called 0.7 structure reveals a plethora of many-body effects, the origin of which continues to be a source of contention in the field of mesoscopic physics. The chapter will proceed with a brief introduction to transport in mesoscopic systems with important length scales in Section 1.08.1.1. Since all 1D devices are derived from 2D systems, we will present a brief introduction to 2D systems in Section 1.08.1.2. Then, in Section 1.08.1.3, we introduce the fundamental concepts of conduction in 1D, which includes an introduction to the density of states and to the Landauer formula for transport in 1D systems, quantum point contacts, and the saddle point potential. In Section 1.08.2, we provide a historical account of the discovery of conductance quantization in n-type GaAs heterostructures. This includes an introduction to the split-gate technique (Section 1.08.2.2) and several other methods to fabricate 1D systems (Section 1.08.2.3), including shallow etching techniques, cleaved-edge overgrowth, V-groove quantum wires, and gated, undoped heterostructures. We then review the outcomes for the quantized conductance plateau in both magnetic field and as a function of source–drain bias in Section 1.08.2.4. This allows us to measure energy-level subband spacings and the g-factor in 1D systems. In Section 1.08.2.5, we review the appearance of a feature at 0.7(2e2/h) seen in high-quality electron systems, before finishing with a discussion of the Kondo model and zero-bias anomaly (Section 1.08.2.6).
In Section 1.08.3, we mirror the discussion on electron systems by considering ballistic transport in 1D hole systems. In particular, we highlight the important role that spin–orbit coupling and the large effective mass have on both the single-particle properties and many-body effects in ballistic hole channels.
1.08.1.1 Important Length Scales and Mesoscopic Systems As semiconductor devices scale down in size, there reaches a limit where device behavior no longer scales with dimension but novel effects start to occur. These effects come into play when the dimensions of the device become comparable to an important length scale, such as the mean free path or the Fermi wavelength. At this point, the system enters what is called the mesoscopic regime, where the wave character of the charge particle comes into play and the kinetic energy becomes quantized. This quantization comes about purely from the spatial confinement of the charge carriers. This confinement can occur in just one spatial dimension to form a 2D film or in two spatial dimensions to form a 1D wire or in all three spatial dimensions to form a quantum-dot or an artificial atom. In each case, the dimensionality of charge motion is the number of spatial directions in which the electron eigenstates are free to evolve and thereby to transport charge. Understanding mesoscopic systems requires an understanding of several key length scales. These are summarized as follows. 1.08.1.1.1
Important length scales The three most important length scales in mesoscopic systems are: the Fermi wavelength, the mean free path, and the phase coherence length. Fermi wavelength, F: The Fermi wavelength for 2D systems, F, is the de Broglie wavelength of electrons at the Fermi energy and is given by F ¼
2 ¼ kF
rffiffiffiffiffi 2 h ¼ pffiffiffiffiffiffiffiffiffiffiffiffi ns 2m EF
where kF is the Fermi wave number, ns is the sheet carrier concentration, m is the effective mass, and EF is the Fermi energy. At low temperatures, current in devices is passed by carriers having an energy close to the Fermi energy, so that the Fermi wavelength becomes the relevant wavelength. Carriers with less energy have longer wavelengths, but they do not contribute to the conduction. The lowest energy
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
mode in a quantum wire is half the Fermi wavelength; therefore, a wire must be at least this wide in order to see conductance quantization. In GaAs/ AlGaAs heterostructures F 400 A˚ for an electron concentration of ns 3 1011 cm2. For 1D systems kF is given by ns. Mean free path, lmfp: A simple explanation of the mean free path is the average distance a carrier travels before it gets scattered into a different wavevector direction. If we consider an electron moving through a perfect crystalline lattice, the electron moves as if it were in vacuum, but with a different mass. However, if there is any deviation from perfect crystallinity either caused by defects or impurities in the lattice or, from thermal vibrations, then the electron’s path is scattered and it changes its momentum. The mean free path lmfp is given by lmfp ¼ vF
where vF is the Fermi velocity and is the momentum relaxation time. The Fermi velocity is given by VF ¼
hkF m
The mean free path dictates the length a quantum wire must be before ballistic transport is observed. Ideally, the length of the wire should be less than the mean free path. Typically, for high-mobility, modulation-doped GaAs/AlGaAs heterostructures, with sheet carrier densities of 3 1011 cm2, and mobilities of 5 106 cm2 V1 s1, the mean free path can be as long as 100 mm at low temperatures. At room temperature, this value can decrease considerably due to the large amount of electron–phonon scattering. The consequence of such long mean free paths at low temperatures is that it is easy to create devices where the device size has comparable dimensions to the mean free path and the system becomes ballistic.
283
Phase coherence length, l : When electrons are confined spatially, we need to consider the electron as a wave packet with a phase coherence length. Electron motion can no longer be described by a single Schro¨dinger equation but can interact with many other degrees of freedom and exchange their energy. The phase-coherence length is essentially the distance the charge carrier can travel before its phase becomes uncorrelated with its original value. The phase-coherence length can only be destroyed by inelastic scattering events – ones that cause phase randomizing collisions. Typically, this is caused by fluctuating scatterers. Some examples of this include electron–electron interactions, where the mutual Coulomb repulsion of electrons causes them to continually move and act as scattering sites. Phonons or lattice vibrations also cause dephasing, but these typically reduce considerably at low temperatures. Indeed, both electron–electron and electron–phonon scattering decrease at low temperatures. Impurity scattering can also be phase-randomizing if the impurity has an internal degree of freedom that can fluctuate with time, such as magnetic impurities where the spin fluctuates with time. The weak localization correction is only significant when l > lmfp. 1.08.1.1.2
Mesoscopic transport Transport in mesoscopic systems can be divided into three distinct regimes depending on the relative magnitude of the size of the system, L, mean free path, lmfp, and the phase coherence length, l, (see Table 1). In the diffusive regime, the sample size is much larger than the mean free path and the transport is essentially independent of the form of the system. In the quantum coherent regime, the sample size is similar to the mean free path; therefore classical transport begins to break down as edge and
Table 1 Trends in g-factor with increasing ID confinement for quantum wires oriented along different crystallographic axes in parallel and perpendicular magnetic fields.
Regime
Important length scales
Diffusive regime
l < L
Quantum coherent regime Ballistic regime
lmfp < L < l
L< lmfp
Reproduced from Koduvayur, (2008).
Impact on electrical conduction Quantum corrections to the conductivity can exist due to phase coherence within regions of size lø Classical theories of conduction breakdown and the device becomes a more complex quantum system Electrons move through the device like billiard balls on a table and the system behaves in a more simple purely quantum mechanical way.
284 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
boundary effects start to dominate. However, in the ballistic regime, the electron makes a ballistic motion in the system and the system boundary plays the role of the scatterer instead of impurities. Metallic systems are typically diffusive since their mean free paths are of the order of 100 A˚ and Fermi wavelengths of 1–2 A˚, such that quantization of the electron energy levels is not so important. However, for semiconductor systems where mean free paths can be 50 mm long and the Fermi wavelengths are large, 500 A˚, both the boundaries and quantization of the system play important roles. Before introducing conduction in ballistic 1D systems, it is important to briefly consider the 2D systems from which they are typically formed.
1.08.1.2
2D Systems
Two-dimensional electron gases (2DEGs) are important since they represent the conducting layer that is subsequently confined laterally to create 1D systems. A 2DEG occurs when electrons are confined to an interface between two different materials, such as the interface between a thin silicon and silicon dioxide insulating layer or the interface between GaAs and AlGaAs. Here, the vertical confinement of the electron gas means that the spacing of all the energetically accessible particlein-a-box energy levels for modes in the z-direction is greater than the level broadenings and kBT. The density of states in the z-direction is measurably discrete and the z-component of the wave function has a standing wave form. Therefore, the electron cannot move classically in the z-direction and the transport is 2D. The dimension at which this vertical confinement occurs is system specific but is usually of the order of the Fermi wavelength, F, and in a metal-oxide semiconductor field effect transistor (MOSFET) or high-electron-mobility transistor (HEMT) is typically hundreds of angstroms at low temperatures. The most authoritative text on 2D systems is that by Ando et al. (1992). However, a brief summary as to how they are formed will be given here before we consider 1D systems. A 2DEG can be formed in GaAs with the important advantage that it is formed at a crystalline GaAs/ AlGaAs heterointerface. GaAs can be considered as a special case of the more general group of ternary compounds AlxGa1xAs, where x represents the Al mole fraction. As a result, the variation in lattice
constant a is very small, with a ¼ 5.6533 þ 0.0078x (Adachi, 1985), minimizing strain and scattering at GaAs/AlGaAs interfaces. This is in stark contrast to the crystalline-amorphous Si–SiO2 interfaces in Si-MOSFETs. Confining the carriers to a 2D plane is made possible through band-gap engineering. While the lattice spacing in GaAs (5.653 A˚) is very similar to AlAs (5.660 A˚), the band gap between them is very different. If we replace a fraction, x, of the Ga atoms with Al atoms in AlxGa1xAs, we can span a whole range of band gaps from Eg ¼ 1.424 þ 1.247x eV at room temperature. With the use of techniques such as molecular beam epitaxy (MBE), it is possible to engineer the band gap by growing a series of layers with different doping and mole fractions x to form heterojunction devices. When a layer of AlGaAs is grown on top of a layer of GaAs, discontinuities in the conduction and valence bands form. It is possible to trap carriers and thereby form a 2D sheet of carriers at the AlGaAs/GaAs interface. However, by a technique called modulation doping, it is possible to position donors in an AlGaAs layer some distance away from the interface where the 2DEG is formed. The electrons from these donors tunnel to the lower energy state at the interface leaving behind ionized impurity states. Since the electron gas formed is spatially removed from these ionized donors, the scattering from these charged impurities is reduced and the mean free path increases. When the doped layer is n-type, the Fermi energy in this layer is shifted toward the conduction band in analogy to a positively biased surface gate. This forces the bands in the intrinsic layers to bend upward in order to balance the internal electric field and creates a triangular quantum-well in the conduction band at the GaAs/AlGaAs heterointerface. This is shown schematically in Figure 1(a). Conversely, if the doped layer is p-type, the bands in the intrinsic layers bend down, forming a quantum-well in the valence band in which holes accumulate to form a 2D hole system (2DHS) as shown in Figure 1(b). A big advantage of GaAs systems over siliconbased devices is that it is possible to produce samples where the electron mean free path is much larger than that of bulk materials with the same carrier concentration. This makes the ballistic regime readily accessible in GaAs devices. Lowerdimensional systems can then be created by using etching or surface gates to confine the 2D system to
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
(a)
285
(b) n-AIGaAs
AIGaAs
GaAs
p-AIGaAs AIGaAs
GaAs
EF 2DEG 2DHG EF
Figure 1 a) Band diagram of a GaAs/AlGaAs heterostructure with an n-type doping layer and intrinsic AlGaAs spacer layer forming a two-dimensional election gas (2DEG) at the GaAs/AlGaAs interface. b) The analogous band diagram for p-type doping to form a two-dimensional hole gas (2DHG).
one or zero dimensions. This brings us to a more extended discussion of the background theory of 1D systems.
(a)
Diffusive W lϕ
1.08.1.3
1D Systems
When the width of the 2D system is confined laterally to a length scale that is comparable to the Fermi wavelength, then we form a quasi-1D system. Again, we can experience diffusive or ballistic transport depending on the comparison between the length of the wire and the mean free path. In the case where the length of the wire is longer than the mean free path, the electrons suffer many elastic scattering events and the wire is diffusive (see Figure 2(a)). If however the length of the wire becomes comparable to the mean free path, the wire becomes ballistic (see Figure 2(b)). There is also a situation where the width is comparable to the length and is much less than the mean free path, and this corresponds to a quantum point contact (QPC) see Figure 2(c). In ballistic wires or QPCs, the constriction in the 2DEG is narrow enough that the electron wave functions form 1D subbands. In an ideal case, the longitudinal momentum is conserved and no dissipation takes place in the semiconductor. The conductance of such a 1D conductor does not depend on the length of the channel but only depends on the number of 1D modes in the channel. Here, we can no longer define a local conductivity as in the diffusive case and we can no longer directly use Einstein’s relation between the conductivity and the diffusion constant. Instead, however, we must use the Landauer formula, which connects the conductance of a ballistic conductor with a Fermi-level property of such a system.
(b) Ballistic W L << lmfp (c)
Quantum point contact W
W ≈ L « lmfp Figure 2 Electron trajectories in the (a) diffusive and (b) ballistic regimes and in (c) a quantum point contact. Here W is the width of the constriction, L is the length l is the phase coherence length, and lmfp is the mean free path. Modified from van Wees BJ, van Houten H, Beenakker CWJ, et al. (1988) Quantized conductance of point contacts in a twodimensional electron gas. Physical Review Letters 60: 848.
1.08.1.3.1 systems
Quantum transport in 1D
The Schro¨dinger equation for a quasi-1D system is given by
P2 þ V ðyÞ ¼ E 2m
where V(y) is the lateral confinement potential. The electron wave function within the quasi-1D conductor can be written as
286 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures k ðx;nÞ ¼ eikx n ðyÞ
to the conductance of the quasi-1D system, where n is the degeneracy of the subband. For GaAs, n ¼ 2 due to the degeneracy of spin-up and spin-down modes. Landauer (1957) showed that the conductance of an ideal conductor (see Figure 4) could be expressed in terms of the transmission and reflection probabilities of electron waves at a barrier, connected on either side by ideal leads. Here, the nature of transport is discussed in terms of the incident carrier flux from randomizing reservoirs. The conductance is given by
where motion along the wire is described by a plane wave. In real quantum wires, V(y) can be approximated by the general form: V ðyÞ ¼ Ajy j
where ¼ 2 for a simple harmonic oscillator potential or ! 1 for a square well potential. In the case of the harmonic oscillator potentials, the quantized energy levels are equally spaced and have the eigenfunctions shown in Figure 3(a). For > 2 the energy levels become increasingly far apart as their energy increases, whereas for < 2 the energy-level spacings become smaller. These energy levels form the subbands of the 1D system. For an arbitrary potential V(y), the energy-level spacings are given by Ekx ;n ¼ E0;n þ
G¼
where T is the transmission coefficient and R ¼ 1 T is the reflection coefficient. Here, we assume the leads are weakly coupled to the reservoirs and hence the chemical potentials considered are those of the leads, A and B, respectively, giving an infinite conductance for perfect transmission T ¼ 1 and R ¼ 0. However, in an experiment, the leads are connected to reservoirs where the chemical potentials S and D are actually measured. The conductance is then given by the modified Landauer formula (Imry, 1997)
h2 kx2 2m
From this relation, we see that the energy-level spacing is inversely proportional to m. Because of this, it is more difficult to observe conductance quantization in p-type GaAs systems due to the fact that the effective mass of holes in GaAs is 5–6 times greater than the effective mass of electrons. We can treat each subband as an independent 1D system. As a result, the density of states is simply the sum of the density of states for each subband. Figure 3(b) shows the density of states for a square well potential. Moreover, each occupied subband contributes exactly G¼
2e 2 T h R
G¼
I 2e 2 T ¼ S – D h
Here, the conductance takes a finite and universal value for T ¼ 1 defined as the contact conductance. Typically, however, the electrons flow out of one reservoir into the other and as such the chemical potentials in the reservoirs are not well defined. As a consequence, four-terminal measurements are generally made experimentally, two terminals for supplying current and two terminals for measuring
ne 2 h
|ψ(y )|2
1.0 0.8
n=2
n=1
ID DOS (a.u.)
n=3
0.6 0.4 0.2
n=0 y
0.0 0
1
2
3 4 E (meV)
5
6
Figure 3 One-dimensional (1D) subbands: (a) eigenfunctions of a simple harmonic oscillator and (b) density of states for a 1D system with a square well potential.
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
1D conductor μB μD
μ μA
Figure 4 A schematic representation of a one-dimensional (1D) conductor with source and drain reservoirs with their respective chemical potentials.
the voltage. When these reservoirs are weakly coupled to the system, the universal conductance is recovered. Indeed, in early measurements, where the mean free path was much larger than the constriction length, quasi-four-terminal measurements revealed quantized conductance plateau and only a small series resistance arising from the 2DEG contact regions needed to be subtracted (Wharam et al., 1988). If we now consider the conductance of a ballistic 1D system under an applied bias, we can write an expression for the current flow. First, we assume that the voltage applied is small in comparison to the Fermi energy and to the subband spacing of the 1D channel – the linear response regime. The current is then calculated for a simple 1D conductor connected to source and drain reservoirs, which are at the chemical potentials S and D. At zero bias, S ¼ D and the net current flow will be zero. If a voltage is applied, the net current will be I ¼
X n
Z1 –
Z1
egN ðEÞvN ðEÞf ðL ;E ÞdE
0
egN ðEÞvN ðEÞf ðR ;E ÞdE
0
where the first term represents electrons moving from source reservoir to drain reservoir and the second term is the current from drain to source, gN(E) is the 1D density of states, and vN(E) is the group velocity. At zero temperature, the Fermi–Dirac distribution is a step function. The product of the density of states g(E)¼1/(dk/dE) and the velocity v¼(1/h–)dE/dk gives a constant 2/h. Therefore, I becomes 0 1 S ZD X 2e X 2e Z 2e @ I ¼ dE – dE A ¼ ð S – D Þ h h h n n 0 0 X 2e ¼ eV h n
The differential conductance, G, is then given by G¼
dI 2e 2 ¼ N dV h
287
where N is the number of 1D channels. This result, which comes about due to the perfect cancellation of the 1D density of states gN(E) and group velocity vN(E), has been most clearly seen in short (<1 mm) ballistic channels. To date, numerous experimental systems have reported 1D ballistic transport and those in high-quality GaAs/AlGaAs heterostructures are summarized in Sections 1.08.2 and 1.08.3. It is interesting to note that while theoretically one might expect to observe conductance quantization in longer channels, it has been shown that typically channels longer than 10 mm in length do not show ballistic quantization. This arises because the elastic scattering length in the constriction is different from the lmfp of the 2DEG. As the constriction narrows and the carrier density decreases, the Fermi wave vector, kF becomes shorter and the screening of the Coulomb impurities in the doped layer by the electron gas becomes less effective. Consequently, the elastic scattering length may be reduced because the confined electron gas ineffectively screens the potential fluctuations. 1.08.1.3.2 QPCs and saddle point potentials
The above discussion suggests that the conductance through an ideal 1D system increases as a series of perfect step functions as the subbands are occupied. In reality, the conductance plateaus are connected by smooth, monotonic risers. While it is tempting to attribute this to thermal or disorder broadening of the discrete energy levels, it is in fact the result of quantum tunneling and gives important information about the shape of the potential in the wire. This can be understood most clearly by considering a quantum wire with zero length, that is, a QPC. In a QPC, the potential varies smoothly along the length of the QPC and can be well approximated by a saddle point potential (see Figure 5(a)) given by 1 1 V ðx;yÞ ¼ V0 – m !2x x 2 þ m !2y y 2 2 2
where V0 is the electrostatic potential at the center of the saddle point. This equation assumes that modes between V0 and EF are perfectly transmitted while all other modes (i.e., those above EF and below V0) are perfectly reflected. However, this is not true as quantum mechanical tunneling and reflection introduces the possibility of partially transmitted modes. As a result, the conduction of a single mode through the
288 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
4.0
10
0z
2.0
G /e 2/h
3.0
–10 –4 –2
4 x
2
0 2
–2
0 y
4 –4
0 ωy 2 ωx
1.0
4 –2
0
4 2 (E –Vo)/ hωx
6
8
0.0
Figure 5 (a) Schematic diagram of a symmetric saddle point potential. (b) Calculated conductance of a quantum point contact (QPC) as a function of energy E relative to the energy of the longitudinal harmonic oscillator energy, h!x/2 plotted for differently shaped saddle potentials parametrized by the ratio !x/!y (Bu¨ttiker, 1990). Here, spin degeneracy has not been included and as a result the conductance quantization is in units of e2/h. Reproduced with permission from Bu¨ttiker M (1990) Quantised transmission of a saddle point constriction. Physical Review B 41: 7906. Copyright (1990) by the American Physical Society.
wire will depend on a transmission probably matrix Tn,m so that G¼
e2 Tn;m h
where Tn,m depends exponentially on the energy of the mode ("n) and is given by Tn;m ¼ n;m 1=ð1 þ e – "n Þ
The total conductance is therefore given by G¼N
e2 X e2 Tm;n ¼ N T h h
As a result, the shape of the saddle point potential plays an important role in determining the shape of the conductance quantization plateaus. For very short quantum wires (!y/!x < 1), there is very little evidence of conductance quantization. Where the confinement is symmetric (!y/!x ¼ 1), plateaus develop in the conductance and as this ratio increases, the plateaus become flatter and longer (Bu¨ttiker, 1990). 1.08.1.4 Observing Conductance Quantization From the above discussion, it is evident that many factors influence whether or not conductance quantization can be observed in 1D systems. These include disorder, confinement potential,
temperature, and effective mass and are summarized briefly here for clarity. Disorder plays a pivotal role by determining all the important length scales. It can be directly correlated to the mobility of the system and therefore the mean free path. Disorder also sets the phase coherence length as a larger number of scattering events increases the probability of a carrier losing its phase memory. However, disorder also influences the Fermi wavelength. This is because disorder determines the minimum density at which a mesoscopic system can operate and hence the maximum Fermi wavelength. As a result, the disorder of a mesoscopic system sets the upper limits of both the length and width of a quantum wire. Confinement potential is also a crucial factor. As we have seen, the relative curvature of the confinement potential parallel and perpendicular to the wire will influence the shape of the conductance plateaus – the length of the steps and the sharpness of the risers. The shape of the confinement potential (i.e., parabolic or square well) determines the relative spacing between energy levels while the strength of the confinement potential will determine the absolute energy-level spacing. The energy-level spacing is also affected by the effective mass of the carriers. Having a small effective mass increases the energy-level spacing. Maximizing the energy-level spacing is particularly important for the observation of conductance
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
quantization to mitigate the effects of thermal broadening, which smears out the plateaus. It follows that lower temperatures are also preferable for the observation of conductance quantization. Today, conductance quantization is readily observable in GaAs due to the development of high-purity heterostructures, nanolithography techniques, and milli-Kelvin measurements. In the next section we review the development of some of these techniques and the insight into 1D ballistic transport that has been gained as a result.
1.08.2 Ballistic Conduction in n-type GaAs-Based 1D Systems 1.08.2.1
Introduction
Ballistic transport was first observed in 1D devices based on GaAs/AlGaAs n-type heterostructures. One of the reasons for this is that GaAs-based heterojunctions have long mean free paths and are generally easier to fabricate. In addition, the smaller effective mass of electrons increases the subband spacing, thereby enhancing quantization effects. Today, advancing growth and fabrication techniques have made it possible to readily achieve long mean free paths and hence obtain 1D ballistic conductance quantization. Moreover, high-purity 1D systems have become a fertile playground for the study of unique manybody phenomena indicative of spontaneous spin polarization at zero magnetic field and Kondo-like physics. In Section 1.08.2.2, we describe the developmental steps that lead to the first observations of ballistic conductance plateaus. In Section 1.08.2.3, we survey several of the most commonly used techniques for fabricating ballistic 1D systems, while the effect of a source–drain bias and in-plane magnetic field will be reviewed in Section 1.08.2.4. Many-body physics in 1D systems will be addressed in the Sections 1.08.2.5 and 1.08.2.6 is where we take a detailed look at the 0.7 structure and zero-bias anomaly and review the theories that have emerged to explain their origins.
289
the dimensionality from 3 to 2 dimensions. Fowler used an n-type silicon MOSFET, with two p-type implanted regions patterned either side of a 1 mm channel covered by a metal top gate. The p-type implanted regions fully depleted the lightly doped n-type channel. However, by applying a positive bias to the top gate, it was possible to induce a 1D conducting channel. The devices, however, were highly disordered and the authors were restricted to looking at the transition from 2D to 1D variable range hopping. Since that time, there have been many investigations of how to controllably confine the width of the conducting channel. The first variable width system was demonstrated in GaAs-based heterostructures by Thornton et al. (1986). Here, they were able to look at the transport characteristics of a 2DEG in a narrow channel formed by the split gate of a GaAs–AlGaAs heterojunction field effect transistor, as shown in Figure 6. A split gate (SG) is literally a gate that has a gap in it so that the two sides can be biased independently. By applying a negative voltage to these gates, they first deplete the carriers underneath them and then start to deplete the carriers laterally. In this way, the width of the 1D channel could be varied from 0.4 mm down to 0 mm. Thornton et al. investigated the conductance in the channel as a function of magnetic field for different temperatures, as shown in Figure 7. They were able to show that, at small sample widths, the magnetic field behavior did not fit standard models of 2D conductance but did fit 1D models. As such, they were able to demonstrate a transition from 2D to 1D conductance as they reduced the width of the conducting region. However, since the dimensions of their device were still relatively large (0.6 mm wide and 16 mm long) compared with the mean free path, they did not observe ballistic transport through their 1D system. Split-gate electrodes 2DEG
1.08.2.2 The SG Technique and the First Observations of Quantized Conductance Initial attempts to laterally confine transport in a 2D system to fabricate 1D systems were in silicon in the 1980s by Fowler et al. (1982). These experiments were guided by earlier GaAs experiments by Pepper (1978) in which a Schottky gate had been reverse biased to reduce the thickness of channel, thereby observing a change in
Figure 6 A schematic of a split-gate (SG) device showing how applying a negative voltage to surface gates called splitgate electrodes depletes the underlying 2D electron gas (2DEG) to form a one-dimensional (1D) channel. Initially the gate voltage depletes the 2D until a 1D channel is formed.
290 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
0.41 K 0.46 K
4.0 × 10–7
0.56 K
0.6 × 15μ Gap
G(B) – G(0) (Ω–1)
3.0 0.6 K 1.0 K 2.0
1.0
0.0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Magnetic field (T)
0.14
Figure 7 Conductance as a function of magnetic field indicated by crosses. The solid lines indicate the best fit to one-dimensional (1D) weak localization theory at different temperatures. The inset shows the gate defining the narrow channel in the underlying heterojunction. From Thornton TJ, Pepper M, Ahmed H, Andrews D, and Davis GJ (1986) One dimensional conduction in the two dimensional electron gas of a GaAs-AlGaAs heterojunction. Physical Review Letters 56: 1198, Fig. 1.
Following these initial results, two groups pursued this technique to make 1D systems in materials in which the elastic mean free path was much longer than both the length and the width of the 1D channel. As a result, the first evidence for ballistic conductance quantization was independently observed by both the Delft–Philips group led by van Wees et al. (1988) and the Semiconductor Physics group, Cambridge University led by Pepper and co-workers (Wharam et al., 1988). Both of these results will be discussed here for completeness. In the Wharam paper (Wharam et al., 1988), the SGs defined a channel that was 0.4 mm long and 0.5 mm wide and 0.7 mm below the surface. Due to the spreading resistance of the electric fields below the gates and the negative voltage applied to the SG, it was possible to reduce the width of the channel and therefore depopulate the 1D subbands. The mobility of the 2DEG was between 2.5 105 and 1 106 cm2 V1s1 for carrier concentrations between 2 and 5 1011 cm2. At low temperatures, the elastic mean free path is given by lmfp
pffiffiffiffiffiffiffiffi h 2n ¼ e
So, for these samples, this corresponded to a mean free path in excess of 2 mm with a Fermi wavelength
of 50 nm. At low temperatures, the longitudinal resistance showed quantized conductance steps as the voltage on the SG was made more negative (see Figure 8): R¼
h ; 2e 2 N
G¼
2e 2 N h
where N is the number of 1D subbands in the constriction. As the split-gate voltage was made increasingly negative, the plot showed an initial rise of the resistance at Vg ¼ 0.4 V due to the depletion of the electrons under the gate to create the narrow channel. This was then followed by the continual depopulation of the 1D channels giving steps in the resistance on the order of h/2e2. The measurements were taken in a quasi-four-terminal configuration with the contacts several microns away. As a result, the resistance of the 2DEG regions at either end contributed a series resistance to the overall resistance of the sample. When the series resistance was subtracted, the plateaus were found to be quantized to within 1%. Generally, the advantage of using a four-terminal resistance measurement is that one can minimize the resistance of the 2DEG region, which is in series with the 1D channel. It is worth mentioning, however, that at present quantized conductance steps cannot be used as a conductance standard. One of the key
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
291
5 1/3
1/4 3 1/5 1/6 1/7 1/8 1/9 1/10
R (kΩ)
R (h/2e 2 )
4
2
1
0 –3
–1
–2 Vg(V)
Figure 8 The channel resistance as a function of split-gate voltage for two different carrier concentrations in a high-mobility twodimensional electron gas (2DEG), showing quantization of the resistance in units of h/2e2. The dimensions of the SG were 0.5 mm wide and 0.4 mm long and measurements were taken at T 0.1 K. Different carrier concentrations were achieved by illumination. The inset shows a schematic diagram of the device used in this work. From Wharam DA, Thornton TJ, Newbury R, et al. (1988) one-dimensional transport and the quantisation of the ballistic resistance. Journal of Physics C: Solid State Physics 21: L209, Fig. 1.
10 Conductance, e 2/πh
reasons for this is that the resistance of the 2DEG contact regions always needs to be considered, unlike in the case of the quantized Hall resistance observed in 2D GaAs/AlGaAs heterostructures. The importance of having a long mean free path was also highlighted in the point contact devices made by the van Wees group (van Wees et al., 1988), who used a 2DEG with an electron density of 3.6 1011 cm2 and a mobility of 8.5 105 cm2 V1 s1 giving a mean free path of 8.5 mm and Fermi wavelength of 42 nm. Their devices were 250 nm wide and they also observed conductance plateaus (see Figure 9). At Vg ¼ 0.6 V, the electron gas underneath the gate was depleted and then conduction was only observed through the point contact until Vg ¼ 2.2V where the channel was completely pinched off. Again, a series resistance (400 ) was subtracted to reveal clear plateaus at integer multiples of 2e2/h. Both these results are well understood as the transmission of spin-degenerate 1D subbands (Landauer, 1957; Imry, 1997). The experimental realization of the SG technique marked an important milestone in the field of mesoscopic physics. Using Schottky gates fabricated on top of a GaAs/AlGaAs heterostructure, one is able to pattern the underlying 2D electron system into any desired shape such as a 1D channels, quantum-dots, quantum billiards, and many others.
8 6 4 2 0
–2
–1.8
–1.6 –1.4 Gate voltage (V)
–1.2
–1
Figure 9 The conductance as a function of gate voltage for a quantum point contact defined in a high-mobility two-dimensional electron gas (2DEG), showing quantized plateaus in units of e2/h. The measurement temperature T is 0.6 K. From van Wees BJ, van Houten H, Beenakker CWJ, et al. (1988) Quantized conductance of point contacts in a two-dimensional electron gas. Physical Review Letters 60: 848, Fig. 2.
1.08.2.3 Other Methods to Fabricate 1DEGs Several other methods have been shown to produce ballistic 1DEGs providing more information on the importance on the geometric nature of the source and drain contacts to the wires. They include shallow etching techniques, cleaved-edge
292 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
overgrowth wires, V-groove quantum wires, and electrostatically induced 1D systems, and are briefly reviewed here. 1.08.2.3.1
Shallow etching techniques Another method for the fabrication of 1DEGs was developed by Scherer et al. (1987). Here, they laterally patterned a 2DEG with low-energy ion-beam-assisted etching to form channels as low as 75 nm wide. They noted that there was no loss in electron mobility by the creation of the wires and that this technique would permit investigation of devices where the wave nature of electron transport would be dominant. Kristensen and co-workers then went on to use shallow-etching techniques to form three different types of quantum point contacts (see Figure 10). In their type-I sample, the 1D constriction was formed by shallow wet etching as shown in Figure 10(a). The etched constriction was then covered by a 10-mm-wide, 100-nm-thick Ti/Au top gate. Negatively biasing the top gate simultaneously reduced the density in the wire and laterally confined the wire. In their type-II and type-III devices (Figure 10(b)), etched trenches define the quantum point contact within two large areas of 2DEG. In this case, the 2DEG region can be used as side gates to alter the width of the conducting channel. In type II (III) devices, the trenches were etched 60 nm (90 nm) to remove the donor layer.
Interestingly, the confinement strength of these shallow-etched samples is much larger than that of a typical SG device, resulting in large 1D subband spacings of 20 meV. In these devices, therefore, the 1D conductance plateaus were observed to persist up to a relatively high temperature of 30 K. These shallow-etched QPCs form a very good system for studying both quantized 1D conductance plateau and the 0.7 structure at elevated temperatures (see Section 1.08.2.5). 1.08.2.3.2
Cleaved-edge overgrowth wire Yacoby and co-workers (Yacoby et al., 1996) pioneered a new technique to fabricate 1D wires using cleaved-edge overgrowth by MBE and a novel in situ contacting scheme. They first grow a modulation-doped GaAs quantum-well embedded between two thick AlGaAs layers with a doped GaAs top layer as shown in Figure 11(a) . In this case, the resulting 2DEG is 500 nm below the top surface. A long and narrow tungsten stripe is then evaporated over the top of the heterostructure that is ultimately used to gate the wire. The quantum wire itself is prepared by cleaving the sample in ultrahigh vacuum as shown in Figure 11(a) and then overgrowing the smooth cleavage plane with a second modulation doping scheme (see Figure 11(b)). As shown in Figure 11(b), electrons are introduced at the edge of the quantum-well,
(b)
(a)
500 nm
AlGaAs
Ti/Au top gate
GaAs
Quantum wire Si donors
AlGaAs
GaAs
Si donors
Quantum wire Side gate
Figure 10 (a) Scanning electron microscopy (SEM) pictures of the shallow-etched quantum point contacts. (a) Type I devices. The quantum point contact (QPC) is formed by shallow wet etching, 60 nm deep. The etched walls are shaped as two back-toback parabolas. The picture was recorded before covering the etched constriction with a 10-mm-wide, 100-nm-thick Ti/Au top gate. (b) Type II and III devices. Two semicircular-shaped, etched trenches define the QPC and two large areas of two-dimensional electron gas (2DEG), which are used as side gates. In type II devices, the trenches are etched 60 nm deep to remove the donor layer. In type III devices, the trenches are etched 90 nm to the AlGaAs/GaAs heterointerface, and subsequently covered with AlGaAs by molecular beam epitaxy (MBE) regrowth. From Kristensen A, Bruus H, Hensen AE, et al. (2000) Bias and temperature dependence of the 0.7 conductance anomaly in quantum point contacts. Physical Review B 62: 10950, Fig. 1.
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
Preliminary measurements of the two-terminal conductance of a cleaved-edge-overgrown 1D wire showed deviation from multiples of the universal values of 2e2/h by as much as 25%. For a long time, it was not clear what caused this deviation. However, in 2000, de Picciotto and co-workers from the same group solved this puzzle by demonstrating that the observed nonuniversal quantized conductance steps were caused by electron scattering across the 2D–1D interface (de Picciotto et al., 2000).
Cleave
W gate (a)
Z
X
Y Vτ = 0V
(b)
(d)
V
Vτ = VD (e)
(c) I DS
T S
S T
Vτ
293
Vτ > VD
V DS (f)
Figure 11 Fabrication procedures of preparing cleavededge overgrown wires. (a) A GaAs heterostructure with a buried two-dimensional (2D) electron system and tungsten surface gate is cleaved in situ. (b) A second modulationdoped heterostructure scheme is then grown with molecular beam epitaxy (MBE) on the cleaved edge. (c) The 1D system is formed in the quantum-well along the cleaved edge by using the tungsten gate to deplete the 2D electron system while maintaining the 1D modes along the edge as shown in (d)–(f). The 2DESs either side of the depleted region act as source and drain contracts to the 1D wire. From Yacoby et al. (1996), Fig.1.
with one and more confined edge states along the cleavage. However, there is strong overlap between the 2DEG and the edge states coupling both systems along the entire edge as shown in Figure 11(c). The tungsten gate (T) is then used to decouple the edge states from the 2DEG. By negatively biasing the top gate (T), the 2DEG directly below it is depleted while the edge states remain. The 2DESs either side of the top gate then act as source and drain contacts to the edge states to the 1D wire. The side gate (S) is primarily used to vary the electron density along the edge. Figures 11(d)–11(f) show a sequence of schematic cross sections of charge distribution in the wire region for different top-gate voltages, VT. As VT is biased increasingly negative, the 2DEG is separated from the 2D sheets that connect, through the edge states, to the 1D wire. At this point, the 1D wire becomes strongly confined in two dimensions.
1.08.2.3.3
V-groove quantum wires Another unique route to fabricating ballistic, 1D wires was taken by the Kapon group (Dwir et al., 1999), who pioneered the fabrication of V-groove quantum wires using organo-metallic vapor deposition on corrugated substrates. Figure 12(a) shows a cross-sectional transmission electron microscopy (TEM) image of such a quantum wire device. The charge distribution due to the modulation doping is shown by the white crosses. In this structure, a top quantum-well and sidewall quantum wires are present. To measure the conductance of the quantum wire, an S-shaped mesa was developed, as shown in Figure 12(b). Here, the groove sidewalls are used as electron reservoirs. To isolate the V-groove quantum wire section between the source and drain contacts, two negatively biased Schottky metal gates of length 2 mm are deposited so that one (left) depletes only the sidewalls whereas the second (right) depletes both the sidewalls and the quantum wire. The conductance was shown to display a step-like dependence on gate voltage but with deviations from the universal quantized conductance values. Using this V-groove architecture, the authors were able to control the strength of the lateral confinement of the wire, thus allowing a comparison between nearly adiabatic and more abrupt transitions from the wire to the reservoirs. They were able to show that in the limit of strong confinement, there was poor coupling between the 1D state of the wire and the 2D states of the electron reservoirs, which lead to the suppression of the conductance steps from 2e2/h (Kaufman et al., 1999). 1.08.2.3.4 Gated, undoped (induced) heterostructures
Kane et al. (1998) demonstrated that semiconductor– insulator–semiconductor field effect transistor (SISFET) heterostructures can be used to fabricate ultra-low disorder n-type quantum wires. The heterostructure has an AlGaAs layer grown on top of an intrinsic GaAs substrate. The structure is then capped with a highly doped layer of GaAs
294 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
(a)
+ Ionized impurities – Eelectrons
(b) Gate1: Sidewall
Sidewalls
Source
QWR
50 nm
Drain QWR
Gate2: Sidewalls with QWR
Figure 12 (a) Cross-sectional transmission electron microscopy (TEM) images of the quantum wire ridge region, charge distribution due to doping is schematically shown. Taken from Fig. 1 in Electron transport in AlGaAs/AlGaAs V-groove quantum wires. (b) A tilted scanning electron microscopy (SEM) image of the S-shaped device. Black arrows indicate the trajectory of the current flow. Two metallic gates of 0.25 mm width each are deposited, one on the sidewall area and the other on the full sidewall–wire–sidewall path. Taken from Fig. 1 (b) in Conductance quantization in V-groove quantum wires, (a) From Dwir B, Kaufamn D, Berk Y, Rudra A, Paleski A, and Kapon E (1999) Electron transport in GaAs/AlGaAs V-groove quantum wires. Physica B 259–261: 1025. (b) From Kaufman D, Berk Y, Dwir B, Rudra A, Palevski A, and Kapon E (1999) Conductance quantization in V-groove quantum wires. Physical Review B 59: R10433.
that can act as an in situ top gate. Positively biasing the gate induces an electron gas at the intrinsic GaAs–AlGaAs interface. A quantum wire can be formed by patterning the in situ top gate into three distinct regions as shown in Figure 13. This is usually done by a combination of electron-beam lithography and wet-etching techniques. In this way, the central region of the top gate can be used to induce a 1D electron system while the outer regions can be biased negatively to control the lateral confinement of the wire. The key advantage of induced, undoped quantum wires is that they have ultra-low levels of disorder. The fact that there is no modulation doping means that scattering from remote-ionized impurities is
(a)
essentially eliminated and the epitaxial GaAs– AlGaAs interface minimizes interface roughness scattering. Several groups have used the undoped heterostructure to understand quantum transport in lowdisorder quantum wires. Reilly et al. (2001) used induced quantum wires to study many-body effects as a function of wire length. Typically, they noted that for these induced architectures, a change in density always coincided with a change in the shape of the quantum-well and it was difficult to disentangle the contributions from each effect. More recently, Sarkozy et al. (2009) have also used induced heterostructures to study many-body effects in lowdisorder quantum wires. In particular, they looked
Side gates: Vs < 0 (b)
Top gates: VT > 0 + + + Barrier Wire
Figure 13 (a) Schematic cross section of an semiconductor–insulator–semiconductor field effect transistor (SISFET) quantum wire device. The metallically doped cap is patterned into three regions that act as independent surface gates. Positively biasing the middle gate induces a one-dimensional interacting election system (1DES) directly beneath the gate at the GaAs/AlGaAs interface while a negative bias on the outer gates controls the lateral confinement potential. (b) Scanning electron micrograph of an induced quantum wire device fabricated by electron beam lithography and wet etching. From Kane BE, Facer GR, Dzurak AS, et al. (1998) Quantized conductance in quantum wires with gate-controlled width and electron density. Applied Physics Letters 72: 3506.
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
at the zero-bias anomaly, which is believed to result from Kondo-like physics (see Section 1.08.2.6). 1.08.2.4 Energy-Level Splitting and the 1D g-Factor 1.08.2.4.1 systems
Zeeman splitting in 1D electron
In Section 1.08.2.2, we introduced the first observation of ballistic conductance in 1D systems. At zero magnetic field, the conductance steps are quantized in units of 2e2/h, where the factor 2 comes from electron spin degeneracy. When a large magnetic field is applied in the plane of the 1D channel, the electron spin degeneracy is lifted and the conductance plateaus become quantized in units of e2/h. This effect is called Zeeman splitting in 1D electron systems. The first Zeeman splitting in a 1D electron system was observed by Wharam et al. (1988), as seen in Figure 14. Here a magnetic field of 13.6 T was applied in the plane and parallel to the 1D subbands of a high-mobility 1 106 cm2 V1 s1 2DEG at T 0.1 K. The magnetic field lifted the spin degeneracy, doubling the number of plateaus observed in the quantized resistance. These steps occur at R ¼ h/2(N þ 1/2)e2. Patel et al. (1991a) extended the parallel field work of Wharam et al. (1988) and measured the transconductance dG/dVg of a 1D channel as a function of SG voltage at various magnetic fields and at various source–drain biases, as shown in Figure 15. The positions of the peaks correspond to the Fermi
R(h/2e 2)
1400
1/10
1300
1/11
1200 1100
Resistance (Ω)
1500 1/9
1000 –1.5
–1.0
–0.5
Vg(V)
Figure 14 The channel resistance of a 1D electron system as a function of applied gate voltage when a magnetic field of 13.6 T is applied parallel to the channel. The measurement temperature T is 0.1 K. The spin splitting gives rise to additional quantized plateaus at R ¼ h/(21e2), h/ (23e2), and an incipient plateau at h/(19e2). From Wharam DA, Thornton TJ, Newbury R, et al. (1988) One-dimensional transport and the quantisation of the ballistic resistance. Journal of Physics C: Solid State Physics 21: L209, Fig. 1.
295
energy crossing the spin-split subbands. By measuring the transconductance, it is possible to get a more accurate position of the conductance steps. If we consider the lowest trace for Figure 15(a), we see the transconductance of the 1D channel versus gate voltage for B ¼ 0. The position of the n ¼ 1 and n ¼ 2 conductance plateaus occur when dG/dVg is zero. As a parallel magnetic field is applied to the current direction, a linear splitting of the transconductance peaks becomes apparent. 1.08.2.4.2 Source–drain bias spectroscopy
The quantized conductance plateaus also split linearly with the application of a large bias between source and drain. The effects of a source–drain bias on 1D subbands were also investigated by Patel et al. (1991a, 1991b), as shown in Figure 15(b). They measured a linear splitting of the transconductance peaks both with increasing magnetic field and for increasing source–drain bias. Increasing the source–drain voltage lifts the momentum degeneracy and splits each conductance plateau (in the differential conductance vs. gate voltage trace) into two. Again, a linear splitting in the transconductance peaks is observed, this time as Vsd increases. Source–drain bias spectroscopy is based on a model developed by Glazman and Khaetskii (1989) and is now a standard method for measuring subband spacings in 1D systems . The method adds a direct current (DC) bias to the alternating current (AC) excitation voltage in order to change the relative chemical potentials, , at the source and the drain of the quantum wire. When no DC bias is applied, the differential conductance follows the standard quantized conductance relation G¼
N X dI e2 2n ¼ h dV n¼0
where N is the number of subbands occupied. However, when a DC bias equivalent to the subband energy spacing is applied, the differential conductance becomes G¼
N X dI e2 ¼ ð2n þ 1Þ dV h n¼0
Applying a source–drain bias is known to select different numbers of channels from the source and drain, and is expected to give rise to a half-integer plateau in zero magnetic field. In other words, the
296 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
(a) 12
(b) 10 B = 15T
8
B = 10T
6 B = 5T
4 2 n=1 0 –3.6 –3.4 –3.2
n=2 –3
B = 0T
–2.8 –2.6 –2.4
Vg (V)
B = 0T
8 dG /dVg (a.u.)
dG /dVg (a.u.)
10
Vsd (mV)
6
1.5
4
1.0
2
0.5
n=1 0 –3.6 –3.4 –3.2
n=2 –3
0.0
–2.8 –2.6 –2.4
Vg (V)
Figure 15 (a) Transconductance as function of gate voltage at different magnetic fields. (b) Transconductance as a function of gate voltage at different source drain biases for B ¼ 0 T. From Patel NK, Nicholls JT, Martin-Moreno L, et al. (1991a) Properties of a ballistic quasi-one-dimensional constriction in a parallel high magnetic field. Physical Review B 44: R10973, Fig. 2.
conductance is quantized in odd integer multiples of e2/h. Therefore, by measuring at what DC bias the conductance plateau becomes quantized in odd integer multiples of e2/h, it is possible to determine the subband energy spacing for each subband. By tuning the source–drain voltage, one can then probe the 1D subband spectrum. It is important to note that these half-integer plateaus at high Vsd are still spin degenerate. Further application of a magnetic field results in the formation of quarter-integer plateaus at 1.25(2e2/h), 1.75(2e2/h), 2.25(2e2/h), 2.75(2e2/h), 3.25(2e2/h), . . . . Thomas et al. (1998b) went on to show that in finite source–drain bias experiments where wellresolved integer and half-integer plateaus were observed quarter-integerplateaus were also seen for every subband even at zero magnetic field.
1.08.2.4.3 in 1D
Determination of the g-factor
The extent to which the spins couple to an external magnetic field is determined by the g-factor, which can be measured by finding the source–drain voltage at which the splitting of the transconductance peaks is equal to the Zeeman splitting in an in-plane magnetic field. At this point, the g-factor is given by the relation eVsd ¼ 2gB SB
where B is the magnetic field, B is the Bohr magneton, and S ¼1=2. As shown in Figure 15, a linear splitting of the transconductance peaks was observed for both increasing magnetic field and for increasing source–drain bias. By comparing the data at high
magnetic fields with those in the presence of source–drain biases, Patel et al. found that for the second and third subband edge, the 1D g-factor was 1.1 – significantly enhanced over the bulk g-factor 0.4. This large discrepancy was attributed to an enhancement of the g-factor due to exchange and correlation mechanisms. This pioneering experiment did not require a perpendicular field as is normally required for the determination of the g-factor for 2DEGs. Thomas et al. (1996) further extended the work of Patel and co-workers and were able to measure the g-factor when the current is either perpendicular or parallel to the applied in-plane magnetic field. Their experimental results are reproduced in Figure 16. For Vg < 4 V, additional spin-split plateau are observed interleaved between those at zero field. For Vg > 4 V, the Zeeman energy is comparable to the subband spacings and both sets of spinsplit plateaus cannot be easily resolved. From the inset (b), we can see gk and g? for all 26 subbands, and it is evident from these results that the g-factor is almost isotropic. It is worth pointing out that when the number of occupied 1D subbands was high, the absolute value of the g-factor was found to be 0.4, that is, close to the bulk value of GaAs. For the last few occupied subbands of the channel, the anisotropy of the confinement potential can be described by a saddle point (Martin-Moreno et al., 1992) and little anisotropy of the in-plane g-factor is observed. However, as the number of 1D subbands decreases, the g-factor was observed to increase gradually. It was suggested that the enhanced g-factor (1) over its bulk value was due to electron–electron interaction effects as the constriction became narrow.
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
• •
20 (b)
g⊥
1
0.5
I 0
10
II 0
•
10 20 Subband Index
Sample A T = 60 mK
5
G (2e2/h)
G (in units of 2e 2/h)
15
g-factor
1.5
1
(a)
0.5
T = 600 mK
•
297
The conductance tends toward 0.5(2e2/h) in a parallel magnetic field (Thomas et al., 1996). The conductance has been observed to tend toward 0.5 2e2/h with both increasing (Reilly et al., 2001) and decreasing density (Thomas et al., 1998a, 2000; Pyshkin et al., 2000). The feature is activated with temperature. Typically, it strengthens with temperature and evolves downward from the 2e2/h plateau as temperature increases (Thomas et al., 1998a). It is often accompanied by a peak in the conductance at zero source–drain bias (Cronenwett et al., 2002).
0 –5.8
0 –6
–5.6 Vg (V)
–5
–4
–3
Gate voltage Vg (V) Figure 16 (I) Gate voltage characteristics at B ¼ 0. (II) The gate characteristics (offset by 0.3 V for clarity) in a magnetic field of 11 T. Insets: (a) detail of the structure at 0.7 2e2/h; (b) the in-plane g-factors as a function of subband index, as obtained from the Zeeman splitting at 8.2 T. From Thomas KJ, Nicholls JT, Simmons MY, Pepper M, Mace DR, and Ritchie DA (1996) Possible spin polarization in a one-dimensional electron gas. Physical Review Letters 77: 135, Fig. 1.
1.08.2.5 Many-Body Physics in 1D – The 0.7 Structure In a very clean 1D channel, a clear plateau-like structure close to (0.7 2e2/h) has been observed at zero magnetic field, which can be seen in the inset (a) to Figure 16 (Thomas et al., 1996). This feature is now well known as the 0.7 structure or 0.7 anomaly, whose conductance value is placed between the spin-degenerate conductance plateau at 2e2/h and the spin-split conductance plateau at e2/h, and cannot be explained within a single–particle picture. The 0.7 structure has been observed in numerous 1D systems with different sample designs establishing that it is a universal effect. Its origin remains the subject of intense experimental and theoretical debate and has focused on zero-field spin polarization (Thomas et al., 1996; Spivak and Zhou, 2000; Bruus et al., 2001; Starikov et al., 2003), spin-density wave formation (Reimann et al., 1999), pairing of electrons (Flambaum and Kuchiev, 2000), singlet– triplet formation (Rejec et al., 2000), Kondo-like interactions (Cronenwett et al., 2002; Meir et al., 2002; Lindelof, 2001), and electron–phonon effects (Seelig and Matveev, 2003). There are some general features of the 0.7 structure that make it unique and intriguing:
1.08.2.5.1
Identifying the 0.7 structure Although the 0.7 structure was observed in some of the earliest experiments of conductance quantization in 1D, it was not appreciated as a phenomenon in its own right until Thomas et al. (1996) performed a detailed study of the anomalous plateau. The initial oversight is likely to be due to the fact that disorder within the channel can cause resonant backscattering and produce similar features at arbitrary conductances. It is therefore necessary to first distinguish the 0.7 structure from these resonances. This is possible because, unlike the 0.7 structure, resonances will change on successive cool-downs, get weaker with temperature, appear in the risers at higher subband indices with increasing amplitude, and will disappear when the channel is shifted laterally. Therefore, it is important when investigating the 0.7 structure to measure not only the temperature and magnetic field dependence but also the ability to laterally shift the channel on the same sample. Liang et al. (2000) performed such a study using a multi-layered gated 1D structure as shown in Figure 17(a). Here, a cross-linked PMMA layer sits between the SG labeled in black and three finger gates (F1, F2, and F3) labeled in gray and acts as an insulating layer, so that by biasing the finger gate voltages, it is possible to control the strength of lateral confinement of the 1D channel. As shown in Figure 17(b), with decreasing negative finger gate voltage, VF2, the 1D conductance steps become less pronounced and finally disappear due to weakening of the lateral confinement strength. The energy separation of the 1D subbands was determined from source–drain bias experiments at various VF2 and demonstrated a good linear fit in Figure 17(c). As the finger gate voltage, VF2 was made more negative the energy spacing between the first and second subband, E1,2 (VSG) decreased, giving rise to the reduction in
298 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
(b)
Device A
(a) SG
G (in units of 2e 2/h)
4
First cooldown T = 1.2 K
3 VF2 = 0 V 2
1 0.05 μm
VF2 = –1.8 V 0
–1.2
–1
0.6 μm
0.3 μm
–0.8
–0.6
–0.4
VSG (V) (c) 2.4
F1
F2
F3
SG
0.8 μm
ΔE12 (mV)
2.1
1.8
1.5
–1.5
–1.2
–0.9
–0.6
–0.3
VF2 (V) Figure 17 (a) A schematic diagram showing a multi-layer-gated structure. The black areas represent the split-gate (SG). The gray lines correspond to three overlaying finger gates F1, F2, and F3. There is a layer of crosslinked polymethylmethacrylate (PMMA) in between SG and F1, F2, and F3 to act as a gate dielectric. (b) Conductance measurements as a function of SG voltage VSG at various finger gate voltages VF2. From left to right VF2¼ 0 to 1.8 V in 0.3 V steps. (c) Measured subband spacing at various VF2. (a) From Liang, (1999), Fig. 1(a). (b) From Liang (1999), Fig. 2(a). (c) From Liang, (1999), Fig. 2(b).
flatness of the conductance plateaus. Using the saddle point potential model (Buttiker, 1990), they estimated the change in lateral confinement strength by a factor of 2 over this range. Therefore, the 0.7 structure persists despite a change in lateral confinement strength by a factor of 2, while the 1D ballistic conductance plateau disappears. This result showed compelling evidence that the 0.7 structure is an intrinsic property of a clean 1D channel and persists over a wide range of lateral confinement potentials. 1.08.2.5.2 In-plane magnetic field behavior of the 0.7 feature
Historically, the 0.7 structure was observable in some of the earliest experiments (van Wees et al., 1988) and first commented on by Patel and co-workers (Patel
et al., 1991a). In Figure 15, Patel et al. observed an additional shoulder-like structure at around Vg¼3.4 V in the transconductance of a high-mobility 1D channel. With increasing applied source drain bias, the shoulder-like structure appeared to split into two. While the structure was observed, its origin could not be explained. It was not until 1996 when Thomas and co-workers performed a systematic study of an ultraclean 1D channel in an in-plane magnetic field that the 0.7 structure was investigated thoroughly. In this work, Thomas et al. used a high-quality 2DEG with a mobility of 4.5 106 cm2 V1 s1 at a carrier density of 1.8 1011 cm2 showing nearly 30 quantized plateaus and a well-defined structure at 0.7(2e2/h). The 0.7 structure was observed to be reproducible on thermal cycling, ruling out the possibility
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
Sample B
G (in units of 2e 2/h)
T = 60 mK 1 B=0T
0.5 B = 13 T
0 –6.8
–6.4 –6.6 Gate voltage Vg (V)
–6.2
Figure 18 Conductance measurements as a function of gate voltage at various in-plane magnetic fields. From left to right: B ¼ 0–13 T in 1 T steps. Curves have been horizontally offset for clarity. From Thomas KJ, Nicholls JT, Simmons MY, Pepper M, Mace DR, and Ritchie DA (1996) Possible spin polarization in a one-dimensional electron gas. Physical Review Letters 77: 135, Fig. 3.
of a universal conductance fluctuation or a scattering effect. Using such a device, Thomas et al. investigated the evolution of the 0.7 structure with increasing inplane magnetic field as shown in Figure 18. The leftmost trace shows a clear structure at 0.7(2e2/h), which by 11 T on the right-hand side has moved down to 0.5(2e2/h). The evolution of the 0.7 structure into the spin-split plateau at e2/h led them to speculate that there could be a possible spin polarization in zero magnetic field. The conclusion that the 0.7 structure was related to spin polarization in the channel was reinforced by the presence of an incipient spin splitting also observed just below the 4e2/h plateau (the 1.7 structure) (Thomas et al., 2000) and an enhancement of the Lande´ g-factor as the carrier density decreased and the 1D bands were depopulated. There were two problems with this proposition. First, the possible existence of a spin-polarized ground state in 1D contradicted a long-standing theory that showed it was impossible to have a spinpolarized ground state in a strictly 1D system (i.e., one that has no 2D leads and where the second 1D subband is infinitely high in energy above the first) (Lieb and Mattis, 1962). However, real quantum point contacts and quantum wires are not strictly 1D, but have finite length and width. In addition, the subband spacing is also finite and they are
299
connected via 2D reservoirs. It has been more recently shown that, for such realistic conditions, a spin-split ground state is possible (Spivak and Zhou, 2000; Starikov et al., 2003). This brings us to the second problem of explaining the 0.7 structure – a complete spin polarization would produce a structure at 0.5(2e2/h). To date, a quantitative explanation of the higher fractional value observed experimentally remains elusive. However, the 0.7 structure has now been widely observed in 1D systems defined in cleaved-edge overgrowth structures (de Picciotto et al., 2004, 2005), induced GaAs electron (Pyshkin et al., 2000), GaAs hole (Danneau et al., 2006b), Si (Bagraev et al., 2002), GaN (Chou et al., 2005), and InGaAs (Simmonds et al., 2008), proving the universal nature of this effect. Hartree–Fock calculations of electrons confined to a cylindrical wire showed that at low electron densities exchange interactions would drive a spontaneous spin polarization (Gold and Calmels, 1996). However, this would give rise to an extra plateau in the conductance at e2/h rather than 0.7(2e2/h). Other theories based on the idea of a spin-split ground state also predicted a plateau at G ¼ 0.5(2e2/h) (Wang and Berggren, 1996). Two groups attempted to use coupled spins to rationalize the fact that the plateau is observed at 0.7(2e2/h) rather than 0.5(2e2/h) by taking a statistical approach (Flambaum and Kuchiev, 2000; Rejec et al., 2000). Both groups developed theories that suggested the electrons couple to form singlet and triplet pairs with different transmission barriers. Since there are three triplet configurations and only one singlet configuration, there is a 3 to 1 probability that a triplet is formed over a singlet. This predicts a plateau at G ¼ 0.75(2e2/h) corresponding to triplet conduction band edge and a second plateau at G ¼ 0.25(2e2/h) corresponding to singlet conduction band edge. Interestingly, plateaus at G 0.25(2e2/h) have been also observed when a source–drain bias is applied and this will be discussed further in Section 1.08.2.5 (Patel et al., 1991b; Thomas et al., 1998b; de Picciotto et al., 2004; Ramsak and Jefferson, 2005; Graham et al., 2006). To further study the role of spin in 1D systems Graham et al. (2003) investigated the behavior of the Zeeman split 1D subbands of different indices in a high parallel magnetic field (see Figure 19). They observed that, as expected, each spin degenerate 1D subband splits into two (see P in the inset) with new conductance plateau appearing at half-integer values of 2e2/h. As Bk increased these half-integer plateaus strengthened while the integer plateaus weakened,
300 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
3
(a)
T = 100 mK
α2
2
α1
B
β
10
B (T)
G (in units of 2e 2/h)
Sample A
B 1 Energy
N+1
Q R
N
P
N–1
Magnetic Field
0 –1.4
α0
–1 –1.2 Gate voltage Vg (V)
Figure 19 Differential conductance G(Vg) traces for different parallel magnetic fields from 0 to 15.6 T in steps of 0.6 T. The traces are offset for clarity. Inset: Schematic energy diagram for a linear Zeeman splitting of onedimensional (1D) subbands and subsequent crossings. From Graham AC, Thomas KJ, Pepper M, Cooper NR, Simmons MY, and Ritchie DA (2003) Interaction effects at crossings of spin-polarized one-dimensional subbands. Physical Review Letters 91: 136404, Fig. 1.
until the point where the Zeeman energy was equal to the subband spacing and the integer plateaus disappeared. This happened at a crossing point, such as Q in the inset. Finally, the half-integer plateaus weakened again as Bk increases further and the integer plateaus reappeared after this crossing (see R in the inset). They considered that the appearance of the 1.5(2e2/h) plateau, which weakened and then evolved into the 2e2/h plateau before reappearing for Bk > 8 T, resembled the evolution of the 0.7 structure to 0.5(2e2/h) with increasing Bk. As a consequence, they have named the 1.5 (2e2/h) plateau, the 0.7 analog. The reappearing 2e2/h plateau carried the opposite spin than before the crossing, while the lowest subband, which does not encounter a crossing, does not change its spin. The evolution of these conductance characteristics is more clearly seen in gray-scale plots of the transconductance as a function of Bk and Vg, as shown in Figure 20. White regions represent the plateaus in conductance and the dark regions correspond to the transitions between plateaus. As Bk was increased from zero, a splitting of the 1D subbands of opposite spin was observed, as shown in Figure 21 for point P. At Bk ¼ 0, the first white plateau, 0, corresponding
–1.25
–1.2 –1.15 Gate voltage Vg (V)
Figure 20 Gray-scale plots of the transconductance, dG/dVg,as a function of Vg and Bk The labels 1, 2, and show the 0.7 analogs at the Zeeman crossing between spin levels 1" and 2#, 1" and 3#, and 2" and 3#, respectively. From Sfigakis F, Graham AC, Thomas KJ, Pepper M, Ford CJB, and Ritchie DA (2008b) Spin Effects in One Dimensional Systems. Journal of Physics C 20: 164213. Fig. 6.
to the 0.7 feature is visible. As Bk was increased, this plateau evolved into the 0.5(2e2/h) and the white region broadened. By Bk ¼ 11 T, the first (N ¼ 1") and second subbands (N ¼ 2#) crossed. After the crossing, the N ¼ 1" showed a discontinuous shift from the crossing point, marked 1. This discontinuity corresponds to the 0.7 analog. This discontinuous shift was also observed at the crossing of the N ¼ 2" and the N ¼ 3# lines marked by , and the N ¼ 1" and the N ¼ 3# lines marked by 2. Interestingly, at these points, there was no anti-crossing of the subbands, but a gap formed abruptly after the crossing. Graham et al. argued that this 0.7 analog is a consequence of the strong exchange interactions and where there is a lifting of the zero-field spin degeneracy in the 0.7 structure, there is also a lifting of the degeneracy of the crossing point for the 0.7 analog (Graham et al., 2003; Graham et al., 2004). In further works using DC bias spectroscopy, they showed that the 0.7 structure and its analogs were caused by the highest energy spin-up subband pinning to the chemical potential, as predicted by Kristensen et al. (2000), together with an abrupt rearranging of
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
1.5
G (2e2/h)
1.0
0.3K 1.5K
0.5
2.7K 5.1K
0.0 0.28
0.30
0.32
0.34
Vgs (V) Figure 21 Conductance as a function of gate voltage Vgs at various temperatures for the first quantized plateau. As the temperature is raised the 0.7 structure emerges. Kristensen A, Bruus H, Hensen AE, et al. (2000) Bias and temperature dependence of the 0.7 conductance anomaly in quantum point contacts. Physical Review B 62: 10950, Fig. 7.
spin-up and spin-down subbands (Graham et al., 2007). The 0.7 analog has since been shown to share the same DC bias, magnetic field, and temperature dependence as the 0.7 structure highlighting the need for a theoretical description of this unusual state that is valid for both features at zero magnetic field.
1.08.2.5.3 Temperature dependence of the 0.7 structure
The 0.7 structure has a characteristic temperature dependence, strengthening as the temperature is increased in contrast to the 2e2/h plateaus (see Figure 21). The 0.7 structure can appear weak at very low temperatures, but typically becomes stronger as the temperature is increased. As the temperature is increased further, the strength of the state stabilizes and then weakens once more, but only after all higher plateaus have been thermally smeared. Kristensen et al. studied the temperature dependence of the 0.7 structure using shallow-etched QPCs described in Section 1.08.2.3.1 (Kristensen et al., 2000). These are ideal for temperature dependence studies of the 0.7 structure since the large 1D subband energy separation in shallowetched QPCs allows studies up to the relatively high temperature of 5 K without appreciable thermal smearing of the quantized conductance steps.
301
At the lowest temperature, the first conductance plateau was broad and flat. With a 1D subband energy separation of 6.5 meV, the thermal smearing of the plateau should be negligible. As the temperature was raised, the conductance plateau remained flat; it however weakened as the temperature was further increased. In contrast, a plateau-like structure emerged and strengthened around the conductance value of 0.7(2e2/h). Following on from this, Kristensen et al. went on to plot the relative conductance suppression. Figure 22(a) shows the graph between 1 – G(T)/G0 and 1/T at the given fixed gate voltage. Interestingly, the linear dependence in the semilogarithmic Arrhenius plot indicates an activated behavior. Figure 22(b) shows the measured activation energy TA as a function of gate voltage. We can see that TA increases from zero to a few Kelvin in the middle of the conductance step. This suggested that the 0.7 structure is associated with thermal depopulation of a subband, which has a gate-voltage-dependent subband edge. Kristensen concluded that these observations ‘‘indicate the importance of interaction effects beyond the simple single-particle subband picture, presumably related to spin polarization.’’ Spontaneous spin-split ground-state models have difficulty in explaining the temperature dependence of the 0.7 structure. The fact that the 0.7 structure may become stronger with temperature suggests that it is not a ground state with a single-particle energy gap, but an activated process with an activation temperature TA. To explain this within the spontaneous spin-splitting paradigm, Starikov et al. (2003) proposed a model in which the polarized ground state is accompanied by metastable states with lower conductivity. Therefore, thermal activation of carriers into the metastable states causes a depression of the conductance plateau at G ¼ 2e2/h to form the anomalous plateau at G ¼ 0.7(2e2/h). Other authors have suggested that the temperature dependence of the 0.7 structure indicates that spontaneous spin splitting is not the underlying mechanism and have looked for an alternative explanation. Seelig and Matveev (2003) proposed a theory based on electron–phonon scattering in the 1D channel. They showed that a significant negative correction to the quantized conductance could occur due to the effect of backscattering of electrons in QPCs by acoustic phonons. In 1998, Kristensen et al. (1998b) also suggested a mechanism whereby scattering from thermally activated plasmons decreased the conductance of the
302 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
(a) –0.4
(b) 20 Vgs = 0.305 V TA = 0.28 K
15 TA (K)
In(1–G/G0)
–0.8
–1.2
10
Vgs = 0.309 V –1.6
5
TA = 1.11 K
–2.0 0.0
0.5
1.0
1.5
2.0
2.5
1/T (K–1)
0 0.30
0.31
0.32
0.33
Vgs (V)
Figure 22 (a) Temperature dependence of the conductance suppression G0-G(T) at fixed gate voltages Vg ¼ 0.305V and 0.309 V, measured on device A. The data show Arrhenius behavior, G(T)/G0 ¼ 1-Cexp(TA/T),with an activation temperature TA. (b) Measured activation energy as a function of gate voltage. (a) From Kristensen A, Bruus H, Hensen AE, et al. (2000) Bias and temperature dependence of the 0.7 conductance anomaly in quantum point contacts. Physical Review B 62: 10950, Fig 8(a). (b) from Kristensen, (2000), Fig 8(b).
G ¼ 2e2/h plateau to form the anomalous plateau at G ¼ 0.7(2e2/h). Cronenwett et al. (2002) noted that the temperature dependence of the 0.7 structure is reminiscent of the Kondo effect in quantum-dots with one unpaired electron. Extending this analogy, they performed temperature dependence and source–drain bias studies and found several other conspicuous similarities. These included a conductance peak at zero source– drain bias. In addition, plots of the conductance along the 0.7 plateau as a function of T collapse onto a single function of empirical Kondo-like form using a single scaling parameter designated the Kondo temperature, TK. Since then, several theories have been developed based on a Kondo explanation for the 0.7 structure, which explain many of the experimental observations (Meir et al., 2002; Hirose et al., 2003; Schmeltzer et al., 2005) (see Section 1.08.2.6). However, Kondo models require that an unpaired electron can become localized in the 1D region to act as a form of magnetic impurity. While it has been shown that such a state is theoretically possible under certain assumptions (Meir et al., 2002), other density functional (Starikov et al., 2003) and Hartree–Fock (Sushkov, 2003) calculations did not confirm a localized spin state in real QPC devices. 1.08.2.5.4 feature
Density dependence of the 0.7
Studies of the carrier density dependence of the 0.7 structure are not straightforward. Initial experiments showed that lowering the carrier concentration from
1.3 1011 to 3 1010 cm–2 shifted the 0.7 structure to 0.5(2e2/h) (Thomas et al., 2000), indicating a complete spin polarization (Wang and Berggren, 1996; Gold and Calmels, 1996). However, subsequent work on induced electron gases showed that the conductance of the 0.7 structure tended toward 0.5(2e2/h) at both the highest and lowest densities, with further reports in an induced 1D wire of length 2 mm showing that the 0.7 structure evolved continuously to 0.5(2e2/h) with increasing 1D density (Reilly et al., 2001). Reilly (2005) showed that the position and strength of the 0.7 structure did not depend on the absolute value of the carrier density but on the relative difference between the potentials in the 1D and 2D regions. Wirtz et al. (2002) performed interesting studies on the hydrostatic effect on the 0.7 structure. Using both pressure and illumination, the electron density in the 2DEG was reduced from 2.14 1011 to 0.61 1011 cm2, and a shift in the conductance of the 0.7 structure toward the spin-split value of e2/h was observed (see Figure 23). The electron density below the SG is lower than in the 2DEG, so it is possible that the exchange energy could be significantly enhanced with the Hartree term being completely collapsed inside the constriction. Theoretically, such an enhancement could lead to a spin polarization. Theoretical results by Bruus et al. (2001) suggested that an increase of the spin gap between the up and down states should result in a shift of the 0.7 structure toward e2/h. Using a cleaved-edge overgrown 1D wire, de Picciotto et al. (2004) observed a 0.7 structure in the
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
Decreasing 2DEG electron density 1.0
G (2e2/h)
0.8 0.6 0.4 0.2 0.0 –2.50
–2.45
–2.40
Vg (V) Figure 23 Conductance measurements at various two-dimensional electron gas (2DEG) densities. From left (ne ¼ 2.14 1011 cm2) to right (ne ¼ 6 1010 cm2). The position of the 0.7 structure is indicated by an open circle in each curve. From Wirtz R, Newbury R, Nicholls JT, Tribe WR, Simmons MY, and Pepper M (2002) Tuning the electron transport properties of a one-dimensional constriction using hydrostatic pressure. Physical Review B 65: 233316, Fig. 4.
differential conductance measurements as shown in Figure 24. It was found that the 0.7 structure existed whenever the kinetic energy provided to the carriers exceeded the Fermi energy. In this regime, the measured conductance exceeded the value calculated from a noninteracting model, suggesting that
4 3 2 1 0 –1 –2 –3 –4
Gate voltage (V)
–3.4 –3.6 –3.8
g∼1
–4 –4.2 g ∼ 0.25 –4.4 g ∼ 0.7 –4.6 –20
g=0
–10 0 10 Source voltage (mV)
20 Δ(g/g ) 0
ΔVg
(V–1)
Figure 24 Transconductance of a one-dimensional (1D) channel as a function of gate voltage and direct-current (DC) bias: The numerical derivative of the differential conductance with respect to gate voltage as a function of gate voltage and DC bias, in a color plot format. The 0.7 structure is indicated by g 0.7. From de Picciotto R, Pfeiffer LN, Baldwin KW, and West KW (2004) Nonlinear response of a clean one-dimensional wire. Physical Review Letters 92: 036801, Fig. 2.
303
electron–electron interactions play an important role in the 0.7 structure observed in their 1D wires. More recent studies of very long 1D wires fabricated by cleaved-edge overgrowth postulated that the 0.7 structure is a property of long wires, where there is both a large aspect ratio and where there are numerous electrons in the wire (up to 100) (de Picciotto et al., 2008). In this study, the authors argued that it is the charge density in the wire that mainly affects the appearance of the 0.7 structure and not the length of the channel. The 0.7 structure was observed to occur at densities low enough such that Fermi energy was lower than the temperature. In general, it is difficult to make a definitive statement about the density dependence of the 0.7 structure and how it fits in with current models. One of the complications in understanding how the 0.7 structure changes with density is to understand experimentally exactly how the density of the 1D region is changed. Typically, a gate is used to change the density within the wire, but this is very sample specific depending on the gate geometry and is also likely to change the shape of the confinement potential and the mismatch between the potential of the 1D and 2D regions. 1.08.2.5.5 Source–drain bias dependence of the 0.7 feature
The effects of a source–drain bias on 1D subbands were discussed in Section 1.08.2.4.2. Increasing the source–drain voltage lifts the momentum degeneracy and splits each conductance plateau (in the differential conductance vs. gate voltage trace) into two. For G 2e2/h in zero magnetic field, half-integer plateaus were observed for every subband. However, Patel et al, also reported that as the source–drain bias was increased, the 0.7 structure evolved into a structure at 0.8–0.85(2e2/h) with another plateau appearing at 0.2(2e2/h) at even higher biases. Initially, this led to suggestions that these additional plateaus were due to a spin degenerate 0.5(2e2/h) feature but with its conductance reduced by some mechanism (Kristensen et al., 2000; Cronenwett et al., 2002; Reilly et al., 2002; de Picciotto et al., 2004; Kothari et al., 2008). Alternatively, it has been described as a fully spinpolarized quarter plateau (Graham et al., 2006; Simmonds et al., 2008). Thomas et al. (1998b) went on to show that in finite source–drain bias experiments where wellresolved integer and half-integer plateaus were observed, quarter-integer plateaus were also seen for every subband even at zero magnetic field. The
304 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
appearance of these quarter-integer plateaus did not change with increasing magnetic field, further supporting the concept of spin polarization at zero magnetic field. More recently, plateaus at 0.25 (2e2/h) and 0.75 (2e2/h), nearly identical to those observed in GaAs quantum wires, have also been observed in nonequilibrium transport studies in In0.75Ga0.25As quantum wires (Simmonds et al., 2008). Their appearance has also been attributed to zero-field spin-polarized quarter-integer plateaus. Sfigakis et al. (2008a) considered that in a single particle picture, the quarter-integer plateau for a given subband should appear at the same time when sweeping the source–drain bias. For the high index subbands, this appeared to be the case. However, as the subband index decreased, the (N þ 0.25) 2e2/h plateau appeared at increasingly higher bias than the (N þ 0.75) 2e2/h. The late formation in source– drain bias voltage of the 0.25(2e2/h) plateau with respect to the formation of the 0.75(2e2/h) plateau was shown to be consistent with the predictions from a spin-gap model. Here, self-consistent calculations with a saddle point model and a density-dependent energy gap between spin subbands were shown to exhibit the same characteristics (Sfigakis, 2005).
1.08.2.5.6 Thermopower, thermal conductance, and shot noise dependence of the 0.7 feature
In addition to conventional transport measurements, the 0.7 structure has also been observed in studies of the thermopower, thermal conductance, and shot noise experiments. While not comprehensive, the following section briefly reviews these experiments. The thermopower of a 1D system, S, is related to the conductance, G: S¼
V 2 kB2 qð1nGÞ ¼ – ðTe þ Tl Þ Te – Tl I ¼0 qm 3e
by the Mott formula (Mott, 1963), where is the chemical potential of the contacts relative to the 1D subbands, Te is the electron temperature, Tl is the lattice temperature, and V is the voltage measured across the constriction. Initial thermopower experiments used a 1D constriction to measure the electron temperature (Appleyard et al., 1998). Here, electrons were heated on one side of a 1D constriction by the application of an electric current. However, the hot electrons are prevented from passing through to the other side by the 1D constriction and a temperature difference is established across the channel. This
results in a potential difference that can be measured. The authors showed that, in a ballistic 1D system, the steps in the conductance between the quantized values gave rise to peaks in the measured thermopower. By measuring the height of the thermopower peaks, it was possible to determine the temperature of the hot electrons in the heating channel. Appleyard et al. (2000) also went on to show the presence of a small shoulder in the thermopower near the 0.7 structure. Application of a magnetic field converted the 0.7(2e2/h) to the spin polarized 0.5(2e2/h) value, accompanied by a decline in the thermopower signal to zero. The observed thermopower signal at 0.7(2e2/h) is consistent with the first subband being split, with the spin# subband being transmitted, and the other pinned near the chemical potential. Since a fully transmitted channel leads to a zero thermopower, the 0.7 structure could be explained by partial transmission of the minority spin" subband. While the Mott formulation does not predict a peak in G at 0.7(2e2/h), it is likely that in the regime of strongly interacting electrons an additional mechanism is required to explain the finite measure thermopower coincident with 0.7 structure. The thermal conductance, , and electrical conductance, G, of noninteracting electrons in quantum wires are known to be related by the Wiedemann– Franz equation K ¼ –
2 kB2 T G 3e 2
As a consequence, if the conductance is quantized in units of 2e2/h, then the thermal conductance will be quantized in units of 2kB2 T =3h. Below 2e2/h, the Wiedemann–Franz equation no longer holds, but at gate voltages near the 0.7 structure, a half plateau in the thermal conductance was observed by Chiatta et al. (2006). Finally, shot noise experiments in a QPC in the regime of the 0.7 structure were also performed. Shot noise is the temporal fluctuation of current resulting from the quantization of charge. DiCarlo et al. (2006) reported simultaneous measurements of the shot noise and DC transport in a QPC. In particular, they focused on the 0.7 structure and its evolution in a parallel magnetic field. They observed a suppression of the noise, compared with that expected for spin degenerate transport, near 0.7(2e2/h) in zero magnetic field in agreement with earlier results by Roche et al. (2004). With increasing Bk, this suppression was observed to evolve into the signature of a spin-resolved transmission. They found
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
quantitatively good agreement between their noise data and the phenomenological model for densitydependent level splitting (Reilly, 2005). Since the first reported observation of the 0.7 structure, its origin has been intensely debated. Throughout the deliberations, both Coulomb and exchange energies have been shown to be very important in quasi-1D systems since they can cause a spontaneous spin polarization. Indeed, the concept of a spin gap has been fairly successful in explaining most of the experimental results of the 0.7 structure (including the temperature dependence, magnetic field, and bias dependence) (Wang and Berggren, 1996; Spivak and Zhou, 2000; Kristensen et al., 2000; Reilly et al., 2002; Berggren and Yakimenko, 2002; Graham et al., 2005; Lassl et al., 2007; Graham et al., 2007). Central to this interpretation is the concept that the spin gap must depend on carrier density (gate voltage), otherwise a quantized plateau at 0.5(2e2/h) would be observed. To date, however, none of the SG models have addressed the appearance of a peak in the differential conductance of the quantum wire for G < 2e2/h at very low temperatures (Cronenwett et al., 2002). This is discussed in the next section. 1.08.2.6 The Kondo Model and Zero-Bias Anomaly 1.08.2.6.1
Overview Although a spontaneous spin-splitting model explains many of the salient features of the 0.7 structure, there sometimes exists a peak in the conductance around Vsd ¼ 0 that cannot be explained within the spin-split paradigm. This conduction peak has been termed the zero-bias anomaly, in analogy to the zero-bias anomaly observed in quantum-dots. The zero-bias anomaly has led to a Kondo-like description of the 0.7 structure that also describes many of its properties – in particular, its unusual temperature dependence. The Kondo model predicts that near pinch-off (Meir, 2008), a 1D system contains a bound state that acts as a magnetic impurity. Without electron–electron interactions, the bound state would block carriers of the same spin from passing through the 1D channel – limiting transport to a single spin channel and reducing the conductivity to 0.5(2e2/h). However, electron–electron interactions cause the bound state to polarize electrons in the leads and thereby screen its effect. At T ¼ 0, the magnetic impurity is screened perfectly by the carriers in the leads and the conductivity reaches the unitary limit, 2e2/h. As the temperature is increased, the magnetic impurity is
305
less efficiently screened, and the conductance is reduced toward its high-temperature limit of 0.5(2e2/ h). This gives rise to the 0.7 structure and provides an explanation for why the plateau at 0.7(2e2/h) is stabilized by temperature. A critical assumption of the Kondo model is that an open 1D system is able to support a bound state. This assumption appears to be robust according to recent spin-density functional theory calculations (Meir et al., 2002; Hirose et al., 2003; Rejec and Meir, 2006) and tunneling spectroscopy measurements between parallel quantum wires (Auslaender et al., 2005; Yoon et al., 2007). From this assumption it follows that (Meir, 2008): 1. The bound state will blockade the current, resulting in a reduced conductance between 0.5(2e2/h) and 2e2/h. 2. As the Fermi level is increased, the Coulomb blockade energy is overcome and the universal value of 2e2/h is recovered. 3. In the presence of a magnetic field, the energy of the localized state splits and the conductance is reduced to 0.5(2e2/h). 4. There is a reduction in shot noise around the 0.7 plateau due to charge being carried by two spin channels, one of perfect transmission and the other with reduced transmission. These predictions are consistent with experimental observations. However, the greatest strength of the Kondo model is arguably its ability to explain the zero-bias anomaly. The zero-bias anomaly was originally studied by Cronenwett et al. (2002), who showed that the 1D zero-bias anomaly shared many of the features of its 0D counterpart including: 1. the collapse of G(T) curves onto a single function with a single scaling parameter designated the Kondo temperature, TK; 2. an exponential dependence of TK on VG; 3. a correlation between zero-bias peak width and TK; and 4. splitting of the zero-bias peak in a parallel magnetic field. Subsequent experiments have confirmed the existence of the zero-bias anomaly as a fundamental property of quantum wires but have found less correspondence with the 0D zero-bias anomaly (Sfigakis et al., 2008a; Sarkozy et al., 2009). In particular, the 1D zero-bias anomaly does not always split with parallel
306 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
magnetic field and the observed splitting is often less than predicted (Chen et al., 2009). Today, there is a growing consensus that bound states can and do exist in quantum wires resulting in Kondo-like physics. However, whether or not this is the origin of the 0.7 structure is still a point of contention as there is mounting evidence to suggest that the 0.7 structure and Kondo-like phenomena co-exist as separate and distinct effects (Sfigakis et al., 2008a). 1.08.2.6.2 Experimental and theoretical evidence for a Kondo model
The detailed studies of the zero-bias anomaly performed by Cronenwett et al. (2002) preceded a concrete theory of Kondo-like physics in 1D and spin-density functional theory calculations demonstrating a stable bound state in QPCs (Meir et al., 2002; Hirose et al., 2003; Rejec and Meir, 2006; Meir, 2008). Cronenwett et al. used surface gate depletion to define a 1D channel in a 2DES and performed source–drain bias measurements on the first three subbands. Figure 25(a) shows the nonlinear differential conductance as a function of source–drain bias for a range of surface gate voltages at base temperature and zero magnetic field. Bunching of the lines along Vsd ¼ 0 indicates the familiar plateaus in the linear conductance at 2e2/h and 2(2e2/h). These plateaus split to form half-plateaus at 1.5(2e2/h) and
3
2.5(2e2/h) that trend upward as Vsd is shifted away from zero (Glazman and Raikh, 1988; Ng and Lee, 1988; Patel et al., 1991b). Below 2e2/h, however, they observed a lot more structure in the nonlinear conductance. The data showed a clear zero-bias peak – the so-called zerobias anomaly. Importantly, there was no bunching of lines along Vsd ¼ 0 and hence, no plateau at 0.7(2e2/h) was observed in the linear conductance. At this low temperature, the zero-bias anomaly appeared to enhance the conductance at Vsd ¼ 0 and wash out the 0.7 structure. Nonetheless, as Vsd was moved away from zero, two distinct plateaus formed at 0.8(2e2/h) and 0.2(2e2/h), which are believed to result from the splitting of spin bands. As the temperature was increased to 600 mK, the zero-bias peak was suppressed as shown in Figure 25(b). Now, a clear bunching of lines at 0.7(2e2/h) around Vsd ¼ 0 is observed. This is a strong indication that the formation of the 0.7 structure coincides with the destruction of the zero-bias anomaly with temperature. The strong correlation between the zero-bias anomaly and 0.7 structure was further exemplified at high-parallel magnetic field. Figure 25(c) shows source–drain bias measurements performed at base temperature and Bk ¼ 8 T. In this case, the zero-bias peak was completely destroyed and the spin-split plateau
(a) T = 80 mK, Bll = 0 T (b) T = 600 mK, Bll = 0 T (c) T = 80 mK, Bll = 8 T 5/2 5/2 2
G (2e2/h)
2
2 3/2 3/2
1
1
1
1
0.8 0.7 1/2 1/2
0 –1
0 Vsd (mV)
1
–1
0 Vsd (mV)
1
–1
0 Vsd (mV)
1
Figure 25 Nonlinear differential conductance as a function of source–drain bias at various surface gate voltages for the first three subbands of a quantum wire. (a) At base temperature and zero parallel magnetic field, there is a strong zero-bias peak in the nonlinear conductance below (2e2/h) – the zero-bias anomaly. (b) Raising the temperature suppresses the zero-bias anomaly and reveals a bunching of lines near 0.7(2e2/h) due to the 0.7 structure. (c) At base temperature and B|| ¼ 8 T, the zero-bias anomaly has vanished and there is a strong spin-split plateau at 0.5(2e2/h). These results demonstrate the intimate relationship between the 0.7 structure and the zero-bias anomaly. Modified with permission from Cronenwett SM, Lynch HJ, Goldhaber-Gordon D, et al. (2002) Low-temperature fate of the 0.7 structure in a point contact: A Kondo-like correlated state in an open system. Physical Review Letters 88: 226805. Copyright (2002) by the American Physical Society.
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
at 0.5(2e2/h) is clearly visible. Again, the appearance of the spin-split subband coincides with the destruction of the zero-bias anomaly with magnetic field. Cronenwett et al. went on to study the temperature dependence of the zero-bias anomaly in detail and showed important similarities between the 1D zero-bias anomaly and Kondo physics observed in quantum-dots (Goldhaber-Gordon et al., 1998; Cronenwett et al., 1998; van der Wiel et al., 2000). Figure 26(a) shows the temperature dependence of zero-bias peaks for five different gate voltages. Plotting the peak height as a function of temperature over a range of gate voltages produced the curves shown in the inset of Figure 26(b). Cronenwett et al. found that these curves could be scaled onto one function using a single scaling parameter TK as shown in the main panel of Figure 26(b). The functional form of the scaled curve agrees with a modified expression of the Kondo conductance:
TK scaled exponentially with gate voltage as shown in Figure 26(c). This is also consistent with TK in quantum-dots, which depends exponentially on the energy of the bound spin ("0) relative to the Fermi energy in the leads ("0 þ U) according to "0 ð"0 þ U Þ TK exp GU
where G is the energy broadening (Haldane, 1978). Moreover, Cronenwett et al. found that the zerobias peak splits with magnetic field and that between 0.7(2e2/h) and 2e2/h, the peak widths increase like 2kTK/e. Both of these observations are consistent with Kondo physics in quantum-dots. However, more recent experiments have shown that the zerobias peak does not always split with magnetic field and that any observed splitting is significantly less than predicted (Chen et al., 2009). Sarkozy et al. (2009) also performed a detailed study of the zero-bias anomaly as a function of magnetic field and temperature and found further discrepancies between the behavior of the zero-bias anomaly and 1D Kondo physics model, namely:
2e 2 1 1 þ G¼ h 2f ðT =TK Þ 2
1. a nonmonotonic increase of TK with surface gate voltage; 2. the FWHM of the zero-bias peak did not scale with Gmax of the peak; and 3. a linear peak splitting with gate voltage at fixed magnetic field.
where f(T/TK) is a universal function for the Kondo conductance. This provided compelling evidence for a Kondo explanation of the zero-bias anomaly and hence the 0.7 structure. Moreover, it was found that (a)
(b) 320 mK 430 mK 560 mK 670 mK
1 10
0.8 0.7 0.6
0.5
0 –1
0.5 0 Vsd (mV)
1
–466 mV
1.0
G (2e2/h)
0.5
G (2e2/h)
0.9 G(2e2/h)
80 mK 100 mK 210 mK
(c) 1.0
1
TK (K)
G(2e2/h)
1.0
307
–488 mV 0.1
0.01
1
T (K)
0.1
1 T/Tk
0 10
0.1 –492
–462 Vg
Figure 26 Temperature dependence data of the zero-bias anomaly by (Cronenwett et al. 2002). (a) Temperature dependence of the zero-bias anomaly at five different surface-gate voltages. As the temperature is increased from 80 to 670 mK, the zero-bias anomaly is destroyed. (b) Zero-bias anomaly peak conductance as a function of temperature for various gate voltages and scaled onto a single functional form using the scaling parameter TK. The inset shows the unscaled data. (c) TK as a function of surface gate voltage, Vg, showing an exponential correspondence between TK and Vg. The plot also shows the linear conductance of the first subband as a function of Vg for 80 mK (solid), 210 mK (dotted), 560 mK (dashed), and 1.6 K (dot-dashed) showing the emergance of the 0.7 structure as the zero-bias anomaly is suppressed.
308 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
While these discrepancies may suggest that the 1D Kondo model is incomplete, the model is still more effective at describing the zero-bias anomaly compared to a spin-split model, which cannot account for the occurrence of the zero-bias anomaly. Guided by the strong experimental evidence, Meir and co-workers developed a comprehensive model for the 0.7 structure based on Kondo physics (Meir et al., 2002; Hirose et al., 2003; Rejec and Meir, 2006; Meir, 2008). A key prerequisite of the theory is the formation of a quasi-bound state in the QPC. This localized state is generated by multiple reflections between the edges of a potential barrier. In the case of a square potential barrier, a particle with higher energy can be reflected as it enters and exits the barrier. Resonant reflections between these two interfaces lead to a quasi-bound state localized within the barrier region. Interestingly, spin-density functional theory calculations indicate that this remains true even for the smooth potential of a QPC, thereby producing a narrow quasi-bound state that may act as a single magnetic impurity. Although the calculations by Meir et al. were perturbative and therefore only semi-quantitative,
1.0
they provided a good qualitative description of many of the important features of the 0.7 structure. Figure 27(a) shows the conductance of the first subband as a function of the Fermi energy ("F) relative to the energy of the bound state ("0) calculated at four different temperatures (Meir et al., 2002). As the temperature increases, a distinct plateau forms at 0.7(2e2/h), in excellent agreement with the experimental data shown in the inset of Figure 27(a). The calculations also show the formation of a strong plateau with increasing parallel magnetic field as shown in Figure 27(b). While this is in qualitative agreement with experimental data shown in the inset, the position of the calculated plateau is significantly lower than that observed experimentally. However, Meir et al. (2002) attribute this quantitative disagreement to the perturbative approach used in their calculations rather than a fundamental difference in the underlying physical mechanism. The Kondo model also provides a very good description of the evolution of the zero-bias anomaly as a function of temperature and magnetic field as shown in Figure 28(a) and Figure 28(b),
(a) Data
1.0 Conductance (2e2/h)
0.5 0.5 VG 1.0
(b) Data 1.0
0.5 0.5 VG 0.0
0.0
0.2
0.4
0.6
0.8
εF/lε0l Figure 27 Linear conductance of the first subband as a function of the Fermi energy "F relative to the energy of the quasi-bound state "0. The calculations by Meir et al. are based on spin-density functional theory calculations of a Kondo model (Meir, 2002). (a) The calculated conductance at 50, 100, 200 and 600 mK. The results agree both qualitatively and quantitatively with the experimental observations of Cronenwett et al. shown in the inset (Cronenwett 2002). (b) The calculated conductance of the first subband in parallel magnetic fields of 0 T, 0.07, 0.12, and 0.4 T (from top to bottom, respectively). The calculations agree qualitatively with experimental observations shown in the inset, and quantitative discrepancies are attributed to the perturbative nature of the calculations. Reproduced with permission from Meir Y, Hinose K, and Wingreen NS (2002) Kondo model for the ‘‘0.7 anomaly’’ in transport through a quantum point. Physical Review Letters 89: 196802 Copyright (2002) by the American Physical Society.
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
1.0
1.0 Data 1.0
0.5 0.5
Data
0.5 1.0 0.5
V
V 0.0 −0.5 Bias voltage V/lε0l
(b) Conductance (2e2/h)
Conductance (2e2/h)
(a)
0.0
309
0.0
0.5
−0.5
0.0 0.5 Bias voltage V/lε0l
0
Figure 28 Calculations of the nonlinear conductance and zero-bias anomaly based on spin-density functional theory calculation of a Kondo model (Meir, 2002). (a) Calculated zero-bias anomaly at for Fermi energies "F ¼ 0.1, 0.03, and 0.01 from top to bottom respectively at four different temperatures T ¼ 60, 100, 200, and 400 mK. The calculations agree both quantitatively and qualitatively by the experimental data taken by Cronenwett, shown in the inset (Cronenwett, 2002). (b) Zero-bias anomaly near the 0.7 structure for different parallel magnetic fields corresponding to a Zeeman splitting of ¼ 0, 0.04, 0.07, and 0.1, where ¼ 0.04. Again, the data agree with the splitting of the zero-bias anomaly peak observed experimentally shown in the inset (Cronenwett, 2002).
respectively. Through this work, combined with experimental studies, there is growing consensus that 1D systems (1.) can support a quasi-bound state, (2.) do display Kondo-like physics, and (3.) the zero-bias anomaly and 0.7 structure are intimately related phenomena based on spin. However, it does not necessarily follow that the 0.7 structure has its origins in Kondo-like physics. There is evidence to suggest that while the zero-bias anomaly and 0.7 structure coexist, their underlying mechanisms are separate and distinct. Using etched channels Sfigakis et al. (2008a) formed open quantum-dot structures that displayed enhanced Kondo behavior while retaining many of the features of the 0.7 structure . These 0D–1D hybrid structures allowed Sfigakis et al. to study the temperature dependence of the 0.7 structure and zero-bias anomaly independently. They found that the two features appeared at different energy scales with the 0.7 structure appearing after the zero-bias anomaly had been destroyed by increasing temperature. In summary the origin of the 0.7 structure remains one of the most intriguing and challenging puzzles in the field of 1D ballistic transport. Many of its properties indicate the presence of spontaneous spin polarization – such as its behavior in magnetic field and source–drain bias and density dependences. However, other properties such as the zero-bias conductance peak and temperature dependence are more commensurate with a Kondo-like model. These conflicting properties suggest that the physics is more complicated than first thought and
that perhaps two separate and distinct phenomena exist near 0.7(2e2/h) concomitantly. In the next section, we review the experiments performed in p-type 1D systems. The unique properties of the valence band provide a different system to probe the 0.7 structure.
1.08.3 Ballistic Transport in p-Type GaAs-Based 1D Systems 1.08.3.1
Introduction
Studying ballistic transport in p-type 1D systems brings with it the advantages of a larger effective mass, strong spin–orbit coupling, and spin 3/2 compared with spin 1/2 for electrons. The large effective hole mass m ¼ 0.2 – 0.4me leads to enhanced manypffiffiffiffi body interactions as parametrized by rs _ m = ns . While 2D electron systems are limited to rs < 5, hole systems can readily achieve rs >> 10. This larger value of rs allows a study of the importance of interaction effects in 1D. The strong spin–orbit coupling makes it possible to manipulate the hole spin with an applied electric field. In addition, the total angular momentum j ¼ 3/2 gives low-dimensional hole systems a much richer spin physics than their electron counterparts (Winkler, 2003). The strong spin–orbit coupling and anisotropic Zeeman splitting in these systems also give important insights into the origin of the 0.7 structure and zero-bias anomaly. The study of hole quantum wires in GaAs is still relatively new
310 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
compared to the broad spectrum of literature on 1D electron systems. However, their complex spin properties and ability to be tuned with external electric and magnetic fields allow unique insights into the spin properties of 1D systems. Progress in the study of 1D GaAs hole systems has been slower for two main reasons. First, the higher effective mass in p-type systems results in smaller subband spacings that require lower temperatures to resolve clear conductance quantization. Second, and most important, fabricating stable ballistic p-type hole wires has proved to be significantly more challenging than is the case for n-type electron wires (Zailer et al., 1994; Daneshvar et al., 1997; Rokhinson et al., 2006). However, recent developments in device fabrication technology and high-purity heterostructures have allowed the fabrication of 1D hole systems that demonstrate clear conductance quantization. This has led to a rapid development in our understanding of both single-particle and many-body phenomena in ballistic 1D channels. In this section, we review the importance of the hole band structure before going on to discuss how different fabrication techniques have allowed progressively more stable and reliable 1D hole systems in Section 1.08.3.2. In Section 1.08.3.3 we review the observation of ballistic transport in 1D hole wires, including measurements of the subband spacing, Zeeman splitting, and measurements of the g-factor anisotropy. Finally, in Section 1.08.3.4 we
(a)
E
discuss the 0.7 structure and the zero-bias anomaly in the context of p-type systems.
1.08.3.2
The structure of the valence band in bulk GaAs systems is more complex than the simple parabolic conduction band for electrons (see Figure 29). In the conduction band, transport is described well by the effective mass approximation due to the fact that the conduction band is, to a good approximation, parabolic and therefore d2E(k)/dk2 is a constant. In contrast, the definition of m in the valence band is complicated by the fact that the bands are nonparabolic and anisotropic around k ¼ 0 (Winkler, 2003). Furthermore, the curvature of the valence band is negative, resulting a negative m. This concept is nonphysical and so we treat these carriers as positively charged particles with a positive mass. Conduction band electrons originate from s-like l ¼ 0 states with spin 1/2, with a quantized projection in any direction of ms ¼ h/2. In contrast, there are three valence band states. In a tight-binding picture, these band states are derived from the three p-orbitals px, py, and pz, and therefore have an orbital angular momentum l ¼ 1. The spin–orbit interaction lifts the sixfold degeneracy of the valence band states at k ¼ 0 leading to a splitting of the valence band states to give four degenerate states of total angular momentum j ¼ 3/2, separated from two degenerate
(b)
Conduction band
E
Conduction band
Eg
K
Eg
Valance band Δ0
The Hole Band Structure
K Valance band
HH ±3/2
Δlh-hh
HH ±3/2
j = 3/2 LH ±1/2
SO
LH ±1/2
}
j = 1/2
SO
Figure 29 (a) Schematic band diagram for bulk GaAs, showing the split-off (SO), light-hole (LH), heavy-hole (HH), and conduction bands. (b) Schematic band diagram for a two-dimensional (2D) hole system (2D HS) where the HH–LH degeneracy is lifted at k ¼ 0 due to the 2D confinement.
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
states of lower energy with j ¼ 1/2. The doubly degenerate states are known as the spilt-off band and have m ¼ 0.17 (Kittel, 1996). The fourfold degenerate states form two band masses, heavy holes (m ¼ 0.5me) with z-component of the angular momentum jz ¼ 3/2 and light holes (m ¼ 0.087me) with jz ¼ 1/2. Confining the GaAs system to 2D lifts the fourfold degeneracy of the heavy hole (HH) and light hole (LH) bands, as depicted Figure 29(b), so that the HH (mJ ¼ 3/2) band becomes the highest energy hole band. Two ladders of energy states are formed as a result of the different effective masses of the bands (Winkler and Nesvizhskii, 1996). However, much of this substructure is hidden from magneto-transport measurements of 2DHSs, which are described primarily by transport through only the highest energy band – typically HH1. In these systems, the Fermi energy is small (EF 1 meV), significantly less than the light-hole heavy-hole splitting (lhhh 10 meV), so that only the heavy-hole band is occupied. Eisenstein et al. (1984) showed that, in 2D asymmetric quantum-wells, a lifting of the spin degeneracy of the HH1 band results in two effective masses, which appear as two distinct frequencies in Shubnikov de Haas oscillations. In contrast, the states in symmetric, square quantum-wells remain doubly degenerate and only one effective mass is present. Even so, assigning an effective mass to 2D GaAs hole systems is nontrivial due to the fact that m is dependent on both the symmetry of the device (Eisenstein et al., 1984; Iye et al., 1986; Pan et al., 2003) and the density (Pan et al., 2003; Noh, 2005). Historically, transport measurements of 2DHSs have been performed on silicon-doped GaAsAlGaAs heterostructures grown on a (311)A-oriented GaAs substrate. The hole mobility in these structures depends on the direction of the current path and is at a maximum for currents parallel to the [233] direction (the fast direction). Between this direction and the [011] direction (the slow direction), the mobility may change as much as 20% (Davies, 1991; Heremans and Santos, 1994). This anisotropy must also be taken into account when calculating transport properties. Recent developments in MBE have allowed the growth of high-quality p-type heterostructures on (100)-oriented GaAs substrates. However, their 1D transport properties are largely unexplored. Further confining the valence band to just one dimension causes a mixing of the HH and LH states
311
(Zu¨licke, 2006). We can consider two limiting case. In the case of very narrow wires, the mj ¼ 1/2 light-hole band becomes higher in energy than the mj ¼ 3/2 heavy-hole band. The other limit is where the 2D confinement from the quantum-well is stronger than the lateral quantum wire confinement. In this case, the ground state remains predominantly mj ¼ 3/2 HH-like. This has important implications for how the holes couple with an external magnetic field as the confinement strength varies. Figure 30 shows the calculated change in g-factor during the 2D-to-1D cross-over for a square wire parallel to the [233]crystallographic axis. In this figure, Wz/Wx is the ratio of the height of the quantum wire (Wz) relative to the width Wx. In the 2D limit, the g-factor for the low-symmetry [311] GaAs is also highly anisotropic. The holes are strongly coupled to an external magnetic field perpendicular to the [311] plane with gz 7. In contrast, holes coupled to a magnetic field in the plane and parallel to the wire have gy 0.6. The holes are even more weakly coupled to a magnetic field in the plane and perpendicular to the wire, with gx 0.2. 1D limit
2D limit 7 6 5 g ∗x,y,z 4 3 2 1 0
g∗z
z x
g∗x
y g∗y
0
0.2
0.4 0.6 Wz / Wx
0.8
1
Figure 30 Calculated g-factors for a rectangular hole quantum wire oriented along [233] as a function of onedimensional (1D) confinement.¯ The g-factors gx and gy correspond to in-plane magnetic fields orthogonal and parallel to the wire, respectively, while gz corresponds to magnetic fields perpendicular to the plane. The 1D confinement is characterized by the ratio of the quantumwell height (Wz) to the width of the wire (Wx). When Wz/Wx ¼ 0, the system is in the 2D limit but tends towards the 1D limit as Wz/Wx ! 1. The calculations predict that increasing the 1D confinement will reduce the out-of-plane g-factor while simultaneously increasing the inplane g-factors, gx and gy. The 1D confinement also affects the g-factor anisotropy. Below Wz/Wx ¼ 0.4 spins couple most strongly to fields parallel to the wire. However, when Wz/Wx > 0.4, fields orthogonal to the wire couple more strongly. From Zu¨licke U (2006) Electronic and spin properties of hole point contacts. Physica Status Solidi (c) 3: 4354.
312 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
1.08.3.3 Fabricating Stable 1D Hole Systems The enhanced m in the valence band implies that resolving conductance plateaus in p-type systems is intrinsically more difficult than for n-type systems. The problem is twofold. First, increasing m reduces the subband energy spacing. As a result, p-type systems must be measured at much lower temperatures to remove the effect of thermal broadening, which smears out the conductance plateaus. Second, increasing m reduces the Fermi velocity and, hence, the mobility of 2DHSs. This makes it more difficult to achieve conduction at low carrier densities, which, in turn, limits the Fermi wavelength. As a result, p-type quantum wires must be made thinner and shorter before conductance quantization can be observed. This has led to ballistic hole transport to be studied primarily in QPCs, that is, quantum wires in which the length tends to zero. However, the need for low measurement temperatures and small quantum wires does not put a practical limit on our ability to fabricate and measure 1D hole systems. Instead, the main hurdle has been obtaining stable currents through the 1D channel. Recently, there has been as significant improvement in the stability of p-type quantum wires, which has opened up the possibility for more detailed studies of the single-particle and many-body phenomena in these systems. As a general rule, the stability of 1D hole systems improves with the mobility of the 2DHS from which they are formed. However, there is increasing evidence to suggest that limiting the scattering caused by remote ionized dopants also improves device stability (Hamilton et al., 2008).
1.08.3.3.1
Etched quantum wires The first observation of conductance quantization in a 1D hole system was made by Zailer et al. (1994). Original data from this work and a schematic of the device structure are shown in Figure 31. In these experiments, Zailer et al. used wet-etching to define two parallel wires in a 2DHS of peak mobility 320 000 cm2 V1 s1 at 50 mK. Using electron beam lithography, Zailer et al. were able to etch two 300nm-wide trenches separated by a 1100-nm gap. A 300-nm dot was then etched between the two trenches to divide the gap into two, 400-nm wide 1D channels. By depositing metal into the etched regions, Zailer et al. formed self-aligned depletion gates that could be used to tune the width of each wire independently. By using the depletion gates to completely pinch off one channel and vary the width of the other channel, Zailer et al. were able to explore the current quantization through a single channel. The wet-etch technique has been used successfully to fabricate n-type quantum wires with very large subband spacings and high stability. However, Zailer et al. found that the current through their p-type devices suffered from random telegraph
8 B = 0T 6 G (e2/h)
The data in Figure 30 show that the g-factors and the g-factor anisotropy change as the hole system approaches the 1D limit. This occurs because the confinement reduces the separation between the HH and LH subbands and allows mixing of these two bands. Most importantly, we see that when Wx > 2Wz, there is a cross-over of the in-plane anisotropy, that is, gx > gy. This cross-over is a result of the LH subband becoming the highest-energy subband and therefore the subband that dominates the transport properties. It is worth noting that here the traditional nomenclature becomes confused as the holes occupying the light-hole subband (characterized by jz ¼ 1/2), in fact, have the greater effective mass in this regime.
Split ga
te
1DHS
1DHS
Split ga
4
te
2
B = 3T
0.4
0.6
0.8
1
1.2
Vg (V) Figure 31 Original data from Zailer et al. showing current quantization in etched one-dimensional (1D) hole systems after averaging over 40 traces. The application of a 3T magnetic field splits the first subband into well-defined steps at e2/h and 2e2/h (Zailer, 1994). (Inset) Schematic of the etched quantum wire device used by Zailer et al. Etched regions define two parallel one-dimensional hole systems (1DHSs), while metal deposited into the etch regions as side gates that can independently control the current through the 1D channels. With permission from, Zailer I, Frost JEF, Ford CJB, et al. (1994) phase coherence, interference and conductance quantization in a confined two-dimensional hole gas. Physical Review B 49: 5101. Copyright (1994) by the American Physical Society.
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
signals. As a result they were forced to average data taken over 40 traces. Nonetheless, they were able to resolve conductance plateaus at G ¼ 2e2/h and 3(2e2/h).
313
12
10
1.08.3.3.2 Surface gate-depleted quantum wires
1.08.3.3.3
Local anodic oxidation In the five years that followed, there was very little work done in this field. However, Rokhinson et al. (2002) fabricated a 1D channel that showed clear but irregular conductance plateaus up to 9(2e2/h). A schematic of the device structure and data from this work are shown in Figure 33. In this device, local anodic oxidation was used to form a 1D constriction in a 2DHS situated just 35 nm from surface. The local anodic oxidation technique uses a biased atomic force microscope (AFM) tip to locally induce an oxidation reaction between the GaAs and native water film. In this way, the AFM tip can be used to oxidize lines in the GaAs substrate down to 10 nm resolution. Rokhinson et al. used the local anodic oxidation technique to pattern their 2DHS into three separate regions. The central
8
G (e2/h)
Daneshvar et al. (1997) fabricated p-type QPCs from a 2D system with a higher mobility: ¼ 1 200 000 cm2 V1 s1 at 1.5 K. These devices used the surface gate-depletion method to define the 1D channel – a method that has also been used with great success in n-type systems. A schematic of the device structure and original data from Daneshvar et al. is shown in Figure 32. Three metal gates were patterned on the surface of the GaAs heterostructure – two SGs separated by a small distance and a third gate that runs between them. Applying a positive bias to the SGs locally depletes the 2DHS directly beneath them and forms a 1D channel. Increasing the bias on the SG constricts and eventually pinches off the 1D channel. The middle gate can be used to vary the density in the 1D channel without affecting the width of the channel. Similar techniques are used routinely in n-type systems to define 1D electron systems with high stability and clear conductance quantization (cf. van Wees et al., 1988). In contrast, Daneshvar et al. were only able to resolve inflections in the conductance up to 5(2e2/h). The ballistic transport was facilitated by the application of an in-plane magnetic field, which lengthened the inflections to form well-defined plateaus and provided a good estimate of the 1D g-factor as a function of subband index.
6 B=0
4
2 B⊥ = 0.7T
0
2.5
2.75
Side gate voltage (V) Figure 32 Original data from Daneshvar et al. showing quantized inflections in the conductance up to 5(2e2/h) in a surface gate-depletion quantum wire (Daneshvar, 1997). A small magnetic field enhances the ballistic transport through the device to produce well-defined and wellquantized conductance plateau. (Inset) Schematic of the surface gate-depletion device architecture used by Daneshvar et al. on a single-layer hole system. The two side gates define and confine the one-dimensional (1D) channel while the middle gate controls the density in the wire. Reproduced with permission from Daneshvar AJ, Ford CJB, Hamilton AR, simnons MY, Pepper M, and Ritchie DA (1997) Enhanced g factor of a one-dimensional hole gas with quantized conductance. Physical Review B 55: R 13409. Copyright (1997) by the American Physical Society.
region formed the source and drain that taper down to the 1D constriction. The 2DHSs either side of the constriction then acts as in-plane gates. Applying a positive bias to these 2DHSs allows the 1D channel to be confined and even completely pinched off. However, the 2D system must be kept close to the surface because the oxidation does not penetrate deeply into the heterostructure. Due to the close proximity to the surface, the mobility of the 2DHS ( ¼ 500 000 cm2 V1 s1 at 0.3 K) was less than half that of Daneshvar et al. Nonetheless, Rokhinson et al. (2004, 2006) used this technique to fabricate 1D hole systems that show well-defined and stable
314 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
10 10
G (2e2/h)
8
In-plane
gate
1 DHS
In-plane
G (2e2/h)
8
gate
6
6 4 2
4
0
2
2.8
3.2
3.6
4.0
Split gate bias (V) 0 –0.8
–0.6 –0.4 –0.2
0.0
0.2
0.4
0.6
Vg1 = Vg2 (V) Figure 33 Original data from Rokhinson et al. showing irregular conductance quantization plateaus up to 9(2e2/h). (Inset) Schematic of the device architecture used by Rokhinson et al. Using local anodic oxidation, a shallow two-dimensional hole system (2DHS) is pattern into three separate regions. The middle region forms the onedimensional (1D) channel while the 2DHSs either side of the channel act as in-plane gates to control the current through the channel. From Rokhinson LP, Tsui DC, Pfeiffer LN, and West KW (2002) AFM local oxidation nanopatterning of a high-mobility shallow 2D hole gas. Superlattices and Microstructures 32: 99.
conductance plateaus at 2e2/h and 2(2e2/h) and have even used these devices to perform systematic studies of the 0.7 structure and the zero-bias anomaly. They have also extended the technique to develop spin filters, which have revealed important information about the origins of the 0.7 structure and are discussed in Section 1.08.3.4.3.
1.08.3.3.4
Bilayer quantum wires In 2006, Danneau et al. applied the surface gatedepletion method to form two hole wires in a modulation-doped double quantum-well. In these heterostructures, two sheets of high-mobility holes are formed in 20–nm-wide GaAs quantum-wells separated by a 30-nm AlGaAs barrier. The barrier is sufficiently thick so that there is no tunneling between the two 2D hole layers. The as-grown hole densities in the two wells were 1 1011 cm2 with peak mobilities in excess of , 106 cm2 V1 s1. Ohmic contacts were made to both 2DHSs in parallel, and an overall back-gate was used to control the hole density in the lower quantum-well. Two parallel quantum wires are defined along the [233] direction by three surface
Figure 34 Conductance quantization observed in a bilayer one-dimensional hole system (1DHS) by Danneau et al. (Inset) Schematic of the surface gate-depletion architecture used by Danneau et al. to define the quantum wires in the bilayer 2DHS. This device was fabricated on a metallically doped substrate that could be used as an in situ back-gate to vary the density in the lower layer. From Hamilton AR, Danneau R, Klochan O, et al. (2008) The 0.7 anomaly in one-dimensional hole quantum wires. Journal of Physics: Condensed Matter 20: 164205.
gates, as shown in Figure 34. A combination of front- and back-gate biases allows the transport properties of the top or bottom quantum wire to be measured independently. A positive bias applied to the two side gates defines the 1D channels in the two quantum-wells. The central top-gate is used to control the density in the upper quantum wire, and the back-gate controls the density in the lower quantum wire. Thus, the top quantum wire can be measured in isolation by depleting the lower wire with the back-gate. In addition, a forth surface gate (not shown) could be used to stop the current flow in the top quantum wire without it being depleted. This allows the lower quantum wire to be measured without having to deplete the upper wire, and can be used to measure the compressibility of 1D systems (Danneau et al., 2006a, 2006b). Using this bilayer heterostructure, Danneau et al. were able to form highly stable and reproducible conductance quantization for the first 11 subbands and a strong 0.7 structure. Figure 34 shows the conductance of the quantum wire in the upper quantum-well as a function of the side-gate bias without any of the resonances that hampered previous studies of lower-mobility hole devices. The bilayer device design has been particularly successful. Danneau et al. subsequently performed
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
1.08.3.3.5 Electrostatically induced quantum wires
Klochan et al. removed the need for dopants by electrostatically inducing a 1D hole system in a GaAs SISFET (Clarke et al., 2006; Klochan et al., 2006) (see Figure 35). This is simply an induced single heterojunction in which the holes are introduced by a negative bias applied to a degenerately doped pþ-GaAs gate electrode, instead of by modulation doping the AlGaAs layer. This approach avoids unwanted scattering from remote ionized impurities. The holes are confined in a triangular potential well, with densities from 1.6 1010 to 1.9 1011 cm2 and mobilities up to 700 000 cm2 V1 s1. To define a quantum wire, the pþ-GaAs layer is divided into three electrically separate gates using electron beam lithography and a shallow wet etch. The hole density in the
e
8 6 G (2e2/h)
detailed studies of the subband energy spacings and g-factor anisotropy in their bilayer 1D hole systems. In addition, the enhanced stability of these devices allowed for the first time an in-depth look at the temperature- and magnetic-field dependencies of the 0.7 structure and the associated zero-bias anomaly (Danneau et al., 2008). However, it is still unclear why the bilayer device design has been so successful compared to single-layer devices that have also used surface gate depletion to define the 1D channel. Indeed, the peak mobility of each of the two layers of the bilayer devices is significantly lower than the mobility of the single-layer 2DHS of Daneshvar et al. One possibility is that remote ionized dopants cause instabilities in the 1D channel. In the devices of Daneshvar et al., the quantum-well was symmetrically doped – with the 2DHS sandwiched between two layers of remote ionized dopants. The bilayer system is also symmetrically doped; however, the top 2DHSs screens the lower 2DHSs from the upper band of remote ionized dopants and vice versa. This effectively halves the impact that the remote ionized dopants have on each layer. It is not possible to rule out that geometric properties of the wire (such as the dimensions of the 1D channel and the 2D-to-1D transition) may have contributed to the enhancement of the channel stability. However, recent experiments by Klochan et al. (2006) have confirmed that by completely removing remote ionized dopants, the stability of the 1D channel is improved dramatically.
lit
at
G
Sp
315
Mid
dle
Ga
te
e
1D
HS
lit
at
G
Sp
4 2 0 0.0
0.8 0.4 Side gate bias (V)
1.2
Figure 35 Conductance quantization observed in an electrostatically induced 1DHS by Klochan et al. (Inset) Schematic of the one-dimensional (1D) semiconductor– insulator–semiconductor field effect transistor (SISFET) device architecture. Electron beam lithography was used to pattern the metallically doped capping layer into three independent surface gates. The middle gate is negatively biased to form the 1D channel while the side gates are used to control the 1D confinement. From Hamilton AR, Danneau R, Klochan O, et al. (2008) The 0.7 anomaly in onedimensional hole quantum wires. Journal of Physics: Condensed Matter 20: 164205.
1D quantum wire and the 2D reservoirs to the left and right of it are controlled with a negative bias applied to the central top-gate. At the same time, a positive bias applied to the two side gates controls the effective width of the 1D wire formed 200 nm below the gate. As a result, they were able to fabricate a 1D hole system with remarkable stability – even very close to pinch-off. The device shows seven clean conductance steps, with additional structure below the first plateau. These devices were found to be exceptionally stable, without the hysteresis or instabilities that occur in most modulation-doped devices. Remarkably, Klochan et al. demonstrated that the current through their 1D channel remained constant – even for conductances as low as 0.3(2e2/h).
1.08.3.3.6 wires
Cleaved-edge overgrowth hole
To date, almost all experiments on ballistic 1D hole systems have been performed on QPCs fabricated in heterostructures grown on the (311)A GaAs. A notable exception to this trend are the p-type wires produced by Pfeiffer et al. (2005), who used cleavededge overgrowth to fabricate 2-mm-long, p-type
316 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
ce
Conductance (2e2/h)
4
rfa
Finger gate
u )S
10
(1
QW 1DHS
Doped AlGaAs
3 gexp 2 2 gexp gexp 0
1.5
2 2.5 Gate voltage (V)
Figure 36 Conductance quantization observed in a 2-mm-long quantum hole wire fabricated using cleaved-edge overgrowth by Pfeiffer et al. In this case the conductance is quantized in units of 0.77(2e2/h) as a result of 1D–2D wave function mismatch (Peiffer, 2005). (Inset) Schematic of the cleaved-edge overgrowth quantum wire. A quantum-well heterostructure grown in [100] GaAs is cleaved along the [110] crystallographyic plane. Additional layers of modulation-doped AlGaAs are grown on the newly cleaved surface, creating an enhancement in the carrier density in the quantum-well along the cleaved edge. Surface gates are then biased such that the two-dimensional hole system (2DHS) in the quantum-well is depleted but a 2-mm-long 1D channel remains along the cleaved edge. Reproduced with permission from Pfeiffer LN, de Picciotto R, West KW, Baldwin KW, and Omony CHL (2005) Ballistic hole transport in a quantum wire. Applied Physics Letters 87: 073111, of American Institute of Physics.
quantum wires from a carbon-doped heterostructure grown on a (100) GaAs substrate. A schematic of the device and the conductance plateau observed is shown in Figure 36. A highmobility ( ¼ 1 500 000 cm2 V1 s1) 2DHS was grown on a (100) GaAs substrate. A series of 13 parallel metallic finger gates were then patterned on the surface of the heterostructure – each gate being 2 mm wide and separated from each other by 2 mm. The heterostructure was then cleaved through the finger gates such that a clean [011] surface was formed along the cleaved edge. A second MBE growth step was then used to grow a modulationdoped AlGaAs heterostructure on the [011] surface – perpendicular to the original 2DHS quantum-well. The additional doping along this edge increases the electron density in the quantum-well near the cleaved-edge interface. As a result, applying a positive bias to a finger gate-depletes the 2DHS directly beneath it, but leaves a 2 -mm-long, 1D conducting channel along the cleaved-edge interface. Increasing the bias on the finger gate constricts the wire and ultimately pinches off the channel. These devices showed well-defined and reproducible conductance plateaus for the three lowest
subbands. However, the plateaus were quantized in units of 0.77(2e2/h) as opposed to 2e2/h. de Picciotto et al., (2000, 2001) explained this discrepancy as a competition between 1D–2D wave function mismatch and the inevitable residual disorder along the ungated regions. 1.08.3.4 Ballistic Transport in Hole Quantum Wires In recent years, the marked improvement in the stability of p-type quantum wires and QPCs has allowed single-particle and many-body phenomena in 1D hole systems to be explored in much greater detail. 1.08.3.4.1
Source–drain bias spectroscopy One of the defining factors regarding the clarity of the conductance plateau is the energy spacing between the 1D subbands, which is related to the strength of the confinement potential. We have seen that experiments on n-type wires defined by wet etching tend to have a very strong confinement potential (Kristensen et al., 1998a, 2000; Skaberna et al., 2000; Ramvall et al., 1997; Regul et al., 2002).
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
This results in large subband spacings, up to 20 mV. As a result, conductance quantization can be observed in these devices at temperatures up to tens of kelvin (Kristensen et al., 1998a, 2000; Skaberna et al., 2000; Ramvall et al., 1997) before thermal broadening destroys the conductance plateaus. On the other hand, 1D wires defined by surface gate depletion exhibit subband spacings of the order of 1 or 2 mV and so require much lower measurement temperatures to resolve the conductance plateaus (Daneshvar et al., 1997; Thomas et al., 1995; Pyshkin et al., 2000; Cronenwett et al., 2002). Another distinguishing feature is that the subband spacings in etched QPCs show little variation between energy levels (Kristensen et al., 2000; Regul et al., 2002), whereas the subband spacings in surface-gated systems tend to decrease rapidly for higher energy levels (Daneshvar et al., 1997; Thomas et al., 1995; Pyshkin et al., 2000; Cronenwett et al., 2002). To date, there have been only two studies to measure the subband energy spacing in 1D hole systems using source–drain bias spectroscopy. One study considered a single hole wire and the other a bilayer hole system. Daneshvar et al. (1997) applied the source– drain bias spectroscopy technique to their single 1D hole system defined by surface gate depletion. The subband energy spacings measured were observed to decrease from 0.27 mV to 0.09 mV for the first five subbands. These spacings are significantly smaller than the subband energy spacings measured in n-type devices defined by the same technique. However, this is consistent with the fact that the effective mass of holes is approximately 6 times larger than that of electrons and therefore we expect the subband energy spacing to be reduced by a factor of 6 for p-type systems. Danneau et al. (2006a, 2006b) also measured the subband spacings of a p-type bilayer QPC using source–drain bias spectroscopy. They found that in the bilayer device, the subband spacing ranged from E1,2 ¼ 0.41 mV to E5,6 ¼ 0.28 mV for the first five subbands, with very little variation in the three highest subband spacings. The larger subband spacings, and the small variation between energy levels in the bilayer devices is more consistent with that observed in etched QPCs than in surface-gated devices. This may indicate a stronger confinement potential set up by the two QPC gates, the midline gate and the in situ back gate. Despite the fact that both devices used surface gate depletion, the source–drain bias spectroscopy revealed that the bilayer system had a much stronger confinement potential, which is consistent
317
with the observation of very well-defined conductance plateaus in these bilayer systems. 1.08.3.4.2 Zeeman splitting and the g-factor anisotropy
For hole quantum wires, the Zeeman splitting of the 1D subbands is highly dependent on the orientation of the magnetic field. For quantum wires fabricated on (311)A heterostructures, the interplay of the confinement potential and the crystal anisotropy implies that the Zeeman splitting is much larger when the magnetic field is applied parallel to the quantum wire (along [233]), than when it applied perpendicular to the quantum wire (along [011]). This is quite different to 1D electrons, where the Zeeman splitting has been shown to be isotropic and independent of the direction of the in-plane field. The way in which 1D hole systems couple with an in-plane magnetic field is dependent on both the direction of the magnetic field and the orientation of the wire relative to the [233] and [011] crystallographic axes. This means that there are four possible g-factors that must be considered: 1. 2. 3. 4.
Wire k [233]and B k [233]; Wire k [233]and B k [011]; Wire k [011]and B k [233]; and Wire k [011]and B k [011].
As early as 1997, it was shown that the g-factor in a 1D hole systems increases with 1D confinement (Daneshvar et al., 1997). However, it was not until very recently that the g-factor anisotropy was studied in detail along the two major orthogonal directions. Danneau et al. (2006a) studied the g-factor anisotropy in a quantum wire oriented parallel to the [233] axis and found that increasing the 1D confinement increased the g-factor for in-plane magnetic fields oriented both parallel and perpendicular to the wire. For magnetic fields parallel to the wire, there was a dramatic 40% increase in g-factor as the wire was confined to a single subband. For fields perpendicular to the wire, the g-factor showed a more modest 15–20% increase. Danneau et al. suggested that the enhanced sensitivity to in-plane magnetic fields was due to the the 1D confinement causing the total angular momentum quantization axis to rotate into the plane and parallel to the wire. As a result, the spins couple more strongly to in-plane magnetic fields and in particular to fields parallel to the wire. However, experiments by Koduvayur et al. (2008) suggested that the total angular moment quantization
318 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures Table 2 Important length scales in mesoscopic systems
Wire k [233] Wire k [011]
Bk[233]
Bk[011]
Nonmonotonic increase 100% increase
Little to no increase (g 0) Little to no increase in g-factor
axis always rotates toward [233], regardless of the orientation of the wire. In these experiments, they compared the g-factor anisotropy for all four combinations of wire and magnetic field orientations. Their results are summarized in Table 2. For a quantum wire oriented along the [233] direction Koduvayur et al. observed that for fields parallel to the wire, the g-factor increased rapidly as the number of subbands in the wire decreased from 6 to 4 but then dropped abruptly for the third subband. Conversely, for fields perpendicular to the wire, Koduvayur et al. recorded g-factors very close to zero that showed no significant change with increasing 1D confinement. This result disagrees with recent theoretical results (Zu¨licke, 2006) that predict that the 2D g-factor along the [011] direction (which has a finite value of g 0.3) will only increase with 1D confinement. For a quantum wire oriented along the [011] direction, the observations are consistent with experiment. For magnetic fields oriented along the [233] direction, the g-factor increases monotonically as the number of subbands decreases. In addition, for fields parallel to the [011] direction, Koduvayur et al. observed a constant but finite value g ¼ 0.3 – very close to the g-factor expected in the 2D limit. These results indicate that even when the wire is oriented along the [011] axis, 1D confinement causes the quantization axis to align itself along the [233] direction, leading Koduvayur et al. to conclude that ‘‘. . .gfactor anisotropy is primarily determined by the crystalline anisotropy of spin–orbit interactions.’’ This rational may also explain the unusual nonmonotonic behavior of the g-factor in the wires oriented along the [233] direction. However, it is not clear why this random fluctuation in g-factors has not been observed previously. Although, it has been predicted by Csontos and Zu¨licke (2007) that spin–orbit coupling may introduce large and apparently random variations in the g-factor for different subbands. These theoretical predictions of Csontos et al. are based on perfectly cylindrical or square-shaped wires as opposed to the square well and saddle potentials in practical devices. Moreover, the theory calculates the g-factors for wires of constant width, while for
transport measurements, accessing a different subband requires a change in the confinement potential. Despite a series of investigations, there is still much to be understood about the g-factor in 1D hole systems, how it changes with a magnetic field, and how it can be tuned with 1D confinement and spin–orbit coupling.
1.08.3.5 The 0.7 Structure in 1D Hole Systems One of the most direct benefits of improved device stability in 1D hole systems has been the ability to resolve and study the 0.7 structure in 1D holes – one of the most intensely studied many-body phenomena in n-type 1D systems. One might expect to see a significant qualitative or quantitative difference between the nand p-type 0.7 structures given that holes have much greater effective mass, particle–particle interaction strength, spin–orbit coupling, and effective spin-3/2. However, to date, experiments have shown that the 0.7 structure in 1D hole systems behaves almost identically to the analogous structure in n-type systems. In particular, the anomalous 0.7 plateau: 1. appears at a conductance value of 0.7(2e2/h); 2. shows activated behavior, becoming stronger with increasing temperature; 3. typically coincides with a zero-bias anomaly; and 4. moves toward 0.5(2e2/h) in the presence of an inplane magnetic field. Interestingly, the behavior of the plateau in a magnetic field shows the same anisotropy as the conductance plateaus at higher subbands, providing further evidence that the origin of the 0.7 structure is fundamentally spin-based. Furthermore, it was found that the zero-bias anomaly mirrors the 0.7 structure anisotropy – disappearing in an in-plane magnetic field at the same time the 0.7 structure is reduced to 0.5(2e2/h). This highlights a close interdependence of these two phenomena. Recent experiments have used spin–orbit coupling to probe the level of spin-polarization around 0.7(2e2/h). The results from these experiments indicate that the anomalous plateau at
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
0.7(2e2/h) does indeed coincide with spontaneous spin-polarization (Danneau et al., 2006b, 2008; Rokhinson et al., 2004).
(a) 2.0 T = 20 mK T = 200 mK T = 320 mK T = 550 mK T = 650 mK
One of the main differences between electron and hole quantum wires is the much larger effective interaction strength rs in holes, which might be expected to have implications for the 0.7 structure. In contrast to n-type systems, there have been relatively few studies of the 0.7 structure in p-type 1D systems. However, Danneau et al. (2006b) performed a comprehensive study of the properties of the 0.7 structure observed in bilayer 1DHSs. These studies showed that despite rs in the electron and hole samples differing by a factor of 5 or more, many of the zero-field properties of the 0.7 structure were identical and, therefore, were independent of carrier type. This suggests that the 0.7 structure may be more sensitive to the potential landscape and/or the disorder environment than to the strength of the interactions. The temperature dependence of the 0.7 structure was observed to be very similar to that in n-type quantum wires. Danneau et al. showed an enhancement of the 0.7(2e2/h) plateau with increasing temperature up to 650 mK (see Figure 37(a)). In contrast to the single particle plateaus, the 0.7 structure became more prominent as the temperature was increased, and moved closer to 0.5(2e2/h). As the temperature was increased further, the strength of the state weakened once more, but only after all the single-particle plateaus had been thermally smeared. This activated behavior was studied in detail in n-type systems where it was shown by Cronenwett et al. (2002) to have similarities to the 0D Kondo effect. The Kondo model was also used to explain another peculiar feature of the 0.7 structure – a peak in the conductance around zero source–drain bias, also known as the zero-bias anomaly. Danneau et al. confirmed the existence of the zero-bias anomaly in 1D hole systems, as shown in Figure 37(b). Moreover, it was shown that in analogy to n-type systems, the p-type zero-bias anomaly was also destroyed with increasing temperature and correlated well with the simultaneous enhancement of the 0.7(2e2/h) conductance plateau. This is evident from the missing zero-bias peak in Figure 37(c).
1.0
0.5 0.0 3.5
3.6
3.7
3.8
Vsg (V)
(b)
(C)
1.0 G (2e2/h)
Characterizing the 0.7 structure
G (2e2/h)
1.5
1.08.3.5.1
319
0.5 T = 20 mK
T = 320 mK
B=0T 0.0 –0.5
0.0 Vsd (mV)
0.5
–0.5
0.0
0.5
Vsd (mV)
Figure 37 (a) Evolution of the 0.7 structure in a bilayer quantum wire as a function of temperature. The data show that the 0.7 structure has the same activated behavior as observed in n-type systems – becoming more evident as the temperature increases. Conversely, conductance plateaus for higher subbands become less well defined as a result of thermal broadening. b) Source–drain bias spectroscopy at T ¼ 0 showing the presence of a strong zero-bias anomaly below the first subband. c) Increasing the temperature destroys the zero-bias peak. From Danneau R, Klochan O, Clarke WR, et al. (2008) 0.7 structure and zero-bias anomaly in ballistic hole quantum wires. Physical Review Letters 100: 016403.
1.08.3.5.2 g-Factor anisotropy of the 0.7 structure and zero-bias anomaly
The similarities between the 0.7 structure observed in n- and p-type ballistic wires diverge with the application of an in-plane magnetic field. It is through this difference that we have gained valuable information about the origin of the 0.7 structure. Earlier, we showed that the Zeeman splitting of the 1D subbands in hole quantum wires is highly dependent on the orientation of the magnetic field. This extreme anisotropy of the spin splitting in 1D holes can be used to probe the 0.7 structure: if it is related to spin, then it should show an anisotropic response to an in-plane magnetic field as well. Danneau et al. (2008) used in-plane magnetic fields to demonstrate that the 0.7 structure shared the same g-factor anisotropy as that observed in
320 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
higher 1D subbands. The fact that the 0.7 feature always moves toward 0.5(2e2/h) at the same rate that degeneracy of the higher subbands is lifted suggests that the underlying mechanism is the same for both – providing strong evidence that the 0.7 structure is a spontaneous spin-split level. In these experiments, Danneau et al. studied the magnetic field dependence of the 0.7 structure formed in a bilayer 1DHS oriented along the [233] crystallographic axis (see Figures 38(a)–38 (d)). Danneau et al. found that for magnetic fields oriented parallel to the wire, the spins coupled strongly to the external magnetic field. This was evidenced by the fact that the 0.7 structure reacted rapidly to the applied field, reaching 0.5(2e2/h) at approximately 3.6 T. At the same time, the higher subbands also experienced strong Zeeman splitting, becoming fully spin split at 3.6 T. (a)
(b)
2
(e)
BII = 3.6 T g (2e2/h)
g (2e2/h)
However, for in-plane magnetic fields oriented perpendicular to the wire (Figures 38(c) and 38(d)) the case was very different. In this case, the 0.7 structure showed very little change in its absolute position – even at 3.6 T. Likewise, the higher subbands showed no signs of Zeeman splitting at this field. The fact that the 0.7 structure shares the same g-factor anisotropy is an indication that the anomalous plateau at 0.7(2e2/h) is the result of same form of spontaneous spin polarization. In electron systems, it has been shown that the 0.7 structure is accompanied by an enhanced conductance at zero source–drain bias, which falls away rapidly as the magnitude of the source–drain bias is increased. The resulting conductance peak at Vsd ¼ 0 is a characteristic signature of an anomaly in the density of states at the Fermi energy, and shares
1
1.0
0.5
BII = 0T
T = 20 mK
B = 3.6 T II
0 3.4
3.6
3.8
4.2
4.0 Vsg (V)
4.4
4.6 0
(c)
2 BII (T)
3
0.0
4
(d)
2
–0.5 0.0 0.5 Vsd (mV)
(f)
B =4T g (2e2/h)
T
1 B =0T T
g (2e2/h)
1
1.0
0.5 T = 20 mK
B = 3.6 T T
T
4.0
4.2
4.4
4.6
4.8
5.0
Vsg (V)
5.2
5.4
0.0 5.6 0
1
2 B (T) T
0
3
4
–0.5
0.5 0.0 Vsd (mV)
Figure 38 Conductance data from a bilayer quantum wire oriented along the [2 –33] crystallographic axis. (a) Conductance quantization and the 0.7 structure as a function of an in-plane magnetic field parallel to the wire B||. (b) Gray scale of the transconductance in the wire as a function of B||. The dark regions correspond to plateaus in the conductance. These data show that the higher sub–bands couple strongly to B|| and that the 0.7 structure is also strongly coupled – moving to 0.5G0 at 3.6 T. (c) Conductance quantization and 0.7 structure as a function of in-plane magnetic field orthogonal to the wire, B?. (d) Transconductance in the wire as a function of B? showing that B? has very little effect on the higher subbands and similarly has no effect on the position of the 0.7 structure up to 3.6 T. (e) Source–drain bias spectroscopy at B|| ¼ 3.6 T showing that the zero-bias anomaly has not been destroyed by the in-plane magnetic field. (f) Source–drain bias spectroscopy at B? ¼ 3.6 T showing that the zero-bias anomaly has not been destroyed by the in-plane magnetic field. Reproduced with permission from Danneau R, Klochan O, Clarke WR, et al. (2008) 0.7 structure and zero-bias anomaly in ballistic hole quantum wires. Physical Review Letters 100: 016403. Copyright (2008) by the American Physical Society.
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
1.08.3.5.3 structure
Spin polarization near the 0.7
The fact that the 0.7 structure moves toward G ¼ 0.5(2e2/h) in a parallel magnetic field strongly suggests that the origin of the anomalous plateau is a spin-split state due to a spontaneous spin polarization at B ¼ 0. However as discussed for 1D electron wires in Section 1.08.2.5, the origin of the 0.7 structure remains highly debated. A further experiment in 1D hole systems that has provided important information on the 0.7 structure was performed by Rokhinson et al. (2004). In this work two quantum point contacts were separated by a small distance, d, as show in the inset of Figure 39. A magnetic field was applied perpendicular to the plane and used to focus the holes injected through the first QPC into the second QPC. When the magnetic radius of the holes was d/n2 (where n ¼ 1,2,3, . . .) there was a peak in the voltage measured across the second QPC. However, as shown in Figure 39, the first peak is in fact a doublet, corresponding to spin states that have been split by spin– orbit coupling. The doublet is most prominent when both QPCs are gated to pass exactly one spin-degenerate mode. The level of splitting can also be tuned by changing the asymmetry of the injecting QPC’s confinement potential. This is
5
Ginj
1
Gc 3
Magnetoresistance (kΩ)
4 Magnetic focusing
Gdet
2 4
3
2
1
0
–1
–1.0
–0.5
0.0
0.5
1.0
B (T) T
many of the features of the zero-bias anomaly observed in quantum-dots. In their experiment, Danneau et al. went on to investigate the g-factor anisotropy of the 0.7 structure with the presence of a zero-bias anomaly peak. Danneau et al. found that for fields parallel to the wire, the zero-bias anomaly collapsed by Bk ¼ 3.6 T. This field lifts the spin degeneracy of the 1D subbands, and suppresses the zero-bias anomaly. As shown in Figure 38(e), the destruction of the zerobias anomaly coincided with the point at which the 0.7 structure reached 0.5(2e2/h). However, if the same magnetic field was applied perpendicular to the quantum wire, the Zeeman splitting of the integer 1D subbands was much weaker as shown in Figure 38(f), and the zero-bias anomaly was still present. In summary, the zero magnetic field data for electron and hole quantum wires are strikingly similar, with a clear 0.7 structure that becomes stronger with increasing temperature. However in contrast to electron systems the results for p-type systems strongly suggest that the destruction of the zero-bias anomaly is spin related, and that the zerobias anomaly and 0.7 structure are intimately related.
321
Figure 39 (Inset) AFM micrograph of a p-type spin filter composed of two p-type QPCs separated by 0.8 mm. The injector QPCs, with contacts 1 and 2, passes a current of 1 nA while the voltage is measured across the detector QPC using contacts 3 and 4. The main panel shows the magnetoresistance measured across the detector QPC as a function of magnetic field perpendicular to the plane B?. For B? < 0 holes passing through the injector QPC are focused away from the detector QPC and as a result the magnetoresistance is determined primarily by Subnikov-de-Haas oscillations in the two-dimensional hole system (2DHS). However, for B? > 0, holes are focused into the detector QPC, resulting in peaks in the voltage across the QPC. The first and sharpest peak is comprised of a doublet of peaks separated by 36 mT – showing that the different spins in the subbands are indeed resolved and that these devices can be used as spin filters. Reproduced with permission from Rokhinson LP, Larkina V, Lyanda YB, Pfeiffer LN, and West KW (2004) Spin separating in cyclotron motion. Physical Review Letters 93: 146601. Copyright (2004) by the American Physical Society.
analogous to turning on spin–orbit coupling in 2DHSs by asymmetrizing the quantum-well confinement potential. Rokhinson et al. found that as the conductance through the injector QPC approached 0.7(2e2/h), the height of the second peak collapsed as shown in Figure 40. This result indicates that the injector current is predominately spin polarized near 0.7(2e2/h) (Rokhinson et al., 2004). Interestingly, in 1D hole samples that did not show a well-defined 0.7 structure, the doublet remained intact below 0.3(2e2/h). In summary we have reviewed the 0.7 structure and zero-bias anomaly in high-quality hole quantum
322 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures
(a)
1.08.4 Summary
(b) Gi (2e2/h)
Injector QPC1
0
V (μV)
0.97 0.96 0.97 0.92 0.73 0.66 0.68 0.43 0.26
0.5
–2
0.0 0.10
0.15 0.20 Vg (v)
0.1
0.2 B (T)
0.3
T
Gi (2e 2 /h)
1.0
Figure 40 (a) Conductance through the injector quantum point contact (QPC) below the first subband as a function of gate voltage showing a well-defined 0.7 structure. The vertical lines indicate the points at which spin-polarized measurements were performed. (b) Voltage drop across the detector QPC as a function of magnetic field for different injector conductances. The data show that when the conductance through the injector is equal to (2e2/h), the voltage across the detector QPC registers two voltage peaks indicating two closely spaced spin states whose degeneracy has been lifted by spin–orbit coupling. However, when the conductance in the injector QPC is reduced to 0.7(2e2/h), the detector QPC registers only a single voltage peak indicating a single spin state and hence a spin-polarized current. The upper solid and dashed lines compare these two cases directly. Reproduced with permission from Rokhinson LP, Pfeiffer LN, and West KW (2006) Spontaneous spin polarization in quantum point contacts. Physical Review Letters 96: 156602. Copyright (2006) by the American Physical Society.
wires. Despite the fact that interaction effects should be much stronger in these systems, because of the enhanced hole effective mass, the zero magnetic field data for electron and hole quantum wires are strikingly similar. However, unlike 1D electron systems the Zeeman splitting caused by an in-plane magnetic field is highly anisotropic for 1D holes, due to the strong spin–orbit coupling. Both the 0.7 structure and the zero-bias anomaly share this anisotropic response to an in-plane magnetic field demonstrating that they are linked, and spin related. Finally, ballistic hole spin filters were used to study the spin polarization around G ¼ 0.7(2e2/h). These devices were able to resolve two spin states in the 1D subbands. However, near the 0.7(2e2/h) plateau, only one spin state was resolved – indicating that the lowest subband is in fact spin polarized at B ¼ 0. More work is needed to understand the electronic and spin properties of holes confined to 1D systems, but already it is clear that 1D holes have unusual spin properties with no counterpart in spin-1/2 electron systems.
In this chapter we have reviewed the extensive and growing literature on ballistic transport in 1D GaAs heterostructures. Since the first realization of ballistic transport in GaAs quantum wires by van Wees et al. and Wharam et al., the field has blossomed over the past two decades as new experiments, and in particular, the ability to create purer and smaller systems has exceeded new length scales, allowing new device architectures to be probed. Nonetheless, there remains important questions to be answered regarding the nature of transport through these devices, in particular the role of spin and interactions at very low densities. Most notably is the 0.7 structure, whose origin is still a point of contention. Although many experiments suggest that the anomalous plateau originates from a spontaneous spin splitting at zero magnetic field, others suggest that Kondo-like physics may govern the underlying mechanism. It appears that at the date of submission neither mechanism can explain all the experimental observations. With the ability to fabricate both ballistic electron and hole systems with different spin states and phonon coupling, there is still a lot more to learn from 1D systems. Perhaps most importantly, the techniques used to study ballistic transport in 1D have provided a solid foundation for understanding other systems. For example, the work on 1D systems in GaAs has opened the door to studies of 0D structures allowing the fabrication of artificial atoms and coupled quantum-dots. These fields are now expanding to include the study of quantum-dots as a possible route to a solid-state quantum computer. The work has also contributed to an increased understanding of quantum transport in other material systems such as nanowires and carbon nanotubes. (See Chapter 2.03).
References Adachi S (1985) GaAs, AlAs, and AlxGa1 xAs – material parameters for use in research and device applications. Journal of Applied Physics 58: R1. Ando T, Fowler AB, and Stern F (1992) Electronic properties of two-dimensional systems. Reviews of Modern Physics 54: 437. Appleyard NJ, Nicholls JT, Pepper M, Tribe WR, Simmons MY, and Ritchie DA (2000) Direction-resolved transport and possible many-body effects in one-dimensional thermopower. Physical Review B 62: R16275. Appleyard NJ, Nicholls JT, Simmons MY, Tribe WR, and Pepper M (1998) Thermometer for the 2D electron gas using 1D thermopower. Physical Review Letters 81: 3491.
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures Auslaender OM, Steinberg H, Yacoby A, et al. (2005) Spincharge separation and localization in one dimension. Science 308: 88. Bagraev NT, Buravlev AD, Klyachkin LE, et al. (2002) Quantized conductance in silicon quantum wires. Semiconductors 36: 439. Berggren KF and Yakimenko II (2002) Effects of exchange and electron correlation on conductance and nanomagnetism in ballistic semiconductor quantum point contacts. Physical Review B 66: 085323. Bruus H, Cheianov VV, and Flensberg K (2001) The anomalous 0.5 and 0.7 conductance plateaus in quantum point contacts. Physica E 10: 97. Bu¨ttiker M (1990) Quantised transmission of a saddle point constriction. Physical Review B 41: 7906. Chen T-M, Graham AC, Pepper M, Farrer I, and Ritchie DA (2009) Non-Kondo zero-bias anomaly in quantum wires. Physical Review B 79: 153303. Chiatta O, Nicholls JT, Proskuryakov Y, Lumpkin N, Farrer I, and Ritchie DA (2006) Quantum thermal conductance of electrons in a one-dimensional wire. Physical Review Letters 97: 056601. Chou HY, Luscher S, Goldhaber-Gordon D, et al. (2005) Highquality quantum point contacts in GaN/AlGaN heterostructures. Applied Physics Letters 86: 073108. Clarke WR, Micolich AP, Hamilton AR, Simmons MY, Muraki K, and Hirayama Y (2006) Fabrication of induced twodimensional hole systems on (311)A GaAs. Journal of Applied Physics 99: 023707. Cronenwett SM, Lynch HJ, Goldhaber-Gordon D, et al. (2002) Low-temperature fate of the 0.7 structure in a point contact: A Kondo-like correlated state in an open system. Physical Review Letters 88: 226805. Cronenwett SM, Oosterkamp TH, and Kouwenhoven LP (1998) A tunable Kondo effect in quantum dots. Science 281: 540. Csontos D and Zu¨licke U (2007) Large variations in the hole spin splitting of quantum-wire subband edges. Physical Review B 76: 073313. Daneshvar AJ, Ford CJB, Hamilton AR, Simmons MY, Pepper M, and Ritchie DA (1997) Enhanced g factor of a onedimensional hole gas with quantized conductance. Physical Review B 55: R13409. Danneau R, Clarke WR, Klochan O, et al. (2006a) Zeeman splitting in ballistic hole quantum wires. Physical Review Letters 97: 026403. Danneau R, Clarke WR, Klochan O, et al. (2006b) Conductance quantization and the 0.7 2e2/h conductance anomaly in one-dimensional hole systems. Applied Physics Letters 88: 012107. Danneau R, Klochan O, Clarke WR, et al. (2008) 0.7 structure and zero-bias anomaly in ballistic hole quantum wires. Physical Review Letters 100: 016403. Davies AG (1991) The Fractional Quantum Hall effect in High Mobilitity Two-dimensional Hole Gases. PhD Thesis, University of Cambridge. de Picciotto R, Pfeiffer LN, Baldwin KW, and West KW (2004) Nonlinear response of a clean one-dimensional wire. Physical Review Letters 92: 036801. de Picciotto R, Pfeiffer LN, Baldwin KW, and West KW (2005) Temperature-dependent 0.7 structure in the conductance of cleaved-edge-overgrowth one-dimensional wires. Physical Review B 72: 033319. de Picciotto R, Pfeiffer LN, Baldwin KW, and West KW (2008) The 0.7 structure in cleaved-edge-overgrowth wires. Journal of Physics C 20: 164204. de Picciotto R, Stormer HL, Pfeiffer LN, Baldwin KW, and West KW (2001) Four-terminal resistance of a ballistic quantum wire. Nature 411: 51.
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324 Ballistic Transport in 1D GaAs/AlGaAs Heterostructures Kaufman D, Berk Y, Dwir B, Rudra A, Palevski A, and Kapon E (1999) Conductance quantization in V-groove quantum wires. Physical Review B 59: R10433. Kittel C (1996) Introduction to Solid State Physics. New York, NY: Wiley. Klochan O, Clarke WR, Danneau R, et al. (2006) Ballistic transport in induced one-dimensional hole systems. Applied Physics Letters 89: 092105. Koduvayur SP, Rokhinson LP, Tsui DC, Pfeiffer LN, and West KW (2008) Anisotropic modification of the effective hole g-factor by electrostatic confinement. Physical Review Letters 100: 126401. Kothari H, Ramamoorthy A, Akis R, et al. (2008) Linear and non-linear conductance of ballistic quantum wires with hybrid confinement. Journal of Applied Physics 103: 013701. Kristensen A, Bruus H, Hensen AE, et al. (2000) Bias and temperature dependence of the 0.7 conductance anomaly in quantum point contacts. Physical Review B 62: 10950. Kristensen A, Jensen JB, Zaffalon M, et al. (1998a) Conductance quantization above 30 K in GaAlAs shallow quantum point contacts smoothly joined to the background 2DEG. Journal of Applied Physics 83: 607. Kristensen A, Jensen JB, Zaffalon M, et al. (1998b) Temperature dependence of the ‘‘0.7’’ 2e2/h quasi-plateau in strongly confined quantum point contacts. Physica B 251: 180. Landauer R (1957) Spatial variation of current and fields due to localized scatterers in metallic conduction. IBM Journal of Research and Development 1: 223. Lassl A, Schlagneck P, and Richter K (2007) Effects of shortrange interactions on transport through quantum point contacts: A numerical approach. Physical Review B 75: 0453456. Liang CT, Simmons MY, Smith CG, Kim GH, Ritchie DA, and Pepper M (1999) Spin-dependent transport in a clean one-dimensional channel. Physical Review B 60: 10687. Liang C-T, Simmons MY, Smith CG, Kim GH, Ritchie DA, and Pepper M (2000) Spin-dependent transport in clean onedimensional channel. Physical Review B 60: 10687. Lieb E and Mattis D (1962) Theory of ferromagnetism and ordering of electronic energy levels. Physical Review 125: 164. Lindelof PE (2001) Effect on conductance of an isomer state in a quantum point contact. Optical Organic and Inorganic Materials 4415: 77. Martin-Moreno L, Nicholls JT, Patel NK, and Pepper M (1992) Nonlinear conductance of a saddle-point constriction. Journal of Physics – Condensed Matter 4: 1323. Meir Y (2008) The theory of the ‘0.7 anomaly’ in quantum point contacts. Journal of Physics: Condensed Matter 20: 164208. Meir Y, Hirose K, and Wingreen NS (2002) Kondo model for the ‘‘0.7 anomaly’’ in transport through a quantum point contact. Physical Review Letters 89: 196802. Mott NF (1963) The Theory of the Properties of Metals and Alloys. Oxford: Clarendon. Ng TK and Lee PA (1988) On-site repulsion and resonant tunnelling. Physical Review Letters 61: 1768. Noh H (2005) Effective mass of dilute two-dimensional holes in a GaAs hetero-structure. Journal of the Korean Physical Society 47: 272. Pan W, Lai K, Bayrakci SP, et al. (2003) Cyclotron resonance at microwave frequencies in two-dimensional hole systems in AlGaAs/GaAs quantum wells. Applied Physics Letters 83: 3519. Patel NK, Nicholls JT, Martin-Moreno L, et al. (1991a) Properties of a ballistic quasi-one-dimensional constriction in a parallel high magnetic field. Physical Review B 44: R10973. Patel NK, Nicholls JT, Martin-Moreno L, et al. (1991b) Evolution of half plateaus as a function of electric field in a ballistic quasione-dimensional constriction. Physical Review B 44: 13549.
Pepper M (1978) Magnetic localisation in silicon inversion layers. Philosophical Magazine B 37: 187. Pfeiffer LN, de Picciotto R, West KW, Baldwin KW, and Quay CHL (2005) Ballistic hole transport in a quantum wire. Applied Physics Letters 87: 073111. Pyshkin KS, Ford CJB, Harrell RH, Pepper M, Linfield EH, and Ritchie DA (2000) Spin splitting of one-dimensional subbands in high quality quantum wires at zero magnetic field. Physical Review B 62: 15842. Ramsak A and Jefferson JH (2005) Shot noise reduction in quantum wires with the 0.7 structure. Physical Review B 16: 161311. Ramvall P, Carlsson N, Maximov I, et al. (1997) Quantized conductance in a heterostructurally defined GaAs0.25In0.75As/ InP quantum wire. Applied Physics Letters 71: 918. Regul J, Keyser UF, Paesler M, et al. (2002) Fabrication of quantum point contacts by engraving GaAs/AlGaAs hetero-structures with a diamond tip. Applied Physics Letters 81: 2023. Reilly DJ (2005) Phenomenological model for the 0.7 conductance feature in quantum wires. Physical Review B 72: 033309. Reilly DJ, Buehler TM, O’Brian JL, et al. (2002) Density dependent spin-polarization in ultra-low-disorder quantum wires. Physical Review Letters 89: 246801. Reilly DJ, Facer GR, Dzurak AS, et al. (2001) Many-body spin related phenomena in ultra-low disorder quantum wires. Physical Review B 63: 121311. Reimann SM, Koskinen M, and Manninen M (1999) End states due to spin-Peierls transition in quantum wires. Physical Review B 59: 1613. Rejec T and Meir Y (2006) Magnetic impurity formation in quantum point contacts. Nature 442: 900. Rejec T, Ramsak A, and Jefferson JA (2000) Conductance anomalies in quantum wires. Journal of Physics C 12: L233. Roche P, Segala J, Glattli DC, et al. (2004) Fano factor reduction on the 0.7 conductance structure of a ballistic onedimensional wire. Physical Review Letters 93: 116602. Rokhinson LP, Larkina V, Lyanda YB, Pfeiffer LN, and West KW (2004) Spin separating in cyclotron motion. Physical Review Letters 93: 146601. Rokhinson LP, Pfeiffer LN, and West KW (2006) Spontaneous spin polarization in quantum point contacts. Physical Review Letters 96: 156602. Rokhinson LP, Tsui DC, Pfeiffer LN, and West KW (2002) AFM local oxidation nanopatterning of a high mobility shallow 2D hole gas. Superlattices and Microstructures 32: 99. Sarkozy S, Sfigakis F, Das Gupta K, et al. (2009) Zero-bias anomaly in quantum wires. Physical Review B 79: 161307(R). Scherer A, Roukes ML, Craighead HG, Ruthen RM, Beebe ED, and Harbison JP (1987) Ultra-narrow conducting channels formed in AlGaAs/GaAs hetero-structures by low energy. Applied Physics Letters 51: 233. Schmeltzer D, Saxena A, Bishop AR, and Smith DL (2005) Electron transmission through a short interacting wire: 0.7 conductance anomaly. Physical Review B 71: 045429. Seelig G and Matveev KA (2003) Electron-phonon scattering in quantum point contacts. Physical Review Letters 90: 176804. Sfigakis F (2005) Electron Transport in Etched GaAs/AlGaAs quantum Wires. PhD Thesis, University of Cambridge. Sfigakis F, Ford CJB, Pepper M, Kataoka M, Ritchie DA, and Simmons MY (2008a) Kondo effect from a tunable bound state within a quantum wire. Physical Review Letters 100: 026807. Sfigakis F, Graham AC, Thomas KJ, Pepper M, Ford CJB, and Ritchie DA (2008b) Spin effects in one dimensional systems. Journal of Physics C 20: 164213.
Ballistic Transport in 1D GaAs/AlGaAs Heterostructures Simmonds PJ, Sfigakis F, Beere HE, et al. (2008) Quantum transport in In0.75Ga0.25As quantum wires. Applied Physics Letters 92: 152108. Skaberna S, Versen M, Klehn B, Kunze U, Reuter D, and Wieck AD (2000) Fabrication of a quantum point contact by dynamic plowing technique and wet chemical etching. Ultramicroscopy 82: 153. Spivak B and Zhou F (2000) Ferromagnetic correlations in quasi-one-dimensional conducting channels. Physical Review B 61: 16730. Starikov AA, Yakimenko II, and Berggren KF (2003) Scenario for the 0.7-conductance anomaly in quantum point contacts. Physical Review B 67: 235319. Sushkov OP (2003) Restricted and unrestricted Hartree–Fock calculations of conductance for a quantum point contact. Physical Review B 67: 195318. Thomas KJ, Nicholls JT, Appleyard NJ, et al. (1998a) Interaction effects in a one dimensional constriction. Physical Review B 58: 4846. Thomas KJ, Nicholls JT, Pepper M, Tribe WR, Simmons MY, and Ritchie DA (2000) Spin properties of low density one-dimensional wires. Physical Review B 61: R13365. Thomas KJ, Nicholls JT, Simmons MY, Pepper M, Mace DR, and Ritchie DA (1996) Possible spin polarization in a one-dimensional electron gas. Physical Review Letters 77: 135. Thomas KJ, Nicholls JT, Simmons MY, Pepper M, Mace DR, and Ritchie DA (1998b) Non-linear transport in single mode one dimensional electron gas. Philosophical Magazine B 77: 1213. Thomas KJ, Simmons MY, Nicholls JT, Mace DR, Pepper M, and Ritchie DA (1995) Ballistic transport in one-dimensional constrictions formed in deep two-dimensional electron gases. Applied Physics Letters 67: 109. Thornton TJ, Pepper M, Ahmed H, Andrews D, and Davis GJ (1986) One dimensional conduction in the two dimensional electron gas of a GaAs–AlGaAs heterojunction. Physical Review Letters 56: 1198. van der Wiel WG, de Franceschi S, Fujisawa T, Elzerman JM, Tarucha S, and Kouwenhoven LP (2000) Kondo effect in the unitary limit. Science 289: 2105. van Wees BJ, van Houten H, Beenakker CWJ, et al. (1988) Quantized conductance of point contacts in a twodimensional electron gas. Physical Review Letters 60: 848. Wang CK and Berggren KF (1996) Spin splitting of subbands in quasi-one-dimensional electron quantum channels. Physical Review B 54: 14257. Wharam DA, Thornton TJ, Newbury R, et al. (1988) Onedimensional transport and the quantisation of the ballistic resistance. Journal of Physics C: Solid State Physics 21: L209. Winkler R (2003) Spin–Orbit Coupling Effects in TwoDimensional Electron and Hole Systems. Berlin: Springer. Winkler R and Nesvizhskii AI (1996) Anisotropic hole subband states and interband optical absorption in [mmn]-oriented quantum wells. Physical Review B 53: 9984. Wirtz R, Newbury R, Nicholls JT, Tribe WR, Simmons MY, and Pepper M (2002) Tuning the electron transport properties of a one-dimensional constriction using hydrostatic pressure. Physical Review B 65: 233316. Yacoby A, Stormer HL, Wingreen NS, Pfeiffer LN, Baldwin KW, and West KW (1996) Nonuniversal conductance quantization in quantum wires. Physical Review Letters 77: 4612.
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Yoon Y, Mourokh L, Morimoto T, et al. (2007) Probing the microscopic structure of bound states in quantum point contacts. Physical Review Letters 99: 136805. Zailer I, Frost JEF, Ford CJB, et al. (1994) Phase coherence, interference and conductance quantization in a confined two-dimensional hole gas. Physical Review B 49: 5101. Zu¨licke U (2006) Electronic and spin properties of hole point contacts. Physica Status Solidi (c) 3: 4354.
Further Reading Ashcroft NW and Mermin ND (1976) Solid State Physics, p. 319. London: Thomson Learning. Bird JP and Ochiai Y (2004) Electron spin polarization in nanoscale constrictions. Science 303: 1621. Casey HC and Panish MB (1978) Heterojunction Lasers, p. 191. New York, NY: Academic Press. Davies AG (2000) Quantum electronics: The physics and technology of low-dimensional electronic systems into the new millennium. Philosophical Transactions of the Royal Society A 358 151–172. Davies JH (1998) The Physics of Low Dimensional Semiconductors. New York, NY: Cambridge University Press. Datta S (1995) Electronic Transport in Mesoscopic Systems. Cambridge: Cambridge University Press. Ferry DK and Goodnick SM (1997) Transport in Nanostructures. Cambridge: Cambridge University Press. Heinzel T (2003) Mesoscopic Electronics in solid state nanostructures. Weinheim: Wiley-VCH. Ihn T (2004) Electronic Quantum Transport in Mesoscopic Semiconductor Structures. New York, NY: Springer. Imry Y (1997) Introduction to Mesoscopic Physics. New York, NY: Oxford University Press. Kelly MK (1995) Low dimensional physics. In: Low Dimensional Semiconductors, ch. 4, pp. 76–101. New York, NY: Oxford Science Publications. Morimoto T, Iwase Y, Aoki N, et al. (2003) Nonlocal resonant interaction between coupled quantum wires. Applied Physics Letters 82: 2952. Puller VI, Mourokh LG, Shailos A, and Bird JP (2004) Detection of local-moment formation using the resonant interaction between coupled quantum wires. Physical Review Letters 92: 096802. Sfigakis F, Graham AC, Thomas KJ, Pepper M, Ford CJB, and Ritchie DA (2008) Spin Effects in One Dimensional Systems. Journal of Physics C 20: 164213. Smith CG (1996) Low dimensional quantum devices. Reports in the Progress of Physics 59: 235–282. Thomas KJ (1997) Transport Properties of High Mobility One Dimensional Electron Gases. PhD Thesis, Cambridge. Wang CK and Berggren KF (1998) Local spin polarization in ballistic quantum point contacts. Physical Review B 57: 4552. Yacoby A, Auslaender OM, Steinberg H, et al. (2006) Tunneling spectroscopy of quantum wires: Spin-charge separation and localization. Physica Status Solidi (b) 243: 3593.
1.09 Thermal Conductivity and Thermoelectric Power of Semiconductors I Terasaki, Nagoya University, Nagoya, Japan ª 2011 Elsevier B.V. All rights reserved.
1.09.1 1.09.2 1.09.2.1 1.09.2.2 1.09.3 1.09.3.1 1.09.3.2 1.09.3.3 1.09.4 1.09.4.1 1.09.4.2 1.09.4.3 1.09.5 1.09.5.1 1.09.5.2 1.09.5.3 1.09.6 1.09.6.1 1.09.6.2 1.09.6.3 1.09.7 References
Introduction The Boltzmann Equation Equations for Conduction Electrons Equations for Phonons Thermal Conductivity Single-Crystalline Semiconductors Amorphous Solids and Glasses Magnetic Semiconductors Thermoelectric Phenomena The Seebeck Effect The Peltier and Thomson Effects The Galvanomagnetic Effects Thermoelectrics Thermoelectric Devices Thermoelectric Materials Oxide Thermoelectrics Nanostructured Materials Superlattices Nanowires Nanostructured Bulk Materials Summary and Outlook
1.09.1 Introduction The history of mankind tells us that our ancestors recognized the importance of fire from the very early stage, and that they were civilized with one of the important properties of fire, that is, heat. It is, however, a rather recent event in the late nineteenth century that the nature of heat was understood; thanks to thermodynamics, we know that heat is a kind of energy, and obeys the energy-conservation law, known as the first law of thermodynamics. Temperature is a kind of intensive variable for heat, given entropy as the corresponding extensive variable. Except for reversible processes, such as the Carnot cycle, entropy will continue to increase, indicating that heat cannot be fully converted into work. This is known as the second law of thermodynamics. According to the second law of thermodynamics, the more the work done, the more the heat increases.
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326 327 327 329 330 330 333 334 336 336 340 341 342 342 345 347 350 350 352 353 355 356
In the course of recent developments in electronic technology, many semiconductor devices have been produced, complied, and used, which finally generate an enormous amount of unusable heat. Heat management has thus become a serious issue these days, and requires deep understanding of the thermal conduction process in solids. Aside from this, the heat flow itself is a prototypical example of nonequilibrium states of matter, which is of vital importance in the development of nonequilibrium thermostatistical physics. Thermal conduction is a predominant process of heat flow in solids, when compared with radiation and convection and is facilitated by conduction electrons and lattice vibrations (phonons). In certain kinds of magnetic insulators, magnetic excitations (magnons) can carry heat much better than the lattice vibrations can, which can be controlled by external magnetic fields. Electrons can carry electrical current as well, so that there exists a finite coupling between
Thermal Conductivity and Thermoelectric Power of Semiconductors
the thermal current and the electrical current. These are known as the thermoelectric phenomena. By making full use of the thermoelectric phenomena in semiconductors, we are able to convert electric power into heat, and vice versa. Such a technology is called thermoelectrics. In this chapter, we briefly review the thermal conductivity and thermoelectricity of semiconductors. This subject has more than a 100-year history since Drude established his kinetic theory of electron gas in 1900. Partly because of the space constraint, we focus on recent developments and show only minimal set of old data (selected with the author’s biased view). Obviously, the data shown here are far from complete, particularly the old ones. Regretably, it is almost impossible to cite all the preceding studies in order from an enormous numbers of papers. Those who wish to know this area more in detail can refer to many textbooks and reviews available. A good summary of Drude’s theory is given by Ashcroft and Mermin (1976), from which one can grasp gross features of the electron transport. Callen (1985) has explained a basic treatment of nonequilibrium thermodynamics in his unique framework of thermal physics. A standard theory of the transport phenomena in solids is concisely described by Ziman (1976). Thermal conductivity of new complex materials is extensively reviewed by Tritt (2004), and the current status of the thermoelectrics is thoroughly compiled by Rowe (2006). The chapter is organized as follows: In Section 1.09.2, we discuss the Boltzmann equations for electrons and phonons, and derive explicit expressions for thermal conductivity. Then, we take a quick look at the thermal conductivities of various semiconductors in Section 1.09.3. We also discuss the thermoelectric effects in Section 1.09.4, and the thermoelectrics in Section 1.09.5. In Section 1.09.6, the thermal and thermoelectric properties in nanostructured materials are briefly introduced. We show that they raise difficult questions to our understanding of thermal conduction. Finally, we briefly summarize this chapter in Section 1.09.7.
1.09.2 The Boltzmann Equation 1.09.2.1 Equations for Conduction Electrons In the framework of nonequilibrium thermodynamics (see Callen, 1985), the electrical current density j and the thermal current density q are written as functions
327
of the gradient of the chemical potential r and the gradient of the inverse temperature Ñð1=T Þ: j ¼ L11
1 1 Ñ þ L12 Ñ T T
ð1Þ
q ¼ L21
1 1 Ñ þ L22 Ñ T T
ð2Þ
where Lij’ s are called the transport coefficients (the Onsager coefficients). The chemical potential consists of an electrostatic part e ¼ eV and a chemical part c, wher e (<0) is the electron charge. Then, the electric field E is given by 1 E ¼ – ÑV ¼ – Ñð – c Þ e
ð3Þ
However, Ñc cannot be observed separately in real experiments, and is considered to be included in the observed E hereafter (Ashcroft and Mermin, 1976). Boltzmann has given a formula to calculate the distribution function f(k, r) of a particle subject to external fields. f(k, r) should change from its equilibrium value f0, but the deviation will be small enough, if the external fields are weak. In this weak limit, f f0 can be expanded in terms of external fields. To include the nonequilibrium state, Boltzmann has introduced the scattering term describing the change of f(k, r) from f0 through a process that a particle with momentum k is scattered to acquire a momentum of k9. Then, the deviation from the equilibrium should be due to diffusion, external field, and scattering, as expressed by vk ?Ñf þ
e @f E ? Ñk f ¼ h @t scattering
ð4Þ
where Ñk refers to the gradient by the momentum k. The first and second terms on the left-hand side correspond to the diffusion and the acceleration, respectively. For the sake of simplicity, we omitted the contribution from magnetic fields in Equation (4). To find the expressions for electrons, we assume the Fermi–Dirac distribution function for f0, that is, –1 ð" k – Þ þ1 f0 ¼ exp kB T
ð5Þ
where "k is the eigenenergy for the momentum k. The scattering term is, in fact, very difficult to handle strictly, so that the relaxation time approximation is often used, in which the scattering term can
328 Thermal Conductivity and Thermoelectric Power of Semiconductors
be written using the deviation from the equilibrium distribution g ¼ f f0 as @f 1 1 ¼ – g ¼ – ðf – f0 Þ @t scattering
ð6Þ
where is the scattering time which characterizes the timescale of the dissipation of the system. It is originally defined in the form of the inverse number 1/ as the probability per unit time that a particle with a momentum k and an energy ! is scattered to one with a momentum k9 and an energy !9. We further assume that f depends on r only through the space variation of temperature T ðrÞ. Making use of the following expression of the group velocity 1 vk ¼ Ñk"k h
ð7Þ
we obtain g¼
@f0 – @"
h
i "k – vk e Eþ ð – ÑT Þ T "¼"k
1 43
Z
evk gd3 k ¼
1 evk f d3 k 43
ð8Þ
ð9Þ
In deriving Equation (9), we used the fact that the above integral gives zero for f ¼ f0, because the electron moves in random directions. The prefactor of 1/43 ¼ 2/(2)3 represents the density of states for the momentum space times the spin degrees of freedom. Next, we find the expression of the thermal current density in the same framework. Recalling that the chemical potential and the eigenenergy are the free and internal energies per electron, respectively, we can define the heat per electron as "k – , and the thermal current density as q ¼
1 43
Z
ð"k – Þvk gd3 k
ð10Þ
Substituting Equation (8) in Equations (9) and (10), we obtain e j ¼ e 2 Kˆ0 E þ Kˆ1 ð – ÑT Þ T q ¼ e Kˆ1 E þ
1 ˆ K2 ð – ÑT Þ T
Z –
@f0 @"
vk vk ð"k – Þn d3 k "¼"k
vkx vkx vkx vky vkx vkz
1
B C B C C v v v v v v vk vk ¼ B ky kx ky ky ky kz B C @ A vkz vkx vkz vky vkz vkz
ð14Þ
We also note that Equations (11) and (12) are identical to Equations (1) and (2), and Onsager’s relation, L12 ¼ L21, is readily satisfied in Equations (11) and (12). All the parameters are reduced to scalar quantities for a cubic crystal, where the conductivity is given by ¼ e 2 K0
ð15Þ
The rest of the parameters are similarly given by S¼
1 K1 eT K0
ð16Þ
K2 T
ð17Þ
where S is the Seebeck coefficient (thermoelectric power or thermopower) and 9 the thermal conductivity in the absence of electric field. The electron thermal conductivity is always measured for the open-circuit condition of j =0; thus, by eliminating E from Equation (12), we get q ¼ S 2 T ÑT – 9ÑT S2 ¼ 9 1 – ð – ÑT Þ 9
ð18Þ
and the thermal conductivity of conduction electron in real experiments is given by S 2 ¼ 9 1 – 9
ð19Þ
The second term is usually small, and we may often neglect the difference between and 9 unless we treat thermoelectric materials (see Section 1.09.5). Using , S, and 9, we eventually simplify Equations (11) and (12) as j ¼ Eþ Sð – ÑT Þ
ð20Þ
ð11Þ
q ¼ ST Eþ 9ð – ÑT Þ
ð21Þ
ð12Þ
where all the parameters , S, and 9 can be determined by experiments. From Equations (15) and (17), one can find that the conductivity and thermal conductivity are calculated from a similar integral for conventional metals, in which the Fermi energy EF is much larger
where 1 Kˆn ¼ 3 4
0
9 ¼
and calculate the electrical current density as j¼
Note that Kˆn is a second-rank tensor through
ð13Þ
Thermal Conductivity and Thermoelectric Power of Semiconductors
than the thermal energy kBT. By expanding @f0/@" in terms of kBT/EF (the Sommerfeld expansion), one can derive K2 in terms of K1 or as 2 kB2 T 2 kB T 2 K2 ¼ ðE Þ þ O F EF 3 e2
T 2 kB2 ¼ ¼ L0 el 3 e2
ð22Þ
ð23Þ
where we write el, instead of 9, to emphasize that the thermal conductivity discussed here is due to conduction electrons, not due to phonons. The numerical value of L0 is 2.4 108 V2 K2 (155 mV/K)2 is known as the Lorentz number. This relation states that electrons in solids can carry heat as well as electricity, or equivalently, that a metal is a good conductor of electricity and heat. Note that the Lorentz number L0 is given only by the universal constants. This justifies the Wiedemann–Franz law that the ratio of the thermal to the electrical conductivity at fixed temperatures is independent of material parameters. This is known as one of the most remarkable successes of Drude’s theory.
1.09.2.2
Equations for Phonons
The Boltzmann equation can also be applied to the lattice thermal conductivity, the contribution of phonons to the whole thermal conductivity. Since the phonon does not carry charge, we only consider the energy (heat) flow, in which the distribution function at thermal equilibrium is given by the Bose–Einstein distribution as –1 h!k –1 N0 ¼ exp kB T
ð24Þ
where !k is the phonon frequency for the wavevector k (note that the chemical potential is zero for phonons). Similarly in the case of electrons, we write the thermal current density of phonons as q ¼
1 83
Z
h!k N vk d3 k
ð25Þ
where N is the distribution function for phonons in the presence of temperature gradient. We take the relaxation time approximation given by N – N0 ¼
–
@N0 vk ð – ÑT Þ @"
Then, we finally obtain the lattice thermal conductivity as ˆ ph ¼
and obtain the Wiedemann–Franz law given by
ð26Þ
329
1 83
Z
@N0 h!k vk vk – d3 k @" "¼h!
ð27Þ
Let us consider the physical meaning of Equation (27). For the sake of simplicity, we assume cubic symmetry, in which the second-rank tensor of Equation (14) is reduced to the scalar quantity of vk 2 . If we further assume that the velocity and scattering time are independent of k, we may rewrite Equation (27) as ph
" # Z 1 1 @N0 3 ¼ v, h!k – dk 3 83 @" "¼h!
ð28Þ
where we take the thermal average of the velocity as vx2 ¼ vy2 ¼ vz2 ¼ v2 =3
ð29Þ
and introduce the mean free path , ¼ v. Noting that the quantity in the brace is the specific heat of phonons, one can approximate the thermal conductivity as the product of the sound velocity, the mean free path, and the specific heat. First, we show the electron thermal conductivity in the Drude model (Ashcroft and Mermin, 1976). Drude regarded electrons in a metal as ideal gas molecules, and developed a kinetic theory of such gases. Suppose a metal rod is subject to a temperature gradient (one-dimensional (1D) model). The electrons coming from the hot side have a larger thermal energy and equivalently, a larger kinetic energy of ", so that the net heat flow is given as q¼ ¼
nvx ½"ðT ½x – vx Þ – "ðT ½x þ vx Þ 2
ð30Þ
d" nvx ,ðT1 – T2 Þ dT
ð31Þ
where n is the carrier density and v(T) is the thermal velocity at temperature T. The factor of 1/2 represents that the half of electrons are moving toward the position x. We take the thermal average of the velocity of Equation (29) and obtain the expression as 1 1 el ¼ Cel v2 ¼ Cel v,el 3 3
ð32Þ
where Cel ¼ nd"/dT and , are the electronic specific heat per volume and the electron mean free path, respectively. One may notice that this is a special form of Equation (28). The interpretation of this equation is straightforward. A good conductor of heat should satisfy that (1) the number of the heat
330 Thermal Conductivity and Thermoelectric Power of Semiconductors
carrier is large (large Cel), (2) the speed for the heat carrier is high (large v), and (3) the heat carrier travels far (large or ,). In fact, Equation (32) is valid for phonons as well: 1 1 ph ¼ Cph vs2 ¼ Cph vs ,ph 3 3
A¼ ð33Þ
in which Cel is replaced by the phonon-specific heat Cph; vs and are the sound velocity and the phonon scattering time, respectively. To step further, let us go back to Equation (27), and discuss the relaxation time . By definition, is a function of the momenta of a phonon scattered from k to k9. While the conduction electrons in metals have nearly the same energy as EF, the phonons may have different energy ranging from zero to kBT. Thus, it is more reasonable to assume that is dependent on energy, or equivalently, frequency ! through k k9. Eventually, the mean free path of phonons may strongly depend on !, implying that Equation (33) is often oversimplified. It has long been discussed how the scattering time of phonons depends on ! (see Yang, 2004). Usually, we can assume that the scattering events from different origins are additive in the form of scattering rates given as –1 total ¼
X
i – 1
ð34Þ
i
This is known as Matthiessen’s rule in electric resistivity, in which the total resistivity is given by the sum of the resistivity due to impurity scattering and that due to electron–phonon scattering (Ziman, 1976). In the case of phonons, we usually consider three to four kinds of scattering, as pointed out by Callaway (Callaway, 1959; Callaway and v Baeyer, 1960). The first one is the boundary scattering corresponding to the elastic scattering. This term is independent of energy, expressed in terms of a parameter L as B – 1 ¼ v=L
ð35Þ
L corresponds to a characteristic length of the system, which can be the dimensions of the sample, the size of the grains, etc. The second one is the point-defect scattering corresponding to the impurity scattering of electrons. Klemens (1960) pointed out that this scattering time PD is proportional to !4, so that PD – 1 ¼ A!4
Steigmeier and Abeles (1964) calculated the parameter A in a solid solution system. According to their calculation, A is given by
ð36Þ
V X m – mi 2 fi 3 m 4v i
ð37Þ
where V is the volume per atom, fi the volume fraction of the atom i, mi the mass of the atom i, and m the average mass. Thus, the parameter A is large, when the mass difference m – mi is large. The third term is the phonon– phonon (normal) scattering. Since the lattice in real materials deviates from the ideal harmonic oscillator, the restoring force can be nonlinear. In terms of phonons, it corresponds to the phonon–phonon scattering. (Recall that the anharmonicity corresponds to products of four phonon operators like bky bk9y bq9 bq .) This scattering is phenomenologically given as N – 1 ¼ B!a T b
ð38Þ
where a and b are some exponents of the order of unity. These parameters are determined by fitting the thermal conductivity measured in real materials. The last scattering is the phonon–phonon Umklapp scattering. At high temperatures where the momentum change k – k9 is large enough to exceed a reciprocal lattice vector, then phonons are scattered from one Brillouin zone to another. Such scattering is called Umklapp scattering. Slack and Galginaitis (1964) proposed the following formula U – 1 ¼
h 2 D 2 ! T exp – Mv2 D 3T
ð39Þ
where D is the Debye temperature and the Gru¨neisen parameter. Equations (38) and (39) have similar temperature dependence except for the exponential term in the latter, and thus the former is sometimes ignored for the data fitting. At first glance, one may wonder if the thermal conductivity is uniquely fit with the above four scattering times. Fortunately, they affect the curve differently, and the scattering times are reasonably determined by fitting the experimental data. We see how experimental data are explained with these scattering times in the following section.
1.09.3 Thermal Conductivity 1.09.3.1 Single-Crystalline Semiconductors Figure 1 shows the thermal conductivity of pure and doped silicon, silicon carbide, and diamond (Slack, 1964). Below around 20 K, the thermal conductivity
104 10
Diamond
3
102 101 100 10
Pure Si
SiC
Doped Si
–1
100
101 102 Temperature (K)
103
70
Ge(99.99%) (M)
70
Ge(99.99%) (S)
70
Ge(96.3)
70
Ge
na
Ge1 (M)
na
103
Ge1 (S)
70/76
Ge
T3 102
T –1 1
Figure 1 Thermal conductivity of pure and doped silicon, silicon carbide, and diamond. From Slack GA (1964) Thermal conductivity of pure and impure silicon, silicon carbide, and diamond. Journal of Applied Physics 35: 3460–3466.
increases approximately in proportion to Tp (3 > p > 2), takes a broad maximum around 20–80 K, and decreases with temperature approximately inversely proportional to T above 100 K. The temperature dependence is qualitatively understood as follows: according to Equation (33), the thermal conductivity is given by the product of the specific heat, the phonon mean free path, and the sound velocity. In many cases, the sound velocity does not change with temperature very much, and the other two predominantly determine the temperature dependence of the thermal conductivity. As is well known, the specific heat is expected to scale with T3 for T D, and to be 3NkB for T D. In contrast, the phonon mean free path is dominated by impurity scattering at low temperatures, and expected to be weakly dependent on temperature. With increasing temperature, the phonon begins to be scattered by other phonons and/or electrons. In particular, Equation (39) indicates that the phonon–phonon scattering rate is roughly proportional to T for T D, so that the mean free path is inversely proportional to T at high temperatures. Thus, the product of the specific heat and the phonon mean free path is expected to be proportional to Tp (2 < p < 3) for low temperatures and to 1/T for high temperatures (see also Figure 2). The thermal conductivity peak corresponds to the crossover between the two temperature regions. The magnitude of the thermal conductivity is 80 W m1 K1 at room temperature for doped silicon. Since a typical value of thermal conductivity of metals is 10–102 W m1 K1 (Ashcroft and Mermin, 1976), this value is reasonably high for phonon conduction.
331
104
Thermal conductivity (W m–1 K–1)
Thermal conductivity (W m–1 K–1)
Thermal Conductivity and Thermoelectric Power of Semiconductors
10
100 Temperature (K)
Figure 2 Thermal conductivity of isotope-pure and natural Ge. From Asen-Palmer M, Bartkowski K, Gmelin E, et al. (1997) Thermal conductivity of germanium crystals with different isotopic compositions. Physical Review B 56: 9431–9447.
The thermal conductivity of diamond is much higher, owing to higher Debye temperature and higher sound velocity. Because of this property, diamond is expected to be used as a semiconductor wafer of next generation. As discussed in the previous section, the pointdefect scattering given by Equations (36) and (37) depends on the mass difference. This means that the isotope atom can be a scattering center of phonons. Geballe and Hull (1958) found that the isotopically enriched 74Ge (96%) has three times larger thermal conductivity than natural Ge. Asen-Palmer et al. (1997) successfully measured the thermal conductivity of isotope-pure 70Ge (99.99%) as shown in Figure 2. The observed thermal conductivity is eight times larger at maximum than that of natural Ge crystal. Figure 3 shows thermal conductivity of various semiconductors with particularly low thermal conductivity (Kurosaki et al., 2005). They are used for thermoelectric materials, which are discussed in Section 1.09.5. All the materials include heavy ions such as Ag, Tl, Te, and Bi, and consequently have low sound velocities. Other complex materials with low thermal conductivity are summarized in a review by Snyder and Toberer (2008). By comparing data from Figures 1–3, we can say that the magnitude of the thermal conductivity ranges from 101 to 104 W m1 K1. This dynamic range is significant, but much smaller than that of electrical conductivity, in which it ranges from 1020 to 106 S cm1. This implies that neither ideal heat insulator nor heat conductor is available in real materials.
332 Thermal Conductivity and Thermoelectric Power of Semiconductors
Thermal conductivity (Wm−1 K−1)
2.0
1.5
+
+
+
+
AggTITe5 AgTITe Bi2Te3
1.0 +
0.5
TAGS-85
0.0 250 300 350 400 450 500 550 600 650 700 Temperature (K)
Figure 3 Thermal conductivity of various semiconductors. TAGS stands for (GeTe)x(AgSbTe2)1x. From Kurosaki K, Kosuga A, Muta H, Uno M and Yamanaka S (2005) Ag9TITe5: A high-performance thermoelectric bulk material with extremely low thermal conductivity. Applied Physics Letters 87: 061919.
As discussed in the previous section, the scattering time for phonons is described phenomenologically by a sum of different scattering rates given as – 1 ¼ v=L þ A!4 þ B!2 T e – D =T
ð40Þ
The first term corresponds to the boundary scattering, while the second term is due to point-defect scattering and the third term is due to phonon– phonon Umklapp scattering. The parameter A is
further written in terms of the mass difference given by Equation (37). Figure 4 shows an example to what extent Equation (40) explains the thermal conductivity measured in skutterudite antimonides (Yang, 2004). From Equation (40), one can understand that the second term corresponds to _ Tp (p > 2) at low temperatures, while the third term corresponds to _ 1/T at high temperatures. Thanks to the fact that the relevant temperature range is different between the two, the fitting parameters of A, B, and L are fairly uniquely determined. The boundary scattering is responsible for the magnitude of the thermal conductivity at low temperatures. With decreasing temperature, only long-wavelength phonons with an energy of h! smaller than kBT are excited, and consequently the second and third terms in Equation (40) do not contribute at T ¼ 0. Thus, a thermal conductivity would diverge without L. In this respect, L roughly gives an average phonon mean free path at low temperatures, but cannot be predicted from a firstprinciples calculation (Chen 1998; Takahata and Terasaki, 2002). In contrast, the second term dominates a saturated value and exponent in the intermediate temperature range. The coefficient A can be evaluated using Equation (37), but is often determined as a fitting parameter. Meisner et al. (1998) evaluated the parameter A of filled skutterudite Cey(Co.Fe)4Sb12 by regarding this as a solid solution
Thermal conductivity (W m–1 K–1)
150
100
Measured data Boundary only Boundary + point-defect Umklapp
50
0 0
50
100
150
200
250
300
Temperature (K) Figure 4 Lattice thermal conductivity of skutterudite antimonide. The contributions of three scattering mechanisms are separately drawn. From Yang J (2004) Theory of thermal conductivity in crystalline semiconductors. In: Trit TM (ed.) Thermal Conductivity. Kluwer.
Thermal Conductivity and Thermoelectric Power of Semiconductors
of CoSb3 and CeCoFe3Sb12, in which the huge mass difference results in a large A. The third term dominates the decreasing thermal conductivity at high temperatures. For T D, all the phonon modes are thermally excited, and the specific heat is saturated as a classical value of 3NkB. In such conditions, the contribution from !4 does not cause strong temperature dependence; the phonon–phonon scattering rate is proportional to T exp(D/T) T. 1.09.3.2
Amorphous Solids and Glasses
The amorphous solids and glasses are materials in which atoms have a short-range order, but no translational symmetry. As a result, no phonons are well defined, or all the phonon modes are localized at each atoms. Cahill et al. (1992) measured and analyzed the thermal conductivity of various disordered crystals, and showed minimum thermal conductivity, as was originally proposed by Slack (1979). Figure 5 shows the thermal conductivity of amorphous solids ( -SiO2 and CdGeAs2) and disordered solids (Or1Ab33An66 feldspar and Ba0.67La0.33F2.33). All the curves have low magnitudes at room temperature (less than 102 W cm1 K1 ¼ 1 W m K1), and weakly dependent on temperature above 10 K. With decreasing temperature below 10 K, the thermal conductivity 10–1
Thermal conductivity (W cm−1 K−1)
Or1 Ab33 An66 10–2 CdGeAs2 10–3 Ba0.67 La0.33 F2.33 –4
10
α-SiO2 –5
10
10–6 0.1
1.0 10 Temperature (K)
100
1000
Figure 5 Thermal conductivity of amorphous solids ( -SiO2 and CdGeAs2) and disordered solids (Or1Ab33An66 feldspar and Ba0.67La0.33F2.33). From Cahill DG, Watson SK and Pohl RO (1992) Lower limit to the thermal conductivity of disordered crystals. Physical Review B 46: 6131–6140.
333
decreases as T3. The temperature dependence is qualitatively understood as follows: in the present situation, the mean free path is as short as the atomic distance, and consequently independent of temperature. Although the sound velocity cannot be defined in disordered materials, it is expected to be similar to that in crystals. Thus the last term, the specific heat, is the origin of the temperature dependence of the thermal conductivity. Another feature to be pointed out is that all the curves are reasonably close to one another, implying somewhat universal behavior. Cahill et al. (1992) proposed a theoretical expression for the thermal conductivity of disordered solids, by assuming that the thermal energy of each atom is lost during half the period of an oscillation of the atom. Based on the Debye model, the minimum thermal conductivity min is given by min ¼
2 1=3 X 2 Z i =T 3 x nL T x e dx kB vi 6 i ðex – 1Þ2 0 i
ð41Þ
where vi corresponds to three sound modes (two for transverse and one for longitudinal). The cutoff frequency i is given in temperature unit as i ¼ vi
h ð62 nL Þ1=3 kB
ð42Þ
where nL is the density of atoms. Note that Equation (41) contains no free parameters, because vi and nL can be determined by different experiments. Figure 6 shows the calculated min plotted against thermal conductivity measured in various disordered solids at room temperature (Cahill et al., 1992). The consistency between the calculation and measurement is surprisingly well. The figure shows that there exists a lower limit for in solids, and an ideal heat insulator cannot be designed. The concept of minimum thermal conductivity resembles that of minimum electrical conductivity min as suggested by Mott (1972) and by Ioffe and Regel (Gurvitch, 1981), in which the electron mean free path cannot be shorter than the unit cell length or the interatomic spacing. It is now generally agreed that there is no min in real materials, and the electrical conductivity can be infinitesimally small at low temperature in the presence of disorder (Rosenbaum et al., 1980). This is known as the Anderson localization (see Kramer and MacKinnon, 1993), which is understood as an interference between an electron and the corresponding backscattered hole. Similar mechanism is theoretically predicted and experimentally observed in photons,
334 Thermal Conductivity and Thermoelectric Power of Semiconductors
Calculated conductivity (mW cm–1 K–1)
50 ZrO2 : Y 20
NaCl:CN
Si
SiO2
YB66
Ca,KNO3
10
CaF2:La
CdGeAs2
Feldspar
5
Ge
As2S3 Se
BaF2:La
KBr:CN
2 1 1
2 5 10 20 Measured conductivity (mW cm–1 K–1)
50
Figure 6 Minimum thermal conductivity calculated from Equation (41) plotted as the measured thermal conductivity. From Cahill DG, Watson SK and Pohl RO (1992) Lower limit to the thermal conductivity of disordered crystals. Physical Review B 46: 6131–6140.
which is known as the localization of light (Albada and Lagendijk, 1985). Thus, there is no reason to deny the possibility of the localization of phonons, which may give a thermal conductivity much lower than min (Venkatasubramanian, 2000).
1.09.3.3
Magnetic Semiconductors
Thermal conductivity (W m–1 K–1)
As is exemplified in transition-metal oxides, semiconductors including d- or f-electrons show magnetism.
At low temperatures, magnetic moments localized on the ions show a long-range-ordered state such as ferromagnetic, antiferromagnetic, and ferrimagnetic orders. The elementary excitations from such orders are known as magnons, which correspond to phonons in a perfectly ordered lattice. Thus, it should be natural to think that magnons can carry heat as phonons do. Quasi-1D magnetic insulator is particularly important. Theoretically, the energy current operator commutes the Heisenberg Hamiltonian in 1D (Niemeijer and Vianen, 1971; Zotos et al., 1997). This means that the energy current is conserved, and travels ballistically through the sample. In fact, such materials are found to show anomalously high thermal conductivity along the spin chain. Figure 7 shows the thermal conductivity of Sr2CuO3 (Sologubenko et al., 2000a). As shown in the inset, this insulating oxide includes a corner-shared CuO4 plane along the b-axis, and is regarded as a 1D Heisenberg antiferromagnet. The local moment on the Cu2þ behaves as S ¼ 1/2, and the exchange energy J is evaluated to be 1300 K for polycrystals (Ami et al., 1995), and 2000 K for single crystals (Motoyama et al., 1995). Contrary to the large J, the antiferromagnetic transition occurs below 4 K, owing to the strong quantum fluctuation of S ¼ 1/2. As shown in this figure, the thermal conductivity is extremely high along the b-axis at room temperature, while the thermal conductivity along the a-axis is nearly the same as that along the c-axis. This indicates that the magnetic excitation is responsible
100 Cu O Sr
b
c
κs
Chain
a b
10
a
c
10
T –1
Temperature (K)
100
Figure 7 Anisotropic thermal conductivity of the 1D magnetic insulator Sr2CuO3 along the a-, b-, and c-axis directions. From Sologubenko AV, Felder E and Gianno` K, et al. (2000a) Thermal conductivity and specific heat of the linear chain cuprate Sr2CuO3: Evidence for thermal transport via spinons. Physical Review B 62: R6108–R6111.
Thermal Conductivity and Thermoelectric Power of Semiconductors
for this high thermal conductivity. Other 1D spin chains such as Sr14Cu24O41 and SrCuO2 show similarly large thermal conductivity along the chain direction. (Hess et al., 2001; Kudo et al., 2001; Sologubenko et al., 2000b). Figure 8 shows the mean free path of the magnetic excitation evaluated from the Drude-like expression: 1 mag ¼ Cmag vmag ,mag 3
ð43Þ
As pointed out in the previous section, the Drudelike expression for is oversimplified. However, in the case of magnetic excitation, we do not know anything about the scattering process, and are unable to calculate scattering times. The evaluated mean free path is as large as 100 nm at low temperatures, which strongly suggests that magnetic excitations in this material travel over 250 unit cells without scattering. Note that the lattice thermal conductivity is less anisotropic; the anisotropy is basically determined by the anisotropy of the sound velocity (Crommie and Zettl, 1991). Even in chain-like materials such as Sr2CuO3, the chemical bonds perpendicular to the chain are similar to those in the chain. As a result, the restoring force will be more or less isotropic. Even in the presence of van der Waals coupling in one direction, the anisotropic ratio of remains around 5–7 (Terasaki et al., 2004). In this respect, the 300-K anisotropy between a- and b-axis directions is remarkable, which may be used as a heat guide in solids. More anisotropic thermal
335
conductivity is seen in the spin-ladder copper oxide Ca9La5Cu24O41, in which the a-axis thermal conductivity is 20 times larger than the c-axis one at 170 K (Hess et al., 2001). If the magnetic excitation dominates the heat conduction in magnetic materials, one may easily imagine that external fields will change the thermal conductivity. Such thermomagnetic effects are widely seen in magnetic materials such as CuGeO3 (Ando et al., 1998), R2CuO4 (Sales et al., 2004), and (La,Ca)MnO3 (Chen et al., 1997). Figure 9 shows the thermal conductivity of the layered vanadium oxide K2V3O8 (Sales et al., 2002). This particular oxide is an antiferromagnetic insulator with the local moment on V ions aligned along the a-axis. This antiferromagnetic order exhibits a spin-flop transition, in which the critical field is less than 1 T. The thermal conductivity suddenly increases above the critical field, suggesting that the spin orientation determines the heat conduction to some extent. Like spin-valve/giant magnetoresistance effects in multilayered ferromagnetic films (Zutic et al., 2004), the magnetothermal effect observed here may be applied to a thermal spin valve (Sales et al., 2004). At the moment, it is remarkable only at low temperatures at high fields (larger than 1 T). Even if a large response in the spin thermal conductivity were discovered at room temperature, the lattice thermal conductivity would remain unchanged, and overlap the magnetic contribution. Thus, the magnetothermal conductivity could not be large at room temperature, in comparison with magnetoconductivity.
6.5
103
+gμH
H//a, Q//a
Is (Å)
κa (W m−1 K−1)
6 5.5
− gμH H //c, Q//a
5
H = 0.55 T
4.5 4
102 20
40
60 80 100 Temperature (K)
200
Figure 8 Mean free path of magnetic excitation in Sr2CuO3. From Sologubenko AV, Felder E and Gianno` K et al. (2000a) Thermal conductivity and specific heat of the linear chain cuprate Sr2CuO3: Evidence for thermal transport via spinons. Physical Review B 62: R6108–R6111.
3.5
T = 3.7 K H = 0.95 T 0
0.5
1
1.5
2 H (T)
2.5
3
3.5
4
Figure 9 Thermal spin valve in K2V3O8. From Sales BC, Lumsden MD, Nagler SE, Mandrus D and Jin R (2002) Magnetic field enhancement of heat transport in the 2D Heisenberg antiferromagnet K2V3O8. Physical Review Letters 88: 095901.
336 Thermal Conductivity and Thermoelectric Power of Semiconductors
1.09.4 Thermoelectric Phenomena 1.09.4.1
The Seebeck Effect
An electron in solids carries an electrical current as an elementary particle with a negative charge of e (<0). Since an enormous number of electrons are at thermal equilibrium in solids, they carry heat and entropy at the same time. Thus in the presence of temperature gradient, they can flow from a hot side to a cold side to cause an electrical current. This implies a coupling between thermal and electrical phenomena, which is known as the thermoelectric effects. The Seebeck effect is a phenomenon in which electric field (E ¼ – ÑV ) is induced in proportion to applied temperature gradient (ÑT ), defined by ÑV ¼ Sð – ÑT Þ
ð44Þ
where S is called the Seebeck coefficient (thermoelectric power, or thermopower). We first discuss the physical meaning of the Seebeck effect with a classical electron-gas model proposed by Drude. Consider a metal rod subject to a temperature gradient, as shown in Figure 10. Suppose that the temperature at the left side is T1, and at the right side is T2 (T1 > T2). Since the average electron velocity is larger at T1, electrons begin to diffuse from left to right just after the temperature difference is given. Owing to the charge neutrality, the side at T1 is positively charged, whereas the side at T2 is negatively charged. This implies that the metal rod behaves like a capacitor in the presence of temperature gradient, which is the origin of the thermoelectric voltage V. In a steady state, the energy per electron is equal everywhere in the system, that is, "ðT1 Þ þ eV ðT1 Þ ¼ "ðT2 Þ þ eV ðT2 Þ
ð45Þ
is realized, where "(T) is the electron energy at temperature T. In the limit of T1 ! T2, the Seebeck coefficient S ¼ dV/dT reduces to eS ¼
d" dT
ð46Þ
d"/dT is equal to the specific heat per carrier, and Equation (46) further states that the Seebeck coefficient is the entropy per electron. Note that Hot T1
Cold T2
Figure 10 A metal rod subject to the temperature gradient. Reprinted from Sorell CC, Sugihara S, and Nowotny J (eds.) (2005) Materials for Energy Conversion Devices. Cambridge, UK: Woodhead Publishing: Figure 13.4.
the electron specific heat Cel is linear in T R(i.e., Cel ¼ T), R where the entropy is equal to Cel dT/T ¼ dT ¼ Cel. Since the right-hand side is positive, the sign of S is equal to the sign of the carrier, and thus S is negative (positive) for electrons (holes). The Boltzmann equations (11) and (12) give an explicit expression for the Seebeck coefficient. For the sake of simplicity, let us consider a cubic system in which all the transport parameters are reduced to scalars. In the absence of current density, we rewrite j ¼ 0 as j ¼ e 2 K0 Eþ
e K1 ð – ÑT Þ ¼ 0 T
ð47Þ
Then, the electric field E is proportional to the temperature gradient ( – ÑT ) as ÑV ¼ – E ¼
1 K1 ð – ÑT Þ eT K0
ð48Þ
The Seebeck coefficient is written by S ¼
¼
1 K1 eT K0 Z 1 eT
@f0 v2 ð"k – Þd3 k @" "¼"k k Z @f0 v2 d3 k – @" "¼"k k –
ð49Þ
Here, we used the definition of K1 and K0 given by Equation (13). From this equation, one can understand generic features of the Seebeck coefficient. First, this is independent of scattering time in the lowest order approximation. The scattering time appears in the numerator and the denominator at the same time, so that it is cancelled when the k dependence of is neglected. Second, the Seebeck coefficient should be zero when the energy dispersion is symmetric below and above the chemical potential, because the integrand in the numerator is proportional to "k – . Thus, the Seebeck coefficient of conventional metals is small, because the chemical potential is in the middle of the valence band where electrons and holes are almost symmetric. Third, the Seebeck coefficient is usually less anisotropic than the electrical and thermal conductivities. The anisotropy comes from the velocity vk , which simultaneously appears in the numerator and the denominator. This is qualitatively consistent with Equation (46); the entropy per carrier is a scalar quantity by definition, and accordingly it should be isotropic in the lowest approximation.
Thermal Conductivity and Thermoelectric Power of Semiconductors
First, we apply Equation (49) to a degenerate noninteracting Fermi gas. Thus, we assume the energy dispersion as h2 k2 2m
S¼
2 kB kB T 2 kB2 Dð0Þ Cel T¼ ¼ 2 e EF 3 e n ne
ð51Þ
where n is the carrier density and D(0) the density of states at EF (Ashcroft and Mermin, 1976). This expression is a special form of Equation (46), and again shows that the Seebeck coefficient is the entropy per carrier. Figure 11 shows the Seebeck coefficient of Sr1xLaxPbO3 (Terasaki and Nonaka, 1999). The perovskite oxide SrPbO3 is a narrow-gap semiconductor or a semimetal (Itoh et al., 1992), in which electrons can be doped through the substitution of trivalent La3þ for divalent Sr2þ. As shown in the figure, the Seebeck coefficient is linear in T at low temperatures. Using Equation (51), the carrier density is evaluated from the T-linear coefficient of S, which is consistent with calculated values from the La content. Figure 12 shows the T-linear term of the Seebeck coefficient (S/T) plotted as a function of electron specific heat coefficient for various metals (Behnia et al., 2004). The magnitude of S/T is of the order of 102 mV K2 for small , because a universal value of kB/je j ¼ 86 mV K1 is reduced by a factor of kBT/EF as is seen in the electron specific heat. On the
Thermopower (μV K–1)
0
–50 x 0.020 0.012 0.010 0.008 0.006 0.004 0
–100
–150
0
100 200 Temperature (K)
300
Figure 11 The Seebeck coefficient of Sr1xLaxPbO3. From Terasaki I and Nonaka T (1999) Thermoelectric properties of Sr1xLaxPbO3 (x < 0.02). Journal of Physics: Condensed Matter 11, 5577–5582.
CeAl2 CeColn2
CePt2Si2 NaxCoO2 NbSe2 La-214 Sr RuO
1
ð50Þ
and perform the Sommerfeld expansion to O(EF/kBT)2. Then, we get the explicit expression of the Seebeck coefficient as
CeCu2 CeCu2Si2
10
2
0.1 S/T (T = 0) (µV K−2)
"k ¼ "k ¼
337
CeRu2Si2 UPd2Al3
4
0.01 −0.01 Cu
Pd
−0.1
(ET)2Br
Bi-2201
(ET)2Cu(NSC)2 YbAl3 URu2Si2
−1
YbCu2Si2
YbCu4Ag
−10
YbCu4.5 UBe13
1
10
100
1000
γ (mJ mol–1 K–2) Figure 12 The T-linear term of the Seebeck coefficient (S/T) plotted as a function of electron specific heat coefficient for various metals. Reproduced from Behnia K, Jaccard D and Flouquet J (2004) On the thermoelectricity of correlated electrons in the zero-temperature limit. Journal of Physics: Condensed Matter 16: 5187–5198.
other hand, materials with large values show large Seebeck coefficients. Such materials are called heavy-fermion compounds in which localized f-electrons and conduction electrons are coupled with the Kondo effect (Coleman et al., 2001). At low temperatures, the localized f-electrons move coherently with conduction electrons, and the local moments vanish to show the largely enhanced Pauli paramagnetism. In a thermodynamic point of view, magnetic entropy due to the f-electrons is attached to conduction electrons. Thus, the entropy per conduction carrier is enhanced to cause the large Seebeck coefficient. Figure 12 suggests that the electron specific heat and the Seebeck coefficient are similarly enhanced, indicating close relationship between the Seebeck coefficient and the thermodynamic quantities. Phonon drag is a similar phenomenon to the enhanced Seebeck coefficient in heavy-fermion compounds. At low temperatures, phonons and electrons are long-lived, and their distribution functions begin
338 Thermal Conductivity and Thermoelectric Power of Semiconductors
to be nonequilibrium in the presence of temperature gradient, in which electrons and phonons can coherently move through the electron–phonon interaction. As a result, the entropy of phonons overlaps the entropy of electrons, which enhances the thermoelectric power. Ziman (1976) called this situation ‘thermoelectric power of lattice.’ In the case of metals in general, the integrand is finite only in the vicinity of the Fermi energy, thanks to the Fermi statistics. By using the Sommerfeld expansion, one can show that @K0 K1 ¼ kB2 T 2 3 @" "¼
than the bandwidth, it tends to zero as T ! 1. On the contrary, the second term is rewritten with the entropy s through an identity of thermodynamics as follows: –
S¼ – ð52Þ
On the same assumptions, we may associate the Seebeck coefficient with as 2 kB2 T @lnð"Þ 3 e @" "¼
ð53Þ
@f0 e 2 "k vk2 – d3 k @" "¼"k 1 Z S¼ – @f eT eT 0 d3 k e 2 vk2 – @" "¼"k Z
ð54Þ
If we regard e 2 vk2 ð– @f0 =@"Þ as a weighting function, the first term will be reduced to h"ki/eT, where h. . .i means the average with weights of e 2 vk2 ð– @f0 =@"Þ. Since the average of the one-electron energy "k is less
@s @N
ð55Þ E;V
kB @lng ¼ e @N eT
ð56Þ
where g is the total number of configurations (Chaikin and Beni, 1976). Let us calculate the Heikes formula for a simple conductor. Let the carrier number be N for N0 sites; then the total number of configurations g is equal to N CN0
This is known as the Mott formula. Note that the conductivity-like function (") in Equation (53) is a fictitious conductivity that a metal would show, if its Fermi energy were equal to ". Do not forget that (") cannot be observed in real experiments. Thus, the Mott formula should be very carefully applied to the analysis of real experiments (Ziman, 1976). Nevertheless, this formula became quite popular, and many experimentalists applied this to explain the measured data without paying much attention to the meaning of ("). Equation (53) includes logarithmic derivative of the density of states, implying that the Seebeck coefficient is highly sensitive to the shape of the Fermi surface; it can be negative or positive depending on subtle change in the curvature of the valence bands at EF. This is the reason why this quantity has not been investigated extensively in comparison with the Hall coefficient. A complementary picture to the Mott formula is the Seebeck coefficient in the high-temperature limit, where the one-electron energy is much smaller than the thermal energy. We can rewrite Equation (19) as
Thus, the Seebeck coefficient is associated with the entropy per carrier, which is called the Heikes formula written as
2
S¼
¼ T
¼
N0 ! ðN0 – N Þ!N !
ð57Þ
For macroscopically large number of N and N0, the Seebeck coefficient S of Equation (56) is S¼
kB N0 – N kB 1 – p ¼ ln ln N p e e
ð58Þ
where p ¼ N/N0 is the carrier concentration per site. When the spin degrees of freedom are included, N0 is replaced by 2N0, and Equation (58) is modified as S¼
kB 2N0 – N kB 2 – p ln ¼ ln e e N p
ð59Þ
If the on-site Coulomb repulsion is strong, only one electron can stand on each site. Then, the configuration from the spin degrees of freedom equals 2N, and Equation (58) is modified as S¼
kB N0 – N kB 2ð1 – pÞ ¼ ln ln 2 N p e e
ð60Þ
There are a couple of features in the Heikes formula. First, Equation (56) is independent of temperature, because g is simply given by the combination of the sites on which the carriers exist. Second, the Seebeck coefficient increases logarithmically with p for p 1. This means that the Seebeck coefficient does not change dramatically in comparison with conductivity ( _ p). Third, we can derive Equation (56) in the high-temperature limit, whenever one-electron energy is defined. This means that the Heikes formula is correct, even if any types of interaction are present. Figure 13 shows the Seebeck coefficient of doped pentacene films (Mu¨hlenen et al., 2007). The numbers
Thermal Conductivity and Thermoelectric Power of Semiconductors
104
2.8 × 1016 cm–3
500 –S (μV K–1)
103
400 300
100 0
102
1.7 × 1019 cm–3
101
200 1 2 180
200
3 4
5 6
7 8
220 240 260 Temperature (K)
n-type-doped SI
9
100 0
280
Figure 13 The Seebeck coefficient of pentacene. Reproduced from Mu¨hlenen Av, Errien N, Schaer M, Bussac M-N and Zuppiroli L (2007) Thermopower measurements on pentacene transistors. Physical Review B 75: 115338.
in the figure represent different samples deposited on different substrates. The Seebeck coefficient is essentially independent of temperature, and the magnitude is well understood from Equation (56). The Seebeck coefficient of organic conductors is reviewed by Kaiser (2001), in which lightly doped organic conductors show temperature-independent thermopower explained by the Heikes formula. The Heikes formula also explains the Seebeck coefficient of transition-metal oxide semiconductors such as La1xSrxCrO3 (Karim and Aldred, 1979) and La1xSrMnO3 (Palstra et al., 1997). Let us consider the Seebeck coefficient in lightly doped semiconductors. For a lightly doped semiconductor, the chemical potential is pinned near the center of the band gap, and has a gap of 2Eg to the bottom (top) of conduction (valence) band. Substituting Eg for in Equation (49), we get S¼
¼
1 K1 eT K0 Z 1 eT
Eg eT
@f0 2 v ð"k Þ þ Eg Þd3 k – @" k Z @f0 2 3 – v d k @" k
50
300
ð61Þ
ð62Þ
In other words, each thermally excited electron will acquire the activation energy, and carry large entropy. Hence, the Seebeck coefficient is large compared with those of metals. Figure 14 shows the Seebeck coefficient of lightly and heavily doped silicon (Weber and
100 150 200 Temperature (K)
250
300
Figure 14 The Seebeck coefficients of lightly doped silicon. From Weber L and Gmelin E (1991) Transport properties of silicon. Applied Physics A 53: 136–140.
Gmelin, 1991). The Seebeck coefficient of the lightly doped silicon is well described by Equation (62), in which the Seebeck coefficient is linear in 1/T. The magnitude is much larger than kB/jej ¼ 86 mV K1, and the activation energy Eg evaluated with Equation (62) is semiquantitatively consistent with the optical band gap. In the case of degenerate semiconductors, the Seebeck coefficient is essentially the same as that of a noninteracting Fermi gas given by Equation (51). The Seebeck coefficient of the heavily doped silicon shown in Figure 14 exhibits T-linear dependence like those of metals. At high temperatures, minority carriers are thermally excited to decrease the magnitude of the Seebeck coefficient, where the Seebeck coefficient roughly obeys Equation (62). Figure 15 shows the Seebeck coefficient of the skutterudite 300 Seebeck coefficient (μV–1 K)
Thermoelectric power (μV k–1)
600
339
1212OB22 10OB22 2NB13 2NB9
250 200 150 100 50 0 300
p-type CoSb3 400
500 600 700 Temperature (K)
800
900
Figure 15 The Seebeck coefficient of the skutterudite antimonide CoSb3. Reproduced from Caillat T, Borshchevsky A and Fleurial JP (1996) Properties of single crystalline semiconducting CoSb3. Journal of Applied Physics 80: 4442–4449.
340 Thermal Conductivity and Thermoelectric Power of Semiconductors
antimonide CoSb3 with various doping concentrations (Caillat et al., 1996). At low temperatures, the Seebeck coefficients decrease with decreasing temperature, suggesting that the electrical conduction is dominated by degenerate carriers. In contrast, the Seebeck coefficients have peaks and decrease with increasing temperature at high temperatures, showing that the minority carriers begin to be excited thermally. The temperature at which the Seebeck coefficients peak corresponds to a crossover from the degenerate to intrinsic regions.
The Harman method is known as a unique technique to measure thermal conductivity using the Peltier effect (Harman, 1958). Suppose that a sample is hung by electrical leads in vacuum at a temperature T. Upon a steady current I0, the Peltier heat I0 is generated at one current contact to the sample, and is absorbed at the other. As the sample is in vacuum, that is, thermally isolated, the Peltier heat is finally balanced by a back flow of thermal current through the sample, and causes a temperature difference T between the contacts to satisfy K T ¼ I0
The Peltier and Thomson Effects
The Peltier effect is the reverse phenomenon of the Seebeck effect; the electrical current flowing through the junction connecting two materials will emit or absorb heat per unit time at the junction to balance the difference in the chemical potential of the two materials. Thanks to this effect, an electronic refrigerator can be made, which is known as the Peltier cooler. The Peltier cooler has been applied to niche areas such as infrared detectors, CPU coolers, wine cellars, etc., because the cooling power is lower than that of compressor-based refrigerators. This technique is called thermoelectrics, and is discussed in the next section. Similarly to the Seebeck coefficient, the Peltier coefficient is defined as the coefficient of the thermal current Q to the electrical current I, given by Q ¼ I
where K is the thermal conductance of the sample. If the Seebeck coefficient is measured in advance, the Peltier coefficient will be given by ¼ ST. Thus, K can be obtained from the observed T. Figure 16 shows an example of T observed in the Harman method (Satake et al., 2004). The temperatures at the left side (TL) and those at the right side (TR) change with external currents. As shown in the inset, the antisymmetric part of the data is plotted to remove the contribution from the Joule heating (which is proportional to I0 2 ), which is linear in external current, as expected from the Peltier effect. The third thermoelectric effect is the Thomson effect. In the presence of temperature gradient and external current, thermal current generates in proportion to the temperature gradient and external current given as
ð63Þ
qx ¼ – T jx
or equivalently, is defined as the coefficient of the thermal current density to the electrical current density
where the current density is further expressed by the electrical conductivity. The Peltier coefficient is related to the Seebeck coefficient. From Equations (11) and (12), we may associate with S as ¼ ST
2
ð64Þ ΔT = TR − TL (K)
q ¼ j ¼ ð – ÑT Þ
1
ΔT(l 0)−ΔT(–l 0) (K)
1.09.4.2
ð66Þ
@T @x
ð67Þ
4 3 300 K
2 1 0
0
0
5 10 l 0 (mA)
15 100 K
ð65Þ
This relationship is not specific to Equations (11) and (12), but can be demonstrated from the first principle, being known as Onsager’s reciprocal theorem. Thanks to this relationship, the Peltier coefficient has been rarely measured. The amount of heat is difficult to measure precisely than the temperature difference, so that the Seebeck coefficient is much easier to measure.
−1 −10
0 Current l0 (mA)
10
Figure 16 The temperature difference by the Peltier effect in the sample hung in vacuum. From Satake A, Tanaka H, Ohkawa T, Fujii T and Terasaki I (2004) Thermal conductivity of the thermoelectric layered cobalt oxides measured by the Harman method. Journal of Applied Physics 96: 931–933.
Thermal Conductivity and Thermoelectric Power of Semiconductors
where T is the Thomson coefficient. This effect is understood through the temperature dependence of the Seebeck coefficient. Suppose that the temperature of the left half of a rod-like sample is T1, and that of the right half is T2. This is equivalent to two rods with different Peltier coefficients of T1S(T1) and T2S(T2) welded at the edge. Then, the welded junction will have the Peltier heat with respect to external current. Eventually, the Thomson coefficient is written as T ¼ T
dS dT
ð68Þ
The measurement of the Thomson coefficient is important, because the absolute Seebeck coefficient can be obtained from T. Note that the thermoelectric voltage V is experimentally observed in the open-circuit condition as V ¼ Slead ðT0 – T1 Þ þ Ssample ðT1 – T2 Þ
ð69Þ
þSlead ðT2 – T0 Þ ¼ ðSsample – Slead ÞðT1 – T2 Þ
ð70Þ
where T0 is the ambient temperature, T1 and T2 are the temperatures at the edges of the sample. Ssample and Slead are the Seebeck coefficients of the sample and the voltage leads, respectively. This indicates that the Seebeck coefficient is always measured as the difference from the Seebeck coefficient of the leads. Figure 17 shows the Thomson coefficient of various metals (Lander, 1948). As discussed above, the Seebeck coefficient is an entropy per carrier.
Thomson coefficient (micro V K–1)
20 Ag Cu
0
+
Au
–
W Mo
−20
Pt
−40
Pd
−60 300
900
1500
2100
Temperature (K) Figure 17 The Thomson coefficient of metals. Reproduced from Lander JJ (1948) Measurements of Thomson coefficients for metals at high temperatures and of Peltier coefficients for solid–liquid interfaces of metals. Physical Review 74: 479–488.
341
Recalling that the entropy goes to zero at 0 K (the third law of thermodynamics), we can assume that the Seebeck coefficient also goes to zero at 0 K, and calculate the absolute Seebeck coefficient from the measured Thomson coefficient. 1.09.4.3
The Galvanomagnetic Effects
A conduction electron is coupled with external magnetic fields through the Lorentz force, and is consequently deflected along the direction perpendicular to both the current and magnetic field. The deflected electrons are accumulated to one side of the sample to cause the transverse voltage, and cancel the Lorentz force in the steady state. This is known as the Hall effect, and the transverse voltage is called the Hall voltage. Similarly to this, the electron in the presence of temperature gradient will be deflected to the direction perpendicular to both the temperature gradient and the magnetic field, and eventually cause the transverse voltage. This is known as the Nernst effect, a typical galvanomagnetic effect. The Nernst coefficient N is defined as Ey ¼ NBz
@T @x
ð71Þ
where Bz is the external magnetic field applied along the z-direction, and the temperature gradient is along the x-direction. The Nernst effect is exceptionally large in semimetals (Li and Rabson, 1970). This is a similar situation of the classical magnetoresistance which has a large magnitude in semimetals (Clayhold, 1996). In particular, the high-mobility semimetals have the Nernst voltage as large as the Seebeck voltage in magnetic fields of several teslas. Figure 18 shows the Nernst voltage of Bi1xSbx in various magnetic fields (Li and Rabson, 1970). For small magnetic fields less than 1 T, the Nernst voltage exceeds 1 mV K1 at low temperatures. Another class of materials exhibiting large Nernst coefficients is a superconductor in the mixed state. Needless to say, a superconductor is a material that exhibits zero resistivity below the superconducting transition temperature Tc. External magnetic field above the lower critical field Bc1 can penetrate a type II superconductor as quantized magnetic fluxes. This state is called the mixed state, in which the quantized fluxes form a triangular lattice. The flux lattice is squeezed by external current through the Lorenz force, and begins to flow above a critical current density. Similarly to this, the flux can flow
342 Thermal Conductivity and Thermoelectric Power of Semiconductors
2.5 77 K
NB (μV K–1)
2
114 K
1.5
188 K 1
0.5
300 K
Te-doped Bi88Sb12 0 0
0.4
0.2
Nernst voltage is observed from the temperature above which the fluxes can move, to the temperature below which the fluxes can exist. Xu et al. (2000) found the large Nernst signal far above Tc in the high-temperature superconductors, which they ascribed to the phase fluctuation of the superconducting order parameter far above Tc. A similar large Nernst signal is observed in organic conductors (Nam et al., 2007).
0.6 B (T)
0.8
1
1.09.5 Thermoelectrics 1.09.5.1
Figure 18 The Nernst voltage of the high-mobility semimetal Bi1xSbx. From Li SS and Rabson TA (1970) The Nernst and the Seebeck effects in Te-doped Bisingle bond Sb alloys. Solid State Electronics 13: 153–160.
in the presence of temperature gradient (Huebener, 1995) from high- to low-temperature side. According to Faraday’s electromagnetic induction, the voltage will be generated in the direction of B ð – ÑTÞ, which is nothing but the Nernst voltage. Figure 19 shows the Nernst voltage NB in the mixed state of the high-temperature superconductor YBa2Cu3O7 (Huebener, 1995). With increasing temperature, the signal in a fixed field sharply rises above a certain temperature, reaches a maximum, and then decreases slowly. The external field enhances the range in which the Nernst signal is observable. This is qualitatively explained as follows: the quantized fluxes are frozen at sufficiently low temperatures, and cannot exist far above Tc. Consequently, the
Thermoelectric Devices
As discussed in the previous section, the Seebeck and Peltier effects are cross-correlations between electricity and heat carried by an electron in solids. When the Seebeck coefficient and the electrical conductivity are large in a material, such a material can convert electric energy into heat and vice versa. This is schematically illustrated in Figure 20. A rod of a material subject to the temperature gradient causes the thermoelectric voltage V ¼ ST across the sample, where T is the temperature difference between the edges of the sample, and S is the Seebeck coefficient. If the voltage V is sufficiently high, the rod works as a battery, and generates electricity at an external load connected to the rod. In this situation, the thermoelectric voltage corresponds to the opencircuit voltage of the battery, and the resistance of the rod corresponds to the internal resistance of the
Ey/∇xT (μV K–1)
3 H=1T H=2T H=3T H=4T H=6T H=8T H = 10 T H = 12 T
2
1
Current
Sample
0 70
80
90
100
110
Temperature (K) Figure 19 The Nernst voltage of the high-temperature superconductor YBa2Cu3O7 in the mixed state. Reproduced from Huebener RP (1995) Superconductors in a temperature gradient. Superconductor Science and Technology 8: 189–198.
Figure 20 A schematic picture of thermoelectric power generation. Reprinted from Narlikar AV (ed.) (2005) Layered Cobalt oxides as a thermoelectric material in Frontiers in magnetic Materials. pp 327–346. Springer: Berlin: Figure 1.
Thermal Conductivity and Thermoelectric Power of Semiconductors
battery. Such a material is called a thermoelectric material, and the device made from is called a thermoelectric device. One can find a good review on thermoelectrics given by Mahan (1997). Actually, Figure 20 is an oversimplified picture for the thermoelectric device. Since the current lead is usually a good conductor of heat, the heat applied at the left edge will flow through the lead, and will not generate sufficient temperature difference and, in the worst case, may damage the load. To avoid this, one should make a pair of rods as shown in Figure 21. This structure is nothing but a thermocouple, and the heat applied at the junction now flows through the pair of thermoelectric legs. In this respect, the thermoelectric device is a thermocouple with an ability to generate electricity, as shown in Figure 21(b). Obviously, the pair should be a pair of n- and p-type materials to maximize the thermoelectric voltage. Another requirement is that the thermal conductivity should be as low as possible to maximize the temperature difference across the legs. The thermoelectric device does not only generate electric power from heat, but also converts electric power into heat through the Peltier effect, that is, it cools the junction with external current, as shown in Figure 21(a). This is known as thermoelectric refrigeration or Peltier cooling. Thanks to the Onsager relation of Equation (65), the Peltier coefficient is equal to the Seebeck coefficient multiplied by absolute temperature. Thus, materials with the large Seebeck coefficient near or below room temperature can be used for thermoelectric refrigeration. We begin with the heat balance of the thermoelectric cooling device shown in Figure 21(a). Let R, S, and K be the net resistance, the net Seebeck
343
coefficient, and the net thermal conductance of the device, respectively. For the sake of simplicity, we assume that all the parameters of the device are independent of temperature. At the junction (cold side), the pumped heat per unit time QC is given by 1 QC ¼ STC I – RI 2 – K T 2
ð72Þ
where the second term is the Joule heat in the sample (we assume that a half of the heat goes to each side) and the third term is the back flow of the thermal current by the temperature difference. Similarly, at the hot side, the emitted heat per unit time QH is given by 1 QH ¼ STH I þ RI 2 – K T 2
ð73Þ
Thus, the total work per unit time is given by W ¼ QH – QC ¼ ðST þ IRÞI
ð74Þ
Equation (74) means that the work to refrigerate is equal to the voltage drop of the device in addition to the thermoelectric voltage. The heat balance at the hot and cold sides in the thermoelectric power generator shown in Figure 21(b) is similarly given by 1 QH ¼ STH I – RI 2 þ K T 2
ð75Þ
1 QC ¼ STC I þ RI 2 þ K T 2
ð76Þ
By connecting an external load Rext ¼ xR, we find the current I ¼ ST/(1 þ x)R. Then, the output power P is equal to P ¼ IV ¼
(a)
ðST Þ2 x R ð1 þ xÞ2
ð77Þ
(b) p-type
TC
TH
TH
TC xR
QC
QH
QH
QC
n-type Figure 21 A schematic picture of thermoelectric devices: (a) thermoelectric refrigerator and (b) thermoelectric power generator. Reprinted from Sorell CC, Sugihara S, and Nowotny J (eds.) (2005) Materials for Energy Conversion Devices. Cambridge, UK: Woodhead Publishing: Figure 13.2.
344 Thermal Conductivity and Thermoelectric Power of Semiconductors
which takes a maximum Pmax ¼ (ST)2/4R at x ¼ 1, that is, Rext ¼ R. Pmax is determined by the following material parameter: S2 ¼ S2
ð78Þ
which is called the (thermoelectric) power factor. Here, we evaluate the maximum heat absorption of the cooling device under the conditions of constant TH and TC. Then, a necessary condition of dQC/dI ¼ 0 gives the optimum current I0 ¼ STC/R. By substituting I0 into Equation (72), we have 2
QCmax
ðSTC Þ – K T ¼ 2R ðSTC Þ2 – T ¼K 2RK
and rewrite
QCmax
ð80Þ
S2 S2 S2 ¼ ¼ RK
ð81Þ
ZTC 2 ¼K – T 2
ð82Þ
as
QCmax
Thus, the maximum heat absorption is directly proportional to Z (or the power factor) for T ¼ 0. Next, we evaluate the lowest achievable temperature TCmin under the conditions of constant QC and TH. A necessary condition of dTC/dI ¼ 0 gives the optimum current I1 ¼ STCmin =R. By substituting I1 into Equation (72), we have T ¼
ðSTCmin Þ2 QC – 2KR K
ð83Þ
The maximum temperature difference (i.e., lowest achievable temperature) is again directly proportional to Z for QC ¼ 0. In addition, we evaluate the maximum efficiency. The energy conversion efficiency for a cooling device is characterized by the coefficient of performance (COP): ¼
QC STC I – RI 2 =2 – K T ¼ W ðST þ IRÞI
ð84Þ
Taking d /dI ¼ 0, we obtain the optimized current I2: ST I2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 1 þ ZT – 1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TC 1 þ ZT – TH pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ð 1 þ ZT þ 1Þ
ð86Þ
after some calculations. For the power generation, the efficiency is given as W QH
ð87Þ
¼
VI STH I – RI 2 =2 þ K T
ð88Þ
¼
xT ð1 þ xÞT þ ð1 þ xÞ2 =Z þ xT =2
ð89Þ
¼
ð79Þ
Then, we introduce the figure of merit Z defined by Z¼
max ¼
ð85Þ
where T ¼ ðTH þ TC Þ=2. By substituting I2 into , we find
By taking d /dx ¼ 0, we find that the maximum efficiency is max ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ð 1 þ ZT – 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TH 1 þ ZT þ TC
ð90Þ
According to Equations (86) and (90), material properties are associated with the conversion efficiency through ZT. This is reasonable, because the conversion efficiency and ZT are both dimensionless quantities. In this respect ZT is the most important parameter for thermoelectrics, and is called the dimensionless figure of merit. We can take some notes on the above results. First, max given by Equation (86) and max given by Equation (90) are reduced to the Carnot efficiency as ZT ! 1. This is reasonable, because thermoelectric energy conversion is a conversion through the electron transport, which is an irreversible process accompanying the Joule heat. Second, as shown in Figure 22, the efficiency is larger for larger ZT and T. Considering that the conversion efficiency of a solar battery is 10–15%, we think that a similar is expected for practical use. For ZT¯ < 1, Equation (90) can be expanded in terms of ZT¯. In the lowest order, max is max ZðTH – TC Þ=4
ð91Þ
Thus, Z(TH TC) ¼ 0.4 is needed for 10% efficiency. For a larger ZT, max tends to saturate, and 10% corresponds to Z > 3 103 K1 for T ¼ 300 K in Figure 22, which corresponds to ZT ¼ 1.8 at 600 K. Third COP of a commercial refrigerator is 1.2–1.3, which corresponds to ZT ¼ 3–4. Thus, much improvement in ZT is needed to replace a freon-gas refrigerator.
Thermal Conductivity and Thermoelectric Power of Semiconductors
30
Efficiency (%)
Z 1 × 10–3 K–1 2 × 10–3 K–1 3 × 10–3 K–1 4 × 10–3 K–1
20
TL = 300 K TH = ΔT + TL
10
0 0
100
200
300
400
500
600
ΔT (K) Figure 22 The conversion efficiency plotted as a function of temperature difference. The cold temperature is set to 300 K. Reprinted from Sorell CC, Sugihara S, and Nowotny J (eds.) (2005) Materials for Energy Conversion Devices. Cambridge, UK: Woodhead Publishing: Figure 13.2.
1.09.5.2
Thermoelectric Materials
Thermoelectric materials are characterized by ZT, that is, they show large Seebeck coefficient, high conductivity, and low thermal conductivity at the same time. Such requirements are difficult to be satisfied, because the three parameters are functions of carrier concentration, which cannot be tuned independently. Figure 23 schematically shows how the three parameters depend on carrier concentration (Snyder and Toberer, 2008). The Seebeck coefficient S decreases with carrier concentration n, whereas the conductivity 1
σ
S
κ
Parameters (a. u.)
ZT
0.5
0 1018
1019
1020
Carrier concentration
1021
(cm–3)
Figure 23 Conductivity (), the Seebeck coefficient (S), thermal conductivity (), and ZT plotted as a function of the carrier concentration. From Snyder GJ and Toberer ES (2008) Complex thermoelectric materials. Nature Materials 7: 105–114.
345
increases. The former is proportional to – ln n, while the latter is proportional to n. As a result, the carrier concentration takes an optimum value for maximizing the power factor S2. In other words, conventional metals have very small Seebeck coefficient, while conventional semiconductors have very low conductivity. Hence, an optimum carrier concentration is evaluated to be 1019–1020 cm3 that is a typical carrier concentration of degenerate semiconductors. Once the carrier concentration is set to be the optimum value, the only way to maximize the conductivity is to maximize the mobility. In fact, the state-of-the-art thermoelectric materials are high-mobility semiconductors. In the case of degenerate semiconductors, the lattice thermal conductivity is predominant to the electron thermal conductivity. The electron thermal conductivity is easily evaluated from the conductivity through the Wiedemann–Franz law given by Equation (23). Typical conductivity of thermoelectric materials is 500–1000 S cm1 at room temperature, which corresponds to el ¼ 0.4–0.8 W mK1. Good thermoelectric materials show a low thermal conductivity of 2–3 W mK1, so that it has been a central issue how to reduce the lattice thermal conductivity. Figure 24 shows ZT for various thermoelectric materials (Mahan, 1997). Thermoelectric materials so far used for practical applications are Bi2Te3, PbTe, and Si1xGex. They show ZT 1 at optimum temperatures. n-Type BiSb is superior at low temperatures, but has no p-type counterpart. Bi2Te3 shows the highest performance near room temperature, and is used for Peltier coolers commercially available. PbTe shows the highest performance near 700–800 K, and has been used for thermoelectric power generators operating at moderately high temperatures. Si and Ge are poor thermoelectric materials at room temperature because of the high thermal conductivity. A solid solution effectively reduces the lattice thermal conductivity, and ZT is maximized above 1200 K. Si1xGex has been used for the electricity source in space crafts, in which the decay from Pu works as a heat source. Above all, Bi2Te3 is a prototypical thermoelectric material (Goldsmid et al., 1958). Reflecting the high mobility, the resistivity is 1–2 m cm at room temperature. It decreases with decreasing temperature, showing that the carrier concentration is in the range of degenerate semiconductors. The Seebeck coefficient is basically linear in T, indicating that the electrons are degenerate to have the Fermi energy larger than the thermal energy. Above room temperature, in contrast, the Seebeck
346 Thermal Conductivity and Thermoelectric Power of Semiconductors
(a) 1
PbTe
ZT
Bi2Te3
Si1–x Gex 0.5
p-type 0 (b)
BiSb
Si1–x Gex Bi2Te3
PbTe
ZT
1
0.5
n-type 0
0
500
1000
Temperature (K) Figure 24 The dimensionless figure of merit ZT for various thermoelectric materials. Reprinted from Sorell CC, Sugihara S, and Nowotny J (eds.) (2005) Materials for Energy Conversion Devices. Cambridge, UK: Woodhead Publishing: Figure 13.5, and Narlikar AV (ed.) (2005) Layered Cobalt oxides as a thermoelectric material in Frontiers in magnetic Materials. pp 327–346. Springer: Berlin: Figure 3.
coefficient takes a maximum in magnitude, and decreases with increasing temperature. The resistivity also takes a maximum near the same temperature. These results are understood in terms of thermally excited minority carriers. The Seebeck coefficient is 200 mV K1 at room temperature, which is expected to give maximum ZT at room temperature. The thermal conductivity is 2 W mK1 at room temperature, in which the electron thermal conductivity is evaluated from the Wiedemann–Franz law to be 20–30% of the total. The low thermal conductivity comes from the low sound velocity due to heavy atoms, and point defect scattering due to solid solutions. Motivated by the discovery of Bi2Te3 in mid1950s, many thermoelectric materials were synthesized, but ZT did not exceed unity for 30 years or more. The situations changed in the 1990s, since Slack (1995) proposed a concept of phonon glass and electron crystal (PGEC). When an atom is weakly bound in an oversized atomic cage, it will vibrate independently from the host to cause large
local vibrations. This vibration and the atom in the cage are named rattling, and rattler, respectively. Compound with such atoms will have a poor thermal conduction like a glass and a good electric conduction like a crystal, in which ZT is evaluated to be 3–4 at maximum. An experimental manifestation of PGEC is seen in the filled skutterudite CexFe3CoSb12 (Sales et al., 1997). This is the first unambiguous example of ZT >1, which is a prototypical success in the material search in the 1990s. The unit cell of the unfilled skutterudite CoSb3 consists of the eight subcells whose corners are occupied by Co atoms. Six subcells out of the eight are filled with Sb plackets, forming the valence band. According to the band calculation, CoSb3 is a narrow-gap semiconductor with an indirect gap of 0.5 eV (Singh and Pickett, 1994), and the hole mobility of CoSb3 exceeds 2000 cm2 V1 s1 at 300 K (Caillat et al., 1996). In the filled skutterudite CeFe3CoSb12, the two vacant subcells are filled with Ce. In order to compensate the charge valance, six Fe atoms are substituted for the eight Co sites, because Ce usually exists as trivalent. The most remarkable feature of this compound is that filled Ce ions reduce the lattice thermal conductivity by several times lower than that for the unfilled skutterudite CoSb3 (Sales et al., 1997), in which Ce ions seem to work as rattlers. It is still controversial whether the rattling motion really occurs in the filled skutterudite. Although a localized phonon mode, like the Einstein mode, is suggested from the specific heat measurement (Keppens et al., 1998), neutron scattering shows that a localized mode is not anharmonic (Viennois et al., 2004). Geometrically, Ce is located in a wide space, but recent band calculations suggest that the rareearth ions are considerably hybridized with adjacent anions (Harima et al., 2002). The hybridization is regarded to be an origin for unconventional physical properties of this class of materials such as superconductivity in PrCo4Sb12 (Bauer et al., 2002). The rattling motion is more clearly observed in the clathrate compounds A6Ga16Ge30 (A ¼ Sr, Ba, Eu) (Nolas et al., 1998). The crystal structure is shown in Figure 25. The Ge and Ga ions form a complicated network composed of face-shared polyhedra. The Ge–Ge bond angle is close to the angle of sp3 orbitals (120 ), so that the electric properties are expected to be similar to crystalline Ge. There exists a large space in the polyhedron cage, in which Sr, Ba, and Eu are filled to rattle. The neutron diffraction experiment has revealed that Sr ions in the clathrate compound have quadruple minimum in the cage, and
Thermal Conductivity and Thermoelectric Power of Semiconductors
347
Ba8Ni6Ge40
κ L (150 K) (W K−1m−1)
2.0
Ba8Zn8Ge38 Ba8Cu6Ge40
Ba8Al16Si30 Cs8Zn4Sn42
Ba8Ga16Ge30
1.5
Cs8Sn44
Ba8Ga16Si30
Rb8Zn4Sn42 Rb8Ga16Sn38
Sr8Ga16Si30 Sr8Ga16Ge30 Ba8In16Ge30
1.0 β-Ba8Ga16Sn30
Figure 25 The crystal structure of the clathrate compound A6Ga16Ge30. The large dots correspond to A ions, and the Ga and Ge ions form the face-shared polyhedral cages. Reproduced from Suekuni K, Avila MA, Umeo K, et al. (2008) Simultaneous structure and carrier tuning of dimorphic clathrate Ba8Ga16Sn30. Physical Review B 77: 235119: Figure 1(b).
the lattice dynamics is anharmonic (Zuric et al., 2004). The lattice thermal conductivity is as low as that of amorphous Ge, being much reduced from that of crystalline Ge (Nolas et al., 1998). Thanks to this, ZT reaches near unity at high temperature. Figure 26 shows the lattice thermal conductivity of the clathrate compound Sr6Ga16Si30xGex (Suekuni et al., 2007). The lattice thermal conductivity systematically decreases with increasing Ge content, showing the maximum at x ¼ 0 and the minimum at x ¼ 30. This indicates that disorder from the solid solution does not dominate the thermal conductivity. Since
Type 1 clathrates
0.5 1.2
1.3
1.4
n1 p1 1.5
Ba8In16Ge28Sn2 Eu8Ga16Ge30
1.6
1.7
Guest free space (Å) Figure 27 The thermal conductivity as a function of spacing in the rattling site. From Suekuni K, Avila MA, Umeo K, et al. (2008) Simultaneous structure and carrier tuning of dimorphic clathrate Ba8Ga16Sn30. Physical Review B 77: 235119.
the cage size is larger for larger x, one can conclude that the degree of the intracage vibration of Sr reduces the thermal conduction in this system. Figure 27 shows the thermal conductivity plotted as a function of the guest free space for various clathrate compounds (Suekuni et al., 2008). The guest free space is defined as Rcage rhost rion, where Rcage is the radius of the cage, rhost is the covalent radius of cage atom (Ga, Ge, Sn, Si, etc.), and rion is the ionic radius of the rattler (Sr, Eu, Ba, etc.). Regardless of the elements, the thermal conductivity is roughly determined by this length. This universal behavior strongly suggests that rattling can be controlled in this system.
x=0
κ L (W K–1 m–1)
1.09.5.3 1
5
20 26 30
Sr8Ga16Si30–xGex
0.1 1
10
100
T (K) Figure 26 The lattice thermal conductivity of the clathrate compound Sr6Ga16Si30xGex. The dotted curves are the fitting curves. Reproduced from Suekuni K, Avila MA, Umeo K, and Takabatake T (2007) Cage-size control of guest vibration and thermal conductivity in Sr8Ga16Si30xGex. Physical Review B 75: 195210.
Oxide Thermoelectrics
As discussed above, the state-of-the-art thermoelectric materials are Bi2Te3, PbTe, and Si1xGex, all of which are degenerate semiconductors of high mobility. Since Te is scarce in earth, toxic, and volatile at high temperature, the application of Bi2Te3 and PbTe has to be limited to a niche area. In contrast, oxides are chemically stable at high temperature in air, and thus oxide thermoelectrics is expected to be used in much wider area. However, most of the oxide semiconductors show very low mobility, and their use has been thought to be out of the question. Since the large Seebeck coefficient and the low resistivity in a NaCo2O4 (NaxCoO2) single crystal were discovered (Terasaki et al., 1997), some kinds of oxides are found to be thermoelectric materials (Koumoto et al., 2006). Following NaCo2O4,
348 Thermal Conductivity and Thermoelectric Power of Semiconductors
Ca3Co4O9 (Funahashi et al., 2000), (Bi,Pb)2Sr2Co2O8 (Itoh and Terasaki, 2000), TlSr2Co2Oy (He´bert et al., 2001), and (Pb,Co)Sr2Co2Oy (Maignan et al., 2002) have been found to show good thermoelectric performance. Some single crystals show even ZT >1 at 1000 K. As shown in Figure 28, the CdI2-type hexagonal CoO2 layer is common to these cobalt oxides, which should be a key ingredient for the unusually high thermoelectric performance of the layered cobalt oxides. Figure 29 shows the thermoelectric parameters of the three kinds of the layered cobalt oxides. In the upper panel the resistivity of the three cobalt oxides along the in-plane direction is shown. The mostconducting compound is NaxCoO2, where a low
of 200 m cm at 300 K decreases in a metallic-like fashion with decreasing temperature down to 1.5 K. This is as conductive as the superconducting Cu oxides, meaning that the layered cobalt oxide is one of the most conductive layered oxides. The other two compounds show fairly large of 1–10 m cm at 300 K with an upturn below about 50 K. The Seebeck coefficient S of the three cobalt oxides along the in-plane direction is shown in the middle panel. The magnitude of S is as large as 100–150 mV K1 at 300 K, being comparable with that of conventional thermoelectric semiconductors. The thermal conductivity of the three cobalt oxides along the in-plane direction is shown in the lower panel. Clearly, a more complicated block layer shows lower , implying that is predominatly determined by the block layer. High was thus far expected for oxides because of the light mass of oxygen, which is not always true in the layered cobalt oxides.
As an origin of the large Seebeck coefficient, Koshibae et al. (2000) proposed an extended Heikes formula for transition-metal oxides: S¼
kB gA p ln Q gB 1 – p
ð92Þ
where gA and gB are the degeneracy of the electron configuration of A and B ions, Q is the charge difference between A and B ions, and p is the atomic content of the A ion. Since kB ln
gA p gB 1 – p
ð93Þ
is equal to the entropy per carrier, Equation (92) is a special case of Equation (56). Let us apply the above formula to NaCo2O4. Assuming that Na and O exist as Naþ and O2 in NaCo2O4, we expect that Co ions exist as Co3þ and Co4þ with a ratio of Co3þ:Co4þ¼1:1. Then, p for NaCo2O4 is equal to 0.5, and S for p 0.5 is simply reduced to S¼
kB gA ln Q gB
ð94Þ
Magnetic measurements reveal that the Co4þ and Co3þ ions are in the low spin state in NaCo2O4. The configuration of the low-spin-state Co3þ is (t2g)6, whose entropy is zero. On the other hand, the low-spin-state Co4þ has a hole in the t2g states, which is sixfold degenerate (two from spin and three from t2g orbitals) to carry large entropy of kB ln6. Suppose that electric conduction occurs by exchanging Co3þ and Co4þ. Then, a hole on Co4þ
Co SrO
Co
BiO
CaO
Na0.5
Co NaCo2O4
Ca3CO4O9
CaO
BiO
CaO
SrO
Co Bi2Sr2Co2Oy
Figure 28 Crystal structures of the layered cobalt oxides. Reprinted from Sorell CC, Sugihara S, and Nowotny J (eds.) (2005) Materials for Energy Conversion Devices. Cambridge, UK: Woodhead Publishing: Figure 13.8, and Narlikar AV (ed.) (2005) Layered Cobalt oxides as a thermoelectric material in Frontiers in magnetic Materials. pp 327–346. Springer: Berlin: Figure 5.
Thermal Conductivity and Thermoelectric Power of Semiconductors
103
(a)
ρ (mΩ cm)
102
Ca3Co4O9
101 100
Bi2Sr2CO2Oy NaxCoO2
10–1 10–2
(b)
S (μV K–1)
150
Ca3Co4O9 Bi2Sr2CO2Oy
100 50
NaxCoO2
0
(c)
κ (W mK–1)
8
NaxCoO2
6 4
Ca3Co4O9
2 Bi2Sr2CO2Oy 0 0
100
200
300
T (K) Figure 29 Resistivity ( ), Seebeck coefficient (S), and thermal conductivity () of the layered cobalt oxides (Koumoto et al., 2006).
can carry a charge of þje j with entropy of kB ln6, which causes a large Seebeck coefficient of kB ln6/jej (150 mV K1). This is close to the hightemperature value of S. Note that carriers in semiconductors have no such internal degrees of freedom; they can only carry entropy due to their kinetic energy. In this sense, a hole in NaCo2O4 can carry much larger entropy than a carrier in semiconductors, which leads to a new design for thermoelectric materials. Koshibae’s theory has successfully explained the high-temperature Seebeck coefficient of NaCo2O4, but the remaining problem is not trivial. The Seebeck coefficient of NaCo2O4 is 100 mV K1 at 300 K, which is about two-thirds of kB ln6, which means that the large amount of entropy of kB ln6 in the high-temperature limit (104 K) survives down to 102 K. We should also note that the resistivity is expected to be a large, constant value in the high-
349
temperature limit. NaCo2O4 shows good metallic conduction, suggesting that the electronic states should be discussed in an itinerant picture. Singh (2000) pointed out that the large thermopower can be explained from the band calculation. Kuroki and Arita (2007) showed that the band structure of NaxCoO2 is special in the sense that the top of the conduction band is flattened like a pudding mold. Such structure breaks electron–hole symmetry at EF, and gives high Seebeck coefficient with high conductivity. Takeuchi et al. (2004) showed that the Seeback coefficient calculated from the observed photoemission spectra quantitatively reproduces the experimental results. Ishida et al. (2007) measured the temperature dependence of the chemical potential from the photoemission spectra, and found that the electronic states gradually change from a high-temperature localized picture to a low-temperature itinerant picture. Although the layered cobalt oxides are good p-type thermoelectric oxides, they cannot be n-type because the large Seebeck coefficient comes from the combination of Co3þ and Co4þ in the low spin state. We expect that the mixture of Co2þ and Co3þ will be n-type, but such cobalt oxides are small polaron conductors to show poor thermoelectric properties (Kobayashi et al., 2002; Nagao et al., 2007). The dielectric oxide SrTiO3 is a promising n-type thermoelectric oxide. This material is a highmobility semiconductor with a large dielectric constant. Okuda et al. (2001) showed that a partial substitution of La for Sr supplies electrons to cause a large Seebeck coefficient with low resistivity. Unfortunately, the lattice thermal conductivity is one order of magnitude higher than that of conventional thermoelectric materials, and ZT remains 0.1–0.2 at room temperature. Ohta et al. (2005) showed that a partial substitution of Nb for Ti improves the thermoelectric properties, and that ZT reaches 0.37 at 800 K in single crystals, thin films, and polycrystalline samples. Transparent oxides can also be good thermoelectric materials. ZnO (Ohtaki et al., 1996) and In2O3 (Be´rardan et al., 2008) show superior thermoelectric properties above 1000 K. The high mobility of these materials is based on the light effective mass, and the Seebeck coefficient is small compared with the state-of-the-art thermoelectric materials. Another drawback is the high thermal conductivity due to the light mass of oxygen atoms.
350 Thermal Conductivity and Thermoelectric Power of Semiconductors 1 1 @ 2 " a2 ¼ 2 2 ¼ 2 tc cos ðkz aÞ mc h @kz h
1.09.6 Nanostructured Materials Superlattices
Superlattice films are prototypical examples of nanostructured materials in which different substances are epitaxially grown layer by layer. A great success of semiconductor superlattices is exemplified by semiconductor lasers based on double-hetero and/or multiquantum well structures. In those devices, the layer thickness is controlled to be much smaller than the mean free path and the de Broglie length of doped carriers to make electronic states quasi-2D, in which the energy levels are quantized along the cross-plane direction. Concomitantly, the 3D parabolic conduction band is split into subbands shifted by the quantized levels along the cross-plane direction, which enables to tune the emission frequency of photons strictly. This is called quantum confinement. Hicks and Dresselhaus (1993a) theoretically proposed that quantum confinement improves the thermoelectric figure of merit, when the layer thickness is comparable to a few nanometers. They assumed that the scattering time of doped carriers is independent of the layer thickness. If so, the sheet resistance will be independent of layer thickness. Suppose that the layer thickness is reduced by x times. Since the sheet resistance remains unchanged, the resistivity decreases by x, and the electron thermal conductivity increases by x through the Wiedemann–Franz law. Then, the figure of merit ZT ¼ S2/ (el þ ph) is modified as ZT ðxÞ ¼
S2 S2 ¼
=xðxel þ ph Þ ðel þ ph =xÞ
ð95Þ 6.0
This corresponds to the fact that the lattice thermal conductivity is effectively reduced by a factor of x. The second effect is the enhancement of the Seebeck coefficient owing to the confinement. Suppose that the tunneling process is dominant along the cross-plane direction, while the carriers are free along the in-plane direction. Then, the energy dispersion of carriers in the multilayer can be written as "ðkÞ ¼
h2 ðkx 2 þ ky 2 Þ – tc cos kz a 2m
ð97Þ
Accordingly, m c is proportional to 1/a2. S roughly increases with m ¼ ðm x m y m z Þ1=3 , and thus it enhances as a2/3 for a ! 0. Figure 30 shows the calculation of the Bi2Te3 thin films. With decreasing layer thickness, ZT rapidly increases to reach ZT ¼ 6 for a layer thickness of 1 nm. Thermoelectric superlattices have been extensively investigated since the proposal offered by Hicks and Dresselhaus (1993a). Venkatasubramanian et al. (2001) succeeded in fabricating an extremely good thermoelectric device showing ZT ¼ 2.5. Their device shows a cooling temperature of 40 K without heat removal of heat sink at an external current of 3 A. This value is about twice larger than the cooling temperature for a bulk device. They ascribed the high ZT to low thermal conductivity. Figure 31 shows room-temperature thermal conductivity of various thermoelectric thin films (Venkatasubramanian, 2000). A minimum thermal conductivity is observed for the superlattice period of 50 A˚, which is significantly lower than the solid solution of BiSbTe3. They discussed this behavior in terms of the blocking of the low-frequency phonons, that is, the interference of phonons between the superlattice interfaces. Although the phonon blocking is not clarified experimentally, the superlattice structure is effective to reduce the thermal conductivity in other materials. Recently, extremely low thermal conductivity has been reported in W/Se multilayers
ð96Þ
where a is the layer thickness, and tc the cross-plane hopping energy. The effective mass along the cross-plane direction is then evaluated as
Z20T
1.09.6.1
4.0 (1) 2.0
0.0 0.0
(2)
20.0
40.0 60.0 a (Å)
80.0
100.0
Figure 30 The calculated ZT as a function of the layer thickness. The curves (1) and (2) are ZT along the a- and c-axis directions, respectively. Reproduced from Hicks LD and Dresselhaus MS (1993a) Effect of quantum-well structures on the thermoelectric figure of merit. Physical Review B 47: 12727–12731.
Thermal Conductivity and Thermoelectric Power of Semiconductors
0.8
8
0.6
BiSbTe3 alloy
0.5
6
0.4
4
0.3 0.2
Imfp (Å) ( , , )
)
0.7 KL (W/m-K) ( ,
10
Sb2Te3
2
0.1 0
0 0
40
80 120 160 Superlattice period (Å)
200
Figure 31 Thermal conductivity of various Bi2Te3/Sb2Te3 superlattice films. Reproduced from Venkatasubramanian R (2000) Lattice thermal conductivity reduction and phonon localization like behavior in superlattice structures. Physical Review B 61: 3091–3097.
by Chiritescu et al. (2007). The magnitude is less than the minimum thermal conductivity given by Equation (41), and even lower than the thermal conductivity of argon gas. The enhancement of the density of states in superlattices is also experimentally observed. Ohta et al. (2007) showed that titanium oxide superlattice films have a 2D electron gas at the interface, and show five times larger Seebeck coefficient to give ZT ¼ 2.5. Figure 32 shows the room-temperature Seebeck coefficient of Sr(Ti0.8Nb0.2)O3/SrTiO3 superlattice films. When the thickness of the conductive Sr(Ti0.8Nb0.2)O3 layer is larger than four unit cells, the Seebeck coefficient is the same as the
351
bulk value. In the case of one unit-cell films, the Seebeck coefficient reaches 500 mV K1. In this oxide superlattice, a finite cover layer of SrTiO3 is necessary to confine the 2D electron gas. If this thickness is included, an effective ZT remains lower than unity. The third effect of superlattice is energy filtering. As mentioned in Section 1.09.4 the Seebeck coefficient is proportional to an average of the electron energy from the chemical potential "k , so that it is small in the presence of the electron–hole symmetry. The energy filtering is a theoretical proposal that a tunneling barrier is introduced in order that only electrons with higher energy should propagate in the sample. This was originally given as thermionic refrigeration by Mahan et al. (1998), and many researchers tried to design such thermionic devices. Vashaee and Shakouri (2004) showed that the energy filtering is enhanced in the presence of momentum-nonconserved scattering. Figure 33 schematically shows the momentum space for the tunnelling process. An electron energy is distributed between the two sphere shells with radii of kf pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and kf þ 2mkB T =h2 . Suppose that an electron with a lateral momentum k1 tunnels through the
kx V2: k > kb
kf + (2m*kBT/h2)1/2
V1: kz > kb kf 500 1 ML
Sr(Ti0.8Nb0.2)O3/SrTiO3
–S (μV K–1)
kz
300 K
400
300
2 ML
200 8 ML 100
kb
16 ML
ky
4 ML
0 0
1
2 3 4 5 6 Sr(Ti0.8Nb0.2)O3 layer thickness (nm)
7
Figure 32 The Seebeck coefficient of Sr(Ti0.8Nb0.2)O3/ SrTiO3 superlattice films as a function of thickness of Sr(Ti0.8Nb0.2)O3 layer at 300 K. From Ohta H, Kim S, Mune Y, et al. (2007) Giant thermoelectric Seebeck coefficient of a two-dimensional electron gas in SrTiO3. Nature Materials 6: 129–134.
k2 k1
Figure 33 Corresponding wave vectors in the k space, k1, k2, and kb correspond to cross-sectional planes and kf is the radius of the Fermi sphere. V1 is the volume of the electrons that participate in thermionic emission above the barrier if the lateral momentum is conserved. V2 is that volume if the lateral momentum is not conserved. Reproduced from Vashaee D and Shakouri A (2004) Electronic and thermoelectric transport in semiconductor and metallic superlattices. Journal of Applied Physics 95: 1233–1245.
352 Thermal Conductivity and Thermoelectric Power of Semiconductors
neighboring layer to get a lateral momentum k2. Then, the momentum space to conserve total momentum is limited in V1, while the space to allow the violation of the momentum conservation is enlarged to V2. As a result, such momentumnonconserving filters will remain conductivity as high as the bulk value, and enhance the Seebeck coefficient significantly. Zide et al. (2006) experimentally demonstrated the enhancement of the Seebeck coefficient in the superlattice. Figure 34 shows the in-plane and cross-plane Seebeck coefficients in InGaAs/ InAlGaAs superlattices. Normally, Seebeck coefficients are weakly anisotropic because the anisotropic factor is cancelled in the calculation (see Equation (49)). Contrary to this, the observed values are unconventionally anisotropic: the cross-plane data are three times larger than the in-plane ones, which is essentially independent of doping levels. Humphrey and Linke (2005) proposed a new design of thermoelectric superlattice, and showed that the conversion efficiency of this device reaches the Carnot efficiency, which corresponds to ZT ! 1. Their device consists of many quantum-wells, and a small temperature gradient is applied in accordance with implemented variation of chemical potential in order that the electron distribution function –1 " – ðrÞ f0 ¼ exp þ1 kB T ðrÞ
ð98Þ
is constant everywhere in the device. Under such conditions, all the wells are in thermal equilibrium, and the electron transfer is dissipationless.
1.09.6.2
Nanowires
Hicks and Dresselhaus (1993b) performed the same calculation for 1D wires as they did for 2D films and showed that ZT is enhanced with decreasing diameter of the wire. They have further found that the Bi wires change their ground state from semimetal to semiconductor, because the quantum confinement lifts the bottom of the conduction band above the top of the valence band. Figure 35 shows the electronic band diagram of Bi as a function of wire diameter (Lin et al., 2000). For a diameter as large as 200 nm, the band of the holes at the T point is above electrons at the L point. The energy shifts to higher values with respect to electron doping, and electrons and holes coexist up to 40 meV. The energy levels of the L electrons increase roughly in inverse proportion to the wire diameter, while the energy levels of the T holes rapidly decrease below 50 nm. As a result, a transition from semimetal to semiconductor occurs below 49 nm. Heremans et al. (2000) have fabricated Bi nanowire bundles and clarified that the electronic states of Bi nanowire can be regarded as single-band semiconductor. As discussed in Section 1.09.3, silicon is a good conductor of heat, and hence the thermoelectric figure of merit ZT remains a low value of 0.005 at 100
600
400 300 200
Le− (A)
60
Cross-plane
Energy (meV)
Seebeck coefficient (μV K–1)
80 500
T holes
40 −Δ0 = 38
20 49.0 nm 0
In-plane
Le− (B, C)
L holes
EgL = 15
−20
100
−40 0 0
2
6 8 4 Carrier concentration (1018cm–3)
10
Figure 34 The in-plane and cross-plane Seebeck coefficient in the InGaAs/InGaAlAs superlattice films. From Zide JMO, Vashaee D, Bian ZX, et al. (2006) Demonstration of electron filtering to increase the Seebeck coefficient in In0.53Ga0.47As/In0.53Ga0.28Al0.19 As superlattices. Physical Review B 74: 205335.
0
50
100 150 Wire diameter (nm)
200
Figure 35 Calculated energy band diagram of Bi wire plotted as a function of wire diameter. Below 49 nm, the energy level of the holes at T point goes down below the energy level of the electrons at L point. Reproduced from Lin Y-M, Sun X, and Dresselhaus MS (2000) Theoretical investigation of thermoelectric transport properties of cylindrical Bi nanowires. Physical Review B 62: 4610–4623.
Thermal Conductivity and Thermoelectric Power of Semiconductors
1.2 20-nm array (Boukai) 1 10-nm array (Boukai)
0.8
ZT
0.6 0.4 0.2 0 –0.2 50
50–nm wire (Hochbaum) 100
150
200 250 300 Temperature (K)
350
400
Figure 36 Dimensionless figure of merit ZT of Si nanowire samples. From Boukai AI Bunimovich Y, Tahir-Kheli J, Yu J.-K, Goddard WA, III, and Heath JR (2008) Silicon nanowires as efficient thermoelectric materials. Nature 451: 168–171; Hochbaum AI, Chen R, Delgado RD, et al. (2008) Enhanced thermoelectric performance of rough silicon nanowires. Nature 451: 163–167.
room temperature. This difficulty has been overcome by fabricating nanowires. Figure 36 shows the figure of merit of Si nanowires. Hochbaum et al. (2008) prepared free-standing silicon wires with diameters of 50–100 nm by self-organization technique based on selective etching, and found that they exhibit a low thermal conductivity of the order of 1 W mK1. This value is 100 times lower than the bulk value, and ZT increases by almost 100 times, and reaches 0.5 at room temperature. At the same time (in the same journal), another research group reported success in getting high ZT in silicon nanowire arrays. Boukai et al. (2008) fabricated arrays of 20-nm-width nanowires of silicon and measured the thermoelectric properties. Their samples also show extremely low thermal conductivity and some of them show ZT ¼ 1 at 200 K. The 100-times lower thermal conductivity observed in the two independent experiments awaits a theoretical explanation. Obviously, the phonon mean free path is of the order of 10–100 nm for bulk Si, being comparable with the wire diameter in the first experiment. Thus, it is highly nontrivial that the mean free path is reduced to be of the order of 1 nm in nanowire. Contrary to Si, nanowires of other elements exhibit unconventionally high thermal conductivity. Carbon nanotubes are nature-made nanowires (see, Ebbesen, 1994 and Ando, 2005), and their physical properties have been extensively investigated. They have a large structure variation with different radii
353
and chiralities. Since the growth conditions are nearly the same, the purification of a single structure is extremely difficult. The physical properties strongly depend on the structure, and samples can be semiconducting, metallic, and even superconducting (Murata et al., 2008). Owing to this, different values of thermal conductivity are reported from sample to sample. Kim et al. (2001) measured the thermal conductivity of a single carbon nanotube, as shown in Figure 37. They fabricated two microheaters of 10-mm2 area using Si3N4/Si-based technology, and set the sample between them. The measured thermal conductivity is 3000 W mK1 at room temperature. This value is higher than that of diamond, and is attractive for application. They further measured the diameter dependence, and found that the thermal conductivity increases with decreasing diameter. For a 1-nm diameter, the thermal conductivity reaches 8000 W mK1, much higher than other bulk semiconductors. The temperature dependence suggests nearly ballistic conduction below 300 K, which breaks owing to the Umklapp scattering above 300 K. The Seebeck coefficient is also measured for the same sample, and is found to be 42 mV K1 at 300 K and roughly is linear in T. This suggests that the electronic states are basically the same as in metals. 1.09.6.3
Nanostructured Bulk Materials
As discussed above, nanostructuring is a powerful method to control thermal and thermoelectric properties in semiconductors. However, heat is an essentially macroscopic object, and cannot be confined in nanospace. Thus, bulk materials are still indispensable to practical applications for thermal and thermoelectric management. It is thus important to synthesize bulk materials with nanostructure modification. Sharp et al. (2001) pointed out that the mean free paths of phonons are more than one order of magnitude larger than those of electrons, and proposed that only the former can be reduced by properly choosing the domain size of the material. Poudel et al. (2008) succeeded in increasing ZT of Bi2Te3 by 50% by reducing the lattice thermal conductivity using nanostructured samples. Ohtaki et al. (2007) included 100-nm-sized organic beads with ZnO powder, and made ZnO ceramic samples with 100-nm-sized voids by evaporating the beads through sintering. The samples show lower thermal conductivity, and ZT is substantially enhanced at high temperatures.
354 Thermal Conductivity and Thermoelectric Power of Semiconductors
10–8 3000
κ (T) (W/m K)
Thermal conductance (W K–1)
10–7
10–9
2000 1000 0
10
100
200 T (K)
300
100 Temperature (K)
Figure 37 Themal conductivity of a single carbon nanotube. The sample mount is shown in the inset. The lower inset shows the thermal conductivity of carbon nanotubes with different diameters. The solid, dashed, and dotted curves correspond to diameters of 14.80, and 200 nm, respectively. Reproduced from Kim P, Shi L, Majumdar A, and McEuen PL (2001) Thermal transport measurements of individual multiwalled nanotubes. Physical Review Letters 87: 215502.
Hsu et al. (2004) found that a PbTe-based new material AgPb18SbTe20 (named LAST) has a high ZT of 2.2 at 800 K. The mechanism of the high ZT is still controversial, but they attributed this to a special tissue of this material. In this material, nanometer-sized droplets of an Ag–Sb rich phase are precipitated in the host PbTe, and may work as quantum-dots. Preceding this work, Harman et al. (2002) showed that quantum-dots in PbTe-based superlattice enhance S and ZT. Self-organized precipitation of nanosized particles works to reduce the thermal conductivity in GaAs-based semiconductor films. Kim et al. (2006) found that coevaporation of Er and Ga in As atmosphere forms nanosized particles of ErAs in the GaAs films. Figure 38 shows the self-grown ErAs dots (shown as dark spots) in a (In,Ga)As film. Since the particle diameter and the interparticle distance are approximately 1 and 10 nm, respectively, these structures may effectively scatter long-wavelength phonons. They measured the thermal conductivity with and without ErAs particles, and found that the lattice thermal conductivity is significantly reduced. Another type of self-organized nanostructure modification is seen in the layered cobalt oxides
5 nm
Figure 38 Self-precipitated ErAs dots (dark dots) in (In,Ga)As matrix. Scale ¼ 5 nm. Reproduced from Kim W, Zide J, Gossard A, Klenov D, Stemmer S, Shakouri A, and Majumdar A (2006) Thermal conductivity reduction and thermoelectric figure of merit increase by embedding nanoparticles in crystalline semiconductors. Physical Review Letters 96: 045901.
(see Section 1.09.5.3). This class of materials has a layered structure consisting of the CdI2-type CoO2 layer and the NaCl-type block layer. This implies
Thermal Conductivity and Thermoelectric Power of Semiconductors
160
CaCoPb
S300 K (μV K−1)
CaBi 120
CaCo
SrBt SrCo
BaBi
SrTI
80
Sr (this work) 40
CaO2 RS b1
n=2 n=1 n=0
CoO2 b2
0 0.50
0.55
0.60
Misfit ratio (=b2/b1) Figure 39 Room-temperature Seebeck coefficient plotted as a function of lattice misfit ratio b1/b2, where b1 and b2 are the b-axis lengths of the block layer and the CoO2 layer, respectively. The inset shows a schematic drawing of the crystal structure of the layered cobalt oxides. Reproduced from Ishiwata S, Terasaki I, Kusano Y, and Takano M (2006) Transport properties of the misfit layered cobalt oxide [Sr2O2D]0.53CoO2. Journal of the Physical Society of Japan 75: 104716.
that a new thermoelectric oxide can be designed by properly combining the block layer, which is called nanoblock integration by Koumoto et al. (2006). A characteristic feature is that the CoO2 layer and the block layer have a lattice mismatch along the b-axis, and the interface is drastically modified by the misfit structure. Maignan et al., (2002) first recognized that the thermoelectric properties in a certain class of cobalt oxides are controlled by the misfit structure. Ishiwata et al. (2006) and Kobayashi and Terasaki (2006) extended this concept to a whole family of the layered cobalt oxides, and found that the Hall coefficient and the Seebeck coefficient are determined by the lattice ratio of the b-axis (b2) of the CoO2 layer to the b-axis (b1) of the block layer. Figure 39 shows that the roomtemperature Seebeck coefficients of various layered cobalt oxides plotted as a function of b2/b1 (Ishiwata et al., 2006), in which all the data tend to increase with b2/b1. Okada et al. (Okada and Terasaki 2005; Okada et al., 2005) synthesized an isomorphic layered rhodium oxide Bi-M-Rh-O, and found that the Seebeck coefficient is also dominated by b1/b2 (Terasaki et al., 2006).
1.09.7 Summary and Outlook In this chapter, we have briefly reviewed the thermal conductivity and thermoelectric power of various semiconductors including amorphous solids, magnetic
355
oxides, and nanostructured materials. While Drude’s kinetic theory of an electron gas unexpectedly works well in element semiconductors, it does not cover more complicated materials. In particular, localized phonon modes such as the rattling mode let perfect crystals to be glass-like, which is named phonon glass by Slack (1995). Nanostructuring effectively reduces the lattice thermal conductivity of silicon, which can control heat flow in the nanowires. However, our theoretical understanding of thermal conduction is far away from the goal, possibly owing to the fact that heat conduction of phonons cannot be treated in quantum physics in a strict sense. We do not fully understand the difference between sound and heat, both of which are described by lattice vibration. Although some theoretical trials are being developed, the concepts of localization, decoherence, confinement, quantum transport, and so on for phonons are still premature, in comparison with the corresponding ideas for electrons. We have also reviewed thermoelectric phenomena, which arise from the cross correlation between electrical and thermal currents in solids. They can be applied to energy conversion between heat and electricity, which attracts renewed interest owing to pressing needs for energy and environment issues. This technology, called thermoelectrics, requires materials having high thermoelectric power with low thermal conductivity and resistivity. To get high efficiency of thermoelectric energy conversion, various complex materials are designed, synthesized, and identified. For example, skutterudites and clathrates show anomalously low thermal conductivity due to the rattling modes, and the layered cobalt oxides have large thermoelectric power coming from the degeneracy of the spin and orbital configuration on the cobalt sites. Nanotechnology has proved to be powerful for improving the thermoelectric performance of bulk materials. Semiconductor technology has been developed with precise control of the electrical current of doped carriers, whereas the thermal current has been left uncontrolled. But now, through the recent development of nanotechnology, we may be able to control heat in the device; the thermal conductivity of bulk materials can be reduced by 100 times, and the thermoelectric power can be increased by several times. Of course, we cannot make Maxwell’s daemon, but we will be able to manage heat much better than we do at present. In this respect, the author hopes that semiconductor technology of next generation will handle the electrical and thermal currents on an equal basis.
356 Thermal Conductivity and Thermoelectric Power of Semiconductors
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1.10 Electronic States and Transport Properties of Carbon Crystalline: Graphene, Nanotube, and Graphite Y Iye, University of Tokyo, Kashiwa, Chiba, Japan ª 2011 Elsevier B.V. All rights reserved.
1.10.1 1.10.1.1 1.10.1.2 1.10.1.3 1.10.2 1.10.2.1 1.10.2.2 1.10.2.3 1.10.2.4 1.10.2.5 1.10.2.6 1.10.3 1.10.3.1 1.10.3.2 1.10.3.3 1.10.4 1.10.4.1 1.10.4.2 1.10.4.3 1.10.5 1.10.5.1 1.10.5.2 1.10.5.3 1.10.5.4 1.10.6 References
Introduction Carbon Allotropes Bond and Local Coordination Dimensionality Electronic Structures Diamond 2D Graphite (Graphene) Bilayer Graphene Single-Wall Carbon Nanotubes Graphene Nanoribbons Bulk Graphite Transport Properties of Graphene Preparation of Graphene Samples Transport at Zero Magnetic Field Quantum Hall Effect Transport Properties of Carbon Nanotubes Electronic State in SWNTs Electron Transport in SWNTs Multi-Wall Nanotubes Transport Properties of Graphite Graphite Materials and Transport Characteristics Semi-Classical Magneto-Transport Magneto-Quantum Oscillations Magnetic-Field-Induced Electronic Phase Transition Concluding Remarks
1.10.1 Introduction 1.10.1.1
Carbon Allotropes
Carbon is one of the most ubiquitous elements on the earth. It is the basic building ingredient of all the organic molecules that constitute our own body and other biological systems. Indeed, the diversity of composition, structure, and functionality of organic molecules is truly rich and awesome. Carbon, however, is also unique in that even as elementary substance it occurs in vastly different forms. Diamond and graphite are just two allotropes of carbon known since antiquity. Another wellknown allotrope is amorphous carbon materials which are found, together with micro- or nanocrystalline materials of graphite or diamond, in coals and soot. During the last quarter of a century, a few new members of
359 359 360 361 361 361 361 365 365 367 368 369 369 369 370 372 372 372 374 374 374 374 376 377 380 380
carbon allotropes were discovered and created much excitement in the research community. Fullerene molecules C60 and C70 were discovered by Kroto et al. (1985). These were initially identified as distinct peaks in the time-of-flight mass spectrograph of carbon clusters evaporated from a graphite target by laser irradiation. The high stability of a cluster with exactly 60 carbon atoms led these authors to propose the truncated dodecahedron or soccer-ball structure. Owing to the small amount of fullerene molecules that could be obtained by the laser abrasion technique, research into fullerene was not fully developed for the first few years after their discovery, let alone investigation of the properties of fullerene solids. The situation changed in 1990 when Kretschmer and Huffman announced massive
359
360 Electronic States and Transport Properties of Carbon Crystalline
production of fullerene molecules by an arc discharge method (Dresselhaus et al., 1996). The arc discharge method yielded another, no less important, family of allotropes, that is, carbon nanotubes. Iijima (1991), while investigating under an electron microscope the soot deposits on the cathode of his arc discharge apparatus, discovered thin tubes made of graphitic layers. Although one can spot in retrospect a few earlier reports on similar substances (Oberlin et al., 1976), it was the report by Iijima that triggered the subsequent research activities on carbon nanotubes (Saito et al., 1998; Dresselhaus et al., 2001). Whereas the initial discovery was on multi-walled nanotubes, single-walled nanotubes were later prepared successfully (Iijima and Ichihashi, 1993; Bethune et al., 1993). A further surprise was brought about in 2004 by the Manchester University group led by Geim (Novoselov et al., 2004; Geim and MacDonald, 2007; Geim and Novoselov, 2007). By an astonishingly simple peeland-stick technique, they prepared samples of a single-layer sheet of graphite on an oxidized surface of silicon wafer. Although cleavage was routinely done by those who handled graphite samples, it came as a total surprise that samples of single-layer graphite of semimacroscopic size can be obtained by such a method. The discovery of these novel species of carbon allotropes has vastly broadened the horizon of condensed matter physics. The exotic electronic properties of nanotubes and graphenes and their potential applications to future electronics are currently under vigorous study worldwide. The carbon materials listed above and depicted in Figure 1 are representative ones that have been most extensively studied. There are many more exotic allotropes, even limiting ourselves to those purely made of carbon atoms. In addition to C60 and C70, there are socalled higher-order fullerenes such as C84. The crystals made of the fullerene molecules can take different
structures. Under appropriate conditions, fullerenes can form dimers, trimers, and various polymers. One of the most beautiful allotropes is carbon nanopeapods, that is, carbon nanotubes enclosing C60 molecules (Smith et al., 1998). Exploring further beyond the boundary of pure carbon territory, one finds a vast landscape of materials diversity. By using a metal-containing carbon rod in the arc discharge process, metallofullerenes, that is, fullerene molecules encapsulating metal ion(s), can be produced. Nanotubes can accommodate various foreign species (including fullerenes as mentioned above). Extensive studies have been made on graphite intercalation compounds (GICs) (Dresselhaus and Dresselhaus, 1981). A wide variety of guest species (intercalant) can be inserted in the interlayer spacing of graphite in a well-ordered fashion. The resulting GICs are good examples of synthetic metals.
1.10.1.2
Bond and Local Coordination
Carbon atom has four valence electrons in the 2s and 2p atomic orbitals. The carbon–carbon (C–C) bond can be either sp3- or sp2-hybridized bond. The richness of carbon allotropes stems from various combinations of the sp3- and sp2-hybridized bonds. In diamond crystals, each carbon atom is tetrahedrally coordinated. Strong C–C bond is formed with each of the neighboring four atoms by sp3hybridization. The C–C bond length in diamond is 0.154 nm. The crystal structure is cubic with lattice constant 0.357 nm. The diamond lattice can be viewed as consisting of two face-centered cubic 111 (f.c.c.) sublattices of which one is shifted by 444 relative to the other.
Figure 1 Carbon allotropes: diamond, graphite, C60 molecule, and carbon nanotube.
Electronic States and Transport Properties of Carbon Crystalline
Carbon atoms in a graphitic sheet form a honeycomb network so that each atom is trigonally coordinated. Three of the four valence electrons are used for the sp2-hybridized -bonds with the three neighbors. The remaining one is in the -orbital and can be easily delocalized. The C–C bond length in the graphitic sheet is 0.142 nm. The three-dimensional (3D) graphite is made of a stack of graphitic sheets. The ordinary graphite (Bernal graphite) exhibits the stacking order ABAB. . .. A thermodynamically unstable variant called rhombohedral graphite has a stacking sequence ABCABC. . .. In the case of amorphous carbon, the C–C bonds consist of mixture of sp3- and sp2-hybridized bonds. Actually, one important parameter that characterizes the amorphous carbon is the ratio of sp3- and sp2-hybridized bonds. Materials that are high in sp3-hybridized bonds are referred to as tetrahedral amorphous carbon or as diamond-like carbon owing to the similarity of many physical properties to those of diamond. Raman spectroscopy provides a useful means of characterization. The Raman spectrum of diamond shows a sharp peak at ! ¼ 1332 cm1, while the frequency of the Raman-active in-plane mode of graphite is ! ¼1582 cm1. The Raman spectra of amorphous carbon generally exhibit broadened mixture of these two peaks. 1.10.1.3
Dimensionality
The line-up of carbon allotoropes introduced in the previous section provides a set of model systems for low-dimensional physics; that is, diamond (3D), graphite (quasi-2D), graphene (2D), nanotube (1D), and fullerene (0D). Dimensionality is a key ingredient that profoundly affects transport and other electronic properties. Indeed, phenomena intimately linked to the dimensionality have been observed in these systems. 2D electron system created at the semiconductor heterointerface is an experimental stage of many intriguing physics, including quantum Hall effect (QHE) and electron localization. Up until the advent of graphene, the experimental systems were virtually limited to GaAs/AlGaAs and SiMOS. Graphene provides not just another 2D system but a very unique one, as explained in later sections. Carbon nanotubes are regarded as nearly ideal 1D quantum wire. Some of the properties of carbon nanotubes seem to corroborate the Tomonaga–Luttinger liquid (TLL) behavior theoretically developed for interacting 1D electron systems. Nanotube is also used in a device called
361
single electron tunneling (SET) transistor, in which tunneling of individual electrons between the quantum-dot and the leads can be controlled by a gate voltage. A single fullerene molecule trapped in the nano-sized gap between two electrodes can be an ultimate quantum-dot. Thus, graphene, nanotube, and fullerene are regarded as wonderful playgrounds for nanoscience and as precious parts of nanotechnology. In the subsequent sections, some of the basics about the carbon-based electronic systems are explained, and the excitement brought by the research efforts since the late 1950s is conveyed. This chapter mainly focuses on the transport properties.
1.10.2 Electronic Structures 1.10.2.1
Diamond
Since this chapter is principally concerned with the graphitic systems, we only briefly touch upon the electronic structure of diamond. Diamond is a wide gap semiconductor. Figure 2(a) shows the band structure of diamond (Saslow et al., 1966; Saravia and Brust, 1968), which is similar to that of silicon. The conduction-band minima are located along the –X lines of the Brillouin zone. The surfaces of equal energy are ellipsoids with m1 ¼ 1.4 m0 and mt ¼ 0.36 m0. The valence band top is located at the -point. The effective masses of heavy and light holes are mhh ¼ 2.12 m0 and m1h ¼ 0.7 m0. The split-off band is 6 meV below the heavy and light hole bands and the effective mass is m1h ¼ 1.06 m0. The indirect gap is 5.47 eV and the direct gap at the -point is 7.3 eV. 1.10.2.2
2D Graphite (Graphene)
Calculations of the electronic band of 2D version of graphite (i.e., graphene in today‘s terminology) have been made by many authors for over 50 years (Wallace, 1947; Coulson, 1947; Bassani and Parravicini, 1967; Painter and Ellis, 1970; Zunger, 1978). Figure 2(b) shows the calculated band structures of 2D graphite taken from Zunger (1978). The Brillouin zone edges denoted by P and Q in the figure correspond to K and M in the today’s conventional notation. The s-orbital and two in-plane p-orbitals form the sp2-hybridized -bonds which make up the strong honeycomb framework of graphene. The bonding -bands
362 Electronic States and Transport Properties of Carbon Crystalline
(a) k2
Λ1
20
Δ2′ Γ12′
Λ3 Γ
Λ
k2
10
Σ
Δ XZ
W
K
E (eV)
k2
l
L3
Λ3
L1
Λ1
Γ2
Δ5 Δ2′
Γ15
X1 Δ1
Γ25′
0
Δ5
Λ3
L3′
Δ1
X4
–10 Δ2′
Λ1 L1 –20
C
L2′
X1 Δ1
Λ1
Γ1
–30 (b) 10
(a) +
P3+
+ Q2v
(b)
Γ1v
(c)
–
Γ2g
–
Γ2g
–
0 +
Energy (eV)
Γ3u
+ Qlu
+ Γ3u
–10
+
Γ3g
Q2u
–
+ Q2g
+
+ Qlu + Qlg
P3–
–
+
Γ3g –
–20 P1+ P3+
Γ2u
Γlg –30
P
Γ
Γ2u
P1+
–
+
Q2g
P3–
+
Γ3g
Q2u + Q2g
+
Q2u + Qlu
+
P1+
Γ2u
+ P3
+ Γlg
–
Q2u +
Q2g +
P3+ +
Γlg Q
+
Γ3u – Q2g
–
Q2g
–
P3
Γ2g
+ P2
P
Γ
+ Qlu + Qlg
Q
P
Γ
Qlu +
Qlg
Q
Figure 2 The Brillouin zone and electronic band of (a) diamond and (b) two-dimensional graphite. (a) Reprinted with permission from Saslow W, Bergstresser TK, and Cohen ML (1966) Band structure and optical properties of diamond. Physical Review Letters 16: 354. Copyright (1966) by the American Physical Society. (b) Reprinted with permission from Zunger A (1978) Self-consistent LCAO calculation of the electronic properties of graphite. I. The regular graphite lattice. Physical Review B 17: 626. Copyright (1978) by the American Physical Society.
and antibonding 9-bands are shown by three solid curves in the lower part and another three in the upper part of the band diagrams in Figure 2(b). The dashed curves represent the bonding -band and antibonding 9-band formed by the remaining out-of-plane p-orbital. The bonding -band and antibonding 9-band touch with each other at the K-point, where the Fermi level of the undoped graphene resides.
As it constitutes the basis for the electronic structure of all graphitic systems, let us take a closer look at the graphene -band structure. The electronic structure of the graphene can be reasonably well described by a simple tight-binding Hamiltonian for the - and 9-bands. The notations here follow those by Ando (2005). The honeycomb lattice of graphene consists of two interpenetrating triangular sublatteices (referred to as
Electronic States and Transport Properties of Carbon Crystalline
(a)
363
(b) 4
z
M
y
3 Energy (units of γ0)
x
γ1 γ0 Monolayer
Bilayer
(c)
ε
ε
K
Monolayer
κx
K
Γ
2 1 EF
0 −1 −2 −3
κy
κy
K
Γ
K
M
K
Wave vector κx
Bilayer
Figure 3 (a) Left: Honeycomb lattice of graphene. Right: Stacking of bilayer graphene. (b) The - and 9-bands of graphene. (c) Characteristic band structures at the K point for monolayer (left) and bilayer (right) graphene.
A and B sublattices hereafter) as depicted in Figure 3(a) with black and white atoms. Each atom of one sublattice is surrounded by three atoms belonging to the other sublattice, and vice versa. The primitive translation vectors can be taken as a1 ¼ a(1,0) and pffiffiffi 1 3 a2 ¼ a – ; , and the vectors connecting the 2 2 1 nearest neighbor carbon atoms are 1 ¼ a 0; pffiffiffi , 3 1 1 1 1 2 ¼ a – ; pffiffiffi , and 3 ¼ a ; – pffiffiffi , where 2 2 3 2 2 3 pffiffiffi a ¼ 0:142 3 ¼ 0:246 nm is the lattice constant. pffiffiffi 3 2 a . The corresponding The unit cell area is 0 ¼ 2 primitive reciprocal lattice vectors are 2 1 2 2 1; pffiffiffi and a2 ¼ 0; pffiffiffi , and the a1 ¼ a a 3 3 area of the hexagonal first Brillouin zone is 2 2 2 0 ¼ pffiffiffi . 3 a In the tight-binding model, the wave function is expressed as the sum of the atomic orbitals localized at the lattice points:
ðrÞ ¼
X
A ðrÞðr
RA Þ þ
RA
X
B ðrÞðr – RB Þ
ð1Þ
RB
Here, ðrÞ is the pz-orbital of a carbon atom, and RA ðRB Þ specifies the lattice point of the A- (B-) sublattice. We write the transfer integral between the nearest-neighbor carbon atoms as – 0 , and ignore the overlap integral between the A and B sublattices. Then the equation reads "
A ðRA Þ
¼ – 0
3 P
B ðRA
– iÞ
i¼1
"
B ðRB Þ ¼ – 0
3 P
ð2Þ A ðRB þ i Þ
i¼1
Assuming A ðRA Þ ¼¼ fA ðkÞexpði k ? RA Þ Equation B ðRB Þ ¼¼ fB ðkÞexpði k ? RB Þ, becomes "fA ðkÞ ¼ – 0
3 P
fB ðkÞexpð – ik : i Þ
i¼1
"fB ðkÞ ¼ – 0
3 P
and (2)
ð3Þ fA ðkÞexpðþik : i Þ
i¼1
The eigenvalues of this equation are obtained as
364 Electronic States and Transport Properties of Carbon Crystalline sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 3ky a kx a kx a " ðkÞ ¼ 0 1 þ 4cos þ 4cos2 cos 2 2 2
ð4Þ
which is shown in Figure 3(b). Note that " ðkÞ ¼ 0 at the K- and K9-points. In the vicinity of the K-points, the dispersion is linear; that is,
" ðkÞ ¼ hvF jkjj
ð5Þ
for a << 1, where k X k – K is the wave pffiffiffi vector measured from the K-point and vF ¼ 30 a=2h is the group velocity of the electrons (vF ¼ 1106 m s – 1 for graphene). The same relation holds for the K9-point. This can be represented as cone structure
at the corners of the Brillouin zone as shown on the left of Figure 3(c). The corresponding density of states reads Dð"Þ ¼
j"j 2h2 vF2
ð6Þ
This linear dispersion in the vicinity of the Kand K9-points is what makes the graphitic systems (graphene and nanotube) unique in their electronic properties, as will be seen in the following. Within the effective mass approximation, the Schro¨dinger equation for graphene -band electrons can be written as
Hˆmono FðrÞ ¼ " FðrÞ 0 K 1 0 K 1 0 K 1 FA ðrÞ FA ðrÞ F ðrÞ A; FK ðrÞ ¼ @ A; FK ðr Þ ¼ @ A FðrÞ ¼ @ FKB ðrÞ FKB ðrÞ FK9 ðrÞ 0 1 0 0 0 hvF ˆ x – iˆ y B C B C B hvF ˆ x þ iˆ y C 0 0 0 B C C Hˆmono ¼ B B C B C ˆ ˆ þ i 0 0 0 h v F x y B C @ A 0 0 hvF ˆ x – iˆ y 0
ð7Þ
ð7Þ
Here, FAK ðrÞ, etc. are the slowly varying envelope functions. The above matrix equation can be rewritten in the following more compact form: (
hvF ðs?kˆ ÞFK ðrÞ ¼"FK ðrÞ
ð8Þ
hvF ðs ?kˆ ÞFK9 ðrÞ ¼ "FK9 ðrÞ
Here, s ¼ x ;y is the pseudo-spin Pauli matrices (the pseudo-spin corresponding to the A and B sublattice), and kˆ ¼ – ir is the wave number operator. Equation (8) is isomorphic to a relativistic Dirac equation for a particle with vanishing rest mass, known as Wyle‘s equation. In this sense, electrons and holes in monolayer graphene are often referred to as massless Dirac fermions. Here, vF ¼ 1 106 m s – 1 plays the role of the speed of light. The quasi-particles in monolayer graphene can be described by the effective Hamiltonian Hˆmono ¼ hvF
0
ˆ x – i ˆ y
ˆ x þ i ˆ y
0
! ð9Þ
which operates in the pseudo-spin space that describes the amplitude of electron waves on the A and B sublattices. The eigenvalues of this Hamiltonian are given by Equation (5). This form is convenient for highlighting the difference from the bilayer case to be discussed in the next subsection. Although the concept of massless Dirac fermion in solid state is not new but has been invoked in discussing certain aspects of the electronic properties of graphite (McClure, 1956) and other systems such as bismuth (Fukuyama and Kubo, 1970) and HgTe, demonstration of its genuine occurrence with all subtleties has been first achieved in monolayer graphene. However, occurrence of genuine massless Dirac fermions in solid state is not exclusive to graphene. Another example recently elucidated is the zero-gap state in a molecular conductor -(bis(ethylenedithia)tetrathiafulvalene (BEDT-TTF))2I3 under pressure (Tajima et al., 2000), whose electronic structure can be described by a tilted Weyl equation with anisotropic Fermi velocity (Katayama et al., 2006; Kobayashi et al., 2009).
Electronic States and Transport Properties of Carbon Crystalline
1.10.2.3
Bilayer Graphene
In a graphene bilayer with Bernal (AB) stacking shown on the right of Figure 3(a), one of the two sublattices in the top layer comes right above a sublattice in the bottom layer, while the other sublattice does not find such a counterpart. The transfer integral 1 between
365
the vertically aligned atoms gives rise to interlayer coupling that changes the band structure of bilayer graphene from the monolayer case. The Schro¨dinger equation for bilayer graphene (to be contrasted with Equation (7) for the monolayer case) can be written as
K Hˆbi FK ðrÞ ¼ "FK ðrÞ
FK ðrÞ ¼ ðF1B ðrÞ; F1A ðrÞ; F2A ðrÞ; F2B ðrÞÞ 0 1 0 vF ˆ x þ iˆ y 0 0 B C B C B vF ˆ x iˆ y C 0 0 1 B C K C Hˆbi ¼ B B C B C ˆ ˆ 0 v þ i 0 1 F x y B C @ A ˆ ˆ 0 0 0 vF x iy
ð10Þ
ð10Þ
together with a similar expression for the K9 valley. This results in four valley-degenerate bands as shown on the right of Figure 3(c). A symmetric pair of the higher p energy ffiffiffiffiffiffiffi bands are separated by a gap of magnitude 21 at ¼ 0, which originates from the interlayer coupling 1 . The effective Hamiltonian for the lower energy band pair relevant to the transport and other low energy properties can be written as 0
1 2 vF2 ˆ ˆ – i x y B C 1 B C Hˆbi ¼ B C @ v2 A 2 F ˆ x þ iˆ y 0 1 0
ð11Þ
which is to be contrasted with Equation (9) for the monolayer case. The energy eigenvalues of Equation (11) are ! rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1 2 þ ðhvF Þ – 1 "ðkÞ ¼ 4 1 2 ðhvF Þ2 2 12
ð12Þ
Thus, the low energy bands of bilayer graphene are quadratic and meets at k ¼ 0. This dispersion relation can be written in the form of a massive electron "ðkÞ ¼ h2 2 =2m , with the effective mass m ¼ 1 =2vF2 , as opposed to the linear dispersion in the monolayercase.Usingavalue 1 0.39 eVtakenfrombulk graphite and vF 1106ms1, the effective mass is
estimated as m 0.034m. In contrast to the monolayer case, electrons and holes in bilayer graphene act as massive Dirac fermions as shown on the right of Figure 3(c).
1.10.2.4
Single-Wall Carbon Nanotubes
Single-wall nanotubes (SWNTs) are formed by rolling up a sheet of graphene into a nanometer-sized cylinder. Electronic motion along the circumference direction is then subjected to a periodic boundary condition that depends on how the sheet is rolled up. The way a graphene sheet is rolled up to make SWNT is specified by a lattice translation vector, called chiral vector, pffiffiffi 1 3 L ¼ n1 a1 þ n2 a2 ¼ n1 ð1; 0Þ þ n2 – ; 2 2 pffiffiffi 3 1 ¼ n1 – n2 ; n2 2 2
ð13Þ
as shown in Figure 4. The hexagons separated by L in the original graphene sheet are rolled onto one another in the SWNT. The length of L is expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jLj ¼ jn1 a1 þ n2 a2 j ¼ n21 ja1 j2 þn22 ja2 j2 þ2n1 n2 a1 ? a2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ a n21 þ n22 – n1 n2 ,a1 ?a2 ¼ – a2 =2 ð14Þ
366 Electronic States and Transport Properties of Carbon Crystalline
ðr þ LÞ ¼ ðrÞ, or equivalently expðik?LÞ ¼ 1, to the graphene band. (For smaller diameter nanotubes, the curvature of the cylindrical surface becomes non-negligible so that mixing between the -bands and -bands has to be taken into account.) The kx -component of the wave vector is quantized to kx ¼ ð2=LÞs (s : integer), while the ky -component remains continuous, so that the allowed values of k lie on a series of parallel straight lines. The band structure of an SWNT critically depends on whether these lines pass through the K and K9-points of original graphene Brillouine zone. When they do, the phase factor takes the following form:
(a) y′
y
τ3
a τ 2
A
T
τ1
b
η
(ma, mb)
B unit cell
x
η
L
x′
(na, nb)
η
a2 a1
(0,0) (b)
Ky K
η
K′ K
Armchair (η = π/6) Kx
K′
K
2 2 expðiK?LÞ ¼ exp i ðn1 þ n2 Þ ¼ exp i 3 3 2 2 expðiK9?LÞ ¼ exp – i ðn1 þ n2 Þ ¼ exp – i 3 3
Zigzag (η = 0)
η
K′
z B
(c)
y x
φ
ð19Þ
L Figure 4 Construction of a carbon nanotube from a graphene sheet. Chiral vector. Reproduced with permission from Ando T (2005) Theory of electronic states and transport in carbon nanotubes. Journal of the Physical Society of Japan 74: 777.
Since the chiral vector L becomes the circumference, the diameter of the SWNT is d¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L a n12 þ n22 – n1 n2 ¼
ð15Þ
Thus, every SWNT is specified by a set of two integers ðn1 ;n2 Þ. The primitive translation vector T along the length of nanotube is written as T ¼ m1 a1 þ m2 a2
ð16Þ
The condition L ? T ¼ 0 reads m1 ð2n1 – n2 Þ þ m2 ð2n2 – n1 Þ – 0
ð17Þ
which can be solved as m1 ¼
n1 – 2n2 2n1 – n2 and m2 ¼ P P
ð18Þ
where p is the greatest common divisor of n1 – 2n2 and 2n1 – n2 . The unit cell of an ðn1 ;n2 Þ nanotube is the rectangular region defined by T and L. The electronic bands of an SWNT is obtained by imposing a periodic boundary condition
where X ðn1 þ n2 Þðmod3Þ is an integer that takes a value 0 or 1. If ¼ 0, the SWNT is metallic because it has two bands crossing at the K- and K9-points without gap. If ¼ 1, on the other hand, there is a non-zero-gap so that the SWNT is semiconducting. An SWNT with a nonspecial L possesses a helical (chiral) structure. There are two special categories of nonchiral SWNT: zigzag nanotubes with ðn1 ; n2 Þ ¼ ðm; 0Þ and armchair nanotubes with ðn1 ; n2 Þ ¼ ð2m; mÞ. According to the above criterion for metallicity, zigzag nanotubes are metallic when m is devisible by 3 and semiconducting otherwise. By contrast, all armchair nanotubes are metallic because n1 þ n2 ¼ 3m. The periodic boundary conditions for the envelope function are 2 FK ðr þ LÞ ¼ e – iK?L FK ðrÞ ¼ exp – i FK ðrÞ 3 ð20Þ 2 K9 K9 – iK9?L K F ðrÞ ¼ exp i F ðrÞ F ðr þ LÞ ¼ e 3
The extra phase appearing here as an electron travels around the cylinder corresponds to a fictitious magnetic flux of magnitude ¼ – ð =3Þ0 for the K-point and ¼ ð =3Þ0 for the K9-point (0 ¼ h=e being the flux quantum) piercing the tube. The envelope wave function is given by a plane wave FK ðrÞ_ exp – ikx x þ iky y . The kx -component is quantized to
Electronic States and Transport Properties of Carbon Crystalline
kx ¼ v ðnÞ ¼
2 n – ; n ¼ 0; 1; 2; ; L 3
ð21Þ
while the ky-component is continuous. The energy eigenvalues are expressed in terms of the quantum number n that specifies the circumferential mode and the wave number ky ð¼ kÞ for the 1D motion along the length of the nanotube as "ðn; kÞ ¼ hvF
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðnÞ2 þ k2
ð22Þ
Figure 5 shows the band structure in the vicinity of the K-point for ¼ 0 and ¼ þ1 (Ando, 2005). For ¼ 0, the n ¼ 0 subband has a linear dispersion and other subbands are twofold degenerate with respect to the positive and negative values of n. For ¼ 1, on the other hand, the ðn ¼ 0Þ subband has a gap "G vF =3L so that the system is n¼0 ¼ "ð0;0Þ ¼ 4h semiconducting. This sort of subband structure can be probed by optical absorption spectroscopy. Kataura et al. (1999) succeeded in preparing a series of SWNTs with different diameters and demonstrated systematic evolution of the subbands with the nanotube diameter. It was subsequently demonstrated that incorporation of Coulomb effects (excitonic effects) is crucial (Ichida et al., 2002). For a metallic SWNT with linear dispersion, the density of states is given by Dð"Þ ¼
4 hvF
ð23Þ
where the factor 4 comes from the spin and valley degeneracy. This is in contrast to the case of graphene (Equation (6)), that is, 2D system with a linear dispersion, where the density of states vanishes at " ¼ 0. ε
ε n = +2 n = –1
n = +2, –2 n = +1, –1
n = +1 n=0
n=0 k
k n=0 n = +1
n = +1, –1
ν=0
n = –1
ν = +1
Figure 5 Subbands of single-wall carbon nanotube for metallic case, ¼ 0 (left), and semiconducting case ¼ 1 (right). Reproduced with permission from Ando T (2005) Theory of electronic states and transport in carbon nanotubes. Journal of the Physical Society of Japan 74: 777.
367
As mentioned above, the periodic boundary condition (Equation (20)) contains an extra phase that can be viewed as arising from a fictitious magnetic flux of magnitude ð =3Þ0 . When an external magnetic field is applied parallel to the nanotube axis, the Aharonov–Bohm (AB) phase is added to the periodic boundary condition. Equation (21) is then modified to kx ¼ ðnÞ ¼
2 n– þ ; n ¼ 0; 1; 2; L 3 0 ð24Þ
This poses an interesting possibility that the electronic band structure of an SWNT can be tuned by an AB flux due to an external magnetic field parallel to the tube axis. As the AB flux is changed, the energy gap oscillates between 0 and 2hvF =L with period 0 . Namely, a semiconducting nanotube can be transformed metallic at appropriate values of the AB flux where the gap vanishes either at K- or K9-point.
1.10.2.5
Graphene Nanoribbons
Another family of carbon allotropes with a basic structure intimately related with that of nanotubes is graphene nanoribons. A graphene nanoribbon is a strip of graphene with a nanometer-size width, which can be viewed as an SWNT cut along the length and opened flat. The dangling bonds that occur in the carbon atoms on the side edges are terminated, for example, with hydrogen. This imposes different kinds of boundary conditions than the periodic boundary conditions for SWNT. In parallel with the corresponding categories for SWNT, there exist different types of nanoribbons depending on how (along which direction) the ribbon is cut out of the original graphene sheet. Two representative types are armchair and zigzag ribbons, whose atomic arrangements along the edge are the armchair and zigzag patterns, respectively. (It should be reminded that there is a difference in nomenclature between nanotubes and nanoribbons. In the case of nanotubes, armchair or zigzag refers to the atomic arrangement along the chiral vector, that is, along the circumference of the tube. Therefore, if an armchair nanotube is cut along its length to make a nanoribbon, the resulting one is a zigzag ribbon, and vice versa.) Many aspects of electronic properties of graphene nanoribbons have parallels with nanotubes. For
368 Electronic States and Transport Properties of Carbon Crystalline
- and 9-bands meet at the K- and K9-points. Interlayer coupling gives rise to a finite overlap of the - and 9-bands, so that 3D graphite is a semi-metal. The first Brillouin zone of graphite is a hexagonal prism shown in Figure 6. The -band structure of graphite in the vicinity of "F, that is, near the H–K–H zone edge, is described by the Slonczewski–Weiss– McClure (SWMcC) model (McClure, 1957; Slonczewski and Weiss, 1958):
example, an armchair ribbon becomes metallic or semiconducting according to whether the width N of the ribbon is N ¼ 3m – 1 ðm ¼ 1;2;3; Þ or otherwise. One notable difference of nanoribbons from nanotubes lies in the existence of edge states. In particular, the band calculations predict that a zigzag ribbon has a flat (dispersionless) band at " ¼ 0, so that the density of states has a sharp peak at the Fermi level (Fujita et al., 1996). This is in marked contrast with the bulk graphene whose density of states is zero at " ¼ 0. This flat band is due to the edge states constructed from the nonbonding orbitals of the atoms along the zigzag edge. Since the weight of the edge states relative to the bulk states increases with decreasing ribbon width, the former plays an important role in nanometer-sized ribbons. For example, the large density of states at the Fermi level would lead to a large Pauli paramagnetic contribution to the magnetic suseptibility and may even give rise to the so-called flat-band ferromagnetism.
1.10.2.6
0
0
B B0 E2 B H ¼B B H13 H23 @
H13 – H23
H13 H13
1
C C H23 – H23 C C E3 H33 C A H33 E3
E1 ¼ þ 21 cosðkz c Þ þ 25 cos2 ðkz c Þ E2 ¼ 21 cosðkz c Þ þ 25 cos2 ðkz c Þ E3 ¼ 22 cos2 ðkz c Þ pffiffiffi 6 H13 ¼ ð – 0 þ 24 cosðkz c ÞÞeij a 4 pffiffiffi 6 ð0 þ 24 cosðkz c ÞÞeij a H23 ¼ 4 pffiffiffi H23 ¼ 33 cosðkz c Þeij a
The global electronic structure of graphite is given by the 2D graphite model described in Section 1.10.2.2. Graphite, however, is a 3D system consisting of ABAB stacking of graphene sheets with the interlayer spacing c ¼ 0.335 nm. For the detailed band structure near the Fermi level, interlayer coupling plays an important role. In the case of 2D graphite, the (b)
K′
A
H′
H Γ K
K′
A H′ H
H K
M L
ð26Þ
If the parameter 3 which describes the trigonal warping is neglected, the SWMcC model yields the following four eigenvalues:
E
H′
ð25Þ
where
Bulk Graphite
(a)
E1
E1
H
H′
0
EF
E3
2γ2 E2
K
K
H H
Figure 6 The electron and hole Fermi surfaces of graphite located along the H–K–H (and H9–K9–H9) edges of the Brillouin zone.
Electronic States and Transport Properties of Carbon Crystalline
The -band near the H–K–H edge is illustrated in Figure 6. There occur electron pockets centered at the K- and K9-points and hole pockets at the H- and H9-points. The bandwidth along the H–K–H edge is 4 1 1.56 eV and the band overlap that makes the 3D graphite a semimetal is 22 39meV. Reflecting the quasi-2D structure of graphite, its Fermi surfaces are highly elongated along the kz -direction. The effective mass for motion within the basal plane is me ¼ 0:057m for electrons and mh ¼ 0:039mh for holes (Soule et al., 1964). (In the early stage of the electronic structure study of graphite, the assignment of the electron and hole pockets was opposite to what was later established.) The effective mass for the out-of-plane motion is estimated to be on the order of m? ¼ ð5 – 15Þm for both electrons and holes. These values are inferred from the angular dependence of the cyclotron mass mc fitted to the following formula assuming ellipsoidal Fermi surfaces:
1 ¼ mc ð Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cos2 sin2
þ m2k mk m?
ð28Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi The anisotropy ratio m? =mk is 17 for the electron Fermi surface and 12 for the hole Fermi surface (Soule et al., 1964). Direct measurement of the effective masses together with correct assignment of carrier pockets were achieved by cyclotron resonance using circularly polarized microwave (Suematsu and Tanuma, 1972). Detailed studies of magneto-quantum oscillations suggest existence of additional small pieces of Fermi surface. These are so-called minority hole pockets residing at the H- and H9-points. The minority hole Fermi surface is separated from the narrow part (neck) of the Fermi surface for majority holes by the spin–orbit splitting.
1.10.3 Transport Properties of Graphene 1.10.3.1
Preparation of Graphene Samples
Graphene offers a totally new class of 2D electron system (2DES) that exhibits novel properties to be contrasted with those of conventional 2DES realized
in semiconductor heterointerfaces. Graphene samples used in transport experiments are typically prepared by pressing mechanically cleaved pieces of graphite onto an oxidized surface of Si substrate (Novoselov et al., 2005b). With enough skills and persistence, a few micron-sized ultrathin graphite pieces can be identified under an optical microscope. Electrical contacts are made by standard lithographic techniques and metal evaporation. With the underlying Si substrate used as a back gate, the system constitutes a fieldeffect-transistor (FET) structure. Figure 7 shows the sheet resistance of a monolayer graphene sample as a function of the back gate bias Vg . Positive (negative) Vg induces electrons (holes) of density n ¼ Cg Vg =e in the graphene sheet. For a typical thickness 300 nm of the insulating SiO2 layer, the coefficient is Cg =e 8 1014 m – 2 V – 1 . Thus, the system functions as an ambipolar FET with carrier mobilities of the order of 0:1 – 1m2 V – 1 s – 1 . Another method to prepare graphene relies on graphitization of the surface layer of a silicon carbide crystal (Berger et al., 2006). The (0001) surface of a 4H-SiC wafer is vacuum heat treated to form an epitaxial layer of graphene. This technique has a potential advantage of preparing a large area sample of graphene. 1.10.3.2
Transport at Zero Magnetic Field
The unconventional nature of the electronic state in graphene is reflected in various transport properties (Geim and Novoselov, 2007; Castro Neto et al., 2009).
6
ρ (kΩ)
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 1 1 3 > > > E ð þE Þ ðE1 –E3 Þ2 þ ð0 –24 cosðkz c ÞÞ2 ðaÞ2 <2 1 3 4 4 E¼ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > >1 > : ðE2 þE3 Þ 1 ðE2 –E3 Þ2 þ 3 ð0 þ24 cosðkz c ÞÞ2 ða Þ2 2 4 4 ð27Þ
369
4
2
0 –60
T=1K B=0T
–30
0 Vg (V)
30
60
Figure 7 Sheet resistance of a monolayer graphene as a function of back gate bias Vg. Reproduced with permission from Novoselov KS, Morozov SV, Mohinddin TMG, et al. (2007) Electronic properties of graphene. Physica Status Solidi (b) 244: 4106. Copyright Wiley-VCH Verlag GmbH & Co. KGaA.
370 Electronic States and Transport Properties of Carbon Crystalline
The electronic dispersion in monolayer graphene is characterized by the Dirac cones, that is, massless chiral fermions. Indeed, graphene offers a unique opportunity of exploring an analog of quantum electrodynamics (QED) in solid state. Charge carriers in bilayer graphene provide massive chiral fermions that are no less exotic. Massless Dirac fermion described by the Hamiltonian, Equation (9), has a unique feature that the electron state in the conduction band and the hole state in the valence band are interconnected, exhibiting properties analogous to chargeconjugation symmetry in QED. For the case of graphene, this symmetry arises from the honeycomb lattice structure with A and B sublattices. The quasiparticles are described by two-component wave functions which is analogous to spinor wave function in QED, with pseudo-spin specifying the sublattice playing the role of real spin in the latter. The symmetry is such that an electron with energy E propagating in one direction belongs to the same branch of the electronic spectrum as the hole with energy –E propagating in the opposite direction. Electrons and holes belonging to the same branch have pseudo-spin pointing in the same direction (parallel to the momentum for electrons and antiparallel for holes). Chirality, which is defined as projection of pseudo-spin on the direction of motion, is positive for electrons and negative for holes. One of the most counterintuitive properties of graphene is the so-called Klein paradox. It has been theoretically predicted that a relativistic particle can freely penetrate a potential barrier by transforming itself to its antiparticle. In graphene, this is manifested as reflectionless tunneling of charge carriers through a potential barrier by unimpeded transformation between electron-like and hole-like quasiparticles. Namely, an electrostatic potential cannot backscatter a massless Dirac fermion. Another way to understand the suppression of backscattering in graphene is to invoke destructive interference between one backscattering process and its time-revered counterpart on account of Berry9s phase associated with the rotation of pseudo-spin. This is analogous to the anti-localization in the presence of a strong spin– orbit interaction. Charge carriers in graphene can indeed propagate without scattering over large distances of the order of micrometers. Carrier mobilities as high as 1:5m2 V – 1 s – 1 are at room temperature and
20m2 V – 1 s – 1 at helium temperature are currently reported. This means that carrier transport is ballistic on submicron scale even at room temperature. Carrier mobilities currently achieved are presumably limited by impurities, by structural disorder (corrugation of graphene sheet) and by coupling with the substrate, and leave room for improvement. As seen in Figure 7, sheet resistance of graphene increases as Vg ! 0, but remains on the order of several kilo-ohm even right at the charge neutrality point, where one naively expects the system to become nonconducting owing to vanishing carrier density. Conductivity on the order of e 2=h per channel at the charge neutrality point has been theoretically predicted (Fradkin, 1986; Shon and Ando, 1998). This conduction without carriers is yet another counterintuitive property of graphene. The reason for the existing discrepancy between experimentally observed values of the minimum conductivity that scatters around 4e 2=h and the theoretical value 4e 2=h is not clear at the moment. Theoretically, proper treatment of strong electron–electron interaction may be crucial. On the other hand, the fact that a real graphene sample at their charge neutrality point most likely consists of puddles of electrons and holes may significantly affect the experimental results (Adam et al., 2007). Theoretical prediction for the high-frequency conductivity of Dirac fremion in graphene is ¼ e 2=4h, as opposed to 4e 2 =h for the DC conductivity. Optical transmittance given by T X ð1 þ 2=c Þ – 2 becomes T¼
– 2 – 2 e2 1 1 þ 2 ¼ 1 þ 1 – 4hc 2
ð29Þ
where X e 2 =hc 1=137 is the fine structure constant. The opacity of monolayer graphene is given by ð1 – T Þ ¼ 0:023. Visible light transmission measurements on suspended graphene membrane have proved that monolayer graphene absorbs about 2.3% of the incident white light (Nair et al., 2008).
1.10.3.3
Quantum Hall Effect
Application of magnetic field normal to the graphene plane causes quantum oscillations of resistance (Shubnikov–de Haas effect) which evolve to QHE under higher fields (Novoselov et al., 2005a;
Electronic States and Transport Properties of Carbon Crystalline
N ¼ 0 and 1 LLs is E ¼ 400K BðT Þ, which, in conjunction with high mobility, makes it possible to observe QHE even at room temperature. In moderate magnetic fields, each LL has fourfold degeneracy. The spin degeneracy of LL is lifted by a strong magnetic field with g-factor somewhat smaller than 2. For the N ¼ 0 LL, even the valley degeneracy seems to be lifted under high magnetic fields whose mechanism is yet to be elucidated (Zhang et al., 2006). Bilayer graphene has parabolic conduction and valence bands that are degenerate at the charge neutrality point. The LLs of massive chiral quasiparticles are given by
Zhang et al., 2005). The QHE in graphene is one of the most conspicuous phenomena that reveal the exotic nature of the system. Figure 8 shows the characteristic QHE behavior in monolayer and bilayer graphene (Novoselov et al., 2005a, 2006). Here the Hall conductance xy and the longitudinal resistance xx are plotted as a function of the charge density controlled by the gate bias. In both cases, the step of the Hall conductance xy is 4e 2=h. The factor 4 reflects the twofold spin degeneracy and the twofold valley degeneracy. For monolayer graphene, the sequence of Landau levels is shifted by 1/2, compared with the conventional QHE, so that the Hall plateau values are xy ¼ 4 e 2=h ðjN j þ 1=2Þ, N being the Landau level (LL) index . The xx peak at n ¼ 0 (Vg ¼ 0) implies that the N ¼ 0 LL which stays at the charge neutrality point, E ¼ 0 is shared equally by electrons and holes. This is a signature of the massless Dirac fermion described by the hamiltonian, Equation (9). The shift by 1/2 is alternatively viewed as arising from an additional phase , known as Berry’s phase, gained by a quasiparticle as it completes cyclotron motion with fixed chirality. The LL spectrum of a massless Dirac fermion (linear dispersion) is EN ¼ vF
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ehjN jB
371
EN ¼
heB pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N ðN – 1Þ m
ð31Þ
The QH plateau sequence is xy ¼ 4e 2=h int which is the same as the conventional 2DEG with the same spin and valley degeneracy, except that the lowest LLs of the valence and conduction bands (N ¼ 0 and 1) are merged so that the plateau at n ¼ 0 is missing. The gapless electronic band of bilayer graphene is modified by applying an electrostatic bias between the layers. The resulting energy gap of the order of 0.1eV can be tuned by the field effect. In the presence of nonzero gap, the above-mentioned anomaly in the QHE due to the degeneracy of the N ¼ 0 and 1 LLs disappears. To observe this splitting at finite interlayer bias, the charge neutrality point has to be shifted by additional chemical doping (Castro et al., 2007).
ð30Þ
to be contrasted with the ordinary EN ¼ ðheB=mÞðN þ 1=2Þ for a massive particle (parabolic dispersion). Equation (29) implies large separation of low-lying LLs. The gap between the
(b)
(a) 4
+5/2
3
+3/2
2
−1/2 5
0 −4
−2
0
2
4
Charge density (1012 cm−2)
Bilayer 6
1 0
4
−1
−3/2
−2
−5/2
−3
−7/2
−4
ρxx (kΩ)
+1/2
σxy (4e 2/h)
ρxx (kΩ)
10
+7/2
σxy (4e 2/h)
Monolayer
2
−4
−2
0
2
4
0
Charge density (1012 cm−2)
Figure 8 Quantum Hall effect in (a) monolayer and (b) bilayer graphene. Reprinted with permission from Geim AK and MacDonald AH (2007) Graphene: Exploring carbon flatland. Physics Today 60: 35. Copyright (2007) American Institute of Physics.
372 Electronic States and Transport Properties of Carbon Crystalline
1.10.4 Transport Properties of Carbon Nanotubes 1.10.4.1
Electronic State in SWNTs
As mentioned in Chapter 3.02, the electronic structure of SWNT is determined by the chiral vector, that is, how a hypothetical graphene sheet is rolled to make the SWNT. An armchair SWNT which has an electronic structure similar to monolayer graphene is metallic. Zigzag nanotubes are, by contrast, either quasi-metallic or semiconducting depending on the chiral vector. The difference in electronic structures and their correlation with the chirality of nanotube has been beautifully verified by spectroscopic measurements using a scanning tunneling microscope (STM) tip (Issi et al., 1995; Wildoer et al., 1998). Figure 9 shows the calculated and measured spectra for metallic and semiconducting SWNTs (Dekker, 1999). The spectra show the characteristic van Hove singularities of 1D
subbands, and nonzero (zero) density of states at the Fermi level for the metallic (semiconducting) SWNT. As shown by Equation (23), the electronic structure of an SWNT is profoundly affected by an AB flux threading through the tube. Full conversion of a semiconducting nanotube to a metallic one (or vice versa) is beyond experimental reach, because it will require an extremely strong magnetic field in excess of 1000 T for a typical nanotube diameter of 1 nm. Still, a precursory effect of AB flux on the band structure of SWNT has been verified by observation of magnetic-field-induced splitting of the optical spectra under magnetic fields up to 45 T (Zaric et al., 2004). 1.10.4.2
Electron Transport in SWNTs
For electrical transport experiments, it is necessary to make electrical contacts. A usual way to achieve this is to disperse or CVD-grow onto a substrate on which
Differential conductance (normalized)
−1
0 Energy (eV)
Density of states
(9,9)
0.1
0.0
Semiconducting nanotube
3 0 2
1
0
−1
0 Voltage (V)
1
(11,7)
0.1
0.0
1
−1
Differential conductance (normalized)
Density of states
Metallic nanotube
0 Energy (eV)
1
0
1
3
2
1
0
−1
Voltage (V)
Figure 9 Calculated (top panels) and measured (bottom panels) tunnel spectra of a metallic nanotube (left) and a semiconducting nanotube (right). Reproduced with permission from Dekker C (1999) Carbon nanotubes as molecular quantum wires. Physics Today 52: 21.
Electronic States and Transport Properties of Carbon Crystalline
a large array of fine metallic electrodes has been pre-patterned. After depositing the nanotubes, the sample is scanned with an atomic force microscope (AFM) to find nanotubes that by coincidence bridge two or more electrodes. Another more controlled method is to sprinkle nanotubes on an insulating substrate, take a scanning image of the deposited area with an AFM, and use that information to make a suitable electrode pattern by electron beam lithography. A problem here is that it is difficult to prejudge whether the target nanotube is metallic or semiconducting. The contact resistance varies widely depending on the kind of metal used for the electrode. Titanium and nickel are often used to achieve low-resistance contacts, while gold and aluminum tend to result in high contact resistance. The trend seems to be correlated with the carbide formation tendency of the respective metals. The intrinsic conductance of armchair nanotube is expected to be 4e 2 =h. Here, the factor 4 refers to the spin and valley degeneracy. That a metallic SWNT acts as a coherent quantum wire was first demonstrated by Tans et al. (1997) and by Bockrath et al. (1997). They observed and analyzed the Coulomb blockade phenomenon at low temperatures, and found that successively added single electrons are carried by one additional molecular orbital, which implied that electrons in SWNT were delocalized over an appreciable length (a few mm).
2.0
I
373
Figure 10 shows the conductance of an SWNT sample measured with highly transparent (lowresistance) contacts achieved with titanium electrodes (Kong et al., 2001). The value of conductance at low temperatures is close to 4e 2 =h. The conductance shows conspicuous fluctuations as a function of gate bias which is attributed to Fabry–Perot-type interference of ballistic electrons repeatedly reflected at the both ends (contacts) of the SWNT sample. It is remarkable that metallic SWNT can remain metallic at all, if one is reminded of intrinsic fragility of 1D electron systems associated with Peierls instability and strong tendency of localization. The tubular structure of SWNT renders robustness against Peierls distortion. The local defect potential is effectively weakened because it is seen by electrons as averaged over the circumference (White and Todorov, 1998). Suppression of backscattering mentioned earlier in the graphene context also plays a role here (Ando, 2005). Once electron–electron interaction is switched on, a 1D electron system is predicted to behave as a so-called TLL rather than an ordinary Fermi liquid. Signatures of TLL behavior are power-law dependence of the conductance on temperature and bias voltage, which are experimentally observed by Bockrath et al. (1999). The TLL nature of SWNTs is also corroborated by the photoemission spectroscopy (Ishii et al., 2003).
II
III 25 K
Conductance (G0)
20 K 1.5
50 K 100 K
1.0
290 K
0.5
0.0 −20
−10
0
10
20
30
Vg (V) Figure 10 Conductance of a nanotube sample as a function of gate bias at different temperatures. The conductance at low temperatures is close to 2 G0 ¼ 4e2/h in value and exhibits random but reproducible fluctuations. Reprinted with permission from Kong J, Yenilmez E, Tombler TW, et al. (2001) Quantum interference and ballistic transmission in nanotube electron waveguides. Physical Review Letters 87: 106801. Copyright (2001) by the American Physical Society.
374 Electronic States and Transport Properties of Carbon Crystalline
1.10.4.3
Multi-Wall Nanotubes
Multi-wall nanotubes (MWNTs) occur in a wide variety, with respect to the number of wall, their morphology, and various defects including adsorbates. In general, adjacent tube sheets have different chirality, so that electrons in a MWNT move in a relatively strong random potential. This and other source of disorder make the electron mean free path in MWNTs much shorter than in SWNTs, so that diffusive transport is usually observed in MWNTs.
1.10.5 Transport Properties of Graphite 1.10.5.1 Graphite Materials and Transport Characteristics A type of graphite widely used in various experiments is highly oriented pyrolytic graphite (HOPG), which is synthesized by thermal cracking of hydrocarbon and subsequent heat treatment under pressure to improve the quality of c-axis orientation of the crystallites. As they have a variety of uses including monochrometers for neutron diffraction and standard specimens for scanning probe microscope, HOPGs are available in various sizes. The in-plane structure of HOPG is a randomly ordered collection of small (mm size) crystallites. Another type of graphite is single-crystal kish graphite obtained by crystallization of carbon from molten steel. They come in the form of thin flakes that contain fairly large (mm size) single crystallites. As grown, kish graphite contains fairly large amount of iron segregates, which are subsequently removed by chemical processing with chlorine. As stated in Section 1.10.2.5, graphite is a semimetal with equal numbers of electrons and holes. Owing to the small Fermi energy, the carrier concentration shows a significant temperature dependence; ne ¼ nh 1 1025 m – 3 at room temperature, and 3 1024 m – 3 at helium temperatures (Dillon and Spain, 1978). The basal plane resistivity at room temperature is limited by acoustic phonon scattering so that it is relatively sample independent with a typical value 40 m cm. By contrast, the low-temperature resistivity limited by defect scattering is much more sensitive to sample quality. The residual resistivity ratio (RRR) defined as RRR ¼ 300K = 4:2K is typically 3–10 among HOPG samples, but ranges
from 10 to 40 among kish graphite samples. Given the change in carrier density with temperature stated above, the residual mobility ratio 300K = 4:2K is about 3 times as large. The values of electron and hole mobility in best-quality single-crystal samples at low temperatures exceed 100 m2 V1 s1. The out-of-plane resistivity c tends to be even more sensitive to the sample quality. The value of anisotropy ratio c = a at low temperature generally falls between 103 and 105. (For reference, the effective mass ratio is estimated as m? =mk 102 .) It is still unclear whether the lower or higher values of the anisotropy ratio represent the intrinsic behavior. On the one hand, less-well-oriented graphite crystals tend to give lower values. On the other hand, easy-to-cleave graphite crystals are vulnerable to defects such as stacking faults and microcracks which would give higher values. The out-of-plane resistivity often shows a slightly negative (semiconducting) temperature dependence from room temperature down to 30 K and then turns to a positive (metallic) one at lower temperatures. This unusual behavior may be simply a combined effect of the temperature-dependent carrier densities and mobilities. On account of the materials problem as stated above, however, full understanding of the c-axis transport in graphite is still not yet reached (Morgan and Uher, 1981; Brandt et al., 1988; Matsubara et al., 1990; Kopelevich et al., 2003). Recently, Kempa et al. (2002) invoke a field-induced metal–insulator transition for the c-axis resistivity based on their scaling analysis. However, given the above-stated temperature dependence of carrier density, the apparent metallic or insulating behavior of resistivity cannot be taken literally. The validity of proposed scaling analysis over a wide temperature range should be carefully accessed.
1.10.5.2 Semi-Classical MagnetoTransport Magneto-transport properties of graphite are characterized by low carrier density, semi-metallic nature, and high basal-plane mobility (small effective mass). Figure 11(a) shows a magnetoresistance trace under a magnetic field applied perpendicular to the basal plane (Woolam, 1970). We first ignore the oscillatory part and focus on the overall magnetic field dependence. The Lorentz force deflects electrons and holes to the same direction so that their Hall currents tend to cancel out. Since there occurs no
Electronic States and Transport Properties of Carbon Crystalline
(a)
(b)
θ 20
0° 19.8° 29.8°
T = 1.22°K
18 16
Magnetoresistance –105 Δ:p
7.40 n = 1 electron
V,(mV)
14
–6
39.7° 50.1°
12
58.3° 10
6.60
–4
63.5° 68.2°
8 3.643 n = 1 hole 3.412 2.952 n = 2 electron 2.936
–2
6 73.3° 4
77.0° 80.0°
1.90
0
375
Magnetic field, T
2
90.0°
0 4
6
8
10 12 14 16 18 20 22 24 H (kg)
Figure 11 (a) Traces of longitudinal magnetoresitance xx (B) in a natural single crystal of graphite at T ¼ 1.1K showing the Shubnikov–de Haas oscillations due to electrons and holes. (b) The magnetoresistance data at different field angles from the c-axis indicate highly elongated electron and hole Fermi surfaces. (a) Reprinted with permission from Woolam JA (1970) Spin splitting, Fermi energy changes, and anomalous g shifts in single-crystal and pyrolytic graphite. Physical Review Letters 25: 810. Copyright (1970) by the American Physical Society. (b) Reprinted with permission from Soule DE, McClure JW, and Smith LB (1964) Study of Shubnikov–de Haas effet. Determination of the Fermi surfaces in graphite. Physical Review 134: A453. Copyright (1964) by the American Physical Society.
Hall voltage to counteract the Lorentz force, a very large positive magnetoresistance appears. In highquality graphite crystals, xx ðBÞ= xx ð0Þ typically reaches 103 at B 1 T and 104 at 10 T. Since xx xy , the diagonal magneto-conductivity is xx ¼ xx = 2xx þ 2xy 1= xx at all magnetic fields. In a semi-classical transport model, the Hall conductivity of a semi-metal in the high field limit is ideally zero. In actual samples, it takes a nonzero value that reflects a small imbalance between the electron and hole densities, xy ðBÞ – ðne – nh Þje j=B. This imbalance is typically jne – nh j=ðne þ nh Þ 0:3 – 0:03%, and is due to the presence of ionized impurities. For the majority of graphite crystals, the imbalance goes toward the electron excess side, that is, more donors than acceptors. Figure 12(a) shows traces of magnetoresistance in single-crystal (kish) graphite at low temperatures (Iye et al., 1985). The classical part of the diagonal resistivity xx(B) exhibits, instead of the standard
quadratic B-dependence for compensated metals, a B-linear behavior over a wide range of magnetic field (McClure and Spry, 1968; Woolam, 1970). The linear increase of xx(B) tends to saturate at higher magnetic fields (Brandt et al., 1974; Lowrey and Spain, 1977). For some samples, xx(B) even starts to decrease at still higher fields (Iye, 1985; Yaguchi and Singleton, 1998). The characteristic B-linear behavior has been interpreted as due to B-dependent screening of ionized impurity potential which is thought to be the dominant scattering mechanism at low temperatures (McClure and Spry, 1968). The saturation of xx ðBÞ at higher field may be attributed to the socalled magnetic freeze-out effect, that is, localization of the excess carriers to the impurity sites, which is known to occur for example in InSb (von Ortenberg, 1973). In the case of graphite, magnetic freeze-out does not affect the total carrier density so much but changes the ionized impurity scattering centers to neutral ones. The two sets of data in Figure 12 represent two types of kish graphite samples in terms of ionized impurity concentration. Type B
376 Electronic States and Transport Properties of Carbon Crystalline
Graphite
ρxx
150 mk 192 255 330 390 485 600
Type A 0.01 Ω cm
0
0
ρxx
175 mk 257 335 390 465
Type B 0.01 Ω cm
620 0
0
0
50
100
150
200
250
B (kG) Figure 12 Traces of longitudinal magneto-resitance xx (B) in single-crystal (kish) graphite at different temperatures. The two set of data are obtained from samples with different ionized impurity concentrations (Iye, 1985).
sample contains an order of magnitude higher concentration of ionized impurities than type A. That the resistivity saturation is more conspicuous in type A sample seems to corroborate this picture. However, genuine freeze-out that should diminish xy B to zero is not observed in this field range, so that the resistivity saturation may be interpreted as precursor of the freeze-out effect.
1.10.5.3
Magneto-Quantum Oscillations
Since the early days of graphite research, the LL structure of graphite has been calculated (Nakao, 1976; Dresselhaus, 1974), and investigated by various magneto-quantum oscillation measurements (Soule et al., 1964; Woolam, 1970, 1971a) and magneto-optical spectroscopy (Schroeder et al., 1971; Doezema et al., 1979). Figure 13(a) shows a calculated LL structure along the K–H edge of the Brillouin zone for B ¼ 1 T. The trace shown in Figure 11(a) contains two series of Shubnikov-de Haas (SdH) quantum
oscillations: one due to the electron Fermi surface and the other due to the hole Fermi surface (Woolam, 1970). Due to the high carrier mobilities, the SdH oscillations can be seen in graphite from magnetic fields as low as 0.1 T. The SdH effect appears as a series of sharp dips, whose shape reflects that of the 1D density of states associated with the motion parallel to the field. Figure 11(b) shows magnetoresistance traces for different values of field angle from the c-axis (Soule et al., 1964). The shift of the SdH oscillation pattern approximately obeys the cos dependence indicating highly elongated electron and hole Fermi surfaces. In magnetic fields higher than 3 T, the SdH dips split into pairs, reflecting the Zeeman splitting (Woolam, 1970). Minority hole pockets are also identified (Woolam, 1971b). More recently, direct probing of LL density of states is made possible by the use of a low-temperature scanning tunneling spectroscope (LT-STM) (Matsui et al., 2005). Figure 14(a) shows the tunneling spectrum of graphite under perpendicular magnetic fields.
Electronic States and Transport Properties of Carbon Crystalline
(a)
(meV)
(b) 12
γ3 = 0
11
c b a
a
c
b
40 10 9
H = 10 kG (H //c)
0.10 n=1
0.08
b
8
20
a
n=2 σ=+
c
0.04
n=0
a a a a a a a a
b c b c b b
c
b
3 2 1 0 –1 1 2 3 45
c b c b c b c b c b c
c a 6
7
b 8 9 10 11 12 13
–60
c a c a
0.1
0.2
0.3
0.4
ξ
n=1 σ=+
0.02
n=0
n = –1 σ=–
0
σ=+
EF –0.02
EF n = –1 n = –2 σ = + σ=–
–0.04 b
b
b c
0 (K)
Energy (eV)
a
–40
a
4 c
H = 25 T H ||c
σ=–
5
0
σ=–
0.06
7 6
–20
377
–0.06 c
n=0 σ=– n=1 σ=+
–0.08 a
0.5 (H)
–0.10
0
0.1
0.3 0.2 ξ = cokz /2π
0.4
0.5
Figure 13 (a) Landau levels of graphite in a magnetic field of 1T. (b) Landau levels of graphite in a magnetic field of 25 T. Only the lowest spin-split Landau levels of electron and hole bands cross the Fermi level. (a) Reproduced with permission from Nakao (1976) Landau level structure and magnetic breakthrough in graphite. Journal of the Physical Society of Japan 40: 761. (b) Reprinted with permission from Iye Y, Tedrow PM, Timp G, et al. (1982) High-magnetic field electronic phase transition in graphite observed by magnetoresistance anomaly. Physical Review B 25:5478. Copyright (1982) by the American Physical Society.
The LLs are manifest as a series of peaks in the spectra indicated by triangles. The observed spectra show good agreement with the calculated ones with an appropriate choice of the surface potential s , as shown in Figure 14(b). The advent of graphene has brought resurgence of interest in graphite. The electronic properties of graphite are now scrutinized under new light. Luk’yanchuk and Kopelevich (2004, 2006) recently made a new analysis of the SdH and dHvA oscillations in bulk graphite. Based on the values of residual phase, they claim that the spectra indicate coexistence of normal (massive) electrons and Dirac-like (massless) holes. Schneider et al. (2009) made a similar but more precise experiment at 10 mK and measured SdH oscillations up to N 90 quantum numbers. They have shown that the data precisely fit the prediction of the SWMcC model, so that there is no ground for invoking massless Dirac fermions in 3D graphite.
For those electronic properties of graphite associated with energy scale larger than the electron– hole band overlap, j2 j 0:02eV, the Dirac-like spectrum is obviously relevant so that interpretation in such terms is possible, as discussed, for example, in LL spectroscopy (Orlita et al., 2009) and photoemission spectroscopy (Gruneis et al., 2008). For lowenergy-scale properties such as transport, however, the 3D nature of the graphite band is quite essential. 1.10.5.4 Magnetic-Field-Induced Electronic Phase Transition In magnetic fields higher than 7.4 T, both electrons and holes are only in their lowest LLs, that is, the system is in the extreme quantum limit. The magnetoresistance traces in Figure 12 show distinct anomalies in the high-field quantum limit regime. The abrupt increase of resistance is indicative of a phase transition to a new magnetic-field-induced electronic state. As
378 Electronic States and Transport Properties of Carbon Crystalline
(a) 4 B = 6T 5T
dI/dV (nA/V)
3
4T 3T 2T
2 0T 1 Kish graphite T = 55 mK
0 (b) 30
dI/dV (nA/V)
φs = 17 meV = 0 meV 20
10
Kish graphite B=6T T = 55 mK
0 −200
−100
0 V (mev)
100
200
Figure 14 (a) Tunneling spectrum of graphite in several magnetic fields obtained by a low-temperature scanning tunneling microscope (LT-STM). (b) Comparison of the spectrum with the calculated local density of states curves for two different values of surface potential s. Reprinted with permission from Matsui T, Kambara H, Niimi, Y, Tagami K, Tsukada M, and Fukuyama H (2005) STS observation of Landau levels at graphite surfaces. Physical Review Letters 94: 226403. Copyright (2005) by the American Physical Society.
seen in the figure, the onset magnetic field decreases with decreasing temperature. Figure 15(a) shows the phase boundary of the field-induced phase plotted with T in a logarithmic scale versus 1/B. The data points for the type A sample (crystals with lower ionized impurity concentration) lie on a straight line, which means that the phase boundary of the fieldinduced phase can be empirically expressed as Tc ðBÞ ¼ T expb – B =B c
ð32Þ
for this field range. The Landau subband structure in the quantum limit regime is shown in Figure 13(b) for B ¼ 25 T (Iye et al., 1982). In the quantum limit, the electronic motion in the basal plane (perpendicular to B) is quenched so that the system is reduced to a narrow-band 1D system
along the c-axis (parallel to B). Yoshioka and Fukuyama considered the possibility of Peierls-type instability of such magnetic-field-induced 1D system and proposed that the observed anomaly could be attributed to occurrence of a charge-density-wave instability (Yoshioka and Fukuyama, 1981; Sugihara, 1984; Takada and Goto, 1998). To be precise, two charge density waves associated with two valleys (along the H–K–H and H9–K9–H9 edges) occur outof-phase with each other to reduce the Hartree term of the Coulomb interaction, so that a valley density wave may be a better term for the proposed state. The critical temperature given by Yoshioka and Fukuyama reads
cos2 12 ckF0" 1 Tc ¼ 4:53"F exp – N0" ð"F Þu˜ cos ckF 0"
ð33Þ
Electronic States and Transport Properties of Carbon Crystalline
(a)
280 260 240
220
200
180
160
(b) Kish graphite B || c-axis
2000
α
β′
β1 β2
α′
Resistance ρxx (a.u.)
Tc (mk)
1000
500
Type A
Type B 200
379
1.1 K
0
1.7 K
0
2.2 K
0
3.2 K 0 4.2 K 0 8.0 K
100 0 50 4
0
6
5
10 K 0
10
1/B (×10–3 kG–1)
20
30
40
50
60
Magnetic field (T)
(c)
Temperature (K)
10 8
α α′ YF theory
6 4 2 0 20
30
40
50
60
Magnetic field (T) Figure 15 (a) The phase boundary of the magnetic-field-induced phase (Iye, 1985). (b) Magnetoresistance traces up to 55 T at different temperatures. (c) The phase boundary in a wider range of temperature and magnetic field, showing a re-entrant behavior at higher magnetic fields. (c) Reprinted with permission from Yaguchi H and Singleton J (1998) Destruction of the field-induced density-wave state in graphite by large magnetic fields. Physical Review Letters 81: 5193. Copyright (1998) by the American Physical Society.
where N0" ð"F Þ is the Fermi-level density of states of the lowest spin-up electron subband, and u˜ is a parameter representing the effective electron– electron interaction strength. This is an analog of a Bardeen-Cooper-Schrieffer (BCS) -type formula for pairing instability, Tc "F exp½ – ð1=N ð"F ÞV Þ . Note that the empirical formula, Equation (31), reflects the fact that N0" ð"F Þ_ B. The lower critical temperature for the type-B samples is attributed to the pair breaking effect of ionized impurity (Iye et al., 1984; Iye, 1985). Studies of neutron-irradiated graphite also corroborate this conclusion (Yaguchi et al., 1999). Figure 15(b) shows magnetoresistance traces up to
55 T measured in a pulse magnet (Yaguchi and Singleton, 1998). At the highest field, the resistance comes back to the trace extrapolated from the lowfield region, indicating a reentrant behavior. Figure 15(c) is a phase diagram over a wider temperature and magnetic field range. The dashed curve represents the prediction of Yoshioka– Fukuyama theory, Equation (32). Thus, the valleydensity-wave phase exists only at temperatures below 10 K. The disappearance of the densitywave phase at higher field is naturally understood since the ð0; " Þ subband relevant to the instability becomes unoccupied.
380 Electronic States and Transport Properties of Carbon Crystalline
Application of hydrostatic pressure p modifies the -band structure by changing the 2-parameter: 2 ðpÞ ¼ ð1 þ pÞ2 ð0Þ
ð34Þ
with the coefficient 0:03. According to Equation (32), the increased -band width affects the transition temperature by changing both "F and N0" ð"F Þ. The observed change of Tc with pressure is found to agree with eq.(32) (Iye et al., 1990).
1.10.6 Concluding Remarks In this sketchy overview of the crystalline carbon materials, we have tried to convey the ever-fascinating variety of electronic transport properties that unfolds in materials just consisting of carbon atoms. As usually the case for this sort of review article, there exists a huge volume of relevant literature so that it is by no means possible to give credit to all those who contributed to this field. The author apologizes for many omissions. Carbon is indeed a fascinating element. Carbon materials span all the dimensions: 3D (diamond), quasi-2D (graphite), 2D (graphene), 1D (nanotube), and 0D (fullerene). The honeycomb structure with C–C double bond gives rise to delocalized -electrons whose unique behavior is at the heart of this field. Back in the 1960s, semi-metal bismuth with its low melting point and low carrier density was the material of choice for electronic transport experiments. Graphite was of much interest in the same context, but research on graphite was sometimes hindered by the difficulty in material preparation. The situation was much improved when HOPG samples became available. Around 1980, research on GICs flourished. The discovery of fullerene in 1985 and that of nanotube in 1991 opened new frontiers. For the development of nanotube research, popularization of techniques facilities that enables to handle the nanoscale materials played a great role. Now the advent of graphene has opened yet another new field of science and technology. What makes graphene especially attractive is that it can be put in an FET structure and various parameters can be tuned. Under the guiding light of 2DEG physics and technology developed in semiconductor heterostructures, graphene research is the most rapidly developing field. (See Chapters 1.01, 1.02, 1.05, 1.06, 1.11 and 1.12).
References Adam S, Hwang EH, Galitski VM, and Das Sarma S (2007) A self-consistent theory for graphene transport. Proceedings of the National Academy of Sciences of the United States of America 20: 18392. Ando T (2005) Theory of electronic states and transport in carbon nanotubes. Journal of the Physical Society of Japan 74: 777. Ando T, Zheng Y, and Suzuura H (2002) Dynamical conductivity and zero-mode anomaly in honeycomb lattices. Journal of the Physical Society of Japan 71: 1318. Bassani F and Parravicini GP (1967) Band structure and optical properties of graphite and of the layer compounds GaS and GaSe. Nuovo Cimento B 50: 95. Berger C, Song Z-M, Li X-B, et al. (2006) Electronic confinement and coherence in patterned epitaxial graphene. Science 312: 1191. Bethune DS, Kiang C-H, de Vries MSM, et al. (1993) Cobaltcatalyzed growth of carbon nanotubes with single-atomiclayer walls. Nature 363: 605. Bockrath M, Cobden DH, Lu J, et al. (1997) Single-electron transport in ropes of carbon nanotubes. Science 275: 1922. Bockrath M, Cobden DH, Lu J, et al. (1999) Luttinger-liquid behavior in carbon nanotubes. Nature 397: 598. Brandt NB, Chudinov SM, and Ponomarev YaG (1988) Semimetals 1. Graphite and Its Compounds. Modern Problems in Condensed Matter Sciences, vol. 20. Amsterdam: North-Holland Physics Publishing. Brandt NB, Kapustin GA, Karavaev VG, Kotosonov AS, and Svistova EA (1974) Investigation of the galvanomagnetic properties of graphite in magnetic fields up to 500kOe at low temperatures. Zhurnal Eksperimentalnoi i Teoretitseskoi Fiziki 67: 1136 (Soviet Physics – JETP (1975) 40: 564). Castro EV, Novoselov KS, Morozov SV, et al. (2007) Biased bilayer graphene: Semiconductor with a gap tunable by the electric field effect. Physical Review Letters 99: 216802. Cho S-J and Fuhrer MS (2008) Charge transport and inhomogeneity near the minimum conductivity point in graphene. Physical Review B77: 08142. Coulson CA (1947) Energy bands in graphite. Nature (London) 159: 265. Dekker C (1999) Carbon nanotubes as molecular quantum wires. Physics Today 52: 21. Dillon RO and Spain IL (1978); Galvanomagnetic effects in graphite – II: The influence of trigonal warping of the constant energy surfaces on xx(Bz). Journal of Physics and Chemistry of Solids 39: 923. Dillon RO, Spain IL, Woolam JA, and Lowrey WH (1978) Galvanomagnetic effects in graphite – I: Low field data and the densities of free carriers. Journal of Physics and Chemistry of Solids 39: 907. Doezema RE, Datars WR, Charber H, and Van Schyndel A (1979) Far-infrared magnetospectroscopy of the Landau-level structure in graphite. Physical Review B 19: 4224. Dresselhaus G (1974) Graphite Landau levels in the presence of trigonal warping. Physical Review B 10: 3602. Dresselhaus MS and Dresselhaus G (1981) Intercalation compounds of graphite. Advances in Physics 30: 139. Dresselhaus MS, Dresselhaus G, and Avouris Ph (2001) Carbon Nanotubes: Synthesis, Structure, Properties, and Applications, Topics in Applied Physics, vol. 80. Berlin: Springer. Dresselhaus MS, Dresselhaus G, and Eklund PC (1996) Science of Fullerenes and Carbon Nanotubes. New York: Academic Press.
Electronic States and Transport Properties of Carbon Crystalline Fradkin E (1986) Critical behavior of disordered degenerate semiconductors II. Spectrum and transport properties in mean-field theory. Physical Review B 33: 3263. Fujita M, Wakabayashi K, Nakada K, and Kusakabe K (1996) Peculiar localized state at zigzag graphite edge. Journal of the Physical Society of Japan 65: 1920. Fukuyama H and Kubo R (1970) Interband effects on magnetic susceptibility. II. Diamagnetism of bismuth. Journal of the Physical Society of Japan 28: 570. Geim AK and MacDonald AH (2007) Graphene: Exploring carbon flatland. Physics Today 60: 35. Geim AK and Novoselov KS (2007) The rise of graphene. Nature Materials 6: 183. Gruneis A, Attaccalite C, Pichler T, et al. (2008) Electron– electron correlation in graphite: A combined angle-resolved photoemission and first-principles study. Physical Review Letters 100: 037601. Ichida M, Mizuno S, Saito Y, Kataura H, Achiba Y, and Nakamura A (2002) Coulomb effect on the fundamental optical transitions in semiconductin single-walled carbon nanotubes: Divergent behavior in the small-diameter limit. Physical Review B 65: 241407. Iijima S (1991) Helical nanotubule of graphitic carbon. Nature 354: 56. Iijima S and Ichihashi T (1993) Single-shell carbon nanotubes of 1-nm diameter. Nature 363: 603. Ishii H, Kataura H, Shiozawa H, et al. (2003) Direct observation of Tomonaga–Luttinger-liquid state in carbon nanotubes at low temperatures. Nature 426: 540. Issi J-P, Langer L, Heremans J, and Olk CH (1995) Electronic properties of carbon nanotubes: Experimental results. Carbon 33: 941. Iye Y and Dresselhaus G (1985) Non-ohmic transport in the magnetic-field-induced charge-density-wave phase of graphite. Physical Review Letters 54: 1182. Iye Y, McNeil LE, and Dresselhaus G (1984) Effect of impurities on the electronic phase transition in graphite in the magnetic quantum limit. Physical Review B 30: 7009. Iye Y, McNeil LE, Dresselhaus G, Boebinger G, and Berglund PM (1985) The electronic phase transition in graphite under strong magnetic field. In: Chadi JD and Harrison WA (eds.) Proceedings of the 17th International Conference on the Physics of Semiconductors, p. 981. San Francisco, CA, USA, 1984. New York: Springer. Iye Y, Murayama C, Mori N, Yomo S, Nicholls JT, and Dresselhaus G (1990) Effect of pressure on the highmagnetic-field electronic phase transition in graphite. Physical Review B 41: 3249. Iye Y, Tedrow PM, Timp G, et al. (1982) High-magneticfield electronic phase transition in graphite observed by magnetoresistance anomaly. Physical Review B 25: 5478. Katayama S, Kobayashi A, and Suzumura Y (2006) Pressureinduced zero-gap semiconducting state in organic conductor -(BEDT-TTF)2I3 salt. Journal of the Physical Society of Japan 75: 054705. Kataura H, Kumazawa Y, Maniwa Y, et al. (1999) Optical properties of single-wall carbon nanotubes. Synthetic Metals 103: 2555. Kempa H, Esquinazi P, and Kopelevich Y (2002) Field-induced metal–insulator transition in the c-axis resistivity of graphite. Physical Review B 65: 241101. Kobayashi A, Katayama S, and Suzuura Y (2009) Theoretical study of the zero-gap organic conductor -(BEDT-TTF)2I3.. Science and Technology of Advanced Materials 10: 02409. Kong J, Yenilmez E, Tombler TW, et al. (2001) Quantum interference and ballistic transmission in nanotube electron waveguides. Physical Review Letters 87: 106801. Kopelevich Y, Esquinazi P, Torres JHS, da Silva RR, and Kempa H (2003) Graphite as a highly correlated electron liquid. Advances in Solid State Physics 43: 207.
381
Kroto HW, Heath JR, O’Brien SC, Curl RF, and Smalley RE (1985) C60: Buckminsterfullerene. Nature 318: 162. Lowrey WH and Spain IL (1977) High field galvanomagnetic properties of graphite. Solid State Communications 22: 615. Luk’yanchuk IA and Kopelevich Y (2004) Phase analysis of quantum oscillations in graphite. Physical Review Letters 93: 166402. Luk’yanchuk IA and Kopelevich Y (2006) Dirac and normal fermions in graphite and graphene: Implications of the quantum Hall effect. Physical Review Letters 97: 256801. Matsubara K, Sugiura K, and Tsuzuku T (1990) Electrical resistance in the c direction of graphite. Physical Review B 41: 969. [Erratum: (1992) Physical Review B 49: 1943.] Matsui T, Kambara H, Niimi Y, Tagami K, Tsukada M, and Fukuyama H (2005) STS observation of Landau levels at graphite surfaces. Physical Review Letters 94: 226403. McClure JW (1956) Diamagnetism of graphite. Physical Review 104: 666. McClure JW (1957) Band structure of graphite and de Haas–van Alphen effect. Physical Review 108: 612. McClure JW and Spry WJ (1968) Linear magnetoresistance in the quantum limit in graphite. Physical Review 165: 809. Morgan GJ and Uher C (1981) The c-axis resistivity of highlyoriented pyrolytic graphite. Philosophical Magazine B44: 427. Nair RR, Blake P, Grigorenko AN, et al. (2008) Fine structure constant defines visual transparency of graphene. Science 320: 1308. Nakao K (1976) Landau level structure and magnetic breakthrough in graphite. Journal of the Physical Society of Japan 40: 761. Novoselof KS, Geim AK, Morozov SV, et al. (2004) Electric field effect in atomically thin carbon films. Science 306: 666. Novoselov KS, Geim AK, Morozov SV, et al. (2005a) Twodimensional gas of massless Dirac fermions in graphene. Nature 438: 197. Novoselov KS, Jiang D, Schedin F, et al. (2005b) Two-dimensional atomic crystals. Proceedings of the National Academy of Sciences of the United States of America 102: 10451. Novoselov KS, McCann E, Morozov SV, et al. (2006) Unconventional quantum Hall effect and Berry’s phase of 2 in bilayer graphene. Nature Physics 2: 177. Novoselov KS, Morozov SV, Mohinddin TMG, et al. (2007) Electronic properties of graphene. Physica Status Solidi (b) 244: 4106. Oberlin A, Endo M, and Koyama T (1976) Journal of Crystal Growth 32: 335. Orlita M, Faugeras C, Schneider JM, Martinez G, Maude DK, and Potemski M (2009) Graphite from the viewpoint of Landau level spectroscopy: An effective graphene bilayer and monolayer. Physical Review Letters 102: 166401. Painter GP and Ellis DE (1970) Electronic band structure and optical properties of graphite from a variational approach. Physical Review B1: 4747. Saito R, Dresslhaus D, and Dresselhaus MS (1998) Physical Properties of Carbon Nanotubes. London: Imperial College Press. Saravia LR and Brust D (1968) Band structure and interband optical absorption in diamond. Physical Review 170: 683. Saslow W, Bergstresser TK, and Cohen ML (1966) Band structure and optical properties of diamond. Physical Review Letters 16: 354. [Erratum: (1968) Physical Review Letters 21: 715.] Schneider JM, Orlita M, Potemski M, and Maude DK (2009) A consistent interpretation of the low temperature magnetotransport in graphite using the Slonczewski–Weiss–McClure 3D band structure calculations. Physical Review Letters 102: 166403.
382 Electronic States and Transport Properties of Carbon Crystalline
Schroeder PR, Dresselhaus MS, and Javan A (1971) Highresolution magnetospectroscopy of graphite. In: Carter DL and Bate RT (eds.) Proceedings of the International Conference on the Physics of Semimetals and NarrowBandgap Semiconductors, p. 139. Dallas, TX, USA, 1970, London: Pergamon. Shon NH and Ando T (1998) Quantum transport in twodimensional graphite system. Journal of the Physical Society of Japan 67: 2421. Slonczewski JC and Weiss PR (1958) Band structure of graphite. Physical Review 109: 272. Smith BW, Monthioux M, and Luzzi DE (1998) Encapsulated C60 in carbon nanotubes. Nature 396: 323. Soule DE, McClure JW, and Smith LB (1964) Study of Shubnikov–de Haas effet. Determination of the Fermi surfaces in graphite. Physical Review 134: A453. Suematsu H and Tanuma S (1972) Cyclotron resonances in graphite by using circularly polarized radiation. Journal of the Physical Society of Japan 33: 1619. Sugihara K (1984) Charge-density wave and magnetoresistance anomaly in graphite. Physical Review B 29: 6722. Tajima N, Tamura M, Nishio Y, Kajita K, and Iye Y (2000) Transport property of an organic conductor -(BEDT-TTF)2I3 under high pressure. Journal of the Physical Society of Japan 69: 543. Takada Y and Goto H (1998) Exchange and correlation effects in the three-dimensional electron gas in strong magnetic fields and application to graphite. Journal of Physics: Condensed Matter 10: 11315. Tans SJ, Devoret MH, Dai H, et al. (1997) Individual singlewall carbon nanotubes as quantum wires. Nature 386: 474. von Ortenberg M (1973) On the problem of magnetic freezeout in indium antimonide. Journal of Physics and Chemistry of Solids 34: 396. Wallace PR (1947) The band theory of graphite. Physical Review 71: 622. White CT and Todorov TN (1998) Carbon nanotubes as long ballistic conductors. Nature 393: 240. Wildoer JWG, Venema LC, Rinzler AG, Smalley RE, and Dekker C (1998) Electronic structure of atomically resolved carbon nanotubes. Nature 391: 59.
Woolam JA (1970) Spin splitting, Fermi energy changes, and anomalous g shifts in single-crystal and pyrolytic graphite. Physical Review Letters 25: 810. Woolam JA (1971a) Graphite carrier locations and quantum transport to 10 T (100 kG): Physical Review B 3: 1148. Woolam JA (1971b) Minority carriers in graphite. Physical Review B 4: 3393. Yaguchi H, Iye Y, Takamasu T, Miura N, and Iwata T (1999) Neutron-irradiation effects on the magnetic-field-induced electronic phase transitions in graphite. Journal of the Physical Society of Japan 68: 1300. Yaguchi H and Singleton J (1998) Destruction of the fieldinduced density-wave state in graphite by large magnetic fields. Physical Review Letters 81: 5193. Yoshioka D and Fukuyama H (1981) Electronic phase transition of graphite in a strong magnetic field. Journal of the Physical Society of Japan 50: 725. Zaric S, Ostojic GN, Kono J, et al. (2004) Optical signatures of Aharonov–Bohm phase in single-walled carbon nanotubes. Science 304: 1129. Zhang Y, Jiang Z, Purewal JP, et al. (2006) Landau-level splitting in graphene in high magnetic fields. Physical Review Letters 96: 136806. Zhang Y, Tan Y-W, Stormer H, and Kim P (2005) Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438: 201. Zunger A (1978) Self-consistent LCAO calculation of the electronic properties of graphite. I. The regular graphite lattice. Physical Review B17: 626.
Further Reading Ando T, Zheng Y, and Suzuura H (2002) Dynamical conductivity and zero-mode anomaly in honeycomb lattices. Journal of the Physical Society of Japan 71: 1318. Castro Neto AH, Guinea F, Peres NMR, Novoselov KS, and Geim AK (2009) The electronic properties of graphene. Reviews of Modern Physics 81: 109.
1.11 Angle-Resolved Photoemission Spectroscopy of Graphene, Graphite, and Related Compounds T Sato and T Takahashi, Tohoku University, Sendai, Japan ª 2011 Elsevier B.V. All rights reserved.
1.11.1 1.11.2 1.11.2.1 1.11.2.2 1.11.2.2.1 1.11.2.2.2 1.11.2.3 1.11.3 1.11.3.1 1.11.3.1.1 1.11.3.2 1.11.3.3 1.11.4 1.11.4.1 1.11.4.1.1 1.11.4.1.2 1.11.4.2 1.11.4.2.1 1.11.4.2.2 1.11.4.2.3 1.11.5 1.11.5.1 1.11.5.2 1.11.6 References
Introduction Angle-Resolved Photoemission Spectroscopy Basic Principle Determination of Band Dispersions Layered materials Three-dimensional materials Experimental Apparatus Graphite Band Structure Fermi surface Edge-Localized States Many-Body Interactions Graphite Intercalation Compounds C8A (A ¼ K, Rb, Cs) Band structure Interlayer band C6Ca Band structure Fermi surface Superconducting gap Graphene Single-Layered Graphite: Dirac Fermion? Comparison with Graphite Concluding Remarks and Summary
1.11.1 Introduction Carbon-based nanomaterials, such as fullerenes and carbon nanotubes, have attracted much attention from both basic- and application-scientific point of views. In an old textbook of solid-state physics, it is stated that carbon takes only two solid-state forms, diamond with the sp3 coordination and graphite with the sp2 coordination. This famous and instructive phrase had to be faced with the appropriate correction when fullerenes (C60), which involve intermediate coordination of sp3 and sp2, were discovered in the solid-state phase (Kroto et al., 1985). Subsequent discovery of carbon nanotubes (Iijima, 1991) has further widened the concept of solid-state carbon as well as opened a way to the industrial application such as an electron emitter in displays.
383 384 384 385 385 386 387 389 389 393 394 395 397 397 398 399 399 400 400 402 403 403 405 407 407
Recently, single-layer graphite, named graphene, has been a target of intensive theoretical and experimental studies, since it is expected that graphene would possess several superior physical properties such as the high carrier mobility compared with silicon due to the peculiar electronic structure originating in the perfect two-dimensional electronic structure. The discovery of superconductivity at an unexpectedly high temperature in a new category of graphite intercalation compounds (GICs) such as C6Ca (Weller et al., 2005) has revived the attention to the intercalation to control the electronic property and achieve a higher superconducting transition temperature. Thus, the old famous phrase in the textbook should be corrected as ‘‘Carbon takes a variety of solid-state forms with a variety of attractive properties.’’
383
384 Angle-Resolved Photoemission Spectroscopy
1.11.2 Angle-Resolved Photoemission Spectroscopy 1.11.2.1
Basic Principle
ARPES is the most sophisticated method among a variety of photoemission spectroscopies, enabling us to determine experimentally the electronic band structure and Fermi surface of crystals. In ARPES experiments, as shown in Figure 1, we irradiate the crystal surface with vacuum ultraviolet (VUV) light, and measure the kinetic energy of photoelectrons emitted from the crystal surface as a function of two angles (: polar angle, : azimuthal angle).
In comparison with angle-integrated PES, which provides us the information only on the electronic density of states (DOS), we need to measure much more spectra in ARPES measurements, but instead we obtain much more information on the electronic structure as a function of momentum. We show in Figure 2 the energy-conservation diagram in an ARPES process. 1. An electron, which was at first located on the occupied band (Ei), is excited to the otherwise empty band (Ef) by a VUV photon with the energy of h!. The energy conservation is as follows: h! ¼ Ef – Ei
ð1Þ
2. We assume that the excited state is the freeelectron state with the bottom of the energy dispersion at E0 with respect to the Fermi level (EF) in the crystal. We define the momenta parallel and perpendicular to the crystal surface of the photo-excited electron in the crystal as kjj and k?, respectively. Then, the energy of the electron in the excited state in the crystal is described as follows: 2 Ef ¼ h2 ðkjj2 þ k? Þ=2m – E0
ð2Þ
3. We define the kinetic energy of the photoelectron emitted into vacuum as EK; then we have Ef ¼ EK þW
ð3Þ
where W is the work function of the sample (see Figure 2). 4. On the other hand, the momenta parallel and perpendicular to the crystal surface of the
Vacuum Photoelectron e– Energy
Angle-resolved photoemission spectroscopy (ARPES) is now regarded as one of the most essential experimental techniques in solid-state physics. However, the energy resolution of two decades ago was at best about 0.3 eV, which enables only a rough band mapping of valence band structure of the crystal. It is well known that the physical and chemical properties of materials are governed mainly by the electronic structure in a very narrow energy region just at/around the Fermi level (EF). The energy resolution at that time was not enough to observe the many-body interactions in electrons, which appear as quasi-particles produced at/around EF. The discovery of high-temperature superconductors and the subsequent fierce race to observe the superconducting gap have accelerated the improvement of energy resolution, leading to the 1 meV resolution at present, more than two orders better than that of two decades ago. The superconducting gap and its symmetry of high-temperature superconductors have been experimentally elucidated with this ultrahigh-energy resolution. In this chapter, we explain recent ARPES experimental results on graphene, graphite, and GICs and discuss the electronic structure that realizes the novel properties observed in these materials.
Ef photon (h ω)
EK Wave vector
Vacuum level W
z Free-electron band
e-
hω
Fermi level (EF) (Energy zero) Ei
E0
Band
θ Crystal surface
x Figure 1 Schematic view of ARPES.
y
Figure 2 Energy-conservation diagram of the photoemission process.
Angle-Resolved Photoemission Spectroscopy
photoelectron emitted into the vacuum (outside the crystal) are described as follows: pffiffiffiffiffiffiffiffiffiffiffi 2mEK sin pffiffiffiffiffiffiffiffiffiffiffi K? ¼ 2mEK cos Kjj ¼
ð4Þ ð5Þ
where is the polar angle of photoelectron as shown in Figure 1. 5. When the photoelectron is emitted from the crystal into the vacuum through the surface, the momentum parallel to the surface is conserved because the transverse symmetry parallel to the surface is hold (Figure 3). This is the most important assumption in ARPES: kjj ¼ Kjj
ð6Þ
Using Equations (1)–(6), we obtain the relationship between the energy and the momentum of the initial state in the crystal: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hkjj ¼ 2mðEi þ h! – W Þsin qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hk? ¼ 2mððEi þ h! – W Þcos2 þ V0 Þ
ð7Þ ð8Þ
where V0 is defined as the sum of E0 and W (V0 ¼ E0 þ W) and is called the inner potential. By applying these formulas to ARPES experimental results, we are able to map out experimentally the relationship between the initial-state energy (Ei) and the momentum (kjj and k?), namely the band dispersions of electrons in the crystal. 1.11.2.2 Determination of Band Dispersions 1.11.2.2.1
Layered materials As described above, when we measure the polar angle () and the kinetic energy (EK) of photoelectrons, we are able to map out the band dispersions (Ei)
385
of electrons in a crystal as a function of the momentum parallel to the surface (kjj). The schematic diagram is shown in Figure 4. This method appears to be very powerful when the crystal under consideration is a highly two-dimensional layered material. In this case, the energy dispersion perpendicular to the layer is negligibly small and therefore the obtained band dispersions as a function of kjj represent the band structure of the material. Figure 5 shows a set of ARPES spectra of layered compound 1T-VSe2 and the experimental band dispersions obtained from these spectra (Terashima et al., 2003). Several characteristic bands from Se 4p and V 3d orbitals are clearly seen in the experimental band structure. We also find that the V 3d band reaches the Fermi level at midway between and M points in the Brillouin zone, producing a metallic Fermi surface. The band structure calculation is also shown for comparison in Figure 5. The good agreement between the experiment and the calculation indicates that ARPES is able to map out the band structure of layered materials with high precision. The concept of Fermi surface is one of important physical concepts based on which we discuss the electronic properties of materials. By measuring many ARPES spectra all over the Brillouin zone and plotting the spectral intensity at the Fermi level as a function of two-dimensional momentum (wave vector), we are able to map out experimentally the two-dimensional Fermi surface. Figure 6 shows an example applied for 1T-VSe2, where the characteristic two-dimensional cylindrical Fermi surface is clearly mapped out by ARPES (Terashima et al., 2003).
Energy band zII
θ
θ
e–
KII
Crystal surface k
e–
kII
Figure 3 Momentum parallel to the surface of photoelectrons is conserved during the photoemission process.
θ1
EF
θ2 θ3 θ4 θ5 EF Binding energy
Energy
K
Photoelectron intensity
θ0
k =
√2mEK sinθ h
k0 k1 k2 k3 k4 k5 Wave vector
Figure 4 Schematic diagram to show how to determine the E–k relation (band structure) by ARPES.
386 Angle-Resolved Photoemission Spectroscopy
1T-VSe2
V 3dz 2
EF M(L) Binding energy (eV)
Intensity (a.u.)
1
Γ(A)
Se 4pz*
2 Se 4px,y
3 4 5 6
7
Γ A
6 5 4 3 2 1 EF Binding energy (eV)
M L Wave vector
Figure 5 Experimentally determined band structure of quasi-1D layered compound 1T-VSe2 (Terashima et al., 2003).
ky Guideline of Fermi surface M(L)
K(H)
High kx
Intensity
Γ(A)
Low First Brillouin zone
1T-VSe2
Figure 6 Fermi surface of 1T-VSe2 determined by ARPES (Terashima et al., 2003).
1.11.2.2.2
Three-dimensional materials In the case of three-dimensional materials, the situation is complicated because electrons in the crystal have a finite energy dispersion perpendicular to the measured crystal surface. When we change the polar angle () to change the momentum parallel to the surface (kjj) in ARPES experiments, the perpendicular counterpart (k?) is also simultaneously altered according to Equation (8). However, in this case, we do not know explicitly the k? value because V0 is unknown. In other words, even when we fix the kjj value by setting the polar angle at a certain value, we are not able to know which point on the band structure perpendicular to the crystal surface we are actually seeing. Despite this difficulty, we can
determine the three-dimensional band structure of three-dimensional materials in many cases, as explained in the following. It is well known that the escape depth of photoelectrons is very short and comparable to a few atomic layers from the surface. Electrons confined in this very thin layered space are expected to have the momentum substantially broadened perpendicular to the layer (surface) because of the Heisenberg’s uncertainty principle (xp h). In ARPES experiments with VUV light of h! ¼ 20–40 eV, the escape depth of photoelectrons is about 5 A˚, meaning that the momentum (wave vector) perpendicular to the surface has an ambiguity of about 0.2 A˚1. Further, if there are no final-state bands corresponding to
Angle-Resolved Photoemission Spectroscopy
Ef (see Figure 2), the escape depth becomes much shorter (one-half to one-third) because the wave function of the photoelectrons outside the crystal becomes an evanescent wave inside the crystal and cannot enter deep inside the crystal. This situation enhances the broadening of momentum perpendicular to the surface, and finally a large portion of k? is simultaneously seen in ARPES measurements. In other words, when we conduct ARPES measurements at a certain polar angle (), the kjj component is uniquely determined from Equation (7), while the k? component is automatically integrated in all (or a large portion of) Brillouin zone perpendicular to the crystal surface because of the short escape depth of photoelectrons. In general, the DOS is relatively larger on the high-symmetry line in the Brillouin zone in comparison with other portions between high-symmetry lines. When we integrate the DOS along k? at a fixed kjj, the high-symmetry line produces a peak in the ARPES spectrum. This means that the peak position in the ARPES spectrum of three-dimensional materials traces the band dispersion on the high-symmetry line in the Brillouin zone. This indicates that we can compare the band dispersions obtained by ARPES for three-dimensional materials with the band structure calculated along the high-symmetry lines. Figure 7 shows the band structure of LaSb determined by ARPES, compared with the band structure calculation (Kumigashira et al., 1998). LaSb is a typical three-dimensional material with face-centered cubic crystal structure. ARPES measurements were
done along the (010) direction at the (001) cleaved surface. So, kjj and k? are defined along the (010) and (001) directions, respectively. When we compare the experimental band dispersions with the band structure calculated along several high-symmetry lines, we find that the band dispersions along X and XWX high-symmetry lines simultaneously appear in the experimentally determined band structure. This indicates that two different k? points are measured simultaneously at a fixed kjj due to the k? integration effect. As shown above, the band structure of three-dimensional materials can be determined by ARPES in many cases. 1.11.2.3
Experimental Apparatus
Figure 8 shows a schematic diagram of a highresolution ARPES spectrometer. The apparatus consists of mainly four parts: (1) a large electrostatic hemispherical electron energy analyzer with the average diameter of about 40 cm; (2) a microwave-driven discharging lamp to produce high-intensity VUV light; (3) a sample preparation vacuum chamber where samples are prepared by cleaving, sputtering, evaporation, etc.; and (4) an ultrahigh vacuum analyzer chamber where the sample is irradiated by VUV light from the discharging lamp. In addition, the apparatus is equipped with a liquid-helium cryostat to cool down the sample, a sample transfer system, many vacuum pumps, etc. We can use synchrotron radiation light to excite photoelectrons.
e1
EF
Σ
Δ Γ
X
X
X K
Δ
W X
2nd Brillouin zone y(010)
X X
x(100)
Binding energy (eV)
h2 Γ
W
X
e3 e2
X
z(001)
387
a
1.0 c
k
h1 b
j i
2.0
d g f
3.0
1st Brillouin zone 4.0 X Γ
Figure 7 Band structure of LaSb determined by ARPES (Kumigashira et al., 1998).
W
X X
388 Angle-Resolved Photoemission Spectroscopy
Hemispherical electron analyzer
CCD camera Installation chamber
TMP Main chamber
He-flow cryostat
He discharge lamp Monochromator
Cryostat
Manipulator
Sample
Preparation chamber
TMP Radiation shield
TSP
Ion pump
Figure 8 High-resolution ARPES spectrometer.
The recent progress in the energy resolution in ARPES experiments owes mainly to the improvement of electron energy analyzer. Figure 9 shows the
schematic diagram to explain how the energy and the polar angle of photoelectrons are measured with the hemispherical analyzer. By using the
Hemispherical analyzer
Slit
E1
E2
E3 Energy
e θ1
gl An θ2
MCP
Electron lens
Screen Photoelectron
hω θ2
θ1
Sample Figure 9 Two-dimensional detection hemispherical electron analyzer.
CCD camera
Angle-Resolved Photoemission Spectroscopy
15 K 25 K 50 K 75 K 100 K 125 K 150 K 175 K 200 K 225 K 250 K 280 K 300 K 320 K
Intensity (a.u.)
Au
f( ε) =
1 exp[(ε–EF) / kBT ] + 1
f( EF) = 0.5
100
50
EF
–50
–100
Binding energy (meV)
Figure 10 Temperature dependence of photoemission spectrum of gold (Au) in the vicinity of EF.
two-dimensional detecting system with a multichannel plate (MCP) and a CCD camera, we are able to measure simultaneously the energy and the polar angle of photoelectrons as shown in Figure 9. This two-dimensional detection system together with a large size of hemispherical analyzer enables a rapid and precise ARPES measurement to lead to the ultrahigh-resolution measurement. Figure 10 shows
389
the temperature dependence of photoemission spectrum of gold (Au) in the vicinity of the Fermi level measured with the energy resolution of about 1 meV. The spectrum shows a systematic temperature dependence with the common intersecting point at the Fermi level. This behavior is exactly the same as the temperature dependence of the Fermi–Dirac (FD) distribution function. This state-of-the-art experimental result indicates that high-resolution photoemission (ARPES) spectroscopy provide us rich information with which we can discuss the relationship between the electronic structure and the novel properties of materials such as superconductivity.
1.11.3 Graphite 1.11.3.1
Band Structure
Carbon-based materials show a variety of interesting properties owing to the dimensionality, structure, and size. These materials have attracted much attention in both the basic science and the device applications. As shown in Figure 11, the structural
Graphene ribbon
Graphene
Graphite
Nanotube GIC Figure 11 Crystal structures of graphene, graphite, and related materials.
390 Angle-Resolved Photoemission Spectroscopy
base of the sp2-hybridized carbon materials is graphene, single-layered graphite, which is purely two dimensional. As discussed later, this graphene behaves as a zero-gap semiconductor where the mass of carriers is regarded as zero (a Dirac fermion). When the graphene stacks along the out-of-plane direction, it becomes three-dimensional graphite. In contrast to the zero-gap nature of graphene, graphite is a semimetal with a small overlap of C 2p bands due to the finite interaction between graphene sheets in the crystal. Various atoms or molecules can be inserted between graphene sheets of graphite. These compounds are called GICs and are usually metallic and sometimes superconducting. It has been theoretically predicted that by cutting the side of graphite (graphite ribbon), anomalous electronic states called edge-localized states appear in the vicinity of EF (Fujita et al., 1996; Kobayashi, 1993; Nakada et al., 1996; Wakabayashi et al., 1999). Moreover, carbon nanotube, created by smoothly connecting both sides of graphene sheets, shows a variety of interesting properties owing to its chirality. The carbon nanotube would be technologically useful for transistors, power fuel cells, displays, and so on.
In all the carbon-based materials, the key to understand the interesting physical properties lies in the carbon 2p and orbitals whose atomic wave functions are schematically shown in Figure 12(a). The orbital is elongated parallel to the graphite plane while the orbital perpendicular to the plane. The band structure for high-symmetry lines in the Brillouin zone derived from these orbitals are shown in Figure 12(b). Because two carbon atoms are included in the hexagonal unit cell in the case of the AB stacking, we find two bonding bands and four bonding bands below EF. The corresponding antibonding counterparts are also seen above EF ( and bands). The bandwidth along the momentum perpendicular to the plane (kz) is very small compared to that for the in-plane, reflecting the quasi-two-dimensional nature of graphite. As one immediately notices from Figure 12(b), both and bands cross EF at H or K point and produce small hole and electron Fermi surfaces. These small Fermi surfaces are responsible for the electric conduction and hence the semimetallic nature of graphite. Figure 13 shows valence-band ARPES spectra of kish graphite (artificially grown single-crystal (c)
(a)
kz
Carbon atom
π orbital
L
σ orbital kx
M
R S
T
ε P ∧ K
A Δ Q Γ
ky
(b) Energy (Ry)
Energy (eV)
σ*
0.0
0.0
–0.4
π*
–0.8
π
–4.0 –8.0
σ
–12.0 –16.0
–1.2
–20.0 –1.6 –24.0 κ
∧
Γ
Σ
M T
KPH
Q
A
R
L S H
Figure 12 (a) Schematic view of the wave function of and orbitals. (b, c) Calculated band structure (Tatar and Rabii, 1982) and Brillouin zone of graphite.
Angle-Resolved Photoemission Spectroscopy Γ(A)
Γ(A)
(a)
391
K(H)
(b)
M(L)
K(H) M(L)
K(H)
Intensity (a.u.)
Intensity (a.u.)
M(L)
M(L)
Γ(A)
Γ(A)
16
12 8 4 Binding energy (eV)
EF
16
EF 12 8 4 Binding energy (eV)
Figure 13 Valence-band ARPES spectra of graphite measured along two high-symmetry lines: (a) KM (AHL) and (b) M (AL).
graphite) measured at 20 K along (a) KM (AHL) and (b) M (AL) directions. We clearly find highly dispersive bands along both directions. For example, along the KM (AHL) direction (Figure 13(a)), one band has the top of dispersion at the (A) point at 4 eV and disperses toward the higher binding energy on approaching the K(H) point. This band has the bottom at about 12 eV at the K(H) point and then disperses back toward the lower binding energy on approaching the M(L) point. We also find another prominent peak located at about 8 eV at the (A) point. This band rapidly approaches EF on moving toward the K(H) point, and the spectrum shows a clear Fermi-edge cutoff indicative of a tiny hole pocket centered at the K(H) point. On the other hand, along the M (AL) direction (Figure 13(b)), the band does not cross EF but stays at 3 eV around the M(L) point, indicative of the absence of Fermi surface along this direction. These features are commonly observed in previous ARPES results (Takahashi et al., 1985; Santoni et al., 1991). It is also noted that we are able to observe separately the electronic structure along the KM (AHL) and M (AL) directions due to the single-crystal nature of kish graphite, which is not possible with the highly
oriented pyrolytic graphite (HOPG). This suggests that ARPES with a high-quality single crystal is essential to establish the intrinsic electronic structure of graphite. To visualize more clearly the dispersive features in ARPES spectra, we have mapped out the band structure and show the result in Figure 14. The experimental band structure was obtained by taking the second derivative of ARPES spectra and plotting the intensity by gradual shading as a function of wave vector and binding energy. Bright areas correspond to experimental bands. Figure 14 also shows the first-principles band calculation of graphite (Tatar and Rabii, 1982) using the Johnson and Dresselhaus (1973) model (the JD model) with the AB stacking sequence. It is clear that the experimental band structure and the bulk band calculation show a good agreement along both directions. However, we also notice quantitative differences. For example, the band due to the C2p bonding states is located at about 4 eV at the (A) point in both the experiment and the calculation, while the dispersive feature from the (A) point to the K(H) point shows a quantitative deviation. The lower-lying branch of the band (2 band) along the KM (AHL) direction is located
392 Angle-Resolved Photoemission Spectroscopy
(b) EF
EF
4
4
Binding energy (eV)
Binding energy (eV)
(a)
8
12
16
8 High
12
Γ A
16 K H Wave vector
M L
Low Γ A
M L Wave vector
Figure 14 Valence-band structure of graphite along two high-symmetry lines KM (AHL) and M (AL) determined by ARPES, compared with the band-structure calculation (Tatar and Rabii, 1982).
at slightly higher binding energy in the experiment than in the calculation. A similar trend is also seen in the M (AL) direction. The band is located at a slightly higher binding energy in the experiment than in the calculation at the (A) point, whereas both perfectly coincide with each other near EF at the K(H) point in the energy scale of Figure 14. The finite deviation of the energy position of bands between the experiment and the calculation may be due to many-body effects and/or final-state effects (Strocov et al., 2001; Heske et al., 1999). (a)
While the ARPES technique determines the occupied electronic states, the band structure in the unoccupied side is also elucidated by the angle-resolved inverse photoemission spectroscopy (ARIPES) (Scha¨ter et al., 1986; Ohsawa et al., 1987) and angle-resolved secondary-electron emission spectroscopy (ARSEES) (Maeda et al., 1988). In the ARIPES, electrons are impinged into the sample surface and emitted photons are detected. Figure 15(a) shows ARIPES spectra of HOPG as a function of polar angle. The ARIPES spectra show clear
(b)
Intensity (a.u)
Graphite (HOPG) ARIPES
69° 65° 60° 55° 50° 45° 40° 35° 30° 25° 20° 15° 10° 5°
20
10
Ev
0°
0
5 10 15 20 Energy relative to EF
25
K
Γ
M
EF
Figure 15 (a) Angle-resolved inverse photoemission spectra of graphite. (b) Comparison of ARIPES-derived unoccupied band structure with the band calculation.
Angle-Resolved Photoemission Spectroscopy
structures denoted by markers, whose energies and intensities are very sensitive to the change of polar angle. Figure 15(b) shows a comparison of experimentally determined unoccupied band structure with the band calculation (Tatar and Rabii, 1982). An overall agreement between the ARIPES experimental result and the calculation is found as in the case of the occupied band structure, in both the and bands. A similar result has been obtained by comparison with a refined band calculation (Fretigny et al., 1989). All these experimental results demonstrate that the band-structure calculation serves as a good starting point to discuss the electronic structure of graphite. 1.11.3.1.1
Fermi surface Since the graphite is a low-carrier semimetal, we expect the presence of tiny Fermi surface(s). This Fermi surface is successfully determined by the high-resolution ARPES. Figure 16(a) displays the ARPES intensity at EF of kish graphite plotted as a function of two-dimensional wave vector. The spectral weight is integrated within 10 meV with respect to EF and the intensity is plotted by assuming the hexagonal symmetry. The bright area corresponds to the high intensity. Figure 16(b) is
–1.0
an expansion around the K(H) point. As clearly seen in Figure 16(b), the spectral intensity at EF is sharply peaked at the K(H) point, demonstrating the existence of an extremely small Fermi surface at the K(H) point. The volume of the observed small hole-like Fermi surface can be estimated by the following method. We first assume that the hole pocket has an ellipsoidal shape elongated along the kz (KH) axis with a circular cross section in the kx–ky plane. The longer axis (2b) of the ellipsoid is supposed to be just half of the unit cell along the kz axis, namely the distance between the K and H points, being 0.942 A˚1, as predicted from the band calculation (Charlier et al., 1991). We determined the shorter axis (2a) of the ellipsoid as 0.034 0.004 A˚1 by extrapolating the band dispersion near the K(H) point toward EF with a linear function. The estimated volume (4a2b/3) of the hole-like Fermi surface is 5.7 0.7 104 A˚3, which corresponds to the carrier number of 9.2 1.1 1018 cm3. This value is apparently larger than the hole carrier number (2.0 1018) estimated from the measurement of the Hall coefficient (Brandt et al., 1988), possibly due to the finite energy and momentum resolution in the ARPES measurement. kx (Å–1) 0.0
1.0
(a) M(L) K(H)
ky (Å–1)
1.0
Γ(A)
0.0
–1.0
ky (Å–1)
–1.6
(b)
–1.7
–1.8 –0.2
–0.1
393
0.0
0.1
0.2
kx (Å–1) Figure 16 (a) Fermi surface of graphite determined by ARPES. (b) Expansion near the K(H) point.
394 Angle-Resolved Photoemission Spectroscopy
1.11.3.2
Edge-Localized States
It has been theoretically suggested that a graphene ribbon possesses a characteristic electronic state near the Fermi level (EF) called the edge-localized state (Fujita et al., 1996; Nakada et al., 1996; Kobayashi, 1993; Wakabayashi et al., 1999), which is not predicted from the bulk band calculation. This newly created electronic state at the graphite edge has been intensively discussed in relation to the magnetic and transport properties. It has been theoretically predicted that the magnetic property of graphite ribbon strongly depends on the structure of edge (zigzag- or armchair-type) (Fujita et al., 1996; Kobayashi, 1993; Nakada et al., 1996; Wakabayashi et al., 1999), adsorbed atoms, and the stacking sequence of graphite layers (Miyamoto et al., 1999). Indeed, as shown in Figure 17, which plots the calculated energy versus k relation of the bands for graphite ribbons parallel to the ribbon direction (Nakada et al., 1996), we find almost nondispersive bands in the close vicinity of EF in the case of ribbons with zigzag edges (Figure 17(a)), while no such feature is seen in the ribbon with armchair edges (Figure 17(b)). It is also known that adsorption of halogen atoms to graphite ribbon remarkably changes the character of edge-localized states (Kusakabe and Maruyama, 2003; Maruyama and Kusakabe, 2004). Recent scanning tunneling spectroscopy/microscopy studies reported a signature of the edge-localized states near EF in the DOS (Matsui et al., 2005), whereas, until recently, there
have been no ARPES investigations reporting the edge-localized states in the graphite surface. By controlling the cleaving condition of kish graphite under ultrahigh vacuum, anomalous feature is sometimes observed in the ARPES spectra near the K(H) point, which could be ascribable to the edgelocalized states. Figure 18 shows ARPES spectra measured along the cut slightly away from the hole pocket at the K(H) point as shown in the inset. We first notice two prominent structures near EF around kx ¼ 0 A˚1. One of them, which is located closer to EF and forms a sharper peak at about 80 meV, shows a steep dispersion toward high binding energy with gradual broadening on going to kx ¼ 0.1 A˚1. This band is assigned as the band, as described previously. It is noted here that this band does not cross or touch EF because the measurement was done in the cut slightly away from the hole pocket at the K(H) point, as shown in the inset. In addition to the band, we find an anomalous feature (band) at about 130 meV around kx ¼ 0 A˚1. In contrast to the bulk band, this band shows a peculiar energy dispersion asymmetric with respect to kx ¼ 0 A˚1; an almost flat dispersion in the side of kx being positive, 0 to 0.05 A˚1, and a slight upward dispersion toward EF on the side of kx being negative of 0 to –0.10 A˚1. This asymmetric behavior might be due to the
kx (Å–1)
0.10 Zigzag ribbon 1B
(b)
1A
1A 2A 1B
1
2A 2B
1 2 3 4 5
2B
2 3
N–1 N
N A NB
Armchair ribbon
N–1 N
NA N B
Intensity (a.u.)
(a)
Γ(A) 0.05 M(L)
K(H)
(d) –3
(c) –3
0.0
0
E
E
–0.05 0
–0.10 0.6 3 –π
0
π
3 –π
0
Figure 17 (a, b) Zigzag and armchair ribbons. (c, d) Calculated energy dispersion for zigzag and armchair ribbons.
π
EF 0.2 0.4 Binding energy (eV)
Figure 18 Near-EF ARPES spectra of graphite showing the edge-localized state. Inset shows the measured cut in the Brillouin zone (red line near the K(H) point). Red arrows indicate prominent structures around kx ¼ 0 A˚1.
Angle-Resolved Photoemission Spectroscopy
matrix-element effect in the photoelectron exciting process because the polarization of incident light is different between the two cases. We have measured ARPES spectra for several cuts near the K(H) point and found that this anomalous feature appears only around the K(H) point. In Figure 19, we plot the ARPES spectral intensity as a function of binding energy and wave vector, together with the projection of bulk band on the (001) plane (dots) (McClure, 1957). As clearly seen in the figure, the projection of the band shows a good agreement with the hole-like band in the experiment. The anomalous feature is observed well outside the projection of the band, indicating that the dispersion along kz of the band is not responsible for the anomalous feature. The antibonding electron-like band ( band) might be the origin of the anomalous feature. However, as seen in Figure 19, the band is situated well above EF in this momentum region and cannot account for the anomalous flat band below EF. The coupling of electrons with a certain bosonic excitation as observed in cuprate superconductors (Campuzano et al., 1999; Lanzara et al., 2001; Sato et al., 2003) cannot explain this observation. Because the band produced by such an excitation should follow or mimic the original hole-like dispersion (Norman et al., 1997) with a considerable renormalization of the bandwidth near EF. However, the observed anomalous band shows an electron-like character in contrast to the hole-like band. The angleintegrated-type background as the origin of the
Binding energy (eV)
EF
0.2
High
0.4
395
anomalous band is simply denied since such a background peak in ARPES spectrum would show no dispersion, contrary to the small but finite energy dispersion of the band. The edge-localized state predicted from the calculation for graphite ribbons (Fujita et al., 1996; Nakada et al., 1996; Kobayashi, 1993; Wakabayashi et al., 1999) may account for the observed anomalous band. According to the calculations, a zigzag edge of graphite ribbon possesses an almost-flat band in the close vicinity of EF around the K point, in good agreement with the present observation. We speculate that cleaved surface of kish graphite contains similar zigzag edges at the steps. In fact, a recent STS/STM study (Matsui et al., 2005) reported the existence of zigzag steps and the resultant edge-localized states on the graphite surface. It is also revealed that the zigzag step affects the electronic structure substantially away from the step because the edge-localized state is observed on the flat surface about 35 A˚ away from the step. Probably this long-distance effect of the zigzag step enables the observation of the edge-localized states by ARPES, which probes the electronic structure averaged over a wide area of the surface. The possibility of dangling bonds are not ruled out to explain the observed anomalous electronic states (Jensen and Blase, 2004; Hahn and Kang, 1999). It has been theoretically suggested (Jensen and Blase, 2004) that the character of the graphite surface is dramatically affected by the presence of dangling bonds. These dangling bonds are associated to undercoordinated carbon atoms at the point or extended defects at the step edges. According to the STM measurement by Hahn and Kang (1999) such dangling bonds, which are not passivated by adsorbates contribute to the DOS near EF. We speculate that dangling atoms or dimers at armchair or zigzag edges, which are not passivated by the adsorbates, may be formed in the present experiment where we cleave the sample in an ultrahigh vacuum of 2 1011 Torr. In this case, the observed anomalous feature near EF in the ARPES experiment might be explained by the dangling-bond scenario.
0.6
1.11.3.3 Low –0.10
–0.05
0.0
0.05
0.10
kx (Å–1) Figure 19 ARPES-intensity plot near EF of graphite as a function of wave vector and binding energy measured near the K(H) point (see Figure 18), compared with the projection of calculated bands (McClure, 1957).
Many-Body Interactions
ARPES is a strong experimental technique to determine the band structure of solids. By qualitatively analyzing the energy dispersion and the energy width of the peak in the ARPES spectrum, we are able to obtain an insight into the many-body interactions responsible for the anomalous physical
396 Angle-Resolved Photoemission Spectroscopy
properties. For instance, as shown in Figure 20(a), when electrons are strongly coupled to some kind of collective excitations or bosonic modes, a kink appears in the energy dispersion near EF. The energy scale of the kink basically reflects the characteristic energy of the mode, and the effective mass of the quasi-particle band is highly renormalized in the vicinity of EF due to the interaction. The dispersion kink has been studied recently in various materials, such as high-Tc cuprate superconductors (Campuzano et al., 2004; Damascelli et al., 2003), charge-density-wave materials (Valla et al., 2000; Scha¨fer et al., 2003), and surface systems (Valla et al., 1999; Hengsberger et al., 1999). As for the origin of the mode, mainly phonons and magnetic excitations were proposed so far. By obtaining a flat clean surface with a negligible contribution from the edge-localized states, we find (b)
(a) EF
Mode energy
Kink
k
With interaction
Energy
No interaction (bare band)
Intensity (a.u)
kF
2.5 2.0 1.5 1.0 0.5 EF Binding energy (eV) F
(c)
E D C BA
Binding energy (eV)
EF 0.5 1.0 1.5
High
K(H) M(L)
2.0
Γ(A)
2.5 –0.2
–0.1
0.0
0.1
Low
kx (Å–1)
Figure 20 (a) Schematic diagram to explain the kink structure in the energy dispersion. (b) Ultrahigh-resolution ARPES spectra near EF of graphite around the K(H) point and (c) the corresponding ARPES-intensity plot as a function of wave vector and binding energy.
a signature of the strong electron-mode coupling in kish graphite (Sugawara et al., 2007). In Figure 20(b), we plot ARPES spectra of kish graphite near EF measured at 7 K along a cut that includes the K(H) point (line at K(H) in the inset to Figure 20(c)). Figure 20(c) shows the corresponding intensity plot as a function of wave vector and binding energy. We clearly identify in Figure 20(b) a couple of hole-like bonding bands separated by 0.6 eV, which disperse toward EF when approaching the K(H) point. The splitting of bands is created by the AB stacking sequence of kish graphite as seen in the band calculations (Tatar and Rabii, 1982; Charlier, 1991; McClure, 1957) (see also Figure 12). An anomalously sharp quasi-particle peak appears in the upper band at a narrow angle region centered at the K(H) point, while the peak becomes significantly broad and does not show prominent energy dependence when the position of peak exceeds 0.2 eV. We also find a tiny peak slightly away from the K(H) point in the close vicinity of EF, which shows a remarkable resemblance to the spectral line shape of the surface band near EF in Be surface (Hengsberger et al., 1999). The marked sharpening of the spectral line shape as well as the appearance of an additional peak indicates that electrons are coupled to certain collective excitations. It is noted that the observed quasi-particle peak in graphite is as sharp as the nodal quasi-particle peak in hole-doped cuprates at the superconducting state with the d-wave gap opening (Kaminski et al., 2000). Both cases can be explained by the similar phase space argument (Bena and Kivelson, 2005); in graphite, the extremely small Fermi surface limits the phase space available for the scattering, while in the d-wave superconductors, the scattering among point nodes causes a similar effect and reduces the scattering rate dramatically. To elucidate quantitatively the character of low-energy excitations, we fit ARPES spectrum by two peaks corresponding to the upper and lower bands. We simulate ARPES spectrum by two weakly asymmetric Lorentzian peaks together with a broad background multiplied by the FD distribution function at 7 K, and they are convoluted with a Gaussian having an energy width of instrumental resolution (4 meV). Figure 21 shows the result of fitting for representative k points as denoted by arrows A–F in Figure 20(c). It is evident that the calculated spectral functions (solid curve) reproduce satisfactorily the experimental data (open circles) in whole energy region up to 2.5 eV. The
Angle-Resolved Photoemission Spectroscopy
200
100
ImΣ (meV)
80
ReΣ (meV)
150 λ = 0.70 ± 0.08
60
100
40 20
50
Intensity (a.u.)
0 0 1.2
1.0
0.8
0.6
0.4
EF
0.2
Binding energy (eV)
A B C D E F 2.5
2.0
1.5
1.0
0.5
EF
Binding energy (eV)
Figure 21 ARPES spectra near EF of graphite and their numerical fittings. Inset shows the real and imaginary parts of the self-energy of electrons in graphite.
inset in Figure 21 displays the imaginary part of the self-energy |Im (!)|, which is equal to the quasiparticle scattering rate and inversely proportional to the quasi-particle lifetime, obtained by plotting the half-width-at-half-maximum of the peak at the upper band. |Im (!)| shows a sudden drop below 0.18 eV, in accordance with the sharpening of the quasi-particle peak. The sudden drop in |Im (!)| is caused by the coupling between electrons and a collective mode. In the inset to Figure 21, we also plot the real part of the self-energy Re (!). We defined Re (!) as the energy difference between the obtained peak position of ARPES spectrum and the linear bare band that passes through two points at EF and 1.5 eV in the experimental dispersion (see Figure 20(c)). As indicated by circles in the inset to Figure 21, Re (!) shows a distinct enhancement at the binding energy of 0.16 eV, which indicates the mass renormalization of band. The energy position of peak maximum in Re (!) is close to the energy position of the sudden drop in |Im (!)|. We also obtained Re (!) by Kramers– Kronig transformation of Im (!) (triangles) and
397
found a reasonable agreement with Re (!) determined from the fitting of energy distribution curve (EDC), demonstrating that appearance of quasiparticle peak and the renormalization of band are directly connected to each other. To elucidate the character of the mode quantitatively, we apply the Debye model to reproduce the obtained self-energy on an assumption of a linear scattering rate higher than the Debye energy !D to account for the energy dependence of |Im (!)|. The bulk three-dimensional model with !D ¼ 0.175 eV and the coupling constant ¼ 0.70 0.08 well reproduce the experimentally obtained self-energy. The estimated !D is similar to the known Debye energy of graphite (!D ¼ 0.2 eV) (Kittel, 1953), suggesting that electrons are strongly coupled to phonons. It is remarked here that a proper calculation would involve the actual phonon dispersion into the Eliashberg coupling function 2F(!), but even the oversimplified analysis by using the Debye model reasonably reproduces the obtained self-energy. We note that the estimated !D value (0.175 eV) is similar to the highest phonon energy at 0.2 eV determined from the tunneling (Vitali et al., 2004) and the Raman scattering (Kawashima and Katagiri, 1995) experiments. According to the theoretical calculation of phonon dispersion (Siebentritt et al., 1997), the highest branch is attributed to the longitudinal optical (LO) phonon which is essentially nondispersive at 0.2 eV around the zone center. Acoustic shear (SH) phonon also produces a pronounced DOS at 0.18 eV. Thus, the sudden drop in Im (!) could be explained by the coupling of electrons with the SH phonon and/or the LO phonon.
1.11.4 Graphite Intercalation Compounds 1.11.4.1
C8A (A ¼ K, Rb, Cs)
GICs were first reported in 1840 (Schafha¨l, 1840), and the systematic study started in the early 1930s (Dresselhaus and Dresselhaus, 2002). The intercalation of atoms/molecules with host graphite changes various physical properties. Among the most striking properties are the superconductivity in the first-stage alkali metal GIC C8K (Hannay et al., 1965; Koike et al., 1978), first discovered in 1965 (Hannay et al., 1965). The high conductivity of GICs originates in the charge (electron or hole) transfer from the intercalant to the graphite layer. Among all GICs, alkali-metal GICs have been most intensively
398 Angle-Resolved Photoemission Spectroscopy
kz
(b)
(a) C8K
A
kx K
Σ
Z B Λ T ΓΔ Y
(a) C8 K
∼ ΓΚ Ζ
He I
ky
70° Graphene layer kx Unit cell 1st layer α (K) 2nd layer β 3rd layer γ 4th layer δ 5th layer α
Y K
65°
Intensity (a.u.)
K
Γ (Γ) Y Γ
60° 55° 50° 45° 40° 35° 30° 25° 20° 15° 10° 5° 0°
(M) ky
Figure 22 (a) Crystal structure and (b) Brillouin zone of C8K.
1.11.4.1.1
Band structure
Figure 23(a) shows ARPES spectra of C8K measured along the KZ line (KM line of bulk graphite) as a function of polar angle . We clearly identify some characteristic features of the electronic states: (1) the unfolded band, which is located at about 9.5 eV in the normal-emission spectrum, disperses remarkably toward the lower binding energy on increasing , but it disperses back again toward higher binding energy at 60 before reaching EF; (2) the unfolded band, which abruptly appears at about 45 at EF, showing a small but finite energy dispersion with the bottom at 1.0 eV; (3) the folded bands, which appear at 4–7 eV in the spectra at 0–20 ; and (4) a nondispersive band at EF with the largest peak intensity at ¼ 0 with some additional features at ¼ 25–30 . In Figure 23(b), we plot energy positions of peaks and shoulders in the ARPES spectra as a function of the wave vector. We find the folding of energy bands of
10
(b)
5 Binding energy (eV)
EF
EF Binding energy (eV)
studied by ARPES, especially in the 1980s, in C6Li (Eberhardt et al., 1980; McGovern et al., 1980), C8K (Takahashi et al., 1986), C8Rb (Eberhardt et al., 1980; McGovern et al., 1980), and C8Cs (Gunasekara et al., 1987). The crystal structure and Brillouin zone for C8K are shown in Figure 22. The graphite and intercalant layers are arranged in the AAA A stacking sequence in C8K. As shown in Figure 22(b), the modified two-dimensional Brillouin zone for C8K occupies one-fourth of the area of the whole Brillouin zone of graphite due to the reconstruction of Brillouin zone by the periodic potential from the alkali-metal ions.
5
10 Γ M
∼ ∼ C K Γ Κ Y Κ 8 Κ (Graphite) Γ
Y
Γ M
YΓZ Γ
Figure 23 (a) ARPES spectra of C8K and (b) the experimentally determined band structure compared with the band calculation (Ohno et al., 1979).
graphite into the Brillouin zone of the 2 2 superstructure, as clearly seen between two K~ points. This indicates that the surface region of the C8K sample probed by ARPES has the same 2 2 superstructure as in bulk. In Figure 23(b), we also plot the bands calculated by Ohno, Nakao, and Kamimura (the ONK model; Ohno et al., 1979) for comparison. The experimentally obtained band structure reasonably agrees with the band calculation, especially for the and bands. We find in Figures 23(a) and 23(b) that the intensity of the folded bands is much weaker than that of the unfolded ones like in C6Li (Eberhardt et al., 1980; McGovern et al., 1980).
Angle-Resolved Photoemission Spectroscopy
1.11.4.1.2
Interlayer band As seen in Figure 23(b), some discrepancies are also found between the ARPES experiment and the calculation: (1) the so-called K 4s band (interlayer band) in the calculation shows a nearly-free-electron-like dispersion around the point while we find only a nondispersive band at EF in the ARPES experiment; (2) the calculated highest energy level of the band at the point is located at about 7 eV while it is experimentally observed at about 6 eV. As for the interlayer band, it has a large energy dispersion along the interlayer direction (Z) as shown in Figure 23(b). This interlayer dispersion may distort the present ARPES spectra, causing a discrepancy between the experiment and the calculation. Although we do not find a clear signature for the interlayer band, the present experimental data suggest the existence of partially filled band around the point. This is derived from the following two experimental facts. First, the experimentally observed band observed at the K~ point shows a good agreement with that of the ONK model, which predicts the presence of the interlayer band below EF (Ohno et al., 1979), while the band does not show a good agreement with the model predicting the absence of the interlayer band below EF (DiVincenzo and Rabii, 1982). The size of electron pocket in the experiment is similar to that in the ONK model, which suggests an electron charge transfer of 0.6e from potassium atoms (e is the unit charge). In order to accommodate the remaining electronic charge of about 0.4e, another band is necessary somewhere below EF in the experimental band structure. This extra band is supposed to be located around the center of the Brillouin zone according to the theoretical interpretations for C8K (Ohno et al., 1979; DiVincenzo and Rabii, 1982). Second, the nondispersive feature observed at EF shows the largest peak intensity at 0 , and when we increase the polar angle, the peak intensity is again enhanced around the point in the next Brillouin zone. This strongly suggests the existence of a band below EF around the point, which would likely be the interlayer band. It is noted that this interlayer band is more clearly observed in C6Ca as we describe in the next section.
1.11.4.2
C6Ca
The discovery of superconductivity in C6Yb and C6Ca (Weller et al., 2005) in 2005 has revived a considerable interest since their Tc values (6.5 and
399
11.5 K) are significantly higher than those of other GICs. To determine its pairing symmetry, several experimental results have been reported. Although the isotropic s-wave superconducting order parameter within the conventional BCS theory has been inferred from the thermodynamic (Sutherland et al., 2007) and magnetic properties (Lamura et al., 2006), substantial deviations and anomalies beyond the conventional framework have also been suggested by the tunneling spectroscopy (Bergeal et al., 2006; Kurter et al., 2007) and the electron spin resonance experiment (Muranyi et al., 2008), proposing the strong-coupling superconductivity, the anisotropic gap, and/or the multicomponent gaps. The discovery of high-Tc GICs has also stimulated several theoretical proposals for the superconducting mechanism. An earlier band calculation (Csanyi et al., 2005) has proposed an unconventional pairing mechanism via the interlayer band with intermediate bosons such as low-energy acoustic plasmons (Csanyi et al., 2005) and excitions (Allender et al., 1973). Pairing mechanism by the resonating valence bond has also been discussed (Black-Schaffer and Doniach, 2007). Recent first-principles calculations (Boeri et al., 2007; Calandra and Mauri, 2005; Mazin, 2005; Mazin et al., 2007; Sanna et al., 2007) have suggested that the electron coupling to the out-of-plane carbon and intercalant phonons is responsible for achieving higher Tc’s, while a clear demonstration to experimentally distinguish various models has not yet been made. This key issue as well as the discrepancies in the nature of superconducting gap among experiments request further precise investigations on the momentum and Fermi-surface dependence of the superconducting gap. However, such an experiment had hardly been done because of the small superconducting gap as well as the lack of high-quality single crystals. Figure 24 shows (a) crystal structure and (b) Brillouin zone of C6Ca. Since the Ca atom stacks with the rhombohedral coordination with an AAA sequence, the Brillouin zone of C6Ca is different from that of C8A. The projected surface Brillouin zone has a hexagonal shape but is rotated by 30 with respect to the Brillouin zone of C8A (Figure 22(b)). C6Ca single crystals were usually grown by the chemical vapor transport method (Weller et al., 2005), while it is also produced by the alloy method (Emery et al., 2005). The inset in Figure 25 shows a picture of C6Ca single-crystal grown by the alloy method by heating kish graphite mixed with molten
400 Angle-Resolved Photoemission Spectroscopy
(a)
(b) Γ
M
: Ca :C
K L X
T Graphene layer Unit cell
1st layer α 2nd layer β 3rd layer γ 4th layer α
χ X
T
Figure 24 (a) Crystal structure and (b) the Brillouin zone of C6Ca.
FC Moment (e.m.u. cm–3)
0.0 –0.2
1 mm
–0.4 Tc = 11.5 K
–0.6 –0.8
ZFC
H = 10 (Oe)
–1.0 6
8 10 12 Temperature (K)
14
Figure 25 The DC magnetic susceptibility of C6Ca. Inset shows a picture of C6Ca single crystal used for ARPES measurements.
lithium–calcium alloy at around 350 C for 6 days under high vacuum. Figure 25 also shows the magnetic susceptibility of C6Ca single crystal as a function of temperature for zero-field cooling (ZFC) and field cooling at 10 Oe. The sharp drop in the ZFC curve confirms that the crystal exhibits the superconductivity with the onset Tc ¼ 11.5 K with the superconducting volume fraction as high as 90%. While kish graphite is very easy to cleave because of the weak van der Waals coupling among graphite layers, C6Ca is harder to cleave due to more three-dimensional-like bonding character. 1.11.4.2.1
Band structure Figures 26(a) and 26(b) show ARPES spectra of C6Ca measured at 15 K along K and M directions of graphite Brillouin zone, respectively (Sugawara et al., 2008). We clearly find highly dispersive features
in both directions, basically similar to those of kish graphite (see Figure 13). We find a peak in the vicinity of EF showing an electron-like energy dispersion around the K(H) point, which is not observed in kish graphite. We also notice a small but finite peak with the Fermi-edge cut off around the point. This behavior has not been recognized in kish graphite. Figure 27 shows the experimentally determined valence-band structure of C6Ca along high-symmetry lines. Although the overall band structure of C6Ca looks to be similar to that of kish graphite (Figure 14), it is noticed that the band structure of C6Ca is shifted as a whole toward high binding energy by 1.0–1.5 eV, indicating that electrons are certainly doped into pristine graphite by Ca intercalation. A small electron pocket that appears in the vicinity of Fermi level (EF) at the K point in C6Ca is ascribed to the band, which is above EF in pristine graphite and is shifted down below EF by the electron doping (McChesney et al., 2007; Molodtsov et al., 2003; Takahashi et al., 1986).
1.11.4.2.2
Fermi surface Figure 28 shows the Fermi surface of C6Ca determined by plotting the ARPES-spectral intensity at EF as a function of the two-dimensional wave vector. The topology of the Fermi surface in C6Ca is drastically different from that of graphite (Figure 16). We find two different Fermi surfaces, a circular Fermi surface at the point and a triangular Fermi surface at the K point, while only one tiny Fermi surface is observed at the K(H) point in graphite (Sugawara et al., 2006). The triangular Fermi surface at the K point originates in the band and is well explained in terms of the simple rigid shift of the chemical potential due to the electron doping (Tatar
Angle-Resolved Photoemission Spectroscopy
(a)
(b)
M(L)
M(L)
K(H) M K
401
K(H) M Γ(A) K
Γ(A)
Intensity (a.u.)
Intensity (a.u.)
K(H)
M(L)
Γ(A) Γ(A) 16
EF 12 8 4 Binding energy (eV)
16
12
8
EF
4
Binding energy (eV)
Figure 26 Valence-band ARPES spectra of C6Ca measured along (a) the K and (b) M directions in the graphite Brillouin zone (Sugawara et al., 2008).
C6Ca
C6Ca
B′
EF 1 ky (Å–1)
Binding energy (eV)
2 EF
M
B
1.0
K A
0.0
M
Γ
A′
K
High
–1.0 4
–1.0 8
High
0.0
1.0
Low
kx (Å–1)
Figure 28 Fermi surface of C6Ca determined by ARPES.
12
16 M
K
Γ M Wave vector
K
Low
Figure 27 Band structure of C6Ca determined by ARPES. The top panel shows an expansion of the near-EF region.
and Rabii, 1982). In contrast, the small Fermi surface observed at the point is not explained by the simple rigid shift of graphite bands or the folding
of band due to the superstructure of Ca intercalants, because the shape and size of Fermi surface are distinctly different from those of the Fermi surface. On the other hand, the band structure calculations (Boeri et al., 2007; Calandra and Mauri, 2005; Mazin, 2005; Mazin et al., 2007; Sanna et al., 2007) have predicted an additional band at the point in C6Ca (named the interlayer band), which has a sizable contribution from the intercalant states and produces a small free-electron-like spherical Fermi surface at the point. Considering a good agreement in the location
402 Angle-Resolved Photoemission Spectroscopy
and the shape, we assign the small Fermi surface observed at the point to the interlayer band (Ohno et al., 1979) in C6Ca. In fact, the observed circular Fermi surface is not ascribed to the calculated two Fermi surfaces in C6Ca, since one calculated Fermi surface has a hexagonal shape and the other is not centered at the point (Sanna et al., 2007; Mazin et al., 2007).
1.11.4.2.3
Superconducting gap Figure 29 shows ARPES spectra near EF of C6Ca at 6 K measured at two representative Fermi vectors (kF’s) on the circular Fermi surface at the point and the triangular Fermi surface at the K point, respectively, as indicated by circles in Figure 28. Note that the spectral intensity shows a sudden drop at 80 meV on the circular Fermi surface (point A) while a small but clear dip is observed at 170 meV in the spectrum measured on the triangular Fermi surface (point B). We find that the energy position of dip does not vary in the momentum region around the kF point. This is not explained by the presence of another band located around the dip energy, but is reminiscent of the strong manybody effects observed in high-Tc cuprates (Campuzano et al., 2004; Damascelli et al., 2003) and Ba0.67K0.33BiO3 (Chainani et al., 2001). In fact, as seen in Figure 29, the Eliashberg function 2F(!) calculated for C6Ca (Calandra and Mauri, 2005)
C6Ca
Intensity (a.u.)
Tc = 11.5 K
A
α 2F(ω)
1.0
0.4
C(xy)
C(z)
0.5
0.3 0.2 0.1 Binding energy (eV)
Ca (xy)
B
EF
Figure 29 ARPES spectra near EF of C6Ca at 6 K measured at points A and B (see Figure 28), compared to the Elieshberg function 2F(!) (Calandra and Mauri, 2005).
shows dominant peaks at 50–75 and 160–180 meV originating in the out-of-plane and the in-plane vibrations of carbon atoms, respectively, in good agreement with the anomalies observed in the ARPES spectra. This suggests that electrons on the circular Fermi surface at the point are strongly coupled to the out-of-plane vibrations while those on the triangular Fermi surface at the K point are coupled dominantly with the in-plane phonons. This experimentally demonstrates that the interlayer electrons strongly interact with the surrounding lattice unlike in the free-electron picture, as predicted from the first-principles calculations (Sanna et al., 2007; Boeri et al., 2007). The reason why electrons at the K point are only weakly coupled to the out-ofplane phonons may be the antisymmetry of the wave function with respect to the graphene layer (Calandra and Mauri, 2005). On the other hand, the interlayer band forming the circular Fermi surface at the point is expected to be strongly coupled to the out-of-plane phonons due to the symmetry (Boeri et al., 2007). We note that a spectral dip related to the coupling with the Ca vibrations is not clearly observed in Figure 29, possibly because the dip is broad (20–30 meV) compared to the Ca phonon energy (15 meV). Figure 30(a) shows ultrahigh-resolution ARPES spectra near EF of C6Ca measured at two representative k points A and B as shown in Figure 28 at temperatures below/above Tc (6 and 15 K). At point A, the midpoint of the leading edge in the spectrum at 6 K is shifted toward high binding energy by approximately 0.8 meV with respect to EF while the spectrum at 15 K has the midpoint at EF. This clearly indicates the opening/closing of a superconducting gap as a function of temperature. On the other hand, at point B, the midpoint of the leading edge is located at around EF even at 6 K, and the spectrum looks to show a simple temperature dependence to obey the FD function, suggesting the absence or the very small magnitude of the superconducting gap. This marked difference is better illustrated in Figure 30(b) where the spectra at 6 K at points A and B are directly compared. To clarify the possible anisotropy of the gap, we also plot the 6-K spectra measured at points A9 and B9 on the same Fermi surfaces but on different high-symmetry lines (Figure 28), since the anisotropy, if it exists, should be largest between points A and A9 (B and B9). As seen from Figure 30(b), the leading edge at point A9 (B9) almost coincides with that of points A (B). This suggests that the anisotropy of superconducting gap
Angle-Resolved Photoemission Spectroscopy (b)
Intensity (a.u.)
C6Ca Tc = 11.5 K A
T= 6K 15 K
10
5 EF Binding energy (meV)
T=6K
Intensity (a.u.)
(a)
B
10
–5
403
A A' B B'
∆ = 1.8 meV Γ = 0.5 meV
A
5 EF Binding energy (meV)
–5
Figure 30 (a) Superconducting gap of C6Ca measured at representative kF points A and B in Figure 28. (b) Top: direct comparison of ARPES spectrum at T ¼ 6 K measured at four kF points in Fig. 28. Bottom: Numerical simulation of ARPES spectrum at point A.
1.11.5 Graphene 1.11.5.1 Single-Layered Graphite: Dirac Fermion? Graphene has attracted much attention since it shows various interesting physical properties such as the massless Dirac-fermion-like behavior and the unusual quantum Hall effect (Novoselov et al., 2005). The recent success of spin injection at room temperature (Ohishi et al., 2007; Tombros et al., 2007) and the observation of bipolar supercurrent (Heersche et al., 2007) suggest that graphene has a high potential of application to spintronics devices. To reveal the origin and the mechanism of these novel properties, it is indispensable to elucidate the electronic structure near the Fermi level (EF). As shown in Figure 31, it has been theoretically predicted (Painter and Ellis,
20
σ∗
10
π∗ EF
π
σ
–10 –20
K
Γ Wave vector
M
K
Figure 31 Schematic band structure of graphene (Painter and Ellis, 1970).
Binding energy (eV)
is very small in both circular and triangular Fermi surfaces, while the gap value is markedly different between these two Fermi surfaces. To estimate the size of superconducting gap, we numerically fit the spectrum by using the Dynes function (Dynes et al., 1978). The representative result for point A is displayed in Figure 30(b). The estimated gap size at 6 K is 1.8 0.2 and 2.0 0.2 meV at points A and A9, respectively. The reduced gap value of 2(0)/kBTc by assuming the mean-field temperature dependence is 4.1 0.5, indicating the intermediate- to strongcoupling character. The first-principles calculation (Sanna et al., 2007) has predicted a similar gap value. On the other hand, at points B and B9, the estimated superconducting gap size is 0.2 0.2 meV. These results suggest the anisotropy of the superconducting gap of 1 meV reported by the tunneling experiment (Bergeal et al., 2006) is not explained in terms of the anisotropy of interlayer Fermi surface at the point, but is actually influenced by a smaller superconducting gap of other Fermi surfaces, most likely the Fermi surface. The reported deviation of the specific heat (Kim et al., 2006) from the single isotropic s-wave behavior (Sanna et al., 2007; Mazin et al., 2007) would also be explained by this Fermi surface dependence of the superconducting gap. Present results demonstrate that the coupling of electrons in the interlayer band with the out-ofplane phonons is essential to realize the superconductivity in C6Ca, while the simple electron doping into the band of graphene layers may not lead to the superconductivity at such a high temperature. The next challenge is to clarify the role of intercalant (Ca) phonons which have been theoretically proposed to enhance the Tc value (Boeri et al., 2007; Calandra and Mauri, 2005; Mazin, 2005, 2007; Sanna et al., 2007).
404 Angle-Resolved Photoemission Spectroscopy
1970) that the and bands contact each other only at K point of the Brillouin zone in graphene and the energy at this contacting point is called the Dirac energy (ED). It is obvious that ED is equal to EF in pristine (undoped) graphene. To experimentally investigate the electronic structure of graphene by ARPES, a clean surface of graphene has to be prepared under ultrahigh vacuum. There are several methods to fabricate the graphene sample. First is to peel off the bulk graphite many times known as a micro cleaving (Novoselov et al., 2005). Second is to evaporate/dissociate organic materials like ethylene on a clean metal surfaces such as Ni(111) (Nagashima et al., 1994a), TaC(111) (Nagashima et al., 1994b), and Pt(111) (Fujita et al., 2005). Third is to segregate a graphene sheet on SiC(0001) crystal surface by heating the SiC crystal (Vanbommel et al., 1975). First method has a difficulty in obtaining a large surface area suitable for ARPES measurements. In the second method, we expect a strong interaction between the substrate and the graphene film, so that near-EF energy bands of graphene are strongly modulated by the hybridization by the energy bands of the metallic substrate. The third method seems useful for ARPES measurement because the interaction between substrate and the graphene film is much weaker than the second case, and a clean surface with the large area (few mm2) is easily obtained by controlling the annealing temperature
and the temperature gradient on the surface. Here we focus on the ARPES results of graphene film prepared by the third method. Graphene thin films were prepared by annealing an n-type Si-rich 6H-SiC(0001) single crystal with resistive heating in an ultrahigh vacuum of 1 1010 Torr (Bostwick et al., 2007; Zhou et al., 2007, 2008; Rotenberg et al., 2008). By controlling the temperature and the heating time, it is possible to selectively grow single-layer graphene and multilayer graphene with more than 10 sheets (Kusunoki et al., 2000). Graphene films are characterized by the low-energy electron diffraction (LEED) and the energy position of and bands observed by ARPES. Figure 32(a) shows the LEED pattern of graphene grown on 6H-SiC(0001) measured with primary electron energy of 107 eV. The LEED pattern of graphene on SiC(0001) consists of three different spots: (1) the (1 1) spots from the bulk SiC p p substrate (SiC (1 1)); (2) the weak 6 3 6 3 spots which originate in the bulk SiC layer bonded with the interface graphene layer (buffer layer); and (iii) the C (1 1) spots due to the honeycomb atomic pattern of graphene (Ni et al., 2008). Figure 32(b) shows the valence-band ARPES spectra of graphene measured along the KM direction. Several highly dispersive bands are clearly observed, and the overall band dispersion is basically similar to that of bulk graphite (see Figure 14(a)).
Graphene/SiC(0001) (a)
(b) Graphene (1 x 1) 6 3x6 3
M
Binding energy (eV)
(c) EF
Intensity (a.u.)
Si (1 x 1)
π σ
4
K
8
Γ
12
16
Γ
K Wave vector
M
16
EF 4 12 8 Binding energy (eV)
Figure 32 (a) LEED pattern of graphene grown on 6H-SiC(0001). (b) Valence-band ARPES spectra of graphene. (c) Experimentally determined band structure compared with the band calculation of graphene.
Angle-Resolved Photoemission Spectroscopy
We find a single band which rapidly disperses toward EF on approaching the K point. Figure 32(c) shows the ARPES-derived band structure of graphene/SiC(0001) compared with the calculated bands. The experimental band structure was obtained by taking the second derivative of ARPES spectra and plotting the intensity as a function of wave vector and binding energy. As expected, the band structure of graphene consists of the highly dispersive and bands. The overall band structure is well reproduced by the calculation, while some quantitative differences are found in the energy position of the top of band and the bottom of band at the point.
1.11.5.2
Comparison with Graphite
Figure 33 shows the comparison of experimental band structure between (a) graphene/SiC(0001) and (b) graphite/SiC(0001). It is noted that the multilayer graphene sample in this study shows the almost identical band structure to that of bulk kish graphite (Sugawara et al., 2006, 2007). Hence we refer to this multilayer graphene as graphite in this section. As clearly visible in Figure 33, the band structures of graphene and graphite look very similar to each other, consisting of the highly dispersive and bands. The band has the bottom of dispersion at around 8–9 eV at the point and the top at EF at the K point, while the band has the top at around 4–5 eV at point and shows a strong downward dispersion toward the K point. However, detailed comparison reveals several quantitative differences.
Graphite/SiC(0001)
EF
π σ
Binding energy (eV)
Binding energy (eV)
EF
For example, the bottom of band at point in graphene is shifted by about 0.7 eV toward the high binding energy compared to that in graphite. As for the band, the top (bottom) of dispersion at the (K) point in graphene is located at 0.5 eV higher (0.7 eV lower) binding energy than that of graphite. These experimental results indicate that the band width of both and bands is larger in graphene than in graphite. This quantitative difference in the band width may be ascribed to the difference in the lattice constant. Actually, the wave vector of the K point estimated from the present ARPES results is slightly different between graphene and graphite; 1.72 0.01 A˚1 for graphene and 1.70 0.01 A˚1 for graphite. The corresponding in-plane lattice vectors () are 2.44 0.01 and 2.46 0.01 A˚, respectively, showing a good agreement with the reported value (2.46 A˚) (Lukesh and Pauling, 1950) for the case of graphite. The lattice contraction in graphene may be due to a slight mismatch of the lattice constant between graphite and SiC as observed by Raman spectroscopy (Ni et al., 2008). In addition to the and bands, we find a weakly dispersing band around 3–4 eV binding energy in graphene, which is not predicted in the band calculation (Painter and Ellis, 1970) and may be p p due to the 6 3 6 3 structure of the buffer layer (Mattausch and Pankratov, 2007; Emtsev et al., 2008). Figure 34 shows the plot of ARPES intensity around the K point as a function of the wave vector and the binding energy for (a) graphene and (b) graphite. To remove the effect from the FD function, the ARPES intensity is divided by the FD function
(b)
Graphene/SiC(0001)
(a)
4
8
405
π σ
4
8
K(H)
12
12
16
16
M(L)
Γ(A)
Γ
K Wave vector
M
Γ(A)
K(H) Wave vector
M(L)
Figure 33 Comparison of valence-band structure between (a) graphene/6H-SiC(0001) and (b) graphite/6H-SiC(0001).
406 Angle-Resolved Photoemission Spectroscopy
(a)
Graphene/SiC(0001)
EF
Binding energy (eV)
EF
Binding energy (eV)
Graphite/SiC(0001)
(b)
0.2 280 meV
0.4 0.6
kx =
0.8
0.0
0.2 0.4 High
0.6 0.8
kx =
Å–1
0.0Å–1 1.0
1.0 –0.04 0.0 0.04 kx (Å–1)
Intensity (a.u.)
–0.04 0.0 0.04 kx
Intensity (a.u.)
Low
(Å–1)
Figure 34 Comparison of near-EF band structure between (a) graphene/6H-SiC(0001) and (b) graphite/6H-SiC(0001), together with the corresponding energy distribution curve at the K point.
convoluted with the Gaussian which reflects the instrumental resolution. The corresponding ARPES spectra at the K point are also shown for comparison. It is remarked in Figure 34(a) that the Dirac energy ED of graphene is not at EF, but is about 0.4 eV below it, suggesting that the graphene sheet on SiC is doped with electrons, as also evident from the appearance of the band at EF. It is speculated that electrons are transferred from the n-type SiC substrate and/or the buffer layer (Bostwick et al., 2007; Rotenberg et al., 2008; Zhou et al., 2007; Zhou et al., 2008). On the other hand, ED of graphite is at or very close to EF, as seen in Figure 34(b). The band structure of graphite grown on SiC shows a close resemblance to that of bulk kish graphite (see Figure 14) (Sugawara et al., 2007). The energy splitting of band at the K point due to the interlayer coupling between graphene sheets is clearly observed and the energy interval is almost identical to that of kish graphite. A sharp quasi-particle peak appears at EF, and the Dirac energy ED is at or very close to EF as in kish graphite. These experimental results suggest that the graphite sample grown on SiC in the present study consists of a sufficiently large number of graphene sheets and hardly suffers the charge-transfer effect from the substrate (Zhou et al., 2007; Ohta et al., 2007). In contrast, the graphene sheet grown on SiC substantially suffers the charge-transfer effect as evident from the large chemical potential shift as seen in Figure 34(a). Further, very interestingly, there seems to be an energy gap between the and the bands in graphene on SiC, while no or a negligibly small gap is seen in graphite. The size of energy gap
in graphene estimated from the interval between the two peaks in the ARPES spectrum is about 280 meV. This band gap has also been observed (Zhou et al., 2007, 2008) and discussed in terms of the sublatticesymmetry breaking due to the graphene–substrate interaction. On the other hand, it has also been reported that there is no gap opening at ED and the observed small anomaly of band dispersion around ED is explained in terms of the electron–plasmon coupling (Bostwick et al., 2007; Rotenberg et al., 2008). The reason for this difference is unclear at present, while it has also been discussed that the quantitative value of the band gap may depend on the sample preparation process (Zhou et al., 2008). As for the origin of band gap in graphene observed in the present ARPES study, it would be most likely related to the interaction between the graphene sheet and the SiC substrate. The band-gap size decreases on increasing the number of layers (Zhou et al., 2007), and finally vanishes in multilayer graphite (graphite) as shown in Figure 34. As described above, the present ARPES results have revealed that the lattice of graphene grown on SiC(0001) slightly contracts compared with that of bulk graphite, as evident from the larger wave vector of the K point and the resultant larger bandwidth of the and bands in graphene. It is also noted that a graphene sheet grown on TaC(111), which has a relatively large lattice mismatch to the substrate, exhibits a substantially large (1.3 eV) energy gap between the and bands (Oshima and Nagashima, 1997). All these experimental results suggest that the energy gap is caused by the lattice strain due to the lattice
Angle-Resolved Photoemission Spectroscopy
mismatch between the graphene sheet and the substrate, and the gap gradually vanishes as the strain is gradually released at the topmost layers of multilayer graphene. This indicates that the effect from the substrate should be taken into account when graphene grown on a SiC crystal is investigated, and also suggests that the magnitude of the band gap can be controlled by tuning the lattice parameters of the base substrate and resultant hybridization strength between the graphene sheet and the substrate. We note that it is important to grow a graphene film free from the lattice mismatch/strain to investigate the genuine Dirac-fermion-like behavior.
1.11.6 Concluding Remarks and Summary The recent remarkable improvement of the energy and momentum resolutions in ARPES enables us to determine experimentally the band structure and the Fermi surface of various novel materials with great accuracy. In fact, the Fermi surface and the superconducting gap of high-temperature cuprate superconductors have been precisely revealed by ARPES. Further, electrons dressed with interactions, namely quasi-particles, are directly observed by high-resolution ARPES as a small anomaly (kink) in the energy dispersion near EF and have been intensively discussed in relation to the mechanism and the origin of the novel properties such as the high-Tc superconductivity. Theoretical calculations, such as that of the band structure, are directly compared with the experimental results from ARPES experiments to examine the validity of the theory. Thus, ARPES is now regarded as one of the essential experimental techniques in solid-state physics and materials science. On the other hand, the discovery and/or the synthesis of new functional materials have been successively achieved in recent years. In particular, carbon-based new materials such as fullerenes and carbon nanotubes have revived/stimulated the study of the mother material (graphite) and produced additional new functional carbon-based material, grapheme, and high-Tc GICs. In this chapter, we have reviewed high-resolution ARPES studies of graphite, GICs, and graphene. The high-resolution ARPES has revealed that this simple sp2 material exhibits a variety of interesting properties such as the interlayer state, the edge-localized state, the Dirac-fermion, and the Fermi-surfacedependent high-temperature superconductivity. In
407
particular, the direct ARPES observation of the interlayer state and its relationship to the superconductivity in C6Ca has definitely resolved the long-standing problem in the mechanism of superconductivity in GICs. It is expected that high-resolution ARPES will be applied to well-characterized single- or multilayer graphene as well as well-fabricated graphene ribbons to elucidate their novel electronic structures. (See Chapters 1.02, 1.10, 1.12 and 2.05).
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1.12 Theory of Superconductivity in Graphite Intercalation Compounds Y Takada, University of Tokyo, Kashiwa, Chiba, Japan ª 2011 Elsevier B.V. All rights reserved.
1.12.1 1.12.1.1 1.12.1.2 1.12.1.3 1.12.1.4 1.12.2 1.12.2.1 1.12.2.2 1.12.2.3 1.12.2.4 1.12.2.5 1.12.3 1.12.3.1 1.12.3.2 1.12.3.3 1.12.4 1.12.4.1 1.12.4.2 1.12.4.3 1.12.4.4 1.12.5 1.12.6 1.12.6.1 1.12.6.2 1.12.6.3 1.12.7 1.12.7.1 1.12.7.2 1.12.8 1.12.9 References
Introduction Crystal Structure Superconductivity Central Issues Organization of This Chapter First-Principles Calculation of Tc Goal of the Problem McMiland’s and Allen-Dynes’ Formulas for Tc Coulomb Pseudopotential Vertex Corrections and Dynamic Screening Ideal Calculation Scheme Calculation of Tc in the G0W0 Approximation Formulation Comments on the Formulation Assessment: Application to SrTiO3 Density Functional Theory for Superconductors Hohenberg–Kohn–Sham Theorem Gap Equation Applications Basic Problems Experiment on Superconductivity in GICs Standard Model for Superconductivity in GICs Characteristic Features of the System Microscopic Model for Superconductivity Calculation of Tc for Alkali-Doped GICs Superconductivity in Alkaline-Earth GICs CaC6 Other Alkaline-Earth GICs Prediction of the Optimum Tc in GICs Conclusion
411 411 411 412 413 413 413 413 413 414 414 414 414 415 416 416 416 417 417 418 418 419 419 420 421 421 421 422 422 423 423
Glossary Allen-dynes formula In 1975, Allen and Dynes modified the McMillan’s formula for Tc, producing a rather large change in Tc for the electron–phonon nondimensional coupling constant larger than 2. Coulomb pseudopotential A phenomenological parameter to estimate the effect of the Coulomb repulsion on the Cooper pairing by taking into account both retardation and screening effects.
410
Density functional theory (DFT) A theory to describe an interacting many-body system in terms of its density distribution in three dimensions. Eliashberg function In the phonon mechanism, the phonon-mediated attractive interaction is characterized by both the electron–phonon coupling (!) and the phonon density of states F(!), but in calculating Tc only their combination
Theory of Superconductivity in Graphite Intercalation Compounds
(!)2F(!), which is called the Eliashberg function, is important. First-principles calculation A calculation is said to be from first principles or ab initio if it starts directly at the level of established laws of physics which contain no fitting or phenomenological parameters. Graphite intercalation compounds (GICs) Complex materials in which the graphite layers remain largely intact, and the guest element or molecule is inserted between the layers. GW approximation In 1965, Hedin proposed an approximation scheme to calculate the electron self-energy in terms of the one-electron Green’s function G and the effective electron–electron interaction W without any vertex corrections. If the same scheme is applied to superconductivity, it is nothing but the Eliashberg theory for superconductivity. Isotope effect If Tc depends directly or indirectly on the mass M of the ions building up the lattice, we call the dependence the isotope effect. In the standard Bardeen–Cooper–Schrieffer (BCS) theory, Tc changes in proportion to M with the isotope coefficient ¼ 0.5. McMillan’s formula In 1968, McMillan proposed a simple formula for Tc in the phonon mechanism composed of an average phonon energy !0, the electron–phonon nondimensional coupling constant , and the Coulomb pseudopotential ,
1.12.1 Introduction 1.12.1.1
Crystal Structure
For many decades, graphite intercalation compounds (GICs) have been investigated from the viewpoint of physics, chemistry, materials science, and engineering (or technological) applications (Fischer and Thompson, 1978; Kamimura, 1987; Zabel and Solin, 1992; Dresselhaus and Dresselhaus, 2002). Among various kinds of GICs, special attention has been paid to the first-stage metal compounds, partly because superconductivity is observed mostly in this class of GICs, the chemical formula of which is written as MCx, where M represents either an alkali atom (such as Li, K, Rb, and Cs) or an alkaline-earth atom (such as Ca, Sr, and Yb) and x is either 2, 6, or 8.
411
based on the numerical solution of the Eliashberg equation for various systems. Phonon mechanism If the Cooper pairs are created by the attractive interaction induced by virtual exchange of phonons between electrons, the superconductivity is considered to be brought about by the phonon mechanism. Plasmon mechanism In the low-density electron gas, plasmons can play a similar role to phonons in superconductivity. We indicate this role as the plasmon mechanism. Spin-fluctuation mechanism In a system with strong spin fluctuations at either long wavelengths (related to a ferromagnetic instability) or short wavelengths (related to an antiferromagnetic instability), the fluctuations can play a role similar to phonons in superconductivity. We indicate this role as the spin-fluctuation mechanism. Stage In a GIC, not every layer is necessarily intercalated by guest elements or molecules. In the so-called stage 1 GICs like KC8, graphite and intercalated layers alternate, while in stage 2 GICs such as KC24, two graphite layers with no guest material in between alternate with an intercalated layer. Similarly, we can define stage n with n > 2. Vertex function A factor to describe the change in the electron charge during scattering processes due to the modification of the many-body wave function. Its deviation from unity (or the bare coupling) is called vertex corrections.
The crystal structure of MCx is shown in Figure 1(a), in which the metal atom M occupies the same spot in the framework of a honeycomb lattice at every (x/2) layers of carbon atoms. 1.12.1.2
Superconductivity
The first discovery of superconductivity in GICs was made in KC8 with the superconducting transition temperature Tc of 0.15 K in 1965 (Hannay et al., 1995). In pursuit of higher Tc, various GICs were synthesized, mostly working with the alkali metals and alkali-metal amalgams as intercalants, from the late 1970s to the early 1990s (Alexander et al., 1980; Belash et al., 1989; Belash et al., 1990; Dresselhaus et al., 1989; Hannay et al., 1995; Iye and Tanuma,
412 Theory of Superconductivity in Graphite Intercalation Compounds
(a)
(b) CaC6 101
YbC6 LiC2
Tc (K)
d
SrC6
100
KC8 RbC8 C5C8
10–1
10–2 : M
4.0
4.5
5.0
5.5
6.0
d (Å)
: C
Figure 1 (a) Crystal structure of MCx (x ¼ 2, 6, 8). The case of x ¼ 6 is illustrated here, in which the metal atoms, M’s, are arranged in a rhombohedral structure with the stacking sequence, implying that M occupies the same spot in the framework of a graphene lattice at every three layers (or at the distance of 3d with d the distance between the adjacent graphite layers). (b) Superconducting transition temperature Tc observed in the first-stage alkali- and alkaline-earthintercalated graphites plotted as a function of d.
1982; Koike et al., 1978; Kobayashi and Tsujikawa, 1979; Koike et al., 1980), but only a limited success was achieved at that time; the highest attained Tc was around 2–5 K in the last century. For example, it is 1.9 K in LiC2 (Belash et al., 1989). A breakthrough occurred in 2005 when Tc went up to 11.5 K in CaC6 (Weller et al., 2005; Emery et al., 2005) (and even to 15.4 K under pressures up to 7.5 GPa (Gauzzi et al., 2007)). In other alkaline-earth GICs, the values of Tc are 6.5 and 1.65 K for YbC6 (Weller et al., 2005) and SrC6 (Kim et al., 2006), respectively, as indicated in Figure 1(b). Since then, very intensive experimental studies have been made in those and related compounds (Emery et al., 2005; Kim et al., 2007, 2006; Kurter et al., 2007). Theoretical studies have also been performed mainly by making state-of-the-art first-principles calculations of the electron–phonon coupling constant to account for the observed value of Tc for each individual superconductor (Mazin, 2005; Csanyi et al., 2005; Calandra and Mauri, 2005; Sanna et al., 2007). Those experimental/theoretical works have elucidated that, although there are some anisotropic features in the superconducting gap, the conventional phonon-driven mechanism to bring about s-wave superconductivity applies to those compounds. This picture of superconductivity is confirmed by, for example, the observation of the Ca isotope effect with its exponent ¼ 0.50, the typical Bardeen–Cooper–Schrieffer (BCS) value (Hinks et al., 2007).
1.12.1.3
Central Issues
In spite of all those efforts and the existence of such a generally accepted picture, there remain several very important and fundamental questions: 1. Can we understand the mechanism of superconductivity in both alkali GICs with Tc in the range 0.01–1.0 K and alkaline-earth GICs with Tc in the range 1–10 K from a unified point of view? In other words, is there any standard model for superconductivity in GICs with Tc ranging over three orders of magnitude? 2. What is the actual reason why Tc is enhanced so abruptly (or by about a hundred times) by just substituting K with Ca, the atomic mass of which is almost the same as that of K? In terms of the standard model, what are the key controlling physical parameters to bring about this huge enhancement of Tc ? This change of Tc from KC8 to CaC6 is probably the most important issue in exploring superconductivity across the entire family of GICs. 3. Is there any possibility to make a further enhancement of Tc in GICs ? If possible, what is the optimum value of Tc expected in the standard model and what kind of atoms should be intercalated to realize the optimum Tc in actual GICs? In order to provide reliable answers to the above questions, it is indispensable to make a
Theory of Superconductivity in Graphite Intercalation Compounds
first-principles calculation of Tc with sufficient accuracy and predictive power. Recently, such a calculation has been completed by the present author, and based on the calculation, some interesting predictions have been proposed (Takada, 2009a; 2009b). The present chapter not only reports some details of this work on the superconducting mechanism in GICs but also makes a brief summary of the current status of the theories for first-principles calculations of Tc. 1.12.1.4
Organization of This Chapter
This chapter is organized as follows: In Sections 1.12.2–1.12.4, a critical review of the theories for quantitative calculations of Tc is given. More specifically, we make comments on the theories based on the McMillan’s or the Allen-Dynes’ formula employing the concept of the Coulomb pseudopotential in Section 1.12.2. In Section 1.12.3 we explain the theory on the level of the so-called G0W0 approximation, where Tc can be obtained without using , a very important advantage. The same advantage can be enjoyed in the density functional theory for superconductors, which will be addressed in Section 1.12.4. In Sections 1.12.5–1.12.8, a review on superconductivity in GICs is given; starting with a summary of the experimental works in Section 1.12.5, a standard model for considering the mechanism of superconductivity in GICs is introduced in Section 1.12.6. In Section 1.12.7, the calculated results of Tc are given for the alkaline-earth GICs and they are compared with the experimental results. The prediction of the optimum Tc is given in Section 1.12.8. Finally in Section 1.12.9, the conclusion of this chapter is given, together with some perspectives on the researches in this and related fields in the future.
1.12.2 First-Principles Calculation of Tc 1.12.2.1
Goal of the Problem
It would be one of the ultimate goals in the enterprise of condensed matter theory to make a reliable prediction of Tc only through the information on constituent elements of a superconductor in consideration. A less ambitious yet very important goal is to make an accurate evaluation of Tc directly from a microscopic (model) Hamiltonian pertinent to the superconductor. If we could find the dependence of Tc on the parameters specifying the model Hamiltonian, we could obtain a deep insight into the mechanism of superconductivity or the
413
competition between the attractive and the repulsive interactions between electrons. Accumulation of such information might pave the way to the synthesis of a room-temperature superconductor, a big dream in materials science. From this perspective, a continuous effort has been made for a long time to develop a good theory for first-principles calculations of Tc, starting with a microscopic Hamiltonian. 1.12.2.2 McMiland’s and Allen-Dynes’ Formulas for Tc In the phonon mechanism, for example, there has been a rather successful framework for this purpose, known as the McMillan’s formula (McMillan, 1968) or its revised version (the Allen-Dynes’ formula) (Allen and Dynes, 1975; Allen and Mitrovic´, 1982; Carbotte, 1990), both of which are derived from the Eliashberg theory of superconductivity (Eliashberg, 1960). In this framework, the task of a microscopic calculation of Tc is reduced to the evaluation of the so-called Eliashberg function 2F(!) from first principles; the function 2F(!) enables us to obtain both the electron–phonon coupling constant and the average phonon energy !0, through which we can make a first-principles prediction of Tc by additionally introducing a phenomenological parameter (the Coulomb pseudopotential (Morel and Anderson, 1962)) in order to roughly estimate the effect of the Coulomb repulsion between electrons on Tc. At present this framework is usually regarded as the canonical one for making a first-principles prediction of Tc. In fact, the superconducting mechanism of many (so-called weakly correlated) superconductors is believed to be clarified by using this framework, whereby the key phonon modes to bring about superconductivity are identified. We can point out that superconductivity in MgB2 with Tc ¼ 39 K is a good recent example (Bohnen et al., 2001; Choi et al., 2002a; 2002b; Kong et al., 2001) to illustrate the power of this framework. The case of CaC6 has also been investigated along this line of theoretical studies (Mazin, 2005; Calandra and Mauri, 2005). 1.12.2.3
Coulomb Pseudopotential
Nevertheless, this framework is not considered to be very satisfactory, primarily because a phenomenological parameter is included in the theory. Actually, it cannot be regarded as the method of predicting Tc in the true sense of the word, if the
414 Theory of Superconductivity in Graphite Intercalation Compounds
parameter is determined so as to reproduce the observed Tc. Besides, as long as we employ to avoid a serious investigation of the effects of the Coulomb repulsion on superconductivity, we cannot apply this framework to strongly correlated superconductors. Even in weakly or moderately correlated superconductors, this framework cannot tell anything about superconductivity originating from the Coulomb repulsion via charge and/or spin fluctuations (namely, the electronic mechanism including the plasmon mechanism (Takada, 1978, 1993a)). Furthermore, in this framework, we cannot investigate the competition or the coexistence (or even the mutual enhancement due to the quantum-mechanical interference effect) between the phonon and the electronic mechanisms. The validity of the concept of is closely connected with that of the Eliashberg theory itself; the theory is valid only when the Fermi energy of the superconducting electronic system, EF, is very much larger than !0. Under the condition of EF >> !0, the dynamic response time for the phonon-mediated attraction !01 is much shorter than that for the Coulomb repulsion E–1 F , precluding any possible interference effects between two interactions, so that physically it is very reasonable to separate them. After this separation, the Coulomb part (which was not considered to play a positive role in the Cooper-pair formation) has been simply treated in terms of . Thus, for the purpose of searching for some positive role of the Coulomb repulsion, the concept of is irrelevant from the outset. 1.12.2.4 Vertex Corrections and Dynamic Screening Incidentally, in some recently discovered superconductors in the phonon mechanism such as the alkalidoped fullerenes with Tc ¼ 18–38 K (Gunnarsson, 1997; Hebard et al., 1991; Takada and Hotta, 1998; Takabayashi et al., 2009), the condition of EF >> !0 is violated, necessitating to include the vertex corrections in calculating the phonon-mediated attractive interaction (Takada, 1993b). Then, it is by no means clear to treat the overall effect of various phonons in terms of the sum of the contribution from each phonon, implying that the Eliashberg function 2F(!) is not enough to properly describe the attraction because of possible interference effects among virtually excited phonons. As a consequence, will not be simply the sum of i, the contribution from the ith phonon, unless i is small enough to validate the whole calculation in lowest-order perturbation.
In the case of EF !0, another complication occurs in treating the screening effect of the conduction electrons. In the usual first-principles calculation scheme, the static screening is assumed in calculating 2F(!), but it does not reflect the actual screening process working during the formation of Cooper pairs. 1.12.2.5
Ideal Calculation Scheme
In order to unambiguously solve this problem of screening, we may imagine a following ideal calculation scheme for Tc: in the first step, we calculate the microscopic dynamical electron–electron effective interaction V in the whole momentum and energy space. This V is assumed to contain both the Coulomb repulsion and the phonon-mediated attraction on the same footing. Then in the second step, we obtain Tc directly from this V by simultaneously determining the gap function in the whole momentum and energy space, reflecting the behavior of V. If this scheme were developed, we could not only calculate Tc from first principles without resort to but also correctly discuss the competition, coexistence, and mutual enhancement between the phonon and the electronic mechanisms.
1.12.3 Calculation of Tc in the G0W0 Approximation 1.12.3.1
Formulation
Although it is along the royal road in the project of obtaining a reliable method for predicting Tc from first principles, this ideal calculation scheme is extremely difficult to achieve in actual situations, because all the difficulties in the quantum-mechanical many-body problem are associated with it. About three decades ago, the present author, who was a graduate student at that time, was struggling with developing such a scheme without perceiving much of the difficulties intrinsic to the many-body problem. After a yearlong struggle, he managed to propose a rather general scheme to evaluate Tc directly from V without introducing the concept of (Takada, 1978), though it was still at the stage far from the ideal scheme. In a broad sense, this scheme may be called an approach from the weak-coupling limit, corresponding to the G0W0 approximation or the one-shot GW approximation in the terminology of the present-day first-principles calculation community. In the same terminology, by the way, the Eliashberg theory
Theory of Superconductivity in Graphite Intercalation Compounds
corresponds to the GW approximation with respect to the phonon-mediated attractive interaction between electrons. Let us explain this G0W0 scheme here (we employ units in which h ¼ kB ¼ 1). For simplicity, imagine the three-dimensional (3D) electron gas in which an electron is specified by momentum p and spin . If we write the electron annihilation operator by cp, we can define the abnormal thermal Green’s function F(p, i!p) at temperature T by F ðp; i!p Þ ¼ –
Z
1=T
dei!p hT cp" ðÞc – p# i
415
where the gap function p and the pairing interaction K 9p;p9 are, respectively, defined by p X 2"p
Z 0
1
d! ImF R ðp; !Þ
ð5Þ
and Kp;p9 ¼
Z 0
1
"p þ "p9 2 V ðp – p9; iÞ ð6Þ d 2 þ ð"p þ "p9 Þ2
By using Kp;p9 thus calculated, we can determine Tc as an eigenvalue of Equation (4).
ð1Þ
0
with !p being the fermion Matsubara frequency. At T ¼ Tc where the second-order phase transition occurs, this function satisfies the following exact gap equation: F ðp; i!p Þ ¼ – Gðp;X i!pX ÞGð – p; – i!p Þ I˜ðp; p9; i!p ; i!p9 ÞF ðp9; i!p9 Þ Tc !p9
p9
ð2Þ
where G(p, i!p) is the normal Green’s function and ˜ p9; i!p ; i!p9 Þ is the irreducible electron–electron Iðp; effective interaction. Now, in the spirit of the G0W0 approximation, we will replace G(p, i!p) by the bare one G0(p, i!p) X (i!p "p)1 in Equation (2), where "p(¼ p2/2m ) is the bare one-electron dispersion relation with m the band mass and the chemical potential. We will ˜ p9; i!p ; i!p9 Þ is also consider the case in which Iðp; well approximated as a function of only the variables (p p9, i!p i!p9) to write ˜ p9; i!p ; i!p9 Þ ¼ V ðp – p9; i!p – i!p9 Þ Iðp;
ð3Þ
just like the effective interaction in the randomphase approximation (RPA), though we do not intend to confine ourselves to the RPA at this point. By substituting Equation (3) into Equation (2) and making an analytic continuation on the ! plane to transform F(p, i!p) to the retarded function FR(p, !) on the real-! axis, we get a gap equation for FR(p, !). Then, by taking the imaginary parts in both sides of the gap equation and integrating over the ! variable, we finally obtain an equation depending only on the momentum variable p. Concretely, the equation can be cast into the following BCS-type gap equation: p ¼ –
X p9 p9
2"p9
tanh
"p9 Kp;p9 2Tc
ð4Þ
1.12.3.2
Comments on the Formulation
Five comments are in order on this framework: 1. Based on Equations (4) and (6), we can obtain Tc directly from the microscopic one-electron dispersion relation "p and the effective electron– electron interaction V(q, i) with no need to separate the phonon-mediated attraction from the Coulomb repulsion. 2. In spite of the similarity of Equation (4) to the BCS gap equation, artificial cutoffs involved in constructing the BCS model are avoided in the present scheme; natural cutoffs are automatically introduced by the calculation of Kp;p9 defined in Equation (6). 3. Except for the spin-singlet pairing, no assumption is made on the dependence of the gap function on angular valuables in deriving Equation (4), so that this gap equation can treat any kind of anisotropy in the gap function, indicating that it can be applied to s-wave, d-wave, etc. and even their mixture like (sþd)-wave superconductors. 4. As can be seen by its definition, the gap function p in Equation (5) does not correspond to the physical energy gap except in the weak-coupling region. Similarly, Kp;p9 is not a physical entity. Both quantities are introduced for the mathematical convenience so as to make Tc invariant in transforming Equation (2) into Equation (4). The key point here is that we need not solve the full gap equation (2) but a much simpler one (4) in order to obtain Tc in Equation (2). Of course, if we want to know the physical gap function rather than p to compare with experiment, we need to solve the full gap equation, Equation (2), with Tc determined by Equation (4). 5. Historically, Cohen was the first to evaluate Tc in degenerate semiconductors on the level of the G0W0 approximation (Cohen, 1964; 1969).
416 Theory of Superconductivity in Graphite Intercalation Compounds
Unfortunately, the pairing interaction is not correctly derived in his theory, as explicitly pointed out by the present author (Takada, 1980) who, instead, has succeeded in obtaining the correct pairing interaction (Takada, 1978) by consulting the pertinent work of Kirzhnits et al. (1973). 1.12.3.3
Assessment: Application to SrTiO3
In order to assess the quality of this basic framework of calculating Tc from first principles, we have applied it to SrTiO3 and compared the results with experiments (Takada, 1980). This material is an insulator and exhibits ferroelectricity under a uniaxial stress of about 1.6 kbar along the [100] direction, but it turns into an n-type semiconductor by either Nb doping or oxygen deficiency, whereby the conduction electrons are introduced in the 3d band of Ti around the point with the band mass of m 1.8me (me: the mass of a free electron). At low temperatures, superconductivity appears and the observed Tc shows interesting features; Tc depends strongly on the electron concentration n and it is optimized with Tc 0.3 K at n 1020 cm3. Its dependence on the pressure is unsual; Tc decreases rather rapidly with hydrostatic pressures, but it increases with the [100] uniaxial stress.
(a) 1.0
Taking these situations into account, we have assumed that the superconductivity is brought about by the polar-coupling phonons associated with the stress-induced ferroelectric phase transition. Then we have calculated the effective electron–electron interaction V(q, i) in the RPA, in which the longrange attraction induced by the virtual exchange of polar-coupled phonons is included with the longrange Coulomb repulsion on the same footing. By substituting this V(q, i) into Equation (6), we have obtained Tc directly from a microscopic model and the results of Tc are in surprisingly good quantitative agreement with experiment, as shown in Figure 2. This success indicates that the present basic framework including the adoption of the RPA is very useful at least in the polar-coupled phonon mechanism.
1.12.4 Density Functional Theory for Superconductors 1.12.4.1
Recently, much attention has been paid to an extension of the density functional theory (DFT) to treat superconductivity, mainly because it provides another scheme for first-principles calculations of
(b) 0.08 : Experiment
ΔTc (K)
0.1
0.05
0.1
0.5
1
n (1020 cm–3)
Experiment : [100] : [110] : H.P.
0 –0.04 –0.08
19 –3 –0.12 n = 6.3 × 10 cm 30 : [100] : [110] 20 : H.P.
m* = 1.8me 0.01 0.05
0.04 0 –0.04 –0.08 n = 2.5 × 1019 cm–3 –0.12 0.04
5
10
ω t (cm–1)
Tc (K)
ΔTc (K)
SrTiO3 0.5
Hohenberg–Kohn–Sham Theorem
10
0
0.5
1.0 1.5 Stress (kbar)
2.0
Figure 2 (a) Electron density dependence of Tc in SrTiO3 and (b) pressure dependence of Tc (Takada, 1980). Both uniaxial stress along either [100] or [110] direction and hydrostatic pressure (H.P.) are considered. The deviation of Tc, Tc, is plotted as a function of stress in units of kbar. For comparison, corresponding experimental results are also shown, together with the results of the transverse polar phonon energy !t in units of cm1.
Theory of Superconductivity in Graphite Intercalation Compounds
Tc without resort to . We shall make a very brief review of it in this section. It is stated in the basic theorem in DFT that all the physics of an interacting electron system is uniquely determined, once its electronic density in the ground state n(r) is specified. This Hohenberg–Kohn theorem (Hohenberg and Kohn, 1964) implies that every physical quantity including the exchange-correlation energy Fxc may be considered as a unique functional of n(r). The density n(r) itself can be determined by solving the ground-state electronic density of the corresponding noninteracting reference system that is stipulated in terms of the Kohn–Sham (KS) equation (Kohn and Sham, 1965). The core quantity in the KS equation is the exchange-correlation potential Vxc(r), which is defined as the functional derivative of Fxc[n(r)] with respect to n(r), namely, Vxc(r) ¼ Fxc[n]/ n(r). It must be noted that Vxc(r) as well as each one-electronic wave function at ith level with its energy eigenvalue "i in the KS equation have no physical relevance; they are merely introduced for the mathematical convenience so as to obtain the exact n(r) in connecting the noninteracting reference system with the real many-electron system. The Hohenberg–Kohn theorem can be applied to the ordered ground state as well on the understanding that the order parameter itself is regarded as a functional of n(r). In providing some approximate functional form for Fxc[n], however, it would be more convenient to treat the order parameter as an additional independent variable. For example, in considering the system with some magnetic order, we usually employ the spin-dependent scheme in which the fundamental variable is not n(r) but the spin-decomposed density n(r), leading to the spinpolarized exchange-correlation energy functional Fxc[n], based on which the spin-dependent KS equation is formulated. 1.12.4.2
417
potential xc(r,r9) ¼ Fxc[n, ]/ (r,r9) appear in an extended KS equation, which is found to be written in the form of the Bogoliubov-de Gennes equation appearing in the usual theory for inhomogeneous superconductors (de Gennes, 1966). Just as is the case with Vxc(r), xc(r,r9) has no direct physical meaning, but in principle, if the exact form of Fxc[n, ] is known, the solution of the extended KS equation gives us the exact result for (r,r9), containing all the effects of the Coulomb repulsion including the one usually treated phenomenologically through the concept of . As a result, we can determine the exact Tc by the calculation of the highest temperature below which a nonzero solution for (r,r9) can be found. In this formulation, we can write the fundamental gap equation to determine Tc exactly as i ¼ –
X j j
2"j
tanh
"j Kij 2Tc
ð7Þ
where i is the gap function for ith KS level. In just the same way as its energy eigenvalue "i (which is measured from the chemical potential), i is not the quantity to be observed experimentally but just introduced for the mathematical convenience so as to obtain the exact Tc by solving this BCS-type equation, Equation (7). Similarly, the pair interaction Kij, defined as the second-functional derivative of Fxc[n, ] with respect to and , does not have any direct physical meaning, either. We note here the very impressive fact that the final forms for the two gap equations, Equations (4) and (7), are exactly the same, in spite of the fact that they are derived from quite different foundations and reasoning. We also note that because of this similarity, we may judge that, as long as Kij is properly chosen, the physics described by is also included in the framework of DFT for superconductors, at least to the extent that it is included in the G0W0 scheme explained in Section 1.12.3.
Gap Equation
Similarly, in treating the superconducting state in the framework of DFT, it is better to construct the energy functional by employing both n(r) and the electronpair density (r,r9)(X h"(r)#(r9)i) as basic variables, leading to the introduction of the exchangecorrelation energy functional Fxc[n(r), (r,r9)], where a(r) is the electron annihilation operator (Oliveira et al., 1988; Kurth et al., 1999). In accordance with this addition of the order parameter as a fundamental variable to DFT, not only the exchange-correlation potential Vxc(r) but also the exchange-correlation pair
1.12.4.3
Applications
In 2005, this DFT framework was extended to explicitly take care of the phonon-mediated attractive interaction (Lu¨ders et al., 2005) and it has been applied to many superconductors (Sanna et al., 2007; Marques et al., 2005; Floris et al., 2005, 2007; Profeta et al., 2006; Sanna et al., 2006; Floris et al., 2007). In order to perform these calculations for actual superconductors, it is necessary to provide a concrete form
418 Theory of Superconductivity in Graphite Intercalation Compounds
for Fxc[n, ]. In the judgment of the present author, the presently available form for Fxc[n, ] contains the information equivalent to that included in the Eliashberg theory for the part of the phononmediated attraction, indicating that no vertex corrections are considered in this treatment, while for the part of the Coulomb repulsion, it contains only very crude physics; the screening effect is treated in the Thomas–Fermi static-screening approximation, which is nothing but the result of the RPA only in the static and the long-wavelength limit, forgetting the detailed dynamical nature of the screening effect. Mainly for this reason, Tc in the present form of Fxc[n, ] is not expected to be very accurate, even though the calculated results for Tc seem to be in good agreement with experiment. 1.12.4.4
Basic Problems
In relation to the above point, it would be appropriate to give the following comment: in the calculations of the normal-state properties in the local-density approximation (LDA) and generalized gradient approximation (GGA) (Perdew et al., 1997) to DFT, we usually anticipate that errors in the calculated results are of the order of 1 and 0.3 eV for LDA and GGA, respectively. These errors are much larger than those expected in the calculation of quantum chemistry (0.05 eV). In DFT for superconductors, calculations of Tc (which is of the order of 0.001 eV in general) are done simultaneously with those for the normal-state properties. This implies that the errors anticipated for Tc would be very large compared to Tc itself. We should also point out that the present form for Fxc[n, ] is not useful in the discussion of the electronic mechanisms like the plasmon and the spin-fluctuation ones, prompting us to improve on the approximate form for Fxc[n, ]. In addition, there are several problems in the fundamental theory; for example, it is by no means clear whether the second-functional derivative of Fxc[n, ] is a well-defined quantity, in just the same way as we have already experienced in the energy-gap problem (Perdew et al., 1997, 1983; Sham and Schlu¨ter, 1983) in semiconductors and insulators.
1.12.5 Experiment on Superconductivity in GICs From this section, let us get back to the review on superconductivity in GICs. As briefly mentioned in Section 1.12.1, the history of the researches on this
issue extends more than four decades. In 1965, the first report of superconductivity was made for KC8, RbC8, and CsC8 (Hannay et al., 1965) in which Tc was not reliably determined; it depended very much on samples. Subsequent works (Alexander et al., 1981; Chaiken et al., 1990; Iye and Tanuma, 1982; Koike et al., 1978, 1980; Kobayashi et al., 1979; Kobayashi and Tsujikawa, 1981; Pendrys et al., 1981) confirmed the occurrence of superconductivity in KC8 with Tc ¼ 0.15 K, but superconductivity did not appear in RbC8 and CsC8 when Tc was down to 0.09 and 0.06 K, respectively. Later works have found that Tc is actually 26 mK for RbC8, but no superconductivity is found in either LiC6 or the second- or higher-stage alkali GICs, though the calculation of Tc based on the McMillan’s formula (McMillan, 1968) predicted an observable value of Tc even for KC24 (Kamimura et al., 1980; Inoshita and Kamimura, 1981). It seems that the usual first-principles calculation of 2F(!) tends to provide an unrealistically large contribution from the intralayer high-energy carbon oscillations to . This unfavorable tendency in the calculation of seems to prevail even in CaC6 (Calandra and Mauri, 2005). The anisotropy of the critical magnetic field Hc2 was also a matter of interest, drawing attention of both experimentalists (Chaiken et al., 1990; Dresselhaus et al., 1989; Iye and Tanuma, 1982; Koike et al., 1980; Roth et al., 1985) and theorists (Jishi et al., 1991, 1992). Note that the gap function p defined in Equation (4) has nothing to do with the anisotropic behavior of Hc2, though in developing a phenomenological theory (Jishi et al., 1991, 1992) some critical comments were made on the results of p (Takada, 1982) with the assumption that the anisotropy in Hc2 should reflect on p. In search of higher Tc, many attempts have been made to synthesize new GIC superconductors such as NaC2 (Tc ¼ 5K) (Belash et al., 1987), LiC2 (Tc ¼ 1.9K) (Belash et al., 1989), and alkali-metal amalgams such as KHgC4 (Tc ¼ 0.73 K) and KHgC8 (Tc ¼ 1.90 K) (Alexander et al., 1980; Tanuma, 1981; Koike and Tanuma, 1981; Pendrys et al., 1981; Alexander et al., 1981; Iye and Tanuma, 1982), but a larger enhancement of Tc was not achieved until CaC6 was found in 2005 with Tc ¼ 11.5 K (Weller et al., 2005). Subsequently, many works have been done on alkaline-earth GIC superconductors (Emery et al., 2005; Hinks et al., 2007; Kadowaki et al., 2007; Kim et al., 2006, 2007; Kurter et al., 2007; Lamura et al., 2006; Sugawara et al., 2009; Valla et al., 2009), but none ever succeeded in
Theory of Superconductivity in Graphite Intercalation Compounds
synthesizing a new GIC with Tc larger than 15.4 K which was observed in CaC6 under pressures (Gauzzi et al., 2007). Thus, some new idea seems to be needed to further enhance Tc. The present author hopes that the suggestions given in Section 1.12.8 help experimentalists synthesize a new GIC superconductor with Tc much higher than 10 K.
1.12.6 Standard Model for Superconductivity in GICs 1.12.6.1 System
Characteristic Features of the
Basically because GICs are not recognized as strongly correlated systems, the usual ab initio self-consistent band structure calculation is very useful in elucidating the important features of the electronic structures of GICs in the normal state. According to such calculations, it is found that there is no essential qualitative difference between alkali and alkaline-earth GICs (see Figure 3). The main common features among these GICs may be summarized in the following way: 1. In MCx, each intercalant metal atom acts as a donor and changes from a neutral atom M to an ion MZþ with valence Z.
419
2. The valence electrons released from M will transfer either to the graphite bands or to the 3D band composed of the intercalant orbitals and the graphite interlayer states (Posternak et al., 1984; Halzwarth et al., 1984; Koma et al., 1986). We shall define the factor f as the branching ratio between these two kinds of bands. For example, Zf and Z(1 f) electrons will go to the and the 3D bands, respectively. 3. The electrons in the graphite bands are characterized by the 2D motion with a linear dispersion relation (known as a Dirac cone in the case of graphene) on the graphite layer. 4. The dispersion relation of the graphite interlayer band is very similar to that of the 3D free-electron gas, folded into the Brillouin zone of the graphite (Csanyi et al., 2005). Thus, its energy level is very high above the Fermi level in the graphite, because the amplitude of the wave function for this band is small on the carbon atoms. In MCx, on the other hand, the cation MZþ is located in the interlayer position where the amplitude of the wave functions is large, lowering the energy level of the interlayer band below the Fermi level. The dispersion of the interlayer band is modified from that of the free-electron gas because
(a)
(b)
3D band
EF
–1eV
K
F2
D
KF3D
2D band KB (η, η, η,) L
(η, –η, 0) Γ
(η, η, 0) χ
X
Z
(η, 0, 0) Γ
T
KF3D
Kz
Figure 3 (a) Band structure of CaC6 (Calandra and Mauri, 2005). (b) Fermi surface of KC8 (Wang et al., 1991). Both materials are characterized by the common feature that the electronic system is composed of the 2D bands of graphite and the 3D interlayer band.
420 Theory of Superconductivity in Graphite Intercalation Compounds
1.12.6.2 Microscopic Model for Superconductivity With these common features in mind, we can think of a simple model for the GIC superconductors, which is schematically shown in Figure 4(a). Actually, exactly the same model was proposed in as early as 1982 by the present author for describing superconductivity in alkali GICs (Takada, 1982). In order to give some idea about the mechanism to induce an attraction between 3D electrons in this model, let us imagine how each conducting 3D electron sees the charge distribution of the system. First Zþ of all, there are positively charged metallic pffiffiffi 2ions M with its density nM, given by nM ¼ 4=3 3a dx, where
(a)
M Z+ M Z+ M Z+
> M Z+ + Ze– f
e– 1–f
>
M
2D π-electrons
C–δ
d v >
of the hybridization with the orbitals associated with M, but generally it is well approximated by "p ¼ p2/ 2m EF with an appropriate choice of the effective band mass m and the Fermi energy EF. Here the value of m depends on M; in alkali GICs, the hybridization occurs with s-orbitals, allowing us to consider that m ¼ me, while in alkaline-earth GICs, the hybridization with d-orbitals contributes much, leading to m 3me in both CaC6 and YbC6, as revealed by the bandstructure calculation (Calandra and Mauri, 2005; Mazin, 2005). 5. The value of f, which determines the branching ratio Zf : Z(1 f ), can be obtained by the selfconsistent bandstructure calculation. In KC8, for example, it is known that f is around 0.6 (Ohno et al., 1979). On the other hand, f is about 0.16 (Calandra and Mauri, 2005) in CaC6, making the electron density n in the 3D band increase very much. This increase in n is easily understood by the fact that the energy level of the interlayer band is much lower with Ca2þ than with Kþ. The concrete numbers for n are 3.5 1021 and 2.4 1022 cm3 for KC8 and CaC6, respectively, in which the difference in both d and x is also taken into account. 6. As inferred from experiments (Csanyi et al., 2005; Kamimura, 1987; Takada, 1982) and also from the comparison of Tc calculated for each band (Takada, 1982), it has been concluded that only the 3D interlayer band is responsible for superconductivity. Note that LiC6 does not exhibit superconductivity because no carriers are present in the 3D interlayer band, although the properties of LiC6 are generally very similar to those of other superconducting GICs in the normal state.
3D itinerant electrons: m*
M C–δ M
C–δ
C–δ Z+
M
Z+
C–δ Z+
M
C–δ Z+
C–δ
C–δ
M Z+
M
e–
Z+
(δ = Zf /x) (b)
=
Π2D, Π3D
W0
V0
V
+
+
+
Figure 4 (a) Simplified model to represent MCx (x ¼ 2, 6, 8) superconductors. We consider the attraction between the 3D electrons in the interlayer band induced by polarcoupled charge fluctuations of the cation MZþ and the anion C . (b) Diagrammatic representation of the equation in the RPA to calculate the effective electron–electron interaction V(pp9, i), which will be substituted in Equation (6) to evaluate the kernel of the gap equation. Reproduced with permission from Takada Y (2009) Unified model for superconductivity in graphite intercalation compounds: Prediction of optimum Tc and suggestion for its realization. Journal of the Physical Society of Japan 78(1): 013703:1–4.
a is the bond length between C atoms on the graphite layer (which is 1.419 A˚). Note that with use of this nM, the density n of the 3D electrons is given by (1 f )ZnM. There are also negatively charged carbon ions C with the average charge of X fZe/x. Therefore, the 3D electrons will feel a large electric field of the polarization wave coming from oscillations of MZþ and C ions created by either out-of-phase optic or in-phase acoustic phonons. We shall consider the coupling of those phonons with the 3D electrons in terms of the point-charge model, allowing us to write the phonon-exchange polar-coupled interaction W0(q,!) for the scattering of the 3D electrons with momentum and energy transfers of q and ! as W0 ðq; !Þ ¼ V0 ðqÞ
!2p ð1 – f Þ2 !2 – !LA ðqÞ2
þ V0 ðqÞ
2 M þ f M=xM 2p ðM=M ! CÞ
!2 !LO ðqÞ2
ð8Þ
p defined, respectively, as with !p and ! sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 e 2 Z2 nM 4 e 2 Z2 nM p ¼ !p ¼ and ! M MM þ xMC
ð9Þ
where MM and MC are, respectively, the atomic masses of M and C, Mð¼ MM xMC =ðMM þ xMC ÞÞ is the reduced mass of MCx, !LO(q) and !LA(q) are the
Theory of Superconductivity in Graphite Intercalation Compounds
energies of LO- and LA-phonons, respectively, and V0(q) is the bare Coulomb interaction 4 e2/q2. (The subscript ‘0’ indicates that it is the bare interaction to be screened by both 2D and 3D mobile electrons.) Owing to the coupling with valence electrons, both !LO(q) and !LA(q) depend on f, but the f-dependence is not important, if we write the phonon-mediated interaction in terms of the corresponding transverse phonon energies, !TO(q) and !TA(q). Thus, we specify the phonon energies in terms of !TO(q) and !TA(q). In actual calculations, we assume that !TO(q) ¼ !t(¼ constant) and !TA(q) ¼ ctjqj, with !t of the order of 150 K and ct of the order of 105 cm s1 for the oscillation perpendicular to the graphite plane. 1.12.6.3 GICs
Calculation of Tc for Alkali-Doped
By combining this polar-phonon-mediated attractive interaction W0(q, !) with the bare Coulomb interaction between electrons V0(q) on the same footing and considering the polarization effects of both 2D and 3D electrons, we faithfully calculate V(q, !) the effective interaction between 3D electrons in the RPA (see Figure 4(b)). The obtained V(q, !) is put into the kernel, Equation (6), of the gap equation (4) to obtain Tc from first principles. The calculated results for Tc in alkali GICs are plotted as a function of f in Figure 5 to find that the overall magnitude of Tc is in the range of 0.1–0.01 K for f 0.5, which is in good agreement LiC2
1
KC8
Tc (K)
0.001
1.12.7 Superconductivity in AlkalineEarth GICs 1.12.7.1
CaC6
Now let us consider alkaline-earth GIC superconductors. We shall investigate them by adopting the same simple model and by using exactly the same calculation code developed in 1982 in order to see whether the model and therefore the picture on the mechanism of superconductivity successfully applied to alkali GIC superconductors can also be relevant to these newly synthesized superconductors or not (Takada, 2009a; 2009b). The parameters specifying the model will be changed in the following way, if CaC6 is considered instead of KC8: 1. Because the valence Z changes from monovalence to divalence, the attractive interaction W0, which is in proportion to Z2, increases by 4 times. 2. The interlayer distance d decreases from 5.42 to 4.524 A˚, so that the 3D electron density n increases. 3. The factor f to determine the branching ratio decreases from about 0.6 to 0.16. 4. The effective band mass for the 3D interlayer band m increases from me to about 3me. 5. The atomic number of the ion A hardly changes from 39.1 to 40.1. Considering the changes in the parameters, we have calculated Tc for CaC6 as a function of f. The results are plotted in Figure 6, from which we can learn the following points:
Experiment on KC8 RbC8
0.0001 0.00001 0
with experiment. Note that smaller values of Tc are obtained for heavier alkali atoms because of the smaller couplings as characterized by both !p and p . This success indicates that the present simple ! model applies well at least to alkali GIC superconductors.
o
0.1 0.01
Z=1 m* = me
421
CsC8 0.2
0.6
0.4
0.8
1.0
f Figure 5 Calculated results for Tc as a function of the branching ratio f for alkali GIC superconductors in which Z ¼ 1 and m ¼ me. Reproduced with permission from Takada Y (2009) Unified model for superconductivity in graphite intercalation compounds: Prediction of optimum Tc and suggestion for its realization. Journal of the Physical Society of Japan 78(1): 013703:1–4.
1. Overall, Tc becomes higher for smaller f. This can be understood by the fact that the screening effect due to the 2D electrons, which makes the polarcoupled interaction weak, becomes smaller with the decrease of f. 2. The enhancement of Tc by about one order is brought about by doubling Z, if m is kept to be the same value. 3. The enhancement of Tc by about one order is also brought about by tripling m from me to 3me, if Z is taken as Z ¼ 2.
422 Theory of Superconductivity in Graphite Intercalation Compounds
20
20
Experiment on CaC6 (CaC6: under pressure)
CaC6 Z=2 15
o
m* = 4me
Experiment
m* = me 0
0.2
o
5 m* = 2me
0.4
0.6
0.8
1.0
f
0
Figure 6 Calculated Tc as a function of f for m in the range of me–4me with other parameters suitably chosen for CaC6. The experimental result is reproduced well, if we choose m 3me. Reproduced with permission from Takada Y (2009) Unified model for superconductivity in graphite intercalation compounds: Prediction of optimum Tc and suggestion for its realization. Journal of the Physical Society of Japan 78(1): 013703:1–4.
Based on these observations, we can conclude that the enhancement of Tc in CaC6 by about a hundred times from that in KC8 is brought about by the combined effects of doubling Z and tripling m. In this respect, the value of m is very important. Appropriateness of m 3me is confirmed not only from the bandstructure calculations (Mazin, 2005; Calandra and Mauri, 2005) but also from the measurement of the electronic specific heat (Kim et al., 2006) compared with the corresponding one for KC8 (Mizutani et al., 1978).
1.12.7.2
o
10
m* = 3me
5
CaC6: m* = 3.3me CaC6: m* = 2.8me Experiment on YbC6
o
Other Alkaline-Earth GICs
Similar calculations are done for other alkaline-earth GIC superconductors as shown in Figure 7 in which m is determined so as to reproduce EF supplied by the bandstructure calculation. We see that very good agreement is always obtained between theory and experiment, implying that our simple model may be regarded as the standard one for describing the mechanism of superconductivity in GICs. Here a note will be added to the case of YbC6; the basic parameters such as Z, f, and m for YbC6 are about the same as those for CaC6, according to the bandstructure calculation. The only big change can be seen in the atomic mass; Yb (in which A ¼ 173.0) is much heavier than that of Ca by about 4 times, indicating weaker couplings between electrons and polar phonons as just in the case of comparison between KC8 and CsC8. In fact, Tc for YbC6 becomes about one-half of the corresponding result for CaC6, which agrees well with
YbC6: m* = 3.0me
Experiment on SrC6 SrC6: m* = 1.5me
o
10
Tc (K)
Tc (K)
15
Z=2
0.2
0.4
0.6
0.8
1.0
f Figure 7 Calculated Tc as a function of f for alkaline-earth GICs with m determined so as to reproduce EF provided by the bandstructure calculation. Reproduced with permission from Takada Y (2009) Unified model for superconductivity in graphite intercalation compounds: Prediction of optimum Tc and suggestion for its realization. Journal of the Physical Society of Japan 78(1): 013703:1–4.
experiment. One way to understand this difference is to regard it as an isotope effect with 0.5 (Mazin, 2005).
1.12.8 Prediction of the Optimum Tc in GICs As we have observed so far, our standard model could have predicted Tc ¼ 11.5 K for CaC6 in 1982 and it is judged that its predictive power is very high. Incidentally, the author did not perform the calculation of Tc for CaC6 at that time, partly because he did not know of a possibility to synthesize such GICs, but mostly because the calculation cost was extremely high in those days; a rough estimate shows that the processing speed of computers has increased by at least a million times in the past three decades. This significant improvement in computational environments is surely a boost to making such first-principles calculations of Tc as reviewed in Sections 1.12.3 and 1.12.4. In any way, encouraged by this success in reproducing Tc in alkaline-earth GICs, we have explored the optimum Tc in the whole family of GICs by widely changing various parameters involved in the microscopic Hamiltonian. Examples of the calculated results of Tc are shown in Figure 8(a) and 8(b), in which f is fixed to zero, the optimum condition to raise Tc, and d is tentatively taken as 4.0 A˚. From this exploration, we find that the most important parameter to enhance Tc is m. In particular, we need m larger than at least 2me to obtain Tc over 10 K,
Theory of Superconductivity in Graphite Intercalation Compounds
423
irrespective of any choice of other parameters, and Tc is optimized at m near 15me. The optimized Tc depends rather strongly on the parameters to control the polar-coupling strength such as Z and the atomic mass A; if we choose a trivalent light atom such as boron to make !t large, the optimum Tc is about 100 K, but the problem about the light atoms is that m will never become heavy due to the absence of either d- or f-electrons. Therefore, we do not expect that Tc would become much larger than 10 K, even if BeC2 or BC2 were synthesized. From this perspective, it will be much better to intercalate Ti or V, rather than Be or B. Taking all these points into account, we suggest synthesizing three-element GICs providing a heavy 3D electron system by the introduction of heavy atoms into a light-atom polarcrystal environment.
scheme, directly from the microscopic Hamiltonian representing the standard model. By suitably choosing the parameters in the microscopic Hamiltonian, we have found surprisingly good agreement between theory and experiment for both alkali and alkaline-earth GICs, in spite of the fact that Tc varies more than three orders of magnitude. In this way, we have clarified that superconductivity in metal GICs can be understood by the picture that the 3D electrons in the interlayer band supplied by the ionization of metals experience the attractive interaction induced by the virtual exchange of the polar-coupled phonons of the metal ions. We have also predicted a further enhancement of Tc well beyond 10 K, giving some suggestions to realize such superconductors in the family of GICs. By first principles we usually mean the calculations based on not the model but the first-principles Hamiltonian. Thus, it might be considered as inappropriate to call the present G0W0 scheme first principles, but it is not an easy task to specify the key parameters to control Tc by just implementing the calculations based on the first-principles Hamiltonian. We can identify the importance of the parameters, m and Z, only through the calculations based on the model Hamiltonian, leading to a better and unambiguous understanding of the mechanism of superconductivity without probing too much into the very details of each system which sometimes obscure the essence in first-principles approaches. Besides, because of the errors involved in the numerical calculations of normal-state properties as mentioned in Section 1.12.4, more accurate results of Tc will be obtained by way of a suitable model Hamiltonian rather than directly from the first-principles one. As a project in the future, it would be important to construct a more powerful scheme for the firstprinciples calculation of Tc by combining the schemes in Sections 1.12.3 and 1.12.4, based on which we may make more detailed suggestions to synthesize GIC superconductors with Tc much larger than 10 K. (See Chapters 1.10 and 1.11).
1.12.9 Conclusion
References
In this chapter, by taking account of the common features elucidated by both the bandstructure calculation and the various measurements on the normalstate properties, we have constructed the standard model pertinent for the description of the mechanism of superconductivity in metal GICs and then made first-principles calculations of Tc in the G0W0
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(a) 100
(b) f=0
90 80
Z=2 BeC2 f = 0 MgC2
BC2(Z = 3)
Tc (K)
70 60
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CaC6
50 40
YbC6 LiC2(Z = 1)
30 20
SrC6
10 0
BaC6 10 20 30 40 m*/ me
0
10
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Further Reading Bennemann KH and Ketterson JB (eds.) (2008) Superconductivity I and II. Heidelberg: Springer. Emery N, He´rold C, Mareˆche´ J-F, and Lagrange P (2008) Synthesis and superconducting properties of CaC6. Sci. Technol. Adv. Mater. 9: 044102:1–7.
Fiolhais C, Nogueira F, and Marques M (eds.) (2003). A Primer in Density Functional Theory. Heidelberg: Springer. Hedin L and Lundqvist S (1969) Effects of Electron–Electron and Electron–Phonon Interactions on the One-Electron States of Solids. In: Solid State Physics, vol. 23. New York: Academic Press. Tinkham M (1996) Introduction to Superconductivity. Dover: Mineola.
